年扳 - 上海交通大学人工结构及量子调控教育部重点实验室

2018 Annual Report from Key Laboratory of Artificial

Structures and Quantum Control, Ministry of Education

http://klasqc.physics.sjtu.edu.cn/

2019. 4

人工结构及量子调控教育部重点实验室

年报 (卷 X: 2018 年度)

i

目录

一、实验室年度数据简表............................................................................................ 1

二、研究水平与贡献

1、主要研究成果与贡献 .......................................................................................... 4

2、承担科研任务 .................................................................................................... 14

三、研究队伍建设

1、各研究方向及研究队伍 .................................................................................... 18

2、本年度固定人员情况 ........................................................................................ 18

3、本年度流动人员情况 ........................................................................................ 21

四、学科发展与人才培养

1、学科发展 ............................................................................................................ 23

2、科教融合推动教学发展 .................................................................................... 25

3、人才培养 ............................................................................................................ 27

（1）人才培养总体情况 ........................................................................................ 27

（2）研究生代表性成果 ........................................................................................ 31

（3）研究生参加国际会议情况 ............................................................................ 32

五、开放交流与运行管理

1、开放交流 ............................................................................................................ 33

（1）开放课题设置情况 ........................................................................................ 33

（2）主办或承办大型学术会议情况 .................................................................... 35

（3）国内外学术交流与合作情况 ........................................................................ 36

（4）科学传播 ........................................................................................................ 38

2、运行管理 ............................................................................................................ 39

（1）学术委员会成员 ............................................................................................ 39

（2）学术委员会工作情况 .................................................................................... 40

（3）主管部门和依托单位支持情况 .................................................................... 44

3、仪器设备 ............................................................................................................ 45

ii

年度代表性论文：

1. Manipulation of domain-wall solitons in bi- and trilayer graphene,

L. L. Jiang, S. Wang, Z. W. Shi, C. H. Jin, M. Iqbal Bakti Utama, S. H. Zhao, Y. R.

Shen, H. J. Gao, G. Y. Zhang, and F. Wang,

Nat. Nanotechnol. 13, 204-208 (2018) .................................................................... 46

2. The nature of spin excitations in the one-thirdmagnetization plateau phase of

Ba3CoSb2O9,

Y. Kamiya, L. Ge, T. Hong, Y. Qiu, D. L. Quintero-Castro, Z. Lu, H. B. Cao,

M. Matsuda, E. S. Choi, C. D. Batista, M. Mouriga, H. D. Zhou, and J. Ma,

Nat. Commun. 9, 2666-(1-11) (2018) ...................................................................... 52

3. Quasiparticle interference and nonsymmorphic effect on a floating band surface

state of ZrSiSe,

Z. Zhu, T. R. Chang, C. Y. Huang, H. Y. Pan, X. A. Nie, X. Z. Wang, Z. T. Jin, S.

Y. Xu, S. M. Huang, D. D. Guan, S. Y. Wang, Y. Y. Li, C. H. Liu, D. Qian, W. Ku,

F. Q. Song, H. Lin, H. Zheng, and J. F. Jia,

Nat. Commun. 9, 4153-(1-8) (2018) ........................................................................ 63

4. Electrode-free anodic oxidation nanolithography of low-dimensional materials,

H. Y. Li, Z. Ying, B. Lyu, A. Deng, L. L. Wang, T. Taniguchi, K. Watanabe, and

Z. W. Shi,

Nano Lett. 18, 8011-8015 (2018) ............................................................................ 71

5. Realization of interdigitated back contact silicon solar cells by using dopant free

heterocontacts for both polarities,

H. Lin, D. Ding, Z. L. Wang, L. F. Zhang, F. Wu, J. Yu, P. Q. Gao, J. C. Ye, and

W. Z. Shen,

Nano Energy 50, 777-784 (2018) ............................................................................ 76

6. Antiferromagnetic order in epitaxial FeSe films on SrTiO3,

Y. Zhou, L. Miao, P. Wang, F. F. Zhu, W. X. Jiang, S. W. Jiang, Y. Zhang, B. Lei,

X. H. Chen, H. F. Ding, H. Zheng, W. T. Zhang, J. F. Jia, D. Qian, and D. Wu,

Phys. Rev. Lett. 120, 097001-(1-6) (2018) .............................................................. 84

7. Field-driven quantum criticality in the spinel magnet ZnCr2Se4,

C. C. Gu, Z. Y. Zhao, X. L. Chen, M. Lee, E. S. Choi, Y. Y. Han, L. S. Ling, L. Pi,

Y.H. Zhang, G. Chen, Z. R. Yang, H. D. Zhou, and X. F. Sun,

Phys. Rev. Lett. 120, 147204-(1-6) (2018) .............................................................. 90

iii

8. Tunable quantum spin liquidity in the 1/6th-filled breathing Kagome lattice,

A. Akbari-Sharbaf, R. Sinclair, A. Verrier, D. Ziat, H. D. Zhou, X. F. Sun, and J. A.

Quilliam,

Phys. Rev. Lett. 120, 227201-(1-6) (2018) .............................................................. 96

9. Transitions from a Kondo-like diamagnetic insulator into a modulated

ferromagnetic metal in FeGa3-yGey,

Y. Zhang, J. S. Chen, J. Ma, J. M. Ni, M. Imai, C. Michioka, Y. Hadano, M. A.

Avila, T. Takabatake, S. Y. Li, and K. Yoshimura,

PNAS 115, 3273-3278 (2018) ............................................................................... 102

10. Quasiparticle interference on type-I and type-II Weyl semimetal surfaces: a review,

H. Zheng and M. Z. Hasan,

Advances in Physics: X 3, 1466661-(569-591) (2018) ......................................... 108

11. Perovskite/c-Si tandem solar cells with realistic inverted architecture: achieving

high efficiency by optical optimization,

L. X. Ba, H. Liu, and W. Z. Shen,

Prog. Photovolt: Res. Appl. 26, 924-933 (2018) (Cover paper) ........................... 131

12. Controllable rotational inversion in nanostructures with dual chirality,

L. Dai, K. D. Zhu, W. Z. Shen, X. J. Huang, L. Zhang, and A. Goriely,

Nanoscale 10, 6343-6348 (2018) (Back cover paper) .......................................... 141

13. Temperature gradient-induced instability of perovskite via ion transport,

X. W. Wang, H. Liu, F. Zhou, J. Dahan, X. Wang, Z. P. Li, and W. Z. Shen,

ACS Appl. Mater. Interfaces 10, 835-844 (2018) ................................................. 147

14. Boosting supercapacitive performance of ultrathin mesoporous NiCo2O4 nanosheet

arrays by surface sulfation,

Y. X. You, M. J. Zheng, D. K. Jiang, F. G. Li, H. Yuan, Z. H. Zhai, L. Ma and

W. Z. Shen,

J. Mater. Chem. A 6, 8742-8749 (2018) ............................................................... 157

15. Magnetic and structural transitions tuned through valence electron concentration in

magnetocaloric Mn(Co1−xNix)Ge,

Q. Y. Ren, W. D. Hutchison, J. L. Wang, A. J. Studer, and S. J. Campbell,

Chem. Mater. 30, 1324−1334 (2018) .................................................................... 165

16. Lithium ion intercalation in thin crystals of hexagonal TaSe2 gated by a polymer

electrolyte,

iv

Y. S. Wu, H. L. Lian, J. M. He, J. Y. Liu, S. Wang, H. Xing, Z. Q. Mao, and

Y. Liu,

Appl. Phys. Lett. 112, 023502-(1-5) (2018) (Editor’s Pick) ................................... 176

17. Approximating quantum many-body wave functions using artificial neural

networks,

Z. Cai and J. G. Liu,

Phys. Rev. B 97, 035116-(1-8) (2018) .................................................................... 181

18. Terahertz streaking of few-femtosecond relativistic electron beams,

L. R. Zhao, Z. Wang, C. Lu, R. Wang, C. Hu, P. Wang, J. Qi, T. Jiang, S. G. Liu, Z.

R. Ma, F. F. Qi, P. F. Zhu, Y. Cheng, Z. W. Shi, Y. C. Shi, W. Song, X. X. Zhu, J.

R. Shi, Y. X. Wang, L. X. Yan, L. G. Zhu, D. Xiang, and J. Zhang,

Phys. Rev. X 8, 021061-(1-9) (2018) .................................................................... 189

19. Discovery of slow magnetic fluctuations and critical slowing down in the

pseudogap phase of YBa2Cu3Oy,

J. Zhang, Z. F. Ding, C. Tan, K. Huang, O. O. Bernal, P. C. Ho, G. D. Morris, A. D.

Hillier, P. K. Biswas, S. P. Cottrell, H. Xiang, X. Yao, D. E. MacLaughlin, and

L. Shu,

Sci. Adv. 4, eaao5235-(1-7) (2018) ....................................................................... 198

20. Fidelity susceptibility of the anisotropic XY model: the exact solution,

Q. Luo, J. Z. Zhao, and X. Q. Wang,

Phys. Rev. E 98, 022106-(1-7) (2018) ................................................................... 205

21. Lattice distortion effects on the frustrated spin-1 triangular- antiferromagnet

A3NiNb2O9 (A=Ba, Sr and Ca),

Z. Lu, L. Ge, G. Wang, M. Russina, G. Guenther, C. R. dela Cruz, R. Sinclair, H. D.

Zhou, and J. Ma,

Phys. Rev. B 98, 094412-(1-10) (2018) ................................................................. 212

22. Understanding shear-induced sp2-to-sp3 phase transitions in glassy carbon at low

pressure using first-principles calculations,

L. B. Wen and H. Sun,

Phys. Rev. B 98, 014103-(1-7) (2018) ................................................................... 222

23. Realization of the high-performance THz GaAs homojunction detector below the

frequency of Reststrahlen band,

P. Bai, Y. H. Zhang, X. G. Guo, Z. L. Fu, J. C. Cao, and W. Z. Shen,

v

Appl. Phys. Lett. 113, 241102-(1-5) (2018) ............................................................ 229

24. Formation of qualified BaHfO3 doped Y0.5Gd0.5Ba2Cu3O7-δ film on CeO2 buffered

IBAD-MgO tape by self-seeding pulsed laser deposition,

L. F. Liu, W. Wang, Y. J. Yao, X. Wu, S. D. Lu, and Y. J. Li,

Appl. Surf. Sci. 439, 1034-1039 (2018) .................................................................. 234

25. Existence of electron and hole pockets and partial gap opening in correlated

semimetal Ca3Ru2O7,

H. Xing, L. B. Wen, C. Y. Shen, J. M. He, X. X. Cai, J. Peng, S. Wang, M. L. Tian,

Z. A. Xu, W. Ku, Z. Q. Mao, and Y. Liu,

Phys. Rev. B 97, 041113(R)-(1-5) (2018) ............................................................... 240

26. A novel seed/buffer-layer construction for enlarging c-directional growth sector in

high performance YBa2Cu3O7−δ bulk,

J. Qian, L. T. Ma, G. H. Du, H. Xiang, Y. Liu, Y. Wan, S. M. Huang, X. Yao, J.

Xiong, and B. W. Tao,

Scr. Mater. 150, 31-35 (2018) ................................................................................. 245

27. Growth and structural characterisation of Sr-doped Bi2Se3 thin films,

M. Wang, D. J. Zhang, W. X. Jiang, Z. J. Li, C. Q. Han, J. F. Jia, J. X. Li, S. Qiao,

D. Qian, H. Tian, and B. Gao,

Sci. Rep. 8, 2192-(1-7) (2018) ................................................................................. 250

28. Cavity optomechanical spectroscopy constraints chameleon dark energy scenarios,

J. Liu and K. D. Zhu,

Eur. Phys. J. C 78, 266-(1-9) (2018) ....................................................................... 257

29. Multiphoton-resonance-induced fluorescence of a strongly driven two-level system

under frequency modulation,

Y. Y. Yan, Z. G. Lu, J. Y. Luo, and H. Zheng,

Phys. Rev. A 97, 033817-(1-17) (2018) .................................................................. 266

30. Exotic odd-even parity effects in transmission phase, (Andreev) conductance, and

shot noise of a dimer atomic chain by topology,

B. Dong and X. L. Lei,

Ann. Phys. 396, 245-253 (2018) ............................................................................. 283

1

一、实验室年度数据简表

实验室名称人工结构及量子调控教育部重点实验室

研究方向 (据实增删)

研究方向 1 人工材料物性的计算研究与结构设计

研究方向 2 半导体量子结构与量子过程调控

研究方向 3 高温超导材料生长调控与机理

研究方向 4 表面和界面量子现象与调控

研究方向 5 小量子系统凝聚态理论

实验室主任

姓名沈文忠研究方向半导体量子结构与量子过程调控

出生日期 1968-5-22 职称教授任职时间 2009 年-今

实验室副主任

姓名贾金锋研究方向表面和界面量子现象与调控


实验室副主任

姓名朱卡的研究方向小量子系统凝聚态理论


实验室副主任

姓名钱冬研究方向表面和界面量子现象与调控


学术委员会主

任

姓名甘子钊研究方向高温超导材料生长调控与机理

出生日期 1938-4-16 职称教授（院士）任职时间 2009 年-今

研究水平

与贡献

论文与专著

发表论文 SCI 74 篇 EI 0 篇（未统计）

科技专著国内出版 0 部国外出版 1 部（章节）

奖励

国家自然科学奖一等奖 0 项二等奖 0 项

国家技术发明奖一等奖 0 项二等奖 0 项

国家科学技术进步奖一等奖 0 项二等奖 0 项

省、部级科技奖励一等奖 0 项二等奖 0 项

2

项目到账总经费

3207.5 万元纵向经费 3108.75 万元横向经费 98.75 万元

发明专利与成果转化

发明专利申请数 15 项授权数 6 项

成果转化转化数 2 项转化总经费成果年度产值：

超过 6.2 亿

标准与规范国家标准 0 项行业/地方

标准 0 项

研究队伍

建设

科技人才

实验室固定人员 42 人实验室流动人员 19 人

院士 1 人千人计划长期 2 人

短期 0 人

长江学者特聘 5 人

讲座 1 人国家杰出青年基金 6 人

国家“万人计划” 2 人青年长江 1 人

青年千人计划 7 人国家优秀青年基金 0 人

其他国家、省部级人才计划

32 人自然科学基金委

创新群体 1 个

教育部创新团队 2 个科技部重点领域创新团队 1 个

国际学术机构任职

姓名任职机构或组织职务

刘荧美国物理学会（APS）会士

贾金锋《Surface Review and Letters》副主编

马红孺《Chinese Physics Letters》副主编

贾金锋《Advanced Quantum Technologies》编委

贾金锋《Nature quantum materials》编委

贾金锋《Condensed Matter》编委

沈文忠《Nano-Micro Letters》编委

沈文忠《Frontiers in Energy》编委

刘灿华美国物理联合会（AIP）

AIP Publication

China Advisory

Board

朱卡的《EPJ Quantum Technology》编委

朱卡的《Scientific Reports》编委

3

刘荧《Chinese Physics B》编委

沈文忠 International Photovoltaic Science and

Engineering Conference (PVSEC)

International Advisory

Committee Member

沈文忠 International Conference on Silicon

Photovoltaics

Reviewing Committee

Member

访问学者国内 1 人国外 2 人

博士后本年度进站博士后 3 人本年度出站博士后 1 人

学科发展

与人才培

养

依托学科 (据实增删)

学科 1 物理学学科 2 材料科学学科 3

研究生培养在读博士生 71 人在读硕士生 26 人

承担本科课程 2062 学时承担研究生课程 588 学时

大专院校教材 0 部

开放与

运行管理

承办学术会议国际 1 次国内

(含港澳台) 3 次

年度新增国际合作项目 6 项

实验室面积 2320m2 实验室网址

http://klasqc.physics.sjtu.edu.cn/

主管部门年度经费投入 (直属高校不填)万元依托单位年度经费投入 130 万元

4

二、研究水平与贡献

1、主要研究成果与贡献

上海交通大学“人工结构与量子调控”教育部重点实验室建设项目于2009

年2月获批启动，2012年6月顺利通过教育部的验收，正式成为教育部重点实验室。2015年度数理、地学领域教育部重点实验室五年工作评估中被评为优秀类实验室。

实验室获批建设以来，从国家高新技术需求和学科前沿的有机结合点出发，针对人工电子/光子结构体系及其相应的量子调控中的重大基础科学问题，选取已在人工结构及量子调控领域有雄厚工作基础和条件、可望在国际科技竞争中占有一席之地的有限目标作为突破口，形成了五个特色鲜明的研究方向：（1）人工材料物性的计算研究与结构设计，（2）半导体量子结构与量子过程调控，（3）高温超导材料生长调控与机理，（4）表面和界面量子现象与调控，（5）小量子系统凝聚态理论。成立以来实验室围绕人工电子/光子结构，以人工结构设计、构造与组装、特异性能表征及应用、量子过程调控、原型器件与理论分析这一系统研究工作为主线，不仅在拓扑绝缘体量子现象、半导体量子器件、高温超导材料物理和小量子系统凝聚态基础理论等方面取得一批国际学术界领先的基础研究成果，而且成功开拓相关第二代高温超导带材和高效硅基太阳电池技术的产业化应用，已经成为国内外有显著特色的人工结构及量子调控领域创新研究基地。

实验室依托于上海交通大学物理与天文系凝聚态物理国家重点学科，已形成一支相对稳定、学术水平高、具有创新意识和团队精神的学术队伍。近年来，实验室从学科建设和队伍建设实际出发，按重点领域和优先次序，持续对学科和人员结构进行优化。本年度，贾金锋教授入选 2018 年第三批国家“万人计划”科技创新领军人才；钱冬教授入选“2018 年科技部中青年科技创新领军人才”并入选 2019 年第四批国家“万人计划”科技创新领军人才；史志文特别研究员入选“上海市千人计划”；郑浩特别研究员荣获“求是杰出青年学者奖”并入选2018 年度上海市“曙光学者”计划。实验室整体人才队伍不断壮大，学科布局和人员梯队更加合理。至 2018 年底，实验室在职人员有固定人员 42 人、行政服务人员 5 人，其中正教授 18 人，40 岁以下研究骨干 12 人。此外，还有兼职教授、访问学者及博士后等流动人员 19 人。

固定人员中包括中国科学院院士一人（雷啸霖）、国家“千人计划”入选者二人（刘荧、顾威）和一批优秀学术带头人。学术带头人中六人获国家杰出青年科学基金（沈文忠、贾金锋、郑杭、马红孺、刘荧、王孝群），六人为教育部“长

5

江学者奖励计划”特聘/讲座教授（沈文忠、贾金锋、马红孺、姚忻、刘荧、钱冬），五人为“百千万人才工程”国家级人选（郑杭、马红孺、沈文忠、贾金锋，王孝群），两人曾入选“教育部跨世纪优秀人才计划”（朱卡的、马红孺），两人入选国家“万人计划”科技创业领军人才（贾金锋、钱冬），一人入选中组部“拔尖人才计划”、科技部中青年科技创新领军人才（钱冬）；此外，还有二人入选上海市“领军人才计划”（马红孺、贾金锋）。

在中青年学术骨干中，有七人入选国家“青年千人计划”（罗卫东、李耀义、马杰、郑浩、张文涛、史志文、王世勇），三人入选上海市“千人计划”（李贻杰、郑浩、史志文），一人入选教育部“青年长江学者”（刘灿华）、四人入选“教育部新世纪优秀人才”计划（董兵、刘世勇、钱冬、刘灿华），三人入选上海市“曙光学者”（钱冬、刘灿华、郑浩），二人入选上海市“东方学者”(钱冬、史志文)，三人入选上海市“浦江人才”计划（董兵、刘灿华、管丹丹），二人入选上海市“启明星”计划（钱冬、蔡子）。

贾金锋教授带领的“新型量子材料物理和器件”研究团队入选 2015 年度国家自然科学基金委创新研究群体和 2016 年科技部创新人才推进计划重点领域创新团队。沈文忠教授带领的“半导体量子结构与量子过程调控”群体为教育部“长江学者与创新团队发展计划”2005 年创新团队（2013 年获滚动支持）。王孝群教授领衔的“计算物理方法的发展及其在新奇量子效应研究中的应用”群体入选2007 年度教育部“长江学者和创新团队发展计划”创新团队。实验室部分学术带头人参与的“人工微结构科学与技术协同创新中心”入选教育部“2011 计划”（南京大学为牵头单位）。

2018 年度，实验室人员年度发表国际学术专著一部（章节）；SCI 论文 74

篇，其中实验室人员以通讯作者完成 53 篇（其中三篇为综述文章）。年度论文中包括在影响因子 8.0 以上的国际一流期刊上正式发表论文 18 篇，其中为主发表 13 篇，合作发表 5 篇。

为主发表的 13 篇高水平论文包括：Nature Nanotechnology 一篇(最新影响因子 37.49) ，Advances in Physics: X 一篇综述文章（最新影响因子 30.917），Nature

Communications 三篇（最新影响因子 12.353），Nano Energy 一篇（最新影响因子 13.12），Nano Letters 一篇 (最新影响因子 12.08)，Physical Review Letters 二篇（最新影响因子 8.839），Journal of Materials Chemistry A 一篇（最新影响因子 9.931），Chemistry of Materials 二篇（最新影响因子 9.89），ACS Appl. Mater.

Interfaces 一篇（最新影响因子 8.097）。另外，在国际光伏科学与技术领域最高水平学术刊物 Progress in Photovoltaics: Research & Applications（最新影响因子

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6.456）上发表两篇论文，其中一篇为封面论文。以上重要论文，实验室均为第一作者或通讯作者单位。

与其他科研机构合作发表的 5 篇高水平论文包括：PNAS 一篇（最新影响因子 9.504），Physical Review X 一篇（最新影响因子 14.385），Physical Review

Letters 一篇（最新影响因子 8.839），Science Advances 二篇（最新影响因子11.511）。

本年度，实验室在开放运行与交流方面取得重要成果。以上 16 篇高水平论文中包括访问学者、美国田纳西大学副教授周海东来访实验室期间完成的工作在《Physical Review Letters》上发表论文二篇。实验室分别为通讯作者单位和第四完成单位。

此外，2018 年以第一完成单位完成四项工作的成果论文已分别被 Nano

Energy（最新影响因子 13.12）、Nano Letters (最新影响因子 12.08)、Physical

Review Letters（最新影响因子 8.839）、Carbon（最新影响因子 7.08）接收；一项合作工作成果论文被 Nature Physics（最新影响因子 22.727）接收。以上五篇论文于 2019 年初正式发表。

2018 年 11 月，美国科睿唯安(Clarivate Analytics)在线公布了全球 2018 年“高被引科学家(2018 Highly Cited Researchers)”名单。上海交通大学共入选 12

人，实验室学术带头人贾金锋教授、钱冬教授入选。其中，钱冬教授为连续第二年入选全球高被引科学家榜单。

2018 年度，实验室组织大型全国性学术会议一次(第十四届中国太阳级硅及光伏发电研讨会)；合作组织全国性学术会议一次 CCMP 2018（第四届凝聚态物理会议）；组织国际研讨会一次；主持行业协会年会一次。实验室固定人员参加国际会议 26 人次，其中作邀请报告 15 人次；参加国内会议 30 余人次，作邀请报告 21 人次。研究生参加国际会议 5 人次，国内会议 35 人次；指导研究生获权威会议优秀论文奖 1 篇。培养研究生获“国家奖学金”等各类奖励 16 人次；指导学生获上海高校学生创造发明“科创杯”三等奖及第四届中国“互联网+”大学生创新创业大赛铜奖；指导学生获全国大学生物理学术竞赛二等奖 1 项，上海市大学生物理学术竞赛特等奖、三等奖各 1 项。获上海交通大学卓越教学奖 1

人，上海交通大学教书育人奖三等奖 1 人，烛光奖一等奖 1 人；主持国家级教学项目 1 项，参与 1 项。本年度新增国际交流项目 6 项，申请国家发明专利 15 件，获国家发明专利授权 6 件；年度经费到款 3207.5 万元。

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各方向年度主要科研内容和成果如下：

研究方向一：人工材料物性的计算研究与结构设计

该方向本年度的主要进展包括：

1）2018 年我们采用有限元分析和第一性原理计算相结合的多尺度方法，完

成了金刚石对顶砧（DAC）能产生最高压力的计算工作，对传统 DAC 是否能产

生 500GPa 或以上压力给出了一个比较合理的理论解释。这是个很重要的问题，

特别是对利用 DAC 高压合成固态金属氢有重要意义，因为固态金属氢的相变点

发生在至少 500GPa 的压力下。但文献中只有早期的一些计算报道，有的计算认

为 DAC 最高能产生 900GPa 的压力，有的计算则认为最高只能产生 300GPa 的压

力。这些理论计算显然和现在的实验结果矛盾，金刚石对顶砧产生的压力早已突

破 400GPa，但远不能达到 900GPa。我们的有限元分析和第一性原理多尺度计算

结果表明，如果对金刚石对顶砧的表面结构进行优化设计，有可能使产生的压力

达到 500GPa 或以上。计算研究结果 2018 年底已在网上正式发表【Carbon 144, 161

(2019) 】。

2）研究了 Ba3CoSb2O9 体系中零场基态反铁磁 120°结构的量子效应与磁场下的“上-上-下”量子态相互作用的相变，通过测量中子衍射和非弹性中子散射实验外加磁场下的磁结构和自旋波，观察到强量子效应导致的磁结构、自旋波以及多磁子连续谱随磁场的变化，明确了由非线性自旋波理论发展的准经典模型的适用范围，不仅在理论上为 Ising 模型及现有的量子态修正研究进行了推广和改进，并为该体系后续实验的晶格调整、掺杂、原子取代、外场调控以及相关的量子理论模型的构造等等研究提出了切实可行的方向【Nature Communications 9, 2666

(2018)】。

3)生长并研究了三阶钙钛矿氧化物（二层非磁性层+一层磁性层） A3NiNb2O9

(A= Ba, Sr, Ca)的结构及相关动力学物性。我们发现虽然 A3NiNb2O9 保持非共线120°反铁磁相为基态，但是随着 A 位离子半径的减小(Ba2+->Ca2+)，磁性 Ni2+离子的晶格从等边三角形（A=Ba）转变为等腰三角形（A=Sr 和 Ca）, 进而在高温诱发一个自旋在 c 方向有倾角的反铁磁过渡相。在线性自旋波理论模拟了非弹性中子散射数据的基础上，揭示了三角晶格中超超交换作用与面间相互作用的竞争关系。这一研究表明，利用在三角晶格中非磁性原子调节磁性原子的物性的方法是切实可行的；为后续的极端条件(高压强、强磁场)量子调控打下基础【Phys. Rev.

B 98, 094412 (2018)】。

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研究方向二：半导体量子结构与量子过程调控

在太阳能光伏科学技术与新型光电子器件的应用基础研究工作中，本年度在

半导体纳米结构的可控制备、光电特性量子调控及新型太阳电池的应用基础研究

方面取得较大进展。取得的重要阶段性突破包括：1）创新提出并实现了一种新

型背结太阳电池—采用 PEDOT:PSS/Si 异质结作为空穴传输层(HTL)和 MgOx/Si

异质结作为电子传输层(ETL)的非掺杂异质结全背太阳电池。创新使用掩埋 ETL

方法，不仅简化了电池结构和制造过程，使其完美适用于溶液法 HTL 的制备

(PEDOT:PSS 膜)，同时保证了 gap 区域的钝化。电池实验最高效率为 16.3%，理

论模拟有望实现超过 22%的转换效率，证明了使用溶液法制造高效非掺杂异质结

全背太阳电池的可行性。研究成果发表在【Nano Energy 50, 777-784 (2018)】。2）

首次提出了一种结合时域有限差分法和光路分析法的杂化计算方法，系统地计算

了倒置式钙钛矿/晶硅两端叠层太阳电池基于各真实参数的性能结果。从光电角

度分析了这类电池的电流损失，提出了提高电池效率的细节优化方案，指出了改

进前表面金字塔绒面的钙钛矿/晶硅两端叠层电池效率可达 29%以上。研究成果

以封面论文形式发表在【Progress in Photovoltaics: Research & Applications 26,

924-933 (2018)】。3）在太阳电池产业化开发方面，优化了背面结硅异质结电池

中各层材料的工艺，阐明了背面结硅异质结电池工艺窗口宽，适于规模量产的根

本原因，提出直接铜金属化技术应用在背面结异质结电池上获得了 22.06%的效

率，研究成果发表在【Progress in Photovoltaics: Research & Applications 26,

385-396 (2018)】；实现了一种改善金刚线切割多晶硅太阳电池外观并实现准全

向宽光谱响应的技术方法，通过改进金属辅助化学腐蚀技术，有效消除多晶晶花、

改善电池外观，在电池背表面引入 SiO2/SiNx 叠层钝化膜，有效地提升电池背表

面的钝化效果，两者相结合实现了高效金刚线切割多晶硅太阳电池的大规模生产，

电池平均转换效率与传统电池相比增幅达 1.2%（绝对值），研究成果发表在【Solar

Energy Materials and Solar Cells 179, 372-379 (2018)】。

在低维量子材料光学研究方面获得两项新突破，包括：1）实现纳米探针操

纵石墨烯堆叠畴壁。拓扑位错和层错在很大程度上影响着晶体材料的性质和功能。

比如，金属材料中位错的密度决定着材料的机械强度；双层石墨烯堆叠畴壁会局

域地改变石墨烯的能带结构并出现沿着畴壁的拓扑边缘态。人们已经在控制材料

位错来改变其特性方面做了很多尝试，但是操作单个位错来改变局部的材料性质

一直是一个很大的挑战。运用扫描近场光学技术首次实现了双层和三层石墨烯中

单个堆叠位错的纳米操纵，可以对石墨烯中的位错进行移位、拆分甚至完全擦除，

从而获得完美的没有位错的大面积单晶石墨烯。该发现为石墨烯在纳米光电子器

件领域的应用提供了技术保障，另一方面，该发现也为研究石墨烯堆叠畴壁中丰

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富的物理提供了一个平台。该成果发表在【Nature Nanotechnology 13, 204

(2018) 】。2）发展了一种高频交流电阳极氧化纳米加工技术。基于局域阳极氧

化的扫描探针纳米刻蚀技术原则上可用于多种材料的纳米加工，然而对于微纳尺

度小样品由于需要预先加工微纳电极，导致其实际应用受到很大限制。发展了一

种全新的局域阳极氧化加工技术，该加工不需要预先加工微纳电极，可以直接对

目标材料进行微纳直写加工。其工作原理是：用高频交流电替换直流电，电流可

穿过基底绝缘层。因此，只需在针尖和基底之间加电压即可实现对目标材料的局

域阳极氧化加工刻蚀。不仅可以加工导电材料，还可以加工绝缘材料；加工精度

~10 nm，方便、快捷。该成果发表在【Nano Letters 18, 8011 (2018)】。

本方向其他进展还有：1）分级的 MoS2/Ni3S2 核壳结构纳米纤维在碱性介质中用于高效和稳定的水全分解。非贵金属电催化剂代替 Pt 催化剂用于高效的析氢反应被高度期望着通过水分解实现 H2 的可持续生产。我们采用一种简单方法在石墨烯包覆泡沫镍上制备出氨插层 MoS2 纳米片修饰的 Ni3S2 核壳纳米纤维结构，这种分级纳米结构提供了充足的析氢活性位点。此外，DFT 计算表明插入的氨离子提高了 MoS2 对氢的化学吸附作用，从而进一步提高了析氢反应活性。在 1M KOH 电解质中，合成的 MoS2/Ni3S2 纳米纤维阴极在 10mA·cm-2 的电流密度下的过电势为 109mV。结合镍铁层状双氢氧化物阳极，在 1 M KOH 电解质中实现了长达 100h 的稳定的完全水分解，在 10mA·cm-2 的电流密度下表现出 1.59V

的低电压。该成果已发表在【Materials Today Energy 10, 214-221 (2018)】。2）对高性能同质结太赫兹探测器进行了优化，并通过一个简单的谐振腔设计，使得该探测器的响应率达到了 6.8A/W,在同类的半导体太赫兹探测器中达到了很优的性能【Appl. Phys. Lett. 113, 241102-(1-5) (2018)】。另外，针对同质结太赫兹探测器，进行了上转换成像方面的研究，设计了上转换器件的结构，并通过黑体观察到了成像光斑。

研究方向三：高温超导材料生长调控与机理

第二代高温超导带材研究方面本年度主要开展了 C276 金属基带上超导薄膜

的人工钉扎、准多层膜和表面处理及纳米化研究工作，具体研究成果包括：1）

研究了 BaHfO3（BHO）、BaZrO3（BZO）、CeO2 和 SrTiO3（STO）第二相纳米

颗粒人工微结构与超导性能之间的关系，阐明了 BHO 和 BZO 为最佳掺杂相。

通过种子层技术提高了 5%BHO 掺杂的 YGBCO 超导薄膜的临界电流密度和磁通

钉扎强度，在 4.2 K，0-9T 和 65 K，0-9T，磁通钉扎力密度 Fp 的最大值分别为

860 GN/m3 和 15.8GN/m3，与没有种子层的 5%BHO 掺杂 YGBCO 超导薄膜的

Fp4.2,max=577.6GN/m3，Fp65,max=11.7GN/m3 比较，分别有了 49%和 35%的提高。2）

通过纳米掺杂 CeO2 夹层，优化了 CeO2 夹层的 YGBCO/CeO2 多层膜，不但拓宽

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了此多层膜工艺，而且还能作为有效的磁通钉扎中心，进而提高其在磁场下的性

能。3）通过对 REBCO 超导薄膜在合适的溶液中浸泡和电化学进行表面处理，

发现能够提高 REBCO 超导薄膜的超导转变温度，这为以后继续提高超导转变温

度提供了思路，奠定了基础。4）开展了 REBCO 超导薄膜表面 Ag 纳米颗粒修饰

的研究工作。实验结果表明在 REBCO 超导薄膜表面沉积银纳米颗粒，既可以提

高超导转变温度，又可以提高临界电流。5）在产业化方面，公里级带材临界电

流达到 300 安培以上，在中国率先实现了 YBCO 高温超导带材的商业化生产与

销售，YBCO 带材性能指标处于世界先进行列。

超导物理机制研究和器件应用方面，本年度在高性能超导块材制备，薄膜微

结构人工调控及超导薄膜热稳定性等方面取得重要结果。利用面内取向为 45°的

薄膜籽晶并结合具有潜在[110]边界的缓冲层，发展了一种新颖的籽晶/缓冲层结

构，有效地增大了块材中的 c 畴生长区。由新型结构诱导的样品的超导性能比传

统样品提高了约 10%, 研究结果发表在【Scripta Materialia 150, 31-35 (2018)】。

通过研究表面张力、液体组分、薄膜籽晶热稳定性三者关系的研究，引入富钡相

缓冲层材料，用于诱导生长掺银的 NdBCO 体系的块材，有效提高了薄膜籽晶的

热稳定性，研究结果发表在【Scripta Materialia 152, 154-157 (2018)】。为了将初

始阶段的溶液状态由不确定态调节到平稳的确定态，提出引入调控因子—初始保

温时间，并结合之前已成熟采用的溶剂质量、熔化时间等变量控制，实现了溶液

状态的精细调控，从而首次在 (110) NdGaO3 基板上外延出 a 轴晶粒镶嵌在 c 轴

膜上的复合人工微结构 YBCO 膜，获得了优异的超导应用性能，研究结果发表

在【J. Appl. Cryst. 51, 714-179 (2018)】。证实了不同材料的基板并不会影响含有

缓冲层薄膜的热稳定性，但是中间层的厚度对薄膜的热稳定性有重要影响，通过

优化中间层的厚度，可以降低薄膜的界面能，从而提高其热稳定性，研究结果发

表在【J. Am. Ceram. Soc. 101, 1704-1710 (2018)】。提供优质晶体，参与超导机

制合作研究，合作论文发表在【Sci. Adv 4, eaao5235 (2018)】上。

本研究方向还在二维晶体器件的电输运性质、电荷密度波和超导等量子有序态等方面取得一些系列进展。1）二维材料 TaSe2 的 Li 离子插层及量子输运研究。课题组在 2018 年初发表的工作中，实验基于机械解理的不同厚度的 TaSe2 的场效应晶体管器件，揭示了离子液体 gating 过程中的 Li 离子插层的微观过程，对二维材料 2H 相 TaSe2 中的电荷密度受 Li 离子插层的调制行为做了深入研究。该成果发表于【Appl. Phys. Lett. 112, 023502 (2018)】。值得一提的是，该文章被APL 选为 Editor’s Pick, 作为推荐文章，在 APL 主页及其 Facebook，tweeter 等被重点宣传。2018年下半年，课题组在对该体系的后续工作中研究了其超导和CDW

与层厚的关系，获得了两者的反关联证据，工作发表于【Nanotechnology 30,

11

035702 (2019)】。2018 年课题组的一个重要工作是对二维 TaSe2 的量子输运的研究。TaSe2 具有强自旋轨道耦合，在输运中体现为弱反局域化。锂离子插层改变了自旋轨道散射和非弹性散射的强弱，从而使得出现弱反局域化到弱局域化的过度的现象。通过调节栅极电压，可以将自旋弛豫长度从 20nm 增长到 60nm，而使电子关联长度从 300nm 减少到 13nm。这些结果说明离子插层具有调节自旋弛豫性质的功能，并与该二维体系中超导温度的提高有潜在的联系。目前该结果已经投稿。2）准一维半金属 Ta2NiSe7 的电荷密度波（CDW）研究。课题组在前期对 Ta2NiSe7 材料的工作基础上【Appl. Phys. Lett. 111, 052405 (2017)】，进一步发现了该材料表现出特异的磁阻各向异性，并揭示了各向异性磁阻可以用作准一维电荷密度波材料的敏感的探测手段。【Appl. Phys. Lett. 113, 192401 (2018)】。3）Ca3Ru2O7 的能带依赖的电输运研究。本课题组近期发展了具有“k 选择”性的热电输运测量。该测量方法从传统的热电效应测量出发，通过对单晶样品沿不同方向切割加工，是的测量时的温度梯度沿着不同的晶体方向。从玻尔兹曼定态方程出发，可以发现，热电势信号中始终存在 vkcosk 的一项，即具有不同费米速度 vk 的载流子的贡献总是耦合到一个 cosk 系数。基于这样的考虑，当沿着不同晶体方向外加温度梯度时，测量到的就是具有不同 vk 的载流子的贡献。在Ca3Ru2O7 中，我们通过“k 选择”性的热电测量，很明显的证明了沿着 M 和 M’两个方向分别存在电子和空穴型的费米口袋，对于解决 Ca3Ru2O7 的低温电子结构这一长期悬而未决的问题提供了重要实验证据。【Phys. Rev. B 97, 041113(R)

(2018)】。

研究方向四：表面和界面量子现象与调控

本年度表面和界面量子现象与调控方向针对拓扑量子材料开展了深入研究，

取得创新性成果如下：

1）拓扑节线半金属 ZrSiSe 中的“半缺失”Umklapp 散射。ZrSiSe 是新一代

拓扑半金属—狄拉克节线半金属的代表材料，其晶体结构具有的非点式空间群，

这决定了其表面必然存浮动带型非传统表面态。实验室成员成员利用扫描隧道显

微镜对真空中原位解理的 ZrSiSe（001）面进行单缺陷准粒子干涉测量。发现在

Zr 缺陷上存在违反传统倒逆散射定义的“半缺失”Umklapp 散射，结合计算和

理论分析，发现这个效应来源于其晶体结构的非点式对称性产生的浮动带表面态，

并适用于具有非点式对称性晶体结构的材料。同时在四重对称性的晶体表面发现

具有二重干涉信号的 Si 缺陷，并且随着能量增加恢复四重对称性，在计算中考

虑其非点式对称的晶体结构后，理论与实验现象符合。这个结果首次揭示了“半

缺失”Umklapp 散射，这一浮动带表面态的标志性物理性质，为理解拓扑节线半

金属奠定了基础。研究成果发表于【Nat. Comm. 9, 4153 (2018)】。

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2）利用 FeNi/FeSe/STO 薄膜的界面自旋耦合确定 FeSe 薄膜的磁性基态。单层 FeSe/STO 薄膜呈现出最高 109K 的超导行为，迄今为止其物理起源不清楚。更有意思的是，单层 FeSe 的拓扑性质也可能是非常特殊的。如果认为 FeSe 薄膜中存在长程反铁磁序，那么单层 FeSe 超导体就可以成为一个二维拓扑超导体。体材料 FeSe 中并没有观测到过反铁磁有序，普遍认为 FeSe 中不存在长程反铁磁序。但是，理论计算建议在 FeSe/STO 界面反铁磁有序可能会被大大增强。实验室成员针对这个问题开展了细致研究。制备了 FeNi/FeSe/STO 这样的二维异质结薄膜，利用界面上的磁交换相互作用来研究 FeSe/STO 非超导母体体系中是否可能存在反铁磁序。高精度的测量发现，在非超导 FeSe/STO 母体中出现磁交换偏置的现象，确凿无疑地说明单层 FeSe/STO 中确实可以存在反铁磁序，反铁磁序的相变温度在 140K 以上。这一结论支持单层 FeSe 薄膜超导体是一种拓扑超导体。研究成果发表【Phys. Rev. Lett. 120, 097001 (2018)】。

研究方向五：小量子系统凝聚态理论

1）带有z场调制的强驱动两能级系统：多光子共振诱导的荧光光谱。用反旋

波杂化旋转波的方法研究非共振和多光子共振情况下频率调制的强驱动两能级

荧光光谱。通过联合幺正变换和Floquet理论，CHRW方法可以从有效哈密顿量

中高效地给出双模Floquet态和准能量，与此形成对比，利用通常的广义Floquet

理论求解双色驱动两能级系统原始哈密顿量求解能量态和准能量效率较低。我们

根据上述求解的结果，利用驱动两能级系统的主方程来说明自发辐射，求得耗散

双色驱动两能级的时间演化动力学，稳态和荧光谱。论文发表在【Phys. Rev. A 97,

033817 (2018)】。

2）共同欧姆谱热浴中两杂质自旋玻色子模型量子相变。综合运用多元变分

计算（Davydov-1）研究了共同玻色环境中两量子比特系统的基态相变（欧姆谱

下两自旋玻色子模型）。应用超过上万个变分参数数值变分方法，研究了基态能

量、自旋磁化和自旋相干性以及与欧姆浴有关的观测值，研究发现线性离散化的

数值变分结果与对数网格离散化精确对角化和变分计算结果相比显示出更高的

精度。在弱隧穿中得到了临界耦合强度αc≈0.31。极限下，它与ED结果αc=0.26

相当，非常接近但不同于数值重整化群的结果NRG(0.18)和QMC的结果（0.22）

和其他数值研究（0.5、0.16和0.125）。在这些早期工作中对转变点的估值可能

是由过渡点的不正确标准引起的。本工作提供了文献中关于临界耦合强度争论的

解决途径。而且在很强的反铁磁耦合区中欧姆谱情况下基态相变属于一级相变，

不同于铁磁区（K<0）或没有自旋耦合(K=0)的情况下的Kosterlitz–Thouless 相变。

此外，我们还研究了隧穿常数和偏置场ε影响，并给出了相图。结果发表在【Ann.

der. Phys. 530, 1800120 (2018)】。

13

3）低维拓扑材料是近年来凝聚态领域研究的热点课题。有实验证据表明一

维体系中也发现拓扑性质，比如已经测量到了一维系统中电子的量子化的Zak相

位。这些实验刺激了基于Su-Schrieffer-Hegger模型的dimer原子链的研究复兴。该

原子链的一个重要理论预言是存在分布在两端的零能边界态，而且它们之间的耦

合强度依赖于原子链长。如何从实验上测量到该零能边界态成为一个重要的问题，

而输运测量毫无疑问是所有测量手段中反映边界态信号的最重要的方法。我们的

理论工作应用非平衡格林函数方法仔细研究了一维dimer原子链在连接正常电极

和/或超导电极情况下的电子输运性质，指出了当存在与不存在零能边界态是电

子输运性质的明显不同，包括线性电导，透射相移及非平衡散粒噪声。我们发现：

在拓扑相下（存在零能边界态），与非拓扑相相比其透射相移，线性电导和/或

线性Andreev电导，以及电流噪声随原子链长表现出完全相反的奇偶宇称变化；

当原子链一端与超导电极相联时，零能边界态能够控制Andreev束缚态的形成，

并导致Andreev非线性电导表现出零偏极大；特别奇怪的是零能处透射相移，在

拓扑相下相移表现出2pi的连续变化，而非拓扑相下相移仅有pi的变化。这可能是

一个关键性的证据【Ann. Phys. 396, 245 (2018)】。

4）研究了相互作用量子点连接一个超导端和两个铁磁导体端系统的电子隧穿性质。对这样的杂化三端结构，偏压加到两铁磁电极中的一端(源)，而另一端(漏)和超导端均接地。可以定义两种不同的电导，一是源端电流与偏压的比值，称为局域电导，另一是漏端电流与偏压的比值，为非局域电导。量子点中由于超导邻近效应会产生 Cooper 对，同时由于强 Coulomb 相互作用及与两电极的隧穿连接会导致 Kondo 效应。我们主要讨论量子点中超导邻近效应与 Kondo 效应间的合作与竞争及其对量子点局域及非局域线性电导的影响。我们发现：当超导杂化效应较强(即超导端的隧穿概率大于量子点与源与漏两端的隧穿概率)时，在Kondo 区会出现非局域电导为负的现象；当两端铁磁导体的极化率较大时 Kondo

区以及电导为负的区域会变小；当量子点能量偏离 Kondo 区时，超导端的作用会被屏蔽。

14

2、承担科研任务

实验室面向国家在人工结构和量子调控领域的重大需求，着力解决拓扑绝缘

体相关量子现象、半导体量子器件物理和高温超导电性机理领域的关键科学问题，

并开拓相关研究成果的转化和应用，承担了一批重要的科研项目。

2018 年度启动的主要项目为郑浩特别研究员主持的自然科学基金重大项目

课题:“新型拓扑超导体和马约拉纳准粒子的实验室研究”（560 万元）。2018

年度在研的其他重要项目（开始时间为 2018 年之前）还包括：王孝群教授为首

席科学家的国家重点研发计划“量子调控与量子信息”重点专项“量子自旋阻挫

体系和自旋液体中的新奇量子效应及调控研究”；国家重点研发计划重点专项课

题两项：“拓扑二维体系的界面量子调控”、“界面和拓扑超导研究”，钱冬教

授和刘灿华教授分别为课题负责人；贾金锋教授主持的自然科学基金委创新群体

项目“新型量子材料物理和器件”、自然科学基金重点项目“人造拓扑超导体与

Majorana 费米子的研究”；钱冬教授主持的基金委大科学装置联合基金重点项目

“大能隙量子自旋霍尔效应薄膜研究”；沈文忠教授主持的企业重大合作项目（中

天光伏材料有限公司）；李贻杰教授主持上海市科委高新技术产业化项目“第二

代高温超导带材性能提升关键工艺及产线智能化控制技术开发”；承担中组部“青

年千人计划”六项、上海市“东方学者”项目一项，主持国家自然科学基金面上

项目 16 项，青年项目 1 项。

这些重大项目的启动与设立，将促进实验室创新性研究的充分开展，有利于

在科学前沿领域实现重点突破。同时，充足的科研经费也为实验室新一年度的研

究任务的顺利执行提供有力的保障。

在执行国家/省部级重大基础研究任务的过程中，实验室重视项目的过程管

理，项目负责人注重科研项目完成质量和效益，取得了很好的成效。其中，王孝

群教授为首席科学家的量子信息与量子调控重大研究专项在 2018 年举行的项目

中期评估中获得“优秀”评价。同时，实验室紧紧围绕国家光伏、超导材料领域

重大战略决策，大力推动具有前瞻性、颠覆性、引领性技术创新成果的转化，为

相关产业转型升级提供新技术、新产品，为企业跨越式发展提供战略支撑。本年

度与上海超导科技股份有限公司、中天科技、协鑫集成、神舟新能源、隆基乐叶

光伏、东方日升等国内外知名企业继续保持密切合作，有力地推动了科学技术的

产业化。实验室成果转化的中天光伏材料有限公司的“太阳电池背板产品”销售

额近年内快速增长，2018 年度销售额达 6.2 亿人民币。

15

在项目申请方面，2018 年新立项项目 20 余项（大部分项目开始时间为 2019

年），合同经费总额近 1200 万元。主要包括：沈文忠教授主持的自然科学基金重点项目：“新型钙钛矿/硅异质结两端叠层太阳电池物理与器件研究”（直接经费 310 万）和国家重点研发计划项目课题：“双面电池先进结构设计仿真和表面钝化技术研究”（课题经费 467 万）。此外，李正平担任国家重点研发计划项目课题“电池表面低复合钝化技术、新型 PN 结/背场结构的设计制备技术”的参与单位负责人（负责经费 64.8 万）；王孝群教授主持的“中国-韩国凝聚态物理前沿研讨会”项目受 2018 年度国家自然科学基金委员会与韩国国家研究基金会联合资助；贾金锋教授申请 2018 年度国家自然科学基金委员会与香港研究资助局联合科研资助合作研究项目获批设立（课题经费 100 万）；李贻杰教授的基金委联合基金项目、刘灿华教授的面上项目、邢辉副研究员的青年项目获批立项；郑浩特别研究员的上海市“曙光学者”计划、史志文特别研究员的上海市“千人计划”以及 10 余项与企业合作的横向课题等项目都将于 2019 年启动。

2018 年在研各类项目 80 余项，合同总金额超过 1.36 亿元，其中超过 100

万的主要项目（所有自然科学基金面上、青年项目作为一个项目）17 项，合同经费共 1.11 亿元。年度科研经费实际到款 3207.5 万元，其中超过 90%来至合同金额超过 100 万元的国家和省部级重大项目以及重大横向课题。年度新增各类科研项目 21 项，其中主要项目（开始时间为 2019 年）4 项，合同金额 1012 万元。

16

本年度承担主要任务（合同经费超过 100 万元）如下：

序

号项目/课题名称编号负责人起止时间

经费 (万元)

类别

1

量子自旋阻挫体系和自旋液体中的新奇量子效应及调控

研究

2016YFA0300500王孝群（项目首席）

2016-2021 3200 科技部国家重点研发计

划项目

2 新型量子材料物理和器件

/ 贾金锋 2016-2018 1200 国家自然科学基金委创新研究群体

3

拓扑二维体系的界面量子调

控 / 钱冬 2016-2021 847

科技部国家重点研发计划项目课题

4 界面和拓扑超

导研究 / 刘灿华 2016-2021 730

科技部国家重点研发计划项目课题

5

新型拓扑超导体和马约拉纳准粒子的实验

室研究

/ 郑浩 2018-2022 560 国家自然科学基金重大

项目

6

人造拓扑超导体与 Majorana费米子的研究

/ 贾金锋 2017-2021 310 自然科学基金重点项目

7

大能隙量子自旋霍尔效应薄

膜研究 / 钱冬 2017-2020 267

基金委大科学装置联合基金重点项

目

8 青年千人计划

启动资金 / 李耀义 2016-2018 200+100

中组部青年千人计划及校内配套


启动资金 / 马杰 2016-2018 200+100



启动资金 / 史志文 2017-2019 200+100



启动资金 / 张文涛 2017-2019 200+100


12 青年千人计划 / 王世勇 2017-2019 200+100 中组部青年

17

启动资金千人计划及校内配套


启动资金 / 郑浩 2016-2018 200+100


14

“东方学者”岗位计划特聘教授资助经费

/ 史志文 2016-2018 100 上海市“东方学者”

15 光伏材料产学

项目合作 / 沈文忠

2013-2016(合同延至2019 年)

500

中天光伏材料有限公司企业横向合

作项目

16

第二代高温超导带材性能提升关键工艺及产线智能化控制技术开发

16521108302 李贻杰 2016-2018 250 上海市科委高新技术产业化项目

17

2018 年度在研自然科学基金面上项目 16

项，青年项目 1项

/ 沈文忠

等 1322+32

自然科学基金面上项目、

青年项目16+1

以下为年度新增项目

18

双面电池先进结构设计仿真和表面钝化技

术研究

2018YFB1500501 沈文忠 2019-2021 467 科技部国家重点研发计划项目课题

19

新型钙钛矿/硅异质结两端叠层太阳电池物理与器件研究

11834011 沈文忠 2019-2023 310 自然科学基金重点项目

20

在拓扑晶体绝缘体/超导体异质节中寻找

Majorana 零能模

1181101036 贾金锋 2019-2022 100

NSFC 与香港研究资助局联合科研资助合作研

究项目

21

2019 年自然科学基金联合项目 1 项、面上项目 1 项，青年项

目 1 项

李贻杰刘灿华邢辉

2019-2022 145 自然科学基

金

18

三、研究队伍建设

1、各研究方向及研究队伍

研究方向学术带头人主要骨干

1、人工材料物性的计算研究与结构设计孙弘、马红孺王孝群、顾威

罗卫东、马杰、蔡子

2、半导体量子结构与量子过程调控沈文忠

郑茂俊、史志文、张月蘅、刘洪、徐林、

李正平、潘葳

3、高温超导材料生长调控与机理姚忻、刘荧

李贻杰刘林飞、邢辉

4、表面和界面量子现象与调控贾金锋、钱冬

刘灿华

管丹丹、李耀义、张文涛、郑浩、王世勇

5、小量子系统凝聚态理论雷啸霖、郑杭朱卡的、董兵

王沁、吕智国、罗旭东、刘世勇、丁国辉

2、本年度固定人员情况

序号姓名性别年龄最后学位

类型技术职称

在实验室工作期限

1 沈文忠男 50 博士研究人员教授 2010 年至今

2 雷啸霖男 80 学士研究人员教授

(院士) 2010 年至今

3 刘荧男 56 博士研究人员教授 2012 年至今

4 王孝群男 56 博士研究人员教授 2013 年至今

5 马红孺男 58 博士研究人员教授 2010 年至今

6 郑杭男 67 博士研究人员教授 2010 年至今

7 贾金锋男 51 博士研究人员教授 2010 年至今

8 顾威男 49 博士研究人员教授 2015 年至今

19

9 孙弘男 61 博士研究人员教授 2010 年至今

10 朱卡的男 58 博士研究人员教授 2010 年至今

11 姚忻男 63 博士研究人员教授 2010 年至今

12 李贻杰男 56 博士研究人员教授 2010 年至今

13 钱冬男 41 博士研究人员教授 2010 年至今

14 董兵男 50 博士研究人员教授 2010 年至今

15 郑茂俊男 56 博士研究人员教授 2010 年至今

16 王沁男 58 博士教学为主教授 2010 年至今

17 袁晓忠男 56 博士教学为主教授 2010 年至今

18 刘灿华男 42 博士研究人员教授 2010 年至今

19 郑浩男 38 博士研究人员特别研究员 2016 年至今

20 王世勇男 32 博士研究人员特别研究员 2017 年至今

21 罗卫东男 40 博士研究人员特别研究员 2013 年至今

22 张文涛男 33 博士研究人员特别研究员 2015 年至今

23 史志文男 35 博士研究人员特别研究员 2015 年至今

24 李耀义男 34 博士研究人员特别研究员 2015 年至今

25 马杰男 34 博士研究人员特别研究员 2015 年至今

26 蔡子男 34 博士研究人员特别研究员 2017 年至今

27 邢辉男 35 博士专职科研助理研究员 2015 年至今

28 刘世勇男 42 博士研究人员副教授 2010 年至今

20

29 吕智国男 43 博士研究人员副教授 2010 年至今

30 罗旭东男 44 博士研究人员副教授 2010 年至今

31 李晟男 42 博士教学为主副教授 2010 年至今

32 张月蘅女 44 博士研究人员副教授 2010 年至今

33 徐林男 46 博士研究人员副研究员 2010 年至今

34 丁国辉男 48 博士研究人员副教授 2010 年至今

35 刘洪男 40 博士专职科研副研究员 2011 年至今

36 刘林飞女 38 博士专职科研副研究员 2010 年至今

37 管丹丹女 36 博士专职科研副研究员 2013 年至今

38 潘葳女 37 博士研究人员讲师 2010 年至今

39 李正平男 42 博士专职科研助理研究员 2013 年至今

40 蒋立峰男 41 博士管理人员讲师 2010 年至今

41 赵西梅女 44 硕士管理人员讲师 2010 年至今

42 蒋震宗男 40 学士技术人员工程师 2010 年至今

以下为行政人员

43 徐秀琴女 64 学士行政人员 2010 年至今

44 韩辉女 36 硕士行政人员 2013 年至今

45 纪敏捷女 34 硕士行政人员 2015 年至今

46 程莹女 34 硕士行政人员 2016 年至今

47 黄彬彬女 33 学士行政人员 2018 年至今

21

3、本年度流动人员情况

姓名性别年龄从事

专业

技术

职称

来自国家

工作单位在实验室工作期限

1 Anthony J. Leggett

男 78

凝聚态物理

教授，诺贝尔物理学奖获得者，美国科学院、美国知识学会、美国艺术与科学学院院士，俄罗斯科学院外籍院士，英国皇家学会、美国物理学会、美国物理联合会会士，英国物理

学会荣誉院士

美国

伊利诺伊大学厄巴纳—香潘恩

分校

2013 年至今每年一个月

2 周海东男 40 材料

科学副教授美国田耐西大学

2017 年起每年 1 个月

3 任杰男 41 物理学副教授中国常州理工大学 2018 年起

每年 1 个月

以上为兼职教授(访问学者)，以下为博士后研究人员：

4 Daniel Crow

男 33 凝聚态物理导师：刘荧美国 2017 年至今

5 钟思华男 31 凝聚态物理导师：郑茂俊中国 2015 年至今

6 顾亮亮男 32 凝聚态物理导师：雷啸霖中国 2016 年至今

7 Anthony Charles Hegg

男 30 凝聚态物理导师：王孝群美国 2017 年至今

8 黄易珍女 34 凝聚态物理导师：蔡子中国 2017 年至今

9 李传维男 33 凝聚态物理导师：王孝群中国 2017 年至今

10 王暾男 30 凝聚态物理导师：沈文忠中国 2018 年至今

22

11 唐天威男 29 凝聚态物理导师：钱冬中国 2017 年至今

12 王国华男 33 凝聚态物理导师：钱冬中国 2016 年至今

13 任清勇男 32 凝聚态物理导师：马杰中国 2017 年至今

14 Sudeshna

Sen 女 32 凝聚态物理导师：顾威印度 2017 年至今

15 王顺权男 35 凝聚态物理导师：沈文忠中国与企业合作培养 2017 年至今

16 杨黎飞男 33 凝聚态物理导师：沈文忠中国与企业合作培养 2017 年至今

17 王锐男 32 凝聚态物理导师：王孝群中国 2017 年至今

18 苏威男 32 凝聚态物理导师：王孝群中国 2018 年至今

19 Waqas

Mahmood

男 35 凝聚态物理导师：董兵巴基斯坦

2018 年至今

23

四、学科发展与人才培养

1、学科发展

实验室所依托的上海交通大学物理与天文学院凝聚态物理学科 2002 年被教育部批准为国家重点学科，但当时主要研究方向为凝聚态物理理论、软凝聚体物理理论与实验、半导体光电子物理和超导单晶生长等方面，研究力量还是比较单薄和分散的。2006 年初我国明确将“量子调控与未来信息科学技术基础”列入国家中长期科学和技术发展规划纲要（2006-2020）重点研究领域，当时我们就认识到量子调控势必承载在人工结构新材料上，人工结构及量子调控研究会是凝聚态物理、材料物理、电子信息最活跃的前沿领域，因为它不仅能为未来信息、材料科学奠定新的物理基础，也能为量子新物态、新材料、新器件的发展提供新思路，将是 21 世纪高新技术发展的重要基础。

在这样的背景下，凝聚态物理国家重点学科从国家需求和学科前沿的有机结合点出发，针对人工电子/光子结构体系及其相应的量子调控中的重大基础科学问题，选取已在人工结构及量子调控领域有雄厚工作基础和条件、可望在国际科技竞争中占有一席之地的有限目标作为突破口。结合理论物理学科，将凝聚态物理理论方向集中于人工材料物性的计算研究与结构设计和小量子系统凝聚态理论领域；引进优秀人才将半导体光电子物理和超导单晶生长方向分别拓展为与半导体量子器件密切相关的半导体量子结构与量子过程调控和高温超导材料生长调控与机理研究领域。2009 年 2 月我们的“人工结构及量子调控”教育部重点实验室获批建设；2010 年起又进一步根据国际上拓扑绝缘体研究热潮，引进了表面和界面量子现象与调控优秀研究团队，2012 年 6 月实验室顺利通过教育部组织的验收。通过验收几年以来，实验室着重加强了杰出人才的引进和培养，凝聚研究方向和研究内容，已经逐步成为国内外有显著特色的人工结构及量子调控领域创新研究基地。在 2015 年度数理、地学领域教育部重点实验室五年工作评估中被评为优秀类实验室。

近年来，在“新型量子材料物理与器件”国家自然科学基金委创新研究群体项目、“量子自旋阻挫体系和自旋液体中的新奇量子效应及调控研究”国家重点研发计划重点专项、科技部重点领域创新团队项目等国家重大科研、团队建设项目的带动上，上海交通大学将实验室作为凝聚态物理国家重点学科建设的主要载体，加大投入，重点予以建设。随着“985—新一轮学科建设”、 “青年千人计划启动经费及校内配套”、 “东方学者”“上海市千人计划”等人才项目的展开，以及新进的多位海外归国优秀中青年科研人员的全职投入，实验室呈现出良好的发展态势，并瞄准国际学术前沿和国民经济发展需求对研究方向的原有布局进行了完善和加强。目前实验室已形成了五个研究方向，七支各有特色的研究团

24

队，具有开放民主、紧密协作的学术氛围和团队文化。已在拓扑绝缘体量子现象、半导体量子器件、高温超导材料物理和小量子系统凝聚态基础理论等方面取得一批国际学术界领先的基础研究成果，同时成功开拓相关第二代高温超导带材和高效硅基太阳电池技术的产业化应用。

“人工结构及量子调控”教育部重点实验室的建设支撑了上海交通大学凝聚态物理国家重点学科的发展，并有力地推动了与我校理论物理、材料物理和微电子与固体电子学等学科的交叉与合作，为我校物理学一级学科几年来的快速发展做了重要贡献。在 2017 年公布的教育部第四轮一级学科整体水平评估中，上海交通大学物理学被评为 A 类学科，在参评的 127 所高校中排名并列第三位。上海高校高峰高原学科建设计划是上海市教委公布的《上海高等学校学科发展与优化布局规划（2014—2020 年）》中的重点任务，旨在引导高校结合经济社会发展需求，通过重点突破，以点带面，优化上海高校学科布局结构，整体提升上海高校学科建设水平；以“国家急需、世界一流”为根本出发点，培养一流创新人才，加速建立能够冲击世界一流的新优势和新实力。2018 年底公布的上海市高峰高原学科中，上海交通大学物理学入选Ⅱ类高峰，随着该计划的稳步推进以及上海交通大学对凝聚态物理学科的重点建设，将为实验室的发展带来新的机遇。

25

2、科教融合推动教学发展

本实验室的教学工作着眼于国家发展和人的全面发展需要，坚持传授知识、

培养能力、提高素质协调发展，着力提高学生的学习能力、实践能力和创新能力。

实验室学术带头人在潜心科研的同时，也时刻铭记为社会培养、输送高素质人才

是高等学校教师的主要任务。因此，实验室所有教授均担任了本科生、研究生基

础课程或主要专业课程的主讲教师；共 6 位教授、副教授担任教学为主岗位，不

安排科研任务；新进以科研为主的中青年教师，也必须承担本科生、研究生的培

养工作。本年度，共有 22 位教师担任本科教学工作，开设 27 门课程，授课 2062

学时；16 位教师担任研究生教学工作，开设 21 门研究生专业课，授课 588 学时；

13 位教师被列入上海交大致远学院物理学师资队伍；指导 6 人完成本科生毕业

设计。其中，在史志文特别研究员的指导下，本科生张怡然、李宏元分别进入加

州理工学院和加州大学伯克利分校攻读博士学位。在实验室学术带头人开设的课

程中，不仅有基础理论知识课程，还有专门介绍凝聚态学科前沿热点、最新成果

的《固体物理专题》、《物理研究实践》、《专业物理实验》等专题课程；针对

不同的学生类型，部分课程采用双语教学或者全英语授课。本年度，袁晓忠教授

荣获上海交通大学第五届卓越教学奖（全校仅 5 名）；吕智国副教授获上海交通

大学教书育人奖三等奖、烛光奖一等奖，并主持和参与国家级教学项目各一项，

校级项目 3 项。

“本科生研究计划（简称 PRP）”是上海交通大学为培养具有“宽厚、复合、

开放、创新”特征的高素质创新人才而制定的一项教学改革，实验室结合本领域

的研究情况，年度内设立适合本科生完成的 PRP 项目 5 项（其中一项为国家级

项目），将基础理论与科学实践相结合，大大拓宽了本科生的科学视角，也为他

们研究生阶段更快熟悉科研工作打下了坚实的基础。

在教学实践中，实验室教师积极探索并建立以问题和课题为核心的教学模式，

倡导以本科学生为主体的创新性实验改革，鼓励学生开展课外科技、实验和创新

实践活动，积极参与到国际前沿学术研究中。本年度，实验室学术带头人李贻杰

教授指导王伟、郑通、刘顺帆获第二十四届上海高校学生创造发明“科创杯” 发

明创新三等奖及第四届中国“互联网+”大学生创新创业大赛上海交通大学校内

选拔赛二等奖、上海赛区铜奖。

中国大学生物理学术竞赛（CUPT）是实践国家教育中长期发展规划纲要的

重要大学生创新竞赛活动，是国内具有重要影响力的大学生物理竞技赛事之一，

被列入中国物理学会物理教学指导委员会的工作计划。潘葳老师带队参加 2018

年第九届中国大学生物理学术竞赛（CUPT），荣获上海大学生物理学术竞赛特

26

等奖及三等奖各一项，华东赛区比赛二等奖，中国大学生物理学术竞赛全国二等

奖。

以上成果的取得，表明了实验室引导大学生进行创新性科学实践的教学工作

是卓有成效的。

27

3、人才培养

（1）人才培养总体情况

“围绕提高自主创新能力、建设创新型国家，以高层次创新型科技人才为重

点，努力造就一批世界水平的科学家、科技领军人才、工程师和高水平创新团队，

注重培养一线创新人才和青年科技人才，建设宏大的创新型科技人才队伍。”是

《国家中长期人才发展规划纲要》对高等院校人才培养方向的指导方针。实验室

成立以来以培养拔尖创新人才为宗旨，以“知识探究、能力建设、人格养成”三

位一体为理念，以“创新性、多元化和国际化”为驱动，形成了有自身特色的人

才培养模式。年度代表性举措及成果包括：

1、坚持“少而精”的教学模式，培养精英型科技人才

实验室将研究生的培养目标定位为物理学基础学科培养一批精英型人才，因

此，长期以来坚持“少而精”的教学模式。由一批热爱教育事业、学术造诣深厚、

具有国际视野的导师，对有志于攀登世界科学高峰的优秀学生予以精心的专门指

导。要求研究生不仅应具有扎实的科学文化知识、精良的专业技能、高尚的道德

情操、健康的身体及心理素质，而且应该具有适应科学技术不断发展、解决实际

问题的能力及创新能力。在导师的全心投入和重点指导下，实验室培养了一批掌

握本领域坚实的基础理论和宽广的专门知识，掌握解决实际问题的先进方法和现

代技术手段，了解本专业的国内外现状和发展方向，勇于在学术前沿深入探索的

优秀研究生代表。研究生以第一作者在 Science、Nature Materials 等顶尖学术期

刊上发表一批高水平论文，已成为实验室科研工作的中坚力量。

本年度，实验室博士研究生朱锋锋同学的博士论文《新颖量子材料的角分辨

光电子能谱研究》获得上海交通大学第二届优秀博士学位论文奖（全校 13 篇）。

朱锋锋同学在实验室学术带头人贾金锋教授和钱冬教授的指导下，从事表面与界

面效应及物理方向的研究。他的关于锡烯薄膜的实验实现的重要研究成果 2015

年以第一作者发表于国际顶级期刊《Nature Materials》上。在博士研究生期间，

他累计完成学术论文 17 篇，其中第一作者 4 篇。曾获得博士生国家奖学金、人

工微结构科学与技术协同创新中心英才奖一等奖等各类奖励。目前朱锋锋同学在

德国亥姆霍兹研究中心做博士后，继续相关领域的研究工作。

2、注重应用型人才培养

科学技术的进步最终要体现在对生产力的推动上，科研成果的转化与应用离

不开技术、应用型人才的培养。实验室根据部分研究方向与产业化应用紧密结合

28

的特色，有针对性对部分学生制定了特殊的研究生培养方案，要求高年级硕士研

究生和博士生在结束基础理论课程后，必须有一半时间深入企业了解研究领域的

产业化流程与标准，了解行业发展的瓶颈，并且在实验室获得的研究成果必须经

过企业的中试生产线的验证。在太阳能光伏和第二代高温超导带材领域，实验室

与国内外知名企业密切合作，一方面实验室为企业培训和输送了一批具有专业知

识背景的人才，另一方面，企业为青年教师、研究生提供了科研成果测试和应用

的实践平台。这样的举措，使得科研成果更加贴近产业化应用，更加符合企业需

求，能更好的服务于国民经济。

实验室主任、国内光伏科学与技术领域著名学者沈文忠教授课题组长期以来

把科研工作的目光聚焦于新型太阳电池的应用基础研究，以是否具有产业化应用

前景，是否有利于企业产业升级作为研究生科研工作的重要评价标准。本年度，

课题组优化了背面结硅异质结电池中各层材料的工艺，阐明了背面结硅异质结电

池工艺窗口宽，适于规模量产的根本原因，提出直接铜金属化技术应用在背面结

异质结电池上获得了 22.06%的效率，研究成果发表在【Progress in Photovoltaics:

Research & Applications 26, 385-396 (2018)】；实现了一种改善金刚线切割多晶硅

太阳电池外观并实现准全向宽光谱响应的技术方法，通过改进金属辅助化学腐蚀

技术，有效消除多晶晶花、改善电池外观，在电池背表面引入 SiO2/SiNx 叠层钝

化膜，有效地提升电池背表面的钝化效果，两者相结合实现了高效金刚线切割多

晶硅太阳电池的大规模生产，电池平均转换效率与传统电池相比增幅达 1.2%（绝

对值），研究成果发表在【Solar Energy Materials and Solar Cells 179, 372-379

(2018)】。以上两项太阳电池产业化开发方面的研究成果的第一作者分别为沈文

忠教授指导的企业博士后杨黎飞和博士生庄宇锋，相关研究也大部分在合作的光

伏企业完成。

3、增加良性竞争机制，探索创新型人才联合培养的模式

实验室还积极探索和实践与国内外科研机构间联合培养创新人才的新途径，

取得了优异成绩。本年度，博士生王闻捷同学赴澳大利亚国立大学，林豪、吴飞

同学赴中国科学院宁波材料技术与工程研究所进行长期交流学习。2019 年初，

博士生丁东赴挪威能源技术研究院短期交流。

本年度，沈文忠教授指导博士生林豪同学创新提出并实现了一种新型背结太

阳电池—采用 PEDOT:PSS/Si 异质结作为空穴传输层(HTL)和 MgOx/Si 异质结作

为电子传输层(ETL)的非掺杂异质结全背太阳电池。创新使用掩埋 ETL 方法，不

仅简化了电池结构和制造过程，使其完美适用于溶液法 HTL 的制备(PEDOT:PSS

29

膜)，同时保证了 gap 区域的钝化。电池实验最高效率为 16.3%，理论模拟有望

实现超过 22%的转换效率，证明了使用溶液法制造高效非掺杂异质结全背太阳电

池的可行性。研究成果发表在【Nano Energy 50, 777-784 (2018)】。本成果大部

分的测试和实验工作在中国科学院宁波材料技术与工程研究所完成。林豪同学和

沈文忠教授指导的另外一位硕士研究生吴飞同学（共同第一作者）在宁波材料所

期间的工作成果论文也已被《Nano Energy》接收并发表【Nano Energy 58, 817-824

(2019)】。

“人工微结构科学与技术协同创新中心(CICAM）”是由本实验室为主体的

上海交通大学凝聚态物理研究团队与南京大学（牵头单位）、复旦大学、浙江大

学、中国科技大学及中科院合肥物质科学研究院共六家凝聚态物理研究领域的科

研人员构成的科研机构，成立于 2014 年。CICAM 将六家科研机构的研究生队伍

纳入了统一培养和奖励体系，鼓励和推动六家单位各课题组研究生间的交换、交

流与合作、竞争。为表彰和激励该中心所属研究组的优秀博士研究生，CICAM

每年度设立三项奖励计划，包括优秀博士后奖、研究生英才奖和研究生入学奖。

本年度，实验室有 3 名在读博士研究生获得 CICAM 优秀研究生奖励计划英才奖

二、三等奖。

4、建立具有国际化视角的教学科研一体化教师队伍，坚持以人为本的教育理念

教师队伍水平的高低直接决定了人才培养的高度。教师的视野，决定了他施

教的广度和深度。为学生提供国际化教育，培养拥有全球视野的创新型人才是世

界一流大学的共识。依托单位上海交通大学正在朝着“双一流”大学的建设目标

迈进，实验室以此为契机，近年来在吸引和培养高水平中青年科研人员方面取得

长足进步。实验室 42 名固定人员中，有 25 人拥有超过一年的海外高水平科研机

构工作、学习的经历；2014 年起，新引进的年轻人员均具有世界一流大学博士

学位和学术工作经历，在学科前沿领域开展创新性研究，取得重要的研究成果，

具有很强的学术潜力，研究工作至少达到了世界一流大学助理教授水平。以“青

年千人”计划为主的海外归国青年学者已经在实验室承担越来越重要的研究工作，

成为实验室科研工作可持续发展的重要保障。目前实验室有中科院院士和“千人

计划”3 人，国家“万人计划”2 人，有“长江学者”“杰青”12 人；有“青年千

人”“青年长江学者”“上海市千人”等 32 人的各类国家/省部级中青年人才计

划获得者，已形成一支老中青结合的具有国际化视野的高水平人才梯队。

实验室骨干人员既是一名科研工作者也是人民教师，承担科研和教学两项基本工作。如何把科研和教学两项任务有机的结合起来，建立一支既具备高水平的

30

科研能力，又具备良好的教学能力的教学科研一体化教师队伍，是实验室人才培养的关键。实验室根据各位教师的特点制定不同的分工，取长补短，互相合作。并关注每位教师个体的发展，将教师个人特长、利益目标和实验室整体利益目标相结合。承认能力差异，在管理上重心下移，考核和激励不作一刀切，不作硬性规定。使得每位教师充分发挥其特长和自身价值，快乐工作，真正实现教育以人为本的宗旨。本年度实验室科研为主的中青年教师培养方面，培养钱冬教授入选“2018 年科技部中青年科技创新领军人才”计划并入选 2019 年第四批国家“万人计划”科技创业领军人才，史志文特别研究员入选上海市“千人计划”，郑浩特别研究员荣获“求是杰出青年学者奖”并入选 2018 年度上海市“曙光学者”计划；派出年轻教师刘洪副研究员、管丹丹特别副研究员、钟思华博士前往法国、美国、瑞士科研机构访学。在教学方面，袁晓忠教授获得上海交通大学第五届卓越教学奖；吕智国副教授获 2018 年度上海交通大学教书育人奖三等奖、烛光奖一等奖；吕智国副教授主持及参与国家级教学项目各一项；多位老师参与校级教学改革和教学研究项目 10 余项。

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（2）研究生代表性成果

在研究生培养过程中，实验室十分重视学生的创新能力、独立工作能力的培养，始终坚持高标准、严要求，并制定了规范的研究生管理条例和学术论文发表条例。研究生已成为实验室科研工作的中坚力量，其中优秀研究生代表有：

1、半导体量子结构与量子过程调控方向在沈文忠教授的带领下，本年度在太阳能光伏领域获得重大进展，团队研究生做出了重要贡献。其中，沈文忠教授指导的博士生林豪同学一年内以第一作者和共同第一作者在国际能源顶尖期刊 Nano

Energy（影响因子 13.12）上发表论文 2 篇，并获协鑫集成一等奖学金，其为第一作者在第 14 届中国太阳级硅及光伏发电研讨会上提交的论文被评为大会优秀论文；沈文忠教授指导的另一名博士研究生巴理想同学以第一作者在国际光伏科学与技术领域最高水平学术刊物 Progress in Photovoltaics: Research &

Applications（影响因子 6.456）上发表封面论文一篇，并获研究生国家奖学金、协鑫奖。郑浩、吕镱同学也获研究生国家奖学金；庄宇峰、王鑫、丁东获协鑫集成一等奖学金；吴飞、王星之分获欧普泰及华为一等奖学金。团队中张月蘅副教授指导的博士研究生白鹏同学以第一作者两年内发表了包括一篇综述在内的 3

篇 SCI 论文，申请 3 项国家发明专利，并获校优秀博士生奖学金。

2、表面和界面量子现象与调控研究方向在贾金锋教授的带领下，形成了严谨求实、着重创新的良好学术氛围。2018 年度，贾金锋教授和郑浩特别研究员指导的博士研究生朱朕同学以第一作者在 Nature Communications（影响因子 12.353）上发表论文。钱冬教授指导的博士生姜文翔同学年度内参加两次国际会议，在APS March meeting 2018 作口头报告。2018 年 3 月，上海交通大学公布了第二届优秀博士论文奖，全校共评选出 13 篇优秀论文。钱冬教授指导的博士生朱锋锋的博士论文《新颖量子材料的角分辨光电子能谱研究》入选。

3、其他优秀研究生成果还有：李贻杰教授指导王伟、郑通、刘顺帆获第二十四届上海高校学生创造发明“科创杯” 发明创新三等奖及第四届中国“互联网+”大学生创新创业大赛上海交通大学校内选拔赛二等奖、上海赛区铜奖。其中，王伟同学年度内申请国家发明专利 4 项，他也获得了研究生国家奖学金。史志文特别研究员的硕士研究生骆兴东同学获得 1984 级校友奖学金，博士研究生胡成同学获人工微结构科学与技术协同创新中心优秀研究生英才奖二等奖；刘荧教授指导的博士生杨宇森、顾威教授指导的博士生邹龙获优秀研究生英才奖三等奖。

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（3）研究生参加国际会议情况

序号参加会议形式学生姓名硕士/博士参加会议名称及

会议主办方导师

1 口头报告姜文翔博士 APS March meeting 2018 American Physical Society

钱冬

2 Poster 白鹏博士 EMN meeting on Terahertz

（2018）张月蘅

3 Poster 姜文翔博士 M2S-2018

Chinese Academy of Sciences 钱冬

4 Poster 李云龙博士 M2S-2018


5 Poster 景强博士 M2S-2018


33

五、开放交流与运行管理

1、开放交流

（1）开放课题设置情况

实验室于 2012 年通过教育部验收以后，遵循《教育部重点实验室建设与运行管理办法》的规定，充分开放运行，建立访问学者制度，设立开放课题，吸引优秀人才开展合作研究。2013 年度开始设立实验室开放课题。在实验室网站专门设立了开放课题专栏，公开接受课题申请。

实验室的开放课题特别面向国内优秀的年轻学者，希望能为他们明确研究方向、加快科研启动提供帮助。通过验收以来，经学术委员会审核通过，已为国内青年学者设立开放课题 11 项，其中 2013 年度 4 项，2014 年度 1 项，2016 年度4 项，2018 年 2 项资助经费共 44 万元，课题执行期为两年。在这些开放课题的资助下，2013-2018 年度共发表了包括 Phys. Rev. Lett.、Nanoscale 在内的 18 篇高水平 SCI 论文。

2018 年新设的开放课题二项，分别资助了淮海工学院宋晓敏（讲师）和苏州科技大学戴璐（副教授）。此外，因课题工作出色，2018 年对电子科技大学周海平副教授的开放课题给予后续资助。

2018 年度，开放课题申请者共发表标注实验室名称的 SCI 论文 3 篇，列表如下：

1. A comparative study on the direct deposition of uc-Si:H and plasma-induced recrystallization of a-Si:H: Insight into Si crystallization in a high-density plasma, H. P. Zhou, M. Xu, S. Xu, Y. Y. Feng, L. X. Xu, D. Y. Wei and S. Q. Xiao, Appl. Surf. Sci. 433, 285-291 (2018).

2. Low-temperature synthesis of graphene by ICP-assisted amorphous carbon sputtering, X. Ye, H. P. Zhou, L. Levchenko, K. Bazaka, S. Y. Xu, and S. Q. Xiao, ChemistrySelect 3, 8779–8785 (2018).

3. Interface properties of ITO/n-Si heterojunction solar cell: Quantum tunneling, passivation and hole-selective contacts, X. M. Song, M. Gao, Z. G. Huang, B. C. Han, Y. Z. Wan, Q. Y. Lei, and Z. Q. Ma, Solar Energy 173, 456-461 (2018).

开放课题的设立，为实验室与国内各单位学者间创造了学术接触、交流和讨论的良好环境。有利于拓宽原有研究方向的学术空间，有利于在学术方向上的集思广益、优势互补，形成创新机制，并有效提升了实验室在领域内的影响力和知名度。

34

实验室开放课题列表：

序号课题名称经费额度

承担人职称承担人单位课题起止

时间

1

宽禁带窗口层硅基异质结太阳电池界面钝化与载流子输运机理研究

4 万周海平副教授电子科技大学

2016.7-

2020.6

（延长资助）

2

ITO/n-Si 异质结中超薄界面层的性质与功能研究

4 万宋晓敏讲师淮海工学院 2018.7-

2020.6

3

纳米双手性多层螺旋带生长机理和机械性质的研究

4 万戴璐副教授苏州科技大学 2018.7-

2020.6

35

（2）主办或承办大型学术会议情况

序号会议名称主办单位名称会议主席召开时间参加

人数类别

1

2018 第十四届中国太阳级硅及光伏发

电研讨会

（14th CSPV）

半导体量子结构与量子过程调控团队（上海交通大学太

阳能研究所）、中国可再生能源学会、浙江大学、中山大学

石定寰

朱俊生

沈文忠

（会议副主席、秘书长）

2018.11.

8-10 1000 全国

2

第四届凝聚态物理会议

(CCMP 2018)

上海交大凝聚态物理研究所暨人工结构及量子调控教育

部重点实验室、复旦大学、中国科学院物

理研究所

陈澍

贾金锋

沈健

2018.7.

5-8 1000 全国

3

International

Workshop: New

Developments in

STM on Surfaces of

Functional Materials

上海交大凝聚态物理研究所暨人工结构及量子调控教育部重点实验室/李政

道研究所

贾金锋

顾威

等

2018.8.

26-28 100

国际研讨会

4

2018 年上海市太阳能学会年会暨先进技术集成研讨会

半导体量子结构与量子过程调控团队（上海交通大学太

阳能研究所）、协鑫集成科技股份有限

公司

沈文忠 2018.7.27 150 国内研讨会

36

（3）国内外学术交流与合作情况

实验室坚持请进来和走出去相结合，积极开展与国内外科研机构的合作与交

流，取得了一批实质性的合作成果，国内外的学术地位与影响力正在稳步提升。

本年度，继续邀请 2003 年诺贝尔物理学奖获得者、著名物理学家 Anthony J.

Leggett 教授担任兼职教授；邀请美国田耐西大学副教授周海东、常州理工大学

任杰副教授来实验室做访问学者，获实质性科研成果，2018 年发表物理顶尖刊

物《Physical Review Letters》论文二篇【Phys. Rev. Lett. 120, 227201 (2018)】【Phys.

Rev. Lett. 120, 147204 (2018)】。派出年轻教师刘洪副研究员、管丹丹特别副研

究员、钟思华博士前往法国、美国、瑞士科研机构访学。派出博士生王闻捷前往

澳大利亚国立大学交流学习。邀请国内外著名专家定期作学术报告，本年度共邀

请 35 位海内外著名学者作专题学术报告。

年度内，组织大型全国性学术会议一次(第十四届中国太阳级硅及光伏发电

研讨会)；合作组织全国性学术会议一次 CCMP 2018（第四届凝聚态物理会议）；

组织国际研讨会一次；主持行业协会年会一次。实验室固定人员参加国际会议

26 人次，其中作邀请报告 15 人次；参加国内会议 30 余人次，作邀请报告 21 人

次。研究生参加国际会议 5 人次，国内会议 35 人次；指导研究生获权威会议优

秀论文奖 1 篇。

国内交流方面，实验室骨干团队继续与南京大学（牵头单位）、复旦大学、

浙江大学、中国科技大学及中科院合肥物质科学研究院五家单位联合深入开展教

育部“2011 计划”— “人工微结构科学与技术协同创新中心(CICAM）”项目

的科研工作。项目针对后摩尔时代人类信息技术可持续发展的迫切需求，以新型

微结构材料中的量子调控科学与技术为核心，推动信息载体和信息处理手段从经

典到量子系统的演变，力争为新一代信息技术革命奠定材料和器件物理基础，已

取得了阶段性成果。

国际交流方面，实验室继续与美国、英国、日本、法国、韩国、加拿大等

10 余个国家的高水平的科研机构保持密切的学术合作。实验室在研各类国际合

作项目 9 项，包括：顾威教授与美国佛罗里达亚州立大学, 美国高磁实验室的合

作项目：“Nature of Mott transition in real materials”；顾威教授与美国露路易斯安

那州立大学、中科院物理所的合作项目：“Nature of Mott transition in real materials”；

史志文特别研究员与美国加州大学伯克利分校的合作项目：“低维材料的近场光

学研究”；马杰教授与美国田纳西大学、美国橡树岭国家实验室、美国国家计量

标准局合作的项目：“量子自旋液体合作研究”等。

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2018 年度，沈文忠教授新增中国与罗马尼亚科技合作委员会第 43 届例会交

流项目“增加光伏发电自有率的热能转换与储存系统”，并受项目资助出访罗

马尼亚；新增“中国-挪威”合作项目，受挪威方邀请，将于2019年初派出博士

研究生赴挪威能源技术研究院(Institute for Energy Technology)交流。王孝群教授

主持的“中国-韩国凝聚态物理前沿研讨会”项目受 2018 年度国家自然科学基金

委员会与韩国国家研究基金会联合资助；并申请获得 2019 年《亚洲自旋阻挫与

自选液体前沿研讨会》、2020 年《国际自旋阻挫与自选液体前沿研讨会》的举

办权。贾金锋教授与香港科技大学的 2018 年度国家自然科学基金委员会与香港

研究资助局联合科研资助合作研究项目获批立项。在 APS March Meeting 举行期

间，贾金锋教授作为中方代表之一参加了美国洛杉矶举行的“2018 年中美物理

学会负责人早餐会”；王孝群教授作为 IAUP C20 的成员（Fellow），参与相关

活动，并作为组织者之一，组织筹办 C20-2019 年在香港中文大学的国际学术大

会。

沈文忠教授带领的太阳能光伏研究团队与新南威尔士大学光伏与可再生能源工程学院、嘉兴市政府共同创建的嘉兴光伏高新技术产业园区光伏产业创新研究中心于 2016 年 6 月正式运营，研究中心拥有场地 2000 多平方米，其中办公、展示、培训场地 400 多平方米，试验测试场地 1600 多平方米。2017 年初嘉兴光伏众创空间在研究中心挂牌，该国际合作项目建设在 2018 年度继续深入推进，成为我国光伏行业产学研结合的示范基地和创业孵化器，研究中心还立项了多个有产业化前景的光伏项目。

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（4）科学传播

作为依托高等院校的科研单位，实验室不仅肩负科研与教学两项重要任务，也承担着进行科学传播的社会责任。本年度实验室的科学传播主要面向青少年，包括大学生和中学生，主要举措与成果如下：

1、定期邀请物理学各个领域的著名大师、学者做科学报告，向实验室研究生介绍物理学各分支的发展历史和最新研究进展。

2、实验室继续成为上海电力学院数理学院的“本科生科学认识实践”课程的合作基地。2018 年 12 月，接待该校本科生 80 余人进行科学实践，实验室助理研究员李正平博士专门作学术报告。这些活动，给予校外大学生利用实验室先进平台进行科研基本训练的机会，也让他们近距离的接触到凝聚态物理学科的学术前沿，以吸引更多优秀学生致力于物理学。

3、2018 年 10 月，第 35 届全国中学生物理竞赛决赛在上海交通大学举行，上海交通大学物理与天文学院承担了考务工作。实验室骨干教师刘世勇副教授、丁国辉副教授参加决赛命题，5 位教师参加了实验考试的监考工作。比赛结束后，实验室学术带头人、李政道研究所副所长、讲席教授顾威为参赛选手作的题为《衍生现象——物理世界的社会行为》的学术报告。

4、为深入学习贯彻十九大精神，提高党员活动参与率，提升学生党建活力，2018

年 8 月 28 日，实验室组织研究生党支部成功开展了“共行计划”暑期社会实践活动，组织支部党员赴上海航天汽车机电股份有限公司开展支部共建，双方围绕支部建设进行了充分的交流，并计划以新能源学科为纽带，开展一系列共建活动。此外，同学们参观了生产车间和装配一线，近距离感受了解新能源光伏重要零部件的加工过程，对科研成果的转化有了更直观的认识。

5、2018 年 11 月 18 日，实验室组织研究生党支部开展了“共行计划”社会实践活动暨第六期学生党员先锋论坛，组织实验室党员学生赴上海友兰科技有限公司开展支部共建，深入生产研发中心，近距离接触光触媒技术发展的尖端力量，对光触媒应用以及光触媒行业有了更直观的了解和认识。上海友兰科技有限公司始建于 2014 年，由中国科学院光催化博士尹浩创立，目前已在室内空气净化技术和消费产品领域推出高科技产品与服务，是阿里巴巴、金地集团、万科物业、华润地产等的战略合作单位。尹浩博士是本重点实验室筹建以来毕业的一批硕士研究生，导师为沈文忠教授。

6、2018 年 11 月 13 日，物理与天文学院青志队在上宝中学举办了科普讲座活动，此次科普活动由实验室的研究生贺飞同学担任主讲老师。

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2、运行管理

（1）学术委员会成员

序号姓名性别职称年龄所在单位是否外籍

1 甘子钊

(主任) 男

教授

(院士)80 北京大学否

2 沈学础

(副主任)男

教授

(院士)80 复旦大学否

3 薛其坤

(副主任)男

教授

(院士)55 清华大学否

4 祝世宁

(副主任)男

教授

(院士)69 南京大学否

5 陶瑞宝男教授

(院士)81 复旦大学否

6 孙昌璞男教授

(院士)56 北京计算科学研究中心否

7 闻海虎男教授 55 南京大学否

8 陆卫男研究员 56 中科院上海技术物理研

究所否

9 陈鸿男教授 58 同济大学否

10 张文清男研究员 52 南方科技大学否

11 雷啸霖男教授

(院士)80 上海交通大学否

12 郑杭男教授 67 上海交通大学否

13 马红孺男教授 58 上海交通大学否

14 沈文忠男教授 50 上海交通大学否

40

（2）学术委员会工作情况

按《教育部重点实验室建设与运行管理办法》的要求，实验室成立了由十四

位知名学者组成的学术委员会，指导实验室的学术方向，评估实验室的研究成果，

审议实验室的重大学术活动和年度工作计划、审批开放研究课题。实验室同时制

定了《人工结构及量子调控教育部重点实验室学术委员会工作条例》，对学术委

员会的组成和相关职能进行规范。

实验室通过验收以来每年均举行学术委员会会议。为了完整的对年度工作进

行梳理后向学术委员会汇报，每年的学术委员会会议均在次年度的4-5月份举行，

2018 年度的学术委员会将于 2019 年上半年举行。

考虑到 2018 年召开的 2017 年度学术委员会可能会受物理楼搬迁的影响（原

计划新理科大楼 2018 年 3 月开始至 6 月底搬迁完成），因此我们将 2017 年度学

术委员会召开的形式由现场会议变为通讯评议。通过函评的方式向学术委员会介

绍实验室的年度工作进展，请学术委员予以评议，并将收到了两位年轻学者对实

验室开放课题的申请提请学术委员会评审。我们的评议评审材料寄出后，收到了

14 位学术委员中的 11 位委员的反馈，超过学术委员会总人数的三分之二。给予

评议和评审的委员包括：主任：北京大学甘子钊院士；副主任：复旦大学沈学础

院士，南京大学祝世宁院士；委员：复旦大学陶瑞宝院士，中科院上海技术物理

研究所陆卫研究员，南京大学闻海虎教授，同济大学陈鸿教授，南方科技大学张

文清教授，上海交通大学的雷啸霖院士、马红孺教授和沈文忠教授。

学术委员会对实验室 2017 年的工作高度评价，11 位委员对实验室总体评价

均为优秀。同时，在“研究队伍建设”和“开放交流与运行管理”等方面也出了

宝贵建议。以下为评议函、评审评议反馈表及学术委员会主任意见：

41

42

43

44

（3）主管部门和依托单位支持情况

本实验室是依托单位上海交通大学校内注册的独立实验室，在实验室用房、

设备管理、人员编制等方面独立统计，并给予专职管理人员名额。上海交通大学

本年度为实验室划拨每年至少 100 万元的基本运行经费（具体额度由考核结果而

定），由实验室主任负责，专款专用，保证了实验室正常运转。同时，近年来，

学校已投入“985”新一轮学科建设经费、“青年千人”配套、教育部“中央高校

改善基本办学条件经费项目”等超过 2000 万元经费为实验室新引进的海外归国

青年科技人才提供科研启动经费。利用这些投入，实验室几年内搭建起了新的具

有国际领先水平的材料生长及测试平台，这些平台的建立，为实验室的可持续发

展奠定扎实的基础。

上海交通大学为实验室提供的科研用房全部集中在上海交通大学闵行校区

物理/物理实验群楼，分别在一、九和十层，总面积约 2320 平方米，下设计算凝

聚态物理实验室、凝聚态光谱与光电子物理实验室、超导和其它功能晶体生长实

验室、高温超导带材实验室、表面和界面量子现象与调控实验室以及太阳能研究

所等。上海超导科技股份有限公司为上海交通大学建造的新理科大楼在 2018 年

度已经完成建造，并开始进行室内装修，实验室将在 2019 年度完成整体搬迁（至

2019 年 3 月，已有部分课题组开始搬迁）。届时本实验室的科研用房面积又将

进一步增加，并且场地布局更加合理，科研环境更加安全规范。

实验室的建设运行中，关系到实验室发展大局的中长期规划通常由学术委员

会及实验室主任、学术带头人共同讨论制定，依托单位配合实施。上海交通大学

在人才引进、团队建设、研究生培养指标、自主选题研究等方面给予了充分的支

持。依托单位在管理上给予的自主权及优先权，为实验室更好的凝练研究方向，

完善学科结构提供了良好的条件。

上海交通大学十分重视对创新基地的培育和考核。学校科学技术发展研究院每年组织专家集中对校内各省部级重点实验室（2018 年为 30 个）进行考核，并将考核结果与实验室运行经费挂钩，促进了校内各实验室间的良性竞争。截至2018 年度，本实验室已经连续六年在上海交通大学校内评估中获评“优秀”。

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3、仪器设备

截至 2018 年底，本实验室共有 20 万元以上的大型设备 69 台（套），总价

值 10544 万元。其中年度新增 20 万元以上设备 3 台（套），共计 628 万元。2018

年度，表面和界面量子现象与调控方向、半导体量子结构与量子过程调控方向的

新测试平台已基本建设完成。其中，表面和界面量子现象与调控方向在郑浩特别

研究员主持下新搭建一套极低温微驱四探针扫描隧道显微镜系统（373 万元）；

半导体量子结构与量子过程调控方向史志文特别研究员课题组新购高分辨原子

力显微镜一套(105 万元)及飞秒激光器 OPO 系统一套(150 万元)。

实验室大型仪器设备均设专职管理员，负责重大仪器设备的登记、使用与维

护。在所制定的《人工结构及量子调控教育部重点实验室管理条例》中，对实验

室的设备购置、大型仪器设备使用与维护做出明确规范。实验室的管理贯彻“三

个整合、两个保证”的原则，即实验设施整合、研究队伍整合、学科交叉整合，

保证稳定的主体研究队伍、保证固定集中的实验研究平台与仪器共享平台，现有

的大部分的平台与上海交通大学其他实验平台实行资源共享。实验室仪器设备在

优先满足本实验室科学研究、教学实验需求的前提下，面向社会、开放使用，以

提高使用效率。

实验室鼓励固定人员积极开发仪器的新功能，表面和界面量子现象与调控方向刘灿华教授近年来利用已有设备完成了基于四电极 STM 的原位双线圈互感技术的设备研制。2015 年，该课题组发明了具有四个电极的 STM 探头从而将一台极低温强磁场 STM 的功能拓展到了物性测量方面，实现了表面电输运测量的功能【Review of Scientific Instruments 86, 053903 (2015)】。2016 年，该实验室进一步在该四电极 STM 探头的基础上，研制出了双线圈互感技术，首次实现了对超导薄膜的抗磁响应进行原位测量【Review of Scientific Instruments 88, 073902

(2017)】。2018 年，该课题组又完成了国家发明专利“一种基于迈斯纳效应的超导薄膜力学特性测量装置及方法”、“一种超高真空样品截断装置”的申请，用有限的经费实现了大型真空互联系统所追求的功能与作用。对已有设备平台进行升级改造和整合，并摸索出具有自主知识产权的材料生长和测试的创新性工艺、创新装置，已成为实验室实验物理基础研究取得重要成果的关键和基础。

年度代表性论文

Lettershttps://doi.org/10.1038/s41565-017-0042-6

1Department of Physics, University of California at Berkeley, Berkeley, CA, USA. 2University of Chinese Academy of Sciences and Institute of Physics, Chinese Academy of Sciences, Beijing, China. 3Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. 4Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China. 5Collaborative Innovation Center of Advanced Microstructures, Nanjing, China. 6Institute of Physics, Chinese Academy of Sciences, Beijing, China. 7Kavli Energy NanoSciences Institute at the University of California, Berkeley and the Lawrence Berkeley National Laboratory, Berkeley, CA, USA. Lili Jiang and Sheng Wang contributed equally to this work. *e-mail: [email protected]; [email protected]

Topological dislocations and stacking faults greatly affect the performance of functional crystalline materials1–3. Layer-stacking domain walls (DWs) in graphene alter its electronic properties and give rise to fascinating new physics such as quantum valley Hall edge states4–10. Extensive efforts have been dedicated to the engineering of dislocations to obtain materials with advanced properties. However, the manipula-tion of individual dislocations to precisely control the local structure and local properties of bulk material remains an outstanding challenge. Here we report the manipulation of individual layer-stacking DWs in bi- and trilayer graphene by means of a local mechanical force exerted by an atomic force microscope tip. We demonstrate experimentally the capabil-ity to move, erase and split individual DWs as well as anni-hilate or create closed-loop DWs. We further show that the DW motion is highly anisotropic, offering a simple approach to create solitons with designed atomic structures. Most artificially created DW structures are found to be stable at room temperature.

Recently, the layer-stacking domain walls (DWs) in graphene have attracted great interest because of their fascinating mechani-cal11, electrical4–9 and optical12 properties. Such topological one-dimensional DWs result from the transition between two different stacking orders (in bilayer AB↔BA, in trilayer ABA↔ABC), through shifting one layer of graphene with respect to its adjacent layer by a single carbon–carbon bond along one of the armchair directions11–13. The relative layer displacement occurring at the DW is known as the displacement vector. Depending on the crystallographic orien-tation and atomic structures inside, the DWs usually exhibit dis-tinct physical properties. Optically, soliton-dependent reflection of two-dimensional graphene plasmons at a bilayer graphene DW has been observed in experiments12. Electrically, the theoretical calcula-tion predicts that electronic transmission across a DW in bilayer graphene strongly depends on their atomic structures14; in trilayer graphene, the DW even produces in-plane metal (ABA)–semicon-ductor (ABC) heterostructures15–21, in which the role of DWs with different atomic structures is also unknown. Experimental explora-tion of the physical properties of different types of DWs is restricted by the determination of the atomic structures of the existing DWs. Directly creating DWs with designed atomic structures can be helpful for the investigation of soliton-dependent physics of the layer-stacking DWs.

Previous studies have shown that high-temperature heating13 or an electric field22 can generate DW motion in graphene layers. However, a controllable way to engineer the DWs into designed structures is still lacking. An alternative way to change the configuration of DWs is applying strain. For example, previous work has demonstrated the possibility to unfold vortices into topological stripes in multiferroic materials using strain23. Here, we demonstrate that the DWs in bi- or trilayer graphene can be moved by mechanical stress exerted through an atomic force microscope (AFM) tip. By controlling the movement of the AFM tip with great precision and flexibility, we realize con-trolled DW manipulation and create DWs with designed structures.

DWs are invisible in conventional AFM topography (Fig. 1a). To image the DWs in situ, we employed near-field infrared nanoscopy measurements4, 12 (based on tapping-mode AFM, see Methods). The near-field infrared image revealed distinct fine features within bi- or trilayer graphene due to the presence of DWs (Fig. 1b). The DW in bilayer graphene is characterized by a bright line between AB- and BA-stacked domains with the same optical contrast. In trilayer graphene, ABA- and ABC-stacked domains have differ-ent infrared responses, thus the DW is identified as the boundary between two regions with different optical contrast.

After locating the DWs, AFM is switched to the contact-mode to perform DW manipulation. To overcome the threshold energy of moving a DW, a blunted tip (Supplementary Fig. 1) and a large force between the tip and sample is used (see Methods). Figure 1c sche-matically shows DW movement using an AFM tip. By sliding the tip across the DW in trilayer graphene shown in Fig. 1d, we moved the DW in the sliding direction by 2 µ m (Fig. 1e).

Versatile DW manipulations are accomplished by our technique, including erasure and split of the DWs, as well as annihilation and creation of closed-loop DWs (Fig. 2). Figure 2a,b shows the DW erasure process in bilayer graphene. By executing successive line scans over the initial DW (Fig. 2a) in the defined area, the DW is erased (Fig. 2b). Similar erasure manipulation can also be employed to eliminate DWs in trilayer graphene (Fig. 2c,d). In the erasure process, the strain stored in the DW gets released at the graphene edge. The intermediate processes of partially erased DWs are dis-played in Supplementary Fig. 2. By pushing the middle section of a DW to a graphene edge, we can split one DW into two discrete ones with opposite displacement vectors (Fig. 2e,f). This is because when the DW hits the graphene edge, part of the DW annihilates at the edge and this induces the initial DW to separate into two sections.

Manipulation of domain-wall solitons in bi- and trilayer grapheneLili Jiang1,2, Sheng Wang1,3, Zhiwen Shi4,5*, Chenhao Jin1, M. Iqbal Bakti Utama1,3, Sihan Zhao1, Yuen-Ron Shen1,3, Hong-Jun Gao2, Guangyu Zhang6 and Feng Wang1,3,7*

© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

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Fig. 1 | Nano-imaging and manipulating DWs in bilayer and trilayer graphene. a, AFM topography image of an exfoliated graphene with both bilayer and trilayer segments on SiO2/Si substrate. b, The corresponding near-field infrared nanoscopy image in which a DW extends through bilayer and trilayer graphene. In bilayer graphene, the DW is characterized by a bright line in a homogeneous background, while in trilayer it is the boundary between two regions with different optical contrast, because the ABA (brighter) and ABC regions have different infrared responses. c, A schematic of DW manipulation using contact-mode AFM. The blue arrow indicates the moving direction of the tip. d, Near-field infrared image of trilayer graphene with a DW between ABA- (brighter) and ABC-stacked regions. e, Near-field infrared image of the reconstructed configuration of the DW after a single-line scan across the DW along the black arrow.

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Fig. 2 | Versatile manipulation of DW solitons in bilayer and trilayer graphene. a,b, Erasure of a DW in bilayer graphene. c,d, Erasure of a DW in trilayer graphene. e,f, Split of a DW in bilayer graphene. g,h, Annihilation of a closed-loop DW within a bilayer graphene. i,j, Creation of a closed-loop DW with an isolated ABA-stacked domain by line cutting. k,l, Creation of a right triangular shape DW (closed-loop) with isolated ABC-stacked domain by scanning along specific zigzag and armchair directions. The red dashed squares represent the scanning area and the black arrows indicate the scanning directions.


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Letters NaTurE NaNoTEcHNoLogy

We can also annihilate and create closed-loop DWs. Annihilation is a process where two DWs with exactly opposite displacement vectors collide with each other and disappear simultaneously. Interestingly, we found annihilation can also occur in a closed-loop DW. Figure 2g,h illustrates such a closed-loop DW in bilayer gra-phene and its annihilation by scanning in the red dashed square area. In fact, such closed-loop DWs are rare in exfoliated gra-phene based on our measurements of several hundreds of samples. Artificially creating loop-shape DWs could be of interest for future electronic transport study of these unusual DW structures. To this end, in Fig. 2i,j we show the creation of a closed-loop DW with an ABA-stacked domain inside by cutting through an existing domain. Figure 2k,l shows another example where a ‘right triangular’ closed-loop DW with an ABC-stacked domain inside is created by scan-ning along specific zigzag and armchair directions of the graphene lattice. We found most artificially created DW structures remained stable at room temperature for several months or even longer (Supplementary Fig. 3).

A careful examination of the DW structures generated by the AFM manipulation revealed that the DW motion exhibits remark-able anisotropy, which allowed us to create DWs with specific atomic structures (shear, normal or mixed) (Fig. 3). The atomic structure of a DW is determined by the angle between its displacement vector and the DW line11–13. For a shear-type DW, its displacement vector is parallel to the DW line and along one of the armchair orientations of the graphene lattice; for a normal-type (tensile or compressive) DW, the displacement vector is perpendicular to its DW line and along one zigzag orientation.

The anisotropic motion depends on the angle (θ) between the tip sliding direction and the displacement vector. It generally fol-lows three rules: (1) sliding the tip across a DW parallel to its dis-placement vector (θ = 0°) generates a strip structure following the scanning line with a curved front; (2) sliding perpendicular to the displacement vector (θ = 90°) generates a rectangular structure with the front along the displacement vector direction (shear segment); (3) sliding along other directions (0° < θ < 90°) generates a triangular

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Fig. 3 | angle-dependent investigation of DW motions in trilayer graphene. a–c, Line scans across a DW with known displacement vector. a, A shear-type DW in trilayer graphene. b, A line scan perpendicular to the displacement vector (θ = 90°, along one zigzag direction) generates a rectangle with a small protrusion in front. c, A line scan parallel to the displacement vector (θ = 0°, along one armchair direction) generates a strip-shaped domain area with curved front. d–g, Line scans across a DW with unknown displacement vector. A line scan along one zigzag direction creates a rectangle (e) and along one armchair direction creates a strip (f), which are similar to the structures in b and c, respectively. However, a line scan along another zigzag direction creates a triangular shape (g), with one side along the scanning line and the other side along armchair orientation. h,i, Line scan along one zigzag orientation across two unconnected DWs. The left DW rearranges into a rectangle with the front segment along the armchair direction. The right one forms a triangle in which one side of the triangle is parallel to the scanning line and the other side is along the armchair direction. The blue and red arrows display the zigzag and armchair directions, respectively, in all images.





structure with one side along the sliding direction and the other side along its displacement vector (shear segment). These rules enable us to determine the displacement vector of any DW in trilayer gra-phene and thus to create DWs with certain atomic structure by con-trolling the sliding direction relative to the displacement vector.

A straight DW in trilayer graphene with known displacement vector is shown in Fig. 3a. It extends directly to a bilayer region above. In bilayer graphene, shear and tensile DWs can be distin-guished by their plasmon reflection interference pattern in the near-field infrared nanoscopy images12. Thus, we know that the DW in Fig. 3a is an almost perfect shear-type (see Supplementary Fig. 4). As a result, its displacement vector is along the armchair orientation indicated by the red arrow in Fig. 3a. Consequently, we obtained the armchair and zigzag orientations of the underlying graphene flake (shown in the inset of Fig. 3a with red and blue lines, respectively). A line scan perpendicular to the displacement vector (θ = 90°, along one zigzag direction indicated by the blue arrow in Fig. 3b) over the shear-type DW creates a rectangular structure with a small protrusion in the front (two long sides being normal-type DWs). The small protrusion in Fig. 3b (also Fig. 3e below) is caused by the unexpected occasional tip movement when it recovers from the ‘lift down’ mode to a regular scanning mode (Supplementary Fig. 5),

which can be eliminated by improving the software design. Subsequently, a line scan parallel to the displacement vector (θ = 0°, along one armchair direction indicated by the red arrow in Fig. 3g) over the newly formed normal-type DW creates a strip structure fol-lowing the scanning line with a curved front (two long sides being the shear-type DWs). This example demonstrates rules (1) and (2).

Then we investigated another DW (Fig. 3d) with unknown dis-placement vector in the same trilayer graphene flake as in Fig. 3a. Its displacement vector should be along one of the three armchair directions. A line scan along one zigzag direction (Fig. 3e) generates a rectangular structure with a small protrusion in front, and a line scan along one armchair direction (Fig. 3f) generates a strip with a curved front, which are quite similar to those observed in Fig. 3b,c, respectively. In addition, a line scan along another zigzag direction results in a triangular rather than a rectangular structure (Fig. 3g). One side of the triangle follows the scan direction and the other side is along the same armchair orientation as in Fig. 3f (rule (3)). These DW motions suggest its displacement vector is along the spe-cific armchair orientation indicated by the red arrow (in Fig. 3f,g) by comparing with the results in Fig. 3a–c. For the three cases in Fig. 3e–g, θ is 90°, 0° and 30°, respectively. Hence, we have created shear-, normal- and mixed-type DWs (Fig. 3g).

The example above also suggests that the anisotropic motion of DWs is not related to the six-fold symmetry of the graphene lattice. To further confirm this, a single-line scan (along the zigzag orienta-tion) is carried out across two unconnected DWs (with indepen-dent displacement vectors) in the same graphene flake (Fig. 3i,h). It creates two different structures: a rectangular structure in the left DW and a triangular structure in the right DW (Fig. 3h). One side of the triangle is again along the AFM scanning direction, and the other oblique side is along the armchair direction (red arrow), con-sistent with it being a shear-type DW. θ for the two DWs should be 90° (left) and 30° (right), respectively. More results are shown in Supplementary Figs. 6 and 7.

The mechanical origin of the movement of the DWs and their anisotropic motions can be qualitatively explained based on a sim-ple strain analysis. In Fig. 4a, we illustrate the DW structures and strain types of different DW segments before (upper panels) and after (lower panels) tip sliding in trilayer graphene. Graphene has a high flexibility of out-of-plane elastic deformation, which can pro-duce a local puckering near the front contact edge between an AFM tip and graphene24. When an AFM tip with a large normal force scans across a graphene surface, the friction exerted by the AFM tip induces a transient local sliding of the top layer atoms underneath the tip, pushing them away from their initial sites. These displaced atoms will later relax to the locally lowest stacking energy sites. As shown in Fig. 4a, the tip scans across the DW from domain 1 to domain 2. The displaced atoms will relax to form the same stacking order as the domain behind the AFM tip (domain 1). Otherwise, a DW would form and it is energetically unfavourable. As a result, the initial DW is moved forward along the AFM scanning direction. The sliding and relaxation processes of the atoms in the DW region are illustrated in Fig. 4b. The yellow dots, orange dots and green dots represent the initial sites (different from Bernal or rhombohe-dral stacking sites), un-equilibrium transient sites and Bernal (or rhombohedral) stacking sites, respectively.

The anisotropy of the reconstructed structures can be under-stood by the simple models in Fig. 4c, assuming that a rectangular structure is created initially in all the three cases and then relaxes to the energetically favourable configurations. The width of the rect-angle is determined by the size of the tip apex (Supplementary Figs. 8 and 9). In the normal-type case (Fig. 4c, left), the front part with normal-type strain evolves into curved DW segment to reduce the total strain energy. In the mixed-type case (Fig. 4c, middle), the top right and bottom right corners shift to the blue dashed line, forming a shear-type DW and also reducing the total length. In the

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Fig. 4 | Schematics of the anisotropic DW motions in trilayer graphene. a, The three types of initial DWs (upper panels) and their reconstructions after line scans across the initial DWs (lower panels). The black dashed arrows represent the tip sliding directions. The red arrows illustrate the displacement vector between the bottom ABA domain (yellow part) and the top ABC domain (pink part). The blue areas are DW regions. A shear segment appears in all three cases. b, The sliding and relaxation processes of the DW atoms underneath the AFM tip. c, The strain types and induced evolvement of the DW configurations in the three different cases. The red arrows illustrate the displacement vector of the DW. In all panels (initial DW): left, normal-type DW; middle, mixed-type DW; right, shear-type DW.


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Letters NaTurE NaNoTEcHNoLogy

shear-type case (Fig. 4c, right), the front segment is a most-stable shear-type DW, which guarantees the reconstructed structure to be stable with a rectangular shape. A shear component is present in all three reconstructed structures. The possible reason is that a shear-type DW has the lowest elastic strain energy per unit length compared with normal- (55% larger) or mixed-type DWs. Hence, a DW tends to have as large a shear component as possible during the reconstructions22,25.

The manipulation of a DW can be used to control local elec-tronic transport properties within different domains and areas closed to the DW in bilayer and trilayer graphene. We studied the nanoscale potential distribution in a trilayer graphene before and after the DW manipulation using scanning voltage microscopy (SVM) (Supplementary Fig.10). The trilayer graphene contains both ABA- and ABC-stacked domains, which exhibits very different resistance in the ABA and ABC domains in the SVM image due to their different electronic structure. We show that DW manipulation can modify significantly the local transport properties and electric field distribution around the DW in this trilayer field-effect device. We expect that our technique to manipulate individual DWs in bi- and trilayer graphene into desired structures with highly stability will enable fundamental understanding of transport properties of different type of DWs, and lead to new ways to build functional devices based on such one-dimensional topological dislocations in atomically thin layered materials.

MethodsMethods, including statements of data availability and any asso-ciated accession codes and references, are available at https://doi.org/10.1038/s41565-017-0042-6.

Received: 21 April 2017; Accepted: 5 December 2017; Published online: 22 January 2018

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13. Alden, J. S. et al. Strain solitons and topological defects in bilayer graphene. Proc. Natl Acad. Sci. USA 110, 11256–11260 (2013).

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acknowledgementsWe acknowledge helpful discussions with M. Asta, D. Chrzan, B. Yacobson, M. Poschmann and R. Zucker. We thank A. Zettl, Y. Zhang, T. Wang and Y. Sheng for their help on sample preparation. The near-field infrared nanoscopy measurements and plasmon analysis was supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division of the US Department of Energy under contract no. DE-AC02-05-CH11231 (Sub-wavelength Metamaterial Program) and National Key Research and Development Program of China (grant number 2016YFA0302001). The bilayer graphene DW sample fabrication and characterization is supported by the Office of Naval Research (award N00014-15-1-2651). L.J. acknowledges support from International Postdoctoral Exchange Fellowship Program 2016 (No.20160080). Z.S. acknowledges support from the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.

author contributionsF.W. and Z.S. conceived the project. F.W., Y.-R.S. and H.-J.G. supervised the project. G.Z. helped to design the study with Z.S. and F.W. L.J., Z.S. and C.J. performed the near-field infrared measurements and DW manipulation work. S.W. and L.J. performed the SVM measurement. L.J. and S.Z. made the FET devices. M.I.B.U. carried out the SEM measurements. L.J., S.W., Z.S. and F.W. analysed the data. All authors discussed the results and contributed to writing the manuscript.

Competing interestsThe authors declare no competing financial interests.

additional informationSupplementary information is available for this paper at https://doi.org/10.1038/s41565-017-0042-6.

Reprints and permissions information is available at www.nature.com/reprints.

Correspondence and requests for materials should be addressed to Z.S. or F.W.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



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MethodsSample preparation. We exfoliated bi- and trilayer graphene from graphite onto Si substrates with 285 nm SiO2 on top.

Infrared nano-imaging of DWs. Our infrared nano-imaging technique was based on tapping-mode AFM26,27. An infrared light beam (λ = 10.6 µ m) was focused onto the apex of a conductive AFM tip. The enhanced optical field at the tip apex interacts with graphene underneath the tip. A MCT (mercury cadmium telluride, HgCdTe) detector placed in the far field was used to collect the scattered light, which carries local optical information of the sample. Near-field images were recorded simultaneously with the topography information during the measurements. In bilayer graphene, the contrast of DWs in near-field images stems from surface plasmon reflection at the DWs; in trilayer graphene, ABA- and ABC-stacked domains give different infrared responses due to their different electronic band structures. DWs are the transitional regions between different stacking domains.

Manipulating DWs with AFM tip. AFM works in a contact-lift mode: during forward scanning, AFM works under the normal contact-mode, where the feedback is on for tracking the topography information; during backward scanning, the feedback is turned off and the z-piezo will move following the topography obtained in the forward scanning but with a set lift height. By controlling the lift height, we can control the magnitude of the normal force between tip and sample, and thus the friction applied to the sample. We applied a large external normal force (typically ~40 µ N, lift-down height around 1 m) between the tip and sample and thus a sufficiently large lateral force28. This value is much larger

than the normal force in a regular contact-mode scanning (~50 nN) and can be controlled by ‘lifting down’ the tip towards the sample with a precise distance. A larger lift-down distance induces a larger normal force by estimating from Hooke’s law Δ=F k x, where k is the force constant of the tip (calibrated by Sader’s method in our experiments29) and ∆ x is the lift-down distance. To enhance the capability to move the DW, we purposely blunted the AFM tip apex to more than 100 nm in diameter (see Supplementary Fig. 1). Combining the increased normal force and the blunt tip, we realized controlled manipulation of DWs in bilayer and trilayer graphene. By controlling the backward line scanning direction, we can control the direction of the force applied to the sample. This contact-lift mode guarantees that the large AFM force is applied only in one direction (that is, the backward scanning direction).

Data availability. The data that support the findings of this study are available from the corresponding author on reasonable request.

References 26. Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared

nano-imaging. Nature 487, 82–85 (2012). 27. Chen, J. et al. Optical nano-imaging of gate-tunable graphene plasmons.

Nature 487, 77–81 (2012). 28. Choi, J. S. et al. Friction anisotropy–driven domain imaging on exfoliated

monolayer graphene. Science 333, 607–610 (2011). 29. Sader, J. E., Chon, J. W. M. & Mulvaney, P. Calibration of rectangular atomic

force microscope cantilevers. Rev. Sci. Instrum. 70, 3967–3969 (1999).


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ARTICLE

The nature of spin excitations in the one-thirdmagnetization plateau phase of Ba3CoSb2O9

Y. Kamiya1, L. Ge2, Tao Hong 3, Y. Qiu4, D.L. Quintero-Castro5, Z. Lu5, H.B. Cao3, M. Matsuda3, E.S. Choi6,

C.D. Batista 7,8, M. Mourigal 2, H.D. Zhou6,7 & J. Ma7,9,10

Magnetization plateaus in quantum magnets—where bosonic quasiparticles crystallize into

emergent spin superlattices—are spectacular yet simple examples of collective quantum

phenomena escaping classical description. While magnetization plateaus have been observed

in a number of spin-1/2 antiferromagnets, the description of their magnetic excitations

remains an open theoretical and experimental challenge. Here, we investigate the dynamical

properties of the triangular-lattice spin-1/2 antiferromagnet Ba3CoSb2O9 in its one-third

magnetization plateau phase using a combination of nonlinear spin-wave theory and neutron

scattering measurements. The agreement between our theoretical treatment and the

experimental data demonstrates that magnons behave semiclassically in the plateau in spite

of the purely quantum origin of the underlying magnetic structure. This allows for a quan-

titative determination of Ba3CoSb2O9 exchange parameters. We discuss the implication of

our results to the deviations from semiclassical behavior observed in zero-field spin dynamics

of the same material and conclude they must have an intrinsic origin.

Corrected: Author correction

DOI: 10.1038/s41467-018-04914-1 OPEN

1 Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan. 2 School of Physics, Georgia Institute of Technology, Atlanta, GA 30332,USA. 3 Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. 4NIST Centre for Neutron Research, National Institute ofStandards and Technology, Gaithersburg, MD 20899, USA. 5Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin, Germany. 6NationalHigh Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA. 7 Department of Physics and Astronomy, University of Tennessee,Knoxville, TN 37996, USA. 8Neutron Scattering Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. 9 KeyLaboratory of Artificial Structures and Quantum Control, Department of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China.10 Collaborative Innovation Center of Advanced Microstructures, 210093 Nanjing, Jiangsu, China. Correspondence and requests for materials should beaddressed to Y.K.(email: [email protected]) or to J.M.(email: [email protected])

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5678

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52

http://orcid.org/0000-0002-0161-8588

http://orcid.org/0000-0002-0161-8588

http://orcid.org/0000-0002-0161-8588

http://orcid.org/0000-0002-0161-8588

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http://orcid.org/0000-0003-2772-8440

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https://doi.org/10.1038/s41467-018-05679-3



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Quantum fluctuations favor collinear spin order in fru-strated magnets1–3, which can be qualitatively differentfrom the classical limit (S→∞)4. In particular, quantum

effects can produce magnetization plateaus5–10, where the mag-netization is pinned at a fraction of its saturation value. Magne-tization plateaus can be interpreted as crystalline states of bosonicparticles, and are naturally stabilized by easy-axis exchange ani-sotropy, which acts as strong off-site repulsion11–14. However, thesituation is less evident and more intriguing for isotropic Hei-senberg magnets, which typically have no plateaus in the classicallimit. In a seminal work, Chubukov and Golosov predicted the1/3 magnetization plateau in the quantum triangular latticeHeisenberg antiferromagnet (TLHAFM), corresponding to anup–up–down (UUD) state5. Their predictions were confirmedby numerical studies6,15–21 and extended to plateaus in othermodels6. Experimentally, the 1/3 plateau has been observed inthe spin-1/2 isosceles triangular lattice material Cs2CuBr422–26, aswell as in the equilateral triangular lattice materials RbFe(MoO4)2(S= 5/2)27–29 and Ba3CoSb2O9 (effective S= 1/2)30–39.Notwithstanding the progress in the search of quantum pla-

teaus, much less is known about their excitation spectra. Giventhat they are stabilized by quantum fluctuations, it is natural toask if these fluctuations strongly affect the excitation spectrum.The qualitative difference between the plateau and the classicalorderings may appear to invalidate spin-wave theory. Forinstance, the UUD state in the equilateral TLHAFM is not aclassical ground state unless the magnetic field H is fine-tuned40.Consequently, a naive spin wave treatment is doomed toinstability. On the other hand, spin wave theory builds on theassumption of an ordered moment |⟨Sr⟩| close to the fullmoment. Given that a sizable reduction of |⟨Sr⟩| is unlikely withinthe plateau because of the gapped nature of the spectrum, a spinwave description could be adequate. Although this may seem inconflict with the order-by-disorder mechanism1–3 stabilizing theplateau40–42, this phenomenon is produced by the zero-pointenergy correction Ezp ¼ ð1=2Þ

Pq ωq þ OðS0Þ (ωq is the spin

wave dispersion), which does not necessarily produce a largemoment size reduction.Here, one of our goals is to resolve this seemingly contradictory

situation. Recently, Alicea et al. developed a method to fix theunphysical spin-wave instability40. This proposal awaits experi-mental verification because the excitation spectrum has notbeen measured over the entire Brillouin zone for any fluctuation-induced plateau. We demonstrate that the modified nonlinearspin wave (NLSW) approach indeed reproduces themagnetic excitation spectrum of Ba3CoSb2O9 within the 1/3plateau30–39. The excellent agreement between theory andexperiment demonstrates the semiclassical nature of magnonswithin the 1/3 plateau phase, despite the quantum fluctuation-induced nature of the ground state ordering. The resulting model

parameters confirm that the anomalous zero-field dynamicsreported in two independent experiments37,39 must be intrinsicand non-classical.

ResultsOverview. In this article, we present a comprehensive study ofmagnon excitations in the 1/3 magnetization plateau phase of aquasi-two-dimensional (quasi-2D) TLHAFM with easy-planeexchange anisotropy. Our study combines NLSW theory within-field inelastic neutron scattering (INS) measurements ofBa3CoSb2O9. The Hamiltonian is

H¼ JPhrr′i

SxrSxr′ þ SyrS

yr′ þ ΔSzrS

zr′

� �þJc

Pr

SxrSxrþc

2þ SyrS

yrþc

2þ ΔSzrS

zrþc

2

� �� hred

PrSxr ;

ð1Þ

where ⟨rr′⟩ runs over in-plane nearest-neighbor (NN) sites of thestacked triangular lattice and c

2 corresponds to the interlayerspacing (Fig. 1a). J (Jc) is the antiferromagnetic intralayer(interlayer) NN exchange and 0 ≤ Δ < 1. The magnetic field isin the in-plane (x) direction (we use a spin-space coordinateframe where x and y are in the ab plane and z is parallel to c).hred= g⊥μBH is the reduced field and g⊥ is the in-plane g-tensorcomponent.This model describes Ba3CoSb2O9 (Fig. 1b), which comprises

triangular layers of effective spin 1/2 moments arising fromthe J ¼ 1=2 Kramers doublet of Co2+ in a perfect octahedralligand field. Excited multiplets are separated by a gap of200–300 K due to spin–orbit coupling, which is much largerthan the Néel temperature TN= 3.8 K. Below T= TN,the material develops conventional 120° ordering with wavevectorQ= (1/3, 1/3, 1)30. Experiments confirmed a 1/3 magnetizationplateau for Hjjab (Fig. 1c)31,33,35,36,38, which is robust downto the lowest temperatures. We compute the dynamical spinstructure factor using NLSW theory in the 1/3 plateau phase.We also provide neutron diffraction evidence of the UUD statewithin the 1/3 plateau of Ba3CoSb2O9, along with maps of theexcitation spectrum obtained from INS.

Quantum-mechanical stabilization of the plateau in quasi-2DTLHAFMs. While experimental observations show thatdeviations from the ideal 2D TLHAFM are small in Ba3CoSb2O9

33,37,39, a simple variational analysis shows that any Jc > 0 isenough to destabilize the UUD state classically. Thus, a naive spinwave treatment leads to instability for Jc > 0. However, the gappednature of the spectrum5 implies that this phase must have a finiterange of stability in quasi-2D materials. This situation mustbe quite generic among fluctuation-induced plateaus, as they

Ba

Co

SbO

b

c

Magnetic field (T)

0

0.5

1.0

1.5

2.0

2.5

0 10 20 30

M (� B

/Co2+

)

Ba3CoSb2O9

a

a

c

b

H || [1,–1,0]

J

Jc

Ce

Co

Ae

Ao

Be

Boc/2

Fig. 1 Stacked triangular lattice and the UUD state. a Spin structure in the quasi-2D lattice. b Crystal structure of Ba3CoSb2O9. c Magnetization curve forHjjab at T= 0.6 K highlighting the 1/3 plateau (the finite slope is due to Van Vleck paramagnetism33)

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04914-1

2 NATURE COMMUNICATIONS | (2018) 9:2666 | DOI: 10.1038/s41467-018-04914-1 | www.nature.com/naturecommunications53


normally require special conditions to be a classical groundstate9,10,43.

To put this into a proper semiclassical framework, we applyAlicea et al.'s trick originally applied to a distorted triangularlattice40. Basically, we make a detour in the parameter space withthe additional 1/S-axis quantifying the quantum effect (Fig. 2).Namely, instead of expanding the Hamiltonian around S→∞for the actual model parameters, we start from the special point,Jc= 0, hred= 3JS, and a given value of 0 ≤ Δ ≤ 1, where the UUDstate is included in the classical ground state manifold. Assumingthe spin structure in Fig. 1a, we define

Sxr ¼ ~Szr ; Syr ¼ ~Syr ; S

zr ¼ �~Sxr ; ð2Þ

for r ∈ Ae, Be, Ao, and Co and

Sxr ¼ �~Szr ; Syr ¼ ~Syr ; Szr ¼ ~Sxr ; ð3Þ

for r∈ Ce, Bo. Introducing the Holstein–Primakoff bosons, a yð Þμ;r ,with 1 ≤ μ ≤ 6 being the sublattice index for Ae, Be, Ce, Ao, Bo, andCo in this order, we have

~Szr ¼ S� ayμ;raμ;r;

~Sþr ¼ ~Sxr þ i~Syr �ffiffiffiffiffi2Sp

1� ayμ;raμ;r4S

� �aμ;r;

ð4Þ

and ~S�r ¼ ~Sþr� �y

for r∈ μ, truncating higher order termsirrelevant for the quartic interaction. We evaluate magnon self-energies arising from decoupling of the quartic term.As shown in Fig. 2b, the linear spin wave (LSW) spectrum for

Jc= 0 and hred= 3JS features two q-linear gapless branches atq= 0, both of which are gapped out by the magnon–magnoninteraction (Fig. 2c). Small deviations from Jc= 0 and hred= 3JSdo not affect the local stability of the UUD state because the gapmust close continuously. Thus, we can investigate the excitationspectrum of quasi-2D systems for fields near hred= 3JS.Figure 2d, e show the spectra for hred shifted by ±10% fromhred= 3JS, where we still keep Jc= 0. For hred < 3JS, a band-touching and subsequent hybridization appear between the

middle and the top bands around q= (1/6,1/6) (Fig. 2d). Forhred > 3JS, a level-crossing between the middle and bottom bandsappears at around q= 0 (Fig. 2e). A small Jc > 0 splits the threebranches into six (Fig. 2f, g). Figure 3a, b show the reduction ofthe sublattice ordered moments for S= 1/2, Jc= 0, 0.09J, and

selected values of Δ. We find δ Sxμ

D E =S≲30% throughout the

local stability range of the plateau. Thus, our semiclassicalapproach is fully justified within the plateau phase. Figure 3c, dshow the field dependence of the staggered magnetization,

MUUD ¼16

SxAe

D Eþ SxBe

D E� SxCe

D Eþ SxAo

D E� SxBo

D Eþ SxCo

D E� �;

ð5Þ

which is almost field-independent; a slightly enhanced field-independence appears for small Δ. Similarly, while the magne-tization is not conserved for Δ ≠ 1, it is nearly pinned at 1/3 forthe most part of the plateau (Fig. 3e, f).

UUD state in Ba3CoSb2O9. Next we show experimental evidencefor the UUD state in Ba3CoSb2O9 by neutron diffraction mea-surements within the plateau phase for field applied along the[1,–1,0] direction. We used the same single crystals reportedpreviously32,37, grown by the traveling-solvent floating-zonetechnique and characterized by neutron diffraction, magneticsusceptibility, and heat capacity measurements. The spacegroup is P63/mmc, with the lattice constants a= b= 5.8562 Å andc= 14.4561 Å. The site-disorder between Co2+ and Sb5+ isnegligible with the standard deviation of 1%, as reported else-where37. The magnetic and structural properties are consistentwith previous reports and confirm high quality of thecrystals30–39. These crystals were oriented in the (h, h, l) scat-tering plane. The magnetic Bragg peaks at (1/3, 1/3, 0) and (1/3,1/3, 1) were measured at T= 1.5 K (Fig. 4a, b). The largeintensity at both (1/3, 1/3, 0) and (1/3, 1/3, 1) confirms the UUDstate at μ0H ≥ 9.8 T32 (Fig. 4c). The estimated ordered moment is1.65(3) μB at 10 T and 1.80(9) μB at 10.9 T. They correspond to 85(2)% and 93(5)% of the full moment33, roughly coinciding with

0 1/3 2/3 1q = (h, h )

0

0.5

1.0

1.5

2.0 b c

q = (h, h )0 1/3 2/3 1

0

0.5

1.0

1.5

2.0

System of interest(quasi-2D; quantum)

3D couplingJc

a

hredMagnetic field

1. NLSW in 2D(↑↑↓ is a CGS)

1/SQuantum effect

hred = 3JSInstability

2. Turn on deviations

Naive classicallimit (S → ∞)

Quasi-2D classical system(↑↑↓ is never a CGS)

0 1/3 2/3 1

q = (h, h )

d

0

0.5

1.0

1.5

2.0

� / J

q = (h, h )

e

� / J

0 1/3 2/3 10

0.5

1.0

1.5

2.0

� / J

q = (h, h, 1)

0

0.5

1.0

1.5

2.0

0 1/3 2/3 1

f

0 1/3 2/3 10

0.5

1.0

1.5

2.0

� / J

� / J� / J

q = (h, h, 1)

g

NLSW (quasi-2D): hred = 3.3JSNLSW (quasi-2D): hred = 2.7JSNLSW (2D): hred = 2.7JS NLSW (2D): hred = 3.3JS

NLSW (2D): hred = 3JSLSW (2D): hred = 3JS

Fig. 2 Scheme of NLSW theory for the 1/3 magnetization plateau in the quasi-2D TLHAFM. a Illustration of the procedure used to compute the spectrum.The filled and open star symbols represent the target quasi-2D quantum system and its naive classical limit, respectively. The spin structures favoredby quantum fluctuation are shown on the Jc= 0 plane. b LSW spectrum along the high-symmetry direction of the Brillouin zone evaluated for S= 1/2,Jc= 0, Δ= 0.85 and hred= 3JS, where the UUD state is a classical ground state (CGS). c–e NLSW spectra for Jc= 0 and Δ= 0.85 with (c) hred= 3JS,(d) hred= 2.7JS, and (e) hred= 3.3JS. f, g NLSW spectra for Jc/J= 0.09 and Δ= 0.85 with (f) hred= 2.7JS and (g) hred= 3.3JS

NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04914-1 ARTICLE




the predicted range (Fig. 3d). This diffraction pattern canbe contrasted with that of the 120° state, characterized by acombination of the large intensity at (1/3, 1/3, 1) and lackof one at (1/3, 1/3, 0). Our diffraction result is fully consistentwith previous nuclear magnetic resonance (NMR)35 andmagnetization measurements31,33.

Excitation spectrum. We now turn to the dynamical propertiesin the UUD phase. Figure 5a–c show the INS intensity I(q, ω)≡ki/kf (d2σ/dΩdEf) along high-symmetry directions. The appliedmagnetic field μ0H= 10.5 T is relatively close to the transitionfield μ0Hc1= 9.8 T32 bordering on the low-field coplanar orderedphase35, while the temperature T= 0.5 K is low enough comparedto TN ≈ 5 K36 for the UUD phase at this magnetic field. The in-plane dispersion shown in Fig. 5a comprises a seemingly gaplessbranch at q= (1/3, 1/3, −1) (Fig. 5c), and two gapped modescentered around 1.6 and 2.7 meV. Due to the interlayer coupling,each mode corresponds to two non-degenerate branches. As theirsplitting is below the instrumental resolution, we simply referto them as ω1, ω2 and ω3, unless otherwise mentioned (Fig. 5).The dispersions along the c-direction are nearly flat, as shown inFig. 5b, c for q= (1/2, 1/2, l) and q= (1/3, 1/3, l), respectively,reflecting the quasi-2D lattice33,37. Among the spin wave modesalong q= (1/2, 1/2, l) and q= (1/3, 1/3, l) in Fig. 5b, c, ω1 forq= (1/2, 1/2, l) displays a relatively sharp spectral line.

As discussed below, most of the broadening stems from thedifferent intensities of the split modes due to finite Jc.Comparing the experiment against the NLSW calculation, we

find that the features of the in-plane spectrum in Fig. 5a areroughly captured by the theoretical calculation near the low-field onset of the plateau in Fig. 2f (hred= 2.7JS ≈ 1.03hred,c1).This observation is in accord with the fact that the applied field(μ0H= 10.5 T) is close to μ0Hc1= 9.8 T32. To refine thequantitative comparison, we calculate the scattering intensityItot q;ωð Þ � γr0=2ð Þ2 F qð Þj j2Pα 1� qαqαð Þg2αSαα q;ωð Þ whereF(q) denotes the magnetic form factor of Co2+ corrected withthe orbital contribution, (γr0/2)2 is a constant, qα ¼ qα= qj j are thediagonal components of the dynamical structure factor evaluatedat 10.5 T; off-diagonal components are zero due to symmetry.Defining the UUD order as shown in Fig. 1a, transverse spinfluctuations related to single-magnon excitations appear in Syyand Szz , while longitudinal spin fluctuations corresponding to thetwo-magnon continuum appear in the inelastic part of Sxx ,denoted as Sjj. Accordingly, Itot(q, ω) can be separated intotransverse I⊥ and longitudinal Ijj contributions. To compare withour experiments, the theoretical intensity is convoluted withmomentum binning effects (only for I⊥) and empirical instru-mental energy resolution. Figure 5d–f show the calculatedI⊥(q, ω), along the same high-symmetry paths as the experi-mental results in Fig. 5a–c, for J= 1.74 meV, Δ= 0.85, Jc/J= 0.09,

Δ = 0.55

Δ = 1.00

Δ = 0.85

Δ = 0.70

Δ = 0.40

Δ = 0.55 (up)

Δ = 1.00 (up)

Δ = 0.85 (up)

Δ = 0.70 (up)

Δ = 0.40 (up)

Δ = 0.55 (down)

Δ = 1.00 (down)

Δ = 0.85 (down)

Δ = 0.70 (down)

Δ = 0.40 (down)

hred / (9JS )

Jc = 0 Jc = 0.09

hred / (9JS )

Jc = 0 Jc = 0.09

0.10

0.15

⏐� ⟨

S�x⟩⏐

/ S

0.20

0.25

0.30

0.78

0.80

0.82

0.84

0.86

MU

UD /

SM

/ S

Δ = 0.55

Δ = 1.00

Δ = 0.85

Δ = 0.70

Δ = 0.40Jc = 0 Jc = 0.09

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.25

0.30

0.35

0.40

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

a b

c d

e f

Fig. 3 Calculated ordered moment within the 1/3 plateau for spin 1/2. Top row: magnitude of the reduction, jδhSxμij, of the sublattice ordered moments(normalized by S) with the designated moment directions (up or down) for (a) Jc= 0 and (b) Jc= 0.09J; for sublattices with up spins, we average δhSxμiover the corresponding two sublattices, discarding the small variance that appears for Jc > 0. Middle row: normalized staggered magnetizationMUUD/S [Eq.(5)] for (c) Jc= 0 and (d) Jc= 0.09J. Bottom row: normalized (uniform) magnetizationM/S for (e) Jc= 0 and (f) Jc= 0.09J. The results correspond to thelocal stability range of the plateau (with the precision 0.001 × 9JS for hred)




and g⊥= 3.95. The agreement between theory and experimentis excellent. When deriving these estimates, J is controlled bythe saturation field μ0Hsat= 32.8 T for Hjjc33. To obtain thebest fit, we also analyzed the field dependence of ω1, ω2, and ω3

(Fig. 6).Remarkably, the calculation in Fig. 5d–f reproduces the

dispersions almost quantitatively. It predicts a gapped ω1 mode,although the gap is below experimental resolution. The smallnessof the gap is simply due to proximity to Hc1. For each ωi, the bandsplitting due to Jc yields pairs of poles ω±

i dispersing with a phasedifference of π in the out-of-triangular-plane direction (Fig. 5e, f).For each pair, however, one pole has a vanishing intensity forq= (1/2, 1/2, l). Consequently, ω1 along this direction is free fromany extrinsic broadening caused by overlapping branches (Fig. 5e),yielding a relatively sharp spectral line (Fig. 5h). The correspond-ing bandwidth ≈ 0.2 meV (Fig. 5b) provides a correct estimatefor Jc. By contrast, for q= (1/3, 1/3, l), all six ω±

i branches havenon-zero intensity, which leads to broadened spectra and lessobvious dispersion along l (Fig. 5c, f).

The field-dependence of ω1–ω3 at q= (1/3, 1/3, 1) is extractedfrom constant-q scans at T= 0.1 K for selected fields 10.5–13.5 Twithin the plateau (Fig. 6a). By fitting the field-dependence ofthe low-energy branches of ω1,2, which become gapless at aplateau edge, we obtain the quoted model parameters. Thefield dependence is reproduced fairly well (Fig. 6b, c), althoughthe calculation slightly underestimates ω3. We find ω1 and ω3 (ω2)increase (decreases) almost linearly in H, while the ω1 and ω2

branches cross around 12.6 T. The softening of ω1 (ω2) at thelower (higher) transition field induces the Y-like (V-like) state,respectively20,35. The nonlinearity of the first excitation gapnear these transitions (visible only in the calculation) is due tothe anisotropy; there is no U(1) symmetry along the fielddirection for Δ ≠ 1.

DiscussionOur work has mapped out the excitation spectrum in the1/3 plateau—a manifestation of quantum order-by-disorder

Inte

nsity

(ar

b. u

nit)

2� (deg.)

120

80

40

25 26 27 28

10 T0 T

a(1/3, 1/3, 0)

2� (deg.)

30 31 32 33

120

80

40

10 T0 T

b(1/3, 1/3, 1)

Experiment (arb. unit)

Sim

ulat

ion

(arb

. uni

t)

5000

10,000

15,000

5000

10,0

00

15,0

00

c

Series of (1/3,1/3,0)Series of (1/3,1/3,1)

Fig. 4 Neutron diffraction data for Ba3CoSb2O9 at 0 and 10 T. (a) q= (1/3, 1/3, 0) and (b) q= (1/3, 1/3, 1) measured at T= 1.5 K. c Comparison of thediffraction intensities between the experiment and the simulation at T= 1.5 K and μ0H= 10 T (the solid line is a guide to the eye). Error bars represent onestandard deviation

0

1

2

3

4

0 1/6 1/3 1/2 2/3

(h, h, –2)

0

1

2

3

4

–2 –3/2 –1

(1/2, 1/2, l )

0

1

2

3

4

I ⊥ (q

,�)

(arb

. uni

t)–2 –3/2 –1

(1/3, 1/3, l )

0

2

4

Exp.Itot

I||

00

2

4 Exp.Itot

I||

cba

d e

g

f h

(1/3, 1/3, –2)

(1/2, 1/2, –2)

h� (

meV

)h�

(m

eV)

h� (meV)

�3�3

�3

�2�2 �2

�1�1 �1

1 2 3 4

�1

�2 �3

�1

�2 & �3

Fig. 5 Excitation spectrum in the UUD phase of Ba3CoSb2O9. a–c Experimental scattering intensity at μ0H= 10.5 T and T= 0.5 K with momentum transfer(a) q= (h, h, −2), (b) q= (1/2, 1/2, l), and (c) q= (1/3, 1/3, l). d–f Calculated transverse part of the scattering intensity, I⊥(q, ω), obtained by NLSWtheory along the same momentum cuts as in (a)–(c) for J= 1.74meV, Δ= 0.85, Jc/J= 0.09, and g⊥= 3.95. The solid lines show the magnon poles. (g)and (h) Energy dependence of the calculated scattering intensity, Itot(q, ω) (solid line), compared with the experiment for (g) q= (1/3, 1/3, −2) and(h) q= (1/2, 1/2, −2) (error bars represent one standard deviation). The longitudinal contribution to the scattering intensity, Ijj q;ωð Þ, is plotted separatelyas a shaded area. The energy of the outgoing neutrons is Ef= 5meV (3 meV) above (below) the dashed line in (a–c), (g), and (h)





effect—in Ba3CoSb2O9. Despite the quantum-mechanical originof the ground state ordering, we have unambiguously demon-strated the semiclassical nature of magnons in this phase. Infact, the calculated reduction of the sublattice magnetization,δSμ= S− |⟨Sr⟩| with r∈ μ, is relatively small (Fig. 3b):δSAe

¼ δSAo¼ 0:083, δSBe

¼ δSCo¼ 0:073, and δSCe

¼ δSBo¼

0:14 at 10.5 T for the quoted model parameters. This is consistentwith the very weak intensity of the two-magnon continuum(Fig. 5g, h). This semiclassical behavior is protected by theexcitation gap induced by anharmonicity of the spin waves(magnon–magnon interaction). We note that a perfect collinearmagnetic order does not break any continuous symmetry evenfor Δ= 1, i.e., there is no gapless Nambu–Goldstone mode.The collinearity also means that three-magnon processes are notallowed44. The gap is robust against perturbations, such as ani-sotropies, lattice deformations40, or biquadratic couplings for S >1/2 (a ferroquadrupolar coupling can stabilize the plateau evenclassically5). Thus, we expect the semiclassical nature of theexcitation spectrum to be common to other 2D and quasi-2Drealizations of fluctuation-induced plateaus, such as the 1/3plateau in the spin-5/2 material RbFe(MoO4)229. Meanwhile, itwill be interesting to examine the validity of the semiclassicalapproach in quasi-1D TLHAFMs, such as Cs2CuBr423, wherequantum fluctuations are expected to be stronger.Finally, we discuss the implications of our results for the zero-

field dynamical properties of the same material, where recentexperiments revealed unexpected phenomena, such as broadeningof the magnon peaks indescribable by conventional spin-wave

theory, large intensity of the high-energy continuum37, andthe extension thereof to anomalously high frequencies37,39. Spe-cifically, it was reported that magnon spectral-line broadenedthroughout the entire Brillouin zone, significantly beyondinstrumental resolution, and a high frequency (≳ 2 meV) exci-tation continuum with an almost comparable spectral weight assingle-magnon modes37. All of these experimental observationsindicate strong quantum effects. Given that the spin Hamiltonianhas been reliably determined from our study of the plateau phase,it is interesting to reexamine if a semiclassical treatment of thisHamiltonian can account for the zero-field anomalies.A semiclassical treatment can only explain the line broadening

in terms of magnon decay44–48. NLSW theory at H= 0 describesthe spin fluctuations around the 120° ordered state by incor-porating single-to-two magnon decay at the leading orderO(S0). At this order, the two-magnon continuum is evaluatedby convoluting LSW frequencies. The self-energies includeHartree–Fock decoupling terms, as well as the bubble Feynmandiagrams comprising a pair of cubic vertices Γ3 � O S1=2

� �45–48,

with the latter computed with the off-shell treatment. The mostcrucial one corresponds to the single-to-two magnon decay(see the inset of Fig. 7a),

Σ q;ωð Þ ¼ 12N

Xk

Γ3 k; q� k; qð Þj j2ω� ωH¼0

k � ωH¼0q�k þ i0

; ð6Þ

where ωH¼0k denotes the zero-field magnon dispersion. We show

the zero-field dynamical structure factor, StotH¼0 q;ωð Þ, at theM point for representative parameters in Fig. 7a–d. TheNLSW result for the ideal TLHAFM (Jc= 0 and Δ= 1) exhibitssizable broadening and a strong two-magnon continuum45–48

(see also Fig. 7e). However, a slight deviation from Δ= 1 rendersthe decay process ineffective because the kinematic condition,ωH¼0q ¼ ωH¼0

k þ ωH¼0q�k , can no longer be fulfilled in 2D for any

decay vertex over the entire Brillouin zone if Δ≲0:9245. Thissituation can be inferred from the result for Jc= 0 and Δ= 0.85,where the two-magnon continuum is pushed to higher fre-quencies, detached from the single-magnon peaks. In fact, thesharp magnon lines are free from broadening. The suppression ofdecay results from gapping out one of the two Nambu–Goldstonemodes upon lowering the Hamiltonian symmetry from SU(2) toU(1), which greatly reduces the phase space for magnon decay.The interlayer coupling renders the single-magnon peaks evensharper and the continuum even weaker (Fig. 7c, d).To determine whether the anomalous zero-field spin dynamics

can be explained by a conventional 1/S expansion, it is crucialto estimate Δ very accurately. Previous experiments reportedΔ= 0.95 (low-field electron spin resonance experimentscompared with LSW theory33) and Δ= 0.89 (zero-field INSexperiments compared with NLSW theory37). However, theNLSW calculation reported a large renormalization of themagnon bandwidth (≈ 40% reduction relative to the LSWtheory)37, suggesting that the previous estimates of Δ may beinaccurate. Particularly, given that Δ is extracted by fittingthe induced gap / ffiffiffiffiffiffiffiffiffiffiffiffi

1� Δp

, the LSW approximation under-estimates 1− Δ (deviation from the isotropic exchange) becauseit overestimates the proportionality constant37.

Figure 7d, f show StotH¼0 q;ωð Þ for Jc/J= 0.09 and Δ= 0.85. Wefind that StotH¼0 q;ωð Þ remains essentially semiclassical, with sharpmagnon lines and a weak continuum, which deviates significantlyfrom the recent results of INS experiments37,39. We thus concludethat the Hamiltonian that reproduces the plateau dynamics failsto do so at H= 0 within the spin wave theory, even after takingmagnon–magnon interactions into account at the 1/S level. Wealso mention that the breakdown of the kinematic condition forsingle-to-two magnon decay also implies the breakdown of the

T = 0.1 K

Energy (meV)

0

1

2

3

4

5In

tens

ity (

arb.

uni

t)

10.5 T

11.25 T

12.0 T

12.75 T

13.5 T

0

Energy (meV)

NLSW

10

Field (T)

0

1

2

3

Ene

rgy

(meV

)

ba

c

11 12 13 15

�3�2�1

14

1 2 30 1 2 3

Fig. 6 Field-dependence of ω1–ω3 at q= (1/3, 1/3, 1) in the UUD phase ofBa3CoSb2O9. a Constant-q scans for the selected values of the magneticfield at T= 0.1 K. The solid-lines show the fitting to Gaussian functions.The locations of the magnon peaks are indicated, where the solid (ω1),dashed (ω2), and dotted (ω3) lines are guides to the eye. b Simulation ofthe line-shape by NLSW theory for J= 1.74meV, Δ= 0.85, Jc/J= 0.09,and g⊥= 3.95 convoluted with the assumed resolution 0.2 meV.c Comparison of the fitted magnon frequencies ω1–ω3 against the NLSWpoles. Error bars represent one standard deviation




condition for magnon decay into an arbitrary number of mag-nons44. Thus, the semiclassical picture of weakly interactingmagnons is likely inadequate to simultaneously explain the low-energy dispersions and the intrinsic incoherent features (such asthe high-intensity continuum and the line-broadening) observedin Ba3CoSb2O9 at H= 0.One may wonder if extrinsic effects can explain these experi-

mental observations. It is possible for exchange disorder toproduce continuous excitations as in the effective spin-1/2 tri-angular antiferromagnet YbMgGaO4

49. However, our singlecrystals are the same high-quality samples reported pre-viously32,37, essentially free from Co2+–Sb5+ site-disorder.Indeed, our crystals show only one sharp peak at 3.6 K in thezero-field specific heat32 in contrast to the previous reports ofmultiple peaks31, which may indicate multi-domain structure.Another possible extrinsic effect is the magnon–phonon coupling,that has been invoked to explain the measured spectrum of thespin-3/2 TLHAFM CuCrO2

50. However, if that effect were pre-sent at zero field, it should also be present in the UUD state. Thefact that Eq. (1) reproduces the measured excitation spectrum ofthe UUD state suggests that the magnon–phonon coupling isnegligibly small (a similar line of reasoning can also be appliedto the effect of disorder). Indeed, we have also measured thephonon spectrum of Ba3CoSb2O9 in zero field by INS andfound no strong signal of magnon–phonon coupling. Our resultsthen suggest that the dynamics of the spin-1/2 TLHAFM isdominated by intrinsic quantum mechanical effects that escapea semiclassical spin-wave description. This situation is analogousto the (π, 0) wave-vector anomaly observed in various spin-1/2square-lattice Heisenberg antiferromagnets51–55, but nowextending to the entire Brillouin zone in the triangular lattice.Given recent theoretical success on the square-lattice56, our

results motivate new non-perturbative studies of the spin-1/2TLHAFM.

MethodsNeutron scattering measurements. The neutron diffraction data under magneticfields applied in the [1,–1,0] direction were obtained by using CG-4C cold triple-axis spectrometer with the neutron energy fixed at 5.0 meV at High Flux IsotopeReactor (HFIR), Oak Ridge National Laboratory (ORNL). The nuclear structure ofthe crystal was determined at the HB-3A four-circle neutron diffractometer atHFIR, ORNL and then was used to fit the nuclear reflections measured at the CG-4C to confirm that the data reduction is valid. Only the scale factor was refined forfitting the nuclear reflections collected at CG-4C and was also used to scale themoment size for the magnetic structure refinement. 14 magnetic Bragg peakscollected at CG-4C at 10 T were used for the magnetic structure refinement. TheUUD spin configuration with the spins along the field direction was found to bestfit the data. The nuclear and magnetic structure refinements were carried out usingFullProf Suite57.

Our inelastic neutron scattering experiments were carried out with the multiaxis crystal spectrometer (MACS)58 at NIST Center for Neutron Research (NCNR),NIST, and the cold neutron triple-axis spectrometer (V2-FLEXX)59 at Helmholtz-Zentrum Berlin (HZB). The final energies were fixed at 3 and 5 meV on the MACSand 3.0 meV on V2-FLEXX.

Constraint on J due to the saturation field. An exact expression for thesaturation field for Hjjc, Hsat, can be obtained from the level crossing conditionbetween the fully polarized state and the ground state in the single-spin–flip sector.From the corresponding expression, we obtain:

J ¼ gjjμBHsatS�1

3þ 6Δþ 2 1þ Δð Þ Jc=Jð Þ ; ð7Þ

where gjj ¼ 3:87 and μ0Hsat= 32.8 T33.

Variational analysis on classical instability of the 1/3 plateau in quasi-2DTLHAFMs. We show that the UUD state is not the classical ground state in thepresence of the antiferromagnetic interlayer exchange Jc > 0. To verify that theclassical ground space for Jc= 0 acquires accidental degeneracy in the in-plane

Inte

nsity

Inte

nsity

0 1 2 3 4 5 6

0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5 6

a b Jc = 0, Δ = 0.85

c d Jc = 0.09J, Δ = 0.85

Jc = 0, Δ = 1

Jc = 0.09J, Δ = 1

Γ3 Γ3*

3

6

5

4

2

1

0

3

6

5

4

2

1

00 0.1 0.2 0.3 0.4 0.5

q = (h, h, 1)

e

f

Jc = 0, Δ = 1

Jc = 0.09J, Δ = 0.85

>3

2

1

0

h� (meV) h� (meV)

h� (

meV

)h�

(m

eV)

SH

=0

(q, �

)to

t

totSH=0 (q, �)

SH=0 (q, �)zz

xxSH=0,L (q, �) + SH=0,L

(q, �)yy

xxSH=0,T (q, �) + SH=0,T

(q, �)yy

Fig. 7 Calculated dynamical spin structure factor of the in-plane 120° state at H= 0. The calculations are made using NLSW theory for spin 1/2. a–d Theresults of the frequency dependence at q= (1/2, 1/2, 1) (M point) for (a) Jc= 0 and Δ= 1 (the ideal TLHAFM), (b) Jc= 0 and Δ= 0.85, (c) Jc= 0.09J andΔ= 1, and (d) Jc= 0.09J and Δ= 0.85. The results are convoluted with the energy resolution 0.015J. The total spin structure factor StotH¼0 q;ωð Þ (solid line)is divided into different components; SzzH¼0 q;ωð Þ and SxxH¼0;T q;ωð Þ þ SyyH¼0;T q;ωð Þ are single-magnon contributions (“T” denotes the transverse part), whilethe longitudinal (L) part SxxH¼0;L q;ωð Þ þ SyyH¼0;L q;ωð Þ corresponds to the two-magnon continuum; single magnon peaks (two-magnon continua) areindicated by arrows (curly brackets), whereas the dashed square brackets indicate anti-bonding single-magnon contributions, which are expected to bebroadened by higher-order effect in 1/S48. The inset shows the lowest-order, O(S0), magnon self-energy incorporating the decay process of a singlemagnon into two magnons. e and f Intensity plots of StotH¼0 q;ωð Þ along the high-symmetry direction in the Brillouin zone for (e) Jc= 0 and Δ= 1 and (f) Jc= 0.09J and Δ= 0.85. J= 1.74 meV is assumed





magnetic field, we rewrite Eq. (1) as

H¼ J2

PΔ

SΔ;A þ SΔ;B þ SΔ;C � hred3J x

� �2� 1� Δð ÞJ P

rr′h iSzrS

zr′ þ Jc

Pr

SxrSxrþc

2þ SyrS

yrþc

2þ ΔSzrS

zrþc

2

� �þ const:;

ð8Þ

where the summation ofP

Δ is taken over the corner-sharing triangles in eachlayer, with r= (Δ, μ) (μ=A, B, C) denoting the sublattice sites in each triangle.This simply provides an alternative view of each triangular lattice layer (Fig. 8a). xis the unit vector in the x or field direction. The easy-plane anisotropy forces everyspin of the classical ground state to lie in the ab plane and the second term in Eq.(8) has no contribution at this level. Hence, for Jc= 0, any three-sublattice spinconfiguration satisfying Szr ¼ 0 and

SΔ;A þ SΔ;B þ SΔ;C ¼hred3J

x; 8Δ; ð9Þ

is a classical ground state, where we momentarily regard SΔ,μ as three-componentclassical spins of length S. Since there are only two conditions corresponding to thex and y components of Eq. (9), whereas three angular variables are needed tospecify the three-sublattice state in the ab plane, the classical ground state manifoldfor Jc= 0 retains an accidental degeneracy, similar to the well-known case of theHeisenberg model (Δ= 1)43. The UUD state is the classical ground state only forhred= 3JS.

The classical instability of the UUD state for Jc > 0 can be demonstrated by avariational analysis. The UUD state in the 3D lattice enforces frustration for one-third of the antiferromagnetic interlayer bonds, inducing large variance of theinterlayer interaction. As shown in Fig. 1a, only two of the three spin pairs alongthe c-axis per magnetic unit cell can be antiferromagnetically aligned, as favored byJc, while the last one has to be aligned ferromagnetically. To seek for a betterclassical solution, we consider a deformation of the spin configurationparameterized by 0 ≤ θ ≤ π at hred= 3JS, such that the spin structure becomesnoncollinear within the ab plane (Fig. 8b). Because the magnetization in each layeris fixed at S/3 per spin, the sum of the energies associated with the intralayerinteraction and the Zeeman coupling is unchanged under this deformation. In themeantime, the energy per magnetic unit cell of the interlayer coupling is varied as

EcðθÞ ¼ 2JcS2 cos 2θ � 2cos θð Þ: ð10Þ

We find that Ec(θ) is minimized at θ= π/3 for Jc > 0, corresponding to a saddlepoint. This is a rather good approximation of the actual classical ground state forsmall Jc > 0, as can be demonstrated by direct minimization of the classical energyobtained from Eq. (1). The crucial observation is that the classical ground statediffers from the θ= 0 UUD state.

NLSW calculation for the UUD state. We summarize the derivation of the spinwave spectrum in the quasi-2D TLHAFM with easy-plane anisotropy [see Eq. (1)].As discussed in the main text, we first work on the 2D limit Jc= 0 exactly at hred=3JS, and a given value of 0 ≤ Δ ≤ 1, which are the conditions for the UUD state to bethe classical ground state. Defining the UUD state as shown in Fig. 1a, we introducethe Holstein–Primakoff bosons, aðyÞμ;r as in Eqs (2)–(4). After performing a Fourier

transformation, aμ;k ¼ 1=Nmag

� �1=2Pr2 μ e

�ik�raμ;r , where Nmag=N/6 is the

number of magnetic unit cells (six spins for each) and N is the total number ofspins, we obtain the quadratic Hamiltonian as the sum of even layers (sublatticesAe–Ce) and odd layers (sublattices Ao–Co) contributions:

H0LSW ¼ H0

LSW;even þH0LSW;odd; ð11Þ

where the constant term has been dropped. Here,

H0LSW;even ¼

S2

Xk2RBZ

ayk� �T

a�kð ÞT� � H0

11;k H012;k

H021;k H0

22;k

!akay�k

!; ð12Þ

with H011;k ¼ H0

22;k , H012;k ¼ H0

21;k , where the summation over k is taken in thereduced Brillouin zone (RBZ) corresponding to the magnetic unit cell of the UUDstate. From now on, we will denote this summation as

Pk . We have introduced

vector notation for the operators

ak ¼aAe ;k

aBe ;k

aCe ;k

0B@

1CA �

a1;ka2;ka3;k

0B@

1CA; ay�k ¼

ayAe ;�k

ayBe ;�k

ayCe ;�k

0BBB@

1CCCA �

ay1;�kay2;�kay3;�k

0BB@

1CCA; ð13Þ

and matrix notation for the quadratic coefficients

H011;k ¼

S�1hred32 J 1þ Δð Þγk 3

2 J 1� Δð Þγ�k32 J 1þ Δð Þγ�k S�1hred

32 J 1� Δð Þγk

32 J 1� Δð Þγk 3

2 J 1� Δð Þγ�k 6J � S�1hred

0B@

1CA;

H012;k ¼

0 � 32 J 1� Δð Þγk � 3

2 J 1þ Δð Þγ�k� 3

2 J 1� Δð Þγ�k 0 � 32 J 1þ Δð Þγk

� 32 J 1þ Δð Þγk � 3

2 J 1þ Δð Þγ�k 0

0B@

1CA;

ð14Þ

with γk ¼ 13 eik�a þ eik�b þ e�ik� aþbð Þ� �

. Similarly, we have

H0LSW;odd ¼

S2

Xk

�ayk� �T

�a�kð ÞT� � �H0

11;k�H012;k

�H021;k

�H022;k

!�ak�ay�k

!; ð15Þ

with

�ak ¼aAo ;k

aBo ;k

aCo ;k

0B@

1CA �

a4;ka5;ka6;k

0B@

1CA;�ay�k ¼

ayAo ;�k

ayBo ;�k

ayCo ;�k

0BBB@

1CCCA �

ay4;�kay5;�kay6;�k

0BB@

1CCA; ð16Þ

and

�H011;k ¼ �H0

22;k ¼0 1 0

0 0 1

1 0 0

0B@

1CAH0

11;k

0 0 1

1 0 0

0 1 0

0B@

1CA;

�H012;k ¼ �H0

21;k ¼0 1 0

0 0 1

1 0 0

0B@

1CAH0

12;k

0 0 1

1 0 0

0 1 0

0B@

1CA:

ð17Þ

The excitation spectrum of H0LSW retains two k-linear modes at k= 0 (Fig. 2b).

Below, we include nonlinear terms to gap out these excitations. At this stage, thenonlinear terms correspond to the mean-field (MF) decoupling of the intra-layerquartic terms. Once we obtain such a MF Hamiltonian with the gapped spectrum,the deviation from the fine-tuned magnetic field hred= 3JS and interlayer coupling(as well as some other perturbation, if any) can be included. Here, the additionalterm contains both LSW and NLSW terms. To proceed, we first define thefollowing mean-fields (MFs) symmetrized by using translational and rotationalinvariance:

ρμ ¼ 1Nmag

Pr2 μ

ayμ;raμ;rD E

0;

δμ ¼ 1Nmag

Pr2 μ

aμ;r� �2 �

0

;

ξμν ¼ 13Nmag

Pr2 μ

Pημν

ayμ;raν;rþημν

D E0;

ζμν ¼ 13Nmag

Pr2 μ

Pημν

aμ;raν;rþημν

D E0;

ð18Þ

where ημν represents the in-plane displacement vector connecting sites r ∈ μ to anearest-neighbor site in sublattice ν. The mean values ⟨...⟩0 are evaluated with theground state of H0

LSW. The MFs for odd (even) layers are obtained from those for

B C

A

B C

A

A B

C

C A

B

xzy

Co

CeBe

Bo

H

Ao

Ae

ba

�

�

�

�

Fig. 8 Classical instability of the UUD state in the quasi-2D lattice. a Three-sublattice structure for a single layer and a decomposition of the intralayerbonds into corner-sharing triangles. b Deformation of the UUD state (seeFig. 1a) parameterized by θ shown in the projection in the ab (or xy) plane;the spins in sublattices Be and Co are unchanged




even (odd) layers as

ρAo¼ ρBe

; ρBo¼ ρCe

; ρCo¼ ρAe

;

δAo¼ δBe

; δBo¼ δCe

; δCo¼ δAe

;

ξAoBo¼ ξBeCe

; ξBoCo¼ ξCeAe

; ξCoAo¼ ξAeBe

;

ζAoBo¼ ζBeCe

; ζBoCo¼ ζCeAe

; ζCoAo¼ ζAeBe

:

ð19Þ

By collecting all the contributions mentioned above, we obtain the NLSWHamiltonian

HNLSW ¼S2

Xk

ayk� �T

�ayk� �T

a�kð ÞT �a�kð ÞT� �

´

Hee;k Heo;k H 0ee;k H0eo;k

Heo;k

� �yHoo;k H0eo;�k

� �TH 0oo;k

H0ee;�k� ��

H0eo;�k� ��

Hee;�k� ��

Heo;�k� ��

H0eo;k� �y

H0oo;�k� ��

Heo;�k� �T

Hoo;�k� ��

0BBBBBBBB@

1CCCCCCCCA

ak�akay�k�ay�k

0BBBB@

1CCCCA;

ð20Þ

where

Hee;k ¼ H011;k þ

�2Jc 0 0

0 2Jc 0

0 0 2Jc

0BB@

1CCAþ S�1

μAeMF þ 2JcρAo

tAeBeMF

� ��γk tCeAe

MF γ�k

tAeBeMF γ�k μBeMF � 2JcρBo

tBeCeMF

� ��γk

tCeAeMF

� ��γk tBeCe

MF γ�k μCeMF � 2JcρCo

0BBBBBB@

1CCCCCCA;

Hoo;k ¼ �H011;k þ

�2Jc 0 0

0 2Jc 0

0 0 2Jc

0BB@

1CCAþ S�1

μAoMF þ 2JcρAe

tAoBoMF

� ��γk tCoAo

MF γ�k

tAoBoMF γ�k μBo

MF � 2JcρBetBoCoMF

� ��γk

tCoAoMF

� ��γk tBoCo

MF γ�k μCoMF � 2JcρCe

0BBBBBB@

1CCCCCCA;

H 0ee;k ¼ H012;k þ S�1

ΓAeMF gAeBe

MF γk gCeAeMF γ�k

gAeBeMF γ�k ΓBe

MF gBeCeMF γk

gCeAeMF γk gBeCe

MF γ�k ΓCMF

0BBB@

1CCCA;

H0oo;k ¼ �H012;k þ S�1

ΓAoMF gAoBo

MF γk gCoAoMF γ�k

gAoBoMF γ�k ΓBo

MF gBoCoMF γk

gCoAoMF γk gBoCo

MF γ�k ΓCoMF

0BBB@

1CCCA;

Heo;k ¼ cosk3

Jc 1þ Δð Þ 0 0

0 Jc 1� Δð Þ 0

0 0 Jc 1� Δð Þ

0BB@

1CCAþ S�1cosk3

tAeAoMF

� ��0 0

0 tBeBoMF

� ��0

0 0 tCeCoMF

� ��

0BBBBBB@

1CCCCCCA;

H0eo;k ¼ cosk3

�Jc 1� Δð Þ 0 0

0 �Jc 1þ Δð Þ 0

0 0 �Jc 1þ Δð Þ

0BB@

1CCAþ S�1cosk3

gAeAoMF 0 0

0 gBeBoMF 0

0 0 gCeCoMF

0BBB@

1CCCA:

ð21Þ

Here the MF parameters are given as follows. First, those associated with theintralayer coupling are

μAeMF ¼ 3J ρBe

� ρCe� 1þΔ

2 ReξAeBe� ReζCeAe

� �� 1�Δ

2 ReξCeAe� ReζAeBe

� �h i;

μBeMF ¼ 3J ρAe

� ρCe� 1þΔ

2 ReξAeBe� ReζBeCe

� �� 1�Δ

2 ReξBeCe� ReζAeBe

� �h i;

μCeMF ¼ 3J �ρAe

� ρBeþ 1þΔ

2 ReζBeCeþ ReζCeAe

� �� 1�Δ

2 ReξBeCeþ ReξCeAe

� �h i;

tAeBeMF ¼ 3J ξAeBe

� 1þΔ4 ρAe

þ ρBe

� �þ 1�Δ

8 δ�Aeþ δBe

� �h i;

tBeCeMF ¼ 3J �ξBeCe

þ 1þΔ8 δ�Be

þ δCe

� �� 1�Δ

4 ρBeþ ρCe

� �h i;

tCeAeMF ¼ 3J �ξCeAe

þ 1þΔ8 δ�Ce

þ δAe

� �� 1�Δ

4 ρCeþ ρAe

� �h i;

ΓAeMF ¼ 3J

21þΔ2 ξCeAe

� ζAeBe

� �þ 1�Δ

2 ξ�AeBe� ζCeAe

� �h i;

ΓBeMF ¼ 3J

21þΔ2 ξ�BeCe

� ζAeBe

� �þ 1�Δ

2 ξAeBe� ζBeCe

� �h iΓCeMF ¼ 3J

21þΔ2 ξ�BeCe

� ξCeAe

� �þ 1�Δ

2 ζBeCe� ζCeAe

� �h i;

gAeBeMF ¼ 3J ζAeBe

� 1þΔ8 δAe

þ δBe

� �þ 1�Δ

4 ρAeþ ρBe

� �h i;

gBeCeMF ¼ 3J �ζBeCe

þ 1þΔ4 ρB þ ρC� �� 1�Δ

8 δB þ δCð Þh i

;

gCeAeMF ¼ 3J �ζCeAe

þ 1þΔ4 ρCe

þ ρAe

� �� 1�Δ

8 δCeþ δAe

� �h i;

ð22Þ

for even layers and

μAoMF ¼ μBe

MF; μBoMF ¼ μCe

MF; μCoMF ¼ μAe

MF;

tAoBoMF ¼ tBeCe

MF ; tBoCoMF ¼ tCeAe

MF ; tCoAoMF ¼ tAeBe

MF ;

ΓAoMF ¼ ΓBe

MF; ΓBoMF ¼ ΓCe

MF; ΓCoMF ¼ ΓAe

MF;

gAoBoMF ¼ gBeCe

MF ; gBoCoMF ¼ gCeAe

MF ; gCoAoMF ¼ gAeBe

MF ;

ð23Þ

for odd layers. Similarly, the new MF parameters associated with the interlayercoupling are

tAeAoMF ¼ Jc � 1þΔ

2 ρAeþ ρAo

� �þ 1�Δ

4 δ�Aeþ δAo

� �h i;

tBeBoMF ¼ Jc

1þΔ4 δ�Be

þ δBo

� �� 1�Δ

2 ρBeþ ρBo

� �h i;

tCeCoMF ¼ Jc

1þΔ4 δ�Ce

þ δCo

� �� 1�Δ

2 ρCeþ ρCo

� �h i;

gAeAoMF ¼ Jc � 1þΔ

4 δAeþ δAo

� �þ 1�Δ

2 ρAeþ ρAo

� �h i;

gBeBoMF ¼ Jc

1þΔ2 ρBe

þ ρBo

� �� 1�Δ

4 δBeþ δBo

� �h i;

gCeCoMF ¼ Jc

1þΔ2 ρCe

þ ρCo

� �� 1�Δ

4 δCeþ δCo

� �h i;

ð24Þ

Figure 9 shows the Δ-dependence of these MF parameters. Because theseMF parameters are real valued, the coefficient matrix in Eq. (20) has the form

HNLSW ¼Pk Qk

Qk Pk

� �; ð25Þ

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.8 0.85

Δ Δ

0.9 0.95 1

a0.7

�MF / J = �MF / JAe Be

�MF / JCe

t MF / JAeBe

t MF / J =t MF / J BeCe CeAe

g MF / JAeBe

gMF

/J =g MF / JBeCe CeAe

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.8 0.85 0.9 0.95 1

b0.4

tMF / JAeAo

tMF / J = tMF / JBeBo CeCo

gMF / JAeAo

gMF / J = gMF / JBeBo CeCo

MF / JCe�MF / J = MF / JAe Be� �

Fig. 9 Δ-dependence of the recombined MF parameters. MF parameters associated with (a) the intra-layer coupling and (b) the inter-layer coupling





with

Pk ¼Hee;k Heo;k

Heo;k Hoo;k

!;Qk ¼

H0ee;k H0eo;kH0eo;k H0oo;k

!: ð26Þ

This form can be diagonalized by a Bogoliubov transformation,

ak�akay�k�ay�k

0BBBB@

1CCCCA ¼

Uk Vk

Vk Uk

� �αk

αy�k

!; ð27Þ

where αk αy�k� �

is the six-component vector comprising the annihilation(creation) operators of Bogoliubov bosons. The transformation matrices satisfyUμκk ¼ ðUμκ

�kÞ� and Vμκk ¼ ðVμκ

�kÞ� . The poles, ωκ,k, are the square-roots of theeigenvalues of S2(Pk + Qk)(Pk−Qk).

When calculating the sublattice magnetization, the reduction of the orderedmoment relative to the classical value S corresponds to the local magnon density.With the phase factors for each sublattice, cAe

¼ cBe¼ �cCe

¼ cAo¼ �cBo

¼cCo¼ 1 (see Fig. 1a), we have

hSxr i ¼ cμ S� hayμ;raμ;ri� �

¼ cμ S� 1Nmag

Xk

Xκ

Vμκk

2 !; ð28Þ

for site r in sublattice μ.The dynamical spin structure factor is defined by

Sαα q;ωð Þ ¼ R1�1 dt2π e

iωt 1N

Pr;r′

e�iq� r�r′ð Þ Sαr tð ÞSαr′ 0ð Þ�

¼ Pnδ ω� ωnð Þ 0jSαqjn

D E 2: ð29Þ

where Sαq ¼ N�1=2P

r Sαr e�iq�r and |n⟩ and ωn denote the nth excited state and its

excitation energy, respectively. The longitudinal spin component is

Sxq ¼ffiffiffiffiNp

3S δq;0 þ δq;Q þ δq;�Q� �

þ δSxq; ð30Þ

with Q= (1/3, 1/3,1) and

δSxq ¼ �ffiffiffiffi1N

r Xμ;k

cμayμ;k�qaμ;k ; ð31Þ

We truncate the expansions of the transverse spin components at the lowestorder:

Syq � �iffiffiffiffiS12

q Pμ

aμ;q � ayμ;�q� �

;

Szq �ffiffiffiffiS12

q Pμ�cμ� �

aμ;q þ ayμ;�q� �

:ð32Þ

The transverse components of the dynamical structure factor,S? q;ωð Þ ¼ Syy q;ωð Þ þ Szz q;ωð Þ, reveal the magnon dispersion,

Syy q;ωð Þ ¼ Pnδ ω� ωnð Þ 0jSyqjn

� 2

� S12

Pκδ ω� ωκ;q

� � Pμ

Uμκq � Vμκ

q� �

2

;

Szz q;ωð Þ ¼ Pnδ ω� ωnð Þ h0jSzqjni

2

� S12

Pκδ ω� ωκ;q

� � Pμcμ Uμκ

q þ Vμκq

� �2

:

ð33Þ

Meanwhile, Sxx q;ωð Þ comprises the elastic contribution and the longitudinalfluctuations,

Sjj q;ωð Þ ¼Xn

δ ω� ωnð Þ 0jδSxqjnD E 2; ð34Þ

which can be evaluated by using Wick’s theorem. The result at T= 0 is

Sjj q;ωð Þ ¼ Θ ωð ÞN�1Xk

Xκ;λ;μ;ν

cμcνReAμν;κλ k; qð Þδ ω� ωκ;�kþq � ωλ;k

� �; ð35Þ

where

Aμν;κλðk; qÞ ¼12

Uμκk�q

� ��Vμλk þ Vμκ

k�q� ��

Uμλk

h iUνκk�q Vνλ

k

� ��þVνκk�q Uνλ

k

� ��h i:

ð36Þ

Data availability. All relevant data are available from the corresponding authorsupon reasonable request.

Received: 8 November 2017 Accepted: 21 May 2018

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AcknowledgementsWe thank H. Tanaka, A. Chernyshev, and O. Starykh for valuable discussions. J.M.acknowledges support of the Ministry of Science and Technology of China(2016YFA0300500) and from NSF China (11774223). Y. K. acknowledges financialsupport by JSPS Grants-in-Aid for Scientific Research under Grant No. JP16H02206. Thework at Georgia Tech was supported by ORAU’s Ralph E. Powe Junior FacultyEnhancement Award (M. Mourigal) and NSF-DMR-1750186 (L. G. and M. Mourigal). C.D. B. acknowledges financial support from the Los Alamos National Laboratory DirectedResearch and Development program and from the Lincoln Chair of Excellence inPhysics. H. D. Z. acknowledges support from NSF-DMR-1350002. The work performedin NHMFL was supported by NSF-DMR-1157490 and the State of Florida. We aregrateful for the access to the neutron beam time at the neutron facilities at NCNR, BER-IIat Helmholtz-Zentrum Berlin and HFIR operated by ORNL. The research at HFIR atORNL was sponsored by the Scientific User Facilities Division (T. H., H. B. C., and M.Matsuda), Office of Basic Energy Sciences, U.S. DOE. Access to MACS was provided bythe Center for High Resolution Neutron Scattering, a partnership between the NationalInstitute of Standards and Technology and the National Science Foundation underAgreement No. DMR-1508249. J. Ma’s primary affiliation is Shanghai Jiao TongUniversity.

Author contributionsY. K. and J. M. conceived the project. H. D. Z. prepared the samples. T. H., Y. Q., D. L.Q., Z. L., H. B. C., M. Matsuda, L. G., M. Mourigal, and J. M. performed the neutronscattering experiments. E. S. C. measured the magnetization. Y. K., L. G., C. D. B., and M.Mourigal performed the NLSW calculations. Y. K., J. M., M. Mourigal, and C. D. B. wrotethe manuscript with comments from all the authors.

Additional informationCompeting interests: The authors declare no competing interests.

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ARTICLE

Quasiparticle interference and nonsymmorphiceffect on a floating band surface state of ZrSiSeZhen Zhu1, Tay-Rong Chang2, Cheng-Yi Huang3, Haiyang Pan4, Xiao-Ang Nie1, Xin-Zhe Wang1, Zhe-Ting Jin1,

Su-Yang Xu5, Shin-Ming Huang 6, Dan-Dan Guan1,7, Shiyong Wang 1,7, Yao-Yi Li1,7, Canhua Liu1,7,

Dong Qian1,7, Wei Ku1,7, Fengqi Song4,7, Hsin Lin3, Hao Zheng 1,7 & Jin-Feng Jia1,7

Non-symmorphic crystals are generating great interest as they are commonly found in

quantum materials, like iron-based superconductors, heavy-fermion compounds, and topo-

logical semimetals. A new type of surface state, a floating band, was recently discovered in

the nodal-line semimetal ZrSiSe, but also exists in many non-symmorphic crystals. Little is

known about its physical properties. Here, we employ scanning tunneling microscopy to

measure the quasiparticle interference of the floating band state on ZrSiSe (001) surface and

discover rotational symmetry breaking interference, healing effect and half-missing-type

anomalous Umklapp scattering. Using simulation and theoretical analysis we establish that

the phenomena are characteristic properties of a floating band surface state. Moreover, we

uncover that the half-missing Umklapp process is derived from the glide mirror symmetry,

thus identify a non-symmorphic effect on quasiparticle interferences. Our results may pave a

way towards potential new applications of nanoelectronics.

DOI: 10.1038/s41467-018-06661-9 OPEN

1 School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China. 2 Department of Physics, National Cheng Kung University,Tainan 701, Taiwan. 3 Institute of Physics, Academia Sinica, Taipei City 11529, Taiwan. 4 College of Physics, Nanjing University, Nanjing 210093, China.5 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 6Department of Physics, National Sun Yat-Sen University,Kaohsiung 80424, Taiwan. 7 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China. These authorscontributed equally: Zhen Zhu, Tay-Rong Chang, Cheng-Yi Huang. Correspondence and requests for materials should be addressed toH.Z. (email: [email protected]) or to J.-F.J. (email: [email protected])


1234

5678

90():,;

63

http://orcid.org/0000-0003-4273-9682

http://orcid.org/0000-0003-4273-9682

http://orcid.org/0000-0003-4273-9682

http://orcid.org/0000-0003-4273-9682

http://orcid.org/0000-0003-4273-9682

http://orcid.org/0000-0001-6603-9926

http://orcid.org/0000-0001-6603-9926

http://orcid.org/0000-0001-6603-9926

http://orcid.org/0000-0001-6603-9926

http://orcid.org/0000-0001-6603-9926

http://orcid.org/0000-0002-6495-874X

http://orcid.org/0000-0002-6495-874X

http://orcid.org/0000-0002-6495-874X

http://orcid.org/0000-0002-6495-874X

http://orcid.org/0000-0002-6495-874X





Research into surface states has been conducted for severaldecades and has recently begun to flourish again due to thediscovery of topologically non-trivial materials1,2. Several

topological surface states have been uncovered with prominentexamples including the spin-momentum locked Dirac cones intopological insulators3,4 and the disconnected Fermi arcs in Weylsemimetals5,6. Many characteristic phenomena, e.g., the prohibi-tion of electron back scattering on a topological insulator sur-face7,8, the tunable mass acquisition of surface fermions in atopological crystalline insulator9 and the electronic sink effect in aWeyl semimetal10–13 have been discerned through quasiparticleinterference (QPI) approaches. These have all been provenadvances in the understanding of the unconventional two-dimensional electron gases. Therefore, the search for new clas-ses of surface states with intriguing physical consequences is aninvaluable endeavor in condensed matter physics.ZrSiSe is a newly discovered non-symmorphic topological

Dirac nodal-line semimetal14–19 and is part of the class ofmaterials which includes ZrSiS, ZrSiSe, and ZrSiTe. Its bulk bandfeatures a linear dispersion in the energy range as broad as 2 eV,much larger than other known Dirac materials and presents theZrSiSe class of materials as an ideal candidate to target newrelated physics17–19. Indeed, a high electron mobility and a but-terfly magnetoresistance was discovered by transport measure-ments20. More importantly, a very recent study revealed anunconventional floating band surface state on ZrSiS but which isalso applicable to ZrSiSe and ZrSiTe. Its origin is directly derivedfrom the non-symmorphic symmetry of the crystal and is distinct

from the well-known Shockley type or dangling-bond type sur-face state21. As demonstrated in Fig. 1a, ZrSiSe is a layeredmaterial and crystallizes into a tetragonal lattice with a spacegroup P4/nmm (#129), which is shared with a broad variety ofquantum materials, e.g., nematic Fe-based superconductorNaFeAs22,23 and heavy-fermion compound with anti-ferromagnetism (AFM) CeRuSiH1.0

24. In the electronic bandstructure of ZrSiSe (Fig. 1b), non-symmorphic symmetry enforcesthe bulk bands to be doubly (quadruply if considering spindegrees of freedom) degenerate along entire X-M line; in otherwords, there exists a Dirac nodal line on the Brillouin Zone (BZ)boundary. On its (001) surface, the symmetry breaking splits atwo-dimensional electronic state from the bulk Dirac band,termed as a floating band. Figure 1c presents the first principlecalculation result where the floating band is highlighted.Obviously, this previously unknown surface state exists in a widerange of P4/nmm symmetric crystals which goes beyond topo-logically non-trivial materials. However, other than the identifi-cation of its origin, little is known about this surface state.Among all surface sensitive measurements, QPI which is

acquired via scanning tunneling microscopy (STM) may be themost direct method to reveal the unique physics of surface states.An ordinary QPI map measures surface standing wave inducedby a number of (usually various types of) point defects. However,the local geometrical and chemical structures of different types ofdefects carry distinctive information.Here, we apply a single-defect induced QPI (s-QPI) approach,

which is of both experimental and theoretical challenge, to

ba

d fe

Γ

1

0

–1

Γ X M M

Si

Zr

Se

c

Γ X Γ

g

1 2 3

High

Low

High

Low

High

Low

Ene

rgy

(a.u

)

Floating band E–E

F (

eV)

1.0

–2.0

–1.0

0.0

Ene

rgy

(eV

)

0.3

–0.3

0.0

dI/dV (nS)

100 mV

M

X

Fig. 1 Structural and electronic properties of ZrSiSe. a Crystal structure of ZrSiSe, which features a non-symmorphic P4/nmm space group. The Si layerserves as a glide mirror plane Mzj 12 1

2 0� �

. The weak Van der Waals interaction between adjacent Se–Zr–Si–Zr–Se quintuple layers provides a naturalcleaving surface between Se surfaces [(001) surface]. Blue, yellow, purple balls stand for Se, Si, Zr atoms, respectively. b Sketch of band structure withouttaking spin–orbit coupling into account. Nodal line bulk state and floating band surface state are plotted in black and red respectively. The non-symmorphicsymmetry in ZrSiSe protects the Dirac nodes located at the X point, as well as generates an unconventional type of floating band surface state, on the(001) surface. Inset is the surface Brillouin zone (BZ) with high symmetry points marked. c Calculated surface band structure of ZrSiSe(001). The floatingband state is highlighted in red. d STM image (0.1 V, 0.2 nA) demonstrating the atomic lattice on ZrSiSe(001) surface. The lattice constant is measured tobe 0.37 nm. Both the inset crystal structure and the STM image show that the surface preserves C4v symmetry. Scale bar stands for 1 nm. e Typical dI/dVspectrum measured on top of a Se atom in a defect free region. f STM image (300mV, 1 nA) showing a large-scale morphology. Arrows indicate twodefects, which apparently break C4v symmetry. Scale bar stands for 5 nm. g dI/dVmap acquired at same region as f. C2v symmetric standing wave patternsaround each defect are clearly discerned




directly measure the interferences at both single Si-defect and Zr-defect sites on the ZrSiSe (001) surface. In addition to the pre-viously insightful QPI discoveries on ZrSiS25,26 and ZrSiSe27, wedirectly identify the characteristic properties of a floating bandsurface state. Moreover, our theoretical analysis reveals theobserved anomalous Umklapp process to in principle exist in abroad class of non-symmorphic crystals.

ResultsRotational symmetry breaking feature. An overview of our low-temperature STM images, spectroscopy and dI/dV map on aZrSiSe (001) surface is shown in Fig. 1. The atomically resolvedSTM image in Fig. 1d clearly shows the square lattice of our highquality ZrSiSe sample, in which the C4v symmetry and the mea-sured lattice constant of 0.37 nm confirms the cleaved surface tobe the (001) orientation. The measured local density of state fromthe dI/dV spectrum (Fig. 1e) exhibits non-vanishing intensity atthe Fermi level, revealing the (semi-)metallic nature of oursample. Interestingly, our STM image (Fig. 1f) and dI/dV map(Fig. 1g) acquired at an energy near the Fermi level demonstratesan unusual ripple pattern. The pattern contains two orthogonalfeatures, each clearly breaking the C4v symmetry of the crystalsurface, and which were not observed on dI/dV maps measuredon a cousin material ZrSiS25,26.

Healing effect. In order to reveal the unique properties of thefloating band surface state, we performed systematic s-QPImeasurements on ZrSiSe(001). Three characteristics stand out inthe voltage-dependent dI/dV maps and their corresponding fastFourier transforms (FFTs). First, in contrast to the C4v patternexpected from the crystalline symmetry, defects showing C2v

symmetric pattern are also found. Figure 2a shows a clearstanding wave pattern around the point defect located at thecenter of the image in the voltage range starting from −50 mV.The wavelength shrinks with elevated bias voltage, thus provingthat the surface quasiparticle possesses an electron like band.Clearly, from the map taken at an energy close to Fermi level, i.e.,50 mV, the wave only propagates along one direction. Second,from both Fig. 2a, b, one can discern that the rotational symmetrybreaking phenomenon gradually disappears at a higher bias

around 400 meV, indicating a healing effect occurring in thesample.

Anomalous Umklapp process. Third, figure 3 shows the expec-ted C4v symmetric s-QPI patterns (see Supplementary Fig. 1 forSTM images of such defects) in which the standing waves pro-pagate equally along two orthogonal directions. Unexpectedly,from the FFT maps in Fig. 3b, one can note that the QPI featuresaround a Bragg point (inside the dotted circle) do not resemblethe central pockets (solid circle). This appears to violate theordinary theoretical understanding of Umklapp scattering andindicates an anomalous structure in the Umklapp process.

DiscussionIt happens that the non-symmorphic symmetry and the specialcrystal structure of the ZrSiSe class of materials are what lead tothe observed QPI features. One unique feature in the structure ofits P4/nmm lattice is the layer dependent location of the rota-tional axis. The ZrSiSe crystal is formed by alternatively stackedSe–Zr–Si–Zr–Se atomic layers. Each layer itself comprises of asquare mesh of atoms which preserves global C4v symmetry.However, the Si layer forms the glide mirror plane, and thus losesC4v symmetry locally at each Si atom site in order to fulfill theMzj 12 12 0� �

operation. The bulk symmetry gives rise to a (001)surface atomic structure (Fig. 1d) where the Se atoms are locatedat the corners of the square surface unit cell, the Zr atom sits atthe face center, and the Si atoms occupy the edge centers. Thisnaturally leads to the appearance of two inequivalent Si positions,each exhibiting only hidden local C2v symmetry. In addition tosingle Si defect, multiple C2v symmetric Si defects may arrangeinto multi-directional (Fig. 1g) or uni-directional (SupplementaryFig. 2) configurations. The latter case, if artificially controllable,will give rise to anisotropic scattering of electrons at the Fermilevel and consequently lead to a two-fold resistance in ZrSiSebased nanostructures.In contrast to the crystal structure analysis, we find that first

principle simulations and theoretical analysis are crucial tointerpret the healing effect occurring in the sample restoring toC4v symmetry and the anomalous Umklapp process. We beginthe discussion by considering s-QPIs on a Zr defect. Havingcarefully identified the floating band from the other states (see

a

b

High

Low

High

Low

–50 mV 50 mV 100 mV 200 mV 400 mV

–50 mV 50 mV 100 mV 200 mV 400 mV

Fig. 2 C2v symmetric interferences on ZrSiSe(001) in single-defect-induced quasiparticle interference (s-QPI) patterns. a and b are voltage-dependentdI/dV maps (18×18 nm2, 400mV, 1 nA) and Fourier transformed (FT) dI/dV maps, depicting the real and reciprocal space s-QPI patterns arising from asingle C2v symmetric defect, respectively. The defect is attributed to a Si vacancy. With increasing bias voltage, the intrinsic C4v symmetry graduallyrecovers. The arrow indicates the wave propagating direction





details in Supplementary Fig. 3), we can now draw a schematicconstant energy contour (CEC) which only exhibits such bands(Fig. 4a). The floating band pockets manifest as four large ringsenclosing the corners of the first BZ. Near a X point, two floatingband contours exist approximately parallel to each other, whichgives rise to a significantly enhanced nesting vector, i.e., Q1 inFig. 4a. Q1 and its C4v rotational partner Q2 together constitutethe bright central square in the s-QPI pattern, which is shown inFig. 4b. In addition to these intra-first BZ scatterings (normalprocesses), inter-BZ scatterings (Umklapp processes) usually alsocontribute to the QPI. For example, a normal scattering vector Q1

followed by a unit reciprocal vector (Gx or Gy) produces a typicalUmklapp process, which should preserve the crystal symmetry.The C4v point group in ZrSiSe should in principle result in thevector Q1+Gx or Q1+Gy generating a QPI contour with theexact shape as Q1, which appears as replica squares at the fourBragg points as shown in Fig. 4d. However, our observationclearly contradicts this ordinary Umklapp process. Concretely,the QPI feature near a Bragg point manifests as a double-parallelarc (Fig. 4b, c) rather than a square. We name this phenomenon,not previously understood, to be an anomalous half-missingUmklapp process, as exactly half of the expected Umklapp pat-tern (a square) is absent in the observation.We carry out a T-matrix based Green’s function approach to

simulate the s-QPI patterns of a single Zr defect based on threedifferent assumptions. We first consider a band structure simu-lation without considering any band unfolding or form factoreffects. Figure 4d, e show such s-QPI patterns which respectivelyallow and forbid inter-BZ scatterings. While both results are ableto capture the square-shaped central feature, they both fail toreproduce the half-missing patterns around the Bragg points. Incontrast in Fig. 4f, by including a sublattice induced form factoreffect and forbidding inter-BZ scatterings, we are able to suc-cessfully and quantitively simulate the entire s-QPI pattern, and,in particular, the double arcs around Bragg points.Based on the above analysis, we now understand that the origin

of these anomalous half-missing Umklapp processes is actually adirect consequence of the non-symmorphic effect on the energyband structure in a P4/nmm crystal. In fact, many importanteffects induced by glide mirror symmetries have been establishedbased on Fe-based superconductors research22,23, but the con-clusions also apply to the all materials which contain Mzj 12 12 0

� �.

We invoke the relevant discovery here to interpret the half-missing Umklapp scattering. Namely, the glide mirror symmetryMzj 12 12 0� �

splits the lattice into two sublattices, where the A and Bsublattices are glide mirror partners to each other. This induces aparticular type of non-trivial form factor, which must beaccounted for in a first principle band calculation. Atomic orbitalscan be divided into even or odd parity under the mirror Mz

operation. By adding a minus sign (a form factor) to odd orbitalson B sublattice but keep all other orbitals intact, the Mz iseffectively absorbed by the wavefunctions. In the new wave-function basis, the fractal translation 1

212 0

� �becomes a good

symmetry, which effectively reduces the original unit cell by halfand thus expands the area of the first BZ by exactly two folds. Inthe reconstructed first BZ (dashed line in Fig. 4a), Q1+Gx

changes from an Umklapp to a normal process, which has noreason to be weak or absent. On the other side, Q1+Gy is nowterminated out of the new first BZ and becomes a real inter-BZscattering, i.e., a real Umklapp process. The suppression orabsence of this feature exactly leads to the half-missing Umklappfeature. More profoundly, it indicates that the floating bandsurface state contains fewer atomic-scale ripples in contrast to thebehavior from a dangling-bond derived surface state. The floatingband state is thus believed to be weakly bounded to the surface,analogous to a Fermi-arc surface state on a Weyl semimetal28.

By introducing the glide mirror symmetry Mzj 12 12 0� �

enforcedparticular type of form factor into our energy band simulations,we can now reproduce the measurements on both C4v and C2v

types of defects by considering different T-matrices which aredirectly derived from the first principle simulations on a singledefect. In Fig. 5a, we present a set of voltage-dependent experi-mental C2v symmetric QPI patterns around uni-directional Sidefects (Supplementary Fig. 2). The adjacent panels in Fig. 5bshow the simulated patterns which also reproduce the healingeffect. The line cuts in Fig. 5c (experimental) and Fig. 5d (theo-retical) demonstrate that our simulation corroborates the mea-surement across a wide energy range, thus proving the robustnessof our theory. The healing effect can be understood by analyzingthe scattering channels. Namely, at a Si-defect, both a directscattering channel between Si-p orbits and an indirect scatteringchannel through p-d orbits coupling (Zr–Si interaction) coexist.Near the Fermi level, the major signal in a C2v s-QPI pattern isdominated by the direct scattering. However, away from the

b

a

High

Low

High

Low

–50 mV 50 mV 100 mV 200 mV 400 mV

–50 mV 50 mV 100 mV 200 mV 400 mV

Fig. 3 C4v symmetric s-QPI patterns on ZrSiSe (001). a and b are voltage-dependent dI/dV maps (18×18 nm2, 400mV, 1 nA) and FT-dI/dV maps arisingfrom a single C4v defect, respectively. The arrows represent the two wave propagating directions. The defect is attributed to a Zr vacancy. Note thescanning directions of all images are rotated π/4 with respect to the images in Fig. 2a for technical reasons. The solid (dotted) circle surrounds the QPIfeature induced from normal (Umklapp) process




Fermi level, the signal from Zr-d orbits, which carry strong C4v

symmetry, becomes enhanced (see Supplementary Figs. 5–7 fordetails), and the interference begins to appear less C2v symmetric.In principle, this healing effect could exist in the entire ZrSiSe-family of topological nodal-line semimetals.In summary, we systemically combined experimental and

theoretical s-QPI techniques to directly visualize the uncon-ventional floating band type of surface state on a topologicalDirac nodal line semimetal ZrSiSe, which features a non-symmorphic space group P4/nmm. Three effects, namely arotational symmetry violation, a healing effect, and a half-missing type anomalous Umklapp process are identified ascharacteristic properties of a floating band. Moreover, the half-missing Umklapp process can be understood as a non-symmorphic effect, which theoretically exists in a broad classof materials whose lattices contain glide mirrors Mzj 12 12 0

� �. One

may potentially be able to deduce the Mzj 12 12 0� �

symmetry

induced phase shift of the electron wavefunction by using theatomic manipulation technique to arrange an array of adatomsinto a particular geometry29. Furthermore, the revealed aniso-tropic charge carrier scattering behavior may provide insightsin the development of new nanoelectronics. Therefore, webelieve our results here are of both fundamental and applica-tional importance.

MethodsSample growth. The single crystalline ZrSiSe samples were synthetized by astandard chemical vapor transport method. A stoichiometric mixture of Zr, Si, andSe powders and the transport agent I2 (5 mg cm−3) were placed at the end of aquartz tube. The quartz tube was then evacuated, sealed and loaded into a hor-izontal tube held at high temperature. The occupied end, which contained thereaction powders, and empty end of the quartz tube were maintained at the hightemperature 950 °C and low temperature 850 °C respectively. The temperaturegradient of tube furnace was maintained for two weeks. The square and rectangularshaped ZrSiSe crystals were formed at the cold end.

f 400 mVd 400 mV

a

Q 1 Q 2

G XG y

Γ

X

M

1st B

Z

Nonsymmorphic reshaped 1st BZ

b

c400 mV

Q1

Q2

GXGy

400 mVe

High

Low

High

Low

High

Low

High

Low

Fig. 4 Non-symmorphic effect on a floating band surface state. a Schematics depicting the anomalous Umklapp process derived from the non-symmorphicP4/nmm group. The blue square surrounds the first surface BZ of ZrSiSe(001), in which only the floating band surface state contours are presented. Q1 andQ2 label two dominate scattering vectors. Gx and Gy represent the reciprocal unit vectors. Normal scattering (Q1) and Umklapp scatterings (Q1+Gx, andQ1+Gy) are expected to generate the same shapes of QPI patterns in a conventional system with C4v symmetry. b Sketch (not to scale) highlighting theQPI features which arising only from the floating band. The artificially added red dots in b, d–fmark Bragg points. The vectors Q1, Q2, Gx, and Gy are definedthe same way and in the same directions as in a, but with different lengths. The central square (denoted by Q1 and Q2) originates from normal scatterings,while the double arcs near Bragg points are induced by Umklapp scatterings. Note the feature at Q1+Gy is absent, which leads to the half-missinganomalous Umklapp process. c The measured C4v s-QPI pattern at 400meV. d (e) is the simulated s-QPI pattern derived from a Zr vacancy by allowing(forbidding) inter-BZ scatterings without considering the non-symmorphic effect. f is same as e but with considering the non-symmorphic effect. From this,it appears that only f reproduces c, especially the half-missing Umklapp process. The Mzj 12 1

2 0� �

symmetry leads to an extension of the first BZ (purpledotted square in a). This non-symmorphic effect naturally induces the half-missing Umklapp interference





STM measurement. The STM/STS measurements were carried out in a scanningtunneling microscope (USM-1600, Unisoku) with an ultrahigh vacuum (basepressure~1 × 10−10 torr). The samples were cleaved in situ at 80 K and thentransferred into the STM head immediately. All the measurements were performedat T= 4.8 K using platinum iridium tips treated with in situ electron-beamcleaning. dI/dV signals were acquired by a lock-in amplifier with modulation of 20mV at 991 Hz. All presented Fourier transformed maps are raw data.

DFT calculations. The first-principles calculations were based on the generalizedgradient approximation30 (GGA) using the full-potential projector augmented-wave method31,32 as implemented in the VASP package33,34. The electronicstructure of bulk ZrSiSe were calculated using a 20×20×10 Monkhorst-Pack k-meshover the Brillouin zone (BZ). We also conducted the calculations of 30-layer ZrSiSeslab using a 20×20×1 Monkhorst-Pack k-mesh. The vacuum thickness was largerthan 2 nm to ensure the separating of the slabs. The spin–orbit coupling wasincluded. We used Zr s, p, and d orbitals, Si s and p orbitals, and Se p orbitals toconstruct Wannier functions without performing the procedure for maximizinglocalization. We combined the bulk Wannier functions and the surface part of slabWannier functions to simulated the surface spectral weight via a semi-infiniteGreen’s function method.

In a ZrSiSe crystal, the glide mirror Mz j 12 12 0� �

guarantees the existence of ABsublattice and enforce the sublattice to precisely locate in the middle of a surfaceunit cell. The non-symmorphic effect induces a non-trivial structure factor which

must be considered in a first principle simulation. In our calculation, we construct aunitary matrix U(k):

UðkÞ ¼eikr1 � � � 0

..

. . .. ..

.

0 � � � eikrn

0B@

1CA

where k is the wavevector, ri is the real space coordinates of ith atom in oneZrSiSe unit cell. By acting this unitary matrix with the Hamiltonian, i.e., U(k) H(k)U+(k), we are able to simulate the non-symmorphic effect. In contrast, a directdiagonalization of H(k) gives rise to the simulated band structure withoutconsidering non-symmorphic effect, which fails to capture the experimental results

Simulation of s-QPI patterns using the T-matrix approach for ZrSiSe. Tosimulate the interference patterns, we adopted the T-matrix approach, which hasbeen widely used in the QPI studies for the surface states on topological materi-als35–38. The retarded surface Green’s function of the system can be written as

G k;ωð Þ ¼ E � Heffs kð Þ� ��1

;

where E= ω+ iη with ω representing energy and η being a small broadening factor

a

b

1

400

200

0

–1 –0.5 0 0.5

1–1 –0.5 0 0.5

c

400

200

0

d

High

Low

High

Low

High

Low

High

Low

50 mV 100 mV 200 mV

50 mV 100 mV 200 mV

Ene

rgy

(meV

)E

nerg

y (m

eV)

–200

–200

Scatter vector Q (2�/a)

Fig. 5 Healing effect on a floating band surface state. a, b The experimental and simulated QPI patterns from Si defects respectively. The patterns showreduced QPI features compared to Fig. 4. Only a subset of Q1, Q1+Gx pockets are prominent, while Q2 counterparts are suppressed by the anisotropicdefect potential. An energy dependent healing effect of the C4v breaking is captured by the simulations. c, d The energy-scattering vector dispersions fromexperiment and calculation, respectively. The dispersions are taken along the diagonal line in a and b. From the comparison between a and b, c and d, wenote that the simulations fit well to the measurements in a large energy range by considering the non-symmorphic effect




and Heffs kð Þ is an effective surface Hamiltonian calculated by semi-infinite Green’s

function method.When the interference due to the presence of a single non-magnetic impurity is

considered, the Fourier transformed impurity-induced local density of states at agiven scattering wavevector q and energy ω can be derived as

ρimpðq;ωÞ ¼i2π

Zd2k

2πð Þ2gimpðk; q;ωÞ;

where the impurity-induced electronic Green’s function gives rise to gimp(k, q, ω) =Tr(G(k, ω)T(k, k + q, ω)G(k + q, ω))− Tr(G(k, ω)T(k, k− q,ω)G(k− q,ω))*.

The T-matrix, T(k, k′,ω), can be expressed as

T k; k′;ωð Þ ¼ 1�Z

d2p

2πð Þ2Vimp p; pð ÞG p;ωð Þ" #�1

Vimpðk; k′Þ:

Note that the impurity potential matrix Vimp(k, k′) is induced by an impurity ora vacancy on the surface and carries k-dependent matrix elements, where k(k′)indicates out-going (in-coming) wavevector. We considered two types of vacancieson the top most surface: (1) Zr and (2) Si vacancies. We modeled a vacancy byremoving all hopping terms associated with the vacancy site. The impuritypotential of vacancy for α atom can be expressed in real space as following:

Vαvac ¼ �

Xi_j2α

cyi Hijcj;

where Hij denotes the hopping amplitude between two orbitals i and j. Thesummation over any one of i or j basis belonging to α atom site is claimed to ensurethat no interactions between the vacancy site α and the surroundings. Vimp(k, k′)was obtained from a Fourier transform. Our final results are obtained by extractingthe signal of Se orbitals on the topmost surface from ρimp(q, ω) by assuming thetunneling currents are only from atoms on the topmost surface in scanningtunneling microscopy measurement.

Data availabilityAll relevant data that support the findings of this study are available from the corre-sponding author upon reasonable request.

Received: 20 March 2018 Accepted: 18 September 2018

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AcknowledgementsWe thank S. Zhang for the helpful discussions. We acknowledge the financial supportfrom National Natural Science Foundation of China (Grant Nos. 11674226, 11790313,11521404, 11634009, U1632102, 11504230, 11674222, 11574202, 11674226, 11574201,U1632272, U1732273, U1732159, 11655002, 1674220 and 11447601), the National KeyResearch and Development Program of China (Grant Nos. 2016YFA0300403,2016YFA0301003, 2016YFA0300500 and 2016YFA0300501), and Technology Com-mission of Shanghai Municipality (Grant Nos. 15JC402300 and 16DZ2260200). Thiswork is supported in part by the Key Research Program of the Chinese Academy ofSciences (Grant No. XDPB08-2), the Strategic Priority Research Program of ChineseAcademy of Sciences (Grant No. XDB28000000). S.-M.H. is supported by the Ministry ofScience and Technology in Taiwan under Grant No. 105-2112- M-110-014-MY3. T.-R.C.is supported from Young Scholar Fellowship Program by Ministry of Science andTechnology (MOST) in Taiwan, under MOST Grant for the Columbus ProgramMOST107-2636-M-006-004, National Cheng Kung University, Taiwan, and NationalCenter for Theoretical Sciences (NCTS), Taiwan.

Author contributionsH. Z. and J.-F. J. oversaw the project. Z.Z. conducted the STM measurement with thehelp of X.-A. N, X.-Z. W, D.-D. G, S. W, Y.-Y. L, C. L. and D. Q. T.-R. C and C.-Y. H.performed the simulations with S.-Y. X, S.-M.H and H.L. H.P and F. S grow the crystals.



https://arxiv.org/abs/1801.09979



Z.-T. J and W.K did the theoretical analysis. All authors discussed the result and con-tributed to the paper writing.

Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-06661-9.

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© The Author(s) 2018



https://doi.org/10.1038/s41467-018-06661-9

https://doi.org/10.1038/s41467-018-06661-9






Electrode-Free Anodic Oxidation Nanolithography of Low-Dimensional MaterialsHongyuan Li,†,‡ Zhe Ying,†,‡ Bosai Lyu,†,‡ Aolin Deng,†,‡ Lele Wang,†,‡ Takashi Taniguchi,§

Kenji Watanabe,§ and Zhiwen Shi*,†,‡

†Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, ShanghaiJiao Tong University, Shanghai 200240, China‡Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China§National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

*S Supporting Information

ABSTRACT: Scanning probe lithography based on local anodicoxidation (LAO) provides a robust and general nanolithographytool for a wide range of applications. Its practical use, however, hasbeen strongly hampered due to the requirement of a prefabricatedmicroelectrode to conduct the driving electrical current. Here wereport a novel electrode-free LAO technique, which enables in situpatterning of as-prepared low-dimensional materials and hetero-structures with great flexibility and high precision. Unlikeconventional LAO driven by a direct current, the electrode-freeLAO is driven by a high-frequency (>10 kHz) alternating currentapplied through capacitive coupling, which eliminates the need of acontacting electrode and can be used even for tailoring insulatingmaterials. Using this technique, we demonstrated flexible nanolithography of graphene, hexagonal boron nitride, and carbonnanotubes on insulating substrates with ∼10-nanometer precision. In addition, the electrode-free LAO exhibits high etchingquality without oxide residues left. Such an in situ and electrode-free nanolithography with high etching quality opens up newopportunities for fabricating ultraclean nanoscale devices and heterostructures with great flexibility.

KEYWORDS: Scanning probe lithography, electrode-free local anodic oxidation, high-frequency ac voltage, graphene,low-dimensional materials

Nanolithography is widely used in fabricating functionaldevices in nanoscience and nanotechnology.1−10 Com-

pared with conventional optical/E-beam lithography withmultiple steps, scanning probe lithography (SPL) based onvarious chemical/physical mechanisms provides a simpler andmore flexible way.2−7 Local anodic oxidation (LAO)lithography has been one of the most robust and versatileSPL methods.10−17 Conventional LAO relies on a spatiallyconfined electrochemical reaction driven by a dc voltageapplied between the tip (cathode) and the sample (anode).The sharp tip apex can cause a localized strong electric field(>107 V/m) in the tip−sample gap, which has two mainfunctions.18 First, it can attract polar H2O molecules from airand form a nanometer-sized water bridge connecting the tipand sample surface.19,20 Second, the strong electric field canhelp generate ions (e.g., H+, OH−, and O2−) by decomposingwater molecules and drive the oxygen-containing radicals (e.g.,OH− and O2−) to the sample surface to achieve oxidation.13,21

Conventional LAO has been demonstrated on variousconductive materials, including Si,22 graphene,13,14,17,23,24

transition metal dichalcogenides (TMDs),25 and carbonnanotubes (CNTs).26 However, the application of conven-

tional LAO is seriously limited due to its complex pretreat-ment, low etching quality, and selection of conductive samples.Especially, to apply a dc voltage to a small sample ofmicrometer size, one needs to first fabricate a microelectrodeconnected to the sample using other nanolithographytechniques, such as E-beam lithography and UV lithography.In addition, patterning via conventional LAO must follow astrict order to ensure the sample is not disconnected from theelectrode, which also limits its application.Here, we report an electrode-free LAO (EFLAO) technique,

which can realize high-quality nanolithography for lowdimensional materials and heterostructures on insulatingsubstrates without microelectrode connection. Unlike conven-tional LAO driven by a dc voltage, the new EFLAO is insteaddriven by a pure ac voltage. With a specially selected substrateconducting high frequency ac current, we are able to performLAO without electrodes connected to the sample. In this letter,we first introduce how the EFLAO works through patterning a

Received: October 17, 2018Revised: November 25, 2018Published: November 30, 2018

Letter

pubs.acs.org/NanoLettCite This: Nano Lett. 2018, 18, 8011−8015

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graphene sample. Then we demonstrate the high quality of thefabricated graphene nanostructures. We further show that thistechnique is also suitable for patterning other low-dimensionalmaterials and heterostructures, such as graphene/hexagonalboron nitride (hBN) heterostructure and CNTs. A simplemodel is proposed to illustrate the mechanism of EFLAO, andsome key factors that impact the etching quality areinvestigated.We first introduce how the EFLAO works through

fabricating a monolayer graphene nanoribbon (GNR) arrayon a SiO2/Si substrate, the schematic of which is shown inFigure 1A. The experiment was performed on a standard

atomic force microscope (AFM) platform in atmosphere. Ahigh-frequency (40 kHz) ac voltage was applied between agold-coated AFM tip and the conductive Si layer of thesubstrate. The tip slowly approached the sample and thenscanned along a designed path to achieve a controllednanoetching. More experimental details are included inMethods. Figure 1B shows the topography of a GNR arrayfabricated by EFLAO, which displays high uniformity andreproducibility (corresponding optical image is shown inFigure S1). Note that no microelectrode is connected to thesample here. The high-frequency ac current can penetrate theSiO2 layer through a capacitive coupling effect, which will bediscussed later.We then demonstrate the high quality of the graphene

nanostructures fabricated by EFLAO. To directly compare theetched edge with an exfoliated natural edge, a taperedgraphene ribbon with both types of edges was fabricated. Asshown in Figure 2A, little distinction in topography can beobserved between the two edges. Moreover, the correspondinginfrared scanning near-field optical microscopic (IR-SNOM,see Supporting Information for more details) image in Figure2B displays almost identical fringes of surface plasmonpolariton (SPP) near both edges. Line profiles of SPP (Figure2C) across the two edges show a symmetric feature, indicatingthat the etched edge is of high quality and comparable to thenatural edge. Otherwise, if there exists graphene oxide areanear the etched edge, the SPP reflection would happen at thegraphene-oxide/graphene interface, and the SPP interferencepattern will shift and show asymmetry. In fact, graphene edgesetched via conventional dc-LAO usually contain oxideresidues,13−15,27 which show either irregular fringes or simply

no IR response (see Supporting Information). Figure 2Dshows the topography of a uniform array of 20 nm-wide GNRs,and the inset shows that those GNRs own very smooth edges.Figure 2E displays an ultrathin graphene nanoribbon withwidth ∼10 nm and edge roughness less than 2 nm, which hasalready reached the resolution limit of our AFM. Owing to itshigh resolution and etching quality, the new technique issuitable for fabricating high-quality nanodevices, such asplasmonic waveguides and Hall bar structures.Furthermore, we show that 1D carbon nanotube and 2D

heterostructures can also be etched using EFLAO. With a 10kHz ac driving voltage applied between the tip and the Si layer,one can directly cut off a nanotube on hBN/SiO2/Si substrate.The obtained segments and the original complete nanotube aredisplayed in Figure 2F. With this technique, one can achievenanotube with desired lengths, which could be useful forfabricating nanotube-based electronic and photonic devices.Next, we show that heterostructures consist of two differentmaterials can also be tailored by EFLAO. We successfullyetched an array of trenches on an hBN/graphene hetero-structure consisting of a 10 nm thick hBN layer on top and amonolayer graphene layer on bottom, as shown in Figure 2G.The etching depth is measured to be 10 nm, revealing that theheterostructure is fully etched through. An additional exampleof etching hBN can be found in Supporting Information. Notethat hBN is an insulator with bandgap of ∼6 eV, which cannotbe etched with conventional dc-LAO. However, a high-frequency ac current is able to penetrate the hBN flake andinduce a novel anodic etching effect. With the assistance of anunderlying graphene layer, a voltage drop can be formedbetween the tip and graphene, and hence a sufficiently strongelectric field is able to be applied to the hBN/water interfaceand enable the anodic etching of hBN. The oxidation products

Figure 1. Illustration of electrode-free local-anodic oxidization(EFLAO). (A) Schematic of the EFLAO. The electrochemicalreaction is driven by an ac voltage applied through the SiO2/Sisubstrate without any electrode directly connected to the sample. Theelectrochemical reaction is localized within the nanosized waterbridge. (B) An array of graphene nanoribbons fabricated usingEFLAO. The inset shows the zoom-in of the white dash box. RH =60%; f = 40 kHz.

Figure 2. Demonstration of the high etching quality and capability ofEFLAO. Topography (A) and IR-SNOM image (B) of a taperedgraphene ribbon with one etched edge and one natural edge. (C) Lineprofiles of plasmons at the line cuts in (B). All the plasmon profilesshow highly symmetric feature, indicating that the sample quality nearthe etched edge is as good as that near the natural edge. (D)Topography of a uniform array of ultrathin GNR with width ∼20 nm.The inset shows the GNRs owns quite smooth edges. (E)Topography of an ultrathin GNR with width ∼10 nm and edgeroughness of ∼2 nm. (F) Topography of nanotube segments cut byFELAO. Cutting positions are labeled by white dash arrows. The insetshows the original complete nanotube. RH = 40%; f = 10 kHz. (G)Topography of an array of hBN/graphene nanotrenches fabricated byEFLAO, the depth of which is ∼10 nm. The inset shows a schematicof the original hBN/graphene heterostructure on SiO2 substrate. RH= 45%; f = 140 kHz.

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of hBN could be NxOy and B2O3. The former as gas can escapeeasily, whereas the latter can be dissolved in water and formH3BO3, which can further vaporize.28 The capability of theEFLAO to pattern 2D heterostructures could largely simplifythe procedure of fabricating devices based on 2D hetero-structures.To confirm that the etching is indeed due to anodic

oxidation, we systematically examined the impact of humidityon the etching effect. Conventional dc-LAO features a strongdependence on relative humidity (RH),10−12 because RH willaffect formation of the water bridge.20 Similar RH-dependenceis observed in our EFLAO experiment. More than 90 etchinglines were performed on a large graphene flake at differenthumidity. Etching results at RH = 10%, 39%, 60%, and 85% areshown in Figure 3A. The etching failed completely at RH =

10% and worked well at RH = 60%, 85%. The statistics of thesuccess rates, defined as the ratio of the successfully etchedlength to the total scanned length, are shown in Figure 3B,which starts from zero for low RH < 25%, and reaches about100% for RH > 45%. The observed RH-dependence confirmsunambiguously that the etching effect results from anodicoxidation.It is noteworthy that a pure ac voltage can also induce an

anodic oxidation. We first point out that the period of the acvoltage applied (∼10 μs) is far longer than the time scale of the

water bridge formation (∼100 ps).19 Therefore, the ac voltage-driven electrochemical reactions can be regarded as aquasistatic process and be simply divided into two stageswithin each ac period. In stage 1, the graphene is positivelycharged (anode). Anodic oxidation of graphene will start whenthe voltage exceeds a threshold value as that in conventionaldc-LAO. This stage is similar to conventional dc-LAO.Sufficient oxidation of graphene will generate gaseous productCO or CO2 and lead to the etching effect. In stage 2, thegraphene is negatively charged (cathode) and the tip ispositively charged (anode). The tip is protected by a layer ofgold from been oxidized, and the graphene may behydrogenated by reactive hydrogen atoms or positive hydrogenions.13 However, the hydrogenation is weak and could becanceled in stage 1, so that the impact of this stage is negligible.Therefore, the net effect induced by a pure ac voltage on thegraphene is anodic oxidation.A significant advantage of the ac-driven EFLAO is its

simplicity with no requirement of prefabricated micro-electrodes, as ac current can flow through dielectric substratesin the form of displacement current. We now introduce howthe ac current is conducted in the EFLAO in detail. Thestructure of graphene/SiO2/Si multilayer shown in Figure 3Ccan be regarded as a capacitor with impedance of 1/j2πf C,where j is the imaginary unit, C is the capacitance, and f is thefrequency. The nanosized water bridge can be regarded as aresistor R. The effect of tip−graphene capacitive impedancecan be neglected here, because it is in parallel with the waterbridge resistor and its value is about 2 orders of magnitudelarger than the water bridge resistance at a typical drivingfrequency of 10 kHz (see Supporting Information for moredetails). The water bridge resistance R and the graphene/SiO2/Si capacitive impedance 1/j2πf C are in series. When anac voltage U is applied between the Si layer and the tip, thevoltage drop across the water bridge UR, which is also thevoltage between the AFM tip and graphene sample, can beeasily calculated as

=+

·π

UR

RU

j fCR 1

2 (1)

UR will monotonically increase with f, and trigger theelectrochemical reaction when it reaches a threshold value.More quantitatively, we provide an estimation on themagnitudes of 1/j2πf C and R. For a 10 μm2 graphene flakeon a 300 nm thick SiO2 layer, 1/j2πf C is in the order of 1010 Ωat f = 10 kHz. The dc resistance of water bridge R is measuredto be ∼1010 Ω when performing EFLAO etching (seeSupporting Information for more details). Theoreticallycalculated value for R using the ideal resistivity of deionizedwater is ∼1011 Ω. The comparison of various impedances at f =10 kHz is shown in Figure 3D. It can be found that R isroughly comparable to 1/j2πf C at f = 10 kHz, which means theapplied voltage can efficiently acts on the water bridge andenable the electrochemical reaction at such a high frequency.The dependence of EFLAO on the driving frequency was

investigated with the results displayed in Figure 3E,F. At f = 0Hz, all etching lines completely failed; at f = 625 Hz, all theetching lines are terminated near the starting points; at f = 2.5kHz, the etching is overall successful but yields many oxideresidues; at f = 20 kHz, all the etching lines had success anddisplay smooth edges without residues. The statistics of thesuccess rate and the amount of residues (see the exact

Figure 3. Principle of EFLAO. (A) AFM images of typical etchingresults obtained at different relative humidity (RH) of air. f = 40 kHz.Scale bar: 400 nm. (B) Statistics for the success rate at different RHvalues. The success rate is close to unity when RH is higher than 50%.Pink dash curve shows a guiding line. The strong RH-dependence ofetching success rate confirms unambiguously that the etching is dueto anodic oxidation (C) Schematic of the equivalent electrical circuitfor EFLAO. The graphene/SiO2/Si multilayer can be regarded as acapacitor. (D) Impedances of the G/SiO2/Si capacitor, water bridgeresistor, and direct tip−graphene (Gr.) contact. For water bridge,both experimental and theoretical values are displayed. (E) AFMimages of typical EFLAO etching results obtained at different drivingfrequencies. RH = 57−60%. Scale bar: 200 nm. (F) Statistics for thesuccess rate and the amount of residues left.

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definition in Supporting Information) shown in Figure 3Freveal that a 10 kHz frequency and above can yield idealetching results with 100% success rate and few residues. Thefrequency-dependent success rate is attributed to frequency-induced change of potential drop UR across the water bridge,which agrees well with our capacitor−resistor model. Thefrequency-dependent generation of oxide residues is morecomplicated and will be discussed later in detail. It isnoteworthy that during the EFLAO etching the AFM tipnever directly contacts the graphene (as illustrated in Figure3C; see experimental evidence in Supporting Information),otherwise the EFLAO will have failed because the tip−graphene contact resistance (∼104 Ω) is too small to obtainsufficient driving voltage.The use of high-frequency ac voltage leads to high etching

quality with no oxide residues, which is another merit of thistechnique. As shown in Figure 2 and Figure 3E,F, the EFLAOetching with a sufficiently high driving frequency (>10 kHz)can result in smooth edges free of oxide residues. One shouldnot simply attribute such a result to a pure frequency effect,because the driving voltage UR is coupled with the frequency faccording to eq 1. In order to distinguish the roles of f and UR,we performed ac-LAO etching on a microelectrode-connectedgraphene sample, which enables independently tuning offrequency f and voltage drop UR. With an increasing f and afixed UR at 10 V, the amount of residues decreases graduallywhile the success rate keeps at 100% (Figure 4A,B). Theincreased frequency also leads to a narrower etching trench.With an increasing UR and a fixed f at 40 kHz, only the successrate increases while all line-etchings are free of residues (Figure4C,D). The two groups of controlled experiments show clearlythat the amount of residues is only related to the drivingfrequency.A likely mechanism for the elimination of graphene oxide

and the narrower etching trench in the high-frequency ac-LAOis again based on a resistor−capacitor voltage divider model, asillustrated in Figure 4E. The impedance of an electrochemicalcell typically results from the electrolyte and the two electrode/electrolyte interfaces. The electrolyte has only resistance,whereas the interfaces own both resistance and capacitance.The capacitance is due to the electric double layer (EDL).29,30

Only the voltage drops at the two interfaces are effective indriving the electrochemical reactions. At high frequency, theeffective impedance of the interfaces will decrease relative toresistance of the electrolyte, and the oxidation process will becut off. This effect dominates for relatively long water bridge,that is, points away from the tip apex, where the electrolyteresistance is large. In other words, it requires the collectivemotion of more ions to establish the electrostatic equilibriumfor long water bridge, which would cost more voltage drop. Forshort water bridge, i.e., points very close to the tip apex, thevoltage drop at the interfaces is still sufficient to drive thereactions. That explains why anodic oxidation etching drivenby high-frequency ac current has a smaller etching width thanthat driven by dc current. Shorter water bridge can also inducestronger electric field, which is beneficial for fully oxidizinggraphene to gaseous CO/CO2. Additionally, even if grapheneoxide is generated during the etching, it can be further oxidizedto gaseous CO/CO2, because oxide residues can still conducthigh-frequency ac current.In conclusion, we have reported a novel electrode-free

anodic oxidation nanolithography driven by high-frequency acvoltage and demonstrated its ability in performing high-quality

tailoring of low-dimensional materials and heterostructures in asingle step with no need of prefabricated microelectrodesconnected to those samples. The nonpretreatment, highetching quality, as well as in situ operation and characterizationof the EFLAO should greatly facilitate its application innanolithography. The EFLAO developed here provides asimple and efficient way to pattern low-dimensional materialsand has the potential to become a standard nanolithographytechnique.

Methods. Silicon-based substrates with a layer of 300 nmthick thermal oxide SiO2 on top were used for the EFLAOetching. To perform etching, an ac voltage with amplitude of10 V and frequency range of 10−1000 kHz (graphene, 40−50kHz (if not specified); graphene/hBN, 140 kHz; carbonnanotube, 10 kHz) is applied between the AFM tip and thesilicon substrate. Then the tip slowly approaches the sample.The AFM is operated in contact mode with a lift-down force of∼1500 nN. The RH of the air is maintained in the range of40−75% (graphene, 55−75%; graphene/hBN, 45%; nanotube,40%) if not specified, and the room temperature is kept at 20°C when conducting the experiments. The tip moving velocitywas kept around 1−4 μm/s during the etching.

Figure 4. Mechanism for elimination of oxide residue in EFLAOetching. AFM images of typical ac-LAO etching results obtained withdifferent driving frequencies at 10 V voltage (A) and with differentvoltages at 40 kHz frequency (C), and their corresponding statisticresults (B,D). The ac-LAO etching was performed by directlyapplying an ac voltage between a microelectrode connected grapheneand the tip to ensure that the voltage and frequency can beindependently controlled. RH = 50−55%. Scale bars: 400 nm (A) and300 nm (C). (E) Schematic of the resistor-capacitor voltage dividermodel and corresponding effective circuit. Red and blue areasrepresent the electric double layers (EDL). The electrolyte has only aresistance whereas the EDL owns both resistance and capacitance. Athigh frequency, longer water bridge (at the left and right side) resultsin larger electrolyte resistance and lower voltage drop at the interfaces.Only the area close to the tip apex can obtain sufficient voltage dropat the interfaces to drive the chemical reactions (central area). Shortergap distance in the central area can also induce stronger electric field,which may be helpful for fully oxidizing graphene.

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■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.nano-lett.8b04166.

(1) Optical image of the GNRs fabricated by EFLAO,(2) IR scanning near-field optical microscopy measure-ment, (3) comparison between ac- and dc-driven LAO,(4) square hBN flake fabricated by EFLAO, (5)estimation of the tip−graphene capacitance, (6)measurement of the water bridge and tip−graphenecontact resistances, (7) method for quantitativelycharacterizing the amount of residues, and (8) evidencefor no direct tip−graphene contact (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] (Z.S.).ORCIDHongyuan Li: 0000-0001-9119-5592Kenji Watanabe: 0000-0003-3701-8119Zhiwen Shi: 0000-0002-3928-2960Author ContributionsH.L. and Z.S. conceived the project. H.L. and B.L. performedthe EFLAO lithography. Z.Y., B.L., and A.D. helped onpreparing the samples. H.L. and Z.S. analyze the data. Allauthors discussed the results and contributed to writing themanuscript.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was mainly supported by National Key Researchand Dev e l opmen t P r o g r am o f Ch i n a (G r an t2016YFA0302001) and National Natural Science Foundationof China (Grants 11574204, 11774224). Z.S. acknowledgessupport from the Program for Professor of Special Appoint-ment (Eastern Scholar) at Shanghai Institutions of HigherLearning and support from the National 1000 Young TalentsProgram and Shanghai 1000 Talents Program.

■ ABBREVIATIONSLAO, local anodic oxidation; EFLAO, electrode-free LAO;hBN, hexagonal boron nitride; CNT, carbon nanotube; GNR,graphene nanoribbon; AFM, atomic force microscope; RH,relative humidity; IR-SNOM, infrared scanning near-fieldoptical microscope

■ REFERENCES(1) Mack, C. Fundamental Principles of Optical Lithography: TheScience of Microfabrication; Wiley: 2007; pp 265−276.(2) Albisetti, E.; Petti, D.; Pancaldi, M.; Madami, M.; Tacchi, S.;Curtis, J.; King, W.; Papp, A.; Csaba, G.; Porod, W.; et al. Nat.Nanotechnol. 2016, 11 (6), 545.(3) Piner, R. D.; Zhu, J.; Xu, F.; Hong, S.; Mirkin, C. A. Science1999, 283 (5402), 661−663.(4) Lyuksyutov, S. F.; Vaia, R. A.; Paramonov, P. B.; Juhl, S.;Waterhouse, L.; Ralich, R. M.; Sigalov, G.; Sancaktar, E. Nat. Mater.2003, 2 (7), 468.(5) Pires, D.; Hedrick, J. L.; De Silva, A.; Frommer, J.; Gotsmann, B.;Wolf, H.; Despont, M.; Duerig, U.; Knoll, A. W. Science 2010, 328,732.

(6) Fenwick, O.; Bozec, L.; Credgington, D.; Hammiche, A.;Lazzerini, G. M.; Silberberg, Y. R.; Cacialli, F. Nat. Nanotechnol. 2009,4 (10), 664.(7) Wei, Z.; Wang, D.; Kim, S.; Kim, S.-Y.; Hu, Y.; Yakes, M. K.;Laracuente, A. R.; Dai, Z.; Marder, S. R.; Berger, C.; et al. Science2010, 328 (5984), 1373−1376.(8) Vieu, C.; Carcenac, F.; Pepin, A.; Chen, Y.; Mejias, M.; Lebib,A.; Manin-Ferlazzo, L.; Couraud, L.; Launois, H. Appl. Surf. Sci. 2000,164 (1), 111−117.(9) Xia, Y.; Rogers, J. A.; Paul, K. E.; Whitesides, G. M. Chem. Rev.1999, 99 (7), 1823.(10) Garcia, R.; Knoll, A. W.; Riedo, E. Nat. Nanotechnol. 2014, 9(8), 577.(11) Kurra, N.; Reifenberger, R. G.; Kulkarni, G. U. ACS Appl.Mater. Interfaces 2014, 6 (9), 6147.(12) Ryu, Y. K.; Garcia, R. Nanotechnology 2017, 28 (14), 142003.(13) Byun, I. S.; Yoon, D.; Choi, J. S.; Hwang, I.; Lee, D. H.; Lee, M.J.; Kawai, T.; Son, Y. W.; Jia, Q.; Cheong, H.; et al. ACS Nano 2011, 5(8), 6417.(14) Masubuchi, S.; Ono, M.; Yoshida, K.; Hirakawa, K.; Machida,T. Appl. Phys. Lett. 2008, 94 (8), 197.(15) Puddy, R. K.; Chua, C. J.; Buitelaar, M. R. Appl. Phys. Lett.2013, 103 (18), 183117−183117.(16) Masubuchi, S.; Arai, M.; Machida, T. Nano Lett. 2011, 11 (11),4542−6.(17) Lee, D. H.; Kim, C. K.; Lee, J. H.; Chung, H. J.; Park, B. H.Carbon 2016, 96, 223−228.(18) Garcia, R.; Martinez, R. V.; Martinez, J. Chem. Soc. Rev. 2006,35 (1), 29−38.(19) Cramer, T.; Zerbetto, F.; García, R. Langmuir 2008, 24 (12),6116.(20) Weeks, B. L.; Vaughn, M. W.; Deyoreo, J. J. Langmuir 2005, 21(18), 8096−8.(21) Kinser, C. R.; Schmitz, M. J.; Hersam, M. C. Adv. Mater. 2006,18 (11), 1377−1380.(22) Calleja, M.; Anguita, J.; Garcia, R.; Birkelund, K.; Perezmurano,F.; Dagata, J. A. Nanotechnology 1999, 10 (1), 34−38.(23) Yong, H.; Kim, K.; Choi, W.; Park, J.; Ahmad, M.; Seo, Y.Carbon 2012, 50 (12), 4640−4647.(24) Tapaszto, L.; Dobrik, G.; Lambin, P.; Biro, L. P. Nat.Nanotechnol. 2008, 3 (7), 397−401.(25) Espinosa, F. M.; Yu, K. R.; Marinov, K.; Dumcenco, D.; Kis, A.;Garcia, R. Appl. Phys. Lett. 2015, 106 (10), 103503.(26) Kim, D. H.; Koo, J. Y.; Kim, J. J. Phys. Rev. B: Condens. MatterMater. Phys. 2003, 68 (11), 113406.(27) Neubeck, S.; Ponomarenko, L. A.; Freitag, F.; Giesbers, A. J.;Zeitler, U.; Morozov, S. V.; Blake, P.; Geim, A. K.; Novoselov, K. S.Small 2010, 6 (14), 1469−1473.(28) Pankajavalli, R.; Anthonysamy, S.; Ananthasivan, K.; Rao, P. V.J. Nucl. Mater. 2007, 362 (1), 128−131.(29) Block, L. P. Astrophys. Space Sci. 1978, 55 (1), 59−83.(30) Bard, A. J.; Faulkner, L. R.; Leddy, J.; Zoski, C. G.Electrochemical methods: fundamentals and applications; Wiley: NewYork, 1980; Vol. 2.

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http://pubs.acs.org

http://pubs.acs.org/doi/abs/10.1021/acs.nanolett.8b04166

http://pubs.acs.org/doi/abs/10.1021/acs.nanolett.8b04166



http://orcid.org/0000-0001-9119-5592

http://orcid.org/0000-0003-3701-8119

http://orcid.org/0000-0002-3928-2960


Contents lists available at ScienceDirect

Nano Energy

journal homepage: www.elsevier.com/locate/nanoen

Full paper

Realization of interdigitated back contact silicon solar cells by using dopant-free heterocontacts for both polarities

Hao Lina,b, Dong Dinga, Zilei Wangb, Longfei Zhangb, Fei Wua, Jing Yub, Pingqi Gaob,⁎,Jichun Yeb,⁎, Wenzhong Shena,⁎

a Institute of Solar Energy, and Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai JiaoTong University, 800 Dong Chuan Road, Shanghai 200240, PR ChinabNingbo Institute of Material Technology and Engineering, Chinese Academy of Sciences (CAS), Ningbo 315201, PR China

A R T I C L E I N F O

Keywords:Heterojunction solar cellPEDOT:PSS hybrid solar cellsInterdigitated back contactDopant-freeCarrier-selective contacts

A B S T R A C T

For crystalline-silicon (c-Si) solar cells, the interdigitated back contact (IBC) structure has been long known as anefficient way to approach the theoretical limit of efficiency. However, the complexity of fabricating this kind ofdevices as well as the high dependence on expensive vacuum systems pose concerns about their commercialpotential. Here, we demonstrate a novel c-Si IBC solar cell featuring dopant-free heterocontacts for both pola-rities, i.e. a solution-proceeded PEDOT:PSS film as hole-transporting layer (HTL) and an evaporated magnesium-oxide film as electron-transporting layer (ETL). Our innovatively buried ETL method provides substantial sim-plification on the architecture and fabrication of the IBC cells and makes it possible to adapt solution-proceededHTLs while keeping good passivation in gap regions. The IBC solar cell shows an efficiency of 16.3%, with apromising short-circuit current density (Jsc) up to 38.4 mA/cm2. A thorough simulation concerning the influenceof pitch size, surface recombination rate (at ETL and gap regions) was conducted, revealing a readily achievableJsc of 41mA/cm2 and a PCE beyond 22%. Our findings demonstrated a feasibility of using solution method tofabricate high efficiency dopant-free IBC solar cells.

1. Introduction

Routine improvements have led to exceptional success of crystallinesilicon (c-Si) solar cells, demonstrated by a new record power conver-sion efficiency (PCE) of 26.7% from an interdigitated back contact (IBC)solar cell combining with advanced heterojunctions (HJs) [1]. The IBCstructure has been long known as an efficient way to avoid shadinglosses and enable full-area passivation on front side because all elec-trodes are placed on the non-illuminated rear side. Meanwhile, the HJscomprising bilayer films of intrinsic amorphous silicon (a-Si:H) anddoped a-Si:H play another important role of passivating contact orcarrier-selective contact (CSC). The two-fold designs of IBC-HJs areresponsible for the highest efficiency by now and could be the possibleroadmap towards 29.4%, a theoretical efficiency limit for single junc-tion c-Si solar cells. However, such solar cells suffer from complexprocessing in patterning discrete contacts to the rear side as well asextremely high facilities investment (more than 4 times to the currentmainstream technique). This severely hinders the industrialization forhigh volume production. In addition, parasitic electrical and opticallosses inherent to the doped layers restrain further promotion on

efficiency. Thus, a few activities have been moved to seeking simplifiedsolutions, such as implementation of high-performance IBC-HJs solarcells via dopant-free manner.

Functional thin films with high/low work function (WF) have thusbeen paid much interest in c-Si solar cells for the formation of dopant-free hole/electron-selective contacts. Most of the functional materialscan be deposited via low-temperature and/or solution-based proces-sing, such as spin-coating or thermal evaporation, providing big po-tentials in both doping elimination and procedure simplification(especially for IBC-HJs) [2,3]. So far, poly(3,4-ethylenediox-ythiophene):polystyrene (PEDOT:PSS) [4,5] and transition metal oxides(TMOs), such as molybdenum oxide (MoOx) [6,7], tungsten oxide(WOx) [8] and vanadium oxide (V2Ox) [9], all with high WFs have beensuccessfully demonstrated as hole-transporting layers (HTLs). Mean-while, low WF materials including titanium oxide (TiOx) [10,11],magnesium oxide (MgOx) [12,13], lithium fluoride (LiFx) [14], etc.have always been served as electron-transporting layers (ETLs). Due toease of processing, tailorable optoelectronic properties, facile integra-tion of conducting polymers [15], PEDOT:PSS/Si heterojunction solarcells, especially for the structure of FrontPEDOT:PSS, have been

https://doi.org/10.1016/j.nanoen.2018.06.013Received 21 April 2018; Received in revised form 28 May 2018; Accepted 5 June 2018

⁎ Corresponding authors.E-mail addresses: [email protected] (P. Gao), [email protected] (J. Ye), [email protected] (W. Shen).

Nano Energy 50 (2018) 777–784

Available online 15 June 20182211-2855/ © 2018 Published by Elsevier Ltd.

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emerging with a fast promotion of PCEs from below 10% to beyond16% [16]. Regardless of this distinct advance, development of IBC-likePEDOT:PSS/Si solar cells is beneficial to circumvent the barriers thatare relevant to the front-sided PEDOT:PSS layer, including parasiticabsorption, poor anti-reflection and inferior coating quality upon the Si-textures [17–19]. However, proven feasibility of solution-processedPEDOT:PSS/Si IBC solar cells is pending. Knowledge regarding the rear-sided partial HTLs with PEDOT:PSS and the matched ETLs, as well asthe integration of those contact materials in the c-Si solar cell archi-tectures, is still in its infancy. Therefore, experimental attempts canhelp us to understand this novel device in more details at the aspectsincluding interfaces, contact ratios, processing related issues, etc., di-rectly guiding the evaluations and designs of high-performance dopant-free IBC-HJs cells.

Here, aiming at the achievement of high efficiency IBC-type or-ganic/Si heterojunction solar cells via a low-temperature processingand dopant-free manner, a new device structure with hole-selectivecontacts of PEDOT:PSS/Si and buried electron-selective contacts ofMgOx/Si was developed. With optimizing the contact properties forboth polarities and the pitch ratios between them, our IBC deviceachieved a PCE of 16.3%, with an open-circuit voltage (Voc) of 581mV,a fill factor (FF) of 73.1% and a short-circuit current density (Jsc) of38.4 mA/cm2. This result fully demonstrated that high-performanceIBC-HJs solar cells can be even made of spin-coated and evaporatedmaterials, exempting the heavy dependence on high-temperaturedoping or expensive chemical vapor deposition processes. Furthermore,a prospective PCE exceeding 22% for the PEDOT:PSS/Si based IBC cellwas predicted once the surface recombination rate at ETL/Si interfacecan be reduced below 100 cm/s.

2. Results and discussion

As shown in Fig. 1a, the n-type c-Si wafer with front-sided pyramids-texture was selected as the substrate for construction of our IBC solarcells. On the front surface, Al2O3 and SiNx films were deposited aspassivation and an anti-reflection layer. The PEDOT:PSS/Ag and MgOx/Al structures were interdigitated on the rear surface of the c-Si, servingas hole- and electron-selective contacts, respectively. We wrapped upthe MgOx/Al contacts with a polymer layer before spin-coating thePEDOT:PSS film (Fig. 1b). Thus, the PEDOT:PSS/Ag contacts coveredthe whole rear surface of the device except for the busbar that wasconnected with the MgOx/Al electrode and must be kept open fortesting. Fig. 1c shows a cross-sectional SEM image of one back-contact,while Fig. 1d–f exhibit the corresponding magnified images collectedfrom the white-square regions marked in Fig. 1c (from left to right).From Fig. 1d, one can clearly see that the MgOx layer together with thecapped Al electrode have a total thickness of around 1μm. The MgOx/Al portion was well wrapped by a polymer film. Meanwhile, thispolymer layer does play another two important functions, i.e. isolatingthe HTL and ETL regions with gaps and protecting the underneathAl2O3 layer (pre-deposited for surface protection) from etching duringthe area-opening (for deposition of HTLs). Fabrication process pleaserefer to Fig. S1. The survived Al2O3 layer thus provide sufficiently highquality of passivation to the gap regions, which is crucial to obtain highefficiency IBC solar cells. Fig. 1e and f show a good coverage, even atthe boundary area between HTL and the polymer, of PEDOT:PSS filmon the entirely bared c-Si surface. This is very important for achievinghigh quality of passivation at HTL regions. Due to the shield effectduring the spinning coating process, the thickness of PEDOT:PSS filmnear the gap region is thicker than that on other areas, reaching atabout 100 nm.

According to the location of the PEDOT:PSS film in a device, thehybrid solar cells can be categorized as three types: Front-PEDOT(Fig. 2a), Back-PEDOT (Fig. 2b) and IBC-PEDOT (Fig. 2c). Forstraightforwardly understand the optical and electrical losses of thesethree types devices, the J-V curves and the photovoltaic (PV)

performance are shown in Fig. 2e and Table 1, respectively. One canfind that the Front-PEDOT and Back-PEDOT devices possessed rela-tively higher Voc of about 620mV, compared to that of IBC-PEDOT(581mV). We note that the lower Voc of the IBC-PEDOT is partiallyascribed to high resistivity of Si wafer (1–10Ω cm) we chosen. It wasreported that the Voc of PEDOT:PSS/Si hybrid solar cells has a positivecorrelation with the doping concentration of Si substrates [20]. Whilethe Si substrate with higher bulk resistance is better for construction ofIBC device due to higher lifetime [21]. The other reason that re-sponsible for the high Voc of both Front-PEDOT and Back-PEDOT solarcells is the utilization of a-Si:H(i)/a-Si:H(n) as ETL. Actually, our IBCdevice with ETL of MgOx can only provide a moderate level of passi-vation. Even so, the IBC device still received a high PCE of 16.3% due tothe highest Jsc of 38.4 mA/cm2, in comparison with 31.8 and 34.9mA/cm2 for the Front-PEDOT and Back-PEDOT devices, respectively. Asshown in Fig. 2f, the Front-PEDOT device shows the lowest EQE valueand the highest overall reflection. While the IBC-PEDOT device showsmuch better EQE almost over the whole useful wavelength range, in-dicating superior light harvesting and carrier collection efficiency. TheJsc losses are calculated according to the experimental results (moredetails are shown in Fig. S2) and correspondingly presented inFig. 2d1–3. The optical losses caused by electrode shade, reflection andparasitic absorption are clearly noted. Besides, the recombinationcaused losses are assessed by subtracting the optical losses from thegross Jsc losses (assuming the best Jsc value of 44mA/cm2) [22]. So, therecombinative losses for Back-PEDOT, Front-PEDOT and IBC-PEDOTdevice are estimated as 1.5, 1.9 and 5.5mA/cm2, respectively.

It is well known that a good ETL should has not only good passi-vation, but also low contact resistivity (ρc). The MgOx layer is applied asETL here mainly due to its convenience for processing, good stabilityand moderate passivation [12]. To investigate the ETL of MgOx layerused here, a series of planar Front-PEDOT solar cells with variedthickness of MgOx films were fabricated. The schematic diagram of thiskind of device is shown in Fig. 3a. Two corresponding TEM imagescollected from the black-square regions at the front and the back in-terfaces are exhibited in Fig. 3b and c, respectively. From Fig. 3b, wecan obviously see a thin silicon oxide (SiOx) layer existing between thec-Si and PEDOT:PSS. The presence of SiOx layer has been proven as akey factor for better passivation at the PEDOT:PSS/Si interface [23].While for the vacuum proceeded MgOx, no distinguishable silicon oxidelayer exists at the interface of MgOx/Si (Fig. 3c). Nevertheless, amoderate level of passivation was provided by the MgOx film on the Sisurface, supporting by the minority carrier lifetime mapping on thesymmetric structure of MgOx/n-Si/MgOx (Fig. S3). The average min-ority carrier lifetime of the sample is about 20 μs, which corresponds acalculated surface recombination velocity of 621 cm/s. The Voc and Jscas a function of the thickness of MgOx films are shown in Fig. 3d, whilethe relevant evolutions of FF and ρc are shown in Fig. 3e. One can seethat the Voc (Jsc) increases quickly from 559 (25.8) to 591mV(27.3 mA/cm2) for the thickness of MgOx ranging from 0 to 0.6 nm, andthen keeps near a constant with the thickness up to 1.8 nm. This resultindicates that a moderate passivation of MgOx layer can be quicklyobtained when the thickness is large than 0.6 nm. With further in-creasing the thickness of MgOx film, the contact resistivity was dra-matically increased from 15 mΩ⋅cm2 at 0.6 nm to ~ 1500 mΩ cm2 at3 nm, leading to severe deterioration in the FF. A full trend of J-V curvesalong with the MgOx thickness is shown in Fig. S4. One can see that avery thin MgOx layer, i.e. 0.2, 0.6 or 1.2 nm can help to get Ohmiccontact properties, possibly owing to the Fermi level depinning effect.While the further increase of the thickness will cause a large resistivebarrier for electron tunneling. Therefore, considering the balance be-tween the passivation quality and the resistive losses, a thickness ofMgOx film among 0.6–1.2 nm will give the best PCE (see Table S2).Thus, 1 nm-thick MgOx film is selected for constructing IBC devices.

Planar IBC solar cells with HTL of PEDOT:PSS and ETL of MgOx

were then fabricated, as schematically shown in Fig. 4a. In order to

H. Lin et al. Nano Energy 50 (2018) 777–784

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study the effect of pitch sizes on the PCEs, the ratio of HTL:ETL:gap wasfixed at 48%:32%:20% [24]. The PV performance of our planar IBC-PEDOT cells is listed in Table 2. With decreasing pitch from 1000 to100 μm, the Jsc raises from 28.8 to 35.0 mA/cm2, the Voc decreasesslightly by about 15mV, while the FF almost keeps at a stable value of71%. As a result, the PCE of the device increases from 11.8% to 14.4%.

In order to well understand above-mentioned phenomena, carriertransport properties of IBC device should be investigated. As shown inFig. 4a, since the HTL interdigitated with ETL on the rear side, photo-generated minority carriers (holes in n-type Si) above HTL have a largeprobability be transported to HTL and directly contribute to the Jsc.While holes above the noncollecting region (including ETL and gapportions) have a large probability to be annihilated firstly if the ETLswith poor passivation were used [25,26]. Meanwhile, broad width ofnoncollecting region will extend the average lateral distance for holestransport and increase the probability of recombination [21]. There-fore, increasing the width of ETL region (the width of ETL increase from32 μm to 320 μm along with the increase of pitch in Table 2) will de-crease the final collection probability of holes to HTL and lead to a lowJsc. The relations between PV parameters and the pitch sizes underthree different passivation levels, i.e. poor (SETL = 106 cm/s), moderate

(SETL = 1000 cm/s) and good (SETL = 10 cm/s), were simulated andshowed in Fig. 4b–d, respectively, where SETL is the surface re-combination rate at ETL. From the results, we can clearly see that thePCEs of the devices with poor ETL passivation are always limited by theextremely low Jsc, showing a value below 11% for all the pitches. Whilefor the good ETL passivation, all the PV parameters can be maintainedat a quite high level, with Voc, Jsc, and PCE of 650mV, 38mA/cm2 and20%, respectively. In term of the moderate passivation case with SETL of103 cm/s, the Voc, Jsc and PCE have a significant dependence on thepitches. The Jsc declines with the pitch size very quickly while the Voc

increases slowly, and the best PCE occurs at the smallest pitch. Thesimulated evolution trends for the moderate passivation case are wellconsistent with those of the experimental results of IBC-PEDOT deviceswith 1 nm MgOx film. This is reasonable because the Seff for our 10 nmMgOx on c-Si is around 621 cm/s. We should note here that the simu-lated PCEs are slightly higher than those collected from experimentsbecause the overestimated FF of 80% in simulation.

In IBC-PEDOT cells, the photogenerated carriers that are mainlylocated at the front surface must be transported to the rear side andthen be collected by the HTL and ETL electrodes. This is well differentto the conventional double-sided junction solar cells, in which a

Fig. 1. The structure of PEDOT:PSS/Si based IBC-HJs solar cell. (a) Schematic of the IBC-HJs device. (b) The cross-sectional view of the back-contact region. (c)Corresponding SEM images of the back-contact region. (d–f) Magnified SEM images of the white-square region in (c) from left to right, respectively. Scale bars, 10μmin (c) and 200 nm in (d–f).


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relatively poor rear-sided passivation may not cause significant influ-ence on the Jsc. In IBC-PEDOT, however, lack of or insufficient passi-vation is likely to lead to the decrease of Voc as well as Jsc. Poor pas-sivation in gap regions would result in recombination of carriers beforethey are collected at the junction [27]. The difference in carriertransport channels for the passivation-free and the Al2O3-passivatedgaps are schematically shown in Fig. 5a. In order to quantitativelyanalyze the influence of gap passivation on the PV performance, cor-responding simulation with varied Sgap and SETL are exhibited inFig. 5b, c and d, respectively. As shown in Fig. 5b–d, the importance ofgap passivation on the PV performance is fully displayed, especiallywhen SETL< 103 cm/s. In other words, when the ETL regions arepassivated beyond moderate level, the gap passivation plays a decisiverole in the performance of our IBC-PEDOT device. For example, as SETL= 10 cm/s but Sgap = 104 cm/s, the Jsc, Voc and PCE will still be quitepoor as 29.4 mA/cm2, 548mV and 12.6%, respectively. In addition, wecan draw another conclusion that the Jsc, Voc and PCE all can be kept ata nearly high constant value when SETL< 102 cm/s for each de-termined Sgap, and then decreases quickly when the SETL increasing

from 102 cm/s to 105 cm/s. Except for the planar IBC-PEDOT device,the IBC-PEDOT device with pyramids-texture on the front side was alsosimulated and shown in Fig. 5b–d. With the same Sgap of 5 cm/s, ap-plying the pyramids-texture on the front surface will predict a Jsc up to41.6 mA/cm2 and PCE exceeding 22.4%. At last, our best experimentalresults at this stage are marked as yellow stars in Fig. 4b–c, pointing outa relatively large space for promotion of the PCE. Future research willbe emphasized on how to reduce the SETL and improve the FF.

3. Conclusions

In summary, we have fabricated a PEDOT:PSS/Si heterojunction all-back-contacted (IBC-PEDOT) solar cell with efficiency over 16.3%. Wesuccessfully demonstrated a reasonable design of buried ETL methodthat not only substantially simplifies the architecture and fabrication ofback-contacted silicon solar cells, but also makes it possible to adaptsolution-proceeded HTL and keeps a good passivation in the gap region.Although the optimized ETL of 1 nm-thick MgOx film in this work candelivery moderate level of passivation and acceptable contact

Fig. 2. Comparison of the three kinds of PEDOT:PSS/Si heterojunction devices. Schematics of (a) Front-PEDOT, (b) Back-PEDOT and (c) IBC-PEDOT devices. (d1–d3)Corresponding Jsc losses estimated by experimental results. (e) Light J-V curves and (f) Reflection and EQE spectra for the three kinds of devices.

Table 1Photovoltaic characteristics of the three kinds of PEDOT:PSS/Si heterojunction solar cells.

Samplesa Vocb (V) Jscb (mA/cm2) FFb (%) PCEb (%)

Front-PEDOT 0.622 (0.619 ± 0.007) 31.8 (31.9 ± 0.4) 71.8 (70.2 ± 1.7) 14.2 (13.9 ± 0.2)Back-PEDOT 0.617 (0.615 ± 0.006) 34.9 (34.7 ± 0.3) 72.6 (71.4 ± 1.2) 15.6 (15.2 ± 0.4)IBC-PEDOT 0.581 (0.576 ± 0.007) 38.4 (38.4 ± 0.3) 73.1 (71.3 ± 1.4) 16.3 (15.8 ± 0.4)

a Data and statistics based on five cells of each condition.b Numbers in bold are the champion values of each condition.


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Fig. 3. Optimization of MgOx film. (a) Schematic of a planar Front-PEDOT device with rear-sided MgOx film. TEM images for (b) PEDOT:PSS/Si interface and (c) Si/MgOx/Al interface. (d) The Voc and Jsc as a function of MgOx thickness. (e) The FF and ρc as a function of MgOx thickness. The scale bars in (b) and (c) are both 5 nm.

Fig. 4. Influence of intercontact pitch and ETL passivation to photovoltaic properties of IBC-PEDOT solar cells. (a) Schematic structure of the simulated planar IBC-PEDOT device. (b–d) Simulated data of Jsc (b), Voc (c), PCE (d) as functions of pitches and SETL. Sgap = 5 cm/s.


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resistance, it is still insufficient. At last, a thorough simulation of theinfluence of pitch, SETL and Sgap on the PV performance revealed thatthe PEDOT:PSS/Si heterojunction IBC solar cell with pyramids-texturecan be readily pushed to a high level with Jsc exceeding 41mA/cm2 andPCE beyond 22% once the surface recombination rate of ETL can becontrolled below 100 cm/s.

4. Experimental section

4.1. IBC-PEDOT solar cells fabrication

Double-side polished, Czochralski, n-type (1–10Ω cm) wafers with athickness of 250 μm were directly used to fabricate planar IBC-PEDOTsolar cells. Randomly pyramids-textured wafers were prepared throughimmersing one-side of Si wafer into 80℃ mixed solutions with 2.5%KOH and 1.25% isopropanol for 15min, while protecting the other sideby a homemade tool. The processing flow for fabricating IBC-PEDOTcells can refer to Fig. S1. Firstly, after cleaning the wafers by a standardRCA1/2 [28] and removing native silicon oxide by a 4% HF solution, a

15 nm Al2O3 thin film was deposited as passivation layer by atomiclayer deposition (ALD) system, and then an 85 nm-thick SiNx film forpyramid-texture device and a 70 nm-thick Al2O3 film for planar devicewas deposited as anti-reflection layer by plasma-enhanced chemicalvapor deposition (PECVD) and E-beam evaporation, respectively. Thedevices were then annealed at 450℃ in nitrogen atmosphere for 30minto fully activate the passivation capability of ALD-Al2O3 thin films.Secondly, the photoresist (AZ 5214) patterns for ETL/Al were fabri-cated by photolithography and the corresponding Al2O3 film above ETLpatterns was removed by 4% HF. Thirdly, 1 nm MgOx film and 1 μm Alfilm were deposited by E-beam evaporation in sequence, and then theETL/Al electrode was formed after lift-off process using acetone.Fourthly, a 3–4μm photoresist patterns wrapped around the ETL/Alelectrode was formed through photolithography, and the correspondingAl2O3 thin film at the open regions was removed by 4% HF. At last,PEDOT:PSS (PH 1000 from Clevios) solution mixed with Triton-100(0.25%) and dimethyl sulfoxide (5%) was spin coated on the rear side ofdevice at a speed of 3000 rpm and annealed at 120℃ for 10min, afterthat a 200 nm Ag film was deposited on the PEDOT:PSS film by E-beamevaporation.

4.2. Front-back contact solar cells fabrication

Both of Front-PEDOT and Back-PEDOT devices in Fig. 2 used one-side randomly pyramids-textured wafers. After cleaning and removingnative oxide, 5 nm a-Si:H(i) layer and 10 nm a-Si:H(n) were depositedon the polished-side for Front-PEDOT and pyramid-side for Back-PEDOT, respectively, through PECVD system. And then 200 nm Al wasdeposited on the a-Si:H layer by thermal evaporation for Front-PEDOT,

Table 2Photovoltaic performance of planar IBC solar cells with different pitches.

Pitcha (μm) Voc (V) Jsc (mA/cm2) FF (%) PCE (%)

100 0.572 ± 0.015 35.0 ± 0.7 71.7 ± 2.0 14.4 ± 0.3200 0.561 ± 0.011 33.1 ± 0.9 71.1 ± 1.0 13.2 ± 0.5500 0.581 ± 0.013 30.5 ± 1.5 71.7 ± 0.7 12.7 ± 0.91000 0.589 ± 0.006 28.8 ± 1.4 69.4 ± 3.2 11.8 ± 0.7

a Data and statistics based on five cells of each condition.

Fig. 5. Influence of gap passivation to PV properties of IBC-PEDOT solar cells. (a) Schematic illustration of holes transmission above gap region without passivation(left) and with passivation (right). (b–d) Simulation of Jsc (b), Voc (c), PCE (d) as functions of Sgap and SETL. The pitch is 100 μm, the yellow stars representexperimental data.


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while 80 nm In2O3:W (IWO) film was deposited on the a-Si:H layer byreactive plasma deposition system for Back-PEDOT. After that, both ofthem were covered with PEDOT:PSS film. At last, a Ag grid electrode(200 nm) was thermally evaporated on the top side of both devices by ametal mask, and a 200 nm thick Ag film was deposited on the rear sideof Back-PEDOT device. For the planar Front-PEDOT in Fig. 3, the pro-cessing flow is same as pyramid-texture Front-PEDOT solar cell exceptfor the replacement of a-Si:H with E-Beam evaporated MgOx film asETL.

4.3. Characterization

The morphological analysis of the samples was conducted by SEM(Hitachi S-4800) and TEM (Tecnai F20). Light J-V curves of solar cellswere measured under a simulated AM 1.5 spectrum sunlight illumina-tion and with a 0.5 cm2 effective illumination area through a mea-surement mask. The reflectance spectra as well as the EQE were mea-sured on the platform of quantum efficiency measurement (QEX10, PVMeasurements), and we adjusted the beam spot of testing light to0.7×0.7 cm2 as well as added a white light bias of 0.1 Suns when wemeasured EQE. The I-V curves of contact resistance were measured by aKeithley 4200-scs semiconductor parameter analyzer. The minoritycarrier lifetime was measured by a microwave photoconductivity decaysystem (WT-2000 μPCD, Semilab).

4.4. Simulation method

In the simulation, firstly, we utilized Lumerical Finite DifferenceTime Domain (FDTD) software to calculate 2D generation rate map ofthe entire structure of 200μm thick silicon substrate. And there weretwo different top surface structure in simulation, one was pyramids-texture with 15 nm Al2O3 and 60 nm SiNx and another was planarsurface with 85 nm Al2O3. Secondly, the generation rate was introducedinto the Lumerical DEVICE software, and through adjusting the re-combination of each interface, we calculated a series of photovoltaicperformance. In DEVICE simulation, the detailed parameters were set asfollowing: Substrate was n-type silicon with 3ms bulk lifetime and thedopant concentration was chosen as 1015 cm−3. The diffusion para-meters of p++ region were chosen as p-type dopant and the con-centration was set at 1×1016 cm−3. This corresponds a Vbi of 660mVforming at the PEDOT/Si interface [29,30]. While the n++ region wasset as n-type dopant with a concentration of 1×1015 cm−3. The sur-face recombination velocity of HTL/Si interface and front Al2O3/Siinterface were set as 500 and 5 cm/s, respectively, according to theexperimental results in Table S1.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11674225, 11474201, and 61674154), MajorState Basic Research Development Program of China (No.2016YFB0700700), Zhejiang Provincial Natural Science Foundation(LR16F040002).

Appendix A. Supplementary material

Supplementary data associated with this article can be found in theonline version at http://dx.doi.org/10.1016/j.nanoen.2018.06.013.

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Photovoltaic Specialists Conference (PVSC), 2016, pp. 3351–3355.[23] L. He, C. Jiang, H. Wang, H. Lei, D. Lai, Rusli, Photovoltaic Specialists Conference,

42, 2012, pp. 002785–002787.[24] S. Jeong, M.D. McGehee, Y. Cui, Nat. Commun. 4 (2013) 2950.[25] P. Procel, M. Zanuccoli, V. Maccaronio, F. Crupi, G. Cocorullo, P. Magnone,

C. Fiegna, J. Comput. Electron. 15 (2015) 260–268.[26] I. Cesar, N. Guillevin, A.R. Burgers, A.A. Mewe, M. Koppes, J. Anker, L.J. Geerligs,

A.W. Weeber, Energy Procedia 55 (2014) 633–642.[27] P. Spinelli, B.W.H. van de Loo, A.H.G. Vlooswijk, W.M.M. Kessels, I. Cesar, IEEE J.

Photovolt. 7 (2017) 1176–1183.[28] H. Tong, Z. Yang, X. Wang, Z. Liu, Z. Chen, X. Ke, M. Sui, J. Tang, T. Yu, Z. Ge,

Y. Zeng, P. Gao, J. Ye, Adv. Energy Mater. (2018) 1702921.[29] Y. Liu, Z.G. Zhang, Z. Xia, J. Zhang, Y. Liu, F. Liang, Y. Li, T. Song, X. Yu, S.T. Lee,

B. Sun, ACS Nano 10 (2016) 704–712.[30] Z. Yang, P. Gao, J. He, W. Chen, W.-Y. Yin, Y. Zeng, W. Guo, J. Ye, Y. Cui, ACS

Energy Lett. 2 (2017) 556–562.

Hao Lin received his B.S. and M.S. degree in faculty ofScience from Ningbo University, China, in 2010 and 2013,respectively. From 2012 to 2015, he worked in departmentof Physics and Materials Science, City University of HongKong. Currently, he is a Ph.D. candidate at School ofPhysics and Astronomy, Shanghai Jiao Tong University. Hisresearch interests include the solar energy materials, anti-reflection structure and dopant-free all-back-contact solarcells.

Dong Ding received his master degree in condensed matterphysics of School of physics and engineering fromZhengzhou University, China, in 2016. He is currently aPh.D. candidate in the School of Physics and Astronomy atShanghai Jiao Tong University, China. His research inter-ests focus on the simulation of solar cells, including theinterdigitated back contact solar cells and related materials.


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http://refhub.elsevier.com/S2211-2855(18)30409-9/sbref1























































Zilei Wang received his B.S. degree in College of Physicsand Information Engineering at Fuzhou University, China,in 2015. He is involved in a joint program betweenUniversity of Science and Technology of China and NingboInstitute of Materials Technology and Engineering, ChineseAcademy of Sciences (CAS) in Ningbo, China. His researchinterests focus on organic-inorganic hybrid solar cells.

Longfei Zhang received his B.S. degree in school of sciencefrom Shandong University of Technology, Zibo, China, in2016. He is involved in a joint program between Universityof Science and Technology of China and Ningbo Institute ofMaterials Technology and Engineering, Chinese Academyof Sciences (CAS) in Ningbo, China. His research interestsfocus on dopant-free all-back-contact solar cells.

Fei Wu received the Bachelor degree in MinZu Universityof China, in 2016. She is currently a Master candidate in theSchool of Physics and Astronomy, Shanghai Jiao TongUniversity. Her research interests include solar energymaterials and solar cells.

Jing Yu received her B.S. degree in Material Forming andcontrolling Engineering from Shandong University,Shandong, China, in 2014. She is currently working towardthe Ph.D. degree with the Ningbo Institute of MaterialsTechnology and Engineering, Chinese Academy of Science,Ningbo, China. Her research interests focus on high effi-ciency heterojunction solar cells with metal oxides as car-rier-selective contacts.

Pingqi Gao received Ph.D. degrees in Department ofPhysics from Lanzhou University in 2010. From 2007 to2011, he worked in Nanyang Technological University as avisiting researcher and a research staff. In 2013, he joinedNingbo Institute of Materials Technology and Engineering,CAS, as an associate professor and then a professor (2015).His research focus on high efficiency solar cell technology,especially on developing new materials and processes forsolar energy conversion. He has published over 70 journalpapers and serves as an active referee for 20 journals.

Jichun Ye received the B.S. degree in Materials Science andEngineering from University of Science and Technology ofChina in 2001 and the Ph.D. degree in Materials Sciencefrom University of California, Davis, USA in 2005. Hejoined Ningbo Institute of Material Technology andEngineering, CAS, as a professor and Ph.D. advisor sinceAugust of 2012. He was awarded for "Thousand YoungTalents Program of China" in 2012. He has published morethan 60 publications with nearly 500 times citations, ap-plied more than 40 patents (including 10 awarded patents).

Wenzhong Shen received his Ph.D. degree in semi-conductor physics and semiconductor device from ShanghaiInstitute of Technical Physics, Chinese Academy ofSciences, in 1995. Since 1999, Dr. Shen has been withShanghai Jiao Tong University, China, as a full professor inthe School of Physics and Astronomy, where he is currentlythe director of Institute of Solar Energy and Key Laboratoryof Artificial Structures and Quantum Control, Ministry ofEducation.


78483

Antiferromagnetic Order in Epitaxial FeSe Films on SrTiO3

Y. Zhou,1 L. Miao,2 P. Wang,1 F. F. Zhu,2 W. X. Jiang,2 S. W. Jiang,1 Y. Zhang,1 B. Lei,3,4,5 X. H. Chen,3,4,5 H. F. Ding,1,5

Hao Zheng,2,5 W. T. Zhang,2,5 Jin-feng Jia,2,5 Dong Qian,2,5,* and D. Wu1,5,†1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China2Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy,

Shanghai Jiao Tong University, Shanghai 200240, China3Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science

and Technology of China and Key Laboratory of Strongly-Coupled Quantum Matter Physics,Chinese Academy of Sciences, Hefei, Anhui 230026, China

4High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei, Anhui 230031, China5Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

(Received 15 September 2017; published 27 February 2018)

Single monolayer FeSe film grown on a Nb-doped SrTiO3ð001Þ substrate shows the highest super-conducting transition temperature (TC ∼ 100 K) among the iron-based superconductors (iron pnictides),while the TC value of bulk FeSe is only ∼8 K. Although bulk FeSe does not show antiferromagnetic order,calculations suggest that the parent FeSe/SrTiO3 films are antiferromagnetic. Experimentally, because of alack of a direct probe, the magnetic state of FeSe/SrTiO3 films remains mysterious. Here, we report directevidence of antiferromagnetic order in the parent FeSe/SrTiO3 films by the magnetic exchange bias effectmeasurements. The magnetic blocking temperature is ∼140 K for a single monolayer film. Theantiferromagnetic order disappears after electron doping.

DOI: 10.1103/PhysRevLett.120.097001

The pairing mechanism of high-temperature supercon-ductors including cuprates and iron pnictides is one of thebiggest challenges in modern physics. The antiferromag-netic (AFM) interaction has been long thought to correlatewith high-temperature superconductivity [1,2] because thesuperconducting state usually appears after the AFM orderis suppressed [3,4]. The AFM spin fluctuations wereproposed to play an important role in the pairing of ironpnictides [1,3,5]. Among various iron pnictides, FeSe hasthe simplest crystalline structure [6]. The TC value of bulkFeSe is ∼8 K and can increase to ∼37 K under highpressure [7]. Unlike other iron pnictides, bulk FeSe crystalsdo not show AFM order [7] unless a certain pressure isapplied [8–10].

Surprisingly, a single monolayer (1-ML) FeSe filmgrown on a Nb-doped SrTiO3ð001Þ [“STO” will refer toNb-doped SrTiO3ð001Þ] substrate after electron doping(through the annealing process) shows a large supercon-ducting gap (∼20 meV) [11] that survives up to ∼65 K[12,13]. Diamagnetic signals below ∼65 K have also beenreported [14]. Recently, the in situ resistance measurementsshowed that TC value of the 1-ML FeSe/STO film can be ashigh as 109 K [15]. The mechanism of such a high TC valueis still an open question. Calculations have shown that theelectron-phonon coupling is significantly enhanced [16]due to the interfacial effect and therefore enhances the valueof TC in this system [17], but the initial pairing mechanismis still unclear. First-principles calculations have shownthat the FeSe/STO interface could enhance the AFM

interaction, which helps maintain large spin fluctuationsunder heavy electron doping [18]. Magnetic frustrationinduced by the combination of the electron doping and thephonons is another possible mechanism for the super-conductivity [19]. Density functional theory (DFT) calcu-lations have suggested that the magnetic ground state of1-ML FeSe/STO film is AFM [18,20,21]. A recent workalso claimed that 1-ML FeSe/STO could be in AFM orderto form topological superconductivity [22]. Therefore, itis very interesting to study the magnetic ground state ofthe 1-ML film before electron doping. Experimentally, themagnetic state of the FeSe/STO films is barely known.Previous angle-resolved photoemission spectroscopy(ARPES) measurements showed indirect signatures ofthe spin density wave [12], but they are indistinguishablefrom electronic nematicity [23–25]. To determine themagnetic state, regular techniques such as neutron scatter-ing, muon spin rotation, and the Mössbauer effect havelimited sensitivity for ultrathin films. In this Letter, wepresent direct evidence indicating that the magnetic groundstate of the parent 1-ML FeSe/STO film is AFM by usingmagnetic exchange bias effect (MEBE) [26] measurements.FeSe/STO films were grown following previous reports

[9,15]. Films before postannealing are called “as-grown”films. The as-grown films were postannealed at∼500 °C for4–8 h in situ to make them superconducting. Before thefilms were transferred to another chamber to grow anFe21Ni79 layer, a 50-nm-thick Se protecting layer wasgrown. The polycrystalline Fe21Ni79 film was grown on

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the FeSe film at room temperature by e-beam evaporationafter removing the Se protecting layer by properly annealingat 400 °C. The Fe21Ni79 thickness was optimized for MEBEmeasurements. Finally, a 10 nm Au film was deposited toprevent oxidation. ARPES were performed at ARPESBeamline of National Synchrotron Radiation Laboratoryin Hefei and at Beamline 4.0.3 of Advanced Light Source.Magnetic properties were measured by a Quantum DesignSQUID-VSM system. The magnetic field (H) was set tozero in an oscillation mode to reduce the residual field of themagnet before measurements. The residual field was furthercalibrated by a reference sample of Au/Fe21Ni79ð10 nmÞ/STO [see the Supplemental Material (SM) [27]]. Thecoercivity of themagnetic hysteresis loop (MHL) is obtainedby linearly fitting the data points very close to the zeromagnetization points. Details on how to determine thecoercivity with high accuracy and also how to determinethe uncertainty is described in the SM [27].The MEBE is widely used for probing the AFM order in

materials, particularly in thin films [26,28]. The MEBEoccurs in a ferromagnet-antiferromagnet heterostructurewhen it is cooled in an external H through the Neeltemperature (TN) of the AFM layer or is grown in an externalfield. TheMEBE relies on the interfacial magnetic exchangecoupling between the AFM and ferromagnetic (FM) layers.Measurements are on themagnetization (M) of the FM layer.The distinct phenomenon of the MEBE is that the center oftheMHL shifts away from theH ¼ 0 point; i.e., the absolutevalues of the coercive fields for increasing (HCþ) anddecreasing (HC−) fields are different. More importantly,the shifting direction reverses when the cooling field (CF) isreversed, as illustrated in Figs. 1(a) and 1(b).First, we studied the thick FeSe/STO to show the

capability of MEBE measurements on an FeSe system.Polycrystalline Fe21Ni79 is used as the FM layer. Figures 1(d)

and 1(e) present the MHLs and the corresponding enlargedplots of a Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/FeSeð100 MLÞ/STO sample measured at 5 K after field cooling (FC) fromroom temperature. The CF is either þ10 kOe or −10 kOe.The linear background originating from the diamagneticsignal of the STO substrate is subtracted from the raw data(see the SM [27]). In Fig. 1(e), theMHLs shift away from theH ¼ 0 point, and the shifting direction is the opposite of thedirection of the CF. The shift of the MHLs and the reverse ofthe shifting direction upon reversing the CF indicate that theobserved effect is the MEBE. From Fig. 1(e), we obtain themagnitude of the shift—the exchange bias field ðHEBÞ ¼jHC− −HCþj/2 ∼ 28� 2.5 Oe. The observed MEBE per-sists up to about 180 K in this sample.Previous experiments have shown that a capping layer

degrades the superconducting properties of 1-ML FeSe/STO [29]. Even an innocuous FeTe overlayer can hole dopethe system [30], provide coupling to phonon modes [31],or intermix Te and Se atoms [32]. Therefore, it is necessaryto examine the possibility of the observed AFM orderinduced by the Fe21Ni79 overlayer. First, it is impossiblethat the AFM order is induced by Fe21Ni79 via theinterfacial magnetic interaction. Instead, a FM layer canalter a non-FM material to a FM order, known as magneticproximity effect [33–35] (more discussion in the SM [27]).Second, other interfacial effects, such as the selenizedFe21Ni79 film, the interfacial intermixing, and the alter-nation of Se height from the Fe plane [36] could lead to anAFM order. It is crucial to carry out control experiments toverify that the observed MEBE is the intrinsic property ofthe as-grown FeSe/STO films. (i) First, we prepared a sampleof Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/STO [Fig. 2(a)] and con-ducted the same measurements. Figures 2(b) and 2(c) show

FIG. 1. The MEBE in an Fe21Ni79/FeSe/STO film. (a), (b) Theschematic MHLs of the MEBE after the positive and negative FC.(c) Layout of the film. (d) MHLs of the sample in (c) measured at5 K after FC from room temperature to 5 K. (e) The correspond-ing enlarged plots near the zero field.

FIG. 2. Control experiments. (a) Layout of control sample 1.(b) MHLs and (c) the corresponding enlarged plots of the samplein (a). (d) Layout of control sample 2. (e) MHLs and (f) thecorresponding enlarged plots of the sample in (d). (g) Layout ofcontrol sample 3. (h) MHLs and (i) the corresponding enlargedplots of the sample in (g). The loops did not shift in all of thecontrol experiments.


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the MHLs and the enlarged plots. No shift was detectedwithin our experimental uncertainty, which excludes anytechnical problems. (ii) Before the deposition of the Fe21Ni79film, the FeSe/STO film was annealed to remove the Seprotecting layer. Therefore, the Fe21Ni79 film might havebeen selenized by the possible residual Se to form an AFMlayer at the interface and lead to MEBE. To exclude thispossibility, we fabricated a control sample of Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/Seð50 nmÞ/STO [Fig. 2(d)]. Figures 2(e)and 2(f) show the MHLs and the enlarged plots. Clearly, theloops do not shift, suggesting that the selenized Fe21Ni79 filmis notAFM. (iii)We also exclude the possibility that theAFMlayer is caused by the intermixing, alloying, and proximity(polarization) effect between the Fe21Ni79 film and theFeSe film or the change of Se height by using bulk FeSeas a reference. We prepared a sample of Auð10 nmÞ/Fe21Ni79ð1.4 nmÞ/bulk FeSe [Fig. 2(g)]. Here, a 1.4-nm-thick Fe21Ni79 film is used to obtain better a signal-to-noiseratio because the high quality cleaved surface area(∼1 × 1 mm2) of bulk FeSe is much smaller than that of aSTO substrate (∼3 × 3 mm2). The measurement temper-ature (10 K) is kept slightly above the TC value of bulk FeSeto avoid the strong diamagnetic signal of the superconductingbulk FeSe. The cleaved bulk FeSe crystal has a (001) surfacewith Se termination which is the same as the FeSe/STO film.Since both interfaces (the Fe21Ni79/FeSe interface) areidentical, one would expect the presence of the MEBE inthe Fe21Ni79/bulk FeSe sample if the AFM layer wereinduced by the interfacial intermixing, alloying, polarization,or height change of the Se atoms. However, theMEBE is notobserved in this control sample, as shown in Fig. 2(i). Withthe three control experiments above, we conclude that theobserved MEBE most likely originates from the AFM layerin the 100-MLFeSe/STO film. In addition, for completeness,we note that the MEBE can also occur between a hetero-structure of a ferromagnet and a spin glass system [26,28].The spin glass state in the FeSe/STO film is excluded by thethermal remnant magnetization measurements [see theSM [27]].To show the temperature dependence more clearly, we

used the so-called inversionmethod to plot theMHLs. In thismethod, bothM andH of the original loop are multiplied by−1. In other words, we invert the MHL. The new loop iscalled the “inverted loop.” After inversion, theHC− value ofthe original loop reflects from the negative H side to thepositive H side; therefore, we can directly show the differ-ence between HCþ and HC−. Figures 3(a)–3(f) show theenlarged plots with original and inverted loops nearHC� of aAuð10 nmÞ/Fe21Ni79ð0.7 nmÞ/FeSeð100 MLÞ/STO samplemeasured at different temperature after FC. The MEBEgradually becomes weaker with increasing temperature. Thetemperature dependence of HEB is summarized in Fig. 3(g).The blocking temperature TB, where HEB becomes zero, is∼180 K. The value of HEB depends on both the AFM andFM layers, while TB depends mainly on the AFM layer [26].

TB andTN are intimately correlated and, in general,TN ≥ TB[26]. Therefore,we obtained the lower limit ofTN of∼180 Kfor the 100-ML FeSe/STO film. The MEBE is also carriedout on the FeTe/STO film, which possesses a well-knownAFM state [37,38]. The determined TB of ∼75 K is com-parable to the TN of the thick FeTe/MgO film [39] (see theSM [27]).After the demonstration of the capability of the MEBE

study on thick FeSe/STO films, we studied the 1-ML FeSefilm. The as-grown 1-ML FeSe/STO film is nonsupercon-ducting. It becomes superconducting by doping electronsthrough the annealing process [11–15]. We prepared twotypes of samples: Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/as-grownFeSeð1 MLÞ/STO (sample 1) and Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/annealed FeSeð1 MLÞ/STO (sample 2). Our sam-ple 1 is in the “N phase” and sample 2 the “S phase” asdefined by He et al. [13]. A superconducting gap wasobserved on annealed FeSe/STO films by ARPES (see theSM [27]). The MEBE is clearly observed in sample 1 at 5 Kafter FC. Shown in Fig. 4(c), the MHLs shift away from theH ¼ 0, and the shifting direction reverses when the CF isreversed. The shift of the MHL is relatively small (∼5 Oe),but it is still well above the error bar (∼0.5 Oe). By contrast,the MEBE was not detected within our experimental uncer-tainty in sample 2, as shown in Fig. 4(e). Because sample 2 isheavily electron doped and no AFM order exists, we wouldlike to propose that the heavy electron doping by theannealing process destroys the AFM order. Figures 4(f)and 4(g) show enlarged original and inverted plots nearHC�

FIG. 3. Temperature dependence of theMEBE. (a)–(f) Enlargedcurves of original and inverted loops of Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/FeSeð100 MLÞ/STO film at different temper-atures after FC. (g) HEB as a function of temperature. The solidline is a guide for the eye.


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of sample 1 measured at different temperatures. Shown inFig. 4(i), TB is∼140 K,meaning that TN ≥ 140 K for the 1-ML as-grown FeSe/STO film. More data on differentsamples of 1-ML films can be found in the SM [27]. TheTN value of the as-grown 1-ML FeSe/STO film is muchhigher than the reported highest TN value (∼55 K) of bulkFeSe under high pressure [9].The MEBE depends on the competition between the

interfacial energy Jint at the ferromagnet-antiferromagnetinterface and the anisotropy energy KAFMtAFM of the AFMlayer, whereKAFM and tAFM are the anisotropy constant andthe thickness of the AFM layer, respectively. The conditionKAFMtAFM ≥ Jint or tAFM ≥ Jint/KAFM is required for obser-vation of the MEBE [28,40], meaning that a critical AFMthickness is needed for the MEBE [28,40]. In Fe21Ni79/FeSe/STO, Jint is relatively weak because the interfacialcoupling occurs indirectly between the Fe/Ni atoms ofFe21Ni79 and the Fe atoms of FeSe through the Se atoms,which means that the critical AFM thickness in Fe21Ni79/FeSe/STO would be very thin. This is why we can observethe MEBE even in 1-ML-thick FeSe/STO films.Furthermore, we carried out measurements on the as-

grown 2-ML FeSe/STO films. The Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/FeSeð2 MLÞ/STO sample exhibits the MEBE atlow temperature after FC, indicating that 2-ML FeSe/STOfilm also has the AFM order. TB is determined to be∼180 K [Fig. 4(i)], which is larger than that of 1-ML film,as expected due to the increased thickness of the AFM layer[26,28]. Interestingly, the TB value of 2-ML FeSe/STO

films is already similar to that of 100-ML FeSe/STO film,implying that the interlayer magnetic interaction is muchweaker than the intralayer interaction, compatible with thelayered structure of FeSe.Finally, we try to get some insight into why AFM order

exists in FeSe/STO films. Although the reason why bulkFeSe has no magnetic order is under debate [41–45], strongAFM fluctuations have been observed in neutron scatteringexperiments [46,47]. A DFT calculation has suggested thattensile stress could enhance the AFM interaction [18].However, 100-ML FeSe/STO films already have a verysimilar lattice constant to bulk FeSe [12], but we stillobserved the MEBE. Therefore, we think that the straineffect plays a very minor or a negligible role for thick FeSe/STO films. In fact, although thick film and bulk crystal havethe same lattice constant, they have very different micro-scopic properties. First, there are a number of Fe vacanciesand domain walls of nematicity in FeSe/STO films [48].Second, the strength of nematicity in FeSe/STO is muchlarger than that in bulk FeSe [48,49]. Third, a very recentSTM study on FeSe/STO films observed a stripe-type chargeordering that does not exist in bulk FeSe, and the chargeordering is pinned in the vicinity of Fe vacancies, as well asdomain walls of nematicity [48]. The pinned charge order isquantitatively comparable to a magnetic channel predictedtheoretically [50]. In other words, impurities (or defects)could help to pin the magnetic fluctuations and could form arelatively long-range AFM order. The existence of AFMorder could be the reason why superconductivity does not

FIG. 4. The MEBE in 1-ML and 2-ML FeSe/STO films. (a) Layout of the films. (b) MHLs of Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/as-grownFeSeð1 MLÞ/STO film and (c) the corresponding enlarged plots. (d) MHLs of Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/annealedFeSeð1 MLÞ/STO film and (e) the corresponding enlarged plots measured at 5 K. (f)–(h) Enlarged curves of the original andinverted loops of the Auð10 nmÞ/Fe21Ni79ð0.7 nmÞ/as-grown FeSeð1 MLÞ/STO film at three temperatures. (g) HEB of theAuð10 nmÞ/Fe21Ni79ð0.7 nmÞ/as-grown (1 and 2 ML) and annealed 1-ML FeSe/STO films as a function of temperature.


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recover in thick FeSe/STO films. On the other side, it ismuchmore complicated in 1-ML films. Strong tensile stress andimpurities (or defects) coexist [11,15,51]. Tensile stressenhances the interaction, while impurities or defects helpto pin the AFM order, so we cannot rule out any of them for1-ML films. Annealing process can inject electrons intothe first ML. Interactions between local magnetic momentsthrough mobile electrons would prefer the FM state whereHund coupling to dominate. Therefore, the competitionbetween AFM and FM interactions could destroy theAFM order and eventually form superconductivity in1-ML films. There could be other possibilities that coulddestroy the AFM order during the annealing process, andmore theoretical input will be needed to fully understand themagnetic property of as-grown FeSe/STO films in the future.In summary, we observed the AFM order in both

100-ML and 1-ML FeSe/STO films before electron dopingby MEBE measurements. The low limit of the Neeltemperature of 1-ML film is about 140 K. Our findingsprovide very important information for a comprehensiveunderstanding of the novel properties of FeSe/STO films.

We acknowledge Chunlei Gao, Wei Ku, and WeijiongChen for the discussions. This work was supportedby the Ministry of Science and Technology of China(Grants No. 2016YFA0301003, No. 2016YFA0300403,and No. 2017YFA0303202) and the National NaturalScience Foundation of China (Grants No. U1632272,No. 11574201, No. 11521404, No. 11674224,No. 11790313, No. 11727808, No. 11674159, andNo. 51471086). D. Q. acknowledges support from theChangjiang Scholars Program. This research used resour-ces of the Advanced Light Source, which is a DOE Officeof Science User Facility under Contract No. DE-AC02-05CH11231.

Y. Z. and L. M. contributed equally to this work.

*[email protected]†[email protected]

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Field-Driven Quantum Criticality in the Spinel Magnet ZnCr2Se4

C. C. Gu,1 Z. Y. Zhao,2,3 X. L. Chen,1 M. Lee,4,5 E. S. Choi,4 Y. Y. Han,1 L. S. Ling,1 L. Pi,1,2,7 Y. H. Zhang,1,7

G. Chen,6,7,* Z. R. Yang,1,7,8,† H. D. Zhou,9,10,‡ and X. F. Sun2,7,8,§1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory,

Chinese Academy of Sciences, Hefei, Anhui 230031, People’s Republic of China2Department of Physics, Hefei National Laboratory for Physical Sciences at Microscale,

and Key Laboratory of Strongly-Coupled Quantum Matter Physics (CAS), University of Science and Technology of China,Hefei, Anhui 230026, People’s Republic of China

3Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences,Fuzhou, Fujian 350002, People’s Republic of China

4National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4005, USA5Department of Physics, Florida State University, Tallahassee, Florida 32306-3016, USA

6State Key Laboratory of Surface Physics and Department of Physics, Center for Field Theory and Particle Physics,Fudan University, Shanghai, 200433, China

7Collaborative Innovation Center of Advanced Microstructures, Nanjing, Jiangsu 210093, People’s Republic of China8Institute of Physical Science and Information Technology, Anhui University, Hefei, Anhui 230601, People’s Republic of China

9Key laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physicsand Astronomy, Shanghai JiaoTong University, Shanghai 200240, China

10Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA

(Received 19 September 2017; revised manuscript received 8 March 2018; published 5 April 2018)

We report detailed dc and ac magnetic susceptibilities, specific heat, and thermal conductivitymeasurements on the frustrated magnet ZnCr2Se4. At low temperatures, with an increasing magneticfield, this spinel material goes through a series of spin state transitions from the helix spin state to the spiralspin state and then to the fully polarized state. Our results indicate a direct quantum phase transition fromthe spiral spin state to the fully polarized state. As the system approaches the quantum criticality, we findstrong quantum fluctuations of the spins with behaviors such as an unconventional T2-dependent specificheat and temperature-independent mean free path for the thermal transport. We complete the full phasediagram of ZnCr2Se4 under the external magnetic field and propose the possibility of frustrated quantumcriticality with extended densities of critical modes to account for the unusual low-energy excitations in thevicinity of the criticality. Our results reveal that ZnCr2Se4 is a rare example of a 3D magnet exhibiting afield-driven quantum criticality with unconventional properties.


Since the beginning of the century, quantum phasetransition has emerged as an important subject in moderncondensed matter physics [1]. Quantum phase transitionand quantum criticality are associated with qualitative butcontinuous changes in relevant physical properties of theunderlying quantum many-body system at an absolute zerotemperature [1,2]. In the vicinity of quantum criticality, thelow-energy and long-distance properties are controlled bythe quantum fluctuation and the critical modes of the phasetransition such that certain interesting and universal scalinglaws could arise. It is well known that quantum criticalityoften occurs in the system with competing interactionswhere different interactions favor distinct phases or orders.Many physical systems such as high-temperature super-conducting cuprates [2], heavy fermion and Kondo latticematerials [3], Fermi liquid metals with spin density waveinstability [4], andMott insulators have been proposed to berealizations of quantum criticality [1]. For superconductors

and metals, the multiple low-energy degrees of freedom andorders may complicate the critical phenomena and theexperimental interpretation. In contrast, Mott insulatorswith large charge gaps are primarily described by spinand/or orbital degrees of freedom and may have theadvantage of simplicity in revealing critical behaviors.The Ising magnets CoNb2O6 and LiHoF4 in external

magnetic fields realize the quantum Ising model andtransition [5–10]. External magnetic fields in dimerizedmagnets like Han purple BaCuSi2O6 [11,12] induce atriplon Bose-Einstein condensation transition. In a morecomplicated example of the diamond lattice antiferromagnetFeSc2S4 [13–17], it is the competition between the super-exchange interaction and the on-site spin-orbital couplingthat drives a quantum phase transition from the antiferro-magnetic order to the spin-orbital singlet phase [18,19].These known examples of quantum phase transitions instrong Mott insulating materials with spin degrees of



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freedomare described by simple Ising orGaussian criticalitywhere there are a discrete number of critical modes gov-erning the low-energy properties. In this Letter, we explorethe magnetic properties of a three-dimensional frustratedmagnetic material ZnCr2Se4. From the thermodynamic,dynamic susceptibility, and thermal transport measure-ments, we demonstrate that there exists a field-drivenquantum criticality with unusual properties such as aT2-dependent specific heat and temperature-independentmean free path for thermal transport. Our quantum criticalityhas extended numbers of critical modes and is beyond thesimple Ising or Gaussian criticality among the existingmaterials that have been reported before.In the spinel compound ZnCr2Se4, the Cr3þ ion hosts the

localized electrons and gives rise to the spin-3=2 (Cr3þ)local moments that form a 3D pyrochlore lattice. Thereported dielectric polarization [20], magnetization andultrasound [21], and neutron and synchrotron x-ray[22,23] studies have shown that, with an increasingmagnetic field, this system goes from a helix spin stateto a spiral spin state to an unidentified regime and then afully polarized state at the measured temperatures. Twopossibilities have been proposed for this unidentifiedregime: an umbrella state and a spin nematic state[21,24]. Both the umbrella state and a spin nematic statebreak the spin rotational symmetry. We address thisunidentified regime by completing the magnetic phasediagram of ZnCr2Se4 with dc and ac susceptibility, specificheat, and thermal conductivity measurements. We do notobserve signatures of symmetry breaking in the previouslyunidentified regime down to the lowest measured temper-ature. We attribute our experimental results to a quantumcritical point (QCP) between the spiral spin state and thepolarized state and identify the previously unidentifiedregime as the quantum critical regime.The experimental details are listed in Supplemental

Material [25]. The dc magnetization measured at 0.01 Tin Fig. 1(a) shows a pronounced peak at TN ¼ 21 K,corresponding to the antiferromagnetic (AFM) order accom-panied by a cubic to tetragonal structural transition aspreviously reported [22]. With increasing fields, the peakshifts to lower temperatures. The dc magnetization mea-sured at 0.5 K in Fig. 1(b) shows an anomaly nearHC1 ∼ 1.6 T, which is more evident as a peak on thedM=dH curve. As previous studies reported, the magneticdomain reorientation occurs at this critical field HC1 and,above which, the helix spin structure is transformed into atilted conical one [20–23]. Because of the reorientation ofmagnetic domains, the magnetization displays hysteresiswhen the field is ramping down below HC1. This reorienta-tion is also revealed as an irreversibility between the ZFCand FC curves below 8 K for the susceptibility measured at0.01 T, while it is suppressed completely at H ≥ 1.7 T.The real part of ac susceptibility χ0 in Fig. 2(a) clearly

shows two peaks at HC1 and HC2. Here, HC1 is consistent

(a)

(b)

FIG. 1. (a) The temperature dependence of zero field cooling(ZFC) and field cooling (FC) dc magnetizations at differentapplied fields. (b) The dc magnetization measured at 0.5 K and itsdM=dH curve.

FIG. 2. The magnetic field dependence of ac susceptibility atseveral temperatures: (a) the real component; (b) the imaginarycomponent. The inset in (b) shows the enlargement of the high-field data. The arrows indicate the evolution of high-fieldanomalies with increasing temperatures.


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with the HC1 obtained from the magnetization data above.HC2 is consistent with the reportedHC2 value, above whichthe spiral spin structure is suppressed with a concomitantstructure transition from tetragonal to cubic. Meanwhile, asmall bump at HC1, a sharp peak at HC2, and a steplikeanomaly near 9.5 T are clearly seen for the imaginary part( χ00) measured at 7.5 K. This steplike anomaly is inaccordance with the plateau observed from the soundvelocity measurements around 10 T at 2 K, which hasbeen correlated to the onset of a fully polarized magneticphase at HC3 [21]. Upon further cooling, HC3 moves tolower fields and is hardly discernible below 1.5 K from theac susceptibility measurement, while HC2 shifts to higherfields [see the inset in Fig. 2(b)].At a zero magnetic field, the specific heat in Fig. 3(a)

shows a sharp peak at TN ¼ 21 K, which shifts to a lowertemperature with an increasing magnetic field anddisappears completely at 6.5 T. Moreover, a small low-temperature hump around 1–2 K is observed at a zeromagnetic field, which is enhanced with an increasing fieldup to 6.5 Tand then strongly suppressed at 10 T. This kind offield dependence is very different from the usual Schottkyanomaly of magnetic specific heat. Therefore, this anomalycould be originated from the spin fluctuations. It is con-sistent with the recent neutron-diffraction studies that revealbroad diffuse scattering due to spin fluctuations in the long-range-ordered state at temperatures down to 4K [22,23]. Thestrongest hump at 6.5 T suggests stronger spin fluctuationsaround this field. Below 1 K, we tend to fit the heat capacitydata at 6.5 Twith a γTα behavior. The obtained result is T2

down to the lowest temperature of 0.06 K. Here we assume

that the lattice contribution of specific heat at such lowtemperatures is negligible, and then theT2 behavior for 6.5 Tdata is abnormal for a 3D magnet.To further manifest the dynamic properties of the system

under the magnetic field, we carry out the thermal conduc-tivity measurement. As we depict in Fig. 3(b), the thermalconductivity κ at 0 T shows a structural-transition-relatedanomaly at TN ¼ 21 K and a strong weakness of the κðTÞslope around 1 K that should be related to the spin fluctua-tions observed from the specific heat. With an increasingmagnetic field, TN shifts to lower temperatures and dis-appears at H ≥ 5 T. The slope change around 1 K is notsensitive to the magnetic field but diminishes at H ≥ 6.5 T.While the κ mainly shows a gradual increase with anincreasing magnetic field at high temperatures (T > 3 K),its field dependence is complicated at low temperatures(T < 1 K), which is more clearly demonstrated in Fig. 3(c).At 1.95 K, the κðHÞ=κð0Þ curve in Fig. 3(c) shows three

weak anomalies at ∼1, 5.5, and 8 T, which correspond toHC1, HC2, and HC3, respectively. At HC1, a spin reorienta-tion appears, which is related to a minimizing of theanisotropy gap and a sudden increase of the AFM magnonexcitations. This could cause an enhancement of magnonscattering on phonons and the low-field decrease of κ. Thesecond anomaly atHC2, which becomes clearer at 0.97 K, isdemonstrated as a diplike suppression of κ and is likely dueto the spin fluctuations at HC2. The third anomaly at HC3,identified as a quicker increase of κ, is apparently due to thestrong suppression of spin fluctuations associated withthe transition or crossover from that unidentified regimeto the fully polarized spin state. The spin fluctuations arestrongly suppressed in the fully polarized spin state, becausethe spin excitation is gapped at low energies. At lowertemperatures that were not accessed in the previous experi-ments [20–23], the anomalies atHC2 andHC3 tend tomerge,consistent with the opposite temperature dependencies ofthese two critical fields observed from our ac susceptibilitymeasurement. In particular, at 0.5 K these two anomaliesmerge into a single one at 6.5 T, and the κðHÞ=κð0Þ curveshows a deep valley at the background of field-inducedenhancement. This is consistent with the specific heat resultshowing that the spin fluctuation is the strongest around6.5 T. As wewill explain in detail, both the specific heat andthe thermal transport results suggest the existence of thequantum criticality at 6.5 T.Before getting onto our interpretation, we here calculate

the phonon mean free path from κ using a standard method[30]. We choose the Debye temperature to be 308 K [31]and assume κ is primarily phononic. The results aredepicted in Fig. 3(d). At 0 T, the phonon mean free path(l ∼ 10−2 mm) is nearly 2 orders of magnitude smaller thanthe sample size (∼1 mm) even at the lowest temperature of0.3 K. This means that the phonon scattering is still activeat such low temperatures. Since the phonon scatteringscaused by phonons, impurities, and other crystal defects are

FIG. 3. (a) The temperature dependence of the specific heat atseveral magnetic fields from 0.06 to 30 K. The dashed linerepresents the T2 dependence. (b) The temperature dependence ofthe thermal conductivity from 0.3 to 30 K at various magneticfields. (c) The field dependence of the thermal conductivity atselected temperatures below 2 K. (d) The calculated meanfree path.


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known to be quenched at low temperatures, there must besome magnetic scattering processes. Also, because of thesmall mean free path l, the magnetic excitations are notlikely to make a sizable contribution to the heat transport.With increasing magnetic fields, l is generally enhanced,indicating a suppression of magnetic scatterings. Under thehighest field of 14 T, the phonon mean free path approachesthe sample size, which indicates the complete suppressionof spin fluctuations in the polarized state. This is consistentwith the gapped spin excitations for the fully polarized spinstate. In contrast, at 6.5 T, l drops back to 10−2 mm sizewith no obvious temperature dependence below 1 K.A detailed H-T phase diagram of ZnCr2Se4 was con-

structed in Fig. 4 by using the phase transition temperaturesand critical fields obtained from our above measurements.By comparing to the reported phase diagram [21], twoimportant new features were observed in this full phasediagram with lower temperatures and higher magneticfields. One is that the phase transition temperature forthe spiral spin structure is suppressed to a zero temperaturewith increasing fields before the system enters the fullypolarized state. Therefore, there is a direct quantum phasetransition between the spiral spin state and the polarizedphase, and this transition is marked as the QCP in Fig. 4.The other one is that the previous unidentified regimebetween the spiral state and the fully polarized state doesnot persist down to the lowest temperature. Note that ourmeasurements were carried out at a much lower temper-ature than the previous reports. Thus, in Fig. 4, thispreviously unidentified regime is naturally identified asthe quantum critical regime that is the finite temperatureextension of the quantum criticality.

Why is the previously unidentified regimenot an umbrellastate or a spin nematic state? As we have pointed out earlier,both states break the spin rotational symmetry, and theformermay break the lattice translation. This is a 3D system,and this kind of symmetry breaking should persist down to azero temperature and cover a finite parameter regime. Thisfinite-range phase is not observed at the lowest temperature.For the same reason, the symmetry should be restored athigh enough temperatures via a phase transition. Such athermodynamic phase transition is clearly not observed inthe heat capacity and thermal transport measurements.The spin spiral state and the fully polarized state are distinct

phases with different symmetry properties. The latter istranslational invariant and fully gapped, while the formerbreaks the lattice symmetry and spin rotational symmetry.There must be a phase transition separating them, and thisquantum phase transition is manifested as the QCP at 6.5 T inFig. 4. What is the property of this criticality? The heatcapacity was found to behave as T2 at low temperatures at theQCP, indicatingamuch larger density of states than the simpleGaussian fixed point. For a Gaussian fixed point, we wouldexpect the heat capacity as T3 up to a logarithmic correctiondue to the critical fluctuations. The T2 heat capacity suggeststhat the low-energy density of states should scale asDðϵÞ ∼ ϵwith the energy ϵ.We know that the nodal line semimetal withsymmetry and topologically protected line degeneracies hasthis extended density of states when the Fermi energy is tunedto the degenerate point [32]. However, our system is purelybosonic with spin degrees of freedom, and there is noemergent fermionic statistics. To support DðϵÞ ∼ ϵ at theQCP, we would have the critical modes be degenerate oralmost degenerate along the lines in the reciprocal space suchthat the current thermodynamic measurement cannot resolvethem. It has been known that the frustrated spin interactionscould lead to such line degeneracies for the critical modes andthe resulting frustrated quantum criticality [33,34]. Thepossibility that infinite modes with line degeneracies becomecritical at the same time is an unconventional feature of thisQCP. These critical modes scatter the phonon strongly andsuppress the thermal transport near the criticality. It will beinteresting to directly probe these degenerate modes withinelastic neutron scattering and explore the fates of the criticalmodes on both sides of theQCP. Our thermal transport resultsalso call for further theoretical effects on the scatteringbetween the extended density of critical modes and thelow-energy phonons near the criticality.Finally, the system displays different lattice structures for

different magnetic phases in the phase diagram. Both thehelix and the spiral spin states have tetragonal structure,while the quantum critical regime and the fully polarizedstate have cubic structure. This is simply the consequence ofthe spin-lattice coupling. The helix and the spiral spin statesbreak the lattice cubic symmetry, and this symmetry istransmitted to the lattice via the spin-lattice coupling. Thequantum critical regime and the fully polarized state are

FIG. 4. TheH-T phase diagram of ZnCr2Se4. “T” and “C” referto the tetragonal and the cubic structure, respectively. “Helix,”“Spiral,” and “FM” stand for the helix spin state, spiral spin state,and spin-fully polarized state, respectively. A QCP is deducedbetween the spiral spin state and the polarized phase. The solid(dashed) boundary refers to an actual phase transition (crossover).The pink region is marked as the quantum critical regime. See themain text for a detailed discussion.


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uniform states and restore the lattice symmetry. The corre-lation between the sound velocity and themagnetic structurein the previous experiments has a similar origin [21].In summary, by completing the H-T phase diagram of

ZnCr2Se4, we demonstrate the existence of the QCP andquantum critical regime induced by an applied magneticphase in this 3D magnet. Our finding of the unconventionalquantum criticality calls for future works and is likely toprovide a unique example of frustrated quantum criticalityfor further studies.

This research was supported by the National KeyResearch and Development Program of China (GrantNo. 2016YFA0401804), the National Natural ScienceFoundation of China (Grants No. 11574323 andNo. U1632275), and the Natural Science Foundation ofAnhui Province (Grant No. 1708085QA19). X. F. S.acknowledges support from the National Natural ScienceFoundation of China (Grants No. 11374277 andNo. U1532147), the National Basic Research Programof China (Grants No. 2015CB921201 andNo. 2016YFA0300103), and the Innovative Program ofDevelopment Foundation of Hefei Center for PhysicalScience and Technology. G. C. thanks the support fromtheMinistry of Science and Technology of China with GrantNo. 2016YFA0301001, the initiative research funds and theprogram of first-class University construction of FudanUniversity, and the Thousand-Youth-Talent Program ofChina. H. D. Z. thanks the support from the Ministry ofScience and Technology of China with GrantNo. 2016YFA0300500 and from NSF-DMR with GrantNo. NSF-DMR-1350002. Z. Y. Z. acknowledges supportfrom the National Natural Science Foundation of China(Grant No. 51702320). M. L. and E. S. C. acknowledgesupport from NSF-DMR-1309146. The work at NHMFLis supported byNSF-DMR-1157490 and theState of Florida.The x-ray work was performed at HPCAT (Sector 16),Advanced Photon Source, Argonne National Laboratory.HPCAT operations are supported by DOE-NNSA underAward No. DE-NA0001974 and DOE-BES under AwardNo. DE-FG02-99ER45775, with partial instrumentationfunding by NSF. The Advanced Photon Source is a U.S.Department of Energy (DOE)Office of ScienceUser Facilityoperated for theDOEOffice of Science byArgonneNationalLaboratory under Contract No. DE-AC02-06CH11357.

C. C. G. and Z. Y. Z. contributed equally to this work.

*[email protected]†[email protected]‡[email protected]§[email protected]

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Tunable Quantum Spin Liquidity in the 1=6th-Filled Breathing Kagome Lattice

A. Akbari-Sharbaf,1 R. Sinclair,2 A. Verrier,1 D. Ziat,1 H. D. Zhou,3,2 X. F. Sun,4,5,6 and J. A. Quilliam1,*

1Institut Quantique and Departement de Physique, Universite de Sherbrooke,2500 boulevard de l’Universite, Sherbrooke, Quebec J1K 2R1, Canada

2Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA3Key Laboratory of Artificial Structures and Quantum Control, Ministry of Education, School of Physics and Astronomy,

Shanghai JiaoTong University, Shanghai 200240, China4Department of Physics, Hefei National Laboratory for Physical Sciences at Microscale and Key Laboratory of Strongly-CoupledQuantum Matter Physics, CAS, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China5Institute of Physical Science and Information Technology, Anhui University, Hefei, Anhui 230601, People’s Republic of China

6Collaborative Innovation Center of Advanced Microstructures, Nanjing University,Nanjing, Jiangsu 210093, People’s Republic of China

(Received 9 August 2017; revised manuscript received 23 March 2018; published 29 May 2018)

We present measurements on a series of materials, Li2In1−xScxMo3O8, that can be described as a1=6th-filled breathing kagome lattice. Substituting Sc for In generates chemical pressure which alters thebreathing parameter nonmonotonically. Muon spin rotation experiments show that this chemical pressuretunes the system from antiferromagnetic long range order to a quantum spin liquid phase. A strongcorrelation with the breathing parameter implies that it is the dominant parameter controlling the levelof magnetic frustration, with increased kagome symmetry generating the quantum spin liquid phase.Magnetic susceptibility measurements suggest that this is related to distinct types of charge order inducedby changes in lattice symmetry, in line with the theory of Chen et al. [Phys. Rev. B 93, 245134 (2016)]. Thespecific heat for samples at intermediate Sc concentration, which have the minimum breathing parameter,show consistency with the predicted Uð1Þ quantum spin liquid.


One of the most sought after magnetic phases is thequantum spin liquid (QSL), wherein spins form a highlyentangled quantum ground state that supports fractional spinexcitations [1]. Two main approaches to the discovery ofQSL materials have been especially fruitful in recent years:spin-1=2 kagome antiferromagnets [2–4] and triangular-lattice antiferromagnets near a Mott transition [5–10].However, much remains to be understood about theseexperimental QSL candidates, and some properties remaindifficult to reconcile with theory [8,11,12]. Hence, the searchfor new QSL candidates based on different mechanisms, forexample, [13,14], remains a valuable pursuit. In particular,systems in which Hamiltonian parameters can be continu-ously tuned may provide a prime opportunity to linktheoretical models to experimental phenomena.In this Letter, we demonstrate that a high degree of

tunability can be achieved with the materialsLi2In1−xScxMo3O8 that incorporate both spin and chargedegrees of freedom. This family of materials consists of a“breathing” kagome lattice (BKL) of Mo ions wherein thetriangles that point upward are slightly smaller than thosethat point downward [13,15,16], with a “breathing ratio”λ ¼ d∇=dΔ. In these particular materials, the lattice is1=6th filled, with one unpaired electron for every threeMo sites, and its insulating character is ensured by strong

next-nearest-neighbor interactions (V1 on up triangles andV2 on down triangles) [17].As proposed by Sheckelton et al. [17] for LiZn2Mo3O8

(LZMO), a similar QSL candidate material [18], a plausiblecharge configuration consists of each electron delocalizedover one “up triangle”, ultimately leading to a triangularlattice of spin-1=2 moments on Mo3O13 clusters, asdepicted in Fig. 1(a). However, it has been pointed out[19] that, due to the large spatial extent of the 4d electrons,the single unpaired electrons may have a nonzero proba-bility of tunneling between adjacent clusters. When λ islarge, the electrons are expected to localize on the smallertriangles, recovering the type-I cluster Mott insulator (CMI)proposed by Sheckelton et al. [17]. When V2 becomescomparable to V1, it is energetically favorable for electronsto collectively tunnel between the small triangles, givingrise to a long range plaquette charge order (PCO), or type-IICMI, as depicted in Fig. 1(b). We show that x inLi2In1−xScxMo3O8 tunes the system from a long rangeordered (LRO) magnetic phase to a QSL phase and proposethat these distinct magnetic phases are a result of thedistinct charge configurations. Although the end points ofthis family (at x ¼ 0 and x ¼ 1) have been studiedpreviously [20–22], we show that intermediate stoichiom-etries are essential to generating a homogeneous QSL.



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Our experimental results agree well with the theoreticalframework developed by Chen et al. [19] and highlight avaluable new system for the study of QSL physics.Polycrystalline samples of Li2In1−xScxMo3O8 were

synthesized by solid-state reaction. A stoichiometric mix-ture of Li2MoO4, Sc2O3, In2O3, MoO3, and Mo wereground together and pressed into 6 mm diameter, 60 mmlong rods under 400 atm hydrostatic pressure which wereplaced in alumina crucibles and sealed in silica tubes at apressure of 10−4 mbar. Finally, the samples were annealedfor 48 h at 850 °C. Powder X-ray diffraction (XRD) patternswere recorded at room temperature with a HUBER imagingplate Guinier camera 670 with Ge monochromatized Cu Kα1 radiation (1.54059 Å). Mo─Mo bond lengths wererefined by the Rietveld method [23] with χ2 in the rangeof 1–2 for all samples. Susceptibility measurements wereperformed at 2 T, and specific heat measurements werecarried out in zero field (ZF), with Quantum Design MPMSand PPMS systems. Muon spin rotation (μSR) measure-ments were carried out at TRIUMF in ZF and longitudinalfield. Measurements in the range from 25 mK up to 3 K wereperformed with the samples affixed to a Ag cold fingerof a dilution refrigerator. Higher temperature measurementswere carried out in veto mode to eliminate the backgroundasymmetry and were used to correct for the backgroundpresent at low temperatures.XRD spectra [24] reveal that as the In ions are replaced

by smaller Sc ions the lattice parameters decrease, and asseen in Fig. 1(c), the ratio a=c varies monotonically with atotal change of about 1.4%. It is important to investigate the

evolution of the breathing parameter with x, and the XRDmeasurements reveal a nonmonotonic behavior of λðxÞ, ascan be seen in Fig. 1(d). The parent compound (x ¼ 0) hasthe highest average degree of asymmetry, whereas at aconcentration of 60% In and 40% Sc (x ¼ 0.6) the lowestdegree of asymmetry is attained. Meanwhile, the reportedstructure of LZMO [17] corresponds to a breathing param-eter of λ ≃ 1.23, making it closer to an ideal kagome latticethan the most symmetric sample in the series studied here.In general, the μSR polarization measured for our

samples shows that the muon spins are influenced by amix of fluctuating and static electron spins, and the dataare fitted with a two-component polarization functionPtot ¼ fPSðtÞ þ ð1 − fÞPDðtÞ, where PSðtÞ is the polari-zation for the fraction f of muons stopping in a staticfraction (ordered or frozen) and PDðtÞ is the contributionfrom regions with dynamic electron spins, either QSL orparamagnetic phases. For the dynamic fraction, PDðtÞ ¼PNðtÞe−t=T1 , where PNðtÞ is a nuclear Gaussian Kubo-Toyabe function and 1=T1 is the spin-lattice relaxation rate.The ZF μSR asymmetry measured at 1.9 K for

Li2InMo3O8 (x ¼ 0) shown in Fig. 2(a) features a slowlydecaying oscillation, demonstrating LRO with well definedinternal fields consistent with NMR measurements of thesame stoichiometry [20]. Here, PSðtÞ for this sample hasthus been fitted to the static Lorentzian Koptev-Tarasovpolarization function. Four distinct frequencies (1.1, 1.4, 2.0,and 3.3 MHz) are extracted, which correspond well to thefour inequivalent oxygen sites. Select polarization curves atdifferent temperatures in Fig. 2(a) show the reduction of theoscillation frequencies (and order parameter) and the appear-ance of a dynamic fraction of the sample as the temperatureis raised. The smallness of the observed frequencies is

(a)

(b)

(c)

(d)

0.552

0.554

0.556

0.558

0.56

0.562

0 0.2 0.4 0.6 0.8 11.255

1.26

1.265

1.27

1.275

FIG. 1. Illustrations of (a) the type-I cluster Mott insulator,where electrons are localized on Mo3 units, leading to 120°antiferromagnetic order and (b) the PCO state. Resonatinghexagons are depicted by dashed circles, and the two spatialconfigurations of the collective tunneling electrons are depictedby the open and full circles. (c) Ratio of lattice parameters, a=cand (d) breathing parameter λ as a function of x. The shadedregion is a guide to the eye. Error bars from the refinement aresmaller than the data points.

0

0.2

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0 1 2 76543 8

(a) (b)

(c)

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1

55 G

100 G

2065 G(d)

0 1 2 76543 8

2080 G

55 G

5.5 G

ZF ZF

FIG. 2. (a) Zero-field muon spin polarization PðtÞ forLi2InMo3O8 (x ¼ 0). (b) Zero-field PðtÞ measured at 25 mKfor LiIn1−xScxMo3O8 for different values of x. Polarization invarious longitudinal fields for (c) x ¼ 0.6 and (d) x ¼ 0.2. Blacklines are fits as described in the text.


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consistent with each spin-1=2 moment being highly distrib-uted over a Mo3O13 cluster, similar to observations insystems of mixed-valence Ru dimers [25].For x ¼ 0.2, x ¼ 0.4, and x ¼ 1, we find an inhomo-

geneous mix of disordered static magnetism (giving aquickly relaxing signal) and a weakly relaxing dynamicfraction as shown in Fig. 2(b). The frozen fractionrepresents 49%, 25%, and 43% of these samples, respec-tively. On the other hand, PðtÞ for x ¼ 0.6 shows noindication of static fields originating from electron spins toas low as 25 mK, which suggests that the entire sample is ina homogeneous QSL phase. In fact, the μSR asymmetryprofile for x ¼ 0.6 is very similar to that of LZMO [26].

To fit the inhomogeneous samples, a Lorentzian Kubo-Toyabe function was used for PSðtÞ [27]. This fitting hasbeen performed in zero and longitudinal field, BL, as shownin Fig. 2(d) and in the Supplemental Material [24]. Thisanalysis conclusively demonstrates that we have correctlyidentified the frozen and dynamic fractions of the sample

since the muon spins are much more quickly decoupledfrom static than dynamic magnetism. For the homogeneousQSL sample, small BL quickly decouples the muon spinsfrom the nuclear moments, but higher field relaxationpersists, indicating that the relaxation is purely of dynamicorigin, as seen in Fig. 2(c). As shown in Fig. 3(a),1=T1ðBLÞ for the QSL fractions is fairly well fit withRedfield theory [28] using a sum of two characteristicfluctuation frequencies. Meanwhile, 1=T1 of the liquidfractions shows relaxation plateaus at temperatures below∼1 K, a common but still poorly understood feature of QSLcandidates [3,29–31].Evidently, the concentration of Sc does not monotoni-

cally change the ratio of static and QSL fractions, but ratherthere is an optimal concentration of x ¼ 0.6 where ahomogeneous QSL is stabilized. The phase diagram as afunction of x, presented in Fig. 4(c), is highly correlatedwith the behavior of the breathing parameter λðxÞ as shownin Fig. 1(d). This suggests that the magnetic phenomenol-ogy of this material is intimately connected to the sym-metry of the BKL and that past a critical value of λ thesystem passes from antiferromagnetic (AFM) to QSL.At critical values of λ, such as for x ¼ 0.2 and x ¼ 1,inhomogeneous phases result.The way in which λ influences the charge degrees of

freedom, and consequently the spins, may be better under-stood with the magnetic susceptibility χ measurements inFig. 4(a). Our measurements of the end points of the series(x ¼ 0 and x ¼ 1) are consistent with previous work [20].For intermediate concentrations, χðTÞ is very different. Forthe homogeneous QSL sample (x ¼ 0.6), χ−1ðTÞ displaystwo apparent linear Curie-Weiss regimes distinguishedby different Curie constants and a smooth crossoverbetween the two regimes. The x ¼ 0.4 and x ¼ 0.8 samplesshow similar behavior [24]. This strong, qualitative change

100

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0.12(a) (b)

FIG. 3. (a) Spin-lattice relaxation rate vs longitudinal field atbase temperature for the liquid phase of several samples, with fitsgiven by Redfield theory with two different fluctuation frequen-cies. (b) Relaxation rate as a function of temperature in longi-tudinal field of 55 G, showing relaxation plateaus typical of QSLmaterials. Curves are guides to the eye.

0

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FIG. 4. (a) Temperature dependent inverse magnetic susceptibility χ−1 for select samples. For x ¼ 0, a sharp feature at the onset ofAFM order is indicated by an arrow at 11 K. Here, χ−1ðTÞ for the homogeneous QSL sample, x ¼ 0.6, shows two apparent Curie-Weissregimes. The fit is described in the text. (b) Magnetic specific heatCM of select samples. The fit to the x ¼ 0 data is a T3 power law plus aCN ∝ T−2 nuclear contribution. The specific heat of x ¼ 0.6 is compared with a T2=3 power law plus nuclear contribution, as well as a T-linear dependence. (c) Magnetic phase diagram for Li2In1−xScxMo3O8. Red squares show the onset of freezing determined by specificheat (for x ¼ 0 and 0.2) μSR (for the remaining samples). The dark red region shows AFM ordering, whereas pink regions show spinfreezing, either spin glass (SG) or disordered antiferromagnetism. The blue region shows the approximate temperature onset of therelaxation plateau in μSR.


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in χðTÞ, even at high temperatures, implies that the effectof x on the magnetic ground state is not simply an effectof disorder.The temperature dependence of the susceptibility has

been a central focus of the discussion surrounding theMo3O13 cluster magnet family. Sheckelton et al. firstreported two Curie-Weiss regimes for the compoundLZMO, where the Curie constant reduces to 1=3 of thehigh temperature value below a crossover at 96 K [17].They attributed this to the condensation of 2=3 of the spinsinto singlets [17,26]. Chen et al. [19] proposed an alter-native theory for the “1=3-anomaly” in χ−1ðTÞ, wherebythe low temperature regime corresponds to plaquette chargeorder. The PCO reconstructs the spinon bands with thelowest band splitting into three sub-bands. The lowest sub-band is completely filled with 2=3 of the spinons, becomingmagnetically inert. The upper sub-band is partially filledwith the remaining 1=3 spinons, and these spinons con-tribute to χ. Chen et al. [19,32] argue that at the crossovertemperature PCO is destroyed, and the full spin degrees offreedom are recovered. However, a transition between thesetwo phases involves a spontaneous breaking of symmetryand should normally give rise to sharp thermodynamicfeatures, the absence of which has been attributed todisorder [19].We propose an alternative mechanism for the 1=3-

anomaly. If the compounds x ¼ 0.6 and LiZn2Mo3O8

are in the strong PCO regime, the energy scale requiredto break the PCO (EPCO ∼ t31=V

22) ought to be significantly

larger than the energy gap ΔE between filled and partiallyfilled spinon sub-bands (which is governed by the next-nearest-neighbor exchange interaction J0), allowing forthermal excitation of spinons across the spinon gap whilepreserving PCO [33]. From a local perspective, eachresonating hexagon in the PCO phase is composed ofthree coupled spins with a Stot ¼ 1=2 ground state manifoldand a Stot ¼ 3=2 excited state. The magnetic susceptibilityfor noninteracting hexagons can be written as

χ0 ¼μ0NAg2μ2β4kBT

1þ 5e−ΔE=kBT

1þ e−ΔE=kBT¼ βðTÞC0

T: ð1Þ

The Stot ¼ 1=2 ground state is doubly degenerate dueto a pseudospin that represents the spatial configurationof entanglement in the resonating hexagon [19]. In a mean-field approximation, the interacting susceptibility thengives χ¼βðTÞC0=½T−βðTÞθW �, naturally leading to twoCurie-Weiss regimes with a ratio of 1=3 between theeffective Curie constants Ceff ¼ βðTÞC0.Equation (1) gives an excellent fit of χ−1ðTÞ measured

for sample x ¼ 0.6, shown in Fig. 4(a), where theparameters extracted from the fit are C0 ¼ 0.264�0.001 emuKOe−1mol−1, ΔE=kB ¼ 109� 1 K, and θW ¼−46.3� 0.5 K. A fit of Eq. (6) to the susceptibility datareported for LiZn2Mo3O8 [17] is also successful (see the

Supplemental Material [24]), with fitting parameters C0 ¼0.277� 0.002 emuKOe−1 mol−1, ΔE=kB ¼ 300� 20 K,and θW ¼ −20� 10 K. The same analysis can also beapplied to other samples that are primarily spin liquids(x ¼ 0.4 and x ¼ 0.8) giving slightly different energy gaps.

The magnetic specific heat, after lattice subtraction, forselect samples is displayed in Fig. 4(b). As expected forLRO, the x ¼ 0 sample displays a peak at TN ≃ 12 K, andthe appropriate power law CM ∝ T3 for gapless magnons.Below 1 K, the specific heat turns upward with a T−2 powerlaw which we attribute to the upper limit of a nuclearSchottky anomaly CN, likely originating from the 95Mo and97Mo hyperfine couplings since the quadrupolar energy of115In is not large enough [22].For samples that are primarily or entirely QSL (x ¼ 0.4,

0.6, and 0.8), there is no sharp peak, and theCMðTÞ is muchshallower. Between 1 and 10 K, CM ∝ T, but below 1 K,CM becomes even shallower than linear. This shallowtemperature dependence of the specific heat in the order-free phase of this series of materials lends further evidencefor a Uð1Þ QSL as predicted [5,19,34]. It can be seen inFig. 4(b) that if we apply the same nuclear contribution tothe specific heat for the x ¼ 0.6 sample as was determinedfor the x ¼ 0 sample; a T2=3 power law provides areasonable fit to the data below ∼2 K. Hence, it is temptingto propose that this intermediate concentration has a Uð1Þspin liquid state, similar to what has been proposed for thetriangular organic QSLs [5–10], although there CM ∝ T isobserved [7,9]. For x ¼ 1, a somewhat steeper CM ∼ T1.4 isobserved similar to the T1.5 power law obtained inRef. [22]. The mixture of QSL and magnetic freezingmay lead to an intermediate temperature dependence.In conclusion, we have demonstrated a high degree of

tunability of the series Li2In1−xScxMo3O8 through isovalantsubstitution of In with Sc. The magnetic phase diagram,Fig. 4(c), shows a strong correlation with the breathingparameter, with a homogeneous QSL phase in the mostsymmetric sample at x ¼ 0.6, suggesting that λ is theprincipal controlling parameter. The nature of χðTÞ alsovaries substantially with x. Notably, in the range of 0.4 <x < 0.8, χ−1ðTÞ is very similar to that of the QSL LZMO,with two apparent Curie-Weiss regimes. This observationfits well with the theory of Chen et al. [19] predicting the1=3-anomaly in the PCO phase, which should be stabilizedby small λ. We propose that the 1=3-anomaly originates fromthermal excitations of the resonating hexagons from theStot ¼ 1=2 ground state to a Stot ¼ 3=2 excited state. Sincesmaller λ and the 1=3-anomaly seem to be associated with aQSL ground state, the spins in the PCO phase appear to bemore frustrated than in the type-I CMI. Indeed, the specificheat of the homogeneous QSL at x ¼ 0.6 has a particularlyshallow temperature dependence, possibly consistent with aUð1Þ QSL [5,34].This work has therefore provided a likely resolution to

the debate surrounding LZMO [17]. An alternative scenario


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to explain the 1=3-anomaly in LZMO has been put forwardby Flint and Lee [35], wherein the electrons are localizedon the up triangles but two thirds of the clusters rotate,generating an emergent honeycomb lattice, thereby leaving1=3 of the spins as weakly connected “orphan” spins.However, we find no natural reason that changes in λwouldencourage rotation of Mo3O13 clusters, and the 1=3 of thespins that remain active at low temperature exhibit astrongly negative Curie-Weiss constant ΘW ≃ −46 K,meaning they cannot be described as orphan spins.Valuable future work on this series could include direct

measurements of charge order with resonant X-ray spec-troscopy, although the changes in local charge density willbe rather small, as well as a search for thermodynamicindications of charge ordering at higher temperatures.Furthermore, it would be interesting to study the parentcompounds under applied pressure instead of chemicalpressure, potentially tuning the system into a QSL phasewithout introducing structural disorder. Indeed the role ofdisorder in either destabilizing or even generating QSL-likephases remains a contentious issue in the field [36].Furthermore, although the model proposed by Chen et al.[19] is consistent with our observations, many assumptionshave been made regarding the appropriate Hamiltonian forthese materials which should be validated with detailedelectronic structure calculations.

We are grateful to the staff of the Centre for Molecular andMaterials Science at TRIUMF for extensive technical sup-port, in particular, G. Morris, B. Hitti, D. Arseneau, andI. MacKenzie. We also acknowledge helpful conver-sations with Y. B. Kim, G. Chen, H.-Y. Kee, M. Gingras,A.-M. Tremblay, F. Bert, and P. Mendels. A. A.-S. andJ. Q. acknowledge funding through NSERC, FRQNT, CFI,and CFREF grants. H. D. Z. acknowledges support from theMinistry of Science and Technology of China with GrantNo. 2016YFA0300500. R. S. and H. D. Z. acknowledgesupport from NSF-DMR with Grant No. NSF-DMR-1350002. X. F. S. acknowledges support from the NationalNatural Science Foundation of China (Grants No. 11374277and No. U1532147) and the National Basic ResearchProgram of China (Grants No. 2015CB921201 andNo. 2016YFA0300103).

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Transitions from a Kondo-like diamagnetic insulatorinto a modulated ferromagnetic metal in FeGa3−yGeyYao Zhanga,b,1, Jie-Sheng Chenb, Jie Mac,d, Jiamin Nie,f,g, Masaki Imaia, Chishiro Michiokaa, Yuta Hadanoh,Marcos A. Avilai, Toshiro Takabatakeh, Shiyan Lie,f,g, and Kazuyoshi Yoshimuraa,j,1

aDepartment of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan; bSchool of Chemistry and Chemical Engineering, ShanghaiJiao Tong University, Shanghai, 200240, China; cKey Laboratory of Artificial Structures and Quantum Control, Shanghai Jiao Tong University, Shanghai200240, China; dSchool of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China; eState Key Laboratory of Surface Physics, FudanUniversity, Shanghai 200433, China; fDepartment of Physics, Fudan University, Shanghai 200433, China; gLaboratory of Advanced Materials, FudanUniversity, Shanghai 200433, China; hGraduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima, 739-8530, Japan; andiCentro de Ciencias Naturais e Humanas, Universidade Federal do ABC, Santo Andre-SP, 09210-580, Brazil; and jResearch Center for Low Temperature andMaterials Sciences, Kyoto University, Kyoto 606-8502, Japan

Edited by Zachary Fisk, University of California, Irvine, CA, and approved February 13, 2018 (received for review August 2, 2017)

One initial and essential question of magnetism is whether themagnetic properties of a material are governed by localizedmoments or itinerant electrons. Here, we expose the case for theweakly ferromagnetic system FeGa3−yGey , wherein these twoopposite models are reconciled, such that the magnetic susceptibil-ity is quantitatively explained by taking into account the effects ofspin–spin correlation. With the electron doping introduced by Gesubstitution, the diamagnetic insulating parent compound FeGa3becomes a paramagnetic metal as early as at y = 0.01, and turnsinto a weakly ferromagnetic metal around the quantum criticalpoint y = 0.15. Within the ferromagnetic regime of FeGa3−yGey ,the magnetic properties are of a weakly itinerant ferromagneticnature, located in the intermediate regime between the localizedand the itinerant dominance. Our analysis implies a potential uni-versality for all itinerant-electron ferromagnets.

spin fluctuations | modulated ferromagnetism | phase transition

Magnetic materials are of particular importance in funda-mental theoretical studies as well as advanced technical

applications due to the subtle correlations within the systems(1–5). Understanding the entanglement of microscopic alignmentof spin moments, i.e., the ingredients of magnetic mechanisms, iscrucially important. Spin fluctuations in many-body systems are ofsuch importance, as they have proven to play key roles leading tothe formation of “strange metal,” non-Fermi liquids, and quan-tum effects extending to high temperatures (6, 7). In the cuprateand iron-based superconductors, the essential pairing interactionis proved to be mediated by the spin fluctuations as a commonthread in the unconventional superconductors (8, 9).

Although ferromagnetism is one of the oldest observed andstudied quantum phenomena, the exact mechanism throughwhich it emerges is not fully understood. Well-established the-ories are restricted to two narrow extremes, i.e., the localizedand itinerant-electron regimes. Although great effort includingHartree–Fock approximation (HFA) and random phase approx-imation (RPA) in magnetic theories has been made to elucidatethe magnetic properties in the intermediate range of the twoopposite extremes (10–14), a successful theory remains elusive.The HFA underestimates the amplitude of fermion thermal exci-tations and the spin density fluctuations due to its oversimplifiedassumption that the thermal spin-flip excited electrons and holesmove independently under a static mean field (15, 16). The RPAtheory in terms of an oscillating molecular field, however, onlytakes into proper account the effects of spin waves as elementaryexcitations around the equilibrium state, missing the correlationsbetween the excited modes of fluctuations (1, 17, 18). A recentpicture of the hybrid nature of localized moments and itinerantelectrons was explored in several systems, on which the hybridmodel in a two-band approximation was proposed to illustratethe magnetism and, in some cases, the origin of superconduc-

tivity (19–24). The self-consistent renormalization (SCR) theoryof spin fluctuations and related theories successfully approachesthe localized regime based on the itinerant picture as an inter-mediate mechanism in the one-band model by mediating themagnetic momentum of itinerant electrons in terms of wave-number-dependent spin fluctuation and generalized dynamicalfluctuations (18, 25–27). Despite this, however, a unified dynam-ical theory is still being debated, particularly due to the limiteddiversity of materials for further study, as well as the difficulty inreconciling these two polar extremes (28).

The heavy-fermion Kondo insulators provide a good platformto explore the physical properties, including magnetic order-ing due to the coupling of the charge dynamics to the com-ponent ordering associated with its related fluctuations duringa metal–insulator transition. The Kondo-insulator-like semicon-ductor, FeGa3, which has a larger pseudogap compared with thetypical Kondo insulators, is such an ideal system owing to itsexpected valence admixture (29, 30). The energy gap of FeGa3

is ∼0.4 eV, and its pseudogap is assumed to be formed by thestrong hybridization between the 3d band of Fe and 4p bandof Ga (31, 32). No magnetic ordering is detected in FeGa3 by

Significance

A unifying mechanism for the origin of ferromagnetism whichtakes into account both well-known and polar extreme the-ories, namely, the localized- and itinerant-electron regimes,remains a longstanding mystery. Here, we study a particularlyinteresting system of FeGa3−yGey , within which the magneticorderings can be adjusted by the Ge-electron filling control,turning from itinerancy to adequate localized through a fer-romagnetic quantum critical transition with increasing y. Byinvolving the effects of the spin fluctuations in general, thetheoretical estimations of the magnetic properties at the inter-mediate state of magnetic orderings formulated in terms ofthe correlation between itinerancy electrons and the spin fluc-tuations are in quantitative agreements with experimentalobservations. Our analysis shows a potential universality toall itinerant ferromagnets.

Author contributions: Y.Z., J.-S.C., C.M., and K.Y. designed research; Y.Z., J.M., J.N., M.I.,Y.H., M.A.A., T.T., and S.L. performed research; Y.Z. analyzed data; and Y.Z. and M.A.A.wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 To whom correspondence may be addressed. Email: [email protected] or [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1713662115/-/DCSupplemental.

Published online March 12, 2018.

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57Fe Mossbauer experiments (32). The nonmagnetic FeGa3 isreminiscent of another Fe-based Kondo insulator, FeSi, whichhas drawn attention for decades. Interesting physical proper-ties, including anomalous Hall effect, strong magnetic resis-tivity, chiral magnetic nature, and reentrant spin glass behav-iors, are induced by electron doping in FeSi, whose novelphenomena are represented by the Dzyaloshinski–Moriya inter-action, conventional isotropic exchange, anisotropic exchanges,the Zeeman interaction under applied field, and cubic anisotropyeffects (33, 34). In this work, we follow the phase transitions byincreasing the Ge doping of FeGa3−yGey , through all of whichthe magnetic properties are significantly affected. The marginalFermi-liquid behavior is observed on the verge of the transi-tion. The magnetic properties in the ferromagnetic region cross-ing over between the localized model and itinerant regimes ofFeGa3−yGey are reconcilably interpreted by mainly taking intoaccount the effects of temperature-dependent spin fluctuationsin general. Spin fluctuations may play a key role in approach-ing a unified theory for localized and itinerant ferromagnets, andFeGa3−yGey seems to be one of the best candidates for probingthe unified theory for itinerant magnetism.

ResultsPhase Transitions. Fig. 1 displays the phase diagram of FeGa3−y

Gey obtained on the basis of the magnetic and transport mea-surements (Figs. S1–S3, and S7). Increasing Ge substitution forGa resulted in FeGa3−yGey turning from a diamagnetic insula-tor into a paramagnetic metal as early as at y = 0.01, and even-tually into a ferromagnetic metal from y > 0.15. The relativelysmall spontaneous magnetic moment PS indicated a weakly fer-romagnetic nature in FeGa3−yGey (Fig. S1); the effective mag-netic moment Peff displayed a weak y dependence as an itin-erant ferromagnet (Fig. S2). The marginal Fermi-liquid statefor composition on the border of d-metallic ferromagnetism isobserved in Figs. S3 and S4, as the electrical resistivity and thespecific heat divided by temperature for y = 0.15 followed theT 5/3 and −lnT dependences, when the temperature was suffi-ciently lower than the characteristic spin fluctuation energy (7,27, 35). The marginal Fermi-liquid behavior was led by the scat-tering of fermions from the spin fluctuation-induced exchangemagnetic field on the verge of metallic ferromagnetism. As themetallization of FeSi1−xGex occurring at x = 0.25 was ordersof magnitude larger than that of y = 0.01 in FeGa3−yGey , and

Fig. 1. Phase diagram of FeGa3−yGey. Open squares represent the ferro-magnetic transition temperature TC, and the bold arrow shows the quan-tum critical point (at y = 0.15). Solid line at y = 0.01 corresponds to the criti-cal transition edge between the Kondo-like insulator and the paramagneticmetal. Solid curve starting from y = 0.15 is the fitting line of the critical tem-perature TC. Color scale represents magnetization of FeGa3−yGey measuredat H = 1 T at various temperatures.

the energy gap of FeSi was ∼1/10th of the gap of FeGa3, theGe substitution considerably affected the electronic state inFeGa3−yGey (36).

Magnetic Orderings and Universality of Spin Fluctuations. Conven-tionally, the critical temperature TC of itinerant ferromagnetsis determined from the M 2 vs. H /M relation. According to themean-effective-field solution of an arbitrary spin Ising model, ifthe Gaussian distribution of exchange coupling intensity is con-siderably greater than the mean value of exchange bonds, Arrottplots (M 2 vs. H /M plots) should show straight lines, and oneplot must pass the origin at TC. In the case of FeGa3−yGey

however, only the Arrott plots for y = 0.32 showed good linearbehavior (Fig. 2A and Fig. S6). The other samples showed con-vex curvatures, even at TC, and the curvature decreased with theincreasing electron doping. In contrast, all of the M 4 plots (Fig.2B and Fig. S7) for ferromagnetic FeGa3−yGey showed good lin-ear behavior, especially at TC, where the M 4 plot passed the ori-gin (0,0), nonsignificant deviations from linear behavior aroundQCP were observed in M 4 plots (Discussion). In these cases, thecritical temperature TC could still be estimated by the low-fielddata of the isothermal Arrott plots, approximating the arbitraryspin Ising model. The QCP at y = 0.15 inferred from the mag-netic measurement as TC reaching 0 was consistent with thatderived from the resistivity and specific heat measurements.

The Rhodes–Wohlfarth and Deguchi–Takahashi plots aredrawn in Fig. 3 A and B, respectively (37, 38). (1/2)PC in Fig.3A represents the effective spin per atom, whose value can bederived from Peff

2 =PC(PC + 2). The largest magnitude of themagnetic ordering parameter PC/PS obtained in this work was2.6 at y = 0.16, corresponding to a weakly itinerant nature. Thesmallest PC/PS of 1.8 at y = 0.32 indicated an adequate local-ized nature within the system, which is comparable with nickel’svalue of 1.5. In the Deguchi–Takahashi plot, where T0 repre-sents the energy width of the dynamical spin fluctuation spec-trum in frequency space corresponding to the stiffness of spindensity in amplitude, the localized nature of electrons becamedominant in the systems if T0 was comparable with TC in magni-tude (18, 38, 39). The right side of the abscissa in Fig. 3B, whereTC/T0∼ 1, represents the localized regime. The left side, whereTC/T0� 1, represents the extreme of itinerancy. Fig. 3B con-sistently shows FeGa3−yGey spread from the localized regimetoward the itinerant one with increasing y , suggesting a mod-ulated state of magnetic moments in the cross-over region ofthe two pole extremes. For magnetic orderings in terms of themodel of closed Kondo–Heisenberg approximation, the increas-ing electron doping caused the effects of the Kondo interaction〈JK

∑i Si · sci〉 to become relatively weaker than the Heisenberg

interaction 〈(JH/z )∑

(ij) SiSj 〉 did in the system, i.e., the itin-erancy acquired from the Kondo effect in d electrons throughintersite exchange became less significant by the continuous elec-tron doping in the FeGa3−yGey .

Importantly, PC/PS of FeGa3−yGey were not described bythe fitting curve and had much smaller values than other ferro-magnetic metals and alloys with the same TC in Fig. 3A. Unlikethe majority of ferromagnetic metals or alloys, FeGa3−yGey con-tains a considerably low effective Fermi energy caused by itssharp density of states at EF (40), resulting in its TC to vary con-siderably less rapidly than the EF and the failure to follow theRhodes–Wohlfarth curve, which well describes the behavior ofmost other metals and alloys (41). However, FeGa3−yGey withvarious amplitudes of dynamical spin fluctuations, correspondingto different T0 values as shown in Table 1, roughly satisfied thegeneralized Rhodes–Wohlfarth theoretical equation, Peff/PS =

1.4× (TC/T0)−2/3 in Fig. 3B, and relatively widely spread alongthe line with its TC increasing from 0 at QCP to a consider-ably high value of 53.1 K. The good fitting of the equation

3274 | www.pnas.org/cgi/doi/10.1073/pnas.1713662115 Zhang et al.103











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A B

Fig. 2. Arrott plots and M4 plots. M2 vs. H/M (Arrott plots) and M4 vs. H/M for FeGa3−yGey with y = 0.14, 0.15, 0.16, 0.20, 0.24, and 0.32, respectively, asA and B. Dashed lines in B are the description of Eq. 4 and should be where M4 plots shown up at the critical temperature TC (text).

for the entire range of weak ferromagnets implies a great relianceon spin fluctuations in reconciling the ferromagnets with differ-ent electron itinerancy from a localized regime to an itinerantregime.

Experiment vs. Theory. Experimental results of inverse suscepti-bilities χ−1 vs. temperature T and those of the theoretical recon-struction are shown in Fig. 4 (see also Fig. S8). The reasonableconsistency between experimental observations and theoreti-cal calculations evidenced the precision of the spin-fluctuationparameters we estimated in this work and also the success of ouranalysis for the modulated ferromagnetic FeGa3−yGey .

DiscussionMagnetic behaviors that are intermediate between localized anditinerant nature in FeGa3−yGey imply great difficulty in explain-ing the magnetic properties within a unified theory. Additionally,celebrated models properly describing the ground state shouldbe taken beyond to involve the temperature-dependent effects

of spin fluctuations. Starting by dealing with the intrinsic freeenergy F in magnetization, which can be expanded in powers ofmagnetization M by tracking the splitting in band calculation:

F (M ,T ) =F (0,T ) +1

2a1(T )M 2 +

1

4a2(T )M 4 + · · ·, [1]

Converted as the magnetic field H -dependent M equation:

H =∂F

∂M= a1(T )M + a2(T )M 3 + a3(T )M 5 + · · ·, [2]

where F (0,T ) is the free energy at M = 0, and ai(T ) are expan-sion coefficients related with the electron density of states and itsderivatives near EF.

The thermodynamic state of the free energy is determinedby the association of the hopping conduction electrons withthe repulsion by electrons with opposite spin directions on site.For an itinerant ferromagnetic system, where its thermodynamic

A B

Fig. 3. Rhodes–Wohlfarth and Deguchi–Takahashi plots. PC/PS vs. TC plot and Peff/PS vs. TC/T0 plot for FeGa3−yGey and various ferromagnets, as Aand B, respectively. Data are reproduced from refs. 42–53. (A) Parameters of FeGa3−yGey do not follow the universal line, and PC/PS of FeGa3−yGey isrelatively small compared with other ferromagnets with same magnitude of TC. (B) Red straight line represents Takahashi’s theoretical line, Peff/PS = 1.4×(TC/T0)−2/3, which roughly describes FeGa3−yGey .

Zhang et al. PNAS | March 27, 2018 | vol. 115 | no. 13 | 3275104


Table 1. Spin-fluctuation parameters

y Peff PC PS TC TA(104) T0(102) F1(105)

0.16 0.71 0.226 0.087 7.2 7.56 1.10 1.390.18 0.74 0.244 0.112 14.3 9.67 1.87 1.330.20 0.79 0.274 0.133 24.8 1.23 2.99 1.350.21 0.80 0.281 0.136 32.6 1.42 3.48 1.590.24 0.90 0.345 0.156 36.4 1.33 4.23 1.110.27 0.91 0.352 0.187 46.9 1.29 4.30 1.030.32 0.96 0.386 0.216 53.1 1.18 4.56 0.73

Spin-fluctuation parameters estimated from magnetic measurements fory = 0.16, 0.18, 0.20, 0.21, 0.24, 0.27, and 0.32. Peff, PS, TC, TA, T0, and F1, rep-resent effective magnetic moment (µB/Fe), spontaneous magnetic momentat ground state (µB/Fe), Curie temperature (K), the width of the distributionof the dynamical susceptibility in the q space (K), the energy width of thedynamical spin fluctuation spectrum (K), and fourth-order expansion coef-ficients of magnetic free energy (K), respectively. 1

2 PC represents effectivespin per atom (µB).

state becomes stable at finite magnetization, its magnetic proper-ties can be described by the linear Arrott plot within coefficientsa1 and a2 neglecting higher power terms, since the conductionelectron density is fairly restricted around the Fermi energy EF

in the ferromagnets, which leads to the famous equation:

M 2(M ,T ) =−a1(T )

a2(T )+

1

a2(T )

H

M (H ,T ). [3]

Numerous systems are governed by Eq. 3. As examples, someweakly ferromagnetic compounds similar to FeGa3−yGey areZrZn2 (42), Sc3In (54), ZrTiZn2 (55), ZrZn1.9 (55), and Ni-Ptalloys (43). However, Arrott plots of ferromagnetic FeGa3−yGey

are not linear around the Curie points, especially when y is posi-tioned close to the critical point of 0.15. This suggests the needfor a higher power term of free energy a3(T )M 5, which is notcontemplated by the ground-state-based magnetic theories suchas HFA or RPA (11, 13, 56). Even in the present form of theSCR theory, the fourth expansion coefficient, a2(T ), is assumedto be temperature-independent, resulting in an inaccurate pre-diction that the spontaneous magnetic moment in ferromagnetsvanishes at the Curie temperature. This also implies the need fora higher power of term a3(T )M 5 in the free-energy function forthe approximation. Inputting all of the dynamical parameters ofai for the M 4 plots at the critical point TC, we have ref. 38:

H /M =T 3

A

2µB[3πTC(2 +√

5)]2

(PS

MS

)5

M 4. [4]

where the spin-fluctuation parameter TA represents the widthof the distribution of the dynamical susceptibility in wave vec-tor space, and MS =N0µBPS is the spontaneous magnetiza-tion in the ground state, with N0 representing the number ofatoms. For y = 0.14, 0.15, and 0.16 in FeGa3−yGey , the smalldeviation from linear of the M 4 plots may be caused by thecomparable a2(T )M 3 terms and a3(T )M 5 terms in the vicin-ity of the QCP, indicating the comparable effects in nonlin-ear couplings of spin fluctuations to the effects of nonneg-ligible temperature dependence in general. For y ≥ 0.16, theterm a3(T )M 4 gradually becomes overwhelming compared with

a2(T )M 2, ( T3Aρ

3

2N50 µ

6B(ρ′2/ρ2−ρ′′/3ρ)[3πTC(2+

√5)]

2M2� 1, where ρ

represents the density of states); hence, the M 4 plots showmuch better linear behaviors than the Arrott plots do, and, syn-chronously, the curvature begins to decrease in the plots with theelectron doping. For ferromagnetic FeGa3−yGey , we observethat the Arrott plots at TC nearly pass the origin, indicatingthe nonnegligible temperature dependence of spin fluctuations

is still considerable, even in the case where their TC vanishes atthe critical point.

Well-established approximations such as HFA and RPA onlydeal with the paramagnetic contributions of spin fluctuationsof elementary excitations; however, for FeGa3−yGey , effects oftemperature-dependent long-range mode–mode coupling spinfluctuations on the thermal equilibrium state is crucial inaccounting for its magnetic properties. We take the quantum sta-tistical mechanical theory of SCR approximation of spin fluctu-ations into consideration, in which two well-known assumptionsare inherited: (i) In the ground state, the magnetic propertiescan be described by the band calculation; and (ii) the effectsof spin–spin couplings can be mainly represented by the secondexpansion coefficient of the free energy. We should mention thatthe theories of spin fluctuations are then in contrast with thephenomenological–theoretical-based technique of the modifiedArrott plot in which arbitrary critical exponents can be applied(18, 57), since the function of free energy in the theories of spinfluctuation is even.

In the weakly ferromagnetic limit of the SCR approximation,the imaginary part of the dynamical spin susceptibility for ferro-magnets is described by the double Lorentzian form in the smallq ,ω region (18):

Imχ(q ,ω) =χ(0, 0)

1 + q2/κ2

ωΓq

ω2 + Γ2q

. [5]

where Γq is the spectral width of the spin fluctuations givenby Γq = (A/C )q(q2 +κ2) = Γ0q(q2 +κ2), and κ2 = %/2Aχ=N0/2Aχ, and it leads to:

P2S

4=

15T0

TAc

(TC

T0

)4/3

, [6]

in weakly ferromagnetic systems.Derived from Eq. 5, the inverse magnetic susceptibility is given

by ref. 38:

y =N0

2TAη2

κ2

κ2 + q2χ−1∼=

F1P2s

8TAη2

{−1 +

1 + νy

c

∫ 1/η

0

dzz 3

[lnu − 1

2u−Ψ(u)

]}. [7]

Fig. 4. Temperature dependence of inverse susceptibility. T dependencesof χ−1 for FeGa3−yGey with y = 0.16, 0.20, 0.24 and 0.32. Black lines andsquares represent experimental results. Red lines represent reconstructedresults based on the theories of spin fluctuations (text).



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With u = z (y + z 2)/t , t =T/TC, ν= η2TA/U , η= (TC/T0)1/3,c = 0.3353. Ψ(u) is the digamma function, and parameterF1 is the mode–mode coupling constant, representing thefourth-order expansion coefficients of magnetic free energy.F1 =N 3

A(2µB)4/ζkB, ζ is the slope of the Arrott plots at low tem-perature. U represents the intraatomic exchange energy, and NA

and kB are Avogadro’s number and the Boltzmann constant (58).Due to the compensation of the increasing thermal amplitude

of spin fluctuation for the suppression of the zero-point spinfluctuation under applied magnetic field with increasing temper-ature, the sum of total spin amplitude squared at finite temper-ature can be treated as nearly conserved, which leads to the fol-lowing equation (38, 39):

F1 =4

15

kBT2A

T0. [8]

The above assumption of total spin amplitude conservationagrees with the first fully quantitative study based on an analo-gous rotationally invariant Hartree approximation of the effectsof spin fluctuations in terms of high-precision studies based onthe de Haas–van Alphen effect in conjunction with semiempir-ical band models or direct experimental measurements such asneutron scattering, as the essential magnetic equations inferredfrom the above two theories are quantitatively consistent undercertain approximations (18, 26, 59, 60). The validity of Eq. 8 isconfirmed by the inelastic neutron scattering or nuclear mag-netic resonance measurements on the archetypal weak itiner-ant ferromagnets that are analogous to FeGa3−yGey (61–65),since the fourth expansion coefficient F1 derived from magneticmeasurements is consistent with the estimated values by usingspin-fluctuation parameters TA and T0 inferred from abovedynamical measurements by applying Eq. 8. The spin amplitudeconservation is also observed in the case of the one-dimensionaland 2D Hubbard’s approximate model (66–68).

The component of the effects of zero-point spin fluctua-tions emphasized in the above assumption for low-temperatureferromagnetic order systems is neglected by the former ver-sions of SCR spin fluctuation theories, in which only the localspin amplitude squared is assumed to be conserved. This dif-ference leads to considerable differences between the currentanalysis and discussions based on former SCR theories. In thiswork, the fourth expansion coefficient F1 under the influenceof the zero-point component of spin fluctuations is assumedto be temperature-dependent and associated with the spectralwidth amplitudes of spin fluctuations. Hence, all necessary spin-fluctuation parameters can be estimated by applying Eq. 8 merelyusing macroscopic magnetization measurements, without theneed of pursuing any dynamical measurements (44, 69). In addi-tion, a sixth expansion term of free energy left absent from theformer SCR theories which is related to the critical tempera-ture TC should also be taken into consideration at relatively

large values of η= (TC/T0)1/3. The importance of this isshown in the magnetic equations as Arrott plots and M 4

plots of FeGa3−yGey . The universally satisfied relation betweenPeff/PS and TC/T0 in the generalized Rhodes–Wohlfarth plotof Fig. 3B is also derived from the total spin amplitude con-servation assumption. The quantitative agreements between thetheoretical reconstruction and the experimental results indi-cate the success in elucidating the magnetization mechanismsfor the intermediate range FeGa3−yGey system. The suc-cessful explanation for the ferromagnetic FeGa3−yGey sys-tem fitting well into the generalized Rhodes–Wohlfarth rela-tion, Peff/PS = 1.4× (TC/T0)−2/3, which describes a largevariety of ferromagnets, indicates a potential universality inquantitatively explaining the magnetism of weakly ferromagneticsystems in a broad TC range by involving the effects of spinfluctuations.

In this work, we have shown that electron doping by Ge substi-tution substantially affects the magnetic ground state and spin–spin correlation in FeGa3−yGey , causing phase transitions andsignificant changes in magnetic orderings, as well as the spinfluctuation mediated marginal Fermi-liquid state on the bor-der of ferromagnetism. We successfully take the temperature-dependent effects of spin fluctuations in general into accountfor the modulated ferromagnetic FeGa3−yGey ranging froman itinerant regime to the adequate localized region, and thetheoretical reconstruction agrees well with the experimentalobservations. Our analysis shows a potential universality forthe entire range of weakly itinerant ferromagnetic systems.FeGa3−yGey should be a promising model system to helpachieve a unified magnetic theory for localized and itinerantelectrons.

MethodsSingle crystals of FeGa3−yGey were synthesized by the Ga self-flux method.Powders of Fe (99.99%), Ge (99.99%), and Ga (99.9999%) ingot with theratio of Fe:Ge:Ga = 1: Y : 9 (0.01≤Y ≤ 3) were loaded and sealed in an evac-uated silica tube. The mixture was melted and homogenized in a furnaceand cooled to room temperature slowly. Excess Ga flux was removed withan aqueous solution of H2O2 and HCl. X-ray diffraction pattern confirmedthe samples are single crystal in FeGa3 type structure without second phase.The chemical composition of FeGa3−yGey was determined by wavelength-dispersive electron microprobe analysis. The magnetization, electrical resis-tivity, and specific heat of FeGa3−yGey were measured by the supercon-ducting quantum interference device, the standard four-probe method ina 4He cryostat and quantum design physical property measurement system,respectively.

ACKNOWLEDGMENTS. We thank Y. Takahashi for commenting on themanuscript and useful discussions. This work is supported by Ministry of Edu-cation, Culture, Sports, Science and Technology of Japan Grants-in-Aid forScientific Research 22350029 and 26410089 and Grants for Excellent Gradu-ate Schools.

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AdvAnces in Physics: X2048, vOL. 3, nO. 1, 1466661https://doi.org/10.1080/23746149.2018.1466661

REVIEW ARTICLE

Quasiparticle interference on type-I and type-II Weyl semimetal surfaces: a review

Hao Zhenga,b and M. Zahid Hasanb

aschool of Physics and Astronomy, shanghai Jiao Tong University, shanghai, china; bdepartment of Physics, Princeton University, Princeton, nJ, UsA

ABSTRACTWeyl semimetals are a new member of the topological materials family, featuring a pair of singly degenerate Weyl cones with linear dispersion around the nodes in the bulk, and Fermi arcs on the surface. Depending on whether the system conserves or violates Lorentz symmetry, Weyl semimetals can be categorized into two classes. Photoemission spectroscopy measurements have confirmed the TaAs class and WTe2 class of Weyl semimetals as type-I and type-II, respectively. This review article aims to elucidate and elaborate on the basic concepts of Weyl semimetals and quasiparticle interference experiments on both type-I and type-II Weyl semimetals. The versatile results which reveal (1) the topological sink effect of the surface carriers, the unique feature of the Fermi arc state; (2) the weakly bound nature of the Fermi arc surface state; (3) the orbital-dependent scattering channels on the surface; (4) a mirror symmetry-protected surface Dirac cone, are summarily discussed. Finally, a perspective toward the future applications of quasiparticle interference techniques on topological materials is presented.

Introduction

In the 1980s, the discovery of the integer and fractional quantum Hall effects, which arise from high mobility two-dimensional electron gasses under high mag-netic field, opened a new era of condensed matter physics [1,2]. The previously successful Landau theory of phase transitions failed to describe these quantum Hall systems, as no symmetry is spontaneously broken. It was later recognized that the nontrivial topological Chern number induced by the Landau quantization of the Block wavefunction could explain the edge conductance in a quantum Hall system [3]. Since the discovery of the quantum Hall system as the first nontrivial

© 2018 The Author(s). Published by informa UK Limited, trading as Taylor & Francis Group.This is an Open Access article distributed under the terms of the creative commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

KEY WORDSWeyl semimetal; Fermi arc; scanning tunneling microscopy; quasiparticle interference

PACS CODES68.35.B- structure of clean surfaces (and surface reconstruction); 68.37.ef scanning tunneling microscopy (including chemistry induced with sTM); 68.47.de Metallic surfaces; 73.20.At surface states, band structure, electron density of states; 73.25.+i surface conductivity and carrier phenomena

ARTICLE HISTORYReceived 5 January 2018 Accepted 14 April 2018

CONTACT hao Zheng [email protected]

OPEN ACCESS

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topological phase, the search for nontrivial topological phases arising from the intrinsic band structure of a crystal without an external high magnetic field has become a vital task for physicists. More than two decades later, the quantum spin Hall effect was experimentally discovered in strong spin-orbit-coupling (SOC) HgTe/(Hg, Cd)Te heterostructures

Theoretical research has revealed that the Z2 topological index in a two-dimen-sional quantum spin Hall system can be generalized to three dimensions, leading to the emergence of the first bulk topologically nontrivial phase, the topological insulator (TI) [7–9]. The topology in a TI manifests itself in its bulk-boundary correspondence. More precisely, the bulk electronic band in a three-dimensional TI is gapped, while the surface is metallic. The TI phase was first discovered in the strong SOC material Bi1-xSbx alloy [10–12], and the Bi2Se3 class of materials, which consist of Bi2Se3, Bi2Te3, and Sb2Te3 [13–17]. The research potential of the latter was immediately realized, as its surfaces possesses only a single spin-mo-mentum locked two-dimensional Dirac cone type electronic band, which can be treated as an unconventional type of two-dimensional electron gas which had not yet been discovered in any real two-dimensional material. Based on this novel surface state, many important effects, e.g. weak anti-localization [18], Landau quantization [19,20], half-integer quantum Hall effect [21], and the generation of Floquet-Bloch states under external excitations [22] have been experimentally explored. Furthermore, by incorporating magnetic order into a TI, it is possible to discern the mass acquisition of the surface Dirac fermions, as well as their hedge-hog spin texture [23,24]. A most remarkable achievement built upon the physics of magnetic TIs is the realization of the quantum anomalous Hall effect [25–27], a milestone of condensed matter physics. Another fascinating research direction is superconducting TIs. By introducing a s-wave pairing to the surface state of a TI, a two-dimensional topological superconductor with unconventional px + ipy superconducting gap symmetry is effectively realized [28–31]. At the boundaries where time reversal symmetry is broken, Majorana-type bound states have been successfully identified [32,33], paving the way toward the realization of non-Abe-lian anyons and topological quantum computation. In an inversion symmetric TI, the topology is characterized by a Z2 index, which is determined by the number of band inversions at Kramers points and the SOC-induced gap in the electronic band. Topological Kondo and Anderson insulators can be categorized in the same class as TIs as they share the same topological invariant Z2 index [34–36]. However, their difference lies in the gap opening mechanism, where the Kondo effect and Anderson localization play the role of SOC in TIs.

The second symmetry-protected topological phase in three dimensions is the topological crystalline insulator (TCI) [37], which has been discovered in the SnTe class of materials [38–41]. Unlike the time reversal symmetry protected phase in TIs, the SnTe class of TCIs are protected by mirror symmetry and are characterized by a mirror Chern number topological invariant. Therefore, a lattice distortion can add a mass term into the surface Dirac cone [42] and may induce a pseudo-magnetic

570 H. ZHENG AND M. ZAHID HASAN

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field to the surface state [43]. Theoretical calculations have predicted a novel large Chern number quantum anomalous Hall effect on magnetic TCIs [44].

The third discovered three-dimensional symmetry-protected topological phase moves beyond the realm of insulators. The topological Dirac semimetal is a three-dimensional counterpart to graphene, which has been discovered to exist in Na3Bi and Cd3As2 [45–53]. The gapless nature of the bulk band of Dirac semimetals brings rich physics into both the two- and three-dimensional elec-tron gases. Indeed, the giant magnetoresistance and chiral anomaly effect were observed in the bulk electron band [54,55], while unconventional quantum oscil-lations were detected in the topological surface state [56,57]. In addition, tip and pressure induced superconductivity in a topological Dirac semimetal has been shown, shedding a light on the realization of a new type of topological supercon-ductor [58–60]. Nanostructures of Dirac semimetals have also revealed significant Aharonov–Bohm oscillations [61].

Beyond the search for new types of symmetry-protected topological phases, the study of Weyl semimetals is strongly driven by the search for the long sought after Weyl fermions. In 1929, H. Weyl found that a massless Dirac fermion can be decomposed into a pair of relativistic particles with opposite chirality; the Weyl fermion. To date, Weyl fermions have yet to be discovered as fundamen-tal particles in high-energy physics. Recently, condensed matter physicists have discovered that in certain crystals that lack either space inversion or time rever-sal symmetry, their low-energy quasiparticle excitations can be described by the Weyl equation [62–65]. Such a crystal has been termed as a Weyl semimetal. The bulk band structure in a Weyl semimetal features pairs of singly degenerate linearly dispersed Weyl cones of opposite chiralities. This chirality can be viewed as the fourth inherent property of a quasiparticle hosted in a crystal other than the charge, spin, and valley degree of freedoms. This leads to many unique Weyl fermion related transport effects such as the Adler-Bell-Jackiw anomaly [66–68], the axial anomaly [69], non-local transport [70,71], and the chirality-dependent Hall effect [72]. These novel phenomena uncover the rich correspondence between high-energy particle physics and low-energy condensed matter physics. In addi-tion, the nontrivial topology in a Weyl semimetal can be characterized by the value of chiral charge, distinct from the topological invariants in TIs (Z2 index) and in TCIs (mirror Chern number). Thus, the Weyl semimetal is identified as a new class of symmetry-protected topological phase. The bulk-boundary correspondence in a Weyl semimetal manifests itself as an open contour on the surface, the Fermi arc surface state. Unconventional quantum oscillations induced by these Fermi arcs have been predicted [56] which serve as a key transport feature of this new type of surface state. Early attempts in the search for a Weyl semimetal in real materials were focused on time reversal symmetry breaking crystals Y2Ir2O7 [73] and HgCr2Se4 [74], TI/trivial insulator multiple layers [75], as well as the solid alloy TlBi(S1-xSex)2 [76] and Hg1-x-yCdxMnyTe [77]. Unfortunately, unusual spin texture, complicated magnetic domain structure, difficulty in preparation, and the

ADVANCES IN PHYSICS: X 571

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extremely strict requirements in fine tuning the stoichiometry in these material systems rendered the realization of these predictions fruitless.

The breakthrough in the search for a Weyl semimetal emerged from the TaAs class of materials [78–81]. The four members of this family share the same space group I41md and a similar electronic band structure which features 12 pairs of Weyl nodes in the bulk bands and multiple Fermi arcs on the surface. Angle resolved photoemission spectroscopy (ARPES) measurements have explicitly proved TaAs [82–85], NbAs [86,87], TaP [88,89], and NbP [90–93] as Weyl sem-imetals by direct observation of the linear dispersion of bulk Weyl cones and the arc-shaped surface states which terminate at the projected Weyl points on the surface. Several key chiral Weyl fermion-induced phenomena have also been observed in electronic transport and optics measurements, such as the extremely large magnetoreresistance [96], the helicity-protected ultrahigh mobility [97], the violation of Ohms law [98], the magnetic tunneling-induced Weyl node annihila-tion [99], the giant anisotropic nonlinear optical response [100], and the optical detection of Weyl fermion chirality [101] have been experimentally discovered. Among these experiments, two exceptionally remarkable achievements lie in the detection of the signature of the chiral anomaly effect [102,103] and the axial–gravitational anomaly effect [104], which may open a new era of table-top experi-mental realizations of high-energy physics. Furthermore, the unusual spin texture of the Fermi arc surface state on TaAs has also been detected [94,95].

Shortly after the discovery of the TaAs class of materials, theory predicted the WTe2 class of layered compounds to be type-II Weyl semimetals [105–108], and identified the TaAs class of Weyl semimetals to be type-I. Type-II Weyl fermions can be viewed as a tilted cone shaped dispersion in momentum space which breaks Lorentz symmetry and thus cannot exist as a fundamental particle in nature. This type of Weyl fermion exists as a unique phenomenon of condensed matter physics. Distinct from type-I Weyl semimetals, many novel effects such as the intrinsic anomalous Hall effect [109], magnetic breakdown and Klein Tunneling effect [110], Landau level collapse effect [111] have been predicted. ARPES measure-ments have been reported on MoTe2 [112–116], WTe2 [117–122], and MoxW1- xTe2 [123,124]. Pump-probe ARPES measurements have discovered Weyl cone–Fermi arc connectivity, thus confirming the Weyl semimetal phase in the material [124]. Furthermore, the spin texture on the surface states of WTe2 [125] and MoTe2 [126] have also been discerned. Meanwhile, transport measurements have dis-covered the anisotropic Adler-Bell-Jackiw anomaly effect in WTe2 crystals [127]. The discovery of MoTe2 to be superconducting whose TC can be enhanced by high pressure or doping with S [128,129] may open a route toward the realization of an unconventional topological superconductor.

From a material science perspective, the type-I Weyl semimetal can be treated as a direct negative band gap semiconductor while the type-II Weyl semimetal features an indirect negative gap as shown in Figure 1. As one might expect, many more type-II Weyl semimetals than type-I exist in real materials. The discovery of


111

the Weyl semimetal phase in TaAs- and WTe2 -class materials has attracted tre-mendous research effort in the search for new Weyl semimetals. Experiments have verified the existence of several new theoretically predicted Weyl semi-metals including LaAlGe [130], TaIrTe4 [131–133], Ta3S2 [134,135], WP2 class [136,137], Mn3Sn class [138–140], and β-Ag2Se [141], which largely belong to the type-II Weyl semimetal category. Research into Weyl semimetals both in the context of new materials and new phenomena is, at present, an extremely dynamic field. This concise review article will focus on the TaAs class and WTe2 class of materials. As discussed above, while ARPES experiments were employed to iden-tify the Weyl semimetal phase in real materials, electronic transport and optical measurements can, for the most part, only detect bulk band-induced effects. As such, directly observing the physics that arises from the Fermi arcs requires a surface sensitive approach. Scanning tunneling microscopy (STM) combined with spectroscopy, which possesses extremely high spatial and energy resolution and probes the surface electronic structure of a crystal, is a natural choice. Here, we summarize the recent advances in STM-based quasiparticle interference (QPI) results on two prototypical type-I and type-II Weyl semimetals. This review article is organized as following: (1) the concept of Berry phase, a building block in the theory of all topological materials, is first introduced; (2) followed by the basics of STM and QPI; (3) and finally, concrete QPI results on both types of Weyl semimetals are presented.

Figure 1. Type-i and type-ii Weyl semimetals (a) A sketch of the e-k dispersion of a type-i Weyl semimetal which can be viewed as a negative direct bandgap semiconductor. The type-i Weyl nodes exist at the crossing points where the conduction band (red) and valence band (blue) dip into each other. A Fermi arc (green dotted line) connects the pair of Weyl nodes. (b) The Fermi surface (Fs) on the surface of a type-i Weyl semimetal as depicted in (a). The Fermi level is located at the energy of the Weyl nodes. Bulk Weyl cones are projected onto the surface as two discrete points (red, blue) connected by a Fermi arc surface state (dotted green line). (c) A sketch of e-k dispersion of an inversion symmetry breaking type-ii Weyl semimetal which can be viewed as a negative indirect bandgap semiconductor. Tilted type-ii Weyl cones form at the intersection of conductance and valence bands. (d) Fs on the surface of a type-ii Weyl semimetal as depicted in (c). inversion symmetry breaking constrains the minimum number of Weyl nodes to be four. here, the Fermi level is assumed to cut at the energy of the lower Weyl node in (c). Projected bulk electron (red) and hole (blue) pockets coexist on the surface Fs and touch at one discrete point, the type-ii Weyl nodes. A Fermi arc (green dotted line) connects one pair of projected Weyl nodes.


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The concept of berry phase, Weyl cone, and Fermi arc

Berry phases were initially introduced to describe the geometrical phase acquired by a particles wavefunction under an adiabatic variation of an applied external field. Later, research showed that the Chern number, an integer number that determines the number of quantum transport channels on the edge of a sample in a quantum Hall regime, is directly related to the Berry phase. Further investi-gations discovered that all symmetry-protected topological phases in condensed matter can be described by the Chern number and Berry phase under various circumstances. We start from the definition of the Berry vector potential in a crystalline solid system:

where |k|. is the Bloch wavefunction in k space. The integration of the Berry vector potential around arbitrary closed c in k space gives the Berry phase.

As we know from electrodynamics, the curl of a magnetic vector potential gives the strength of the magnetic field. Here, we apply the same concept to calculate the curl of the field strength of the Berry vector potential; the Berry curvature. Through Stokes theorem, we have:

Here, the surface S in k space is surrounded by loop c. If the integration is over a closed surface in k-space, for example, a Brillouin zone (BZ) in a two-dimensional material, or a closed surface inside of the BZ in a three-dimensional material, we obtain the Chern number C which determines the number of edge states on the boundary.

We now describe a two-level system with Hamiltonian H = d(kx, ky, kz) ⋅��, where dx,y,z are functions of (kx, ky, kz) and �x,y,z are the Pauli matrices. Its eigenstate intrin-sically possesses a nontrivial Chern number as the Berry curvature is calculated to be ∇

k× A(k) =

⇀

d

2d3. From here, we can recognize this to be a Berry curvature

monopole in k-space, and the Chern number to be the flux of the Berry curvature field on the integration plane.

(1)A(k)= < k|∇

k|k >

(2)𝛾 = ∮ c

A(k)dk

(3)𝛾 = ∮ c

A(k)dk = ∬

s

(∇k× A

(k))dS

k

(4)C =1

2𝜋 ∮ B

(∇k× A(k)) ⋅ dS

k)


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While, historically, the Weyl semimetal was discovered after the topologi-cal insulator, topological crystalline insulator, and Dirac semimetal, the Weyl semimetal is perhaps the simplest symmetry-protected topological phase in a mathematical sense. Unlike the required time reversal symmetry in TIs, mirror symmetry in TCIs, and rotational symmetry in Dirac semimetals, a Weyl sem-imetal only requires translational symmetry to preserved, a condition naturally fulfilled by any crystalline solid. In this sense, a Weyl semimetal may be the most robust symmetry-protected topological phase. A simple model Hamiltonian which describes a Weyl semimetal containing only a single pair of Weyl nodes can be written

In the Hamiltonian, the two energy bands cross each other at the two (0, 0, ± arc-cos(m)) points in momentum space. In the vicinity of these two degeneracy points, we can expand the Hamiltonian in Equation (5) and get h = ± (vxkxσx + vykyσy + vzkzσz). This is nothing but a Weyl fermion with a speed of ⇀v instead of light speed.

The Hamiltonian in Equation (5) can be written in the form H = d(kx, ky, kz) ⋅��. It thus can be topologically nontrivial as its wavefunction may feature nontrivial Chern numbers. Interestingly, if one chooses a (kx; ky) plane located in the range of arccos(m) < kz < arccos(m) as the integration plane of the Berry curvature, a Chern number of 1 is calculated, indicating the existence of one edge mode in this region (a surface state in three dimensions), while integrating on the planes of either arccos(m) > kz or kz > arccos(m) results in a Chern number of zero, describing the absence of an edge mode in this region. It is this kz-dependent Chern number which generates the unique shape of the surface state as shown in Figure 2. The surface state occurs only in the region between the pair of projected Weyl nodes on the surface and has an unclosed contour shape and is thus accordingly called a Fermi arc.

During the discovery of type-II Weyl semimetal, the researchers found that a Weyl cone can be described as H =

∑kiAi,j�j(i = x, y, z; j = 0, x, y, z) with energy,

momentum dispersion as E±(k) =

∑i

kiAi,0 ±

��∑i

�∑j

kiAi,j

�2

= T(k) ± U(k).

T(k) and U(k) can be considered as the kinetic and potential components of the dispersion. T(k) > U(k)titles the Weyl cone, thus leads to the type-II Weyl sem-imetal. The generalized Weyl Hamiltonian is certainly beyond the original Weyl equation H = ±k ⋅ ��, but still has the form of H = d(kx, ky, kz) ⋅��Therefore, both types of Weyl cones posses same Fermi arc type surface state.

(5)H(k)= 2t1 sin

(kx2

)𝜎x + 2t1 sin

(ky

2

)𝜎y + t2(coskz −m)𝜎z


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Basics of quasiparticle interference experiment and theory

Measuring the surface state on a crystalline solid requires a surface sensitive measurement. The principle behind STM lies in measuring the tunneling current between a sharp metal tip brought very close (<1 nm) to a clean crystal surface. The relationship between the tunneling current and the applied voltage and dis-tance between the tip and sample is as follows (under a one-dimensional barrier Bardeen tunneling model at zero temperature):

In most cases, the first derivative (dI/dV) of the tunneling current is proportional to the local density of states of the sample �s. Thus, a dI/dV(x,y) conductance map can directly measure the real space distribution of the local density of states. A sur-face state on a crystal can be treated as a special kind of two-dimensional electron gas which can be scattered at local point defects on the surface such as an atom vacancy or adatom. The incident wave vector (��ki) and reflected wave vector (��kr) can interfere at the site of these local point defects, leading to two-dimensional standing wave patterns. These patterns display different wave lengths at distinct energies, which is known as quasiparticle interference (QPI). A Fourier transform (FT) is applied to the real space dI/dV maps in order to gain insight in momentum space. A FT-dI/dV map plots all allowed surface scattering vectors (Q = ��kr −

��ki

(6)I =4�e

ℏ

eV∫0

�T (� − eV )�S(�)|M(�)|2d�

(7)|M(�)|2 ∝ exp

(−2d

√2m

ℏ�

)

Figure 2. demonstration of the Fermi arc and the topological sink effect. (a) in momentum space, the kz-dependent values of the chern number generate the interrupted shape surface state terminating at the projected position of the Weyl nodes. This is the Fermi arc surface state. (b) By placing a Weyl semimetal under a magnetic field perpendicular to the z direction, a surface electron will travel through a Fermi arc surface state, reach the projected Weyl node, sink into the bulk state, move to the opposite surface, and finish the closed loop. This topological sink effect is the key phenomenon associated with the unclosed shape of the Fermi arc.


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). In combination with theoretical calculations, QPI measurements can provide rich information on the surface band structure both in occupied and unoccupied states, as well as the allowed and forbidden scattering channels of surface states. QPI measurements have been extensively employed in the investigation of surface states in many different topological materials [142–157]. On the surface of a TI, QPI results demonstrate that the back scattering of surface carriers are forbidden at time reversal invariant local defects while they are allowed at magnetic defects, a signature manifestation of the unusual spin-momentum locking in a topolog-ical surface state. On the surface of TCIs, the unconventional orbital texture of the surface Dirac cone at different valleys has been successfully detected by QPI.

When acquiring QPI measurements on a crystal surface, a few technical issues need to be taken into account. On a surface, local defects will also induce a defect state which imposes an additional signal to that of the lattice. When this signal is included in the Fourier transform to Q-space, it can complicate the interpretation of the FT-dI/dV map. However, as QPI patterns measure surface standing waves which usually disperse with energy, defect states tend not to be dispersive and can thus be distinguished from the surface states. Secondly, there are two manners in which to obtain QPI maps: a dI/dV grid, also refer to as constant current tunneling spectroscopy (CITS), and dI/dV maps. A CITS grid is obtained by measuring the energy-dependent dI/dV spectra on each real space point over an entire image, in order to obtain a set of local density of state maps at different energies. A dI/dV map simultaneously measures a dI/dV signal at a single energy, along with a constant current STM image. By varying the set point voltage, a series of dI/dV maps can be individually obtained. Furthermore, as shown above, the tunneling current is the integration of the samples local density of states from the Fermi level to the set point voltage. Even under a small voltage, the variation in the spatial local density of states can induce large changes in the tunneling current. To main-tain a constant current mode, the tip of the STM will withdraw from or approach the sample accordingly. This change in the tip-to-sample distance will affect the strength of the dI/dV signal and is referred to as a set point effect. A typical set point effect manifests as an additional duplicate and weakly dispersed pattern in the FT-dI/dV maps [158]. A dI/dV grid measurement set at large voltages tends to result in a relatively small set point effect.

Theoretical calculations of the QPI of a particular material are based on a den-sity functional theory (DFT) simulated surface electronic band on a slab geometry. An autocorrelation of the surface Fermi surface results in a joint density of state (JDOS). By prohibiting spin-flip scattering, a spin-dependent scattering prob-ability (SSP) is generated and can be applied to interpret the experimental QPI results [159]. A more comprehensive method of simulating the QPI signal is the so-called T-matrix model, which is believed to capture the interference effects on the surface electronic state [160]. In the T-matrix method, the governing equations for the QPI can be written as [161]


116

where k and Q are the wave vector and scattering vector, respectively, G(k,𝜔) is the surface

Green function, T(k,𝜔) is the T-matrix, and Vi is the defect-induced potential.

Quasiparticle interference on type-I weyl semimetal

The TaAs class of materials which includes TaAs, TaP, NbAs, and NbP all share the same face-centered tetragonal crystal structure with space group I41md which lacks inversion symmetry, as well as similar bulk and surface electronic bands. In the bulk electronic structure with no SOC, the conduction and valence bands of these materials intersect on four closed loops in the first bulk BZ. The inclu-sion of SOC gaps out these closed loops, but the bands still intersect at 12 pairs of discrete points, each of which is a pair of Weyl nodes with opposite chirality. On the (001) surface of TaAs, the 24 Weyl nodes project onto 16 points. Eight of these projected Weyl nodes (W1) near the surface BZ boundary (Xand Y) carry a projected chiral charge of ±1. However, as these W1 Weyl nodes are located very near to each other in k-space, the Fermi arcs associated with these Weyl nodes are difficult to distinguish from the trivial surface state. The other eight projected Weyl nodes near the Γ point have a projected chiral charge of ±2 and are referred to as W2. As demonstrated in Figure 3(c), the surface Fermi state of NbP(001) can be grouped into three types of pockets according to their differing contour shapes: a bow tie-shaped contour near X and an elliptical contour near Y which surround W1 Weyl nodes and are dominated by topological trivial states, as well as a tadpole-shaped contour pointing to the Γ point. The head of this tadpole-shaped contour connects one pair of projected bulk W2 Weyl nodes with projected chiral charges of ±2, and is thus identified as the Fermi arc.

As the TaAs class of materials lack inversion symmetry, the P/As surface is dis-tinct from the Ta/Nb surface. In STM, prior to QPI measurements, dI/dV spectra taken on the surface are compared to the theoretical calculated LDOS in order to determine the surface orientation. To date, all reported STM experiments have been performed on the anion-terminated (0 0 1) surface [162–167]. Zheng et. al reports QPI results on the NbP(001) surface [162]. Voltage-dependent dI/dV

(8)QPI(Q,𝜔) =i

2𝜋∫ d2k

(2𝜋)2[B(Q,𝜔

)− B∗(Q,𝜔)]

(9)B(Q,𝜔

)= ��[G(k,𝜔)T(k,𝜔)B(k + Q,𝜔)]

(10)T(k,𝜔

)=

[1 − Vi ∫ d2k

(2𝜋)2G(k,𝜔)

]−1Vi


117

maps show surface standing waves increasing in wavelength, demonstrating the hole-like nature of these states. The main features in the FT-dI/dV map taken near Fermi level (V = −10 mV) in Figure 3(b) can be divided into three groups: (1) a bow tie-shaped contour at the center of the image the end of which is marked by the vector Q1; (2) an elliptical contour in the center of the image and perpendic-ular to the bow tie. The vector Q2 points to its end; (3) a rectangular feature with curved edges in each quadrant of the image. The corner is indicated by Q3. The features at the Bragg spots can be viewed as replicas of the bow tie and elliptical features. A model calculation taking matrix element effects into account (Figure 3(d)) reproduces the features in the experimental QPI pattern well. Owing to this agreement with these theoretical results of the QPI contours (Figure 3(b)) and the surface Fermi surface pocket (Figure 3(c)), the three dominant scattering vectors can be unambiguously identified. Q1 and Q2 arise from the intra-contour scat-tering, while Q3 is the inter-contour scattering. The tadpole to bow tie (S1), and tadpole to a perpendicular tadpole (S2) scattering are absent from experiment, which is attributed to the different orbital character of the contours.

Figure 3. scattering channels on nbP(001) surface. (a) sketch of the QPi pattern on nbP(001). The QPi features can be categorized into three groups, namely the bow tie shaped (red), ellipse (blue) and rounded rectangle contours. (b) experimental QPi pattern measured near the Fermi level. The three dominant scattering vectors are marked. (c) First principle simulation of the surface Fermi surface of nbP(001). The intra-bow tie, intra-ellipse, and inter contour scattering vectors are indicated, corresponding to the three scattering vectors in (b). (d) Theoretically calculated QPi pattern which corroborates the measurement [162].


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Inoue et al. report QPI results on a TaAs(001)-As surface [163], identifying the scattering vector (Q2 in Figure 4(a)) linking the tadpole and elliptical contours. They find the bulk state in the vicinity of the Weyl cones arises mainly from the Ta-d orbital and the surface state mainly from the As-p orbital. Simulated QPI patterns were calculated based on two different geometries, namely on a single surface unit consisting of four Ta-As bilayers, and on only the topmost As-layer. A comparison between the measurement and simulation indicates that only the top As-layer is observed in QPI patterns. This is interpreted as the electron from a Fermi arc surface state (on an As site) moving to the Ta site entering the bulk Weyl cone, leaving the surface and sinking into the bulk. This result serves as an important signature of the unique topological sink effect of a Fermi arc state.

Figure 4. summary of recent QPi results on TaAs and nbP type-i Weyl semimetal surfaces (a) an experimentally acquired QPi pattern on TaAs(001) surface. (b) a theoretical calculation of the QPi pattern with taking into account of the entire surface unite contribution. (c) same as (b) but only considering the topmost As layer. Obviously, (c) fits to (a) better than (b) [163]. (d) an experimentally acquired QPi pattern on TaAs(001) surface from another report [164]. (e) a subtraction of the QPi contour at Bragg points from the central point. it is believed to demonstrate the interference from Fermi arc surface state. (f ) a theoretical calculation of the QPi pattern of (e). (g) a sketch of the two-dimensional dirac cones (upper panel) and the Fermi surface (lower panel) in the first surface BZ of nbP(001) [167]. (h) an experimentally acquired QPi pattern on nbP(001) surface. (i) an energy scattering vector (e-Q) dispersion cut along the dotted line in (h). A dirac cone type feature is revealed.


119

Batabyal et al. also measured the TaAs(001)-As surface QPI [164] and found similar QPI patterns as Inoue et al. as shown in Figure 4(d). Through a weighted subtraction of the bow tie and elliptical features at the Bragg points from the center region, they discovered that the residual QPI features are induced by scattering between the Fermi arc to the nearby bulk state. As these features do not repeat themselves at Bragg points, the authors interpret it as the electronic state which induces these scatterings possessing only long wavelength standing waves in real space rather than atomic scale corrugations. Thus, they postulate that the Fermi arc surface state is weakly bonded to the lattice in contrast to a topologically nontrivial surface state.

Zheng et al. have theoretically predicted two-dimensional Dirac cone states on the surface of NbP(001) which are protected by the X − Γ − X mirror symmetry [167]. A pair of surface Dirac cone states is located near the point in the surface BZ, where the Dirac nodes are about 300 meV above the Fermi level. As the energy moves toward the Fermi level, the two Dirac cones expand in size and eventually merge and evolve into the bow tie-shaped contour at the Fermi level. The Dirac nodes exist in the unoccupied states, and thus cannot be accessed by conventional ARPES measurements. Energy resolved QPI measurements were able to detect the surface Dirac cone. As shown in Figure 4(i), an energy scattering vector (E - Q) dispersion measurement on the NbP(001) surface reveals a Λ-shaped feature in the center, proving the existence of the predicted mirror protected Dirac cone surface state.

Quasiparticle interference on type-II weyl semimetal

It was only recently that type-II Weyl semimetals were predicted. To date, the most intensively studied type-II Weyl semimetals are those in the WTe2 class of transition metal dichalcogenides which includes WTe2, MoTe2, and their alloy MoxW1-xTe2. Both WTe2 and MoTe2 feature type-II Weyl cones in their bulk bands when they crystallize in the Td structure polymorphy. Although the number of Weyl nodes (4 or 8) is sensitive to tiny variations in the lattice parameters used in first principle calculations, all simulations agree that the Weyl cones are located above the Fermi level. Therefore, it is beyond the capability of conventional ARPES measurements to detect the Weyl cone and the Fermi arc state at the energy of the Weyl node. However, QPI measurements do not suffer such constraints [168–171]. Figure 5(a) demonstrates the entire Fermi surface of MoxW1-xTe2 (001) surface. The type-II nature of the Weyl cones results in the coexistence of projected bulk and surface states. The bulk state consists of one dog-bone-shaped hole pocket in the centere of the BZ and two elliptical electron pockets near the left and right edges. The projected Weyl nodes (black and white dots in Figure 5(a)) are located at the boundary between the hole and electron pockets. The two bright yellow semi-circular contours in Figure 5(a) are surface states, where the middle seg-ments are the Fermi arc surface states. Theoretical analysis has revealed that the


120

Fermi arc surface states are derived from the Mo-d orbitals. In other words, the MoxW1- xTe2 alloy may enhance the surface electron scattering at metallic sites, consequently enhancing the Fermi arc interference signal. The theoretical QPI calculated by considering both the projected bulk and surface states and generates a pattern consisting of seven pockets (Figure 5(b)), while a surface state only QPI consists of only three pockets (Figure 5(d)). The experimental QPI is consistent with the result from Figure 5(d), indicating that the projected bulk states do not factor into the QPI. This is interpreted as a surface electron which is sitting on a Fermi arc moving to the projected Weyl cone pocket and sinking into the bulk, and thus does not contribute to the surface standing waves, the signature of the topological sink effect in a type-II Weyl semimetal. Due to the tilted nature of the type-II Weyl cone, the area of the projected Weyl bulk states in a type-II WSM is

Figure 5. QPi results on WTe2 class of type-ii Weyl semimetal surfaces. (a) calculated "complete” Fermi surface of a MoxW1-xTe2(001) surface. it contains both the bulk and surface state. (b) QPi pattern derived from (a), presenting only the intra-BZ scattering. (c) similar to (a), but with only the surface states taken into consideration. The Fermi arc is the central segment of the semicircular contours. (d) QPi pattern derived from (c) [168]. (e) The theoretically simplest Fermi arc QPi pattern [159]. (f ) experimental QPi data on MoxW1-xTe2(001), corresponding to(d). (g) experimental QPi data on a WTe2(001) surface [170]. (h) experimental QPi data on a MoTe2(001) surface [169].


121

much larger than in type-I, which exists at a single discrete point. Therefore, the topological sink effect is in principle much more pronounced on a type-II WSM. Both measured QPI patterns on WTe2 and MoTe2 in Figure 5 manifest themselves as three pocket structures as observed in MoxW1-xTe2, indicating the robustness of the topological sink effect in this class of materials.

Furthermore, simulations considering only the surface state (Figure 5(c)) reveal that on the (0 0 1) surface of MoxW1-xTe2, an ideal WSM is approximately realized, which would lead to the simplest Fermi arc QPI (Figure 5(d)). The theoretical sim-ulation in Figure 5(d) is based on a simple analytical model without considering material specific parameters and shows only a single pair of Fermi arcs. The QPI features a butter y shape with three QPI pockets, corroborating the measurements on MoxW1-xTe2. It thus renders this class of materials as an ideal real material for the investigation of Fermi arc interference patterns.

Perspective and outlook

In the research of surface electronic states, ARPES possesses a unique position due to its distinct capability for directly detecting the dispersion of surface bands. However, conventional ARPES is only capable of probing the occupied states and is furthermore incompatible with magnetic field. On the other hand, scan-ning tunneling microscopy QPI measurements provide only indirect information on the surface band structure but are not restricted to the occupied states and are well suited for magnetic field-dependent measurements. Weyl semimetals can be divided into two categories depending on whether they are inversion or time reversal symmetry breaking. The aforementioned TaAs and WTe2 classes of materials both belong to the inversion breaking Weyl semimetal. Recently, a time reversal symmetry breaking Weyl semimetal phase was predicted in ferro-magnetic Heusler half metals [174,175]. Interestingly, magnetization of the sam-ples along distinct crystalline directions generates different band topologies and consequently distinct surface Fermi arc states. This exotic topological response to external field is unfortunately not accessible by ARPES. Due to the non-vanishing bulk electronic state, the electronic transport measurements may also fail to track such surface effects due to a lack of surface sensitivity. QPI measurements may then be the most suitable approach for probing such phenomenon. In fact, most phenomena arising from topological surface electronic states on time reversal symmetry breaking Weyl semimetals are possibly most readily discerned and discovered via this strong experimental technique of QPI measurement.

Acknowledgments

We thank S. Zhang, S. Huang, and S. Xu for the helpful discussions. H.Z. acknowledges the financial support from National Natural Science Foundation of China (grant nos. 11674226, 11790313) and the National Key Research and Development Program of China (2016YFA0300403). M.Z.H. acknowledges the Gordon and Betty Moore Foundations EPiQS


122

Initiative through Grant GBMF4547 and U.S. National Science Foundation (NSF) grant no. NSFDMR-1006492.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11674226], [grant number 11790313]; National Key Research and Development Program of China [grant numer 2016YFA0300403]; Gordon and Betty Moore Foundations [grant number GBMF4547]; and U.S. National Science Foundation (NSF) [grant number NSFDMR-1006492].

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130

PHOTOVOLTAICSPROGRESS IN

ISSN 1062–7995 VOLUME 26 • NUMBER 11 • NOVEMBER 2018

RESEARCH AND APPLICATIONS

wileyonlinelibrary.com/journal/progressinphotovoltaics

Received: 23 October 2017 Revised: 10 May 2018 Accepted: 16 May 2018

DOI: 10.1002/pip.3037

R E S E A R CH AR T I C L E

Perovskite/c‐Si tandem solar cells with realistic invertedarchitecture: Achieving high efficiency by optical optimization

Lixiang Ba1 | Hong Liu1 | Wenzhong Shen1,2

1 Institute of Solar Energy and Key Laboratory

of Artificial Structures and Quantum Control

(Ministry of Education), School of Physics and

Astronomy, Shanghai Jiao Tong University,

Shanghai 200240, People's Republic of China

2Collaborative Innovation Center of Advanced

Microstructures, Nanjing 210093, People's

Republic of China

Correspondence

Hong Liu, Wenzhong Shen, Institute of Solar

Energy and Key Laboratory of Artificial

Structures and Quantum Control (Ministry of

Education), School of Physics and Astronomy,

Shanghai Jiao Tong University, Shanghai

200240, People's Republic of China.

Email: [email protected]; [email protected].

cn

Funding information

Natural Science Foundation of China, Grant/

Award Numbers: 61234005, 11674225 and

11474201

924 Copyright © 2018 John Wiley & Sons, L

Abstract

Many theoretical analyses for perovskite/c‐Si monolithic tandem solar cells (TSCs)

have shown optical optimization and high efficiency limits, but they use many ideal-

ized assumptions and draw some unpractical conclusions for experiments. In this

work, we have introduced a composite method combining the finite difference time

domain and light path analysis for the first time. By using this method, we have sys-

tematically calculated perovskite/c‐Si monolithic TSCs with inverted architecture

based on realistic solar cell parameters. Theoretical results have demonstrated very

good match of the experimental external quantum efficiencies of both subcells. More

importantly, from optical and electrical point of view, we have analyzed current losses

of suchTSCs and proposed detailed optimization for achieving high efficiency. Finally,

we have presented improved configuration of perovskite/c‐Si monolithic TSCs with

addition of pyramids structure in front surface, which can effectively increase the

tandem cell efficiency to 29.05%. This work can be served as a practical guidance

for the realization of high‐efficient perovskite/c‐Si monolithic TSCs.

KEYWORDS

FDTD, inverted configuration, light path analysis, optical and electrical optimization, perovskite/c‐

Si monolithic TSCs

1 | INTRODUCTION

Crystalline silicon (c‐Si) solar cells occupy an important position in

photovoltaic market (over 90%) because of its low cost, high effi-

ciency, and mature industrialization. The world record efficiency of

26.6% reported by Yoshikawa et al1 is extremely close to the Shock-

ley‐Queisser efficiency limit, so further improvement becomes very

difficult. In recent years, many groups have studied perovskite

because of its high absorption coefficient, sharp absorption edge,

and tunable bandgaps. The efficiencies have increased from 3.8%2 in

2009 to 22.1%3 in 2017, but further enhancement also faces difficul-

ties. c‐Si and perovskite have bandgaps of 1.1 eV and 1.5 to 2.3 eV,

respectively, which are suitable for spectrum matching so as to break

the limit for even higher efficiencies. Some theoretical calculations

have assessed efficiency limits of perovskite/c‐Si tandem solar cells

(TSCs) with >30%.4-7 Tandem solar cells can be fabricated mainly with

2 different configurations: mechanically stacked (4‐terminal) or

td. wileyonlinelibr131

monolithically integrated (2‐terminal) tandems. Compared with 4‐ter-

minal configuration,8-11 2‐terminal configuration has less complexity

and better feasibility in application fields.

To date, many experimental works12-16 have been contributed to

reduce current losses and enhance the best matched short‐circuit cur-

rent density in the perovskite/c‐Si monolithic TSCs. The first experi-

mental record was 13.7% by Mailoa et al12 in 2015, and soon

increased to 23.6% by Bush et al14 in 2017 with reduction of parasitic

absorption and recombination in different layers and interfaces by

using more suitable materials and fabricating thinner carrier transport

layers. Nevertheless, even the best optimized record has not yet

exceeded that for pure silicon solar cells1 and far below its theoretical

prediction.5 Therefore, more realistic approach for the mechanism

study would be necessary to understand such difference and search

for any possibilities to change that. Previously, many theoretical

works4-7,17,18 have thoroughly elucidated light trapping in top cell

and optimized front surface textures such as pyramids and inverted

Prog Photovolt Res Appl. 2018;26:924–933.ary.com/journal/pip

http://orcid.org/0000-0003-1118-0054




https://doi.org/10.1002/pip.3037

http://wileyonlinelibrary.com/journal/pip

BA ET AL. 925

nanopyramids by using the finite difference time domain (FDTD) or

transfer matrix method (TMM). Shi et al5 reported perovskite/c‐Si

monolithic TSCs with inverted nanopyramids and achieved current

matching by adding a well‐designed intermediate contact layer as

reflector for short wavelengths. Santbergen et al4 simulated mono-

lithic perovskite/c‐Si tandem devices with different textured configu-

rations and achieved the best matched short‐circuit current density

by tuning interlayer/burial layer refractive index. However, these

results directly ignored parasitic absorption of some layers,4 used

experimentally unreasonable thickness of perovskite layer,6 or

overlooked some electrical properties.5 More detailed theoretical

calculation is probably necessary to include those factors, which could

have played important role and cannot be simply approximated.

In this study, we have introduced a composite method combining

FDTD and light path analysis together for the TSCs or devices for the

first time. The combination, in addition with effective long wavelength

modification, can hopefully resolve the problems induced by the vast

scale difference between grid cells and devices using either FDTD or

analytical calculation alone. Comparing with other methods such as

FIGURE 1 A, Schematic drawing of perovskite/c‐Si monolithic TSCs withP, and H denote the pyramid base angle, period, and height, respectively.λ = 1100 nm, P = 5.0 μm, and H = 3.0 μm. C, Illustration of the absorption pperovskite/c‐Si monolithic TSCs (the dotted lines) and the measured EQEwileyonlinelibrary.com] 132

TMM, FDTD is easier to solve problems and get visual results in time

domain.19,20 We have thoroughly calculated perovskite/c‐Si mono-

lithic TSCs absolutely relying on reliable experimental thicknesses

and materials. Simulated results have shown quite good match with

the latest and best experimental report of external quantum efficien-

cies (EQE) of both subcells.14 We have carried out detailed optimiza-

tion for achieving high efficiency in perovskite/c‐Si monolithic TSCs,

including the cell configuration, thickness and bandgap of perovskite

layers, and current loss in different layers. It is found that there are sig-

nificant current losses in surface reflection of TSCs and parasitic

absorption of indium tin oxide (ITO) layer, which could be the main

ways for the further improvement in experiments. Finally, we have

further suggested that addition of pyramids structure in front surface

can effectively increase theTSC efficiency of 23.6% to 29.05%, which

has given light to the experimental research of real applicable high‐

performance perovskite/c‐Si tandem cells. This work could hopefully

facilitate a more detailed understanding of the optoelectronic mecha-

nisms of perovskite/c‐Si monolithic TSCs and more significant

improvement of their performance in application.

flat front surface and pyramid‐textured rear surface. The parameters α,B, Angular distribution and the 3‐dimensional vector distribution forrocess of the silicon layer. D, Simulated absorptance of both subcells inof both subcells (the curves) [Colour figure can be viewed at

http://wileyonlinelibrary.com

926 BA ET AL.

33

2 | METHODS AND VALIDATION

The simulated monolithic TSCs considered here are the current world

record perovskite/c‐Si TSCs with an efficiency of 23.6%.14 The solar

cell, of which a schematic drawing is shown in Figure 1A, consists of

a 150‐nm‐thick lithium fluoride (LiF) antireflective layer, a 150‐nm‐

thick transparent conductive oxide layer of top ITO with a carrier

concentration of 2.0 × 1020 cm−3, a 10‐nm‐thick electron transport

layer of PC60BM, a 464‐nm‐thick perovskite layer (Cs0.17FA0.83Pb

(Br0.17I0.83)3, with a bandgap of 1.63 eV), a 28‐nm‐thick hole trans-

port layer of NiO, a 20‐nm‐thick transparent conductive oxide layer

of ITO with a carrier concentration of 5.0 × 1020 cm−3, and a

280‐μm‐thick single‐side‐textured (SST) c‐Si/amorphous silicon

heterojunction solar cell. This silicon solar cell includes a polished

front surface, a micron‐sized pyramid‐textured rear surface (ca.

5.0 μm), a 300‐nm‐thick silicon nanoparticle (NP) layer (with a refrac-

tive index of 1.4), and a 200‐nm‐thick silver back layer. Other

amorphous silicon thin film layers whose thicknesses were less than

10 nm were eliminated from our optical model. The thicknesses of

all layers were all acquired from Bush et al,14 and the refractive

indexes and extinction coefficients of all materials were obtained

from the recent literatures.21-25 As a highly insulating layer, LiF had

negligible influence on the total absorption because of the extremely

low extinction coefficients and high work function.21 Therefore, we

fixed the thickness (150 nm), refractive index (1.39), and extinction

coefficient (approximate 0) of LiF layer.

First, we used FDTD simulations package in the Lumerical FDTD

Solutions software (version 8.17.1072, 2017a) to perform the optical

calculations and then to compute the TSC electrical characteristics.

FDTD is a time domain method by using finite difference approxima-

tions to solve Maxwell's equations.19,20,26,27 Except of neglecting

diffraction and local electric field effects, it is more intuitive than other

techniques and easier to get visual results which are good for design-

ing and analyzing simulated structure.28-30 We set corresponding

parameters in FDTD as follows. The incident light plane wave was

set to have a spectrum close to AM 1.5G (with λ between 300 and

1200 nm) and was oriented in the negative z‐direction (see Figure 1

A). The thicknesses, refractive indexes, and extinction coefficients of

different layers were set by adding corresponding structures and

materials in FDTD. We obtained the normalized reflectance R (λ) and

the transmittance T (λ) into c‐Si using frequency‐domain transmission

monitors set on the top surface of the total solar cell and on the inter-

face between ITO and silicon. We used the “power absorbed” (Pabs)

analysis group in the FDTD package to get the absorptance of specific

layers including the PC60BM, perovskite, NiO, top ITO, and ITO layers

by surrounding corresponding layers. Perfectly matched layer bound-

ary conditions were used in the z‐direction, and periodic boundary

conditions were used in the x‐y directions.

We then directly acquired the reflectance and the absorptance in

top layers from the frequency‐domain transmission monitor and the

Pabs analysis group, respectively. The electron extraction was effective

and sufficient because of appropriate work function resulting in

forming an accumulation layer between ITO and PC60BM,31-33 so we

assumed internal quantum efficiency of unity in the simulated mate-

rials. We can thus obtain the short‐circuit current density (Jsc) by1

integrating the photon flux of the AM 1.5G solar spectrum with the

corresponding absorptance. The Jsc was calculated using Equation (1):

Jsclayerð Þ ¼ q

hc∫λEAM1:5G λð ÞPabs layerð Þ λð Þdλ; (1)

where EAM1.5G(λ) is the incident photon energy flux and q is the elec-

tron charge.

We can calculate the Jsc of most layers including the PC60BM,

perovskite, NiO, top ITO, and ITO layers, but the Jsc of the silicon layer

cannot be obtained in this way. This is because the difference in scale

between the top and bottom cells (1 vs. 280 μm) makes the Pabs calcu-

lations in the silicon layer prohibitively memory intensive or even

makes it not accomplishable. Shi et al5 obtained the Jsc of silicon by

subtracting the Jsc of the top layers from the full‐spectrum current

density calculated with an internal quantum efficiency of 1, but the

simulated Jsc of silicon with this method is larger than that measured

experimentally especially for λ > 1000 nm because of the ignorance

of the rear surface reflection for long wavelengths. Gee et al34

obtained the absorptance of silicon by assuming that the rays inside

the silicon followed a random angular distribution. Nevertheless, a

random angular distribution cannot be used to correctly describe the

real light path here, as the incident light and reflected light paths are

still fairly vertical.

Here, we proposed a hybrid method of FDTD with light path

analysis to effectively simulate some special device structures, which

cannot be directly treated by FDTD. Indeed, we used a frequency‐

domain transmission monitor (Rext) positioned at the interface

between ITO and silicon to get the light incident into silicon bottom

cell. After absorbing by silicon, some of photons arrived in the silicon

rear surface. Then, we used a frequency‐domain transmission monitor

positioned at the top of the silicon back surface (red dotted line shown

in Figure 1A) to better understand the light path inside the silicon

layer. By converting from vector coordinates to angles measured from

the vector directions to the positive z‐axis, we can get the angular

distribution of the light reflected by the back surface of silicon. The

3‐dimensional vector distribution for λ = 1100 nm along with the

angular distribution is presented in Figure 1B. The reflected light with

the angular distribution comes back toward the internal front surface

of the silicon layer with a transmittance (Tθ) and a reflectance (Rint

(θ,θ′)). By iteratively summing the absorptance over the different pos-

sible reflection angles, we can calculate a more accurate absorptance

value in the silicon layer. The absorptance of silicon is given as34

PabsSið Þ ¼ 1−Rextð Þ× 1−T0ð Þ½

þ∑θ;θ′ T0Rbr θð Þ 1−Tθð Þ þ T0TθRbr θð ÞRint θ;θ′� �

1−Tθ′ð Þ þ⋯h ii

;

(2)

where, as illustrated in Figure 1C, Rext is the normalized reflectance at

the initial interface, Tθ is the transmittance of an incident light at angle

θ propagating from the top to the bottom surface of silicon layer, Rbr

(θ) is the normalized rear surface reflected angular distribution, and

Rint (θ,θ′) is the normalized front surface reflected angular distribution

for an incident light at angle θ. The incident light for λ < 1000 nm is

rapidly absorbed and does not reach the rear surface, so the calcula-

tion is only valid for long wavelengths.

BA ET AL. 927

34

Next, we used the Shockley diode model to study overall perfor-

mance of realistic perovskite/c‐Si solar cell. This model has been used

and verified in many papers35-38 and can give correct relationships of

electrical characteristics. The open‐circuit voltage (Voc) calculated from

the Jsc by the Shockley diode equation is given as

Voc ¼ kbTq

lnJscJ0

þ 1

� �; (3)

where kb is the Boltzman constant and T is the room temperature

(298 K). J0 is the diode saturation current density which can be

obtained from experimental results: for different top perovskite cells,

J0(Perovskite) was derived from the current density‐voltage curve of

the perovskite solar cells acquired from recent literatures,13,14,39,40

while J0(Si) = 8.51 × 10−12 mA/cm2 was derived from the world record

silicon heterojunction solar cell reported by Taguchi et al41 with

Voc = 0.75 V and Jsc = 39.5 mA/cm2. The fill factor (FF) was calculated

using the well‐established expression.42

FF ¼Voc−

kbTq

lnqVoc

kbTþ 0:72

� �

Voc þ kbTq

: (4)

The efficiency η of the simulated solar cell was obtained by

η ¼ FF×Jsc×Voc

0:1W=cm2(5)

Finally, we show in Figure 1D the plot of the simulated absorp-

tance of both subcells in the monolithic TSCs, together with the mea-

sured EQE of both subcells.14 We have already assumed that every

absorbed photon generates a hole‐electron pair, so the simulated

absorptance is equal to the EQE. It is clear that the simulated absorp-

tance is very close to experimental EQE especially at wavelengths

ranging from 400 to 1200 nm. The main differences between the

absorption and EQE curves are at wavelengths ranging from 800 to

1000 nm. One possible reason is from the thin amorphous silicon thin

film layers (normally ~5 nm) being omitted in the simulation for simpli-

fication, which may increase the reflectance value. Secondly, it may be

induced by the difference between the chosen thickness of silicon in

the simulation and the real value in the experimental samples.14 The

proposed theoretical value may be lower than the real experimental

thickness so that the simulated absorptance of silicon subcell is lower

than the experimental EQE. By integrating the absorptance and EQE

spectra over the AM 1.5G spectrum, we found that the simulated

perovskite top cell and silicon bottom cell generated 18.9 and

18.1 mA/cm2, respectively, which is very close to the measured results

of 18.9 and 18.5 mA/cm2. The differences between the absorptance

and EQE yield less than 2% in the difference of Jsc, which is quite

small for the simulation. The calculated cell's electric parameters

(Jsc 18.1 mA/cm2; Voc 1.66 V; Eff. 23.7%) have little difference with

realistic experimental parameters (Jsc 18.5 mA/cm2; Voc 1.62 V; Eff.

23.6%),14 and can already prove the validity of the calculation

method.43 The reflection and parasitic absorption in the top ITO layer

both play an important role in the current loss. Their optimization will

be discussed in Section 5. 1

3 | PEROVSKITE/C‐SI MONOLITHIC TSCSWITH VARYING PYRAMID SIZES IN REARSURFACE

As a general principle, the geometry of the reflective rear surface

determines the absorptance for long wavelengths in a silicon cell.

Optimizing the pyramid‐textured rear surface is thus a premise for

achieving maximum efficiency in perovskite/c‐Si TSCs. Baker‐Finch

et al44 reported that the characteristic base angle α shown in

Figure 1A of the pyramid texture was close to 50° to 52°. In addition,

Shi et al5 reported the relationship of reflectance at fixed period (P) or

height (H), that was, P had an influence on the position of reflectance

minimum, while H mainly influenced the magnitude of reflectance. We

optimized the size (P and H) of pyramid‐textured rear surface within

that base angle α range after making a trade‐off between the effects

of P and H. Figures 1B and 2A to C show the angular distributions

of the light reflected by the pyramid‐textured rear surfaces in

the perovskite/c‐Si TSCs with 3 different pyramid sizes (for

λ = 1100 nm): P = 5.0 μm and H = 3.0 μm in Figure 1B, planar in

Figure 2A, P = 1.5 μm and H = 0.9 μm in Figure 2B, and P = 2.5 μm

and H = 1.5 μm in Figure 2C, respectively. In comparison with a planar

rear surface in Figure 2A, a pyramid‐textured rear surface increases

the average reflected angle, which increases absorptance, as the light

path is longer. The average reflected angles of the 3 different sizes

of pyramid‐textured rear surfaces are 29° in Figure 2B, 45° in

Figure 2C, and 43° in Figure 1B. Therefore, the best size to achieve

highest average reflected angle is P = 2.5 μm and H = 1.5 μm.

Figure 2D shows the reflectance of the perovskite/c‐Si TSCs with a

planar rear surface and the 3 different pyramid‐textured rear surfaces.

For λ = 1100 nm, the highest reflectance is 0.975, with the planar rear

surface, and the lowest reflectance is 0.92, with P = 5.0 μm and

H = 3.0 μm. It is obvious that a larger size leads to a lower reflectance.

As a conclusion, we need to find the right trade‐off between the

reflectance and absorbed light path to get the maximum absorptance.

By using Equation (2), we calculated the absorptance of TSCs with

different rear surface textures in Figure 2E. The Jsc calculated using

Equation (1) from 1050 to 1200 nm are, respectively, 1.16 mA/cm2

for planar, 2.14 mA/cm2 for P = 1.5 μm and H = 0.9 μm, 2.30 mA/

cm2 for P = 2.5 μm and H = 1.5 μm, as well as 2.28 mA/cm2 for

P = 5 μm and H = 3 μm. Compared with the planar rear surface, the

Jsc of the pyramid‐textured rear surface with P = 2.5 μm and

H = 1.5 μm can be increased by a factor of 2 from 1.16 to 2.30 mA/

cm2. The best geometric parameters were found to be P = 2.5 μm

and H = 1.5 μm, different from those used in experiment with ca.

P = 5.0 μm and H = 3.0 μm. That is, we can still optimize the size of

pyramids of back surface in Bush et al14 to get the best absorptance

at long wavelengths.

To prove the veracity of our simulation, we further compared with

the EQE results, reported by Werner et al,13 of perovskite/c‐Si TSCs

on double‐side‐polished (DSP) and SST silicon bottom cells without

antireflective layer. The detailed structures are as followed: ITO

(150 nm)/Sprio‐OMeTAD (150 nm)/perovskite (MAPbI3, 300 nm)/

PC60BM (20 nm)/ITO (30 nm)/silicon (300 μm)/ITO (100 nm)/Ag

(150 nm) with a micron‐sized (ca. P ~5 μm) pyramid‐textured SST

and a DSP rear surface. We show in Figure 2F the plot of the

FIGURE 2 A‐C, Angular distributions in the perovskite/c‐Si monolithic TSCs with planar and 2 different pyramid sizes (for λ = 1100 nm): planar,P = 1.5 μm and H = 0.9 μm and P = 2.5 μm and H = 1.5 μm. D, E, Reflectance and absorptance of the perovskite/c‐Si monolithic TSCs with planarand 3 different pyramid sizes (for wavelengths between 1050 and 1200 nm). F, Measured (red line) and the simulated (black line) absorptance ratioof SST and DSP TSCs [Colour figure can be viewed at wileyonlinelibrary.com]

928 BA ET AL.

measured absorptance ratio (red line) of SST and DSP TSCs, together

with the simulated ratio (black line). For wavelengths between 1050

and 1120 nm, the simulated and measured ratios are extremely

consistent. For wavelengths between 1120 and 1150 nm, the simu-

lated ratio is higher than the measured ratio, which is because of the

extinction coefficient of silicon in the bandgap (1.124 eV) being not

accurate enough.

35

4 | PEROVSKITE/C‐SI MONOLITHIC TSCSWITH VARYING BANDGAPS ANDTHICKNESSES OF PEROVSKITE LAYERS

In this section, to achieve the best matched short‐circuit current den-

sity, we have simulated various perovskite/c‐Si monolithicTSCs with 6

different bandgaps (Eg) of perovskite layers from 1.51 to 2.30 eV and

12 different perovskite layer thicknesses (dp) from 200 to 750 nm. The

thicknesses of the other layers were kept as in the previous section, as

shown in Figure 1A. To calculate the absorption, we used the refrac-

tive indexes and extinction coefficients of perovskites with different

bandgaps given in Ndione et al,24 and calculated the absorption curve

using FDTD. The bandgaps were from perovskites of different compo-

sitions, and the samples were tested with reproducible experiments.24

In short wavelength, the extinction coefficients remain almost

unchanged for different bandgaps, while the extinction coefficients

in long wavelength are negatively proportional with the bandgaps.

We then calculated the Jsc(Perovskite) and the Jsc

(Si) using Equation (1).

The results presented in Figure 3A, B reveal that, in general, a larger

dp leads to a larger Jsc(Perovskite) in the top cell and a smaller Jsc

(Si) in

the bottom cell. Meanwhile, a larger Eg results in a smaller Jsc(Perovskite)

in the top cell and a larger Jsc(Si) in the bottom cell. Therefore, there

exists a region in which the Jsc(Perovskite) in the top cell can be equal

to the Jsc(Si) in the bottom cell. 1

To further elucidate the light splitting mechanism, we have pro-

vided the wavelength dependence of the overall absorptance on the

parameters dp and Eg, respectively. Figure 3C, D shows the absorp-

tance of perovskite and silicon layers for various Egs (from 1.51 to

2.30 eV) at fixed dp (450 nm). An increase in Eg results in a decrease

in absorptance for a fixed wavelength. Therefore, the Eg determines

the maximum absorptance of the perovskite layer at fixed dp. When

Eg is larger than 1.8 eV (700 nm), it is impossible to equalize

Jsc(Perovskite) and Jsc

(Si) by tuning dp. Figure 3E, F illustrates the absorp-

tance of the perovskite and silicon layers for different dps (from 200 to

500 nm) at fixed Eg (1.62 eV). Obviously, at wavelengths ranging from

300 to 500 nm, the absorptance of the perovskite layer is not depen-

dent on dp. This is because most of the light is absorbed by the top

ITO layer and the perovskite layer. So, it does not reach the bottom

of the perovskite layer. At wavelengths ranging from 500 to 700 nm,

an increase in dp leads to an increase in absorptance for the perovskite

layer, which indicates that we can equalize Jsc(Perovskite) and Jsc

(Si) by

tuning dp at fixed Eg. The dotted curves in Figure 3A, B represent

the regions where Jsc(Perovskite) and Jsc

(Si) are matched. Along these 2

curves, the common value of Jsc(Perovskite) and Jsc

(Si) is equal to ca.

18.3 mA/cm2, slightly larger than the experimental result of

18.1 mA/cm2.14

5 | PEROVSKITE/C‐SI MONOLITHIC TSCSWITH OPTIMIZING CURRENT LOSSES

In Section 4, we have achieved the best matched short‐circuit current

density by tuning perovskite layer thicknesses (dp) and bandgaps (Eg).

However, the cell surface reflection and parasitic absorption in the

ITO layer still play an important role in the current loss. Therefore,

we analyzed the current loss in different layers of the best‐matched

perovskite/c‐Si TSCs discussed in Section 4. As shown in Figure 4A,


FIGURE 3 A, B, Contour of the short‐circuit current density of perovskite layer (Jsc(Perovskite)) and silicon layer (Jsc

(Si)) with different bandgaps (Eg)and thicknesses (dp), respectively. The black dotted curves mark that the common value of Jsc

(Perovskite) is equal to Jsc(Si). C, D, Absorptance of

perovskite layer and silicon layer for various Egs (from 1.51 to 2.30 eV) at fixed dp (450 nm). E, F, Absoprtance of the perovskite and silicon layersfor different dps (from 200 to 500 nm) at fixed Eg (1.62 eV) [Colour figure can be viewed at wileyonlinelibrary.com]

BA ET AL. 929

36

obviously, the cell surface reflection plays the most important roles in

current loss and the minimum is ca. 6.9 mA/cm2. The second loss

comes from the parasitic absorption in the top ITO layer (ca.

2.1 mA/cm2). Besides, the values of parasitic absorption in PC60BM,

NiO, and other layers are ca. 0.5, 0.2, and 0.1 mA/cm2, respectively.

These losses in surface reflection and top ITO layer will induce ca.

6% of efficiency drop compared to the whole, which can explain the

limit of current experimental result as long as they keep flat in front

surface. Therefore, optimizing the cell surface reflection and the para-

sitic absorption in the top ITO layer is the best way to enhance the

efficiencies of the monolithic TSCs.

In general, the most effective way to improve reflection is to tex-

ture the front surface of bottom silicon solar cells,45 because the thick-

ness of top perovskite solar cell is too thin compared to the common

size of pyramid texture. Therefore, we added a pyramidal front surface

into our structure as shown in Figure 4C (the thicknesses and mate-

rials of others layers were kept identical to those in Figure 1A). We

optimized the sizes of pyramid‐textured front surface (period (Ptop))1

and rear surface (period (Pbottom)). The base angle α is within the base

angle range discussed in Section 3. In Figure 4B, we show the short‐

circuit current loss caused by reflection (Jsc(R)). The minimum is

reached when Ptop and Pbottom are equal to ~1 and ~2.5 μm, respec-

tively, as marked by a dotted oval in Figure 4B. Compared with the

value 6.9 mA/cm2 of a structure with flat front surface as shown in

Figure 4A, the minimum short‐circuit current loss can be reduced to

2.9 mA/cm2 in the optimized pyramid‐textured front and rear surfaces

of the best‐matched perovskite/c‐Si TSCs.

Holman et al22 reported that the top ITO layer, which serves as an

antireflection coating, should have a uniform thickness for a given

structure. The thicknesses of top ITO layers are both 150 nm in arti-

cles reported by Werner et al13 and Bush et al.14 In addition, the

absorptance of ITO is mainly determined by extinction coefficients

which can be easily tuned by carrier density.22,46 Therefore, we just

optimized the current loss in the top ITO layer by tuning the carrier

densities of the top ITO layer but not changing the thicknesses. The

refractive indexes and extinction coefficients of different carrier


FIGURE 4 A, Current losses of the best current matched perovskite/c‐Si monolithic TSCs with flat front surface. B, Contour of the short‐circuitcurrent density caused by reflection (Jsc

(R)) with different Ptops and Pbottoms. C, Schematic drawing of perovskite/c‐Si monolithic TSCs withpyramid‐textured front and pyramid‐textured rear surfaces. The parameters α, Ptop, and Pbottom denote the pyramid base angle and top and bottompyramid periods, respectively. D, Absorptance of different top ITO layers with carrier densities ranging from 2.5 × 1019 to 6.0 × 1020 cm−3. E,Current loss in top ITO layers with various carrier densities from 2.5 × 1019 to 2.0 × 1020 cm−3 [Colour figure can be viewed at wileyonlinelibrary.com]

930 BA ET AL.

densities of ITO materials were obtained from the recent literature.22

Figure 4D shows the absorptance of different top ITO layers with

carrier densities ranging from 2.5 × 1019 to 6.0 × 1020 cm−3. We can

easily conclude that, for wavelengths between 300 and 500 nm, a

higher top ITO carrier density leads to a lower absorptance of the

top ITO layer, but for wavelengths between 500 and 1200 nm, it is the

opposite. We also can see that the absorptance at wavelengths

between 300 and 500 nm is much higher than that at wavelengths

between 500 and 1200 nm. Hence, there is a best top ITO carrier den-

sity for the lowest current loss in the top ITO layer. We calculated the

current loss in different top ITO layers with various carrier densities

from 2.5 × 1019 to 2.0 × 1020 cm−3 as shown in Figure 4E. The lowest

current loss in the highest conductivity top ITO layer is ca. 0.9 mA/

cm2 with a corresponding carrier density of 5.0 × 1019 cm−3. After this

optimization, we can further increase short‐circuit current density by

1.2 mA/cm2 (absolute).

TABLE 1 Efficiencies and the corresponding parameters of the bestcurrent matched perovskite/c‐Si monolithic TSCs with flat and pyra-mid‐textured front surfaces

Texture Material Eg (eV)dp(nm)

Jsc (mA/cm2)

Voc

(V) η (%)

Flat FA0.85Cs0.15PbI3 1.51 236 18.25 1.53 21.13MAPbI3 1.56 272 18.26 1.75 24.28FA0.85Cs0.15PbBrI2 1.62 418 18.26 1.66 23.96FA0.85Cs0.15Pb (Br0.4I0.6)3

1.76778 18.25 1.82 25.28

Pyramid FA0.85Cs0.15PbI3 1.51 240 20.51 1.53 24.44MAPbI3 1.56 290 20.47 1.76 27.96FA0.85Cs0.15PbBrI2 1.62 470 20.45 1.67 26.54FA0.85Cs0.15Pb (Br0.4I0.6)3

1.76820 20.44 1.83 29.05

37

6 | DISCUSSION

In previous sections, we have optimized the size of pyramid‐textured

rear surface, the thickness (dp) and bandgap (Eg) of the perovskite

layer, the current losses in the top ITO layer, and the reflection,

respectively. We have also calculated the electrical characteristics of

the best current matched perovskite/c‐Si monolithic TSCs as shown in

Table 1.

On the upper part of Table 1, the monolithic TSCs, as shown in

Figure 1A, have the following structure: flat front surface/LiF

(150 nm)/top ITO (150 nm, with a carrier concentration of

2 × 1020 cm−3)/PC60BM (10 nm)/perovskite (material, Eg and dp as

seen in Table 1)/NiO (28 nm)/ITO (20 nm, with a carrier concentration1

of 5.0 × 1020 cm−3)/c‐Si (280 μm)/silicon NP (300 nm)/Ag (200 nm)/

pyramid‐textured rear surface (Pbottom ~2.5 μm). The best efficiency

is 25.28% when the material of the perovskite layer is FA0.85Cs0.15Pb

(Br0.4I0.6)3 (Eg = 1.76 eV). Its absorption and reflection characteristics

are shown in Figure 5A. The best‐matched current Jsc is ca.

18.25 mA/cm2. Besides, the current losses in the top ITO layer,

PC60BM, and reflection are ca. 2.13, 0.52, and 6.81 mA/cm2, respec-

tively. Compared with the experimental efficiency of 23.6%,14 we

can increase the efficiency by 1.68% (absolute) through optimizing

the material of the perovskite layer. In addition, we can easily fabricate

the corresponding devices by using the method reported by Bush

et al14 and only simply changing the material and thickness of the

perovskite layer.

After addressing all the optimizations, on the lower part of Table 1

, the monolithic TSCs, as shown in Figure 4C, have the following struc-

ture: pyramid‐textured front surface (Ptop ~1 μm)/LiF (150 nm)/top

ITO (150 nm, with a carrier concentration of 5 × 1019 cm−3)/PC60BM



FIGURE 5 Simulated absorptance and reflectance of the best current matched perovskite/c‐Si monolithic TSCs with (A) flat and (B) pyramid‐textured front surfaces (perovskite material is FA0.85Cs0.15Pb (Br0.4I0.6)3), respectively [Colour figure can be viewed at wileyonlinelibrary.com]

BA ET AL. 931

(10 nm)/perovskite (material, Eg and dp as seen in Table 1)/NiO

(28 nm)/ITO (20 nm, with a carrier concentration of 5.0 × 1020 cm−3)/c‐Si (280 μm)/silicon NP (300 nm)/Ag (200 nm)/pyramid‐textured

rear surface (Pbottom ~2.5 μm). Accordingly, the best efficiency is

29.05%, and the best‐matched current Jsc is ca. 20.44 mA/cm2. Its

absorption and reflection characteristics are shown in Figure 5B. We

can enhance the efficiency by 3.77% (absolute) compared with the

best flat front surface result of 25.28%. In this case, the current losses

in the top ITO layer, PC60BM, and reflection are ca. 1.73, 0.57, and

2.59 mA/cm2, respectively.

As for the preparation method,14 the deposition methods of ITO,

PC60BM, and LiF, such as sputtering, pulsed chemical vapor deposi-

tion, and atomic layer deposition, can be easily transferred to the fab-

rication process of pyramid‐textured perovskite/c‐Si TSCs, as shown

in Figure 4C. However, it is difficult to fabricate a uniform perovskite

layer above a pyramid‐textured front surface using traditional

methods, such as 1‐step precursor solution deposition,47 2‐step

sequential deposition,48 and dual‐source vapor deposition.49 There-

fore, we have to find a new method to achieve it. Fortunately, there

may be some newly developed methods to achieve it, such as an

electric field‐assisted reactive deposition approach reported by Zhou

et al50 and a solvent‐free deposition method reported by Chen

et al.51 Therefore, we can be confident that efficiencies above 29%

can be achieved in perovskite/c‐Si monolithic TSCs.

*Correction added on 2 July 2018, after first online publication: the issue and

page details for these references have been corrected.38

7 | CONCLUSIONS

In summary, we have introduced a composite method combining

FDTD and light path analysis together for the tandem cell or devices.

We have presented the optimized results of perovskite/c‐Si mono-

lithic TSCs with flat and pyramid‐textured front surfaces from optical

and electrical point of view by using this method. We have found an

optimized set of pyramid parameters Pbottom ~2.5 μm in rear surface

that enhances the absorption in long wavelengths (λ > 1000 nm).

The best‐matched short‐circuit current density in a planar inverted

structure is restricted by parasitic absorption in top ITO (ca. 2.1 mA/

cm2) and in cell surface reflection (ca. 6.8 mA/cm2). By using the opti-

mized carrier density of top ITO (5 × 1019 cm−3) and sizes of pyramids

in front surface of bottom silicon solar cells (Ptop ~1 μm), the parasitic

absorption in top ITO and in reflection is reduced to 1.7 and 2.5 mA/

cm2. Finally, the best‐calculated efficiency of 29% is achieved at1

1.76 eV perovskite bandgap and FA0.85Cs0.15Pb (Br0.4I0.6)3 perovskite

material in monolithic configuration. These results will provide useful

guidelines for the realization of high‐efficient perovskite/c‐Si mono-

lithic TSCs.

ACKNOWLEDGEMENT

This work was supported by the Natural Science Foundation of China

(11474201, 11674225, and 61234005).

ORCID

Lixiang Ba http://orcid.org/0000-0003-1118-0054

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page details for these references have been corrected. 140

How to cite this article: Ba L, Liu H, Shen W. Perovskite/c‐Si

tandem solar cells with realistic inverted architecture: Achiev-

ing high efficiency by optical optimization. Prog Photovolt Res

Appl. 2018;26:924–933. https://doi.org/10.1002/pip.3037

https://doi.org/10.1002/pip.3037

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Showcasing collaborative research from the Suzhou University

of Science and Technology, Suzhou, China, the University of

Oxford, Oxford, UK, the Chinese University of Hong Kong,

Hong Kong SAR, China, the Shanghai Jiao Tong University,

Shanghai, China and Donghua University, China.

Controllable rotational inversion in nanostructures with dual

chirality

Dai, Zhang, Goriely, and co-workers report the controllable

rotational inversion in the helices with dual chirality: from gourd/

cucumber tendrils to helical nanobelts. A peculiar rotational

inversion of overwinding followed by unwinding, observed in

some gourd and cucumber tendril perversions, not only exists in

the transversely isotropic dual-chirality helical nanobelts, but also

in the binormal/normal ones whose rectangular sections close to

a square.

As featured in:

rsc.li/nanoscale

See Lu Dai, Li Zhang et al. , Nanoscale , 2018, 10 , 6343.

Nanoscalersc.li/nanoscale

ISSN 2040-3372

COMMUNICATION Zhenyu Zhang, Nan Jiang et al. In situ TEM observation of rebonding on fractured silicon carbide

Volume 10 Number 14 14 April 2018 Pages 6217–6760

Nanoscale

PAPER

Cite this: Nanoscale, 2018, 10, 6343

Received 4th December 2017,Accepted 19th February 2018

DOI: 10.1039/c7nr09035h

rsc.li/nanoscale

Controllable rotational inversion in nanostructureswith dual chirality†

Lu Dai, *a,e Ka-Di Zhu,b Wenzhong Shen,b Xiaojiang Huang,c Li Zhang *d andAlain Gorielye

Chiral structures play an important role in natural sciences due to their great variety and potential appli-

cations. A perversion connecting two helices with opposite chirality creates a dual-chirality helical struc-

ture. In this paper, we develop a novel model to explore quantitatively the mechanical behavior of normal,

binormal and transversely isotropic helical structures with dual chirality and apply these ideas to known

nanostructures. It is found that both direction and amplitude of rotation can be finely controlled by design-

ing the cross-sectional shape. A peculiar rotational inversion of overwinding followed by unwinding,

observed in some gourd and cucumber tendril perversions, not only exists in transversely isotropic dual-

chirality helical nanobelts, but also in the binormal/normal ones when the cross-sectional aspect ratio is

close to 1. Beyond this rotational inversion region, the binormal and normal dual-chirality helical nanobelts

exhibit a fixed directional rotation of unwinding and overwinding, respectively. Moreover, in the binormal

case, the rotation of these helical nanobelts is nearly linear, which is promising as a possible design for

linear-to-rotary motion converters. The present work suggests new designs for nanoscale devices.

Introduction

Chiral structures play a prominent role in many natural andtechnology processes ranging from protein configuration,1,2

development, to nanomechanics.3,4 Helices with given chiralityare critical elements in a host of applications at the nanoscaleas they provide simple springs and, more importantly, a directway to convert linear motion to rotational motion androtational motion to linear motion. An example of this conver-sion process is the functionalized helical micro-/nano-swim-mers, which are optimized to have a pure rotation translationalong their helical axis.5 Such swimming robots are a promis-ing tool for single-cell-targeted drug, DNA, and enzyme deliv-ery in vitro as well as in vivo.6–8 Conversely, in the transform-ation from linear motion to rotational motion, elasticity playsa key role. Yet, the linear regime of simple springs is limiteddue to torsional lock-up: as a spring is pulled in simple exten-sion, it quickly stiffens due to its inability to untwist without

one of the ends turning. An elegant solution to this mechani-cal problem, first proposed in ref. 9, is to design a “twistlessspring” by using filaments that exhibit both left and right chir-ality connected by a short inversion called a “perversion”, aterm introduced by the mathematician J. B. Listing to describethe reversal of one chiral structure into another.10

This kind of helix with dual chirality was first described inplant physiology in a letter of André-Marie Ampère.11,12 ThenCharles Darwin pointed out that a tendril with perversioncreates a twistless flexible elastic structure connecting a climb-ing plant to its support13 (see ref. 14 for historical details).Inspired by the tendrils, it is found that an inverted structurecan be created through an instability in a filament with intrin-sic curvature under tension by either decreasing the tension orincreasing the intrinsic curvature.15 The structure emerginghas dual chirality and, due to its particular cancellation oftwist, has an excellent mechanical behavior of tension-exten-sion close to an ideal linear Hookean response.9 Moreover, ahelical structure with dual chirality has a remarkablerotational property during extension: it is reported that someyoung and old cucumber tendril coils unwind and overwindwith axial extension, respectively.16

There is yet another peculiar rotational behavior of heliceswith dual chirality. As presented in Fig. 1(a), we find that someyoung gourd and cucumber tendrils, always overwind in thebeginning of axial extension, and then unwind when elonga-tion is further increased (ESI Movie S1† for gourd and ESIMovie S2† for cucumber). In this experiment, we use a slow

†Electronic supplementary information (ESI) available. See DOI: 10.1039/c7nr09035h

aSchool of Mathematics and Physics, Suzhou University of Science and Technology,

Suzhou 215009, China. E-mail: [email protected] of Physics and Astronomy, Shanghai Jiao Tong University,

800 Dongchuan Road. Minhang District, Shanghai 200240, ChinacCollege of Science, Donghua University, Shanghai 201620, ChinadDepartment of Mechanical and Automation Engineering, The Chinese University of

Hong Kong, Shatin NT, Hong Kong SAR, China. E-mail: [email protected] Institute, University of Oxford, Oxford OX2 6GG, UK

This journal is © The Royal Society of Chemistry 2018 Nanoscale, 2018, 10, 6343–6348 | 6343141

www.rsc.li/nanoscale

http://orcid.org/0000-0003-4145-7667

http://orcid.org/0000-0003-1152-8962

http://crossmark.crossref.org/dialog/?doi=10.1039/c7nr09035h&domain=pdf&date_stamp=2018-04-04

axial loading (of about 0.3 cm s−1 and 0.05 coil length persecond) so that the tendril is in a quasi-static equilibrium atall times. Fig. 1(b) displays the rotation states of a gourdtendril coil during axial loading. The gourd tendril coil firstoverwinds when the elongation increases to 1.6 cm, and thenunwinds during the rest of the loading process. Interestingly,this non-monotonic behavior, known as the twist-stretchcoupling, also exists in the microscopic single-chirality DNAmolecules.17,18 As extension increases, each point on the DNArotates around the axis by first overwinding around it (addinga further twist in the spring) and then unwinding it (henceremoving the twist).19,20

Helical rods can be classified into three types: transverselyisotropic helices (with rotationally invariant sections such asthe squares and circles), and normal and binormal heliceswith non-rotational invariant sections (such as the rectangularor elliptical cross-sections).21 In this paper, we study therotational and extensional behaviour of nanohelices with dualchirality that have either transverse isotropy or are composedof normal and binormal helices (referred to here as “nano-belts”). A binormal dual-chirality nanobelt can be fabricatedvia a strain-driven self-rolling mechanism.22 These structuresare known to unwind during the axial extension.23 A normaldual-chirality nanobelt can be realized by 3D direct laserwriting, which has been used to print single normal nano-helices.24 A transversely isotropic cellulosic micro/nano-fiberwith dual-chirality can be produced by electrospinning in

liquid crystalline solutions.25 Under electronic beam exposure,a suspended cellulosic fiber exhibits unwinding and overwind-ing behavior.26 Therefore, it is of great practical significance toprovide an accurate theoretical description of the mechanicalproperties for the normal, binormal, and isotropic dual-chiral-ity nanohelices.

In this paper, we provide a theoretical basis for the mech-anics of normal, binormal, and transversely isotropic helicalnanostructures with dual chirality by employing a generalextensible rod theory. We show that by modifying the shape ofthe cross-section, one can tune the rotational properties ofdirection and amplitude of these structures to obtain a linearrotational response under extension. In particular, a controlla-ble rotational inversion can be obtained from the dual-chiralitynanohelices of transverse isotropy, as well as of normal/binor-mal in a narrow region defined by the aspect ratio of the rec-tangular cross-section.

Modeling

The general set-up of our model is shown in Fig. 2(a)–(d). Weassume that a rod with width w and thickness t (w > t ) rolls upinto a uniform helical structure with dual chirality HPI, withradius a0, pitch b0, and N0 helical turns. This structure isslowly loaded by an axial force F, while the ends are preventedfrom rotating. In this process, the structure is transformedinto another helical structure with dual chirality HPF, with aradius a, pitch b, and N helical turns. The director basis Di (i =1, 2, and 3) consisting of the normal, binormal, and tangentvectors of HPI and the director basis di of HPF are described bytheir Euler angles (φ0, θ0, ψ0) and (φ, θ, ψ), respectively.27,28 We

Fig. 1 (a) A gourd tendril coil. (b) The rotation states of a gourd tendrilcoil during an axial loading process.

Fig. 2 Schematic illustration of the dual-chirality nanohelices with therectangular cross-section HPI of (a) binormal and (c) normal nanohelices.The configurations of the corresponding elongated nanohelix with thedual chirality HPF of (b) binormal and (d) normal after loading by thetensile force F along the helical axes. The corresponding cross-sectionsof the whole (e) binormal and (f ) normal nanohelices with dual chiralityat the mirror symmetry axes.

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6344 | Nanoscale, 2018, 10, 6343–6348 This journal is © The Royal Society of Chemistry 2018142

can use the mirror symmetry of these structures to our advan-tage by modeling their response as two helical springs whereone end is free to rotate. We follow the terminology of ref. 29and denote S to be the arc length along the fixed referenceconfiguration HPI and s the arc length along the deformed con-figuration HPF. The corresponding derivatives are defined

by cð�Þ ¼ @ð�Þ=@S and ð�Þ�¼ @ð�Þ=@s. Based on the general elastic

rod theory, the derivation processes for the radii a0 and a,pitches b0 and b as well as the loading force F, the torquealong the helix axis M of each helix of HPF are the same asthose of a loaded helical structure with two ends restrictedfrom winding,29 except that in this situation ψ ≠ ψ•0 due to thefact that N ≠ N0. Therefore, the radius and pitch are given by:

a0 ¼ sin θ0ψ0

; b0 ¼ 2π cos θ0ψ0

: ð1Þ

a ¼ 1ψ

FE3

cos θ þ 1� �

� FE1

cos θ� �

sin θ;

b ¼ 2πψ

FE1

sin2 θ þ FE3

cos θ þ 1� �

cos θ� �

:

ð2Þ

For a helix with N helical turns of HPF, the number of turnsN is given by:30

N ¼ ψ

2πl0; ð3Þ

where l0 ¼ N0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2πa0Þ2 þ b02

qis the length of the coil wire of HPI.

The balance of force and moment connect the axial force Fand axial moment M to the deformed shape of the helicalstructure by the following two relations:

1E3� 1E1

� �cos θ sin θF2 þ sin θF � C ψcos θ � ψ0cos θ0ð Þψ sin θ

þ EI1 1� 1� Δð Þδi2½ � ψ sin θ � ψ0 sin θ0ð Þψ cos θ ¼ 0;

ð4Þ

M ¼ EI1½1� ð1� ΔÞδi2�ðψ sin θ � ψ0 sin θ0Þ sin θ

þ Cðψ cos θ � ψ0 cos θ0Þ cos θ;ð5Þ

where Δ ≡ I2/I1, i = 1 for a normal (i = 2 for a binormal) helixand I1 = w3t/12 and I2 = wt3/12 are the moments of inertia of arectangular cross-section. δi2 is the Kronecker delta. E1 = KGtw,E3 = Etw and C = 4GI1I2/(I1 + I2) according to the scaled tor-sional stiffness.31 K is the Timoshenko shear coefficient andrelated to Poisson’s ratio ν through K = (5 + 5ν)/(6 + 5ν).32

E and G = E/2(1 + ν) are the Young’s and shear moduli of thematerial, respectively.33

It is of particular interest to look at the case M = 0 for aloaded helical structure, corresponding to the case where oneend is free to rotate. From (4), we obtain

ψ ¼ EI1½1� ð1� ΔÞδi2�sin θ0 sin θ þ C cos θ0 cos θEI1½1� ð1� ΔÞδi2�sin2 θ þ C cos2 θ

ψ0: ð6Þ

Fig. 2(e) and (f) are the cross-sections of the binormal andnormal helical structures with dual chirality at the mirror sym-metry axes, respectively. The perversions before and after

loading are presented by the solid and dashed rectangles,respectively. Φ is the rotation angle of the free end of thehelical structure, i.e., the rotation angle of perversion:

Φ ¼ 360°� ðN � N0Þ: ð7ÞThe spring constant of each helical structure of HPF is

deduced from (1)–(4), in the linear limit h = dF/d(Nb):

hS ¼ � P1P32ðP1P4 þ P2Þ ;

P1 ¼ 1E3� 1E1

� �sin θ � cos2 θ

sin θ

� �F2 � cos θ

sin θF þ Q6

2cos θsin θ � cos2 θ

sin θ

� �ψ2

þ 2sin θQ6ðQ4Q5 þ Q3Q6Þ

Q42 � Q3

sin θ

� �ψ0ψ

þ sin θQ5ðQ4Q5 þ Q3Q6Þ

Q42 ψ0

2; P2 ¼ Q2 sin θ; P3 ¼ 1Q2l0

;

P4 ¼ �Q1

Q2;

where,

Q1 ¼ 1E3� 1E1

� �cos2 θ þ 1

E1;

Q2 ¼ 21E3� 1E1

� �F cos θ þ 1;

Q3 ¼ EI1½1� ð1� ΔÞδi2� sin θ0 sin θ þ C cos θ0 cos θ;

Q4 ¼ EI1½1� ð1� ΔÞδi2�sin2 θ þ C cos2 θ;

Q5 ¼ � EI1½1� ð1� ΔÞδi2�sin θ0cos θsin θ

þ C cos θ0;

Q6 ¼ 2ðEI1½1� ð1� ΔÞδi2� � CÞcos θ:

ð8Þ

Since the two opposite-handed helical structures of HPF areconnected in series, the spring constant of HPF is:

hP ¼ hS2: ð9Þ

Using (1)–(9), we can obtain the radius a, pitch b, N helicalturns and the spring constant hP of the helical structure withdual chirality HPF from the known radius a0, pitch b0, thenumber of turns in each helix N0 of HPI and the loading force F.

Results

In order to understand the mechanical behavior of nanohe-lices with dual chirality, we analyze a rolled-up nanohelix. Thestrain-induced self-scrolling mechanism is a highly controlla-ble fabrication method that allows to create dual-chiralitynanohelices with adjustable helix angles. Fig. 3(a) shows that abinormal dual-chirality nanohelix is fabricated from the sym-metric V-shaped SiGe/Si/Cr nanobelt, which leads to a left andright-handed arm having the same geometry parameters. The8/10 nm thick SiGe/Si hetero-structures with approximately40% Ge in the SiGe layer were epi-grown by chemical vapordeposition (CVD) on the Si(110) substrates. The 13 nm thickamorphous Cr layers were deposited by e-beam evaporation.The details of the SiGe/Si/Cr pattern fabrication and the wet

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chemical etching for the subsequent underetching aredescribed elsewhere.22,34 The binormal nanohelix with dualchirality has a radius a0 = 1.18μm, pitch b0 = 4.47μm and N0 = 6.The insets present the V-shaped mesa designs of 60°, aswell as the rolling direction of the helix as indicated with awhite arrow. In the following calculations, we use the para-meters of this fabricated SiGe/Si/Cr nanohelix, including thearea of the cross-section, the radius, the pitch, the number ofturns, and the material parameters. (As presented in ESIFig. S1,† we provide another SiGe/Si/Cr binormal dual-chiralitynanohelix as an example to quantitatively analyse the mechan-ical properties of rotation.)

By pulling both ends of a nanohelix with dual chirality, thecentral part of the perversion performs a rotary motion,16,23

which makes it a perfect material for a linear-to-rotary motionnanometer converter. Fig. 3(b) presents the rotation angle ofperversion versus the axial elongation for the fabricated binor-mal SiGe/Si/Cr nanohelix with a red curve. The modelingresults are deduced from (1)–(6) with the geometry parametersas well as the material parameters of ESiGe = 161.2 GPa, νSiGe =0.27,34 ESi = 168.9 GPa, νSi = 0.36,35,36 and ECr = 377 GPa, vCr =0.31.37 The SiGe/Si/Cr binormal nanohelix unwinds whileextending axially and rotates 358°, ca. 1 turn, when it isstretched to 160% of its original length. The unwindingrotation direction of this binormal nanohelix is marked withred arrows in Fig. 2(b) and (e). Remarkably, during the firstturn, the rotation angle and the axial elongation are very closeto a linear relation and the corresponding linear-to-rotary ratiois approximately 597° per unit length. This kind of linearrotation has been observed in a loading experiment of theSiGe/Si nanohelix.23 As shown in Fig. 3(b), we also study thelinear-to-rotary motion of a normal dual-chirality helical nano-

helix with the same parameters as those of the binormal SiGe/Si/Cr nanohelix. It is interesting to compare the binormal andnormal nanohelices. The normal helix overwinds in thereverse direction and has a larger amplitude of rotation: itoverwinds to 1116°, i.e. 3.1 turns, when the elongation reaches60% (see Fig. 2(d) and (f)). It is notable that the normal helixdeviates from the linear behavior in the elastic regime by only10%. Therefore, the binormal nanohelices with dual chiralityare a more appropriate choice for a linear-to-rotary motionnanometer converter in 3-D scanning probe microscopes ormicrogoniometers.

Fig. 3(c) illustrates the axial load versus elongation of theSiGe/Si/Cr binormal nanohelix with dual chirality and thecorresponding normal helix in the region of 60% elongation,using (1)–(6). All the loading forces of binormal/normal nano-helices are divided by their respective maximum loading forcein this region. We observe that the binormal nanohelix withdual chirality is stretched linearly with the loading force,unlike the normal helix.

We further describe in Fig. 3(d) how the spring constantdepends on the elongation for both the SiGe/Si/Cr binormaland normal nanohelices with dual chirality in the region of60% elongation, obtained using (7) and (8). The spring con-stant of the binormal nanohelix remains constant with a valueof 0.012 N m−1, which will facilitate the actuation of themotion converters. A spring constant of the same magnitudeof order as 0.012 N m−1 has been measured in a SiGe/Si nano-helix.23 In contrast, the normal nanohelix has a wide springconstant change from 0.014 N m−1 to 0.379 N m−1, increasing27 times under load. Therefore, we conclude that binormalnanohelices with dual chirality are more appropriate for high-resolution force measurement in nanoelectromechanicalsystems.

Since normal and binormal helices exhibit opposite rotarymotions, what is the behavior of a transversely isotropic rod(created with a square or circular section)? Fig. 4(a) presentsthe rotation angle Φ of perversion versus the axial elongationfor a transversely isotropic dual-chirality nanohelix with asquare cross-section, derived from (1)–(7). In the loadingprocess, the dual-chirality nanohelix with a square cross-section exhibits the rotational inversion: it first overwinds to18°, i.e. 0.05 turns, when the elongation increases to 22%;then unwinds to 73°, i.e. 0.2 turns, when the elongationincreases to 60% as shown in Fig. 4. Therefore, the rotationproperty of transversely isotropic dual-chirality nanohelices isdisplayed in different stages with different axial loads.Interestingly, this rotational inversion is similar to the one weobserved in some gourd and cucumber tendrils (Fig. 1(b)). Wenote that all transversely isotropic rods will behave, as expectedfrom the general theory and illustrated for a fabricated SiGe/Si/Cr nanohelix transforms with a circular cross-section (inset ofFig. 4(a)).

We further identify the range of the aspect ratio of the rect-angular cross-section η = w/t for the nanohelices with dualchirality that acquires the characteristic of rotational inversion.Fig. 4(c) shows how the rotation direction of perversion

Fig. 3 (a) SEM image of a binormal SiGe/Si/Cr dual-chirality helicalnanohelix formed by a symmetric V-shaped mesa with both ends fixedto the Si(110) substrate. The inset shows the mesa design and the rollingdirection of the helix as indicated with a hollow arrow. (b) Rotation angleof perversion, (c) axial load, and (d) spring constant versus axial elonga-tion of the fabricated binormal nanohelix as well as a normal one withthe same parameters.

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depends on η during the axial loading, based on (1)–(6). Thedashed line of η = 1 represents the dual-chirality nanohelixwith a square cross-section. The areas above and below thedashed line indicate the binormal and normal helical nano-helices with dual chirality, respectively. We note that therotational inversion of overwinding followed by unwindingonly happens in a very narrow region of 1 < η < 1.35 for thebinormal nanohelix with dual chirality and 1 < η < 1.6 for thenormal one.

Fig. 5 illustrates the rotation angle versus the elongation for1 ≤ η ≤ 50. The area between the two red dotted dashed curvesof η = 1.35 and η = 1.6 is the region of rotational inversion asshown in Fig. 5(b); while the rest is the region of unidirectionalrotation. A binormal dual-chirality nanohelix with η ≥ 1.35 or anormal one with η ≥ 1.6 will only unwind or wind, respectively,during the whole loading process. According to the colourmap,

the uni-directional rotational behavior of perversion is affectedsignificantly by the aspect ratio η when it is smaller than 10:for an elongation of 60%, the rotation angle varies from 186°to 352° with η increasing from 1.35 to 10 (binormal), and from198° to 1049° with η increasing from 1.6 to 10 (normal).However, when the value of η exceeds 10, the relationshipbetween the elongation and the rotation angle is close tolinear for the binormal nanohelices with dual chirality. We seefrom this analysis that both direction and amplitude ofrotation can be finely adjusted by changing the shape of thecross-section for a dual-chirality helical micro-/nano-structuremade out of the determined material.

Conclusions

We have shed light on the important mechanical properties ofhelical nanostructures with dual chirality by using a generalelastic rod theory that include bending, torsion, twist, exten-sion, and shear. Our model was used to analyze the behaviorof a SiGe/Si/Cr dual-chirality nanohelix. It reveals that thetransversely isotropic nanohelix always overwinds initially inaxial extension, and then unwinds for larger tension. We alsoobserve that this kind of rotational inversion exists in somegourd and cucumber tendrils. Importantly, we find that arotational inversion region defined by the aspect ratio of rect-angular cross-sections η is given by: 1 < η < 1.35 and 1 < η < 1.6for binormal and normal nanohelices with dual chirality,respectively. Beyond this narrow region, the binormal andnormal nanohelices with dual chirality only unwind and over-wind, respectively. It is found that for the normal dual chiralitynanohelices, the rotation angle of perversion, the loading forceand spring constant all increase substantially and nonlinearlywith extension; while binormal dual chirality nanohelices withη > 10 rotate and stretch both linearly with loading force, and aspring constant for η = 50 as small as 0.012 N m−1. Therefore,these remarkable mechanical properties suggest that binormalnanohelices with dual chirality would be excellent linear-to-rotarymotion converters. This work provides a theoretical frameworkfor further experimental investigation on helical structures withdual chirality, as well as their applications in novel helical devicesand micro-/nano-electromechanical systems.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China under Grant No. 11547042, 11274230,11574206, 11474201 and 11674225. L. Z. thanks thefinancial support, by the Early Career Scheme (ECS) with theProject No. 439113, from the Research Grants Council (RGC)of Hong Kong SAR, and the Science, Technology andInnovation Committee of Shenzhen Municipality (SZSTI)

Fig. 4 (a) Rotation angle Φ of perversion versus axial elongation of thetransversely isotropic dual-chirality nanohelices with square and circlecross-sections. (b) The rotational inversion process of a transversely iso-tropic dual-chirality nanohelix with the square cross-section, as well asits cross-section at the mirror symmetry axis. (c) The rotational inversionregion defined by the aspect ratio of the rectangular cross-section η.

Fig. 5 Rotation angle perversion versus elongation for 1 ≤ η ≤ 50.

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Fundamental Research and Discipline Layout project (No.JCYJ20170413152640731). We thank Qianqian Wang (TheChinese University of Hong Kong) for simulated three-dimen-sional helices.

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Temperature Gradient-Induced Instability of Perovskite via IonTransportXinwei Wang,†,§ Hong Liu,*,†,§ Feng Zhou,† Jeremy Dahan,‡ Xin Wang,† Zhengping Li,†

and Wenzhong Shen*,†

†Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Institute of Physics and Astronomy,Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, People’s Republic of China‡MINES Paris Tech, 60, Boulevard Saint-Michel, 75272 Paris Cedex 06, France


ABSTRACT: Perovskite has been known as a promising novel material forphotovoltaics and other fields because of its excellent opto-electric properties andconvenient fabrication. However, its stability has been a widely known hauntingfactor that has severely deteriorated its application in reality. In this work, it has beendiscovered for the first time that perovskite can become significantly chemicallyunstable with the existence of a temperature gradient in the system, even attemperature far below its thermal decomposition condition. A study of the detailedmechanism has revealed that the existence of a temperature gradient could induce amass transport process of extrinsic ionic species into the perovskite layer, whichenhances its decomposition process. Moreover, this instability could be effectivelysuppressed with a reduced temperature gradient by simple structural modification ofthe device. Further experiments have proved the existence of this phenomenon indifferent perovskites with various mainstream substrates, indicating the universalityof this phenomenon in many previous studies and future research. Hopefully, thiswork may bring deeper understanding of its formation mechanisms and facilitate the general development of perovskite towardits real application.

KEYWORDS: perovskite, stability, temperature gradient, ion transport, photovoltaics.

1. INTRODUCTION

Perovskite solar cells (PSCs) have attracted increasingattention because of their impressive high efficiency, lowcost, and fabrication requirement.1−5 The recent certifiedrecord has reached 22.1%,6 showing potential competenceeven with high-performance crystalline Si solar cells. Never-theless, the instability of perovskite has been for long time awell-known obstacle toward real applications.7−9 It actuallyencompasses the phenomena of following aspects: (1)deformation or decomposition of perovskite under certainconditions (chemical and physical environments, substrate);10

(2) drastic decrease of performance when expanding to a scalelarger than 1 cm2;11 and (3) reproducibility issues.12 Amongthese problems, the first issue is the most essential one, wheremany opinions have been proposed and lots of efforts havebeen devoted to resolve it.Depending on the origin, the instability of perovskite can be

induced by light, electricity, chemicals, and heat. According tocurrent mainstream opinions, the photo instability of perov-skite is normally weak under standard illumination.13 Thethermal instability has not been intensively discussed, as thenormal fabrication is carried out under low temperature (<150°C). It is widely accepted that thermal annealing at about 70−100 °C would be necessary for the good crystallization of

perovskite and for optimization of PSCs.14−17 The electricalinstability (or the “electrical hysteresis”) is known as a normalphenomenon for unmodified perovskite that the reverse andforward scanning results in different J−V curves, which isreversible and has been claimed to be due to different carriermobilities.18,19 The chemical instability has been mostintensively discussed because it can directly lead to actualdecomposition of the perovskite and is irreversible in mostcases.20 This instability has significantly influenced the realapplication of not only perovskite itself but also somepromising candidates for other parts of the cell, namely, theelectrodes, the hole-transporting layer (HTL), and theelectron-transporting layer (ETL). For instance, ZnO, whichhas 1 order higher mobility and a lower synthesis temperaturethan TiO2, was regarded as a good substitutive ETL materialfor PSCs.21,22 However, many researchers have demonstratedthe dramatic decomposition of the CH3NH3PbI3 perovskitefilm on ZnO ETL during the annealing process (heatingtemperature >80 °C).23,24 Hence, most high records have beeneither based on modified TiO2 or other ETL materials.

Received: November 22, 2017Accepted: December 19, 2017Published: December 19, 2017

Research Article

www.acsami.orgCite This: ACS Appl. Mater. Interfaces 2018, 10, 835−844

© 2017 American Chemical Society 835 DOI: 10.1021/acsami.7b17798ACS Appl. Mater. Interfaces 2018, 10, 835−844

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To resolve the instability problem of perovskite, tremendousefforts have been devoted, with significant advances achievedmostly based on doping chemicals in the perovskite layer,incorporating functional layers, and interface engineeringbetween different layers (ETL, perovskite, and HTL).20,25−28

Nevertheless, the real solution for this issue would rely on theunderstanding of the detailed mechanisms. In early research,composition of moisture, oxygen, and heat (sometimes withlight too) was widely considered the main reason fordecomposition of perovskite,7,20,29 whereas counterexamplescould also be found; for example, MAPbI3 has shown stabilityin aqueous solution under certain conditions.30 For ZnO, somepreliminary research related the less stability of perovskite onZnO to the existence of hydroxyl or acetate ligands in it.31,32

More recently, studies have revealed a strong relationshipbetween the perovskite instability and ion migration, which hasalso indicated that the moisture, oxygen, and heat might not bealways necessary for the decomposition of perovskite.33−36 Forinstance, different groups reported mass transport of extrinsicparticles (Au) or ions (Li+, etc.) from the adjacent layer intoperovskite, which leads to degradation of the PSC.35,37,38 Inanother research, Poglitsch and Weber reported that theCH3NH3

+ cation (MA+) could not be fixed in the crystalstructure because of the disordered orientation and mobility ofthe cation in the crystal.39 Nevertheless, the factors that induceand influence the transport are still not clear, and a clear andconsistent picture for those processes still remains unknown(key necessary parameters, species and their interactions, andany probable simple solutions), which requires moreexperimental and theoretical works for an answer.In this work, we have discovered that perovskite could

thermally decompose below the critical temperature that hasbeen previously claimed to be stable. A perovskite sampleplaced above a hot plate has much faster degradation rate thanthat in an oven at the same temperature (100 °C), even withN2 protection and isolation from light. A detailed experimentalstudy that followed has revealed a decisive role of thetemperature gradient in this phenomenon. It also proves thatmoisture and oxygen are not always necessary for theinstability of perovskite. Further characterization has suggesteda strong relationship of this phenomenon with the migration ofionic species driven by the temperature gradient other thanthermal diffusion under a concentration gradient. Taking theexample of perovskite/ZnO, the thermal transport of Zn2+ intothe perovskite layer under an asymmetric temperature field cansignificantly influence the balance of the decompositionreaction of perovskite, so that its degradation process is greatlyaccelerated. We have found that such a process can beeffectively suppressed when the temperature gradient wasreduced by simple addition of a thermal reflector above theperovskite layer during the plate heating. Moreover, furtherexperiments have shown that such a temperature gradient-driven chemical instability of perovskite widely exists in otherperovskites (such as FAPbI3) on various substrates (e.g., TiO2,SnO2, and PCBM (the short abbreviation for the fullerenederivative [6,6]-phenyl-C61-butyric acid methyl ester)) as welland can also be physically suppressed by the reduction of thetemperature gradient. This has proved the thermal migration ofions as an important factor for instability of perovskite in manyprevious studies and possible developments in the future. It hasalso suggested a simple but effective way of stabilizing PSCsregardless of detailed types of materials and systems.

2. EXPERIMENTAL SECTION2.1. Preparation of Substrates. Fluorine-doped tin oxide

(FTO) glass (Materwin, 14 Ω·sq−1, 2.2 mm thick) was cleanedsequentially by ultrasonic bath with deionized water (18.2 MΩ·cm),acetone (Sinopharm, AR, ≥99.5%), isopropanol (Sinopharm, AR,≥99.7%), and ethanol (Sinopharm, AR, ≥99.7%) for 30 min, driedwith nitrogen (99.99%), and then cleaned by an ultraviolet-ozonecleaner (Hwotech, BZS250GF-TS). To prepare ZnO by the sol−gel(SG) method, a mixture was made with 0.15 M zinc acetate dehydrate[Zn(CH3COO)2·2H2O, Aladdin reagent, 99.995%] and 0.15 Mdiethanolamine (Aladdin reagent, ≥99.7%) in 1-butanol (Aladdinreagent, ≥99.7%). It was stirred until transparent and then aged for 24h. Afterward, the solution was spin-coated onto FTO glass at 2000rpm for 30 s in a spin coater (KW-4B, Jing Lin) three times, withheating on a hot plate (C-MAG HS7, IKA) at 120 °C for 5 minbetween each time. Finally, the products were transferred to a muffleoven (F48020-33-80CN, Thermo Scientific) and annealed at 350 °Cfor 1 h. ZnO was also fabricated by magnetron sputtering (MS) in ahigh vacuum chamber (CLUSTER-PECVD350, Xinlantian Techni-cal) under 100 W ratio frequency sputtering for 7 min. The TiO2substrates were prepared by the spin-coating of 0.15 M titaniumdiisopropoxide bis(acetylacetonate) solution (Materwin, in 1-butanol) on the FTO glass under the same condition as that forZnO and annealing at 500 °C for 15 min. For SnO2, 0.15 M SnCl4·5H2O (Aldrich, 98%) was dissolved in 30 mL of ethanol (Sinopharm,AR, ≥99.7%) with stirring for 2 h to form the sol solution. Afterfiltration, the solution was deposited on the FTO glass by three turnsof spin-coating (3000 rpm, 30 s for each turn with heating on a hotplate at 120 °C for 5 min between each two turns) and then annealedin air at 450 °C in the oven for 10 min. The PCBM substrate wasfabricated by spin-coating (2000 rpm for 30 s) with 30 mg mL−1

PCBM (Aldrich, 99.5%) chlorobenzene solution (from Materwin) onFTO and annealing at 100 °C on a hot plate for 20 min.

2.2. Fabrication and Treatment of Perovskite. For MAPbI3,60 μL of perovskite precursor solution [461 mg PbI2 (Aldrich,99.999%)], 159 mg of CH3NH3I (from Materwin), and 78 mg ofdimethylsulfoxide (Aladdin reagent, ≥99.7%) dissolved in 600 mg ofN,N-dimethylformamide (Aldrich, 99.8%) was spin-coated on ZnO(or TiO2) at 4000 rpm for 30 s, with 0.5 mL of diethyl ether(Sinopharm, AR, ≥99.5%) being slowly dripped down in 10 s. ForFAPbI3, 60 μL of FAPbI3 perovskite precursor solution (1.0 M,Materwin) was spin-coated on TiO2 at 4000 rpm for 30 s. Both kindsof perovskite products were then placed in a N2-protected glovechamber for 24 h until their colors turned to reddish brown. Forinstability experiments on perovskite, the samples were directly placedon a hot plate and a muffle oven with temperature already havingreached 100 °C. For stability experiments on PSCs, the perovskite wasannealed in the oven at 100 °C for 30 min.

2.3. Assembling of PSCs. For the growth of Spiro-OMeTAD,73.3 mg of 2,20,7,70-tetrakis(N,N-di-p-methoxyphenylamino)-9,90-spirobiuorene (Materwin) in 1 mL of chlorobenzene was doped with17.5 μL of Li-TFSI (520 mg mL−1 in acetonitrile) (Aladdin reagent,99%) and 28.8 μL of 4-tert-butylpyridine (Aladdin reagent, 96%).Such a solution (50 μL) was spin-coated on the perovskite layer at4000 rpm for 30 s. Finally, an 80 nm Au layer was deposited on theSpiro-OMeTAD layer by thermal evaporation. All PSCs were withoutencapsulation and any post-treatment.

2.4. Characterization of Materials and Devices. Themorphological and structural analyses of the samples wereinvestigated by a field-emission scanning electron microscope (ZeissUltra Plus) and an X-ray polycrystalline diffractometer (D8ADVANCE Da Vinci, Bruker, Germany), respectively. The profileanalysis was examined by time-of-flight secondary-ion massspectrometry (ToF-SIMS, IonTOF, Germany). The real-time voltagemeasurement was done by an electronic sourcemeter (Keithley,2400). The real-time temperature of the sample was detected by twoinfrared sensors (MI3, Raytek). The PSCs were characterized bystandard procedures afterward.40−42 The J−V characteristics weremeasured by a solar simulator (Newport, Oriel Sol-2A) under 1 sun

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DOI: 10.1021/acsami.7b17798ACS Appl. Mater. Interfaces 2018, 10, 835−844

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(AM 1.5, 100 mW cm−2) at room temperature (RT, 25 °C) with 10samples under each condition. The scanning voltage was from −0.3 to1.2 V (forward) and 1.2 to −0.3 V (reverse) with a rate of 100 mV s−1

and a 10 ms time delay. The size of PSCs was 2 × 2 cm2, with aneffective area of 0.15 cm2 defined by a mask covered on top. Astandard reference Si solar cell (effective area 2 × 2 cm2, certified byVLSI Standards Inc.) was applied for the calibration of the simulatorand calculation for the spectral mismatch, with a mismatch factor M =1.003. The external quantum efficiency (EQE) spectrum wasmeasured by a quantum efficiency/IPCE system (PV MeasurementsInc., QEX10) in the wavelength range of 300−850 nm at RT.

3. RESULTS AND DISCUSSIONThe temperature gradient-driven instability was first observedduring our optimization experiment on the thermal treatmentof perovskite on the ZnO substrate, and then systematicexperiments were designed and carried out. First, ZnOsubstrates were fabricated by the SG method on the FTOglass, and then perovskite was grown on the substrates by one-step spin-coating.43 The annealing processes were carried outon a programmable hot plate in a glove chamber and a muffleoven, separately, both set at 100 °C. To minimize the influencefrom the temperature ramping in the plate heating, the samplewas only placed on the heating plate after the platetemperature had actually reached 100 °C. The setups wereprotected from the influence of moisture and oxygen by a N2atmosphere and isolated from light to exclude possibleinvolvement of photocatalytic effect brought by the ZnOsubstrate. In the first method, the color of the sample started tochange at 3 min and turned completely yellow in 5 min, asshown in Figure 1a,b (with the photograph shown in the

inset). This result indicated significant decomposition ofperovskite even with nitrogen protection under such acondition. In the second method, the perovskite remainedblack even after 1 h of continuous heating (Figure 1c), whichmeans perovskite remained quite stable under this condition,though on the ZnO substrate that was previously considered tofacilitate perovskite instability.21,44 This quite distinctivebehavior has been confirmed by the X-ray diffraction (XRD)results (Figure 1d). On the one hand, the peak position at12.7° of the sample by plate heating was very similar to that of

crystallized PbI2 ([001] direction). On the other hand, thesample with oven heating for 1 h showed strong peaks at 14.1°,20.1°, 28.7°, and 32.0°, corresponding to the [110], [112],[220], and [310] directions of the typical crystallizedtetragonal perovskite, respectively. The different performanceof the assembled PSCs shown in Figure 1e (for plate-heatedand oven-heated samples, the η under reverse scanning are13.3 and 1.1%, respectively; other parameters and EQE curvesare in Table S1 and Figure S1 in the Supporting Information,respectively) also demonstrated the consequence that wasinduced by this effect.According to some opinions from previous studies, the

origin of the instability of perovskite on ZnO should beattributed to the reactions between perovskite and the contentsof the substrate, for instance, the hydroxyl and acetate groupson the surface and in the bulk of the ZnO layer.45−47

Therefore, a second experiment was carried out to evaluate theinvolvement of these species in the decomposition ofperovskite in plate heating. The ZnO substrates weresynthesized in two routes: (1) MS (with the ZnO substratedenoted as ZnOMS) in vacuum to rule out the involvement ofcommon ligands (OH− and COOH−); (2) SG method in air(the corresponding ZnO layer is denoted as ZnOSG) forcomparison. The result is shown in Figure 2. According to theSEM images, the surfaces of ZnOSG and ZnOMS appeared quitesimilar, and both were covered by many small inlands. For X-ray photoelectron spectroscopy (XPS) results, in ZnOSG, the O1s core level spectrum contains three peaks at 530.0, 531.2,

Figure 1. Effect of different heating methods on perovskite/ZnO. (a−c) Scanning electron microscope (SEM) images and real photographsof CH3NH3PbI3/ZnOSG (a) heated on the hot plate (100 °C, 3 min),(b) heated on the hot plate (100 °C, 5 min), and (c) heated in theoven (100 °C, 1 h). (d) XRD patterns of PbI2/ZnO (RT) andCH3NH3PbI3/ZnOSG heated in the oven and on the hot plate. (e) J−V characteristics of the assembled PSCs.

Figure 2. Further study on the thermal instability perovskite. (a,b) O1s XPS core level spectra of ZnOSG and ZnOMS, respectively, withSEM images as insets. (c) XRD pattern, photograph, and SEM imageof perovskite/ZnOMS annealed on the hot plate (100 °C, 5 min). (d)Temperature vs time at different positions of the sample in the plateheating. (e) Simulation of the temperature gradient near the surface(∼500 nm below the top surface) of the sample as the heating timeincreases. (f) Time dependence of voltage over the sample (taking thebottom as the positive position of the voltage and the upper side asgrounded).



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and 532.4 eV, corresponding to the Zn−O bond in thewurtzite structure and oxygen vacancies or defects in thepresence of oxygen-deficient and chemisorbed functionalgroups such as hydroxyl and acetate ligands, respectively.48,49

Meanwhile, no peak at 532.4 eV has been observed in ZnOMS,which means no detectable existence of hydroxide and acetateligands can be found, which is reasonable because the vacuumsystem contained only very little water vapor. However, theperovskite on ZnOMS turned yellow soon after plate heating forabout 5 min, with a distinct characteristic peak of PbI2 in XRD(Figure 2c). The total survival time of perovskite on ZnOMGwas short similar to that on ZnOSG, compared to the muchlonger survival time with oven heating (Figure 1c).Furthermore, stronger perovskite stability has been foundonSG-fabricated TiO2 (shown in Figure S2 in the SupportingInformation), which also contains hydroxyl groups. Therefore,the chemisorbed groups (hydroxyl and acetate ligands, etc.)mentioned in previous reports are seemly not a decisive factorin this experiment.23,47,50,51

Because the samples were all identical with exclusion of theillumination and the chemical circumstance, the most probablefactor that can cause such a difference should be thetemperature distribution in the two heating systems. On theone hand, constant heating is symmetrically applied to thesample in the oven, so the temperature difference over thesample will soon become small after the heating started. Onthe other hand, constant heating is applied only from thebottom (plate), so the cold side always exists due to thedissipation to the atmosphere, and the temperature differencewill be comparatively larger than that in the oven heating (canbe seen from the experimental measurement shown in Figure2d). We have also carried out some calculation on the detailed

temperature distribution according to the basic model and theboundary temperature condition measured in the experiment(more detailed description can be found in Figure S3 in theSupporting Information). The calculated result also showsquite a high inhomogeneity in the plate heating but quite ahomogeneous distribution in the oven. Moreover, theevolution of the temperature gradient (Figure 2e, at thedepth of ∼500 nm, close to the thickness of the perovskite andZnO substrate) shows that the temperature gradient in plateheating very rapidly rises up to a high positive value (whichmeans decreasing temperature from the bottom to the top),whereas it has only a much smaller negative value in the ovenheating.To study the exact effect that the asymmetry of heat brings

to the perovskite stability, further work was carried outphysically and chemically. The first suspected factor was thecommonly known thermoelectric effect that exists on PNjunctions or heterojunctions.52,53 It means that this instabilitycould have been possibly caused by electrochemical decom-position of perovskite influenced by the thermoelectric voltageof the heterojunctions in the perovskite/ZnO structure. So, atime-dependent series voltage measurement was carried outwith an nV voltmeter under different conditions.54,55 As shownin Figure 2f, the upper surface was grounded, and the voltagewas measured at the FTO side. It can be seen that the voltagedrop across the sample was positive in both cases. In the oven-heating, the U−t curve was quite flat, with the amplitude in therange of 10−2 mV through the heating process. On thecontrary, under the plate-heating condition, there was a drasticincrease of voltage at about 100 s and reached the maximumamplitude of around 1.1 mV at about 180 s. The emergencetime of the voltage peak has been quite comparable to the one

Figure 3. Detailed analysis on the formation and composition of CH3NH3PbI3/ZnO under different heating conditions. (a−c) SEM of samplestreated by (a) RT for 24 h; (b) oven heating, 100 °C, 1 h; (c) plate heating, 100 °C, 3 min (insets showing the top view). (d−f) SIMS results underconditions of (a−c), respectively, sputtered by Cs+. (g−i) SIMS signals under conditions of (a−c), sputtered by O2−. (j−l) Comparison of SIMSsignals from Zn species under condition of (a−c), respectively, sputtered by Cs+.



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for the significant decomposition of perovskite by plateheating. It can be seen that even the highest value of voltagein the plate heating (100 mV) is much smaller than that isnormally needed for the electrochemical decomposition ofperovskite (normally in the range of 101 V), according to ourprevious research.56 Therefore, the thermoelectric effect hasseemly not been the main origin of this instability.Furthermore, as shown in Figure 2d, the temperaturedifference over the sample had a rapid decrease only in theinitial 10 s and then became quite stable, much earlier than theemergence of the voltage peak. This nonsynchronized behaviorindicates that the change in the temperature difference ingeneral seems not to be the direct reason for the voltage peak.Instead, it may have been the consequence of other processesrelated to the decomposition reactions of perovskite under theinfluence of the temperature gradient. A more soundexplanation for this might have been the collective motion ofionic species that were involved in the decomposition reactionsof perovskite, according to some previous review works thathave discussed the possibility of the perovskite instabilitygenerated by the ion-drifting processes.29,57

To investigate more details in this process, chemical analysiswould be necessary to monitor the key difference between thetwo heating methods. Hence, ToF-SIMS was applied to detectthe spatial distribution of chemical compounds after differentheating processes by sputtering the sample with high-energycesium and oxygen ion beams and analyzing the correspondingnegatively and positively charged debris, respectively. Theperovskite samples were taken after oven heating (100 °C) for1 h and heterogeneous heating (100 °C) for 3 min. A controlsample was prepared in the glove chamber for 24 h at RT forcomparison. As shown in Figure 3a−c, the average size ofgrains is small in the RT sample but larger for the heatedsamples, which is a well-known effect that larger crystallizedgrains can be formed by higher activation energy induced bythermal heating.58,59 For plate-heated samples, certain damageappears from the cross-sectional view. The distribution of thekey components is presented in Figure 3d−f (sputtered bycesium) and Figure 3g−i (sputtered by oxygen). Thesputtering time can be well-proportional to the depth of thesample that the ion beams has reached in different times of thesputtering, based on the principle of SIMS. On the basis of thedistributions of PbI3

−, ZnO−, and SnO−, the graphs can bedivided into three zones corresponding to the spatial positionsof the perovskite, ZnO, and FTO layers. Two significantphenomena could be noticed. First, compared to the sampleswith oven heating and without heating, the MA+ ionspenetrated much deeper into the layer of the ZnO in thesample with plate heating (if considering the sputtering rateconstant, such depth was almost one-third of the ZnOthickness). Second, there are significantly stronger zinc iodide(ZnIx

−, x = 1, 2, and 3) signals detected in the perovskite layerin the plate-heated sample, which can evidently prove theexistence of ZnI+, ZnI2, and ZnI3

− complexes, respectively.60 Itclearly indicates that there has been opposite migrations of zincand MA species from the bottom-up and top-down directions,respectively. Moreover, taking a closer look, the distribution ofZnIx species seemed to obey a certain sequence with increasingx index, as the distance is closer to the upper surface ofperovskite (Figure 3j−l). It appeared that after ZnI+ wasformed (by reactions between the transported Zn2+ and I−),they combined with more I− to form ZnI2 and then ZnI3

−

sequentially, as they approach the upper surface. Significant

migration of alien species into the neighboring layers wasdetected in the plate-heating system, whereas no detectabletrace of such a phenomenon can be observed in thehomogeneous heating system or in the RT situation.From the above results, we can try to draw a rough sketch of

the mechanisms along with some theoretical explanations. As itis well-known, the particle flux in thermomigration can bedescribed by equation (in one-dimensional system)61,62

ikjjjj

y{zzzzJ J J D C

Q CRT

Tx xd m 2= + = − ∇ −*

∇(1)

where J is the total flux of particles, Jd is the part driven bydiffusion from the concentration gradient, and Jm is the fluxdriven by the thermal gradient, together with Q* the activationenergy, D the diffusion coefficient, and C the particleconcentration. In plate heating, there is always a hot side(FTO glass) by continuous heating from the bottom and acold side (perovskite) by the dissipation process to theatmosphere, and therefore, the temperature gradient would besignificant throughout the process. First, the Zn−O bond canbe broken because of thermal excitation of Zn atoms by theheat coming from the hot side.63 Second, these Zn2+ ions canbe transported to the colder side, that is, the perovskite layer,and the flux will be proportional to the temperature gradientaccording to the second term of eq 1. One would naturally alsonotice the influence of the concentration gradient in the firstterm of eq 1. However, experiments have shown that thetransporting process in the oven would be much weaker thanon the plate, where the situation of the first term would bemore or less similar. Therefore, the role of the concentrationgradient in the comparison of two different heating methodswould be accordingly much less important than the thermalgradient. According to the currently accepted opinion, thedecomposition of perovskite consists of the followingreactions29,64

CH NH PbI CH NH I PbI3 3 3 3 3 2↔ + (2)

CH NH I CH NH I3 3 3 3↔ ++ −(3)

Equations 2 and 3 stand for the decomposition of MAPbI3and MAI, respectively, which can take place subsequently inthe normal decomposition process of perovskite. The invadingZn2+ ions can easily combine with I− and thus accelerate thedecomposition of MAI in the second step.57,60,65 Therefore,the overall decomposition of perovskite would be accelerated.Zinc iodide can penetrate deeper into the perovskite andcombine with more I− ions when being driven by the thermalgradient. The production of zinc iodide species can bedescribed as below21

xI Zn ZnIxx2 2+ →− + −

(4)

In the meantime, the boosted perovskite dissociation canincrease the production of the MA+ cation. Such an increase ofthe MA+ concentration can significantly enhance its diffusioninto both the upper side of perovskite and the ZnO side andleave detectable traces in the SIMS spectroscopy that followed.Considering the positive sign of the voltage peak shown inFigure 2e and the dominance of ZnI3

− in the zinc iodidespecies shown in Figure 3, the main content of the ioniccurrent is likely the MA+ cation that moves from the top to thebottom, and ZnI3

− from the bottom to the top. Moreover,according to the second term in eq 1, the thermomigration is



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also proportional to the local concentration of transportedspecies. Because the bonding energy of the Zn−O (≤226 kJmol−1) bond is much smaller than that of the Ti−O (666.5 kJmol−1) bond, the decomposition process in the TiO2 substratewould be much weaker.63,66 Therefore, the thermomigration ofionic species by the temperature gradient would be smaller inTiO2, which explains the relatively higher stability ofperovskite/TiO2 under the same condition (as can be seenin Figure S2). Moreover, the activation energy and thetemperature are also noticeable factors in the same term.Hence, altering the structural parameter of the substratematerial, changing its chemical composition, or tuning theambient temperature may also influence the thermal stability ofperovskite-on-substrate in this aspect.From the above discussion, we can more or less conclude

the relationship of the significant instability of perovskite/ZnOfor enhancement of ion migration induced by the temperaturegradient. To confirm this, an experiment was further carriedout to investigate the consequence of suppressing thetemperature gradient. As is well-known, Al foil can effectivelyreflect infrared radiation and therefore increase the temper-ature of the atmosphere nearby. In this experiment, an Al foilwas simply fixed on top of a sample, and then the whole setup(shown in Figure S4 in the Supporting Information) washeated on the hot plate. A control experiment was also carriedout without the Al cover on top of the perovskite. As shown inFigure 4, on the one hand, the sample without the Al coverturned yellow in 5 min. On the other hand, the perovskite withthe Al cover on top remained entirely black even after 30 min.According to the SEM images, certain damage emerged in thesample without the cover, whereas the one with the Al coverwas still quite uniform. The XRD in Figure 4f shows asignificant peak at 14.1°, 20.1°, 28.7°, and 32.0° but with theabsence of peak at 12.7° (corresponding to PbI2) in the secondsample. This result has clearly shown much improved stabilityof crystallized perovskite by simply utilizing the Al foil as asimple heat reflector into the system. Besides, it has confirmedagain the important role of temperature gradient in theinstability of perovskite and its dominant influence in the

transport process, compared to the concentration gradient-driven diffusion, as discussed in eq 1.Nevertheless, in real application, though the stability of

perovskite may also be improved using oven heating during thefabrication stage (Figure 1), a significant temperature gradientcan widely exist in the solar cells under sunlight illumination.Therefore, we also carried out similar experiments on PSCs(with classical FTO/ZnO/CH3NH3PbI3/Spiro-MeOTAD/Auarchitecture). The PSCs were heated on the 100 °C hot platein the glove chamber in the N2 atmosphere without and withAl cover on top. The J−V characteristics were taken at differenttime stages in 24 h, with the representative result shown inFigure 4k (more statistical details can be found in Tables S2−S4 and Figure S5 in the Supporting Information). The test wasrun ten times for each condition to be more statisticallycertain. For identical comparison (Al cover can block somelight), both heating processes were isolated from illumination.For the PSC without the Al cover, the sample began to turnyellow in the first 30 min. In 3 h, the perovskite layer wassignificantly damaged in continuity and uniformity, whereasseveral significant yellow regions appeared in the sample(Figure 4h). Meanwhile, the PSC with the Al cover had nearlyno significant change either in the morphology or color. Forconversion efficiency, on the one hand, the PSC without thecover showed drastic degradation versus time, with η droppedover 50% after the first hour and to almost zero after 24 h. Onthe other hand, the PSC with the cover has even showed aninitial increase of conversion efficiency in 3 h and then quite aslow degradation. As shown in Figure 4k, the PSC built onZnO still retained the efficiency of 10.6% (77.4% of initialvalue) even after 100 °C heating treatment for 24 h. Thisexperiment has also shown that the temperature gradient-induced thermal instability also exists in annealed crystallineperovskite, though with a longer lifetime compared to the onewithout annealing, as shown in the result in Figure 1. Besidebetter crystalline conditions, the longer lifetime in PSCs couldalso have been facilitated by the blocking effect on the heat andleakage of MA by the HTL layer and the glass above it.

Figure 4. Stabilization of perovskite and assembled PSCs by reducing the temperature gradient: (a) Brief description of the decompositionmechanism. (b) Photograph and (c) SEM of the sample heated on the hot plate at 100 °C without cover. (d) Photograph and (e) SEM of sampleheated on the hot plate at 100 °C with aluminum foil cover. (f) XRD of samples without and with Al cover. (g) SEM and real photograph of PSCbefore the stability test. (h,i) PSCs being heated on the hot plate at 100 °C for 3 h without and with Al cover, respectively. (j) Efficiency of PSCs vsheating time. (k) J−V characteristics of PSCs with and without Al cover heated at 100 °C for 24 h (control device was kept at RT for 24 h).



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Until now, we have discussed the situations for perovskite onthe ZnO substrate. In the current research, other candidateshave been more frequently used because of their better stability(such as TiO2, SnO2, and PCBM), though normally moreexpensive than ZnO.67 Therefore, similar instability experi-ments have also been performed on these three mainstreamETL substrates. As widely known, the performance ofperovskite is very sensitive to the inhomogeneity inside it, sothe time when the perovskite layer turned yellow (correspond-ing to the decomposition of perovskite and formation of PbI2)is an adequate critical point for instability of perovskite, nomatter how large the yellow area is. The results are listed inTable 1 (detailed photographs of the time series experiment

can be found in Figures S6−S8 in the SupportingInformation). It can be seen that the distinctive stability ofperovskite depending on the heating methods can exist onTiO2, SnO2, and PCBM. On the hot plate, the survival time ofperovskite is almost 1 order of magnitude shorter than in theoven. Furthermore, the addition of the heat-reflecting cover ontop of the perovskite can also significantly enhance its stabilityon those substrates. Such experiments have further proved thatthe temperature gradient-induced instability can take place onalmost all mainstream ETL substrates. In those processes, themass transport of particles (however, the type of transportedparticle will depend on the detailed substrate) would also playan important role in the generation of instability. What is more,we also performed the stability test of FAPbI3 on the TiO2substrate and discovered a similar phenomenon, where FAPbI3decomposed in only 5 min on the hot plate but remainedalmost unchanged in 3 h in the oven (details can be found inFigure S9 in the Supporting Information). Meanwhile, theaddition of the thermal reflection cover could significantlystabilize it on the hot plate (lifetime >30 min). It furthersupported that such an instability can take place on other typesof perovskite as well if they chemically follow a similar way ofdecomposition like MAPbI3 and could also be stabilized bysimply reducing the temperature gradient. Predictably, suchphenomena may also very probably exist in the more recentlydeveloped p−i−n-structured perovskite solar cells, with theperovskite layer grown on HTLs, and could be even strongerthan that on ETLs because of their (NiOx, CuS, CuSCN,PEDOT:PSS, etc.) good ionic-transporting ability.68−71

Finally, according to this investigation, there are thefollowing key processes involved in the stability of perovskite:(1) existence of ions or neutral particles that can enhance thedecomposition of perovskite, which could have already existedor be generated by intrinsic or thermoexcitation in the layersadjacent to perovskite; (2) effective mass transport of suchspecies in the adjacent layer through the interfaces and theninto the perovskite layer; and (3) reaction of such species with

the local perovskite. To stabilize the perovskite, one can eitheravoid the generation of deteriorating particles adjacent toperovskite, block the pathway of mass transport, lower themobility of particles, reduce the force that drives the transport,or weaken the reaction of those species with local perovskite.Looking back into all stabilization efforts on the perovskite, theabove routes have been normally realized by passivation ofperovskite by chemical doping and blockage effect by interfaceengineering.22,23,31,46,51,57,72,73 Nevertheless, perhaps it will alsobe a good trial to stabilize the perovskite by weakening thetransport itself, according to this paper.

4. CONCLUSIONSIn conclusion, this work has proved the existence of a new typeof chemical instability of perovskite, caused by the temperaturegradient, and studied its origin and detailed mechanism.Temperature gradient can greatly deteriorate the stability ofthe as-formed perovskite and the crystalline one in PSCs at lowtemperature even with protection from moisture, oxygen, andlight, whereas the hydroxyl and acetate groups near theinterface or the direct reaction with the adjacent layer areseemly not necessary conditions for the perovskite instability.Further mechanism study has revealed that the temperaturegradient appeared as the main driving force for the masstransport of extrinsic ionic species relevant to the decom-position reaction of perovskite from the neighboring layers.Moreover, we have proved the significant enhancement of thestability of perovskite by reducing the temperature gradientwith the addition of a covering structure. In the perovskite/ZnO system, the survival time extended from 2−3 min to 1order of magnitude longer at 100 °C heating in air and hasshown even better effect in the application of PSCs. Eventually,the universality of such a phenomenon has also been foundexisting in many other systems with various perovskites andETL substrates, and predictably may also exist with HTLsubstrates (NiOx, CuS, CuSCN, PEDOT:PSS, etc.). In general,this work may give some light to the study on the stabilizationof perovskite and the development of applicable perovskitedevices via different approaches.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acsami.7b17798.

Voltage value between the TiO2 layer and the perovskitelayer versus time under different heating conditions;stability experiment of perovskite on TiO2 substrates andthe corresponding analysis; approximate simulation oftemperature distribution for samples; schematic illus-tration of the experimental setup for reducing thetemperature gradient; stability test of MAPbI3 on TiO2,SnO2, and PCBM and of FAPbI3 on TiO2, currentdensity−voltage characteristics of each treatment (PDF)

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: [email protected] (H.L.).*E-mail: [email protected] (W.S.).ORCIDHong Liu: 0000-0002-2241-1199

Table 1. Thermal Stability of Perovskite Films onMainstream Substrates under Different Conditionsa

method ZnOTiO2(h)

SnO2(h)

PC60BM(h)

hot plateheating

withoutcover

3 min 4 2 3

with cover 45 min >32 28 32oven heating 2.5 h >32 >32 >32aThe stability is represented by the time it takes for the emergence ofyellow color in the sample heated at 100 °C, with N2 protection andisolation from illumination.



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http://pubs.acs.org/doi/abs/10.1021/acsami.7b17798




http://orcid.org/0000-0002-2241-1199


Author Contributions§X.W. and H.L. contributed equally to this work.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported by the Natural Science Foundationof China (61234005, 11204176, and 11474201), the NationalMajor Basic Research Project (2012CB934302), and theFundamental Research Funds for the Central Universities fromthe Ministry of Education (China).

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aKey Laboratory of Articial Structure and Q

School of Physics and Astronomy, Shanghai

PR China. E-mail: [email protected]; FabCollaborative Innovation Center of Advan

Nanjing, 210093, PR ChinacSchool of Chemistry and Chemical Tech

Shanghai, 200240, PR China

† Electronic supplementary informa10.1039/c8ta01442f

Cite this: J. Mater. Chem. A, 2018, 6,8742

Received 10th February 2018Accepted 11th April 2018

DOI: 10.1039/c8ta01442f

rsc.li/materials-a

8742 | J. Mater. Chem. A, 2018, 6, 874

pacitive performance of ultrathinmesoporous NiCo2O4 nanosheet arrays by surfacesulfation†

Yuxiu You,a Maojun Zheng, *ab Dongkai Jiang,a Fanggang Li,a Hao Yuan,a

Zhihao Zhai,a Li Mac and Wenzhong Shena

Surface functionalization is as an effective way to modulate the electrochemical or photoelectrochemical

properties of nanomaterials. Sulfated ultrathin mesoporous NiCo2O4 nanosheet arrays are fabricated based

on a convenient galvanic displacement process, exhibiting stimulated chemical reactivity and boosted

supercapacitive performance. This method not only realizes synchronization of synthesis and surface

functionalization but also readily tailors the functionalizing degree of the sulfate-ion and surface

reactivity of NiCo2O4 through adjusting the addition amount of the sulfur source. Moderately sulfated

NiCo2O4 exhibits a capacitance activation in the first 2100 cycles and achieves the highest specific

capacitance of up to 1113 F g�1, an increase of 57% over that of pristine NiCo2O4 during 5000 cycling

tests at a high current density of 5 A g�1. Additionally, the sample displays an outstanding cycling

performance with 166% capacitance retention. On the basis of structural characterization and surface

chemical analysis, this research puts forward a scientific explanation for significantly boosting the

electrochemical performance by surface sulfation. In addition, we present a facile route for fabricating

sulfated metal oxides without post-processing for energy conversion and storage fields.

1. Introduction

The increasing energy crisis associated with environmentalpollution and global warming1–3 compels humanity to searchfor clean and renewable power sources as well as efficientenergy storage technologies.4,5 In recent years, supercapacitors,one of the energy-storage devices, are of considerable interest byvirtue of their long cycling life, fast charge–discharge processand pollution-free operation.6–8 In comparison with variouselectrode materials including carbon,9 polymers10 and transi-tion metal hydroxides,11 mixed transition metal oxides (MTMO)coupled with different metal species possess unique featureswith high capacitance, electronic conductivity and rich redoxreactions, playing promising roles in supercapacitors.12–14

Among these MTMO materials, NiCo2O4 has been widelyinvestigated due to its appealing characteristics of low cost,nontoxicity, natural abundance and superior supercapacitiveperformance.12,15,16 To promote the electrochemical

uantum Control, Ministry of Education,

Jiao Tong University, Shanghai, 200240,

x: +86-021-54741040

ced Microstructures, Nanjing University,

nology, Shanghai Jiao Tong University,

tion (ESI) available. See DOI:

2–874915

performance, most studies have been devoted to fabricatingvarious nanostructures with a large surface area or goodconductive composites, for instance Fe-doped NiCo2O4 micro-spheres@nanomeshes,17 NiCo2O4 tetragonal microtubes,18

NiCo2O4 hollow spheres,13 NiCo2O4@Ni3S2 core/shell nano-thorns6 and TiN@NiCo2O4 coaxial nanowires.19 Nevertheless,these strategies still do not lead to a satisfactory capacitybecause the insufficient surface reactivity is a hindrance to fullyrealize its storage capability. The charge storage mechanism ofNiCo2O4 mainly based on redox reactions is closely related tothe surface chemistry and electronic structure of the material.Surface functionalization can modulate the surface electronicstructure and chemical environment and then boost the elec-trochemical performance of active materials.

Surface functionalization has been widely used for carbonelectrode materials;20–22 however, there are still very few reports onsurface functionalization of transition metal oxides for super-capacitors.23 Bai et al. reported NiO nanobers functionalizedwith citric acid as a supercapacitor electrode which exhibiteda much higher specic surface area and specic capacitance(336 F g�1) than those of pristine NiO (136 F g�1).24 Recently, Xiaand co-workers found that phosphate ion functionalized Co3O4

nanosheets could signicantly increase the number of surfaceactive sites and reduce charge transfer resistance, leading togreatly improved capacitive performance (up to approximatelyeightfold enhancement of specic capacitance).23 Lu et al.synthesized phosphorylated TiO2 nanotube arrays greatly

This journal is © The Royal Society of Chemistry 20187

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boosting sodium storage.25 As one kind of efficient functionali-zation strategies, sulfation has been successfully used in variouscatalyst elds, effectively enhancing catalytic reactivity.26–32 Gaoet al. illustrated that sulfated oxides displayed high thermosta-bility, very strong acidity and high catalytic reactivity.26 Liu et al.synthesized sulfated porous Fe–Ti bimetallic solid superacidcatalysts, which showed high reactivity for efficient photochem-ical oxidation of azo dyes under visible light.28 As noted above, weanticipate that sulfation can endow ultrathin mesoporousNiCo2O4 nanosheet arrays with higher surface chemical reactivity.To the best of our knowledge, until now, sulfated NiCo2O4 appliedin supercapacitors has not been reported.

In this work, we develop a facile galvanic displacementapproach to synthesize sulfated ultrathin mesoporous NiCo2O4

nanosheet arrays. Moderately sulfated NiCo2O4 exhibits anincreased capacitance far exceeding that of pristine NiCo2O4

(�57%) at a high current density of 5 A g�1 and a superiorcycling stability with 166% capacity retention aer testing for5000 cycles. This remarkable supercapacitive performance canbe ascribed to several merits. The unique ultrathin mesoporousnanosheet arrays provide a large surface area for faradaicreactions and short channels for ion diffusion and chargetransfer. Sulfation improves the surface reactivity and electrodekinetics and introduces highly active sites, bringing about lessenergy requirement for faradaic reactions. Herein, we not onlypropose a facile route to effectively functionalize metal oxidesfor energy conversion and storage but also establish a basiccomprehension of the correlation between sulfation and thecapacitive enhancement of NiCo2O4.

2. Experimental section2.1 Materials and methods

Nickel foam with an area of 1 cm � 4 cm pressed by a 2 MPaforce was cleaned in order with acetone and 2 M HCl in anultrasound bath for 15 min and rinsed with ethanol and DIwater several times. The sulfated ion functionalized NiCo2O4

was synthesized in a galvanic cell system with two half-cells. TheNi foam immersed into a mixed solution of 5 mM NiCl2, 10 mMCoCl2 and 0.25 M or 0.5 M thiourea was externally connected toa certain amount of an Al sheet dipped into 0.25 M NaOH witha copper wire. In addition, a saturated KCl salt bridge was usedto connect the two half-cells. The experiment was conducted atroom temperature for 1 h. The grown sample was washed withDI water and ethanol several times and then dried in air at 60 �Cfor 12 h. The dried sample was then annealed at 250 �C for 2 h inair. Without the addition of thiourea, the pristine NiCo2O4 canbe fabricated under the same experimental conditions. Forsimplication, the sample grown in 0.25 M and 0.5 M thioureais denoted as S0.25M and S0.5M, respectively. The weight of allsamples was measured before deposition and aer annealing,and the mass of active materials was 0.5 mg cm�2.

2.2 Structural characterization

Powder X-ray diffraction (XRD) was performed using a RigakuUltima IV X-ray diffractometer with a Cu Ka radiation source.


X-ray photoelectron spectroscopy (AXIS ULTRA DLD, Kratos,Japan) was also carried out to analyze the surface chemicalcomposition and valance states of the samples. The details ofthe structure and morphology were observed by eld emissionscanning electron microscopy (FESEM, Zeiss Ultra Plus) andtransmission electron microscopy (TEM) (JEOL JEM-2100F withan acceleration voltage of 200 kV).

2.3 Electrochemical measurements

We researched the electrochemical properties of all samples onNi foam as the working electrode in a three-electrode installa-tion with a SCE reference electrode and a Pt plate counterelectrode in 2 M KOH. The cyclic voltammetry (CV) and elec-trochemical impedance spectroscopy (EIS) tests were performedon a CHI760B electrochemical workstation. The galvanostaticcharge–discharge (GCD) measurements were conducted ona LAND CT-2001A. The calculation equation of specic capaci-tance is as follows:

C ¼ It

mDV(1)

where C (F g�1) is the specic capacitance, I (mA) is the currentdensity, t (s) is the discharge time, m (mg) is the quantity ofactive materials, and DV (V) is the voltage window.


The ultrathin mesoporous NiCo2O4 nanosheet arrays weresynthesized by a solution-based galvanic displacement methodand then annealed at 250 �C for 2 h in air, as illustrated inFig. 1a. The details of the preparation process can be found inthe Experimental section. The component of the as-depositedsamples is indexed to Co–Ni layered double hydroxides from theXRD spectra in Fig. S1.† The XRD spectra in Fig. 1b show thatthe peaks at 31.2�, 36.5�, 59.2�, and 65.1� of pristine andsulfated NiCo2O4, excluding two intense peaks originating fromNi foam, can be well indexed to the (220), (311), (511), and (440)crystal planes of the spinel NiCo2O4 cubic phase (JCPDS no. 20-0781), respectively. Compared with the pristine NiCo2O4, thefunctionalized NiCo2O4 without any impurity peaks and shi-ing peaks is proven to be of high purity.

X-Ray photoelectron spectroscopy (XPS) of each element andits tting results are further shown in Fig. 2a–d. The Ni 2p and Co2p spectra are both tted with two spin–orbit doublets andshakeup satellites. The XPS spectra of Ni 2p display two ttingpeaks at 854.1 eV and 871.5 eV, which indicate the presence ofNi2+.33 The spin–orbit peaks of Ni 2p3/2 at 855.7 eV and Ni 2p1/2 at873.2 eV correspond to the Ni3+ cation, which is in majority in Niatoms in the crystal lattice. In the Co 2p spectra, two peaks withbinding energies of 779.6 eV and 794.6 eV belong to the Co3+

cation and the other peaks at 781.1 eV and 796.3 eV are indexedto the Co2+ cation.34 The O 1s spectra show four different oxygencontributions, three peaks of which, at 529.5, 531.8 and 532.9 eV,are associated with the metal oxide, number of defect sites andwater adsorbed on the surface, respectively.35 The O 1s spectra ofpristine and sulfated NiCo2O4 both have a peak at 531.00 eV,originating from OH� and SO4

2� groups, respectively.36–38 The

J. Mater. Chem. A, 2018, 6, 8742–8749 | 87438


Fig. 1 (a) Schematic of synthesizing pristine and sulfated NiCo2O4 nanosheet arrays on Ni foam; (b) XRD spectra of synthesized sulfatedNiCo2O4, S0.25M and S0.5M.

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peak intensity at 531.0 eV improves with increasing sulfatecontent , implying that the OH� group on the NiCo2O4 surfacehas been replaced by the SO4

2� group during the surface sulfa-tion process. The core-level S 2p spectra in Fig. 3d show a broaderpeak centered at 168.5 eV, which can be attributed to SO4

2�.27,28

Moreover, the peak intensity of SO42� is proportional to the

added thiourea concentration, which is consistent with Fig. 2c.

Fig. 2 The XPS spectra of NiCo2O4, S0.25M and S0.5M: (a) Ni 2p spectr

8744 | J. Mater. Chem. A, 2018, 6, 8742–874915

The eld-emission scanning electron microscopy (FESEM)images of the as-prepared NiCo2O4 nanosheet arrays are given inFig. 3. Fig. 3a–c present the pristine NiCo2O4, S0.25M and S0.5Mwith a similar porous array micro-structure formed by crossstacking of numerous independent mesoporous nanosheets.These independent nanosheets with clearly observed mesoporeshave a planar gauze-like morphology with several hundred

a, (b) Co 2p spectra, (c) O 1s spectra, (d) S 2p spectra.



Fig. 3 The low- and high-magnified SEM images of (a and d) NiCo2O4, (b and e) S0.25M and (c and f) S0.5M. Insets in the upper-row figures showthe different distributions of NiCo2O4, S0.25M and S0.5M nanosheets on Ni foam.


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nanometers in lateral dimension. This open porous architecturecan provide abundant surface reactive sites and enough ionstorage space for fast faradaic reactions.39 However, there are stillsome differences in morphology and distribution between thethree different samples, the crook degree and the number ofwhose nanosheets is in proportion to the added amount ofthiourea as shown in the insets. The introduction of thiourea tothe precursor solution is benecial to form the nanosheet.

The transmission electron microscopy (TEM) measurementswere further employed to investigate the structure of thesynthesized NiCo2O4 nanosheets. From Fig. 4, the nanosheetsof pristine NiCo2O4 are composed of lots of nanoparticles witha size of about 12 nm consistent with the result calculated usingthe Scherrer equation (see the ESI†). The mesopores betweenthese nanoparticles mainly range from 2 to 5 nm, resulting fromthermal annealing of the precursor. Unlike pristine NiCo2O4,the nanoparticles in S0.25M have no obvious edge and tend tointerconnect, suggesting that the surface is reconstructed dueto the introduction of the sulfate ion. Besides, the well-deneddiffraction rings displayed in the selected-area electrondiffraction (SAED) patterns of NiCo2O4 and S0.25M both illus-trate their polycrystalline nature. In accordance with the XRD

Fig. 4 The TEM images and the SAED pattern (insets) of (a) NiCo2O4, (b


results, these rings can be satisfactorily ascribed to the (220),(311), (400) and (440) crystal planes of the cubic NiCo2O4 phase,indicating that sulfation will not change the crystal structure.

The pristine and sulfated NiCo2O4 nanosheet arrays on Nifoam were used as binder-free electrodes for supercapacitors toevaluate their electrochemical properties. Fig. 5 shows therepresentative cyclic voltammogram (CV) curves at varioussweep rates in the range from 5 to 100 mV s�1 in the potentialwindow of 0–0.6 V vs. SCE. A pair of distinct redox peaks of theCV curves is derived from reversible faradaic reactions, clearlyrevealing the conventional faradaic behaviors of this battery-type electrode.4,40 The reversible redox reactions in sulfatedNiCo2O4 are inferred as the following equations:

NiCo2O4�x(SO42�)x + OH� + H2O 4 NiO1�y(SO4

2�)yOH

+ 2CoO1�(x�y)/2(SO42�)(x�y)/2OH + e� (2)

CoO1�(x�y)/2(SO42�)(x�y)/2OH + OH�4

CoO2�(x�y)/2(SO42�)(x�y)/2 + H2O + e� (3)

Signicantly, with the gradual increase of scan rates, theshape of these CV curves can be maintained well except a slight

) S0.25M and (c) S0.5M.

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Fig. 5 CV curves of (a) NiCo2O4, (b) S0.25M and (c) S0.5M at scan rates ranging from 5 mV s�1 to 100 mV s�1; (d) CV curves of NiCo2O4, S0.25Mand S0.5M at 50 mV s�1.


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shi of the peak position, suggesting that the fast redox reac-tions occur in the electrode. As can be seen from Fig. 5d, thearea relationship of CV curves at a scan rate of 50 mV s�1

between pristine and sulfated NiCo2O4 is AS0.5M > AS0.25M >Apristine NiCo2O4

.The galvanostatic charge–discharge measurements were

carried out with the potential window between 0 and 0.53 V (vs.SCE) to investigate the specic capacitances of the electrodes.Consistent with the CV measurements, pronounced voltageplateaus corresponding to the redox couple can be observedfrom the galvanostatic charge–discharge curve proles asshown in Fig. 6a. Furthermore, the symmetric shapes of thesecurves imply good reversibility of redox reactions.41 The speciccapacitance as a function of current density for pristine andsulfated ion functionalized NiCo2O4 is plotted in Fig. 6b.Remarkably, NiCo2O4, S0.25M and S0.5M deliver speciccapacitances of 390.56, 524.52, and 656.6 F g�1 at a currentdensity of 1 A g�1, respectively. Surprisingly, the speciccapacitance increases rapidly as the current density increases,which can be attributed to the capacitance activation. When thecurrent density increases from 1 to 40 A g�1, the speciccapacitance retention is calculated to be 116, 144 and 138% forNiCo2O4, S0.25M and S0.5M, respectively.

The cycling stability of NiCo2O4, S0.25M and S0.5M wasshown in Fig. 6c. All the samples exhibit capacitance activationin the rst 1000 cycles. Due to the porous array architecture,with the gradual penetration of electrolyte into the interiorstructure, the inner active material is activated and participatesin redox reactions, increasing the capacitance gradually.42–44

Aer 1000 cycles, the specic capacitance of S0.5M decreasesrapidly while the S0.25M presents the highest specic

8746 | J. Mater. Chem. A, 2018, 6, 8742–874916

capacitance of 1113 F g�1 until the 2100th cycle, which is 1.57times higher than that of pristine NiCo2O4 with the highestcapacitance of 707 F g�1. Moreover, the NiCo2O4, S0.25M andS0.5M, respectively, show a capacitance retention of 169, 166and 112% aer 5000 cycles. Accordingly, moderate sulfationcan greatly improve the capacitive performance for NiCo2O4.

In order to investigate the inuence of long-term cyclingtests on the morphology and structure, SEM and TEMmeasurements are further conducted for all samples aer 5000cycles. From Fig. S2 and S3,† it can be seen that all samplesmaintain a good porous-array structure and ultrathinmorphology except for some slightly collapsed nanosheets,implying the negligible inuence of activation and charge–discharge cycle process on the structure and morphology. The S2p core-level XPS spectra of the sulfated samples aer beingfully activated are further measured to investigate the reason forthe capacitance decrease of S0.5M (see Fig. S4†). Aer beingfully activated, S0.5M loses almost all sulfate ions while S0.25Mretains the majority of its sulfate ions, indicating that excesssulfate ion functionalization gives rise to the unstable surfacechemistry or electronic structure which results in easy and rapidloss of sulfate ions. Thus, a higher level of sulfation causes morecapacitance decrease aer the activation process.

To further research the resistive capacity of the activematerials, electrochemical impedance spectroscopy was per-formed at an open circuit potential in the frequency range of0.1–100 kHz. From the Nyquist plots of the different samples inFig. 6d, the equivalent series resistances of NiCo2O4, S0.25 andS0.5M were 0.58, 0.6 and 0.55 U, respectively, implying the lowinternal resistance of the electrode and good adhesion to thesubstrate of the active materials.45 Compared with pristine



Fig. 6 (a) The galvanostatic charge–discharge curves at various current densities of S0.25M; (b) the specific capacitances at different currentdensities, (c) cycling performance at a current density of 5 A g�1 and (d) Nyquist plots of NiCo2O4, S0.25M and S0.5M


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NiCo2O4, S0.25M and S0.5M exhibited smaller semicircles andmore vertical lines, indicating the enhancement in chargetransfer and ion diffusion.1,46 In comparison with S0.25M,S0.5M shows a smaller slope and a larger semicircle, whichindicate that excess sulfate ions prevent the decrease of electrontransmission and ion diffusion resistance and facilitate fastredox reactions.

To comprehend the correlation between surface sulfation andenhanced electrochemical performance, we further researchthe surface electronic structure and chemical reactivity. The

Fig. 7 Schematic of the functionalized surface and redox reaction proc


introduction of sulfate ions changes the chemical bonds andcorresponding electronic chemical environment of metal sites,thus signicantly promoting surface reactivity.23 The length ofthe Co–O bond of about 1.855 A is shorter than that of Co–SO4,which is about 2.0 A. Furthermore, the Pauling electronegativityof –SO4 is calculated to be 3.225,47 smaller than that of the Oelement (3.44). The covalent character of the Co–SO4 bond isfurther estimated to be 63.6%, higher than that of the Co–O bondwhich is about 54.4%. The sulfate functionalized NiCo2O4 witha longer bond and higher covalent character needs less energy for

ess in the sulfated NiCo2O4 nanosheet arrays.

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redox reactions to occur than that required before functionali-zation, thus enhancing surface reactivity and electrode kinetics.Fig. 7 gives the schematic illustration of the functionalizedsurface, redox reaction process and electronic structure of thefunctionalized NiCo2O4. The ultrathin mesoporous morphologyand open porous arrays provide a large surface area with lots ofactive sites and “ion reservoir” providing enough electrolyte ionsfor fast redox reactions at high current densities. The synergisticeffect of structure and moderate sulfation facilitates electrontransmission and improves surface reactivity, thus resulting inhigh capacity and excellent stability.

4. Conclusion

In summary, we fabricate sulfated ultrathin mesoporousNiCo2O4 nanosheet arrays on Ni foam using a facile self-assembly technique without conventional post-processing.Adjusting the addition amount of the sulfur source can readilytailor the functionalized degree of the sulfate-ion and surfacereactivity of NiCo2O4. The surface reactivity of the NiCo2O4

nanosheet arrays is boosted by surface sulfation. Benetingfrom the enhanced surface activity, electrode kinetics andunique nanosheet array structure, the moderately functional-ized NiCo2O4 (S0.25M) achieves a fascinating electrochemicalperformance with excellent stability and much higher capaci-tance far exceeding that of the pristine sample at a high currentdensity of 5 A g�1. Given the excellent electrochemical perfor-mance and facile synthesis process, we believe this sulfationstrategy is a promising route to fabricate enhanced-perfor-mance electrodes for energy conversion and storage.

Conflicts of interest

There are no conicts to declare.

Acknowledgements

The authors gratefully thank the support of this work by theNational Natural Science Foundation of China (grant no.11174197 and 11574203) and the support for SEM tests from theCenter for Advanced Electronic Materials and Devices (AEMD)of Shanghai Jiao Tong University.

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J. Mater. Chem. A, 2018, 6, 8742–8749 | 87494


Magnetic and Structural Transitions Tuned through Valence ElectronConcentration in Magnetocaloric Mn(Co1−xNix)GeQingyong Ren,*,†,⊥,#,|| Wayne D. Hutchison,*,† Jianli Wang,*,‡,¶ Andrew J. Studer,§

and Stewart J. Campbell†

†School of Physical, Environmental and Mathematical Sciences, The University of New South Wales, Canberra at the AustralianDefence Force Academy, Canberra, Australian Capital Territory 2600, Australia‡Institute for Superconductivity and Electronic Materials, University of Wollongong, Wollongong, New South Wales 2500, Australia§Australian Centre for Neutron Scattering, Locked Bag 2001, Kirrawee DC, New South Wales 2232, Australia⊥School of Physics and Astronomy and #Key Laboratory of Artificial Structures and Quantum Control, School of Physics andAstronomy, Shanghai Jiao Tong University, Shanghai 200240, China||Collaborative Innovation Center for Advanced Microstructures, Nanjing 210093, China¶College of Physics, Jilin University, Changchun 130012, China


ABSTRACT: The structural and magnetic properties ofmagnetocaloric Mn(Co1−xNix)Ge compounds have beenstudied. Two responses to the increase of valence electronconcentration on substitution of Ni (3d84s2) for Co (3d74s2) inthe orthorhombic phase (Pnma) are proposed: expansion ofunit-cell volume and redistribution of valence electrons. Wepresent experimental evidence for electronic redistributionassociated with the competition between magnetism andbonding. This competition in turn leads to complex depend-ences of the reverse martensitic transformation temperature TM(orthorhombic to hexagonal (P63/mmc)) and the magneticstructures on the Ni concentration. Magnetic transitions fromferromagnetic structures below x = 0.50 to noncollinear spiralantiferromagnetic structures above x = 0.55 at low temperature (e.g., 5 K) are induced by modification of the density of states atthe Fermi surface due to the redistribution of valence electrons. TM is found to decrease initially with increasing Ni content andthen increase. Both direct and inverse magnetocaloric effects are observed.

■ INTRODUCTIONThe structural and magnetic properties of MnCoGe-basedalloys have been studied extensively in recent years;1−5 thisstems mainly from their potential application as magneticcooling materials based on the magnetocaloric effect (MCE).There are two stable crystallographic structures, correspondingto the nominal high-temperature hexagonal (Hex) phase withNi2In-structure (P63/mmc) and the nominal low-temperatureorthorhombic (Orth) phase with TiNiSi-type structure (Pnma)(see Figure 1 and Figure S1).6 It was reported that bothcollinear ferromagnetic (FM) and noncollinear spiral anti-ferromagnetic (spiral-AFM) structures may exist in theorthorhombic phase at low temperature.7−10 The presence offerromagnetic or spiral antiferromagnetic structures in theorthorhombic phase would allow scope for a direct (positive)MCE11−13 or inverse (negative) MCE.14−16 It is thereforeimportant to study the mechanism for the magnetic transferbetween FM and spiral-AFM in the orthorhombic phase.One mechanism for a noncollinear magnetic structure is the

competition between symmetric superexchange and the

antisymmetric Dzyaloshinsky−Moriya (DM) interaction,17,18

such as in CuB2O7,19 LiFeAs2O7,

20 and FeBO3.21 The DM

interaction is a relativistic correction to the usual superexchangedue to spin−orbit coupling.18,22 One fundamental macroscopicfeature of this antisymmetric coupling takes place in non-centrosymmetric magnetic crystals.23,24 Another possiblereason derives from competition between interactions such asRuderman-Kitel-Kasuya-Yosida (RKKY) in itinerant sys-tems25,26 or Heisenberg exchanges in localized momentsystems.27,28 Different from the DM interaction that dependson lattice symmetry, these interactions depend strongly on thedistances between the magnetic moments. The RKKYinteraction is long-range interaction,29 with oscillation of themagnetic interactions being responsible for a spiral structure.30

In the Heisenberg model, frustration between nearest-neighbor

Received: November 29, 2017Revised: January 23, 2018Published: February 12, 2018

Article

pubs.acs.org/cmCite This: Chem. Mater. 2018, 30, 1324−1334

© 2018 American Chemical Society 1324 DOI: 10.1021/acs.chemmater.7b04970Chem. Mater. 2018, 30, 1324−1334

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(NN) and next-nearest-neighbor (NNN) exchange interactionscan likewise cause a spiral structure.31,32

An alternative explanation for a noncollinear spiral-AFMbuilds on the stability of the band structure. Earlier reports inMn-based orthorhombic systems indicated that a high densityof states at the Fermi surface (N(EF)) makes FM unstable whilea noncollinear AFM can reduce N(EF).

33,34 Such lower N(EF)in a noncollinear AFM compared with a FM state is linked tothe modification of Fermi surface topology which has beendiscussed by Lizarraga et al.35 This discussion is based on thepossibility of hybridization of the spin-up and spin-downchannels for a noncollinear spin configuration. The opening ofa band gap at the Fermi level due to the hybridization wouldlower the total energy over a ferromagnetic configuration. It isconsidered that a noncollinear structure can be obtained bymodifying the Fermi surface in any element or compound,especially in a ferromagnet with nesting features between thespin-up and spin-down Fermi surface,35−37 e.g., throughapplication of high pressure38 or modification of the constituentelements.39 Such noncollinear structures generally correspondto the systems exhibiting small unit-cell volumes and smallmagnetic moments, e.g., in Fe−Ni Invar alloys40 and zinc-blende MnAs.36

In the case of MnCoGe-based compounds, it was reportedthat the valence electron concentration e/a (VEC, defined asthe average number of valence electrons per atom) has a stronginfluence on the magnetic properties.41,42 In the present work,

the magnetic transfer between ferromagnetism and spiralantiferromagnetism in the orthorhombic structure of MnCoGewill be tuned by modifying VEC through partial substitution ofNi (3d84s2) for Co (3d74s2). A comprehensive set of 15Mn(Co1−xNix)Ge compounds with a broad Co/Ni ratio (0.12≤ x ≤ 1.00) have been prepared and investigated over thetemperature range 5−450 K using magnetization, X-ray andneutron diffraction measurements. In addition, the relationshipof the VEC with magnetic structures and structural transitionswill be discussed from the aspects of electronic redistributionand unit cell volume, which is associated with the modificationof the density of states at the Fermi surface (N(EF)).

■ EXPERIMENTAL SECTIONSynthesis. Fifteen polycrystalline Mn(Co1−xNix)Ge samples with x

= 0.12, 0.14, 0.16, 0.18, 0.20, 0.30, 0.40, 0.50, 0.55, 0.58, 0.60, 0.70,0.80, 0.90, and 1.00 (∼2 g each sample) were prepared by arc meltingstoichiometric amounts of Ni, Co, Ge and Mn (purities >99.95 wt %)under an argon atmosphere on a water cooled Cu hearth. The oxidizedsurface of the Mn flakes was first removed using diluted nitric acid.The ingots were melted four times to ensure homogeneity with 3%excess Mn added to compensate for mass loss during the meltingprocess. The ingots were wrapped with Ta foil and sealed in quartzglass tubes under vacuum. The samples were then annealed in ahorizontal tube furnace at 850 °C for 7 days. Finally, the samples werequenched into iced-water and ground into powder using an agatemortar/pestle for the measurements.

Figure 1. Crystal structures and Co/Ni−Ge networks in the Mn(Co0.40Ni0.60)Ge sample with (a−c) a Ni2In-type hexagonal structure (P63/mmc)and (d−f) a TiNiSi-type orthorhombic structure (Pnma). Only the Mn atoms in the first unit cell are shown. Both structures are built from two-dimensional layers (indicated by I, II, ...) of edge-sharing six-membered Co/Ni−Ge rings. (a, d) Three-dimensional views, (b, e) two-dimensionalviews projected on the (110) plane and ac plane, and (c, f) III and IV layers projected on the ab and bc planes for the hexagonal and orthorhombicstructures, respectively. The numbered Mn atoms are used as guides for the rotation of the networks. The annotations ′ and ″ in a−c are used asindicators of the atoms on the inner plane and the next-inner plane, respectively. The Co/Ni−Ge bonds between the two-dimensional layers areseparated into two groups, marked as [A] and [B], respectively. Bonds [B] were broken during the martensitic transformation from the hexagonalstructure to the orthorhombic structure. The figures are created based on the Rietveld refinement results of the X-ray diffraction pattern forMn(Co0.40Ni0.60)Ge at room temperature (∼295 K, see Figure 2).

Chemistry of Materials Article

DOI: 10.1021/acs.chemmater.7b04970Chem. Mater. 2018, 30, 1324−1334

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X-ray Diffraction. The quality and phases of all samples werechecked by X-ray diffraction measurements at room temperature(∼295 K) using a PANalytical diffractometer with CuKα radiation. Inaddition to the neutron diffraction measurements outlined below, thestructures and related martensitic reverse transformation temperatureswere investigated by variable temperature X-ray diffraction measure-ments over the temperature range from 20 to 310 K in 10 K steps. Aholding time of 10 min was used at every temperature step to ensurethermal equilibrium.Magnetic Measurements. Magnetic measurements were per-

formed on the polycrystalline samples using a Quantum Designphysical property measurement system (PPMS). The magnetizationversus temperature at constant-field sweeps were performed in anapplied magnetic field of 0.01 T with 5 K steps over a temperaturerange of 5−300 K or 5−320 K. Two measurement sequences wereused: heating after zero-field cooling (ZFC) and field cooling (FC) ata rate of 0.7 K/min. The field-dependent isothermal magnetizationswere recorded in fields up to 5 T in steps of 0.2 or 0.25 T around themagnetic phase transitions temperatures in 5 K steps, using thestandard isothermal process.43

Neutron Powder Diffraction. Neutron powder diffractionexperiments were performed on Mn(Co1−xNix)Ge samples with x =0.14, 0.40, 0.55, 0.58, 0.60, 0.90, and 1.00 using the high-intensitypowder diffractometer WOMBAT at the OPAL Reactor (LucasHeights, Australia) with a neutron wavelength of 2.4143 Å.44 Acontinuous detector bank of 160° × 120 mm high was used. Variabletemperature measurements were carried out in zero magnetic fieldusing a top loading cryofurnace over the temperature range 5 to 450 K.Refinements of both the X-ray diffraction and neutron patterns wereperformed using the Rietveld method and irreducible representationtheory using the program FullProf.45,46

■ RESULTS AND DISCUSSIONStructural Description. X-ray diffraction patterns of

Mn(Co1−xNix)Ge (x = 0.14 to 1.00) at room temperature arepresented in Figure 2. All of the samples were found to exhibitthe TiNiSi-type orthorhombic structure (Pnma) and/or theNi2In-type hexagonal structure (P63/mmc). Examples of therefinements of the X-ray diffraction patterns are given in FigureS2. There are no discernible impurities (neutron diffractionmeasurement with higher resolution demonstrates a small

fraction of the impurity phase MnNi1.25Ge0.75 (≤2(1) wt %) inthe samples with x = 0.90 and 1.00). The fraction of theorthorhombic phase at room temperature initially decreaseswith Ni content and then increases as shown in the inset ofFigure 2. As noted the hexagonal and orthorhombic structuresof the Mn(Co0.40Ni0.60)Ge at room temperature are shown inFigure 1. It is considered that the Co/Ni (2d site) and the Ge(2c site) atoms form a three-dimensional five-connected(3D5C) network in the hexagonal structure.47 This networkcan be viewed as being built from two-dimensional flat layers ofedge-sharing six-membered Co/Ni−Ge rings. The Co/Ni−Gehoneycomb is stitched together by Mn atoms. X-ray diffractionof the MnCoGe single crystal indicated that Co atoms havemuch larger thermal vibrations in the hexagonal structure(along the ahex-axis) than in the orthorhombic structure.6 Thislarge vibration amplitude of Co brings about higher entropy inthe high-temperature hexagonal structure and hence leads to adiffusionless and displacive structural transition into the low-temperature orthorhombic structure. During this transition, theadjacent two-dimensional Co/Ni−Ge layers in the hexagonalstructure glide toward opposite directions along ⟨11 0⟩ asindicated by the bold arrows in Figure 1. As a result of therelease of the Co thermal vibration, bonds between the Co/Ni−Ge layers are varied dramatically and half of them (shortdashes in Figure 1e, labeled as [B]) are broken. The flat Co/Ni−Ge layers are wrinkled and become pseudozigzag layers inthe orthorhombic structure. Finally the 3D5C Co/Ni−Genetwork in the hexagonal structure changes as a 3D4C Co/Ni−Ge network in the orthorhombic structure, in agreement withthe description by Landrum et al.48

As described previously,49 the relationships between thelattice parameters of the two structures depicted in Figure 1 canbe derived as aorth ≈ chex, borth ≈ ahex, corth/√3 ≈ ahex, and Vorth/2 ≈ Vhex. The lattice parameters as functions of Ni content x forthe orthorhombic phase in Mn(Co1−xNix)Ge (x = 0.12−1.00)as determined from Rietveld refinements of the X-raydiffraction patterns (Figure 2) are shown in Figure 3. Whilethe value of borth is found to decrease from 3.817(1) Å for x =0.12 to 3.760(1) Å for x = 1.0 (Figure 3b) on substitution of Niof atomic radius 1.21 Å for Co of atomic radius 1.26 Å, aorth and

Figure 2. X-ray diffraction patterns of Mn(Co1−xNix)Ge samples (x =0.14 to 1.00) at room temperature (∼295 K). The Miller indices ofreflections from the hexagonal and orthorhombic structures areindexed with and without asterisks, respectively. Inset are the fractionsof the hexagonal and orthorhombic phases as a function of Ni contentx (x = 0.12 to 1.00). Interestingly, it is noted that the orthorhombicphase fraction decreases from 98(3) wt % (x = 0.12) to 13.5 (5) wt %(x = 0.40) and then increases from 12.3(6) wt % (x = 0.58) to 99(4)wt % (x = 1.00).

Figure 3. (a−c) Lattice parameters and (d) unit-cell volumes for theorthorhombic phase in the Mn(Co1−xNix)Ge (x = 0.12 to 1.00) fromthe room temperature X-ray diffraction patterns of Figure 2. The fulllines are guides to the eye with the dashed lines indicating the slightdiscontinuities in aorth, borth, and corth around x = 0.55 as discussed inthe text. The errors bars are as shown (in some cases the uncertainty issmaller than the symbol).



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corth increase from 5.981(1) Å to 6.046(1) Å and from 7.064(1)Å to 7.095(1) Å (Figures 3a, c, respectively). Such pronouncedanisotropy leads to the interesting variation in unit cell volumeVorth shown in Figure 3d. Vorth does not follow Vegard’s law butrather exhibits a Λ-like dependence on the Ni content x:increases first with Ni content to x ≈ 0.55 and then decreases.In addition, it is noted that slight discontinuities are alsoobserved in aorth, borth, and corth around x = 0.55, as indicated bythe dashed lines in Figure 3a−c.Magnetization. Magnetization curves in an applied

magnetic field of 0.01 T for Mn(Co1−xNix)Ge with x = 0.14,0.40, 0.55, 0.60, and 1.00 are shown in Figure 4a (for more

information, see Figure S3). The samples with x = 0.14, 0.40,and 0.50 exhibit ferromagnetic-like behavior below themagnetization transition temperatures. In contrast, the magnet-ization curves for the samples with x = 0.55, 0.58, and 0.60 haveantiferromagnetic-like features at low temperature followed by atransition to ferromagnetism at 155(4) K, 210 (4) K and245(4) K, respectively. As evident in Figure S3 the magnet-izations for Ni-rich samples with x ≥ 0.70 are significantlyreduced compared with the magnetizations of the othersamples. As outlined below, analysis of the neutron diffractionpatterns demonstrate that for the Mn(Co1−xNix)Ge with x ≥0.7, the antiferromagnetic phase transforms directly toparamagnetic phase at the corresponding Neel temperatureTN (the transition temperatures are listed in Table S1).The magnetic transitions from ferromagnetic structure to

antiferromagnetic structure with increasing Ni content x at 5 Kare illustrated by the isothermal magnetizations curves in Figure4b. As evident in Figure 4b, whereas the magnetic isotherm for

the Mn(Co1−xNix)Ge with x = 0.40 increases rapidly withmagnetic field, metamagnetic behaviors are observed for thesamples with x = 0.55, 0.60, and 1.00. Such metamagneticbehaviors indicate the rotation of the antiferromagneticmoment toward a ferromagnetic alignment under appliedmagnetic field. In addition, a higher magnetic field is needed toalign the antiferromagnetic moments with increasing Nicontent, e.g., the ferromagnetic alignment occurs at μ0H ≈6.0 T for x = 1.00 compared with μ0H ≈ 1.2 T for x = 0.60.This indicates that the antiferromagnetic phase becomesincreasingly stable with the increased Ni content.The magnetic transitions in the Mn(Co1−xNix)Ge samples

with x = 0.40 to 0.60 occur between ∼280 K and ∼310 K.Trung et al. have demonstrated that magnetic transitionsoccurring in the temperature range 275 to 345 K in MnCoGe-based compounds generally coincide with a structuraltransition, and that such coincidence provides scope for theformation of magneto-structural transitions from ferromagneticorthorhombic structure (FM-Orth) to paramagnetic hexagonalstructure (PM-Hex) and hence an associated large MCE.11

Therefore, it is also expected that a magneto-structuraltransition and large MCE will be observed in the Mn-(Co1−xNix)Ge samples with x = 0.40 to 0.60. More detailsabout the magnetic transition temperatures obtained for thepresent Mn(Co1−xNix)Ge samples are summarized in Table S1.It is also noted that the thermal hysteresis, ΔThys, i.e., thedifference in magnetic transition temperature on warming andcooling, decreases from 15(2) K for x = 0.40 to 5(2) K for x =0.60. Such decrease of thermal hysteresis is advantageous forimproving the energy efficiency in magnetocaloric materials.50

Magnetic Structures. As demonstrated from the magnet-ization measurements, Mn(Co1−xNix)Ge with Ni concentra-tions in the range x = 0.14 to x = 1.00 exhibit ferromagneticand antiferromagnetic behaviors. The magnetic structures ofthese compounds have been investigated in three differentregions delineated by the magnetic phase transitions: sampleswith x ≤ 0.50 show a single FM/PM transition; samples with0.55 ≤ x < ∼ 0.70 have an AFM/FM transition and a FM/PMtransition; samples with x ≳ 0.70 exhibit a single AFM/PMtransition. It is reported that the Ni atoms do not carry amagnetic moment in MnCoGe/MnNiGe-based alloys.8 Inaddition, following our previous study of (Mn0.98Fe0.02)CoGeand Mn(Co0.96Fe0.04)Ge, the magnetic moment on Co atomswas shown to be too small to be resolved by neutron powderdiffraction measurements.4,5 Therefore, in the present workonly the magnetic moments on the Mn sublattice of theMn(Co1−xNix)Ge compounds are considered in the analysis ofthe neutron diffraction patterns. In addition, the Ni/Cooccupancies obtained through the Rietveld refinements of theneutron diffraction patterns match well with the nominalcompositions of the samples (see Table S2).

1. Magnetic Structures for Mn(Co1−xNix)Ge (x = 0.14 and0.40). The neutron powder diffraction patterns for Mn-(Co0.86Ni0.14)Ge and Mn(Co0.60Ni0.40)Ge are shown in FiguresS4a, b. Both samples have the orthorhombic structure at lowtemperatures, followed by a structural transition to the high-temperature hexagonal structure at 370(2) K and 299(1) K,respectively. No satellite peaks are observed in the diffractionpatterns, precluding an incommensurate structure in these twosamples. Analysis of the magnetic structure was carried outusing irreducible representation theory as described in ourprevious report.5 (More details about the analysis of the

Figure 4. (a) Temperature dependences of the magnetization for theMn(Co1−xNix)Ge samples, x = 0.14, 0.40, 0.55, 0.60 and 1.00, in anapplied field of μ0H = 0.01 T on warming after zero-field cooling(ZFC, solid symbols) and field cooling (FC, open symbols). (b) Fielddependence of the magnetization for samples at 5 K with x = 0.40,0.55, 0.60, and 1.00.



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neutron diffraction patterns are given in Section 4.2 in theSupporting Information).Both Mn(Co0.86Ni0.14)Ge and Mn(Co0.60Ni0.40)Ge have

ferromagnetic structures below the Curie temperatures,345(5) K and 280(4) K, respectively (see Figures S3 andS4). The magnetic moments on the Mn sublattice align alongthe corth axis (FMc) in Mn(Co0.86Ni0.14)Ge as shown in Figure5a; the magnetic moments on the Mn atoms are 3.4(1) μB at 5

K. Interestingly, it is found that further doping of Ni causes areorientation of the magnetic moments on the Mn atoms in theorthorhombic structure. For example, the 3.5(1) μB magneticmoments in Mn(Co0.60Ni0.40)Ge lie along the borth axis (FMb)at 5 K as shown in Figure 5b. Moreover there are otherdifferences apparent: Mn(Co0.86Ni0.14)Ge has separated mag-netic and structural transitions at 345(5) K and 370(2) K,whereas Mn(Co0.60Ni0.40)Ge has only one FM-Orth/PM-Hexfirst order magneto-structural transition (FOMST) at 299(1) Kwith a full width at half-maximum of 13(1) K.

2. Magnetic Structures for Mn(Co1−xNix)Ge (x = 0.55, 0.58and 0.60). The Mn(Co1−xNix)Ge samples with x = 0.55, 0.58,and 0.60 are found to exhibit similar magnetic structures andphase transitions. The neutron powder diffraction patterns forMn(Co1−xNix)Ge with x = 0.60 are shown in Figure 6a with thediffraction patterns for Mn(Co1−xNix)Ge with x = 0.55 and0.58 shown in Figures S4c, d. The Mn(Co0.40Ni0.60)Ge sampleis taken as an example for discussion of these magneticstructures.From the neutron data of Figure 6a it is concluded that the

orthorhombic structure dominates in Mn(Co0.40Ni0.60)Gebelow 312(1) K. In addition, a group of satellite peaks isobserved below 245(5) K, indicating a possible incommensu-rate AFM structure. Enhancement of several nuclear scatteringpeaks (e.g., (103) peak around 2 theta ∼66.5° in Figure 6a; alsosee Figure S6 for variation of the (101)orth peak intensity withtemperature) as well as disappearance of the satellite peaks areobserved above 245(5) K in Mn(Co0.40Ni0.60)Ge. According tothe magnetization data in Figure 4a, this increase of the peakintensity may be the result of a magnetic transition to aferromagnetic structure. On further warming the Mn-

Figure 5. Magnetic structures in the orthorhombic phase of (a)Mn(Co0.86Ni0.14)Ge and (b) Mn(Co0.60Ni0.40)Ge at 5 K. Each samplehas a ferromagnetic structure: the magnetic moments in Mn-(Co0.86Ni0.14)Ge point to corth, corresponding to the irreduciblerepresentation model Fz of Γ3; and the magnetic moments inMn(Co0.60Ni0.40)Ge point to borth, corresponding to the irreduciblerepresentation model Fy of Γ5.

Figure 6. Neutron diffraction patterns (λ = 2.4142 Å) for (a) Mn(Co0.40Ni0.60)Ge and (c) MnNiGe over the temperature range 10−450 K. Most ofthe peaks are identified as the orthorhombic structure (Miller indices without asterisk) and the hexagonal structure (Miller indices with asterisks). Asmall fraction of the impurity phase MnNi2Ge (≤2(1) wt %) is observed in MnNiGe, as indicated by triangles in c. The orange curves indicate theevolution of the magnetic satellite peaks. Some satellite peaks which are too small to be discerned are not marked. The green circle around 2θ ≈66.5° in (a) indicates the enhancement of the diffraction intensity on the (103) peak of the orthorhombic structure in Mn(Co0.40Ni0.60)Ge above245(5) K. (b, d) Magnetic structures in the orthorhombic phase at 5 K for Mn(Co0.40Ni0.60)Ge and MnNiGe, respectively. The propagation vectorsare 0.247(1) and 0.184(1) along aorth for both samples. θ angles are 70 and 0°, corresponding to cycloidal-spiral (CS) structure and a simplecycloidal (SC) structure, for Mn(Co0.40Ni0.60)Ge and MnNiGe, respectively.



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(Co0.40Ni0.60)Ge sample transforms to the hexagonal structureat 312(1) K.The propagation vectors for Mn(Co0.40Ni0.60)Ge below

245(5) K were determined using the K-search softwareembedded in FullProf. The results indicate that the magneticmoments propagate along the aorth-axis with the magnetic peaksfound to be well matched using a conical structure. A schematiccoordinate system for the arrangement of a magnetic momentin this conical model in which the magnetic moment μ lies on acone with a half angle of ψ is given in Figure S7. The sphericalangles of the spiral (cone) axis l of the first atom are θ and φwith respect to spherical coordinates, with determination of theparameters ψ, θ and φ given in Section 4.4.1 of the SupportingInformation.The Rietveld refinement results for the neutron diffraction

pattern of Mn(Co0.40Ni0.60)Ge at 5 K are shown in Figure S10a.The optimal parameters for the conical structure are ψ = 90°, θ= 70°, and φ = 0°, corresponding to a flat cycloidal-spiral (CS)structure. The propagation vector is determined as kx =0.247(1) along the aorth axis. The configuration of this magneticstructure is shown in Figure 6b, and the refinement results aresummarized in Table 1. Rietveld refinements of the diffractionpatterns above 245(5) K demonstrate a ferromagnetic structurewith the magnetic moment pointing along borth (FMb), similarto the magnetic structure in Mn(Co0.60Ni0.40)Ge. As anexample, the refinement results for the neutron diffractionpattern of Mn(Co0.60Ni0.40)Ge at 280 K are summarized inTable 1.3. Magnetic Structures for Mn(Co1−xNix)Ge (x = 0.90 and

1.00). Satellite peaks are also observed in the neutron powderdiffraction patterns for Mn(Co0.10Ni0.90)Ge below 350(5) K(Figure S4e) and MnNiGe below 360(5) K (Figure 6c). In thiscase, there is no AFM/FM transition. The Orth/Hex structuraltransition temperatures TM are also much higher: 442(5) K forMn(Co0.10Ni0.90)Ge and >450 K for MnNiGe compared withMn(Co1−xNix)Ge compounds (x = 0.12 ≤ x ≤ 0.80).The Rietveld refinement for MnNiGe at 5 K is shown in

Figure S10b resulting in the values ψ = 90°, θ = 0°, φ = 0° andkx = 0.184(2). This is a simple cycloidal (SC) magneticstructure with magnetic moments in the ab plane as shown inFigure 6d. Clearly there are differences in the magneticstructures of Mn(Co0.40Ni0.60)Ge and MnNiGe at 5 K. In fact,in the Mn(Co1−xNix)Ge samples the angle θ of the spiral axis l

ranges between 0° and 90° with Ni content x and withtemperature in a slightly irregular manner (see Figure S9; asdoes the propagation vector discussed below). For x = 0.55 to0.60, θ generally decreases with increasing temperature,whereas for x = 0.90 and 1.00, remarkably, θ starts at zero atlow temperature and flips to 90° and 70° near 18 K and 220 K,respectively.

4. Variation in Propagation Vector kx. Close comparison ofthe neutron diffraction patterns of Mn(Co0.40Ni0.60)Ge inFigure 6a and those of MnNiGe in Figure 6c, reveals that thesatellite peaks move toward the nuclear peaks in Mn-(Co0.40Ni0.60)Ge with increasing temperature whereas thesatellite peaks shift away from the nuclear peaks in MnNiGe.A detailed comparison of the propagation vector kx for differentNi content x as functions of temperature is shown in Figure 7a.The propagation vector kx decreases with increasing temper-ature in the samples with x = 0.55, 0.58, and 0.60 while incontrast, kx increases with temperature for samples with x =0.90 and 1.00. Given the strong dependence of the rotationangle (propagation vector) in a spiral structure on the latticeparameters,51−53 the different temperature dependences of themagnetic propagation vectors (kx, 0, 0) in Mn(Co1−xNix)Gewith x ≥ 0.55 are considered as the results of differentanisotropic thermal evolutions of the lattice parameters. Asshown in Figure 7(b), the Mn(Co1−xNix)Ge samples with x =0.55, 0.58, and 0.60 have different temperature dependences ofthe aorth/(corth/√3) ratios from those with x = 0.90 and 1.00.More information about the analysis of the neutron patterns isgiven in the Supporting Information, e.g., temperaturedependences of the magnetic moments are shown in FigureS11.

Magnetic Phase Diagram. On the basis of the findingsfrom the present magnetization, variable-temperature X-ray andneutron diffraction studies, we have constructed a magneticphase diagram for annealed Mn(Co1−xNix)Ge as shown inFigure 8. Although the incommensurate magnetic structureshave different θ in the samples with x = 0.55, 0.58, 0.60, 0.90,and 1.00 these can be considered as a uniform phase with aspiral structure (spiral-Orth). TM is dependent on the Nicontent x: TM initially decreases with x up to ∼0.50, and thenincreases with x from 0.55 onward. In contrast to the magneticphase diagram of Nizioł et al.,10 (for which martensiticstructural transition temperatures TM ≳ 430 K are reported)

Table 1. Crystallographic and Magnetic Data of Mn(Co1−xNix)Ge with x = 0.14, 0.40, 0.60, and 1.00 as Determined fromRietveld Refinements of the Neutron Diffraction Patterns at 5 K (also at 280 K for x = 0.60)a

Mn(Co0.86Ni0.14)Ge Mn(Co0.60Ni0.40)Ge Mn(Co0.40Ni0.60)Ge MnNiGe

5 K 5 K 5 K 280 K 5 K

Orth. Hex. Orth. Hex. Orth. Hex. Orth. Hex. Orth. Hex.

space group Pnma P63/mmc Pnma P63/mmc Pnma P63/mmc Pnma P63/mmc Pnma P63/mmca (Å) 5.921(1) 4.073(22) 5.956(1) 4.072 (12) 5.982(1) 4.070(16) 6.010(1) 4.076(5) 6.025(1) 4.066(20)b (Å) 3.793(1) 3.757(1) 3.738(1) 3.763(1) 3.708(1)c (Å) 7.049(1) 5.304(45) 7.049(1) 5.282(26) 7.058(1) 5.304(36) 7.060(1) 5.367(12) 7.061(1) 5.324(30)V (Å3) 158.29(4) 76.19(9) 157.70(87) 75.84(48) 157.81(6) 76.10(66) 159.65(4) 77.23(21) 157.76(4) 78.88(4)phase fraction (wt%)

99(3) 1(1) 99(3) 1(1) 99(2) 1(1) 98(2) 2(1) 98(1) 1(1)b

magnetic state FMc FM FMb CS FMb SPmoment (μB) 3.4(1) 3.5(1) 3.2(1) 2.0(1) 3.32Rp, Rwp 3.41, 4.45 3.60, 4.65 2.63, 3.43 2.34, 3.06 2.70, 3.63R exp, χ2 1.99, 5.00 1.69, 7.58 1.62, 4.46 1.59, 3.72 1.45, 6.23aFMc, FMb, CS, SP represent paramagnetic, ferromagnetic along corth axis, ferromagnetic along borth axis, and cycloidal-spiral and spiral structures,respectively. bMnNiGe sample has ∼2(1) wt % of MnNi1.25Ge1.75 impurity.



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our results show that the structural transition temperature TM

for a broad Ni concentration range, ∼0.20 < x ≲ 0.65, is lowerthan ∼350 K. This is likely to be attributed to the differentsample preparations and heat treatment procedures. Throughcomparing TM in Figure 8 with the thermal evolution of themagnetic moments in Figure S11, it can be concluded that afirst-order magneto-structural transition, FM-Orth/PM-Hex,exists within the Ni range ∼0.20 < x ≲ 0.65. As indicated inFigure 8, this range of Ni concentrations has a broadtemperature window of ΔTMCE = 290(1) K to 345(4) K forobservation of a magneto-structural transition. Large directMCE values are expected in this temperature window. Theoccurrence of the spiral-AFM/FM transitions around TN

SP−FM inMn(Co1−xNix)Ge with ∼0.55 < x ≲ 0.75 is also noted with aninverse magnetocaloric effect expected in this region.54

Stabilities of Magnetic Phases. In the orthorhombicstructure (Pnma) of MnCoGe/MnNiGe-based compounds,each Mn atom has four Mn nearest-neighbors with twocharacteristic Mn−Mn separations, d1 and d2, as depicted inFigure S1. Given the centrosymmetric TiNiSi-type structure,the Dzyaloshinsky−Moriya interaction cannot be the origin ofthe spiral structure in the orthorhombic phase of Mn-(Co1−xNix)Ge. As an alternative explanation, it was consideredthat the occurrence of a spiral structure in this system is a resultof the variation of the Mn−Mn distance, especially d1,

2,3,55 e.g.spiral-AFM occurs when d1 ≳ 2.95 Å in MnNi(Ge1−xSix)

2 andd1 ≳ 3.37 Å in MnCo(Ge1−xPx).

3 The evolution of the d1 andd2 Mn−Mn distances in Mn(Co1−xNix)Ge with x at 5 K areshown in Figure 9a with their temperature dependences shown

Figure 7. (a) Variation in the propagation vector versus temperaturefor the incommensurate spiral structures in the Mn(Co1−xNix)Gesamples. The temperature at the end point of each curve is the Neeltemperature of the corresponding sample. (b) Thermal evolution ofthe lattice parameter ratio aorth/(corth/√3) for the set of Mn-(Co1−xNix)Ge samples indicated.

Figure 8. Partial phase diagram for the annealed Mn(Co1−xNix)Ge(0.14 ≤ x ≤ 1.00) samples. TC

Orth‑c, TCOrth‑b, and TN

Orth are the Curietemperatures for the ferromagnetic orthorhombic phase with magneticmoment along the corth axis and borth axis and the Neel temperature ofthe spiral orthorhombic phase, respectively. TM is the martensiticstructural transition temperature from the orthorhombic to thehexagonal structures. TC

Orth‑c‑b and TNSP‑FM are the magnetic transition

temperatures from the corth axis ferromagnetic structure to the borth axisferromagnetic structure and from the spiral structure to ferromagneticstructure (magnetic moment aligning along the borth axis), respectively.PM, FMc, FMb, and SP represent paramagnetic, ferromagnetic alongcorth axis, ferromagnetic along borth axis, and spiral structures,respectively. ΔTMCE is the temperature window for magneto-structuraltransition and large magnetocaloric effect in annealed Mn(Co1−xNix)Ge samples. The dashed lines represent extrapolations of transitionsboundaries.

Figure 9. (a) Variation in the Mn−Mn nearest-neighbors distances, d1and d2, with Ni content x in the orthorhombic phase ofMn(Co0.86Ni0.14)Ge at 5 K. (b) Variation of the magnetic momenton the Mn atoms of Mn(Co1−xNix)Ge alloys in the orthorhombicphase with Ni content x at 5 K.



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in Figure S13. According to this criterion, the spiral-AFMshould become more stable at high temperature due to thethermal expansion of d1. However, the spiral-AFM wastransferred to the FMb structure with increasing temperaturein the samples with x = 0.55, 0.58, and 0.60. Therefore, thissimple interatomic criterion cannot properly explain for theoccurrence of the spiral-AFM in Mn(Co1-xNix)GeHere, we give a brief discussion on the origin of noncollinear

spiral structure based on the experimental results. We attributethe occurrence of the spiral-AFM to the “self-adjustment” ofthe system to accommodate increasing valence electronconcentration (VEC, the average number of valence electronsper atom) due to the replacement of Co (3d74s2) by Ni(3d84s2). The unit cell volume Vorth plotted in Figure 3d has amaximum for midrange Ni content: Vorth increases initially withincreasing x and then decreases for x ≥ 0.60. This deviationfrom the prediction of Vegard’s law indicates that expansion ofthe unit cell volume is one mechanism for MnCoGe-basedcompounds to readjust to the increased number of valenceelectrons. In addition, crystal orbital overlap population(COOP) calculations for the TiNiSi family of compoundsdemonstrated that extra valence electrons introduced by dopingwould enter into the bonds between Ti (corresponding to Mnin MnCoGe) and Ni−Si networks (corresponding to Co−Genetworks in MnCoGe) and hence reduce the impact on theNi−Si network and stabilize the crystal structure to a TiNiSi-type structure.48 By analogy, doping MnCoGe would stabilize aTiNiSi-type structure. This redistribution of valence electronswould influence the density of states at the Fermi surface,N(EF), and lead to a noncollinear structure whose occurrencecan reduce the number of bands crossing the Fermi energyEF.

33

All of the anomalous behaviors in the structural and magneticproperties of Mn(Co1−xNix)Ge can be explained by thecompetition between the two mechanisms for MnCoGe-based compounds to accommodate extra valence electrons asintroduced above: the variation of the unit cell volume and theredistribution of valence electrons. It should be noted that thisanomalous variation in unit cell volume with increasing Nicontent, occurs despite the increased fraction of the smaller Niatoms (radius ∼1.21 Å) compared with the Co atoms (radius∼1.26 Å). Addition of further valence electrons to MnCoGesettle in Mn(Co1−xNix)Ge by, first, expanding the Vorth withinlow Ni content samples as demonstrated by the expansion ofVorth in Figure 3d. The accumulation of the valence electron inthe Co/Ni−Ge networks undermines the stability of theorthorhombic structure and TM decreases with x in theapproximate region x ≈ 0.15−0.55, as illustrated in Figure 8.However, the redistribution of the valence electrons in the casesof x ≥ 0.55 leads to the occurrence of the spiral-AFM and themodification of N(EF). This abrupt change in N(EF) isevidenced by the drop of the magnetic moment on Mnatoms, μMn, as shown in Figure 9b. As a result of the electronicredistribution, the expansion pressure from the increasedvalence electrons on Vorth and the impact of valence electronson the stability of the orthorhombic structure are both reduced.Therefore, Vorth decreases but TM increases above x = 0.55 asobserved in Figures 3d and 8. Such decreases in the values ofμMn and Vorth in Mn(Co1−xNix)Ge are similar in behavior to theoccurrence of a noncollinear structure in other systems.36,40 Inparticular, for zinc-blende MnAs36 and Fe- Ni Invar alloys,40 anoncollinear structure has been accompanied by smaller unit-cell volumes and smaller magnetic moments compared with the

unit-cell volumes and magnetic moments for a collinearstructure.The electronic redistribution and the occurrence of the

noncollinear spiral-AFM around x = 0.55 are also accompaniedby the change in the bond strength which causes the abruptjump of the lattice parameters of the orthorhombic structure asshown in Figure 3a−c. Similar competitions between magnet-ism and bonding due to redistribution of electronic densitywere also observed in MnFe(P,Si,B).57 In addition, strongmagneto-volume effects are observed via changes in the latticearound the magnetic transition for the sample with x = 1.00 asshown in Figure 10. To estimate the spontaneous magneto-

striction that causes the changes, the phonon contribution tothe lattice thermal expansion was calculated using theGruneisen-Debye model,58,59 with a Debye temperature of320 K56 and extrapolated from the temperature region aboveTN. The fit to the Gruneisen−Debye model is shown as thedashed line in Figure 10. The magnetic contributions areevident and the change in the unit cell volume around the Neeltemperature TN = 360(5) K amounts to 0.66%.The competition between the two mechanisms is also

important with regard to temperature dependence of themagnetic structures. In the samples with x = 0.55, 0.58, and0.60, the expansion of the unit cell volume Vorth reduces thepotential to change the band structure, and therefore the low-temperature spiral-AFM transfers to the FMb at high temper-atures, as shown in Figures 6a and 8 and Figure S4c, d.However, in the case of x = 0.90 and 1.00, the further increasesof the valence electron concentration cannot be accommodatedmerely by adjusting the unit cell volume through thermalexpansion. To maintain the lowest total energy, the spiral-AFMconfiguration remains stable up to the transition to aparamagnetic phase.

Magnetocaloric Effect. According to our findings assummarized in the phase diagram (Figure 8), direct andinverse magnetocaloric effects are expected in Mn(Co1−xNix)Ge compounds, at the martensitic transition temperatures TM

Figure 10. Lattice parameters and unit-cell volume for MnNiGesample (TN = 360(5) K) determined from refinement of the neutronpowder diffraction patterns. The dashed line on the unit-cell volumecurve represents the phonon contribution to the thermal expansion(θD = 320 K).56



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(around room temperature), and around the spiral-AFM/FMtransition TN

SP−FM (below ∼250 K), respectively. The directMCE values around room temperature are of greater interestwith practical application in mind and are shown in Figure 11

(for inverse MCE, please see Section 6 in the SupportingInformation). The peak values of the entropy changes for theseries of Mn(Co1−xNix)Ge compounds are − ΔSMpeak (μ0ΔH = 5T) = 14(2) J kg−1 K−1, 13(2) J kg−1 K−1, 15(2) J kg−1 K−1,14(2) J kg−1 K−1, and 14(2) J kg−1 K−1 at 292(2) K, 284(2) K,283(2) K, 285(2) K, and 302(2) K for x = 0.40, 0.50, 0.55, 0.58,and 0.60, respectively. Note that, these values are larger than10(2) J kg−1 K−1 at 297(1) K in as-prepared (Mn0.98Fe0.02)-CoGe;4 11(2) J kg−1 K−1 at 299(1) K in as-preparedMn(Co0.96Fe0.04)Ge;

5 and 13(2) J kg−1 K−1 at 314(1) K inannealed (Mn0.96Ni0.04)CoGe.

60 However, the refrigerationcapacities (RCs, the calculation method is given in theSupporting Information) with μ0ΔH = 5 T in Mn(Co1−xNix)-Ge (0.40 ≤ x ≤ 0.60) are found to be significantly smaller thanthe values in as-prepared (Mn1−xFex)CoGe,

4 as-preparedMn(Co1−xFex)Ge

5 and annealed (Mn1−xNix)CoGe.60 These

diminished RC values are due to the relatively smaller magneticmoment in Mn(Co1−xNix)Ge compared with these otherMnCoGe-based systems (see Table S4 and Figure S15). Inaddition, the results reported here provide experimentalevidence for the redistribution of valence electrons andmodification of the density of states at the Fermi surface.Following the discussion of Boeije et al.,57 the electroniccontribution may also play an important role in themagnetocaloric effect in MnCoGe-based compounds, similarto the Fe2P-based compounds57 and FeRh.61

■ CONCLUSIONSOur detailed investigations of an extensive series of Mn-(Co1−xNix)Ge (0.12 ≤ x ≤ 1.00) compounds provide insightinto their crystal and magnetic structures. Our findings have ledto a new explanation for the occurrence of spiral ground state inMn-based orthorhombic alloys (Pnma).It is concluded that the crystal and magnetic structures can

be tuned by changing valence electron concentration throughthe substitution of Ni (3d84s2) for Co (3d74s2). With increasingNi concentration, the ferromagnetic moments along corth on the

Mn sublattice in the orthorhombic phase turn first to the borthaxis due to anisotropy at 5 K. Further doping of Ni contentleads to a noncollinear spiral antiferromagnetic structure wherethe magnetic moments propagate along aorth. The occurrence ofthe spiral antiferromagnetic structure is accompanied by abruptchanges in lattice parameters, reduction of the unit cell volumeVorth, reduction of the magnitude of the magnetic moments onMn atoms and discontinuities in the lattice parameters. Thesephenomena are associated with the competition betweenmagnetism and bonding, and occur as a result of thecompetition between the expansion of unit-cell volume andthe redistribution of the valence electrons, which is associatedwith modification of the density of states at the Fermi surface.The competition between the expansion of unit cell volume

and the redistribution of the valence electrons also causes adecrease of the martensitic transformation temperature TM asthe Ni concentration in Mn(Co1−xNix)Ge increases up to x ≈0.50 and then an increase of TM for Ni concentrations above x≈ 0.55. In particular a first-order magneto-structural transitionand large direct magnetocaloric effect in Mn(Co1−xNix)Ge isformed within ∼0.20 < x ≲ 0.65 while the magnetic transitionfrom the spiral antiferromagnetic structure to the ferromagneticstructure provides scope for a region of inverse magnetocaloriceffect within ∼0.55 < x ≲ 0.75.

■ ASSOCIATED CONTENT

*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.chemma-ter.7b04970.

Relationship between Mn−Mn and Co/Ni−Ge net-works, analysis of X-ray diffraction patterns, magnet-izations, additional neutron diffraction patterns, modelsused for magnetic structure analysis, Rietveld refinementsfor neutron diffraction patterns, atomic occupancies,temperature dependences of the magnetic moments, theMn−Mn atomic distances and the lattice parameters, andanalysis of MCE (PDF)

■ AUTHOR INFORMATION

Corresponding Authors*E-mail: [email protected] (Q.Y.R.).*E-mail: [email protected] (W.D.H).*E-mail: [email protected] (J.L.W).

ORCIDQingyong Ren: 0000-0002-3163-9320NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported in part by grants from the AustralianResearch Council: Discovery project DP110102386 and LIEFgrant LE1001000177. QYR is grateful to the UNSW Canberrafor a Research Training Scholarship and the support from theNational Natural Science Foundation of China (Grant No.11774223). The authors thank Professor J. M. Cadogan,UNSW Canberra, for helpful discussions, particularly hisassistance with aspects of the Rietveld refinements.

Figure 11. Isothermal magnetic entropy changes for Mn(Co1−xNix)Gewith x = 0.40, 0.50, 0.55, 0.58, and 0.60 for magnetic changes of μ0ΔH= 2 T (dashed lines with open symbols) and μ0ΔH = 5 T (solid lineswith solid symbols).



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Lithium ion intercalation in thin crystals of hexagonal TaSe2 gatedby a polymer electrolyte

Yueshen Wu,1 Hailong Lian,1 Jiaming He,1 Jinyu Liu,2 Shun Wang,1,a) Hui Xing,1,b)

Zhiqiang Mao,2 and Ying Liu1,3,4,c)

1Key Laboratory of Artificial Structures and Quantum Control and Shanghai Center for Complex Physics,School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China2Department of Physics and Engineering Physics, Tulane University, New Orleans, Louisiana 70118, USA3Department of Physics and Materials Research Institute, Pennsylvania State University, University Park,Pennsylvania 16802, USA4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

(Received 8 October 2017; accepted 23 December 2017; published online 10 January 2018)

Ionic liquid gating has been used to modify the properties of layered transition metal

dichalcogenides (TMDCs), including two-dimensional (2D) crystals of TMDCs used extensively

recently in the device work, which has led to observations of properties not seen in the bulk. The

main effect comes from the electrostatic gating due to the strong electric field at the interface. In

addition, ionic liquid gating also leads to ion intercalation when the ion size of the gate electrolyte

is small compared to the interlayer spacing of TMDCs. However, the microscopic processes of

ion intercalation have rarely been explored in layered TMDCs. Here, we employed a technique

combining photolithography device fabrication and electrical transport measurements on the thin

crystals of hexagonal TaSe2 using multiple channel devices gated by a polymer electrolyte LiClO4/

Polyethylene oxide (PEO). The gate voltage and time dependent source-drain resistances of these thin

crystals were used to obtain information on the intercalation process, the effect of ion intercalation, and

the correlation between the ion occupation of allowed interstitial sites and the device characteristics. We

found a gate voltage controlled modulation of the charge density waves and a scattering rate of charge

carriers. Our work suggests that ion intercalation can be a useful tool for layered materials engineering

and 2D crystal device design. Published by AIP Publishing. https://doi.org/10.1063/1.5008623

Atomically thin crystals of layered transition metal

dichalcogenides (TMDCs) were found to show physical

properties including basic band structure which are drasti-

cally different from those of the bulk,1,2 making mechanical

exfoliation preparation of such atomically thin crystals a

powerful route for material discovery. Ionic gating was used

to tune the properties of these interesting materials further,

leading to remarkable success.3–7 By supplying an exceed-

ingly large number of charge carriers, the physical properties

of these layered compounds were shown to be greatly tun-

able. The ionic gating involves both electrostatic tuning of

the surface charge carriers through an electric field and ionic

intercalation of the layered compound in the presence of a

gate electrolyte with small ion size. The latter possibly modi-

fies the interlayer coupling in these few-atomic-layer

TMDCs. While its ability to manipulate the properties of

these atomically thin crystals were well recognized,3,4 the

underlying physical processes of ion intercalation were not

fully resolved. Understanding how the ion intercalates in

atomically thin layered TMDCs, including the kinetics,

stages, interstitial sites of occupation, and the distribution of

the ions, is important for materials engineering and the

designing of functional electronic devices.

As a form of tantalum diselenide, 2H-TaSe2 features a

hexagonal crystal structure with a stack of one Ta layer sand-

wiched between two Se layers in an ABAB stacking. 2H-

TaSe2 is a superconductor with an intrinsic transition tempera-

ture of 0.15 K.8–10 It also features an incommensurate charge

density wave (ICDW) transition at 120 K followed by a com-

mensurate charge density wave (CCDW) transition at

90 K.11–16 In the bulk form, intercalating 2H-TaSe2 by various

ions, such as Fe, Ni, or Pd, into 2H-TaSe2 was found to lead to

changes in both superconducting and CDW properties of this

compound.17–22 X-ray diffraction (XRD)23 and M€ossbauer

spectroscopy24 studies revealed that, for nearly stoichiometric

bulk LiTaSe2, Li ions occupied octahedral sites between TaSe2

layers with the valence of Ta changing from Ta4þ to Ta3þ,

suggesting that each Li ion contributes one electron to the

host material. Optical transmission studies showed that the Li

intercalated 2H-TaSe2 is semiconducting, consistent with the

results of theoretical calculations.18,25

Atomically thin single crystals of 2H-TaSe2 were

found to exhibit properties not seen in the bulk. For exam-

ple, weak antilocalization was observed and attributed to a

strong spin-orbital coupling.26 Thermal conductivity was

seen to be suppressed from that in the bulk.27 In all these

studies, however, the process of Li ion intercalation and its

effect on the electrical transport in the thin crystal were not

studied in detail. In this work, we attempted to address

these issues by employing a technique combining photoli-

thography device fabrication and electrical transport

a)Present address: MOE Key Laboratory of Fundamental Physical Quantities

Measurements, School of Physics, Huazhong University of Science and

Technology, Wuhan 430074, China.b)Electronic mail: [email protected])Electronic mail: [email protected]

0003-6951/2018/112(2)/023502/5/$30.00 Published by AIP Publishing.112, 023502-1

APPLIED PHYSICS LETTERS 112, 023502 (2018)

176

https://doi.org/10.1063/1.5008623

https://doi.org/10.1063/1.5008623

https://doi.org/10.1063/1.5008623

https://doi.org/10.1063/1.5008623



http://crossmark.crossref.org/dialog/?doi=10.1063/1.5008623&domain=pdf&date_stamp=2018-01-10

measurements on thin crystals of 2H-TaSe2 featuring mul-

tiple channels gated by a polymer electrolyte.

Bulk single crystals of 2H-TaSe2 were grown by a

chemical vapor transport method. Atomically thin crystals

of 2H-TaSe2 were obtained by mechanical exfoliation

and transferred onto a 300-nm-thick SiO2/Si substrate by

Polydimethylsiloxane stamp.28 The crystal thickness was

estimated by a color code that infers the thickness of the crys-

tal based on the color and faintness of the color calibrated

using an atomic force microscope. The device pattern was

defined by photolithography, with the contacts prepared by

the deposition of a 10-nm-thick Ti and a 100-nm-thick gold

film in series, followed by a lift-off process. To probe the

effect of ion intercalation on the electronic properties of the

atomically thin crystals of TaSe2, it is useful to distinguish the

contribution of the electric field gating and ion intercalation

on the sample. For electrostatic gating, Li ions form electron

double layers at the surface, which affects the channel

resistance through only the top layer of the crystal due to elec-

trostatic screening of a metallic sample. However, the interca-

lation of Li ions, which in our devices enter the crystal from

the edges and are inserted between the atomic layers, affects

the properties of the entire crystal. We developed a technique

to study the ion intercalation process by directing ions to enter

from the selected part of an atomically thin crystal and moni-

tor the ion diffusion in the crystal through the measurement of

the resistances of spatially separated channels as a function of

gate voltage and time. As shown in Figs. 1(a) and 1(b), this

technique features a photoresist window that covers only part

of a 10-nm-thick crystal, allowing Li ions to enter the crystal

from the uncovered sides.

The polymer electrolyte was prepared by mixing LiClO4/

PEO, 0.3 g and 1 g, respectively, which was dissolved in 15 ml

anhydrous methanol, followed by stirring for 10 h at the tem-

perature of 50 �C. A droplet of the electrolyte was applied on

the device covering both channel and gate leads. The entire

sample was baked at 97 �C for an hour to remove residual sol-

vent in high vacuum (10�5 Torr) just before measurements.

We ramped the gate voltage (VG) at 330 K with a constant

sweeping rate at 0.5 mV/s. The gate voltage dependences of

the channel resistance between neighboring voltage leads, R12,

R23, R34, and R45, shown in Figs. 1(c)–1(f), were simulta-

neously measured as the gate voltage was swept. Previous

studies show that the ions cease to diffuse in this polymer elec-

trolyte below 280 K.29,30 For the channel resistance as a func-

tion of temperature measurements, a wait time of 90-min was

used to allow the ions to diffuse into the crystal before the

device was cooled down to lower temperatures. After all the

measurements, no obvious chemical reaction was observed in

our sample as shown in the lower panel of Fig. 1(b).

The gate voltage dependence of resistance for source-

drain channels of fully covered parts of the crystal, R34

[Fig. 1(e)] and R45 [Fig. 1(f)], as well as that of the uncovered

part R12 [Fig. 1(c)], was obtained at 330 K. Interestingly, the

covered and uncovered parts of the crystal were found to

respond to the gate voltage similarly, which indicates that the

ion intercalation plays a dominant role in determining the

channel resistance. At low gate voltage in the uncovered part,

the resistance increased slightly with increasing gate voltage,

suggesting a weak intercalation effect mixed with electrostatic

gating. In this regime, it is difficult to determine the weight of

both effects since the large carrier density in metallic TaSe2

makes the electrostatic gating effect much less pronounced

than that found in semiconducting MoS2.31 It was also found

that the hysteresis in the R vs. VG curves was larger in the cov-

ered channels than that of the uncovered ones, indicating that

the Li diffusion in the crystal is significant even during gate

voltage sweep. Very recently, ultrafast Li diffusion in bilayer

graphene was found,32 which suggests that transport measure-

ments may be used to study the ion diffusion in a 2D crystal.

In the present work, the channel resistances of the spatially

separated pairs of the covered part of the device were seen to

start to increase in sequence [Fig. 1(g)]. If the onset of the

resistance increase was used to mark the start of the Li

FIG. 1. (a) Schematic of the multi-channel field-effect transistor device

gated by a polymer electrolyte. The photoresist covers the part of the crystal.

(b) Optical image of the device prepared using a 10-nm-thick 2H-TaSe2

single crystal before (upper panel) and after (lower panel) ionic gating. The

photoresist window is outlined by the dashed line. (c-f) Values of R12, R23,

R34, and R45 as a function of VG (see text). (g) Time dependent resistance

changes of R12, R23, R34, and R45 at VG ¼ 1:4 V.

023502-2 Wu et al. Appl. Phys. Lett. 112, 023502 (2018)

177

intercalation and the covered part of the device can be approx-

imated as a one-dimensional diffusion model, we can estimate

the diffusion coefficient of the Li ion at the starting of interca-

lation using the equation D ¼ ðx24�x2

3Þ

2Dt , as done previously.33,34

Here, the length xi is the distance of the i-th voltage lead away

from the photoresist window. It appears that the Li ions took

roughly Dt¼ 3.5 h to diffuse from voltage leads 3 to 4, so that

D is approximately 2:8� 10�11cm2=s at 330 K. This shows

that the diffusion constant D can be measured in our device

using the electrical transport measurements. However, it is

important to note that D is affected by many factors, including

Li concentration, VG, and the chemical structure of the elec-

trolyte,35,36 making the precise determination of the diffusion

constant difficult.

The physical properties of this thin crystal of 2H-TaSe2

were found to evolve continuously as the Li ions intercalate

it. In Fig. 2, the evolution of the temperature dependence of

the sample resistance of an uncovered sample taken at a

fixed gate voltage is shown. During the measurements of Rvs. T curves, the gate voltage is ramped to a desired value at

330 K, and the sample was cooled down from 330 to 2 K at a

constant rate of 1 K/min. The sample was warmed back up at

the same rate to 330 K, with the channel resistance measured

during the warming up as well. The values of the channel

resistance obtained during the cooling down and warming up

were found to differ only above a relatively high tempera-

ture, T � 280 K, suggesting that the Li ions were frozen in

below this temperature, consistent with previous studies.29,30

The R vs. T curves, which were obtained as the sample is

cooled down, revealed a continuous evolution of behavior as

the gate voltage was varied. It is seen that, at small gate vol-

tages, the sample exhibits metallic behavior over the entire

temperature range, with the R vs. T curve shifting to higher

value as VG increases. A hump was found in the R vs. Tcurve around 120 K [Fig. 2(a)], which marks the CDW tran-

sition in this 10-nm thick crystal of TaSe2. It becomes

weaker and eventually disappeared with increasing gate volt-

age, suggesting the suppression of CDW by the Li ion inter-

calation. There are earlier efforts in modulating CDW by

ionic liquid gating. For example, in TaS2 electron double

layer transistors, the transition temperature of CDW can be

tuned from 190 K to 140 K due to the change in Fermi sur-

face topology by carrier doping.37 The other effect was

found in Cu doped TaS238 in which the disruption of CDW

coherence is induced by a random distribution of ions. As

the gate voltage increased further, the overall behavior of

resistance was found to show a decrease overall with increas-

ing VG, as shown in Fig. 2(b). In addition, a crossing of the Rvs. T curves taken at different VG was found. Similar features

were found in Li intercalated bulk crystals of 2H-TaS2.39

Both observations suggest that the processes are more com-

plex than simple addition of charge carriers to the system.

Finally, at the highest values of VG, the channel resistance

was found to rise rapidly with increasing VG, with an upturn

in the channel resistance in the low-temperature limit for

the highest VG [Fig. 2(c)]. A resistance anomaly around

200–250 K was also observed, above which the resistance

slope dR/dT increased slightly.

We propose that the evolution of the electronic transport

properties revealed in Fig. 2 is linked to the distribution of the

Li ions during the intercalation process. Upon entering the

crystal, Li ions will adopt several possible sites between

layers featuring different coordination environments. Based

on the theoretical calculation and experimental studies in the

bulk,23,25 the most favorable intercalation sites between the

layers are the octahedral sites, which will be taken up the first,

as shown schematically in Fig. 3(c). The next favorable ones

are tetrahedral sites [Fig. 3(e)]. Even at the initial process of

the Li ion intercalation on the octahedral sites, ordered ion

distribution may be formed due to ionic Coulomb repulsion

FIG. 2. (a)–(c) R vs. T curves with an up-sweep of VG up to 4.1 V. The

hump around 120 K signaling the onset of CDW was found to become

weaker and eventually disappear at low gate voltage. A resistance anomaly

around 200–250 K was also observed at high gate voltages.

FIG. 3. VG dependences of R300K (a) and RRR at different locations (b). The

evolution of behavior is marked by different colors and correlated with the

ion distribution during the ion intercalation process. (c)–(e) Schematics of

randomly distributed Li on octahedral (o) sites, an ordered structure of Li on

octahedral (o) sites, and an ordered structure of Li on octahedral sites

together with a randomly distributed Li on tetrahedral (t) sites.


178

[Fig. 3(d)]. For Li ions occupying the octahedral sites at the

lowest values of VG, the electrons will tend to be bound to the

positive charged ions, which will reduce the number of the

hole carriers (electron doping). If these positively charged

ions are randomly distributed, it will tend to enhance scatter-

ing of conducting carriers. Once ordered ion distribution

occurs, however, the scattering of carriers will change.

Finally, as tetrahedral sites started to be occupied, which

appeared to contribute different scatterings of charge carriers,

distinct features in R vs. T including a negative dR/dT at low

temperatures for the highest VG and a bump near 200–250 K

were found even though the precise origin of these features

has not yet been identified.

The values of the room-temperature channel resistance,

R300K, and residual resistance ratio, RRR¼R330K/R2K, are

plotted against VG in Figs. 3(a) and 3(b). As VG increases,

R300K is seen to first stay roughly a constant, increasing only

slightly as VG increases. At VG ¼ 1.6 V, R300K is seen to drop

suddenly. Above VG ¼ 2.5 V, R300K was found to increase

sharply as the gate voltage was increased further. A similar

behavior was seen in the RRR vs. VG plot. Consistent behav-

iors of RRR vs. gate voltage were found at different locations

of the sample, with minor differences in the detailed behav-

ior of resistance and RRR, which could be due to unavoid-

able inhomogeneity of Li occupation. It is interesting to note

that the suppression of CDW, the increase in R300K, and the

decrease in RRR appear to be correlated with the Li ion

distribution proposed above. In particular, the randomly dis-

tributed Li ions in octahedral sites is transformed into an

ordered structure when the concentration of Li ions is suffi-

ciently high [Fig. 3(d)], as seen in other intercalated TMDCs

with a crystal structure identical with 2H-TaSe2 which fea-

ture staging and disorder-order transition.40 Decreasing RRRat the highest VG values may indicate that the scattering

from impurity is enhanced. It is likely that some Li ions are

trapped in tetrahedral sites instead of octahedral sites, lead-

ing to enhanced scatterings of charge carriers [Fig. 3(e)]. It

should be noted that a previous study of bulk PdxTaSe222

showed that a CDW phase was found to be suppressed by

ion intercalation. There RRR was found to decrease mono-

tonically as the ion concentration increased. The maximum

doping level was 0.14 in PdxTaSe2. No evidence for Pd ions

to form an ordered structure in such a concentration level

was found. The increase in the Pd ion concentration leads to

a decreased RRR. These differences suggest that not only

carrier doping but also scattering from ions affects the physi-

cal properties of thin crystals of TaSe2. It is likely that the Li

concentration in our devices was higher than that of Pd in

PdxTaSe2. Unfortunately, it is difficult to estimate the Li

concentration by the Hall measurement in the current work

due to the multiband nature of TaSe2. More experiments are

needed to confirm the picture proposed above.

In summary, we developed a technique to study Li inter-

calation and its effect on the physical properties of atomically

thin crystals of 2H-TaSe2 by electrical transport measure-

ments using polymer electrolyte gated 2D-crystal devices fea-

turing covered and uncovered channels. We found that the Li

ion diffusion coefficient is 2:8� 10�11cm2=s at 330 K. The

effect of ion intercalation on the properties of the thin crystals

of TaSe2 and possible correlation between the ion occupation

of allowed interstitial sites and the device characteristics

were demonstrated. A gate voltage controlled modulation of

the charge density waves and a scattering rate of charge car-

riers established in this work may help advance the practical

use of 2D crystal materials.

We acknowledge useful discussions with Dr. Zhe Wang.

The work done in China was supported by the MOST of

China (Grant Nos. 2015CB921104 and 2014CB921201), the

National Natural Science Foundation of China (Grant Nos.

11474198, 11521404, and 91421304), and the CAS/SAFEA

international partnership program for creative research teams

of China. The work at the Penn State was supported by NSF

under Grant No. EFMA1433378 and at Tulane supported by

the U.S. Department of Energy under EPSCoR Grant No.

DESC0012432 with additional support from the Louisiana

Board of Regents.

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https://doi.org/10.1080/13642818708220162

https://doi.org/10.1016/0167-2738(84)90087-0

PHYSICAL REVIEW B 97, 035116 (2018)

Approximating quantum many-body wave functions using artificial neural networks

Zi Cai*

Key Laboratory of Artificial Structures and Quantum Control, Department of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China

Jinguo LiuBeijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

(Received 25 April 2017; revised manuscript received 19 November 2017; published 9 January 2018)

In this paper, we demonstrate the expressibility of artificial neural networks (ANNs) in quantum many-bodyphysics by showing that a feed-forward neural network with a small number of hidden layers can be trained toapproximate with high precision the ground states of some notable quantum many-body systems. We considerthe one-dimensional free bosons and fermions, spinless fermions on a square lattice away from half-filling, aswell as frustrated quantum magnetism with a rapidly oscillating ground-state characteristic function. In the lattercase, an ANN with a standard architecture fails, while that with a slightly modified one successfully learns thefrustration-induced complex sign rule in the ground state and approximates the ground states with high precisions.As an example of practical use of our method, we also perform the variational method to explore the ground stateof an antiferromagnetic J1-J2 Heisenberg model.

DOI: 10.1103/PhysRevB.97.035116

I. INTRODUCTION

A central challenge in quantum many-body physics is devel-oping efficient numerical tools for strongly correlated systems,whose Hilbert space dimensionality grows exponentially withthe system size, so does the information required for character-izing a generic state of the system. However, for many physicalsystems of practical interest, the ground states may have asimplified structure, thus, it can be appropriately approximatedusing an exponentially smaller number of parameters than thatrequired for characterizing generic states. Typical examplesinclude low-dimensional strongly correlated systems, whoseground states can be represented in terms of matrix productstates by taking advantage of limited entanglement entropyin the ground states [1–3]. For higher-dimensional systems, adifferent routine involving stochastic sampling applies for cer-tain types of strongly correlated systems with positive-definiteground states, with quantum Monte Carlo (QMC) algorithmproviding a well-controlled method to evaluate the physicalquantities of interest based on an exponentially small fractionof all possible configurations [4–6]. In spite of the remarkableachievement of these numerical methods, developing a generalstrategy to represent typical many-body wave functions thatbypasses the exponential complexity remains a formidable, ifnot impossible, task and is a question of principal interest inthe condensed matter community.

In general, a many-body wave function can be expanded interms of a set of orthogonal bases (e.g., the Fock basis) and canbe fully characterized by a function where one feeds in a basisand gets the output the set of corresponding coefficients in thewave function. Consider a spin lattice system as an example; in

*[email protected]

this case, an arbitrary wave function can be expanded as � =∑σ C[σ ]|σ 〉, where |σ 〉 = |σ1 . . . σL〉 are {Sz

i } eigenstatesspanning the Hilbert space of the spin configurations. There-fore, the task amounts to efficiently approximating the functionC[σ ] using an exponentially smaller number of parametersthan the Hilbert space dimensionality ∼2L. Among the existingmodern techniques used for approximating functions, artificialneural networks (ANNs), as a powerful tool for data fittingand feature extraction, have not only achieved remarkablesuccesses in machine learning and cognitive science fieldsin the past decades [7,8], but have also recently attractedconsiderable attention of researchers in the condensed mattercommunity. Applying machine learning methods to problemsin condensed matter physics is not only interesting in its ownright [9–12], but it may also potentially provide new ideasand have practical applications for solving complex physicsproblems, such as identifying classical and quantum phasesof matter and locating the phase transition points [13–21],categorizing and designing materials [22–24], improving ex-isting numerical techniques [25–29], and even developing newmethods in the quantum many-body physics [30,31]. Owing toits tremendous capability in function approximations, ANNscan also be considered as novel representations of many-bodywave functions [30,32–35], e.g., in a seminal work, Carleoet al. proposed a new kind of variational wave functions usingthe restricted Boltzmann machine [30].

Even though it can be proven mathematically that ANNscan in principle approximate any smooth function to anyaccuracy [36–38], what matters in practice is the efficiencyof the method: the amount of resources an ANN needs toapproximate a given multivariable function. In this paper,we will demonstrate the expressibility of neural networks inapproximating and characterizing quantum many-body wavefunctions using some notable examples of physical interest.

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ZI CAI AND JINGUO LIU PHYSICAL REVIEW B 97, 035116 (2018)

For a wave function with completely random coefficients,the information encoded in it cannot be compressed; thus, anexponentially large number of parameters are needed. The goalof this paper is to express the ground states of several notablemany-body systems in terms of neural networks of feasible sizeand a small number of hidden layers, and, most importantly,networks learn in a reasonable time. The success of thisapproach relies on the specialty of the ground state comparedto a generic eigenstate, where the underlying physical lawsencoded in the ground state’s wave function can be extractedby the neutral networks through a large amount of training.

In the following, we show which ground-state wave func-tions can be efficiently expressed by a simple neural network,and for those who cannot, how the ANN’s architecture shouldbe modified to achieve this goal. The scaling of the computationresources with the system size is also investigated. Specialattention is devoted to wave functions with the “sign problem,”where the function C[σ ] may alter its sign even for a localchange in the input [σ ], and thus cannot be considered a“smooth” function. The sign problem not only hinders theapplication of QMC methods, but also make it more difficult fora simple ANN to learn a wave function. However, as we haveshown, this problem can be circumvented by dividing the taskinto two subtasks and by designing ANNs with correspondingarchitectures. This is a typical example that illustrates theneural network’s ability of extracting the physics laws, evenfor those too complex to be written explicitly.

The rest of the paper is organized as follows: First, we intro-duce the structure of the ANN, then use it to investigate somenotable examples, including one- (1D) and two-dimensional(2D) free bosons and fermions, whose exact ground statesare compared to those predicted by the ANN. To determinewhether the ANN approach can work for large systems, weadopt the importance sampling algorithm to calculate physicalquantities instead of wave functions, and compare them to theexact ones. Then, we attack the most difficult part: approx-imating the ground state of a frustrated quantum magnetismwhose characteristic function can dramatically change its sign,thus being very different from a smooth function, which makesit extremely difficult for a regular ANN to approximate. Inspite of this, we find that a slight modification of neuronsin the ANN allows to capture the sign rule of the frustratedquantum magnetism, even at the phase transition point. Finally,we discuss the practical application of this method based onthe variational method.

II. METHODS

Before we proceed further to discuss specific examples,let us describe the details of the ANNs and the optimizationmethods we will use. We consider a fully connected feed-forward neural network consisting of an interconnected groupof nodes (neurons) with a stacked layered structure, andits expressibility is encoded in sets of adaptive weights ofconnections between neurons in adjacent layers. As shown inFig. 1, we consider a four-layer ANN, with two hidden layers(each containing Nb neurons) that are sandwiched betweenthe input layer [accepting the Fock basis ([n] or [σ ])] and theoutput layer that output the corresponding coefficient predictedby the ANN (CP [σ ]). A neuron can be considered to be an

FIG. 1. (Left) The structure of the feed-forward ANN we usedto approximate quantum many-body ground-state wave functions;(right) three types of neurons with different activation functions.

elemental processor:

o[n+1]i = f

⎛⎝∑

j

W[n]ij o

[n]j + b

[n]i

⎞⎠, (1)

where o[n]j is the output of the j th neuron of the nth hidden

layer, W[n]ij denotes the connection weight between the nth

and (n − 1)th hidden layers, and b[n]j is the bias in this neuron.

The activation function f (x) can be any smooth nonlinearfunction. However, as we show below, choosing a propernonlinear function may significantly increase the ANN’sefficiency for approximating certain target functions. Weintroduce a fidelity function

F = 1 −∣∣∣∣∣∑

σ

〈C∗T [σ ]CP [σ ]〉

∣∣∣∣∣ (2)

to measure the difference between the target function CT [σ ]and the one predicted by the ANN CP [σ ]. Here, we shouldpoint out that to calculate this quantity, the target functionsare known in advance. We will discuss how to generalize thecurrent method to explore new quantum many-body systemswith previously unknown ground states in Sec. V. The problemnow reduces to an optimization problem with the goal offinding the minimum of F in the landscape of ANN parameters(weights of the connection and bias, which are denoted as{W } in the following). In the following, we adopt the ANNconstruction methods and the optimization techniques that arereadily available in the machine learning libraries TensorFlow[39], with the training time measured in the units of T0,corresponding to the period of a single optimization iterationthat depends on the details of the ANN; more details aboutthe initialization and training can be found in the Appendixes.

III. FREE BOSONS/FERMIONS SYSTEMS

The first class of wave functions that we investigate are theground states of the simplest many-body systems composed offree bosons or fermions. In spite of their extreme simplicity,these wave functions are perfect touchstones to test the ex-pressibility of the ANN, for two reasons: first, the well-knownanalytic forms of these wave functions serve not only as atarget function during the training process of a neural network,

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APPROXIMATING QUANTUM MANY-BODY WAVE … PHYSICAL REVIEW B 97, 035116 (2018)

but also a tester of the accuracy of its predictions; second, insome cases, e.g., the ground state of 2D free fermions awayfrom half-filling, the wave function is simple but not trivial,in the sense that it is difficult to be characterized using theexisting numerical methods such as the matrix product state(MPS) and the path-integral QMC method, because this wavefunction suffers from the entanglement area law and the signproblem simultaneously.

The examples we studied in this section include the groundstate of free bosons in 1D lattice, and those of free fermionsin 1D and 2D lattices, with the filling factor away fromhalf-filling. For free bosons (FB) in a 1D lattice with length L

and unit filling factor (with the overall number of bosons N =L), the ground-state wave function can be written as �FB =∑

n CFB[n]|n〉 where |n〉 = |n1 . . . nL〉 is the occupation num-ber basis spanning the Hilbert space under the constraintN = L. The analytic form of the multivariable characteristicfunction is CFB[n] = √

L!/n1! . . . nL!/LL/2. The wave func-tion of a free-fermion (FF) lattice system has a similar formwhile ni can only be 0 or 1. We assume there are N particlesand CFF[n] = det[M], where M is an N × N matrix with thematrix elements Mij = fi(xj ), with xj denoting the positionof the j th fermion, and fi(x) denoting the ith single-particleeigenstate, in a 1D chain with a periodic boundary condition(PBC), fi(x) = 1√

Leikix with ki being the ith momentum. In

a 2D lattice, both xj and ki are replaced by a 2D vector x j

and ki . Based on the above exact results, we implemented thetraining process, aiming to minimize the fidelity function F

by adjusting the parameters of the ANN. To avoid the problemof overfitting, the number of the variational parameters inthe ANN was chosen to be on the order of O(N2

b ) ∼ 103,significantly smaller than the typical Hilbert space dimension-ality of the systems we studied here (∼106). As shown inFig. 2(a), for cases of 1D free bosons or fermions, an ANNwith only a few neurons Nb ∼ O(101) can easily approximatethe corresponding target function with an extraordinarily highprecision. For 2D fermions, a direct approximation of CFF[n]based on the ANN with the current structure seems to failbecause the sign of CFF[n] can dramatically change owning tothe fermionic statistics in two dimensions. We will reconsiderthis point later on. To avoid this problem, here we chose

|CFF[n]| instead of CFF[n] as our target function, whichstill allowed us to calculate the average values of diagonaloperators in the Fock basis [e.g., the nearest-neighbor (NN)density correlation Onn = 1

L

∑i〈nini+1〉 and the local density

operator On1 = 〈n1〉]. As shown in Fig. 2(c), an ANN withNb ∼ O(102) can give rise to values of Onn and On1 withprecisions ∼O(10−3). We also notice that the ground stateof interacting quantum models (e.g., the quantum spin model[30] and Bose-Hubbard model [40,41]) have been studied byother machine learning methods, where the ANNs are trainedto minimize the ground-state energy, instead of the fidelity ofthe wave function.

In all of the above-studied cases, the system size is relativelysmall since we want to compare our results with the exactones. In what follows, we show that, in principle, there is nointrinsic difficulty to use ANNs to simulate larger systems,whose Hilbert space dimensionality is much larger than thememory of any computer, thus making it impossible to storethe predicted wave functions and calculate its overlap with theexact ones. As a consequence, we focus on physical quantitiesin the ground states instead of focusing on the wave functionsthemselves, to test the accuracy of the ANN. The key point isthat we approximate the characteristic functions only trainingthe ANN only over a tiny fraction of the entire configurationspace and we assume that the predicted functions are alsovalid for other configurations once the ANN finds the correctform of the target function. Obviously, the efficiency of thefunction approximating strongly depends on the choice oftraining configurations, which need to be “representative” inthe Hilbert space. In other words, the probability of choosinga certain configuration [n] should be proportional to its weight|C[n]|2 in a given wave function, which can be achievedby the importance sampling in the Monte Carlo algorithm.Consider the ground state of a 1D L = 64, N = 31 free-fermion system as an example. For this system, we firstimplemented importance sampling to generate millions of“representative” configurations based on the exact value of|CT [n]|2, and use these generated configurations to train theANN and approximate the target function. After completingthe training, a new set of “representative” configurations weregenerated according to their weights predicted by the ANN

FIG. 2. (a) Fidelity F = 1 − |〈�ED|�ANN〉| as a function of the number of neurons Nb, for 1D free bosons with L = 12, N = 12, and thecorresponding Hilbert space dimensionalityD = 1 352 078 and for 1D free fermions with L = 24, N = 11, andD = 2 496 144. (b) The precisionfunction δ = |OANN − OED|/OED for two different quantities Onn = 1

L

∑i〈nini+1〉 and On1 = 〈n1〉 for 2D free fermions with L = 24(4 × 6),

N = 13, and D = 2 496 144, the inset shows a typical Fock configuration. (c) The precision function δ for Onn = 1L

∑i〈nini+1〉 as a function

of Nb obtained by the importance sampling algorithm for a 1D free fermion, where both the amount of the training and the sampling data arechosen as H = 106 in the simulations of various system sizes; the inset shows δ as a function of L with a fixed H = 106 and Nb = 200.

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|CP [n]|2, and we calculated the value of the physical quantitiesbased on these new configurations and compared them withtheir exact values. The result is shown in Fig. 2(c), wherewe observe that the precision of Onn evaluated based on theabove-described sampling scheme can reach O(10−3). For afixed amount of training and sampling data, the scaling relationbetween the precision and system size is shown in the inset ofFig. 2(c).

IV. NON-POSITIVE-DEFINITE WAVE FUNCTIONS

In general, neural networks perform much better on ap-proximating smooth functions rather than rough ones, whichmay restrict their applicability in quantum many-body physics,because for certain types of important wave functions, theircharacteristic functions may drastically change and even altertheir sign in response to a slight change in the input con-figuration. The sign problem, in its different forms, imposeschallenges on both existing methods (e.g., QMC) and thecurrent neural network function approximating. For example,the ANN with a simple structure illustrated in Fig. 1 fails toapproximate the ground state of one of the simplest models:a 1D antiferromagnetic Heisenberg model. In QMC, the signproblem in such a bipartite lattice can be avoided by performinga basis rotation. In the current method, we attack the problemby separately approximating the amplitude and the sign ofthe target function by two different ANNs. The ANN thatapproximates the amplitude is similar to those studied above,with the only difference that the target function is replaced byits absolute value. This “amplitude ANN” performs as well asthat in the previously described ones in Sec. III.

The difficult part is to approximate the sign function:S[σ ] = (C[σ ]/|C[σ ]| + 1)/2, which takes the values of 1(0)if C[σ ] is positive (negative). For a 1D lattice (or moregenerally, a bipartite lattice) model, it is well known that thesign of C[σ ] obeys the Marshall sign rule: S[σ ] = 1/0 if inσ the total number of down spins in the odd sites is evenor odd. This mathematical theorem enables us to perform abasis rotation in the even or odd sites to eliminate the signproblem in QMC simulations. For the current method, thequestion is without the prior knowledge of the Marshall signrule, whether a ANN can automatically extract it from thetraining set data? Based on our numerical tests, we found thata standard ANN, as the one shown in Fig. 1, fails to extract theMarshall sign rule, but a modified ANN with the activationfunction of neurons in the first hidden layer replaced by acosine function (hereafter denoted as “sign ANN”) succeeds.This modification can significantly increase the efficiencyof the ANN with the accuracy of 100% because the cosinefunction is more capable of capturing the even/odd featuresin the input data. Another important wave function is theground state of the Majumdar-Ghosh model: the dimerized

state � = ⊗ L2i=1

1√2[|↑〉2i−1|↓〉2i − |↓〉2i−1|↑〉2i], with the sign

function S[σ ] = [∏ L

2i=1(Sz

2i−1 − Sz2i) + 1]/2. We found that

this sign rule can also be satisfactorily learned by the signANN with the accuracy of 100%.

In the two examples above, the sign rules are rather simplein the sense that they can be written explicitly in a simple form.However, for a generic wave function with the sign problem,

this is not the case. The sign rule may be too complex to becaptured by programming or designing explicit algorithms,which on the other hand is exactly what ANNs are goodat. Approximating the sign function becomes a classificationproblem, which reminds us of one of the most successfulapplications of ANN: recognizing handwritten digits. In thisclassical problem, the ANN was shown to automatically andsuccessfully infer the rules of classification using the trainingset examples. Here, we adopt a similar strategy to extract theelusive sign rule for the ground state of a frustrated quan-tum magnetism model: the 1D J1-J2 antiferromagnetic (AF)Heisenberg model with the Hamiltonian H = ∑

i[J1 Si Si+1 +J2 Si Si+2] (where both J1 and J2 are non-negative). The twocases studied above are exactly the ground state of this modelin two limits: α = 0 and 0.5 with α = J2/J1.

For a general α, there is no exact solution of the groundstate; thus, we used the Lanczos method to calculate the signfunction for a finite-size system and compared it to the onepredicted by the sign ANN. The accuracy versus α is plottedFig. 3(a) for various Nb, and we observe that in the entire region0 � α � 0.5, the accuracy predicted by the sign ANN is high,reaching 99%, while the minimum of the accuracy correspondsto the phase transition point. Aside from the accuracy, theefficiency of an ANN also depends on the typical training timeand how this time scales with the system size. The “time”evolution of the accuracy during a training process is shownin Fig. 3(b), where we observe that the ANN first experiencesa period of “confusion” with the prediction accuracy P 0.5until a certain timeTc, after which the machine finally learns thecorrect approximation of the exact sign function and the accur-acy will increase rapidly before saturating. This time scale Tc,together with the number of the neurons Nb, can be understoodas the computation resources one needs to capture the sign ruleusing the ANN. The scaling relation between Tc and the systemsize L is plotted in the inset of Fig. 3(b). The system size westudied is relatively small, thus, it is difficult to tell whether Tc

scales with L in a polynomial or exponential manner, whichremains an open question.

Now, we discuss more details of the ANN. First, eventhough throughout this paper we choose a ANN with atwo-hidden-layer structure, one may wonder whether furtherincreasing the number of the layers can improve the predictionaccuracy or not. To address this issue, we calculate P fordifferent ANN with different hidden layer (up to three) withfixed iteration steps. As shown in Fig. 2(c), we found that, atleast for this example, even though an ANN with two hiddenlayers indeed performs much better than that with only onehidden layer, further increasing the number of layers doesnot significantly improve the performances. We also checkthe correlation between the erroneously assigned signs for acertain coefficient and the coefficient’s absolute value. To dothat, we define a parameter ε to measure the ratio between theaverage absolute value of the coefficient with the erroneouslyassigned sign and that of the whole training set:

ε =1D′

∑σ ′ |C[σ ′]|

1D

∑σ |C[σ ]| , (3)

where [σ ′] denotes the set of input basis with the erroneousassigned sign, with the dimensionality D′, while [σ ] denotes

035116-4184


0.0 0.1 0.2 0.3 0.4 0.50.95

0.96

0.97

0.98

0.99

1.00

P

α

Nb=200 Nb=120 Nb=60

(a)

FIG. 3. (a) The accuracy function P (α) of the sign rule in the ground state of 1D J1-J2 model with L = 24 and various Nb, with the insetshowing the modified ANN used to capture the sign rule with neurons in the first hidden layer replaced by those with the cosine activationfunctions f (x) = cos(πx). (b) The evolution of P during the training with fixed Nb = 200 and various systems sizes, with the inset showinga typical “confusion” time Tc in the training as a function of system size with Nb = 200. (c) The accuracy P predicted by the sign ANN as afunction of the layer of the ANN with a fixed Nb = 120. (d) The ratio between the average absolute value of the coefficient with the erroneouslyassigned sign and that of the whole training set with Nb = 120. (e) The fidelity as a function of Nb for the amplitude ANN. (f) The fidelityin the J1-J2 model and various α as a function of Nb predicted by the combination of the sign (with Ns

b = Nb) and amplitude ANNs (withNa

b = Nb/2). For (c)–(f), the system size L = 24.

all the training bases with the dimensionality D. As shown inFig. 3(d), we can find that ε ∼ O(10−2) � 1, indicating thatthe erroneous predictions of the sign ANN tend to occur forthose input bases whose coefficients have small absolute value.

By combining the results of the amplitude [as shown inFig. 3(e) for examples] and sign ANNs, we can calculatethe ground state of this frustrated quantum magnetic model,which agrees very well with the exact results, as shown inFig. 3(f). In summary, by dividing the ANN into two parts withdifferent architecture, we can approximate the ground state ofa frustrated quantum magnetism with high precisions. Thisstrategy may shed light on using ANNs to solve the complexquantum many-body systems with sign problems.

V. VARIATIONAL RESULTS

In all the cases studied above, because the target functions(the ground-state wave functions) are given, either analyticallyor numerically, one may expect that the current method wouldnot be useful for exploring new quantum many-body systemswith previously unknown ground states. However, our previousresults have established that the ground states of some quantummany-body systems can be efficiently represented by ANNs,which enables us to consider an ANN as a variational wavefunction for the true ground state of a new system. Theconnection weights and the bias in an ANN are the variationalparameters with respect to which we seek to minimize theexpectation value of the Hamiltonian (energy), instead of thefidelity function as described above. For a given ANN with a

set of parameters {W }, the corresponding variational energyE({W }) can be estimated using Monte Carlo simulations, witha procedure similar to those used for previously studied free-fermion cases in Sec. III. The minimum of E({W }) in the spaceof parameters can be found using the stochastic reconfigurationoptimization method. Similar strategy has been used in adifferent type to ANN: the restricted Boltzmann machine [30]to solve the nonfrustrated quantum magnetic models, e.g., theHeisenberg model and transverse Ising model. For the complexproblems like the frustrated quantum magnetisms, we shallshow that the strategy we proposed by dividing the ANNs intoamplitude and sign parts can also improve the efficiency of thisvariational method.

In the following, we use the ANN combined with variationalMonte Carlo (VMC) methods (the details of the method canbe found in the Appendix) to solve the ground state of a 1DL = 30 J1-J2 AF Heisenberg model. The structure of the ANNwe used in the variational method is shown in Fig. 4(a). Weadopt the strategy proposed in Sec. IV that the ANN has beendivided into the amplitude and sign parts, and the final outputis the product of them. However, one of the crucial differencesis that the output of the sign ANN is a continuous numberinstead of a discrete one. The reason is that a discrete functionis usually nondifferentiable, which may decrease the efficiencyof the variational method. We also notice that the structure ofthe ANN in Fig. 4(a) is much simpler than the one we usedin Sec. IV: there is only one hidden layer instead of two inboth amplitude and sign ANN. We will turn back to this pointlater.

035116-5185


(a)

FIG. 4. (a) The structure of the ANN we used in the variationalmethod calculations. (b) The “time” evolution of the excess energy persite �e = (Ev − E0)/L during the training process for a 1D J1-J2 AFHeisenberg model with L = 30 using the trial wave function as shownin (a) with the neuron number in the hidden layer of the amplitudeANN Nb = 30.

Since the target wave function is unknown, the cost functionto be minimized during the training process is the variationalenergy of the system, which can be evaluated by the VMCmethod for a given trial wave function (ANN). In Fig. 4(b),we plot the “time” evolution of the excess energy per site�e = (Ev − E0)/L during the training process, where Ev

is the variational energy evaluated by VMC and E0 is theexact ground-state energy calculated by the density matrixrenormalization group method. As shown in Fig. 4(b), an ANNwith a simple structure shown in Fig. 4(a) can give rise to avalue of the ground-state energy with precisions ∼O(10−3).

It is interesting to notice that compared to the ANN weused in Sec. IV, the structure of the ANN we used in thisvariational method is much simpler (only one hidden layer),but its performance is even better. There are several reasons forthis counterintuitive fact: (i) In the variational method, the signand amplitude ANN are trained as a whole, while in Sec. IVthey are trained separately. However, for the sign ANN, anextremely high accuracy may not be very necessary for the finalresults, since as shown in Fig. 3(d) the erroneous predictionsof the sign ANN tend to occur for those input bases whosecoefficients have very small absolute value, which give verylittle contribution to the final results, especially to the energy.(ii) In all the sections except Sec. V, during the optimizationprocess the training data sets are divided into N batchesand the optimization is performed batch by batch, while inthe variational method, all the entries in wave function areoptimized simultaneously. (iii) In Sec. IV, we choose the costfunction as the overlap of the wave functions (fidelity), whichis a global quantity, thus is usually difficult to be minimized,while in the variational method, we only need to minimize alocal quantity: the variational energy.

VI. CONCLUSION AND OUTLOOK

In this paper, we demonstrated the powerful applicabilityof a simple ANN in approximating the ground-state wavefunctions of some notable quantum many-body systems. Eventhough an ANN with a simple structure can already approxi-mate some of them with a high precision, there is still a longway to go before this method can learn to solve problems thatremain inaccessible by any other well-established numericalmethod, and the efficiency improvement plays a key role inthis process.

Some avenues along this line suggest the directions forfuture studies. First, as with many existing numerical meth-ods, imposing the Hamiltonian symmetries on the ANN willsignificantly improve its efficiency. Consider a 2D systemfor example. An ANN with translational symmetry does notonly reduce the number of the variational parameters andtraining data, but also learns the lattice geometry from thebeginning. Inspired by the impressive success of deep learningtechniques, we expect that an ANN can be more powerfulwhen networks are made deeper. In numerical simulations, wefound that even though an ANN with two hidden layers indeedperforms much better than that with only one hidden layer,further increasing the number of layers does not significantlyimprove the performances. One of the possible reasons isthat for an ANN with more hidden layers, even though itsexpressibility may be more powerful, the computational costto find the optimal representation is significantly increasedsince the landscape in the parameter space is more complex.Incorporation of “deep learning” into our simulations remainsan open issue and deserves further studies.

A fully connected ANN used in our simulations may containmany redundant connections that complicate the optimizationprocess. Adopting ANN with more advanced architectures(e.g., convolutional neuron networks) may help to avoid thisredundancy, thus likely significantly improve its efficiency.Recently, the relation of the quantum entanglement and therange of the connections in the ANN have been built, and weexpect the convolutional neuron networks may work well forthose ground states with short-range entanglement [16]. Lastbut not least, it is known that the artificial neural network,in general, is a heuristic algorithm whose efficiency largelydepends on the designer’s experience and intuition; however,to make this approach valuable in the physics, a systematicunderstanding of its validity and limitation, at least for thisconcrete problem, is still needed, which may be also beneficialto the artificial intelligence community.

ACKNOWLEDGMENTS

We wish to thank L. Wang and J. Carrasquilla for fruit-ful discussions. Z.C. is supported by the National KeyResearch and Development Program of China (Grant No.2016YFA0302001), the National Natural Science Foundationof China under Grant No.11674221, and the Shanghai Rising-Star Program. J.G.L. is supported by the National NaturalScience Foundation of China under Grant No. 11774398.

APPENDIX A: DETAILS OF THE OPTIMIZATIONALGORITHM

During the training process of the artificial neural network,one needs to adjust the parameters in ANN to minimize thedistance function, which turns to a multivariable optimizationproblem. In our simulations, we use two optimization tech-niques depending on the concrete problems: the stochasticgradient descent (SGD) [8] and an adaptive learning rateoptimization algorithm: adaptive moments (Adam). Here, weonly explain the details of the SGD method, and the Adam op-timization algorithm has been explicitly illustrated in Ref. [42].

035116-6186


In our simulation, each input data is composed of the basisand the corresponding coefficient (label) calculated by exactdiagonalization (ED) or other methods (e.g., {[σ ],C[σ ]}). Fora state with large Hilbert space dimension D, the typicalcoefficient is pretty small C[σ ] ∼ O(1/

√D). In general, the

nonlinear activation function is not sensitive in the regimewhere the input/output is too small, therefore, to make the ANNmore efficient, we multiply the target function by a factor of√D, thus C[σ ] ∼ O(1). This renormalization does not change

the results of any physical observable, but can significantlyimprove the efficiency of the fitting. The input data set arerandomly divided into two groups: 80% of them are used fortraining and the rest are testing data sets, and we choose arandom set of ANN parameters W0 as the initial parameters.The training set of data are randomly reshuffled and dividedintoN batches, each of which containM data. In each batch, thetraining data are labeled as {σ (1), . . . σ (M)} with correspondingtarget C[σ (i)].

After the initializations are finished, we start the optimiza-tion process. In each step of the SGD update, we choose onebatch and calculate the gradient estimate in the parameterlandscape W:

g = ∇W1

M

M∑i=1

L(Cp[σ (i)],C[σ (i)]), (A1)

where Cp[σ ] is the coefficient predicted by the ANN andL(Cp[σ ],C[σ ]) = (C[σ ] − Cp[σ ])2 is a function of W, de-noted as loss function. Once we obtain the gradient, theparameter W is updated as

W ← W − εg, (A2)

where ε is the parameter controlling the learning rate, whichgradually decreases over time. The above optimization pro-cesses continue until all the batches are chosen, then oneiteration of the training is finished. In our simulations, thetraining time is measured in the unit of the time of a singleiteration T0. The typical training time ranges from 102 ∼103 T0 depending on the convergency of the problems. TheSGD method is the most used optimization technique formachine learning. In our simulations, we use the GPU to speedup the computational efficiency of the training. A set of rawdata generated during the training process can be found in theSupplemental Material [43].

APPENDIX B: DETAILS OF THE VARIATIONAL METHOD

In this Appendix, we will show the variational analysis ofusing the artificial neural network to explore new ground states.In this case, the target function is previously unknown, there-fore, during the training process, what we need to minimize isnot the distance function F , but the variational energy (the ex-pectation value of the Hamiltonian over the ANN trial wavefunction), which can be calculated using the variational MonteCarlo method, as we will show in the following. In general,

a Hamiltonian can be split into the diagonal and nondiagonalparts: H = T + V in the basis |σ 〉 (or Fock basis). For a giventrial wave function in terms of the ANN with a set of variationalparameter {W }, the expectation value of the diagonal part

V ({W }) = 〈V 〉 =∑[σ ]

pσ {W }〈σ |V |σ 〉, (B1)

where pσ {W } = |CANN[σ ]|2 is the probability predicted by theANN. The summation

∑[σ ] is over the whole Hilbert space,

whose dimensionality exponentially grows with the systemsize. Following in the spirt of Monte Carlo, the summationover the whole Hilbert space can be replaced by the summationover those important configurations:

V ({W }) ∑[σ ]

pσ {W }Z{W } 〈σ |V |σ 〉 (B2)

with Z{W } = ∑[σ ] pσ {W }, and [σ ] is an exponentially small

fraction of the whole configuration, denoting the set of rep-resentative configurations chosen by the importance samplingusing Metropolis algorithm according to its probability pσ . Theexpectation value of the off-diagonal term can be evaluated in asimilar way, without loss of generality, we assume T = |σ ′〉〈σ |with |σ 〉 = |σ ′〉, thus,

T ({W }) ∑[σ ]

pσ {W }Z{W } × CANN[σ ′]

CANN[σ ]〈σ ′|T |σ 〉, (B3)

where the observable we need to calculate during the samplingis not only 〈σ ′|T |σ 〉, but 〈σ ′|T |σ 〉CANN[σ ′]/CANN[σ ]. Bycombining Eqs. (B2) and (B3), we can obtain the variationalenergy for a given trail ANN wave function. To minimizethe variational energy, one needs to calculate the derivativeof H ({W }) with respect to {W }, and the variational parameterscan be determined by solving the equation

∂H ({W })∂wi

= ∂V ({W })∂wi

+ ∂T ({W })∂wi

= 0. (B4)

Notice that an ANN is a combination of a set of nonlinearfunctions whose explicit forms are already known; as a conse-

quence, one can easily obtain its derivative ∂C{W }ANN[σ ]∂wi

, thus,

∂V ({W })∂wi

=∑[σ ]

pσ {W } × ∂C{W }ANN[σ ]

∂wi

2〈σ |V |σ 〉C

{W }ANN[σ ]

,

∂T ({W })∂wi

=∑[σ ]

pσ {W } × 〈σ ′|T |σ 〉C

{W }ANN[σ ]

×[

∂C{W }ANN[σ ′]∂wi

+ ∂C{W }ANN[σ ]

∂wi

C{W }ANN[σ ′]

C{W }ANN[σ ]

].

By performing the importance sampling, one can calculate the∂V ({W })

∂wiand ∂T ({W })

∂wi, then substitute them into Eq. (B4) to solve

the variational parameters.

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Terahertz Streaking of Few-Femtosecond Relativistic Electron Beams

Lingrong Zhao,1,2 Zhe Wang,1,2 Chao Lu,1,2 Rui Wang,1,2 Cheng Hu,3,4 Peng Wang,5 Jia Qi,5 Tao Jiang,1,2

Shengguang Liu,1,2 Zhuoran Ma,1,2 Fengfeng Qi,1,2 Pengfei Zhu,1,2 Ya Cheng,5,6 Zhiwen Shi,3,4 Yanchao Shi,7 Wei Song,7

Xiaoxin Zhu,7 Jiaru Shi,8 Yingxin Wang,8 Lixin Yan,8 Liguo Zhu,9 Dao Xiang,1,2,10,* and Jie Zhang1,2,†1Key Laboratory for Laser Plasmas (Ministry of Education), School of Physics and Astronomy,

Shanghai Jiao Tong University, Shanghai 200240, China2Collaborative Innovation Center of IFSA (CICIFSA),

Shanghai Jiao Tong University, Shanghai 200240, China3Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education),

School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

5State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics,Chinese Academy of Sciences, Shanghai 201800, China

6State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China7Science and Technology on High Power Microwave Laboratory,

Northwest Institute of Nuclear Technology, Xi’an, Shanxi 710024, China8Department of Engineering Physics, Tsinghua University, Beijing 100084, China

9Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang, Sichuan 621900, China10Tsung-Dao Lee Institute, Shanghai 200240, China

(Received 19 January 2018; revised manuscript received 15 April 2018; published 8 June 2018)

Streaking of photoelectrons with optical lasers has been widely used for temporal characterization ofattosecond extreme ultraviolet pulses. Recently, this technique has been adapted to characterize femto-second x-ray pulses in free-electron lasers with the streaking imprinted by far-infrared and terahertz (THz)pulses. Here, we report successful implementation of THz streaking for time stamping of an ultrashortrelativistic electron beam, whose energy is several orders of magnitude higher than photoelectrons. Such anability is especially important for MeV ultrafast electron diffraction (UED) applications, where electronbeams with a few femtosecond pulse width may be obtained with longitudinal compression, while thearrival time may fluctuate at a much larger timescale. Using this laser-driven THz streaking technique, thearrival time of an ultrashort electron beam with a 6-fs (rms) pulse width has been determined with 1.5-fs(rms) accuracy. Furthermore, we have proposed and demonstrated a noninvasive method for correction ofthe timing jitter with femtosecond accuracy through measurement of the compressed beam energy, whichmay allow one to advance UED towards a sub-10-fs frontier, far beyond the approximate 100-fs (rms) jitter.

DOI: 10.1103/PhysRevX.8.021061 Subject Areas: Atomic and Molecular Physics,Interdisciplinary Physics, Photonics

I. INTRODUCTION

Ultrafast phenomena are typically studied with a pump-probe technique in which the dynamics are initiated by apump laser and then probed by a delayed pulse [1]. Becauseof the Ångstrom-scale wavelength, both electrons andx-rays have been used as the probe pulses for watchingatoms in motion during structural changes [2–4]. With the

advent of ultrashort lasers, the temporal resolution in suchexperiments depends primarily on the pulse width andarrival time jitter of the probe pulse. Currently, the brightesthard x-ray pulse is provided by free-electron lasers (FELs)[5–7] and, with the tremendous efforts that have beendevoted, it is now possible to produce a subfemtosecondx-ray pulse [8,9] with its arrival time determined withfemtosecond precision [10–13]. In contrast, for electronprobes as in ultrafast electron diffraction (UED) [14,15],the long-standing goal to deliver a few-femtosecond high-brightness electron beam with a well-characterized arrivaltime still remains quite challenging.In UED, the shortest electron pulse width is mainly

limited by Coulomb repulsion [16]. In the past twodecades, many methods have been developed to mitigatethis effect, e.g., reducing the beam propagation length


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[2,17], reducing the number of electrons per pulse [18–20],increasing the electron beam energy [21–28], and com-pressing the beam with a radio-frequency (rf) buncher[29,30]. Among all these approaches, bunch compressionprovided the most significant advance that enabled a newclass of experiments [31,32]. In this method, the elongatedelectron beam is first sent through a rf buncher cavity,where the bunch head is decelerated to lower energy, whilethe bunch tail is accelerated to higher energy; after passingthrough a drift, the bunch tail with higher energy will catchup with the bunch head, leading to a longitudinally com-pressed isolated bunch. When the rf buncher cavity isreplaced by a laser, the energy modulation at opticalwavelength may lead to the formation of attosecondelectron bunch trains that can also be applied to a certaintype of experiments [33,34].While the rf buncher technique has been widely used in

the UED community and, very recently, a relativisticelectron beam as short as 7 fs (rms) has been produced[30], it is also realized that the space charge force inducedpulse broadening was solved at the cost of increasingtiming jitter. This is because the phase jitter in the rf cavityleads to beam energy jitter, which is further converted intotiming jitter at the sample [19,35]. Similar jitter sourcesexist for FELs as well [10–13]. Such timing jitter, ifnot measured and corrected, will limit the temporalresolution in pump-probe applications to a similar level.Unfortunately, time-stamping techniques developed forhigh-charge GeV and low-energy keV electron beamscannot be easily implemented for MeV UED beams. Forinstance, the arrival time of a GeV beam has been measuredwith the electro-optic sampling (EOS) technique [10,36],but it is difficult to reach high temporal resolution whenapplying this technique to a MeV UED facility, where thebeam charge is relatively low. Time stamping of keVelectron beams with a laser triggered streak camera hasbeen used to correct timing jitter, but the accuracy is on theorder of tens of femtoseconds [37]. Very recently, it hasbeen shown that a buncher cavity powered by a laser-drivenTHz pulse may allow one to compress a keV beam withoutintroducing such jitter [38]. However, it requires a veryintense THz source in order to apply this scheme to theMeVelectron beam. It has been demonstrated recently thata MeV electron beam can also interact with a THz pulseeffectively through the inverse FEL mechanism for beamacceleration and manipulation, but this scheme requires adedicated undulator and careful matching of the phase andgroup velocity of the THz pulse with the electron beam[39]. It should also be noted that a few-femtosecondrelativistic electron beam with intrinsically small timingjitter has also been produced in a laser wakefield accel-erator, where the accelerating gradient is several ordersof magnitude higher than that achieved in rf guns, butthe beam quality and stability still need significantimprovements in order to apply the beam for UEDapplications [40].

Here, we demonstrate a laser-driven THz streakingtechnique with which the arrival time of a 6-fs (rms)ultrashort electron beam was determined with 1.5-fs(rms) accuracy. With the newly developed rf deflector thatprovides about 2.5 fs (rms) temporal resolution, the rfbuncher cavity is optimized to compress the relativisticelectron beam from about 200 fs to well below 10 fs. With anarrow slit to both enhance the local THz field strength andreduce the electron beam intrinsic angular fluctuation,accurate measurement of the relativistic electron beamarrival time is achieved with a THz pulse with moderatefield strength (approximately 100 kV=cm). Because theTHz pulse used for time stamping is tightly synchronizedwith the laser, the measured timing information can bedirectly used for machine optimizations and for correctingtiming jitter in laser-pump electron-probe applications.Furthermore, a noninvasive method for correcting thetiming jitter of a compressed beam through measurementof the compressed beam energy has been proposed anddemonstrated. This noninvasive time-stamping method iseasy to implement and can be applied to both keVand MeVUED to significantly improve the temporal resolution to apotentially sub-10-fs regime.

II. THZ STREAKING DEFLECTOGRAM

The setup for THz streaking of a few-femtosecondrelativistic electron beam is shown in Fig. 1. An approx-imately 50-fs (FWHM) Ti:sapphire laser at 800 nm is firstsplit into two pulses with a 50%–50% beam splitter (BS1).One pulse is frequency tripled to produce an electron beamin a 1.5-cell S-band (2856-MHz) photocathode rf gun. Theother pulse is further split into two parts with a 10%–90%beam splitter (BS2) with the main pulse (approximately2 mJ) used to produce THz radiation through opticalrectification in LiNbO3 crystal [41] and the remaining partfor in situ characterization of the THz pulse at theinteraction region through EOS technique. The approxi-mately 3.4-MeV electron beam is compressed by a C-band(5712-MHz) rf buncher cavity and the electron beam arrivaltime is measured with THz streaking in a narrow slit. In thisexperiment, both the THz and electron beam are running at50 Hz. The electron beam charge is measured to be about30 fC with a Faraday cup.The laser-driven THz streaking measurement combines

the standard streak camera technique with the concept oflaser streaking from attosecond metrology (see, e.g.,Refs. [42,43]). In this experiment, a 250 × 10-μm slitperforated on a 50-μm-thick Al foil with laser machiningis illuminated with a THz pulse with a 30-degree angleof incidence (Fig. 2). The electron beam gets a time-dependent angular kick from the THz pulse when it passesthrough the slit. For electrons that impinge the foil, becauseof multiple scattering, most of them are lost because theirangles are larger than the acceptance of the vacuum pipe,and a small fraction may arrive at the detector, forming a

LINGRONG ZHAO et al. PHYS. REV. X 8, 021061 (2018)

021061-2190

relatively uniform background. The measurement alsoresembles the streaking technique used to measure x-raypulse width and timing jitter in FELs [8,11,44], except that,here, the electron beam angular distribution rather thanenergy distribution is changed by interaction with the THz

pulse, and the electron beam energy is several orders ofmagnitude higher than that of the photoelectrons. Veryrecently, such a scheme has also been applied to keVenergyelectrons for characterizing the electron pulse width andarrival time [38].The THz pulse is produced with an approximately 2-mJ

800-nm femtosecond laser through optical rectification in aLiNbO3 crystal with the tilted-pulse-front-pumping scheme[41], where the pulse front of the laser is first tilted with adiffraction grating and then imaged onto the LiNbO3 withtwo cylindrical lenses (CL1 and CL2 in Fig. 1) formatching the laser phase velocity with the THz groupvelocity. The THz energy is measured to be about 0.5 μJwith a calibrated Golay cell detector. With a THz camera(IRXCAM-THz-384), the THz divergence at the exit of thecrystal was quantified through measurement of the THztransverse size at various positions along the propagationdirection. Then, a focusing system that consists of three off-axis parabolic mirrors was used to guide and focus the THzto the slit. The THz transverse size at the slit is measured tobe about 0.5 mm (rms). The THz waveform measured byEOS [45] is shown in Fig. 3(a), with its correspondingspectrum shown in Fig. 3(b). The THz field strength iscalculated using the known thickness and electro-optic

FIG. 1. THz streaking of a relativistic electron beam experiment setup. The electron beam is produced in a photocathode rf gun byilluminating the cathode with a UV laser and longitudinally compressed with a rf buncher by imprinting a negative energy chirp (i.e.,with the bunch head having lower energy than the bunch tail) in the beam phase space. A set of off-axis parabolic (OAP) mirrors allowstight focus of the THz pulse onto the slit. The electron beam experiences transverse Lorenz force when passing through the slit and theTHz-induced angular deflection is converted into spatial shift at the screen P1 after a drift of 1.8 m. In general, the electron beam isstreaked in the vertical direction, with its time information mapped into spatial distribution on screen P1. Alternatively, the streakedelectron beam may be sent through an energy spectrometer for measuring the longitudinal phase space at screen P2. The slit forstreaking, zinc telluride (ZnTe) crystal for EOS, and a transmission electron microscope (TEM) grid for synchronization are mounted ona remote-controlled manipulator.

FIG. 2. Schematic of the THz-electron interaction at thenarrow slit.

TERAHERTZ STREAKING OF FEW-FEMTOSECOND … PHYS. REV. X 8, 021061 (2018)

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coefficients of ZnTe. It should be pointed out that, becauseof crystal defects and imperfect crystal orientation, thecalculated peak value of about 90 kV=cm should beconsidered as the lower limit of the THz field strength.Alternatively, the upper limit of the THz peak electric fieldis estimated to be about 170 kV=cm using the measuredTHz energy, transverse beam size, and waveform.When this single-cycle THz pulse is focused onto the

narrow slit with its polarization pointing along the slit’sshort axis, near-field enhancement occurs in the slit thatincreases the streaking strength (see, e.g., Ref. [46]). Thesimulated field (using CSTMicrowave Studio) in the centerof the slit is shown in Fig. 3(c), where one can see that,because of transmission resonance, the field strength isincreased by about a factor of 4 and the single-cycle THzpulse becomes multicycle resonating at the wavelength ofapproximately twice the length of the long axis of the slit,i.e., the cutoff wavelength of a 250 × 10-μm rectangularwaveguide.This enhancement also limits the effective THz-electron

interaction within the region of the slit. The simulatedelectron beam centroid deflection for various time delay isfound with the integral of the Lorentz force along the THz-electron interaction region, as shown in Fig. 4(a). Becausethe effective interaction region is much smaller than thewavelength of the oscillation field, the deflection [Fig. 4(a)]just closely follows the electric field [Fig. 3(c)], whichdominates over the magnetic field in our interactionconfiguration.Effective interaction between the THz pulse and

electron beam is achieved when the electron and THzbeam overlap both spatially and temporally in the slit.This is done with the help of the 800-nm laser used for

EOS. First, the ZnTe crystal is put in the center of theinteraction chamber, and the EOS signal is maximizedwhen the 800-nm laser is well overlapped with the THzpulse both in space and time. Then, the ZnTe crystal isremoved from the beam path and a TEM grid is inserted.The BS2 is replaced with a mirror so that the energy ofthe 800-nm laser is sufficient to produce transientplasmas around the interaction point on the TEM grid.The time of the laser and THz pulse is varied with adelay stage until considerable perturbation to the electronbeam transverse profile from the transient electromagneticfield associated with the transient plasma is observed(see, e.g., Refs. [47–49]). Note that the delay stage doesnot change the relative timing between the laser and THzpulse and, with this technique, temporal overlap betweenthe electron beam and the THz pulse is achieved. TheTEM grid is then removed from the beam path and thenarrow slit is inserted. The position of the slit is varieduntil the 800-nm laser passes through the slit. Finally, theelectron beam is steered to pass through the narrow slit,and both spatial and temporal overlap between the THzpulse and electron beam is then achieved. After thisprocedure, the delay between the electron beam and THzbeam is varied and the measured streaking deflectogramis shown in Fig. 4(b), which is in good agreement with

FIG. 4. Simulated beam centroid deflection as a function of thetime delay between the electron beam and THz pulse (a) and themeasured streaking deflectogram as a function of time delay (in67-fs steps) between the electron beam and THz pulse. Themeasured beam centroid deflection at each time delay is shownwith white points. In the measurement, a collimator (3 mmupstream of the slit) is used to reduce the beam size to about20 μm (full width) at the slit, such that the electron beam feels auniform streaking force, and each time slice is integrated over 50single-shot measurements.

FIG. 3. Measured THz waveform (a) at the streaking interactionposition with EOS, the corresponding THz spectrum (b), and thesimulated THz electric field and magnetic field in the center of theslit (c).


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the simulation result. It should be pointed out that thesimulated deflection angle is slightly smaller than theexperimental result, which is probably due to the factthat, in the simulation, the lower limit (90 kV=cm) of thepeak electric field is used.This time-dependent angular streaking allows one to

map the electron beam time information into a spatialdistribution on a downstream screen. Similar to THzstreaking in FELs, the electron beam should overlap withthe THz pulse near the zero crossing of the deflectogram(i.e., the rate of angular change is approximately linear) formeasurement of beam timing jitter and the dynamic rangeof the measurement is limited to half of the wavelengthof the streaking. The maximal streaking ramp [around thet ¼ 4 ps region in Fig. 4(b)] is found to be 5.1 μrad=fs. Theaccuracy of the arrival time measurement is mainly affectedby the fluctuation of the centroid divergence of the electronbeam, resulting in temporal offset in the measurement.Benefiting from the narrow slit, the shot to shot fluctuationof the beam centroid divergence is found to be about7.6 μrad, corresponding to an uncertainty of 1.5 fs in thebeam arrival time determination. It should be mentionedthat, in principle, one may rotate the slit and THzpolarization by 90 degrees to imprint energy modulationin the electron beam and determine the beam arrival timeby monitoring the energy change of the electron beamimprinted by the THz pulse, similar to that used inRef. [39]. However, the accuracy will be much lowerbecause one cannot benefit from the narrow slit (the slitdoes not reduce the beam energy fluctuation) and thebuncher cavity increases the beam energy fluctuation (thebuncher cavity does not increase the beam centroiddivergence fluctuation).

III. TIME STAMPING OF AN ULTRASHORTRELATIVISTIC ELECTRON BEAM

It is worth mentioning that the electron beam temporalprofile can also be retrieved from the broadening of thestreaked beam angular distribution. However, with thebeam intrinsic divergence being approximately 50 μrad,the temporal resolution in the beam temporal profilemeasurement is estimated to be about 10 fs (rms), limitedby the strength of the streaking field. Because the availablec-band rf deflector can provide a much higher resolution, inour experiment, the electron beam temporal profile ismeasured with the rf deflector. The rf deflector is a rfstructure operating in the TM11 mode, which gives thebeam a time-dependent angular kick (i.e., y0 ∝ t) afterpassing through at the zero-crossing phase. The beamangular distribution is converted to a spatial distributionafter a drift section, and the vertical axis on the phosphorscreen (P1 and P2 in Fig. 1) becomes the time axis (y ∝ t).

Figure 5(a) shows measurements of the 30 fC electronbeam temporal profile for various voltages of the bunchercavity, corresponding to no compression (Vb ¼ 0), undercompression (Vb ¼ 0.84 MV), and full compression(Vb ¼ 1.0 MV). The corresponding beam longitudinalphase space [Fig. 5(b)] was measured at screen P2 down-stream of the energy spectrometer. In this measurement,the electron beam is bent in the horizontal direction in theenergy spectrometer, such that the horizontal axis on thephosphor screen P2 becomes the energy axis (x ∝ E).Then, the beam longitudinal phase space is mapped to thetransverse distribution at screen P2. The absolute time iscalibrated by scanning the rf phase and recording thevertical beam centroid motion on the screens (a 1-degree

FIG. 5. Longitudinal compression of a relativistic electron beam to sub-10 fs. (a) Electron beam temporal profile for various bunchervoltages measured with the rf deflector. (b) Corresponding beam longitudinal phase space (bunch head to the left). (c) 100 consecutivemeasurements of the beam profile with rf deflector off (the first 10 shots) and on (the remaining 90 shots). The average beam profileswith the deflector off (black line) and on (white line) are also shown. The number of electrons in the bunch is about 2 × 105.


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change in rf phase corresponds to about a 0.5-ps change intime). The absolute energy is calibrated from the measuredmagnetic field of the energy spectrometer and thedispersion at the phosphor screen.As can be seen in Fig. 5(b), initially the beam longi-

tudinal phase space has a positive chirp (bunch head havinghigher energy than bunch tail), which is caused by the spacecharge force that accounts for bunch lengthening. As thebuncher voltage is gradually increased (Vb ¼ 0.84 MV),the energy chirp is reversed to negative, enabling bunchcompression after a drift. With the buncher voltage set toV ¼ 1.0 MV, the bunch tail exactly catches up with thebunch head, and the shortest bunch length is achieved.Under full compression condition, 100 consecutive mea-surements of the raw beam profile (with vertical axisconverted into time) with the rf deflector off and on areshown in Fig. 5(c), where one can see that the beam hasbeen stably compressed to about 6 fs (rms). In thismeasurement, the voltage of the rf deflector is about1.8 MV and a 20-μm narrow slit is used to improve thetemporal resolution of the beam temporal profile

measurement to about 2.5 fs (rms), as limited by theintrinsic beam size with the rf deflector off. Because boththe beam intrinsic divergence and high order effects in therf deflector contribute to the measured vertical beam sizewith the deflector on [30], the estimated value of 6 fs (rms)should be considered as the upper limit of the bunch length.Though the bunch length is compressed to a few fs, the

timing jitter is likely to be at a much larger timescale. In thisparticular measurement, we ignore variations in the beamtemporal profile and the arrival time of the electron beamunder full compression condition is determined by record-ing the fluctuations of the beam centroid with THzstreaking. One hundred consecutive measurements of beamarrival time with THz streaking are shown in Fig. 6(a), andthe timing jitter at full compression collected over 500 shotsis estimated to be about 140 fs (rms), as shown in Fig. 6(b).Fortunately, such timing jitter can be corrected with femto-second precision [as shown in Fig. 6(a), the jitter for mostof the shots can be corrected with an accuracy betterthan 3 fs], which significantly improves the temporalresolution in laser-pump electron-probe applications.

IV. REAL-TIME NONINVASIVETIME STAMPING

The current setup with a 10-μm slit is best suited formachine optimizations, because the narrow slit reduces theuseful number of electrons in UED. Although, with a moreintense THz source, a wider slit may be used, the dynamicrange (about 600 fs in this experiment) may still hinder itsapplications to cases where the maximal arrival timedifference is larger than half the period of the streakingfield. Motivated by the fact that the timing jitter is primarilycaused by energy variations after the buncher cavity, here,we quantify their correlation and demonstrate that thetiming jitter related to bunch compression may be correctedin a noninvasive way through measurement of the shot-by-shot beam energy fluctuation. In a separate experiment, thephase of the rf buncher was varied, and the measured beamdistribution on screen P2 is shown in Fig. 7(a). In thismeasurement, the electron beam is streaked vertically bythe THz pulse and is bent horizontally by the energyspectrometer. It should be noted that the changing buncherphase is equivalent to scanning the delay time between theTHz pulse and electron beam. This is the main reason thatthe energy-deflection map in Fig. 7(a) is quite similar to thetime-deflection map in Fig. 4(b). Combining the timeinformation from Fig. 4(b) and energy information fromFig. 7(a), the correlation between the arrival time andcentroid energy of the beam is shown in Fig. 7(b), whereone can see that the beam timing jitter Δt is indeed linearlycorrelated with the beam energy jitter ΔE=E, i.e.,Δt ¼ R × ΔE=E, with R determined to be −117 ps, ingood agreement with the momentum compaction of thedrift [50], i.e., R56 ¼ −L=cγ2 ≈ −120 ps, with L ≈ 1.6 m

FIG. 6. Time stamping of a relativistic electron beam with THzstreaking. (a) One hundred consecutive measurements of beamarrival time with THz streaking using the single-valued streakingramp (white curve). Scales on the right correspond to theaccuracy in beam arrival time determination, which dependson the relative time with respect to the zero crossing of thestreaking ramp. The arrival time of the shots within �300 fs isdetermined with an accuracy higher than half the bunch length,i.e., 3 fs (rms). (b) Distribution of the electron beam arrival timecollected over 500 shots. A Gaussian fit (magenta line) to thedistribution within �300 fs yields a timing jitter of about 140 fs(rms) between the electron beam and THz pulse. The distributionwith red color has uncertainty larger than 3 fs and is not used inthe fitting.


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being the distance from the buncher cavity to the THz slitand γ being the Lorentz factor of the electron beam.With the coefficient determined with THz streaking, the

timing jitter may be corrected in a noninvasive way (e.g.,using the undiffracted beam). The dynamical range of thejitter measurement is no longer limited by the wavelengthof the THz pulse and, thus, even a picosecond jitter may becorrected with femtosecond precision. For this method, theaccuracy is limited by the uncertainty of beam energy atthe entrance to the buncher cavity. In our experiment, withthe beam energy stability being about 0.02%, and thedistance between the cathode to the buncher being 0.8 m(corresponding to a momentum compaction of about−60 ps), the accuracy is estimated to be about 12 fs.Note that, for keV UED, where the beam energy stability atthe entrance to the buncher cavity is orders of magnitudehigher, the accuracy of using beam energy jitter to correcttiming jitter should be well below 10 fs.

V. CONCLUSIONS AND OUTLOOK

In conclusion, we have experimentally demonstrated anovel method for the time stamping of relativistic electronbeams. A noninvasive, easy-to-implement method forcorrecting timing jitter with high accuracy through meas-urement of the undiffracted electron beam centroid energyhas also been proposed and demonstrated. Together with

the available few-cycle optical lasers for exciting thedynamics, the demonstrated technique should allow oneto advance UED towards the sub-10-fs frontier. In thefuture, the rf buncher voltage may be increased to produce asubfemtosecond beam. With a stronger streaking field (notethat a LiNbO3-based THz pulse with an electric fieldexceeding 1 MV=cm has been achieved [51]), the reso-lution of the demonstrated method may also be extended towell beyond a subfemtosecond, making attosecond electrondiffraction metrologies capable of visualizing attosecondstructural dynamics within reach.

ACKNOWLEDGMENTS

The authors want to thank S. Li, Z. Tian, and X. Sufor help in THz source design. This work was supportedby the Major State Basic Research Development Programof China (Grant No. 2015CB859700) and by the NationalNatural Science Foundation of China (GrantsNo. 11327902, 11504232, 11655002, and 11721091).One of the authors (D. X.) would like to thank the officeof Science and Technology, Shanghai MunicipalGovernment, for their support (Grant No. 16DZ2260200).

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1State Key Laboratory of Surface Physics, Department of Physics, Fudan University,Shanghai 200433, People’s Republic of China. 2Department of Physics and Astronomy,California StateUniversity, Los Angeles, CA 90032, USA. 3Department of Physics, CaliforniaState University, Fresno, CA 93740, USA. 4TRIUMF, Vancouver, British Columbia V6T 2A3,Canada. 5ISIS Facility, Science and Technology Facilities Council Rutherford AppletonLaboratory, Harwell Science and Innovation Campus, Chilton, Didcot OX11 0QX, UK.6State Key Lab for Metal Matrix Composites, Key Laboratory of Artificial Structuresand Quantum Control (Ministry of Education), Department of Physics and Astronomy,Shanghai Jiao TongUniversity, Shanghai 200240, People’s Republic of China. 7CollaborativeInnovation Center of Advanced Microstructures, Nanjing 210093, People’s Republicof China. 8Department of Physics and Astronomy, University of California, Riverside,Riverside, CA 92521, USA.*Present address: National HighMagnetic Field Laboratory, Tallahassee, FL 32310, USA.†Corresponding author. Email: [email protected]

Zhang et al., Sci. Adv. 2018;4 : eaao5235 5 January 2018 198

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Discovery of slow magnetic fluctuations and criticalslowing down in the pseudogap phase of YBa2Cu3Oy

Jian Zhang,1 Zhaofeng Ding,1 Cheng Tan,1 Kevin Huang,1* Oscar O. Bernal,2 Pei-Chun Ho,3

Gerald D. Morris,4 Adrian D. Hillier,5 Pabitra K. Biswas,5 Stephen P. Cottrell,5 Hui Xiang,6

Xin Yao,6,7 Douglas E. MacLaughlin,8 Lei Shu1,7†

The origin of the pseudogap region below a temperature T* is at the heart of themysteries of cuprate high-temperaturesuperconductors. Unusual properties of the pseudogap phase, such as broken time-reversal and inversion symmetry areobserved in several symmetry-sensitive experiments: polarized neutron diffraction, optical birefringence, dichroic angle-resolved photoemission spectroscopy, second harmonic generation, and polar Kerr effect. These properties suggest thatthe pseudogap region is a genuine thermodynamic phase and are predicted by theories invoking ordered loop currentsorother formsof intra-unit-cell (IUC)magneticorder.However,muonspin rotation (mSR) andnuclearmagnetic resonance(NMR) experiments do not see the static local fields expected for magnetic order, leaving room for skepticism. Themagnetic resonance probes have much longer time scales, however, over which local fields could be averaged by fluc-tuations. The observable effect of the fluctuations inmagnetic resonance is then dynamic relaxation.We havemeasureddynamic muon spin relaxation rates in single crystals of YBa2Cu3Oy (6.72 < y < 6.95) and have discovered “slow” fluc-tuating magnetic fields with magnitudes and fluctuation rates of the expected orders of magnitude that set inconsistently at temperatures Tmag ≈ T*. The absence of any static field (to which mSR would be linearly sensitive) isconsistent with the finite correlation length from neutron diffraction. Equally important, these fluctuations exhibit thecritical slowingdownatTmagexpectedneara time-reversal symmetrybreaking transition.Our results explain theabsenceof static magnetism and provide support for the existence of IUC magnetic order in the pseudogap phase.

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INTRODUCTIONThemysterious pseudogap phase in high-temperature superconductingcuprates has been the subject of an enormous amount of research (1–4)butwith little consensus on either its origin or its role in high-temperaturesuperconductivity. The name arises from the loss of low-lying electronicexcitations below a temperature T*, which depends strongly on the holeconcentration on CuO2 planes. This loss results in considerable modifi-cation of properties connected with these excitations, but their nature hasbeen difficult to determine. Even whether or not T* is in fact a phasetransition temperature has been controversial.

Recent experiments have shown that, consistent with a phasetransition, time-reversal symmetry (5, 6) and spatial rotation and inver-sion symmetries (7–9) are broken below T* in a number of cupratesuperconductors. However, lattice translational symmetry is preserved,indicating that the antiferromagnetism that lies near the pseudogapphase in the temperature-doping phase diagram is not involved. The un-broken translational symmetry has focused attention on the phenomenawithin the crystalline unit cell. In particular, the broken time-reversalsymmetry is predicted by theories that invoke states with intra-unit-cell(IUC) magnetic order (10–12). However, doubt has been cast on the ex-istence of IUCmagnetic order: Local staticmagnetic fields of the expected

magnitude were not observed in muon spin rotation (mSR) (13–18) ornuclear magnetic resonance (NMR) (19, 20) experiments. mSR andNMR are magnetic resonance techniques in which spin probes (im-planted muons or nuclei) are sensitive to magnetic behavior on the local(atomic) scale. mSR in particular is capable of detecting nearby staticmagnetic moments in the range of 0.001 to 0.01 mB and did not do soin cuprate systems where neutron diffraction measurements observedmoments of the order of 0.1 mB.

IUC magnetic order would be retained, and the absent static fieldwould be accounted for, if the ordered moments fluctuate among alter-nate orientations (15, 18, 19). This would average the local fieldBloc(t) atspin-probe sites to zero for time scales longer than a characteristic cor-relation time tc. These fluctuations could be due to finite-size domainsof an ordered phase with different field orientations (21), as seen intunneling spectroscopy (22). For NMR and mSR, the experimental timescale for this averaging is considerably longer (≳10−5 s) than that for theother techniques (≲10−10 s). Thus, all experiments would be consistentif the fluctuations were “slow,” with tc between these limits.

Averaging or “motional narrowing” of Bloc(t) (the term comes fromthe effect of nuclear motion on static NMR line broadening in an ap-plied field) occurs in the rapid fluctuation limit gBrms

loc tc≪1, where g isthe gyromagnetic ratio of the nucleus or muon and Brms

loc ¼ ⟨B2locðtÞ⟩1=2

is the root-mean-square (rms) local field. For themuon,g = gm =8.5156×108 s−1T−1. The IUC-orderedmoments per triangular plaquette obtainedfrom neutron diffraction [0.05 to 0.1 mB in YBa2Cu3O6.66 (5, 6) and pro-gressively lower at higher y] give dipolar field values Bloc = 1 to 10 mT atcandidate muon sites in the unit cell of La2−xSrxCuO4 (15). For thesefields, the above inequality is satisfied for tc≲ 10−6 s, which is in the rangeof experimental consistency. Thus, Bloc(t) gives rise to dynamic or “spin-lattice” nuclear or muon spin relaxation. Itoh et al. (23) have reported“ultraslow” fluctuations in HgBa2CaCu2O6+d that may be of this kind.

In mSR experiments, the motionally narrowed dynamic muon spinrelaxation rate in zero applied field (ZF) is given by lZF ¼ ðgmBrms

loc Þ2tc

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(24, 25). The relaxation rate measured in a longitudinal magnetic field(LF) (that is, themagnetic field parallel to the initial muon polarization)depends on themagnitudeHL of the field, an effect of sweeping themu-on Zeeman frequency through the fluctuation noise power spectrum(24, 25). For Markovian fluctuations with a single well-defined correla-tion time tc, the LF relaxation rate lLF in a field HL is given by the so-called Redfield relation (24)

lLFðHLÞ ¼g2mðBrms

loc Þ2tc1þ ðgmHLtcÞ2

ð1Þ

The dependence of lLF onHL, if observed to be of the form of Eq. 1,provides estimates of tc andBrms

loc .More generally, one expects a decreaseof lLF for gmHL greater than a characteristic fluctuation rate 1/tc, inwhich case tc and Brms

loc from fits of Eq. 1 to the data are heuristic esti-mates of the characteristic time and field scales. The Redfield relationhas been widely applied in mSR to characterize dynamic fluctuatingmagnetic fields in strongly correlated electron systems, for example,in the heavy fermion superconductor PrOs4Sb12 (26).

Here, we report the discovery of slowmagnetic fluctuations in singlecrystals of YBa2Cu3Oy, y = 6.72, 6.77, 6.83, and 6.95 (superconductingtransition temperatures Tc = 72, 80, 88, and 91 K, respectively) via thefield dependence of themSRdynamic relaxation.ConsistencywithEq. 1 isfound, and values of tc and Brms

loc are obtained. We also find maxima atTmag≈T* in the temperature dependences of the rates inZF andLF.Thisis consistent with critical slowing down of magnetic fluctuations near thetransition and demonstrates that these fluctuations are associated withthe IUC order. There is considerable statistical uncertainty because themeasured relaxation rates are near the limit of the technique, but standardstatistical analysis techniques (see below) demonstrate the validity of ourconclusions.

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RESULTSWeperformedLF-mSR experiments on sampleswith y=6.72, 6.77, and6.83. The field dependence of the exponential relaxation rate lLF wasmeasured for these samples at temperatures below T* and above Tcfor 2mT≤m0HL≤ 400mT.Theminimum fieldwas chosen to bemuchlarger than the ~0.1-mT quasi-static field from nuclear dipoles (seeMaterials and Methods and section S1), so that the dipolar field is “de-coupled” (25); that is, the resultant field is nearly parallel to the muonspin, and the dipolar fields do not cause appreciable muon precession.Thus, one observes only dynamic relaxation.


The results are shown in Fig. 1, together with fits of Eq. 1 to thedata. The rates are very small, close to the lower limit accessible to thetechnique, and the statistical uncertainty is large. Control experimentsand precautions taken to minimize systematic errors are discussed insection S2.

In Fig. 1 (A and B), the data lie above the fits for HL ≲ 5 mT. Thismight suggest a logarithmic field dependence as an empirical descrip-tion of the data. However, there is no clear physical mechanism for thisinYBa2Cu3Oy. Inhighly anisotropic tetracyanoquinodimethane (TCNQ)charge transfer salts, nuclear spin relaxation due to diffusion of electronicspin quasi-particles exhibits log H behavior for particular values of theanisotropic hopping rates (27). However, there is no reason to suspectthis diffusive conduction in cuprates, and the higher low-field rates inFig. 1 are more likely to be due to incomplete decoupling for these fields.

Table 1 gives values of tc andBrmsloc from the fits, togetherwith 1 stan-

dard deviation (SD) (1 s) statistical uncertainties. The experimentalvalues of Brms

loc differ from zero by 5 to 9 s individually and ~10 s cu-mulatively; nonzero values are established at this level. Both tc and Brms

locvary smoothly with y. We carried out dipolar lattice sum calculationsfor Bloc, assuming candidate muon stopping sites from the literature(28, 29) and approximating the current loops as point dipoles (30).These yield estimates Bloc ≈ 1 to 1.5 mT for 0.1-mB loop-currentmagnetic moments, of the same order of magnitude as the observedvalues. The calculated values are not changed significantly for the“criss-cross” bilayer loop-current configurations recently reported byMangin-Thro et al. (31).

It is evident that tc falls in the middle of the range of experimentalconsistency discussed above. The observed increase of tc with increasingy could be due to the approach to a quantum critical point as Tmag→0.However, fluctuations of the short-range IUC magnetic order may beassociated with defects (21), in which case tc could depend on samplepreparation and not be an intrinsic property. More work is required toelucidate the nature of the observed fluctuations.

Exponential relaxation is observed in ZF together with the expectedGaussian contribution due to random quasi-static dipolar fields fromnuclear moments (compare section S1). Just above Tc, lZF for y =6.72 (Fig. 2A) is a factor of about 5 higher than lLF above Tc (Fig. 1A).Some of this increase is due to a Lorentzian contribution to the dis-tribution of static fields (13), but some of it is doubtless because of dy-namic relaxation; in ZF, the two are hard to disentangle experimentally(section S1).

The temperature dependence of lwasmeasured in ZF and LF (HL =4mT) for various samples. Figure 2 (A andB) shows lZF(T) and lLF(T),respectively, for y = 6.72, Fig. 2C shows lLF(T) for y = 6.77, and Fig. 2D

10 1000

3

6

9

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2

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6

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2

4YBa2Cu3O6.77

T = 85 K

λ LF(m

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CB

λ LF(m

s–1)

YBa2Cu3O6.72

T = 80 K

HL (mT) HL (mT)

AYBa2Cu3O6.83

T = 93 K

λ LF(m

s–1)

HL (mT)

Fig. 1. Dependenceof the LF exponential relaxation rate lLF(HL) on LFHL inYBa2Cu3Oy. (A) y=6.72, T=80 K. (B) y=6.77, T=85 K. (C) y=6.83, T=93 K. Curves: Fits of Eq. 1 tothe data. A fit for y = 6.72 using Eq. 1 with tc fixed at one order larger than the optimal fit result [tc = 5 (2) ns] is plotted in (A) (dashed curve) for comparison of fit quality.

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shows lZF(T) for y = 6.95. Maxima at Tmag≈ T* and low-temperatureincreases are observed in all samples and fields, the former with statis-tical significance levels of 4 to 5 s individually and ~8 s cumulatively(see Materials and Methods). A relaxation rate maximum is often ob-served at second-ordermagnetic transitions (32), associatedwith criticalslowing down of magnetic fluctuations near Tmag. However, the low-temperature increase is unusual and is discussed below in more detail.

In the phase diagram of Fig. 3, Tmag from our mSR data is plottedversus the hole concentration p (and the oxygen content y) along withTc and T* from other experiments. Values of Tmag are consistent withresults of other experiments: polarized neutron diffraction (5, 6), tera-hertz birefringence (7), resonant ultrasound (33), and second harmonicgeneration (8). [T* is also the temperature around which changes intransport properties (34) and thermodynamic properties (35, 36) fromthose of the strange metal phase begin to be observed.] The observedTmag also corresponds well to the results of recent torquemagnetometrymeasurements, which support a second-order phase transition at T*(9). The inset of Fig. 3 compares the doping dependence of the squareroot I1=2neu of the polarized neutron diffraction cross section (37) with thatof Brms

loc . These quantities are both proportional to the order parameterfor IUC magnetic order, and they follow the same trend.

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DISCUSSIONThe observed increase of motionally narrowed muon spin relaxationwith decreasing temperature below T*, shown in Fig. 2, cannot occurin a transition to a uniform ordered state. It is, however, consistent withlow-frequency fluctuations in domains of IUC magnetic order, ofincreasing magnitude with the increasing order parameter (21).Scanning tunneling spectroscopy experiments (22) have found thesedomains in the pseudogap phase associated with defects, with lineardimensions of ~20 unit cells; these provide a mechanism for the pseu-dogap in the one-particle spectra (21). Other experiments (38, 39) haveobserved mysterious anomalous low-frequency fluctuations in thepseudogap phase ascribed to extended defects.

Phenomena other than IUC magnetism can affect mSR experi-ments. Sonier and co-workers (14, 16) observed features inZFdata fromYBa2Cu3Oy and attributed them to charge and structural inhomogene-ities from lattice changes and CDW formation. Fits of an exponentiallydamped Gaussian Kubo-Toyabe (KT) to data from a nearly fully dopedsample (y = 6.985) (14) yielded correlations between D(T) and l(T),leading to the conclusion that the exponential rate could not bedetermined unambiguously (40). Structure in ZF l(T) with D(T) heldfixed was related to the onset of short-range CDW order (41) for thatdoping.

For slightly lower doping (y = 6.95), however, fits with D(T) a freeparameter (42) yield a clear peak in l(T) at Tmag = 80 K with nocorresponding structure in D(T) at that temperature (compare section


S3). NQR measurements (43) on a sample of similar oxygen contentshowed that CDW order sets in at a considerably lower temperature(~35 K). For y < 6.95, ZF CDW transition temperatures from NMR/NQRand other measurements are significantly lower than Tmag from ourexperiments. For all y, the onset of long-rangeCDWorder is at still low-er temperatures and is only observed in high-applied magnetic fields(20). Recent torque magnetometry measurements also provide strongevidence that CDW order and the pseudogap phase are characterizedby distinct broken symmetries at different temperatures (9). The detailedtemperature-doping phase diagram (Fig. 3) shows that lattice change/CDWand pseudogap phases set in at distinct temperatures and are dis-tinct phenomena.We conclude that there is no evidence for associatingthe maxima in l(T) and its increase at low temperatures with eitherCDW or charge inhomogeneity.

It has been claimed (40) that the maxima for y = 6.72 and 6.77 at~210 and ~165 K, respectively (Fig. 2, A and C), could be due to theonset of thermally activated muon hopping (diffusion) (14); this causesunwanted dynamic relaxation by nuclear dipolar fields. However, in theusual trapping-detrapping scenario (44), the maximum is due to a tem-perature sequence in which the muons are trapped and static at lowtemperatures, begin hopping with increasing temperature, find deepimpurity traps and become static again at an intermediate temperature,and finally detrap definitively at high temperatures. It has been argued(42) that this is highly unlikely in the present case because independentmSRdeterminations (14, 42, 45) of themuonhopping rate using dynam-ic Gaussian KT fits (25) show that hopping is too slow to produce theobserved decrease in l(T) below the maximum (compare section S4).Alternatively, the maximum might be purely dynamic because withincreasing temperature, the muon hopping rate passes through themuon Zeeman frequency. In that case, however, the application ofHL =4 mT field would move the maximum to temperatures where thehopping rate is ~gmHL = 3.4 ms

−1. According to the dynamic KT fits, this

0 100 200 3000

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10

20

30

40

80 120 160 200 2402

4

6

8

10

80 120 160 200 2402

4

6

8

10YBa2Cu3O6.72

HL = 0

λ ZF,L

F(m

s–1)

Tc

T∗

A B

C D

Tc

YBa2Cu3O6.95

HL = 0T

∗

YBa2Cu3O6.77

HL = 4 mTT∗

Temperature (K)

YBa2Cu3O6.72

HL = 4 mTT

∗

Fig. 2. Temperature dependence of the dynamic muon relaxation rate l inYBa2Cu3Oy. (A) y = 6.72, LF HL = 0. (B) y = 6.72, m0HL = 4 mT. (C) y = 6.77, m0HL = 4 mT,(D) y = 6.95, HL = 0. The pseudogap onset temperature T* is shown for each doping.

Table 1. Correlation times tc and rms muon local fields Brmsloc from muon

spin relaxation rates in YBa2Cu3Oy.

y
Temperature (K) tc(ns) Brmsloc (mT)
6.72
80 5(2) 0.92(19)
6.77
85 10(3) 0.87(10)
6.83
93 25(10) 0.37(6)
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is well above 300 K, whereas in our data (Fig. 2B), the position of themaximum is unchanged. We conclude that muon hopping is not theorigin of the maximum for y = 6.72. Note that, for y = 6.67, no maxi-mum in lZF(T) was seen near T* ~200 K (13), which is probably due tosignificant muon hopping at this temperature.

Previous transverse-field (TF) mSR experiments in the pseudogapphase (46) observed exponential relaxation and ascribed it to static spatialinhomogeneity of superconducting fluctuations. Our observed LF-mSRrates (Fig. 1) are an order of magnitude slower than the TF-mSR rates.This is consistent with the assumption that the latter are static and pre-cludes detecting the dynamic relaxation in TF-mSR.

The observed critical slowing down at Tmag ≈ T* (Fig. 2) indicatesthat this temperature marks the onset of broken time-reversal sym-metry, as is also found in each of the four hole-doped cuprate familiesamenable so far to polarized neutron diffraction experiments. The ob-served magnitude of the order parameter of about 0.1mB staggered mo-ment per unit cell has a condensation energy of ~50 J/mol, similar to themaximum superconducting energy in cuprates (47). These propertiesare all consistent with our results. Note that a recent LF-mSR study ofBi2+xSr2−xCaCu2O8+d (45) also reported a quasi-static internalmagneticfield in the pseudogap phase.

Our discovery of fluctuatingmagnetic fields provides an understandingof the absence of static magnetic fields due to IUC magnetic order inYBa2Cu3Oy. The expected fields are present but fluctuating. AlthoughmSR is a point probe in real space and, thus, is not directly sensitive tothe spatial symmetry, our results are strong evidence for IUC order andits excitations and establish them as important for understanding theunusual behavior of cuprates.


MATERIALS AND METHODSSample growth and characterizationHigh-quality single crystals of YBa2Cu3Oy were grown by the top-seeded solution growth polythermalmethod using 3BaO-5CuO solvent(48). AYBa2Cu3Oy single crystal with an ab plane area of 10mm×10mmand c axis length of 8 mm was synthesized with a cooling rate of 0.5 Kper hour in air. The crystal was then cut into small pellets with thick-nesses of 0.55mmand lateral dimensions of 2mm×2mm. Single crystalswith optimalTc = 91 Kwere achieved by annealing at 400°C for 180 hoursin flowing oxygen. A range of oxygen concentrations of YBa2Cu3Oy wasachieved by post-annealing in flowing oxygen at different temperatures asdescribed in the study by Gao et al. (49), resulting in superconductingtransition temperatures between 72 and 88 K.

Figure 4 shows the temperature dependences of the magnetizationand the resistivity in our samples. The data indicate that the supercon-ducting transitions are sharp. The values of T* from the departure of theresistivity from linearity are in agreement with previous results (50).

Muon spin relaxation experimentsIn the time differential mSR technique (51), spin-polarized muons areimplanted into the sample. In the muon decay to a positron and twoneutrinos, the positron momentum is preferentially oriented alongthe direction of the muon spin at the time of decay. The time evolutionof the ensemble muon polarization can therefore be monitored via theasymmetry in positron emission count rates.

Our mSR experiments were performed using the Los AlamosMesonPhysicsFacility spectrometer at theM20beamlineofTRIUMF(Vancouver,Canada) and the MUSR and EMU spectrometers at ISIS, Rutherford

Fig. 3. Phase diagram of pseudogap and charge-density-wave/charge inhomogeneity onset temperatures in YBa2Cu3Oy. Red diamonds: Temperatures Tmag of maximain mSR exponential relaxation rates (Fig. 2). Open green squares: Pseudogap temperatures T* from polarized neutron diffraction (5, 6). Open blue triangles: T* from THz bi-refringence (7). Pink pentagons: T* from resonant ultrasound (33). Orange circles: T* from second harmonic generation (8). Green stars: T* from magnetic torque (9). Magentaleft triangles: Charge-density-wave (CDW) onset temperatures TCDW from NMR (20). Filled blue triangle: TCDW from nuclear quadrupole resonance (NQR) (43). Black squares: TCDWfrom Hall effect (53). Blue circle: TCDW from high-energy x-ray diffraction (54). Black circles: Onset of “charge inhomogeneity” (CDW or lattice change) from mSR experiments (14).Gray points: Superconducting transition temperatures. Inset: Doping dependences of the square root I1=2neu of the polarized neutron diffraction cross section (37) and the rmsmagnitude Brms

loc of the fluctuating local field (Table 1).

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AppletonLaboratory (Chilton,UnitedKingdom).At both facilities,muonswere implanted into the sample with their initial spin polarization Pmperpendicular to the ab plane.

Appropriate functional forms, discussed in section S1, were least-squares fit to the asymmetry data using the MUSRFIT mSR analysisprogram (52). The LF rates were very low and were near the resolutionlimit of the technique. The difference between the rates was small butwas resolved. Note that lLF in Bi2+xSr2−xCaCu2O8+d is also quite smallforHL > 1mT (45). To our knowledge, no LF-mSRdata for other cupratesuperconductors in comparable fieldshavebeen reported.Weneverthelessspeculate that the scatter in Fig. 1was typical for the hole-doped super-conducting cuprate family.

Statistical analysisRandom error in mSR experiments arises from the Poisson distributionof positron count rates. The MUSRFIT analysis software computes thepropagation of this error to that of the parameters of the fit function andreports their probable SD s. All parameter uncertainties quoted in thisarticle are 1 s.


The inverse relative SDs (IRSDs) (the N in “Ns”) for Brmsloc (Table 1)

and for themaxima in the temperature dependences atTmag (Fig. 3) areshown inTable 2. The IRSD forBrms

loc is simply its value divided by its SD.For the maxima at Tmag, baseline points were chosen above and beloweach maximum, and baseline values at intermediate points were esti-mated by linear interpolation. The IRSD of each point is its amplituderelative to the baseline divided by its SD, and the IRSD of themaximumis the square root of the sum of squares of the IRSDs of the points. Thesign of the amplitude was included in this sum to account for negativecontributions. For both Brms

loc and the maxima, the cumulative IRSD isthe square root of the sum of squares of the individual sample IRSDs. Itcan be seen that some individual IRSDs are marginal, but thecumulative values are quite satisfactory.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/1/eaao5235/DC1section S1. Muon relaxation functionssection S2. Control experimentssection S3. Temperature dependence of static Gaussian KT relaxation rate DZF(T)section S4. Muon hopping, superconductivitysection S5. Superconducting fluctuationssection S6. High-temperature relaxationfig. S1. ZF relaxation of the muon asymmetry Am(t) in YBa2Cu3O6.72.fig. S2. Time evolution of the positron count rate asymmetry Am at various temperatures andfields in single-crystal YBa2Cu3Oy.fig. S3. LF muon spin relaxation rates in silver samples.fig. S4. Fits of representative ZF-mSR spectra from YBa2Cu3O6.95 for DZF fixed and free.fig. S5. Temperature dependence of ZF exponential damping rate lZF and static Gaussian KTrate DZF for YBa2Cu3O6.95.fig. S6. Damped dynamic Gaussian KT fit of ZF data from YBa2Cu3O6.72.References (55–58)

REFERENCES AND NOTES1. T. Timusk, B. Statt, The pseudogap in high-temperature superconductors: An

experimental survey. Rep. Prog. Phys. 62, 61–122 (1999).2. M. R. Norman, C. Pépin, The electronic nature of high temperature cuprate

superconductors. Rep. Prog. Phys. 66, 1547–1610 (2003).

40 50 60 70 80 90 100 110

–12

–8

–4

0

0 100 200 3000

2

4

100 150 200 250 300

–1.0

–0.5

0.0

C

B

χ(c

m3 /m

ol)

Temperature (K)

YBa2Cu3Oy

y = 6.72y = 6.77y = 6.83y = 6.95

H = 1 mT

A

y = 6.77

y = 6.77

ab

(10–4

• Ω c

m)

Temperature (K)

Δρab

(10–4

• Ω c

m)

Temperature (K)

T ∗

ρ

Fig. 4. Characterization data fromYBa2Cu3Oy single crystals. (A) Magnetization of YBa2Cu3Oy, y= 6.72, 6.77, 6.83, and 6.95, showing sharp superconducting transitions.(B) Temperature dependence of electrical resistivity rab for y = 6.77 showing a deviation from linearity. (C) Temperature dependence of the deviation Drab from linearresistivity for y = 6.77. The deviation sets in below T* = 156 K, which is consistent with reported results (50).

Table 2. IRSDs of Brmsloc and relaxation rate maxima near Tmag from

muon spin relaxation rates in YBa2Cu3Oy.

y
Brmsloc Relaxation rate maxima (Fig. 2)
IRSD
HL (mT) No. of points IRSD
6.72
4.8 0 7 4.5
6.72
4 11 3.8
6.77
8.7 4 15 5.2
6.83
6.2
6.95
0 6 3.9
Cumulative
11.7 8.8
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3. M. R. Norman, D. Pines, C. Kallin, The pseudogap: Friend or foe of high Tc? Adv. Phys. 54,715–733 (2005).

4. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, J. Zaanen, From quantum matter tohigh-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).

5. B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhès, C. T. Lin, X. Chaud, P. Bourges, Magnetic orderin the pseudogap phase of high-Tc superconductors. Phys. Rev. Lett. 96, 197001(2006).

6. L. Mangin-Thro, Y. Sidis, A. Wildes, P. Bourges, Intra-unit-cell magnetic correlations nearoptimal doping in YBa2Cu3O6.85. Nat. Commun. 6, 7705 (2015).

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Acknowledgments: We are grateful to C. M. Varma for proposing these experiments and fornumerous discussions. We thank the support teams at TRIUMF and ISIS for help during theexperiments and J. H. Brewer for discussions and correspondence. Funding: The researchperformed in this study was partially supported by the National Key Research andDevelopment Program of China (nos. 2016YFA0300503, 2017YFA0303104, and 2016YFA0300403),the National Natural Science Foundation of China (nos. 11774061 and 11474060), and theScience and Technology Commission of Shanghai Municipality of China (grant no. 15XD1500200).Research at University of California (UC), Riverside was supported by the UC RiversideAcademic Senate. Work at California State University (CSU), Los Angeles was funded by NSF/


DMR (Division of Materials Research)/Partnerships for Research and Education inMaterials–1523588. Research at CSU-Fresno was supported by NSF DMR-1506677.Author contributions: L.S. and D.E.M. conceived the experiment and wrote the beam-time proposals. J.Z., Z.D., C.T., K.H., O.O.B., P.-C.H., G.D.M., A.D.H., P.K.B., S.P.C., D.E.M., andL.S. performed the mSR measurements, with D.E.M. and L.S. overseeing the experimentalwork. J.Z., H.X., and X.Y. grew the single-crystal samples. J.Z., D.E.M., and L.S. performed the dataanalysis; these authors wrote the manuscript. All authors discussed the data and contributedto the analysis. Competing interests: The authors declare that they have no competing interests.Data and materials availability: All data needed to evaluate the conclusions in thepaper are present in the paper and/or the Supplementary Materials. Additional data related tothis paper may be requested from the authors.

Submitted 10 August 2017Accepted 30 November 2017Published 5 January 201810.1126/sciadv.aao5235

Citation: J. Zhang, Z. Ding, C. Tan, K. Huang, O. O. Bernal, P.-C. Ho, G. D. Morris, A. D. Hillier,P. K. Biswas, S. P. Cottrell, H. Xiang, X. Yao, D. E. MacLaughlin, L. Shu, Discovery of slowmagnetic fluctuations and critical slowing down in the pseudogap phase of YBa2Cu3Oy. Sci.Adv. 4, eaao5235 (2018).
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PHYSICAL REVIEW E 98, 022106 (2018)

Fidelity susceptibility of the anisotropic XY model: The exact solution

Qiang Luo,1,* Jize Zhao,2 and Xiaoqun Wang3,4,†1Department of Physics, Renmin University of China, Beijing 100872, China

2Center for Interdisciplinary Studies, Lanzhou University, Lanzhou 730000, China3Key Laboratory of Artificial Structures and Quantum Control, Ministry of Education, School of Physics and Astronomy,

Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China4Collaborative Innovation Center for Advanced Microstructures, Nanjing 210093, China

(Received 2 March 2018; published 7 August 2018)

We derive several closed-form expressions for the fidelity susceptibility (FS) of the anisotropic XY model in thetransverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are relatedto a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating itsvalidity for characterizing the phase transitions. Moreover, the critical exponents ν associated with the correlationlength in both critical regions are successfully extracted by the standard finite-size scaling analysis.

DOI: 10.1103/PhysRevE.98.022106

I. INTRODUCTION

Quantum phase transitions (QPTs), which are driven solelyby quantum fluctuations and are characterized by drasticchanges in the ground state, are of great interest for the inter-pretation of widespread phenomena in physics [1,2]. Over thepast few decades, numerous theoretical methods for detectingQPTs have been introduced regarding the aspect of quantuminformation sciences [3–9]. The ground-state fidelity F [10]is one of such methods. It measures the overlap between twowave functions of the same Hamiltonian but at different valuesof the control parameter λ. As a result, a notable change in thefidelity is expected to occur at the transition point λc even for afinite-size system. However, the fidelity is sometimes chaoticnumerically in that it depends on the increment of the controlparameter and it vanishes exponentially with an increase ofthe system size. Therefore, the fidelity susceptibility (FS) χF

[11–13], the derivation of the fidelity with respect to λ, is in-troduced to eliminate such drawbacks and turns out to be morepowerful. Technically, the FS is nothing but the nontrivial lead-ing quadratic term of the fidelity, so its divergence at the transi-tion point is reminiscent of the singularity of the latter. The past15 years have witnessed the explosive applications of fidelityand FS to the QPT of various strongly correlated systems[14–25], including the intricate Berezinskii-Kosterlitz-Thouless transition [26–28] and the unconventional topologi-cal phase transition [29–34].

Historically, the plausible evidence of the FS as a probefor quantum criticality is revealed by the similarity betweenthe scaling behavior of the FS at the critical point andthat of the second derivative of the ground-state energy,where the transverse-field Ising model (TFIM) is illustratedas an example [35,36]. Following standard arguments in the


scaling theory of a continuous QPT, one obtains that theFS per site in a d-dimensional system with length L scalesas [37–39]

χF (λ)/Ld ∼ L2/ν−df (|λ − λc|L1/ν ), (1)

where ν is the critical exponent of the correlation lengthand f (·) is a scaling function. It is inferred that near thecritical point the scaling expression behaves as χF (λc )/Ld ∼Lk1 (k1 = 2/ν − d ). Alternately, one may look at χF slightlyaway from the critical point at the thermodynamic limit (TDL),where χF (λ)/Ld ∼ |λ − λc|k2 (k2 = dν − 2). Consequently,the scaling ansatz in the system exhibiting logarithmic diver-gences requires that the absolute value of the ratio k2/k1 is thecritical exponent ν [37–39]. The critical exponent ν is usuallycalculated numerically because of the absence of analyticalexpression for FS. The main obstacle lies in that no algebraictechnique is available to obtain its closed form at finite length.A breakthrough was made on the single-parameter TFIMby Damski and Rams in light of several elegant summationformulas [40–42]. This method, however, is difficult to followand can hardly be generalized to a double-parameter XY modelwhere a quadratic summation is explicitly involved [10,43].Notably, after a proper symmetry analysis, we find that thepartial fraction expansion method can be used to divide thequadratic term into two coupled linear terms. As a result, allthe terms are related to a simple fraction or its derivative(see Appendixes A and B) that can be treated exactly inprinciple.

The remainder of the paper is organized as follows. In Sec. IIwe introduce the anisotropic XY model in the transverse fieldand give the expressions for the FS. In Sec. III the closedform of the expressions are presented in detail and the criticalexponent ν is calculated analytically by the scaling ansatz.Section IV is devoted to a summary.


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QIANG LUO, JIZE ZHAO, AND XIAOQUN WANG PHYSICAL REVIEW E 98, 022106 (2018)

II. MODEL

We calculate analytically the FS of the one-dimensionalspin- 1

2 anisotropic XY model in the transverse field [44]

H = −N∑

n=1

(1 + γ

2σx

n σ xn+1 + 1 − γ

2σy

n σy

n+1 + hσ zn

), (2)

where σαn (α = x, y, z) is the α component of the Pauli operator

acting on site n, γ is the anisotropy parameter at the xy plane,and h is the external field in the z direction. The number ofsites N = 2N is assumed to be even for brevity and bothsymbols N and N are used throughout the paper. Here aperiodic boundary condition (σN+1 = σ1) is imposed so thatEq. (2) can be diagonalized exactly via the Jordan-Wignertransformation. The physical ground state depends strongly onthe proper choice of momentum quantization, which results ina positive- or negative-parity sector [45,46]. The ground statecould be determined by a competition between the vacuumstates of the two parity sectors at given parameters [45].Interestingly, the energy gap, obtained from the two states,shows a rather anomalous behavior [46]. For even N , itis argued that the physical ground state lies in the positivesector at least outside the disordered circle h2 + γ 2 = 1. Inthe circle, the energy gap vanishes rapidly near γ = 0. So forsimplicity we will only consider the momentum quantizationin the positive sector hereafter. The single-particle energiesare given by �k =

√ε2k + γ 2 sin2 k, where εk = cos k − h,

k = (2n − 1)π/N (n = 1, 2, . . . , N ) [45,46]. With cos θk =εk/�k in mind, we have the analytic expressions for the FSas χ (q )(h, γ ) = 1

4

∑Nn=1(∂θk/∂q )2 with q = h and γ [10].

Consequently, the explicit expressions for the FS with respectto the two kinds of QPTs (see below) are [10]

χ (h)(h, γ ) = 1

4

∑k>0

γ 2 sin2 k

[(cos k − h)2 + γ 2 sin2 k]2(3)

and

χ (γ )(h, γ ) = 1

4

∑k>0

sin2 k(cos k − h)2

[(cos k − h)2 + γ 2 sin2 k]2. (4)

Actually, the FS defined above is none other than the diagonalelement of a more general concept, the quantum metric tensor[47]. The expression for the remaining off-diagonal element isshown in Appendix C.

As shown in Fig. 1, the model has a richer phase diagramwhen compared with the TFIM, a special case of Eq. (2) atγ = 1. There are four different phases in the (h, γ ) plane,which are separated by the lines h = ±1 and by the segment|h| < 1, γ = 0. The corresponding QPTs are referred to asthe Ising transition and the anisotropy transition, respectively.The Ising critical lines are the boundaries between ferromag-netic phases and paramagnetic phases, whereas the anisotropytransition separates the ferromagnets with spins in the x and y

directions. Both Ising and anisotropy transitions share the samecritical exponent ν = 1 [48,49], and we intend to calculateit analytically according to expressions (3) and (4) in thefollowing section. There are two fascinating curves in Fig. 1.The circular curve (h2 + γ 2 = 1, dashed pink line) separatesthe regions with oscillatory and nonoscillatory correlations

FIG. 1. Ground-state phase diagram of the anisotropic XY modelin the transverse field at the (h, γ ) plane. There are paramagneticphases with opposing orientation along the z direction when |h| > 1and ferromagnetic phases when |h| < 1. There are two fascinatingcurves in the ferromagnetic phases (see the main text) [44].

asymptotic behaviors, while the parabolic curve (γ 2 ± h = 1,dotted cyan line) is the boundary between commensurate andincommensurate phases [50]. It is worth mentioning that thecritical exponents at the multicritical points obey a differentuniversality [51] and is beyond the scope of the present paper.

III. EXACT SOLUTIONS

A. Ising transition

From Eq. (3) we can rewrite the FS for the XY model withrespect to the external magnetic field h as

χ (h)(h, γ ) = γ 2

4(1 − γ 2)2

N∑n=1

1 − c2n[

c2n − 2h

1−γ 2 cn + h2+γ 2

1−γ 2

]2 , (5)

where cn = cos (2n−1)πN . Hereafter, |γ | < 1 is assumed to avoid

possible ambiguity and we note that our main results remainunchanged for arbitrary γ . For example, our results are stillvalid for |γ | = 1 since the FS is continuous when crossing thelines. The summation in Eq. (5) is not easy to handle directlydue to the existence of the quadratic terms. To eliminatethem, we here employ a factorization method. It can be foundthat when h2 + γ 2 > 1 we can factorize φ(t ) = t2 − 2h

1−γ 2 t +h2+γ 2

1−γ 2 = (t − λ+)(t − λ−), where λ′υs (υ = ±) are the real

roots of the equation φ(t ) = 0 and

λυ = h + υ|γ |√

h2 + γ 2 − 1

1 − γ 2. (6)

The signs of the two roots λυ are the same and are consistentwith the sign of the field h. In addition, their absolute valuesare both larger than 1 so long as |h| �= 1 (otherwise the smallerone equals 1). When h2 + γ 2 < 1, however, the roots arecomplex and the imaginary unit i = √−1 should be involved.By virtue of the partial fraction expansion of the expression

022106-2206

FIDELITY SUSCEPTIBILITY OF THE ANISOTROPIC … PHYSICAL REVIEW E 98, 022106 (2018)

1−c2n

(cn−λ+ )2(cn−λ− )2 , Eq. (5) is recast into the symmetric form

χ (h)(h, γ ) = 1

16(h2 + γ 2 − 1)

∑υ=±

F (h)υ , (7)

where

F (h)υ = (

λ2υ − 1

)L′(λυ ) − 2(λυλυ − 1)

(λυ − λυ )L(λυ ), (8)

with υ the complementary component of the branch υ

(υυ = −1). In Eq. (8), the complex analytic function L(α) =∑Nn=1

1cn+α

(α ∈ C) is introduced and the odevity of it and itsderivative are utilized as well. The highlight of the paper isthat the closed form of L(α) is discovered, making it possibleto have exact expressions for FS. The mathematical details areomitted here and are presented in Appendix A; the relationbetween L′(α) and L(α) is shown in Appendix B. Accordingto the formula presented in Appendix A, it is useful to havegυ = λυ + √

λ2υ − 1 so that

gυ = h + υpυ

√h2 + γ 2 − 1

1 − pυ |γ | , (9)

where pυ = pυ (h) is a piecewise sign function of the field h

and branch υ,

pυ ={υ, |h| > 1

sgn(h), |h| < 1.(10)

The relevant correlation length near the critical point readsξ = 1/| ln g+| [44], indicating that it is the positive branch ofF (h)

υ (υ = 1) that gives rise to the divergence behavior of theFS in the TDL. The explicit expression of F (h)

υ is calculatedanalytically as

F (h)υ = N 2gN

υ(gN

υ + 1)2 + NCυ

2

gNυ − 1

gNυ + 1

, (11)

with the expression

Cυ = pυ

h2 − 1

[υγ 2h√

h2 + γ 2 − 1− h2 + γ 2 − 1

|γ |

]. (12)

We note here the detailed derivations are presented in theSupplemental Material (SM) [52].

We now consider the FS at special values. When γ = 1, theXY model is reduced to the TFIM and the FS has the form

χ (h)(h, 1) = N16h2

×[ NhN

(hN + 1)2+ 1

2

((h2 + 1)(hN − 1)

(h2 − 1)(hN + 1)− 1

)],

(13)

which agrees with the result by finite sums of hyperbolicfunctions [40]. When h = hc, i.e., |h| = 1, the FS is

χ (h)(hc, γ ) =N 2 − 3−γ 2

2γN

32γ 2

+N

[(N + 3−γ 2

2γ

)( 1+γ

1−γ

)N + 3−γ 2

2γ

]16γ 2

[( 1+γ

1−γ

)N + 1]2 . (14)

0.20.0-2

0.5

-1 0.00

1.0

1 -0.22

FIG. 2. Normalized fidelity susceptibility χ (h) with respect to h.It has a sharp extremum at the critical lines |h| = 1 where continuousQPTs with ν = 1 occur.

The first term of Eq. (14) is the leading term, while the secondterm is either exponential decay or linearly increased with thesystem size N , depending on the sign of γ . In the finite-size system, the maximal value of the FS χ (h)(h = hm, γ )is slightly larger than that of χ (h)(h = hc, γ ), but with thequadratic term of N unchanged. Therefore, χ (h)(hm, γ ) �χ (h)(hc, γ ) � N

32γ 2 (N − 3−γ 2

2|γ | ). Considering only the leadingterm, we have

ln

(χ (h)(hm, γ )

N

)= lnN − ln(32γ 2). (15)

In the TDL, the exact expression for the FS can be calculatedby the residue theorem and turns out to be [53]

χ (h)(h, γ )

N = 1

16×

{ 1|γ |(1−h2 ) , |h| < 1

|h|γ 2

(h2−1)(h2+γ 2−1)3/2 , |h| > 1(16)

and the scaling behavior around the critical points is

ln

(χ (h)(h, γ )

N

)= − ln |h − hc| − ln(32γ ). (17)

It can be noticed that the absolute value of the prefactorsof Eqs. (15) and (17) are equal, indicating that the criticalcomponent ν = 1 according to the ansatz (1). The normalizedFS χ (h)(h, γ ) with respect to h in the TDL is shown in Fig. 2.It is found that the FS has a sharp extremum at |h| = 1where a continuous QPT occurs. This result demonstratesconvincingly that FS can be used to characterize the quantumcritical behavior.

B. Anisotropy transition

Calculation for the FS with respect to the anisotropytransition is similar to the Ising transition discussed above.To begin with, we should rewrite the expression (4) as

χ (γ )(h, γ ) = 1

4(1 − γ 2)2

N∑n=1

(1 − c2

n

)(cn − h)2

(cn − λ+)2(cn − λ−)2, (18)

where the λ′υs (υ = ±) are defined in Eq. (6). Once the

partial fraction expansion of the expression (1−c2n )(cn−h)2

(cn−λ+ )2(cn−λ− )2 isobtained, the FS is readily split into two symmetric forms, eachof which could be calculated according to the method shown

022106-3207


in Sec. III A. So we neglect the tedious details and present thefinal result

χ (γ )(h, γ ) = 1

16(h2 + γ 2 − 1)

∑υ=±

F (γ )υ − N

8(1 − γ 2)2,

(19)

where

F (γ )υ =

(g2

υ − 1)2

4g2υ

[N 2gN

υ(gN

υ + 1)2 + N

(Cυ

2+ pυ

|γ |)

gNυ − 1

gNυ + 1

].

(20)

Here pυ , gυ , and Cυ are as defined earlier. Again, the detailedderivations are presented in the SM [52]. It is well knownthat the anisotropy transition occurs when |h| < 1, and in thissegment χ (γ )(h, γ ) reaches its maximum exactly at γ = 0.Precisely, we have

χ (γ )(h, 0) = −N8

+ 1

4

[ N 2gN

(gN + 1)2+ Nh

2i√

1 − h2

gN − 1

gN + 1

],

(21)

where g = h + i√

1 − h2. Physically, χ (γ )(h, 0) is a realexpression with a vanishing imaginary part. This can be seenclearly by parametrizing h = cos θ and θ ∈ (0, π ),

χ (γ )(h, 0) = N

4

[N

cos2[N cos−1(h)]+ tan[N cos−1(h)]

tan[cos−1(h)]− 1

].

(22)

It can be proved that the ratio of the second term to the firstone in the large square brackets is an oscillating functionand is bounded. So the second term can be neglected inthat it contributes the leading term to a prefactor at most.The third term also does not need to be considered for largeenough system size N . In light of the relation χ (γ )(h, γm) =χ (γ )(h, γc ) � N 2

16 cos2[N cos−1(h)/2] , we obtain that

ln

(χ (γ )(h, 0)

N

)= lnN − ln

[16 cos2 N cos−1(h)

2

]. (23)

Similarly, the exact expression for the FS is [53]

χ (γ )(h, γ )

N = 1

16

1

|γ |(1 + |γ |)2, |h| < 1 (24)

in the TDL, indicating that the scaling behavior around thecritical points is

ln

(χ (γ )(h, γ )

N

)= − ln |γ | − 4 ln 2. (25)

Similarly, we are safe to conclude from Eqs. (23) and (25)that the critical exponent ν = 1 according to the ansatz (1).In addition, the normalized FS χ (γ )(h, γ ) with respect to γ

in the TDL is shown in Fig. 3. This result also demonstratesconvincingly that FS can be used to characterize the quantumcritical behavior.

IV. CONCLUSION

We have derived several closed-form expressions for thefidelity susceptibility of the anisotropic XY model in the

0.20.0-2

0.5

-1 0.00

1.0

1 -0.22

FIG. 3. Normalized fidelity susceptibility χ (γ ) with respect to γ .It has a sharp extremum at the segment |h| < 1 and γ = 0 wherecontinuous QPTs with ν = 1 occur.

transverse field after a symmetry analysis. The FS for thespecial case γ = 1 can be recovered and is consistent withthe results obtained by Damski and Rams in light of severalelegant summation formulas [40–42] of the transverse-fieldIsing model. Our method is easy to follow and promises to beuseful to other exactly solvable models, such as the XY modelwith bond-alternated interaction [54] or a staggered field [55].The correlation length critical exponent ν = 1 is calculatedanalytically according to the standard finite-size scaling ansatz.

ACKNOWLEDGMENTS

We thank W.-L. You for some useful discussions. Q.L. wasfinancially supported by the Outstanding Innovative TalentsCultivation Funded Programs 2017 of Renmin University ofChina. J.Z. was supported by the National Natural ScienceFoundation of China (Grant No. 11474029). X.W. was sup-ported by the National Program on Key Research Project(Grant No. 2016YFA0300501) and by the National NaturalScience Foundation of China (Grant No. 11574200).

APPENDIX A: CLOSED-FORM EXPRESSION

Lemma 1. For any nonzero (complex) variable α, the closed-form of the summation

L(α) =N∑

n=1

1

α + cn

, cn = cos(2n − 1)π

N (A1)

can be expressed as

L(α) =

⎧⎪⎨⎪⎩

N√α2−1

βN −1βN +1 , |α| > 1

sgn(α)N2, |α| → 1N√

1−α2 tan[N cos−1(α)], |α| < 1,

(A2)

where N = 2N , β = α + √α2 − 1, and sgn(·) is the sign

function.Proof. The function L(α) is an odd function with respect to

the variable α, which can be verified immediately by noting thesymmetry relation cn = −cN+1−n. The monotonic behavior ofthe function depends highly on α and we will consider the case|α| > 1 first.

022106-4208


FIG. 4. Illustration of the function L(α) for (a) the even case N =6 and (b) the odd case N = 7. In both cases, the solid blue lines arethe curve for L(α), while the dash-dotted black lines represent thediscontinuous points.

To begin with, let ζ = e(π/N )i be the primitive 2N th rootof unity and

� = {ζ 2n−1| − N < n � N} = {ζ±(2n−1)|1 � n � N}be the set of roots for the polynomial zN + 1 = 0. Therefore,for any allowed integer n, cn = ω+ω−1

2 with ω = ωn = ζ 2n−1.Employing the logarithmic derivative with respect to z of theequality ∏

ω∈�

(z + ω) =∏ω∈�

(z − ω) = zN + 1,

we then arrive at the important summation identity

∑ω∈�

z

ω + z= N zN

zN + 1. (A3)

Actually, it is natural for us to define the new variable β =α + √

α2 − 1 so that α = β+β−1

2 . In light of Eq. (A3) we have

L(α) = 1

2

N∑n=−N+1

1

α + cn

= 1

2

∑ω∈�

1β+β−1

2 + ω+ω−1

2

= 1

β − β−1

∑ω∈�

(β

ω + β− β−1

ω + β−1

)

= N√α2 − 1

βN − 1

βN + 1. (A4)

If the hyperbolic functions are involved, Eq. (A4) is equivalentto

L(α) = Nsgn(α)√α2 − 1

tanh[N cosh−1(α)]. (A5)

The exact expression for the case |α| < 1 can be obtainedfrom the former in the spirit of analytic continuation or simplyby the substitution β = α + i

√1 − α2. In this situation we

have

L(α) = N√1 − α2

tan[N cos−1(α)]. (A6)

It should be noted that the point α = 0 is out of our consid-eration in general since whether this point make sense or notdepends on the odevity of N . It is zero when N is even anddiverging for the odd case (see Fig. 4). More generally, there areN discontinuous points in the range of |α| < 1, namely, αdisc =cos 2m+1

N π , m = 0, 1, 2, . . . , N − 1. Therefore, the functionL(α) is continuous at the points |α| = 1 and its values thereofare equal to sgn(α)N2.

Altogether, we finish the full processes of the proof. Weend this Appendix with Fig. 4, which presents the curvature offunction L(α) for even and odd N . �

APPENDIX B: DERIVATIVE RELATION

Lemma 2. For any nonzero (complex) α, the first derivativeof the function L(α) defined in Eq. (A1) is

L′(α) = ∂L(α)

∂α

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

4β2

(β2−1)2

[ N 2βN

(βN +1)2 − N2

(β2+1)(βN −1)(β2−1)(βN +1)

], |α| > 1

−sgn(α)N2(16N2+9)24 , |α| → 1

− L(α)√1−α2

[ Nsin(N cos−1(α)) − α√

1−α2

], |α| < 1,

(B1)

where N = 2N , β = α + √α2 − 1, and sgn(·) is the sign

function.

APPENDIX C: QUANTUM METRIC TENSOR

The quantum metric tensor [47] is a concept stemming fromdifferential geometry and information theory. It describes theabsolute value of the overlap amplitude between neighboringground states. Therefore, like the fidelity susceptibility, themetric also plays a vital role in understanding the quantumphase transition.

For the anisotropic XY model [see Eq. (2)] in the (h, γ )parameter space, the quantum metric tensor is defined as gσσ =14

∑Nn=1(∂θk/∂μσ )(∂θk/∂μσ ), where μ1,2 = h, γ [47]. So the

solely off-diagonal element of the tensor is

χ (hγ )(h, γ ) = γ

4

∑k>0

sin2 k(cos k − h)

[(cos k − h)2 + γ 2 sin2 k]2. (C1)

The method for the calculation of Eq. (C1) has beenexplained in the main text. The detail is omitted here and ispresented in the SM [52]; the result is

χ (hγ )(h, γ ) = sgn(γ )

16(h2 + γ 2 − 1)

∑υ=±

F (hγ )υ , (C2)

where

F (hγ )υ = g2

υ − 1

2pυgυ

[ N 2gNυ(

gNυ + 1

)2 + N2

(pυ

|γ | + Cυ

)gN

υ − 1

gNυ + 1

].

(C3)

Also, the gυ , pυ , and Cυ are defined in Eqs. (9), (10), and (12),respectively.

022106-5209


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Lattice distortion effects on the frustrated spin-1 triangular-antiferromagnetA3NiNb2O9 (A = Ba, Sr, and Ca)

Z. Lu,1 L. Ge,2 G. Wang,3 M. Russina,1 G. Günther,1 C. R. dela Cruz,4 R. Sinclair,5 H. D. Zhou,5 and J. Ma3,6,*

1Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin, Germany2School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

3Key Laboratory of Artificial Structures and Quantum Control, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China

4Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA5Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

6Collaborative Innovation Center of Advanced Microstructures, Nanjing, Jiangsu 210093, China

(Received 11 May 2018; published 11 September 2018)

In geometrically frustrated materials with low-dimensional and small spin moment, the quantum fluctuationcan interfere with the complicated interplay of the spin, electron, lattice, and orbital interactions, and hostexotic ground states such as the nematic spin state and chiral liquid phase. While the quantum phases of theone-dimensional chain and S = 1

2 two-dimensional triangular-lattice antiferromagnet (TLAF) have been morethoroughly investigated by both theorists and experimentalists, the work on the S = 1 TLAF has been limited. Weinduced the lattice distortion into the TLAFs A3NiNb2O9 (A = Ba, Sr, and Ca) with S(Ni2+) = 1, and appliedthermodynamic, magnetic, and neutron scattering measurements. Although A3NiNb2O9 kept the noncollinear120◦ antiferromagnetic phase as the ground state, the Ni2+ lattice changed from an equilateral triangle (A = Ba)into an isosceles triangle (A = Sr and Ca). The inelastic neutron scattering data were simulated by the linearspin-wave theory, and the competition between the single-ion anisotropy and the exchange anisotropy from thedistorted lattice are discussed.

DOI: 10.1103/PhysRevB.98.094412

I. INTRODUCTION

In the geometrically frustrated system, the complicatedinteraction(s) between electron, phonon, spin, and orbitalcan lead to degenerate ground states, which introduce exoticproperties and have attracted a lot of attention over the pastdecades [1–3]. Meanwhile, these degenerate states wouldeasily be held by the symmetrical inconsistency and be de-stroyed by significant quantum fluctuations, not only inducedby the complicated interactions among low dimensionality,geometrical frustration, small spin, and the applied magneticfield, but also modifying the classical Heisenberg model [4,5].The triangular-lattice antiferromagnet (TLAF), one of thesimplest frustrated two-dimensional (2D) materials, has beensuggested to be strongly influenced by the strong quantumspin fluctuations with small effective spin (S = 1

2 or 1) andexhibits a rich variety of interesting physics [6–8]. A strikingexample of these quantum phenomena is the transitionfrom a noncollinear 120◦ spin configuration at 0 T intoa fractional-magnet lattice under a finite range of appliedmagnetic field, such as a collinear up-up-down (uud) phasecorresponding to a magnetization plateau with one-third of itssaturation value [9–13].

Recently, theoretical research indicated the uud state asa commensurate analog of the incommensurate spin densitywave which was predicted and observed for frustrated one-

*Corresponding author: [email protected]

dimensional spin-1 chains and the S = 12 TLAF [14,15],

and suggested the possibility of exotic magnetic excitations.While there has been a great deal of theoretical activity inthe domain, a full consensus on the origin of the uud state(even the ground magnetic state, noncollinear 120◦ at zerofield) has been limited. This is true as well for the state-of-the-art experimental investigation of its spin excitations dueto the lack of triangular-lattice materials. Although the latticedistortion has been believed to influence the quantum effectsin the systems of kagome, square, and triangular motifs viathe antisymmetric Dzyaloshinskii-Moriya (DM) interactionand requires that the dynamic models include the item ofthe lattice contribution [16–20], how the ground state of thetriangular lattice originates from the lattice is still unclear. Ifthe distortion effect were gradually introduced in the system,it would be insightful to obtain the specific physical propertiesas the related lattice is changed.

Ba3NiNb2O9 was one of the first-studied equilateralTLAFs with highly symmetric crystal structure, which wasfree from the antisymmetric DM interaction such that a simpleHamiltonian model was expected to describe the physics ofthis material. The crystal structure is trigonal, P -3m1: (i) thecorner oxygen is shared by the NiO6 and NbO6 octahedra; (ii)Ni ions occupy the 1b Wyckoff sites and form the triangularlattice in the ab plane; (iii) Ba2+ ions build up the unitcell frame and split NiO6 octahedra. As the ab planes areseparated by double nonmagnetic layers consisting of theNb2O11 double octahedra and Ba2+ ions, it is expected thatthe interlayer exchange interaction would be much smallercompared to the nearest-neighbor (NN) exchange interaction




Z. LU et al. PHYSICAL REVIEW B 98, 094412 (2018)

FIG. 1. (a) The neutron powder diffraction (NPD) pattern of polycrystalline Ca3NiNb2O9 performed at 0.3 K. The refinement was doneusing the monoclinic space group, P 121/c1. (b) The NPD patterns taken at 0.3 and 8 K. The (1/3, 1/3, 1/2) and (4/3, 1/3, 1/2) magnetic Braggpeaks at 0.3 K are given based on the hexagonal P -3m1 space group. (c) Temperature dependence of the order parameter in Ca3NiNb2O9.Lines are fits to one (dashed) or two (solid) order parameters. The latter fit the data better suggesting a two-step transition consistent withwhat is observed in the bulk magnetization measurements discussed in the next section. (d) The schematic crystal structure of A3NiNb2O9.(e) Triangle lattice of Ni2+ ions in ab plane of A3NiNb2O9.

in the ab plane of Ni2+ ions and the compound could betreated or approximately treated as a 2D system.

In this paper, we focus on the ground state of the non-collinear 120◦ spin structure reported for the triangular-latticeantiferromagnets A3NiNb2O9 (A = Ba, Sr, and Ca). Sincethe magnetic exchange energies are sensitive to the bondlengths/angles of Ni and O ions, it is interesting to com-pare the family of A3NiNb2O9 (A = Ba, Sr, and Ca) as thestructures are gradually distorted. As Sr2+ and Ca2+ ionsare smaller than the Ba2+ ion, the effect from the latticedistortion is introduced gradually and the A-site effect onthe exchange interactions is observed. In addition, a strongcoupling between the successive magnetic phase transitions

and the ferroelectricity has been observed in Ba3NiNb2O9

and Sr3NiNb2O9 by magnetic and electric bulk measurements[10,21]. Our study on the lattice effect on the triangular latticeshould not only discuss the quantum effect in TLAFs, but alsobe beneficial for the development of multiferroicity theory inlow-dimensional frustrated materials [22].

II. EXPERIMENTAL

Polycrystalline A3NiNb2O9 (A = Ba, Sr, and Ca) sampleswere prepared with the solid state reaction method. Stoichio-metric mixtures of BaCO3/SrCO3/CaCO3, NiO, and Nb2O5

were ground together, and calcined in air at 1230 ◦C for

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LATTICE DISTORTION EFFECTS ON THE FRUSTRATED … PHYSICAL REVIEW B 98, 094412 (2018)

24 h. A commercial SQUID magnetometer (MPMS, Quan-tum Design) and a high-field vibrating sample magnetometer(VSM) were employed to measure the dc magnetization as afunction of temperature and magnetic field. The specific heatwas measured with a physical property measurement system(PPMS, Quantum Design) in two steps. First, the backgroundspecific heat was measured by an empty pucker with N-grease;then, a dense and solid thin plate of the A3NiNb2O9 samplewith a total mass around 10 mg was measured. After that,by subtracting the background specific heat from this totalspecific heat, we obtained the specific heat of the sample.

The neutron powder diffraction (NPD) measurementsdown to 0.3 K were performed using the HB-2A powderdiffractometer at the High Flux Isotope Reactor (HFIR), OakRidge National Laboratory (ORNL), with a 3He insert system.About 5 g of pelletized powder for each sample was loaded ina vanadium can with He exchange gas. Data were collectedat selected temperatures using two different wavelengths λ =1.538 and 2.406 A. The collimation was set as open-open-6′.The shorter wavelength was used to investigate the crystalstructures with the higher Q coverage, while the longer wave-length was important for investigating the magnetic structuresof the material with the lower Q coverage. The Rietveldrefinements on the diffraction data were performed using theprogram FullProf [23].

The inelastic neutron scattering (INS) of polycrystallineA3NiNb2O9 (A = Ba, Sr, and Ca) samples was carried outon the recently upgraded cold-neutron direct-geometry time-of-flight spectrometer NEAT at Helmholtz-Zentrum Berlin[24]. The spectrometer allows us to cover a wide range ofenergy transfers hω and scattering angles, thereby allowingdetermination of a large swath of the scattering intensityS(Q,ω) as a function of momentum transfer hQ and energytransfer hω, where Q is the scattering vector. Around 4 gof each sample was packed in aluminum cans filled with Heexchange gas. Each scan was counted around 6 hours with theincident neutron energy Ei = 3.272 meV.

III. RESULTS

A. Neutron powder diffraction

As shown in Figs. 1(a) and 1(b), low-temperature NPD wasemployed at 8 and 0.3 K to identify the magnetic structureof polycrystalline Ca3NiNb2O9. The space group could beindexed using a monoclinic unit cell, space group P 121/c1(No. 14), with one Ni atom at the 2a (0, 0, 0) site and another

one at the 2d site (1/2, 1/2, 1/2), and the other Ca, Nb, andO atoms at the 8f (x, y, z) site. By comparing the Rietveldrefinement results of Ca3NiNb2O9 patterns at 0.3 and 20 K(20 K data are not presented here), it is suggested that nocrystal structure (P 121/c1) transition was observed down to0.3 K. The (1/3, 1/3, 1/2) and (4/3, 1/3, 1/2) magnetic Braggpeaks were observed at 0.3 K and absent at 8 K with Q ∼0.85 A

−1and 2.0 A

−1, which is consistent with the commen-

surate magnetic wave vector qm = [1/3, 1/3, 1/2] [Fig. 1(b)],suggesting a similar 120◦ spin structure to Ba3NiNb2O9 andSr3NiNb2O9 at 0.3 K [10,21]. The magnetic phase transitionwas displayed by the temperature dependence of the orderparameter (OP), Fig. 1(c), measured as the temperature depen-dence of the intensity of the [1/3, 1/3, 1/2] magnetic peak.

Figure 1(d) presents the schematic crystal structure ofA3NiNb2O9. For the triple-perovskite system A3NiNb2O9,the Ni-triangular layers were split by the nonmagnetic corner-shared Nb2O11 perovskite. At the substitution of the smallerions into the Ba site with Ca2+, the lattice of Ca3NiNb2O9

was distorted from the hexagonal (P -32/m1) to monoclinic(P 121/c1) space group as Sr3NiNb2O9. Therefore, the equi-lateral Ni triangle changed to an isosceles triangle by replac-ing the Ba2+ ions with Sr2+ and Ca2+ ions, Fig. 1(e). The Bacompound exhibited an equilateral Ni triangle with bonds of5.7509 A. For the Sr compound, one longer bond of 5.6387A and two shorter bonds of 5.6339 A were obtained; for theCa compound, one shorter bond of 5.4470 A and two longerbonds of 5.5056 A were observed.

The tolerance factor, t , suggested by Goldschmidt has beenemployed to describe the stability of perovskite phases: thedeviation of t from t = 1 can be applied to estimate theinternal strain in perovskites and oxygen octahedral tilt dueto the misfit of the A and B site ionic radii [25]. The definitionof t is given by

t = RA + RO√2(〈RB〉 + RO)

,

where 〈RB〉 is the average ionic radii for the B-site ions [26],and RA and RO are the A- and O-site ionic radii, respectively.

The crystallographic properties of A3NiNb2O9 (A = Ba,Sr, and Ca) are given in Table I. For Ba3NiNb2O9, thetolerance factor is bigger than 1 while it is smaller than 1for Sr3NiNb2O9. However, the t deviations from t = 1 aresimilar with ∼0.030, which suggests a similar distortion fromthe ideal perovskite for them. Therefore, the crystal structure

TABLE I. Crystallographic properties of A3NiNb2O9 (A = Ba, Sr, and Ca).

Parameters Ba3NiNb2O9 [10] Sr3NiNb2O9 [21] Ca3NiNb2O9

Space group P -32/m1 P 121/c1 P 121/c1

a = 9.7549(5) A a = 9.5695(8) Aa = b = 5.7509(5) A

Lattice parameters b = 5.6387(1) A b = 5.4472(2) Ac = 7.0343(8) A

c = 16.9194(8) A c = 16.7876(3) Aα = β = 90◦ α = γ = 90◦ α = γ = 90◦

Interaxial anglesγ = 120◦ β = 125.066(4)◦ β = 125.718(3)◦

t 1.031 0.972 0.938

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TABLE II. Bond lengths of A3NiNb2O9 (A = Ba, Sr, and Ca).

Bonds Ba3NiNb2O9 [10] Sr3NiNb2O9 [21] Ca3NiNb2O9

〈Ba1/Sr1/Ca1-O〉 2.8695(1) 2.7829(4) 2.7345(3)〈Ba2/Sr2/Ca2-O〉 2.8850(9) 2.8243(4) 2.8689(2)〈Sr3/Ca3-O〉 – 2.8962(6) 2.8560(9)〈Ni1-O〉 2.0901(3) 1.9416(9) 2.1165(9)〈Ni2-O〉 – 2.0449(5) 2.0653(4)〈Ni-O〉 – 1.9933(2) 2.0909(6)〈Nb1-O〉 1.9416(7) 2.0004(7) 2.0005(5)〈Nb2-O〉 2.0662(2) 2.0731(1) 2.0625(8)〈Nb-O〉 2.0039(5) 2.0367(9) 2.0315(6)

distorted from hexagonal to monoclinic structure is expectedto result in a smaller t . Moreover, the angles of the equilateraltriangle in Ba3NiNb2O9 are all 60◦; for Sr3NiNb2O9, they are60.090◦, 59.955◦, and 59.955◦; and for Ca3NiNb2O9, they are58.930◦, 60.535◦, and 60.535◦.

Table II shows the bond lengths of A3NiNb2O9 (A = Ba,Sr, and Ca). Considering the bond length, the six Ni-O bondshad the same lengths of 2.090 A in Ba3NiNb2O9. However,the smaller ionic radius of Nb5+, 0.64 A, compared to 0.69 Afor Ni2+ in octahedral coordination resulted in breathingdistortion (extension or contraction of the Nb-O bond lengths)of the NbO6 octahedra with two different lengths of Nb-Obonds, 1.942 and 2.066 A, respectively. In Sr3NiNb2O9 andCa3NiNb2O9, the A-site ion is too small to hold its oc-cupied cuboctahedral site, and then a combination of thebreathing and tilting distortions both occurred, which canreduce the volume of the interstice and thereby improve thestructural stability. With decreasing the size of the A-site ion,for Sr3NiNb2O9 the average bond lengths of A-O and B-Odecreased, but for Ca3NiNb2O9 the smallest A-site ion the〈A-O〉 and 〈B-O〉 are bigger than those for Sr3NiNb2O9 dueto stronger lattice distortion.

In Fig. 1(c), the fitting of OP was assumed with one or twoorder parameters, shown as dashed and solid lines, respec-tively. TN1 = 4.90 K was obtained with the single-OP fittingwith βone ≈ 0.39. The two-OP fitting gave TN1 = 4.85 K andTN2 = 4.05 K and yielded critical exponents βtwo-1 ≈ 0.34and βtwo-2 ≈ 0.36, respectively, which were a better descrip-tion of the data. It is worth noting that the obtained values

are comparable with those of 2-vector XY (β = 0.35) or3-vector Heisenberg (β = 0.36) models. The latter fit the databetter suggesting a two-step transition consistent with what isobserved in the bulk magnetization measurements discussedin the next section. However, the effect of averaging in powderdiffraction and the limited Q range of the instruments can onlydetermine the 120◦ spin direction projected in the ab planeand cannot determine whether the systems form a coplanarmagnetic structure or not [27]; more details can be analyzedby the dynamics.

B. Magnetic susceptibility and heat capacity

Figure 2(a) shows the specific heat data of the Ca3NiNb2O9

at different fields. A peak was observed around 4.2 K in thezero-field curve. According to the literature on Sr3NiNb2O9

[21] and Ba3NiNb2O9 [10], this peak corresponds to thelong-range magnetic order transition. As the magnetic fieldincreases, the peaks shift to lower temperature and becomebroader. In Fig. 2(b), Cp/T vs T and dCp/dT vs T presenttwo anomalies as TN2 and TN1, respectively, which are similarto Sr3NiNb2O9 and should be related to the distorted lattice.

The temperature dependence of the dc susceptibility ofCa3NiNb2O9 with different magnetic fields is shown inFig. 3(a). As the field increases, the magnetic transitiontemperatures decrease and become broader similarly to thebehavior of the heat capacity anomaly. The data at 0.02 Tfollow the Curie-Weiss law at high temperature (inset). Fittingthe χ (T ) data from 100–300 K with a linear Curie-Weiss law,

FIG. 2. (a) Temperature dependence of the specific heat for the polycrystalline Ca3NiNb2O9 sample at different magnetic fields. The insetis the specific heat at zero field and fit. (b) Temperature dependence of Cp/T and dCp/dT around TN1 and TN2 at zero magnetic field.

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FIG. 3. (a) Temperature dependence of susceptibility around the phase transition at different fields. The inset is temperature dependenceof χ and 1/χ for the polycrystalline Ca3NiNb2O9 sample at H = 0.02 T. The green line is the linear fit of 1/χ from 100 K to 300 K. (b) Thetemperature dependence of d (χT )/dT and χ from 2.5 to 7 K.

we obtain −28 K for the Curie-Weiss temperature (θCW) sug-gesting the dominance of antiferromagnetic (AFM) exchangeinteractions. For Ca3NiNb2O9, S = 1, the effective moment(μeff ) is calculated as 3.16 μB/Ni with a corresponding Landég factor of 2.23, based on μeff = g

√S(S + 1)μB, which is

comparable with those of Ba3NiNb2O9, Sr3NiNb2O9, andother compounds with Ni2+ ions [9,10,21,28,29].

The spin frustration ratio f is defined as the ratio of theabsolute value of the Curie-Weiss temperature (θCW) to tran-sition temperature TN; thus f = |θCW|/TN. For Sr3NiNb2O9

and Ca3NiNb2O9 samples, we chose the higher transitiontemperature, TN1, for calculation. As shown in Table III,the frustration ratio f increases with decreasing the A-siteion radii, and Ca3NiNb2O9 has the strongest frustration andcompeting interactions.

From Fig. 3(b), the χ data exhibit a clear peak at ∼4.8 K,which is consistent with TN1 from the dCp/dT data. Anotherpeak can be observed from the d (χT )/dT at ∼4.2 K, whichis related to a long-range magnetic ordering and correspondsto TN2 defined from the specific heat, Fig. 2(b). It is worthnoting for the Ca compound that the two-step phase transitionis not so strong and sensitive to different measurementtechniques to some extent. For example, the TN2 transitionis more sensitive to thermal measurement while the TN1

transition can be detected more easily by dc susceptibility.Figure 4(a) compares the temperature dependence of Cp/T

for the polycrystalline A3NiNb2O9 (A = Ba, Sr, and Ca)samples at zero magnetic field. The Ba compound exhibitsthe sharpest peak with the lambda-type feature in specificheat data, which results from its single-step phase transitionat zero field. As shown in Fig. 4(b), at low magnetic fields, the

TABLE III. Parameters extracted from the specific heat and dcsusceptibility measurements for A3NiNb2O9 (A = Ba, Sr, and Ca).

Parameters Ba3NiNb2O9 [10] Sr3NiNb2O9 [21] Ca3NiNb2O9

TN1 (K) 4.9 5.5 4.8TN2 (K) – 5.1 4.2θCW (K) −16.4 −21.5 −28μeff (μB/Ni) 3.15 3.21 3.16g 2.23 2.27 2.23f 3.4 3.9 5.83

temperature dependence of d (χT )/dT for the polycrystallineBa3NiNb2O9 sample also shows a sharp peak at TN, while thetwo nearby TN peaks of the Sr and Ca compounds indicate atwo-step transition for them.

Based on the mean-field theory, the Heisenberg Hamil-tonian, J

∑〈i, j 〉(SiSj ), can be approximately related to the

exchange J and the Curie-Weiss temperature θCW, J =−3kBθCW/zS(S + 1). For A3NiNb2O9 (A = Ba, Sr, and Ca),each Ni2+ ion is surrounded by 6 NNs (z = 6) with the inter-action J in the triangular lattice; hence, J/kB = −3θCW/12,and JBa/kB, JSr/kB, and JCa/kB are ∼4.1 K (0.35 meV),5.4 K (0.46 meV), and 7.0 K (0.60 meV), respectively. There-fore, the Sr and Ca compounds exhibit larger J than the Bacompound according to the calculated results.

The magnetic entropy of the A3NiNb2O9 is shown in theinset of Fig. 4(a). The total magnetic entropy suggests that thedegree of disorder of these three compounds is Ca > Sr > Ba,which is consistent with the behavior of the tolerance factor t

in Table I. To extract the magnetic contribution from the totalheat capacity, an equation consisting of the linear combinationof one Debye and several Einstein terms is used to estimate thelattice specific heat,

CL(T ) = CD

[9R

(T

θD

)3 ∫ θD/T

0

x4ex

(ex − 1)2 dx

]

+∑

i

CEi

⎡⎣3R

(θEi

T

)2 exp( θEi

T

)[exp

( θEi

T

) − 1]2

⎤⎦,

where R is the universal gas constant, and θD, θE are the Debyeand Einstein temperatures, respectively. CD , CEi

are the rela-tive weights of the acoustic and the optical phonon contribu-tion of the heat capacity. There are 15 atoms per formula inour system. The best fit to the data from 30 to 250 K resultin one Debye and two Einstein terms with the proportion 1 :5 : 9 for the CD : CE1 : CE2 , and θD = 160 K, θE1 = 255 K,θE2 = 540 K. The magnetic component of the specific heatCm is obtained after subtracting the lattice contribution fromthe data. The magnetic entropy Sm is obtained by integratingCmT throughout the range of temperatures measured [inset ofFig. 4(a)]. The Ni2+ has a spin-1 in this compound. The totalentropy saturated at about 79.1% of the Rln3. The entropy

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FIG. 4. (a) The comparison of the specific heat for A3NiNb2O9 at zero magnetic field. The inset is the magnetic entropy of A3NiNb2O9.(b) The comparison of the d (χT )/dT for A3NiNb2O9 at very low magnetic fields.

loss could be due to the quantum fluctuation in this spinfrustrated system. We also measured the heat capacity of theSr3NiNb2O9 and Ba3NiNb2O9, and then processed the datawith the same method of Ca3NiNb2O9. The total magneticentropy was about 73.9% and 88.7% of Rln3 for Sr3NiNb2O9

and Ba3NiNb2O9, respectively. The total magnetic entropydecreased as Ba > Ca > Sr, which indicated the smallestmagnetic entropy loss of Ba compound due to its smallestlattice distortion. However, for the Sr and Ca compounds, themagnetic entropy losses were dominated by the competitioneffect between lattice distortion and quantum fluctuation.

C. Inelastic neutron scattering

To explore the spin dynamics of A3NiNb2O9 (A = Ba, Sr,and Ca) in detail, we measured the magnetic excitations fromINS spectra. Figures 5(a)–5(c) show the powder spectrumof A3NiNb2O9 (A = Ba, Sr, and Ca) at 1.5 K, below TN .Similarly to the S = 1

2 TLAF compound, Ba3CoSb2O9, bothgapped and gapless modes are observed in the magnetic DOS[27]; meanwhile, unlike Ba3CoSb2O9, there is no obviouscontinuum observed at higher energy, which might be due to

the larger spin moment (S = 1) and the weak signal from thepowder average effect.

The low-energy magnon bandwidth of Ba3NiNb2O9 isaround 1.0 meV, which is lower than those of Sr3NiNb2O9

and Ca3NiNb2O9 (∼1.25 meV). The magnetic DOS of theA3NiNb2O9 (A = Ba, Sr, and Ca) system showed min-

ima around Q ≈ 0.85 A−1

, which notably corresponds tocommensurate magnetic wave vector qm = [1/3, 1/3, 1/2].Moreover, the gaps of this branch at qm = [1/3, 1/3, 1/2]for Ba3NiNb2O9 and Ca3NiNb2O9 were around 0.6 meV,while Sr3NiNb2O9 exhibited a similar gap of around 0.8 meV,Figs. 5(a)–(5c). The magnetic signals were momentum-

dependent due to stronger ridges of intensity at Q ≈ 0.85 A−1

than at Q ≈ 1.5 A−1

and Q ≈ 2.0 A−1

.The INS results are simulated by the linear spin wave

(LSW) theory based on the quasi-2D XXZ Hamiltonian ona vertically stacked triangular lattice, Figs. 5(d)–5(f). Theappropriate Heisenberg Hamiltonian is

H = J

layer∑〈i, j 〉

(Sx

i Sxj + S

y

i Sy

j + �Szi S

zj

) + J ′interlayer∑

〈l, m〉Sl · Sm,

FIG. 5. Powder spectra measured experimentally at 1.5 K on spectrometer NEAT: (a) Ba3NiNb2O9, (b) Ca3NiNb2O9, (c) Sr3NiNb2O9

powder spectra at 1.5 K. The powder-average LSW approximation of (d) Ba3NiNb2O9, (e) Ca3NiNb2O9, (f) Sr3NiNb2O9.

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FIG. 6. Comparison between experiment (red dots) and LSWT (the black line) simulation shown as energy-integrated within 0.2 � E �1.5 meV for (a) Ba3NiNb2O9, (b) Ca3NiNb2O9, (c) Sr3NiNb2O9.

where J and J ′ (J, J ′ > 0) are the intra- and interlayer NNantiferromagnetic exchange energies, respectively. � (0 <

� � 1) is the easy-plane exchange anisotropy from the same120◦ structure as in the Heisenberg case. Although due tothe isosceles triangular geometry in the Sr and Ca com-pounds, it is more intuitive to use two different exchangesfor the intralayer coupling; the [1/3, 1/3, 1/2] wave vectorrequires them to be the same [30]. Table IV is a summaryof the parameters in the Hamiltonian according to the LSWapproximation.

The E-integrated scans are obtained in Fig. 6; INS mea-surements and the powder-averaged LSW approximation havea good agreement: the data of the Ca compound can beexplained as the most Heisenberg-like feature, and the bestfit is obtained.

In the A3NiNb2O9 system, the Sr compound exhibits thelargest intralayer NN exchange (J = 0.36 meV), while theCa compound obtains the largest interlayer exchange (J ′ =0.3J ). Hence, the intra- and interlayer NN exchange sat-isfies these relations, JBa < JCa < JSr and J ′

Ba < J ′Sr < J ′

Ca,respectively. And the mean-field theory does not work forSr3NiNb2O9 and Ca3NiNb2O9, as shown in Table IV. Thebiggest gap for the Sr compound (∼0.8 meV) suggests a largerintraplane NN exchange anisotropy than those of ∼0.6 meVfor the Ba and Ca compounds. When the Ba2+ ion is sub-stituted with the smaller Sr2+ ion, the Ni2+ triangle changesfrom an equilateral triangle to an isosceles triangle, as shownin Fig. 1(e). The smaller A-site ions result in a smaller trianglelattice and larger intraplane exchange anisotropy. When thesmallest Ca2+ ion is on the A site, the lattice distortion isenhanced further, and the interplane interaction was strongenough to affect the intraplane anisotropy, which presentsthe weak quantum effect. From the NPD data, the distancesbetween the triangular ab planes for the Ba, Sr, and Cacompounds are 7.034 A, 6.924 A, and 6.815 A, respectively.

TABLE IV. Summary of the exchange parameters in the Hamil-tonian according to the LSW approximation.

Parameters Ba3NiNb2O9 Ca3NiNb2O9 Sr3NiNb2O9

J MFT (meV) 0.35 0.60 0.46J (meV) 0.30(1) 0.31(1) 0.36(1)J ′/J 0.05(5) 0.30(5) 0.05(5)� 0.70(5) 0.95(5) 0.75(5)

And the ratios of the average intraplane Ni-Ni bond lengthsto these interlayer distances are ∼0.81 for the Ba and Srcompounds, and ∼0.80 for the Ca one. Therefore, the shortestinterlayer distances and the smallest ratios of bond lengthsto plane distances for the Ca compound lead to very stronginterplane coupling with the largest J ′/J = 0.30 and reducethe J slightly to 0.31 meV (<JSr = 0.36 meV). In addition, theCa compound shows a more obvious gapless feature than theSr compound from the fitting. The best-fitting parameters ofeasy-plane exchange anisotropy are � = 0.95 and � = 0.75for Ca3NiNb2O9 and Sr3NiNb2O9, respectively, as shownin Table IV. This indicates that the Ca compound has lesseasy-plane anisotropy compared to the Sr one due to itsmore dispersive feature with a similar bandwidth to the Srcompound.

The competition of the easy-plane exchange anisotropy �

and J ′/J is treated as the dominant effect in the A3NiNb2O9

system. Moreover, we also consider the easy-axis anisotropy,single-ion anisotropy, and DM effects in the system by theLSW approximation: (1) Easy-axis anisotropy can lift theentire band up, create a gap, and modify the ordering wavevector. (2) If single-ion anisotropy exists, the out-of-planecanting would be bigger and towards the collinear struc-ture. (3) DM interactions that stabilize the [1/3, 1/3, 1/2]structure have to be perpendicular to the easy plane and notoverwhelmingly large. Within the LSW approximation, suchDM terms would have very similar impact on the excitationsand therefore are not considered here for simplicity. Sincethe Hamiltonian still has continuous symmetry in the easyplane, and the ground state can break it spontaneously, nomatter how weak the powder spectra are, they will always begapless.

IV. DISCUSSION

To understand the intralayer antiferromagnetic interactionof the Ni2+ ions, we examined the superexchange interactionin the structure using the Goodenough-Kanamori theoreticalframework [31], which discusses the relation betweenthe symmetry of electron orbitals and superexchangeinteraction via a nonmagnetic anion. In Ba3NiNb2O9,two superexchange pathways for the Ni2+ spins in thesame layer are shown in Fig. 7(a), Ni2+-O2−-O2−-Ni2+and Ni2+-O2−-Nb5+-O2−-Ni2+, respectively, by sharing thecorner oxygens of the NiO6 and NbO6 octahedrons. Althoughthe first Ni2+-O2−-O2−-Ni2+ superexchange pathway could

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FIG. 7. (a) Superexchange pathways for FM Ni2+-O2−-O2−-Ni2+ and AFM Ni2+-O2−-Nb5+-O2−-Ni2+ interactions in Ba3NiNb2O9. (b)Orbital configurations of FM interactions through Ni2+-O2−-Nb5+-O2−-Ni2+ superexchange pathway in Ba3NiNb2O9. (c) Superexchangepathways in Sr3NiNb2O9. (d) Superexchange pathways in Ca3NiNb2O9.

be observed as the antiferromagnet, the connection ofO2−-O2− needs to hybridize/distort the s orbital or the otherp orbitals of O2−. For the second superexchange path ofNi2+-O2−-Nb5+-O2−-Ni2+, Fig. 7(b), the superexchangeinteraction between the spins on the dx2−y2 orbitals of theNi2+ ions are considered. From the Rietveld refinements ofneutron powder diffraction, the O2−-Nb5+-O2− bond angle isvery close to 90◦ at 91.13◦, and the Ni2+-O2−-Nb5+ is 180◦.In this case, the spin-1 on the left Ni2+ ion transferred to the2py orbital of the O2− by combining with the 2py orbital ofthe filled outermost Nb5+ 4p orbitals to form a molecularorbital while the spin-2 on the right Ni2+ ion transferred to themolecular orbital composed of the 2px orbital of the O2− andthe Nb5+ ions. According to Hund’s rule, these two spins onthe py and px orbitals must be parallel. Thus, a ferromagneticsuperexchange interaction formed between these twoNi2+ ions after these two spins transferred back to thedx2−y2 orbitals of Ni2+. Therefore, the Ni2+-O2−-O2−-Ni2+exchange had a shorter connection path and thereby a strongerorbital distortion than the one of Ni2+-O2−-Nb5+-O2−-Ni2+.For Ba3NiNb2O9, the distortion effect was not dominant,and the AFM interaction overcame the FM interaction. Asthe magnetic field was applied, the distortion effect wasstrengthened, and the FM interaction could be observed

in the uud phase [10]. Similar competitive superexchangeinteractions in other triangular-lattice models with layeredperovskite structures have been reported. For example, inBa3CoNb2O9, the AFM Co2+-O2−-O2−-Co2+ interactionis stronger than the FM Co2+-O2−-Nb5+-O2−-Co2+interaction, which resulted in a weak AFM interaction[32]. However, in other triangular-magnet systems, such asAAg2M (VO4)2 (A = Ba, Sr; M = Co, Ni) [33], the AFMCo2+-O2−-O2−-Co2+ interaction was weaker than the FMCo2+-O2−-V5+-O2−-Co2+ interaction, and for YCr(BO3)2,the AFM Cr3+-O2−-O2−-Cr3+ interaction was alsoweaker than the FM Cr3+-O2−-Yb3+-O2−-Cr3+ interaction[34].

As shown in Figs. 7(c) and 7(d), in Sr3NiNb2O9 andCa3NiNb2O9, the lattice distortion results in the variationof the Ni2+-O2− bond lengths. Most of the O2−-Nb5+-O2−bond angles are distorted away from the ideal value of 90◦,to as small as ∼76◦ and some as large as ∼104◦. The Ni2+,O2−, and Nb5+ ions are not on the same line. Thus, theFM interactions are influenced by the distortion and leadto stronger resultant AFM interactions. To obtain the uudphase, a higher magnetic field needs to be applied [21].Meanwhile, the polarization of Sr and Ca compounds shouldbe smaller than that of the Ba compound, and the reentrant

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signal in pyroelectric current of the Ba compound is absent[21].

Moreover, only one transition is observed in Ba3NiNb2O9,which is consistent with the easy-plane anisotropy in thisisotropic system. For Sr3NiNb2O9 and Ca3NiNb2O9 withdistorted structures, the easy-axis anisotropy is excluded bythe LSW approximation of the INS measurements while twomagnetic transitions are observed in both compounds. Anextra noncollinear magnetic phase exists at low temperature,which corresponds to the phase transition at TN1. For theS = 1 case, the high-temperature transition from the para-magnet is not clear and a different phase diagram has beenproposed by the different theories; specifically the stripephase is related to a spin reorientation in the honeycomblattice [35], and the zigzagging stripe phase in the isosce-les triangular networks by the harmonic particle interaction[36].

V. CONCLUSIONS

In summary, the lattice distortion effect on the magneticground states of spin-1 TLAFs was investigated by comparingtwo isosceles triangular lattice antiferromagnets Sr3NiNb2O9

and Ca3NiNb2O9 with an equilateral triangular compound

Ba3NiNb2O9. Although the effective magnetic moment acrossthe family is the same, the magnetic frustration of the systemincreases from the equilateral triangle to the isosceles one.Moreover, the two magnetic phase transitions were observedin A = Sr and Ca compared to one in A = Ba. However, thelattice distortion did not tune the easy-plane anisotropy ofthe Ba compound. Instead, the lattice distortion generated anextra competitive magnetic phase with strip configuration atlow temperatures for both Sr and Ca compounds and yieldedlarger interplane exchange energy with greater anisotropy forthe ground state of the spin-1 A3NiNb2O9 TLAF.

ACKNOWLEDGMENTS

J.M. and G.H.W. acknowledge support from National Sci-ence Foundation of China (Grant No. 11774223). We arethankful for support from NSF-DMR through Grant No.DMR-1350002. Research conducted at ORNL’s High FluxIsotope Reactor was sponsored by the Scientific User Facil-ities Division, Office of Basic Energy Sciences, and U.S. De-partment of Energy. The work is supported by the Starting-upFund of Shanghai Jiao Tong University (Shanghai, People’sRepublic of China) and Thousand-Youth-Talent Program ofPeople’s Republic of China.

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Understanding shear-induced sp2-to-sp3 phase transitions in glassy carbon at lowpressure using first-principles calculations

Libin Wen and Hong Sun*

School of Physics and Astronomy and Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education),Shanghai Jiao Tong University, Shanghai 200240, China

and Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

(Received 10 February 2018; revised manuscript received 11 May 2018; published 5 July 2018)

Glassy carbon is a disordered carbon allotrope consisting of sp2-hybridized bonds, which can be transformedto mixed sp2-sp3 forms with completely different mechanical and electrical properties. The transformation froman sp2-rich to sp3-rich structure under pressure has been extensively studied, while the effect of shear strain onthe transformation remains unexplored. In this work, we use a first-principles calculation method to study thephase transitions of glassy carbon under both shear strain and pressure. We find that shear strain can significantlyreduce the external pressure needed to transform the structure from sp2 rich to sp3 rich. Compared with the initialsp2-rich structure, sp3-rich structures recovered to ambient condition have a much higher mechanical strengthand lower electronic density of states near the Fermi level. Our results demonstrate that applying large shear strainis a promising approach for industrial production of superhard amorphous carbon under lower pressures.

DOI: 10.1103/PhysRevB.98.014103

I. INTRODUCTION

Glassy carbon (GC, type I) is a widely used nongraphitizingcarbon material, consisting of randomly distributed fullerenefragments [1,2]. Previous studies suggest that the chemicalbonding of GC is nearly pure sp2 hybridization. Just likeother fullerene-based materials, such as carbon nanotubes andC60, GC is a good precursor for studying the transformationmechanism between various carbon allotropes with differentfractions of sp, sp2, and sp3 bonding. Recently, high-pressureand high-temperature or high-pressure-only experiments onGC have observed significant changes in the mechanical, elec-trical, and optical properties of this material [3–7], which areaccounted for by the diverse hybridization states of carbon. Thesp3-rich amorphous carbon (or diamondlike carbon) can havehigh mechanical hardness, chemical inertness, and low frictioncoefficients, thus making it suitable to use for superhard, wear-resistant, and lubricious coatings, which have wide industrialapplications, including cutting tools, mechanical seals, criticalengine parts, and artificial medical implants [8–10].

Lin et al. [3] reported a fully sp3 bonded superhardamorphous carbon allotrope converted from GC under pressureof about 45 GPa at room temperature. The transformationswere further evidenced by the work of Yao et al. [4], althoughsimilar results were not observed by Solopova et al. [11].Yao et al. explained the contrary results by noticing thedifferent pressure conditions used in the two experimentsand suggested that the shear stress could play an importantrole in the transformation. Although the mechanism remainsunclear, shear-induced phase transitions have been reportedin various systems, including the superhard carbon fromsingle-wall carbon nanotubes [12], wurtzitic boron nitride from

*[email protected]

nanocrystalline hexagonal boron nitride [13], and anisotropicplastic flow from bcc tantalum [14]. These findings indicatethat applying shear strain is a useful method for studyingthe phase transitions under high pressure, especially foramorphous materials. In addition, shear stresses more or lessexist in diamond-anvil cells, which makes studying the effectof shear strain on phase transition necessary. For practical use,shear-driven phase transformation may be realized using ahigh-energy ball mill, which could, in principle, produce largeshear stress and shear strain. Previous experiments [15–17]using a ball mill show that complex phase transformationsoccur during the milling process and various milling effectscan be achieved by properly controlling the ball mill.

Regarding the specific role shear strain plays in the trans-formation of GC, so far only some preliminary assumptionshave been made based on experiments [4]. A reliable studyof the effect of shear strain on the transformation is stilllacking. To fill this blank, a systematic study of shear-strain-induced transformation in GC using numerical calculationmethods is required. This study will provide insight into furtherunderstanding of the transformation mechanism from GC toother carbon phases.

In this work, we investigate the effect of shear strain onthe transformation of GC under pressure using first-principlescalculations. GC structure models were generated by as-sembling multiple fullerene fragments in a cubic box andthen quenched, similar to the stochastic quenching methodreported in earlier works [18,19]. A rotational shear modelwas designed to simulate the rotational diamond-anvil cell(RDAC) for producing large and continuous shearing on thestudied structures. Our results provide theoretical evidencethat shear strain can effectively drive the transformation ofGC from sp2 to sp3 hybridization under lower pressure. Themechanical properties and electronic density of states (DOS)of the produced structures are also calculated and discussed.




LIBIN WEN AND HONG SUN PHYSICAL REVIEW B 98, 014103 (2018)

II. METHOD OF CALCULATION

Every initial GC structure had a 12.6×12.6×12.6 A3

cubicbox supercell containing about 150 C atoms, which periodi-cally repeats itself to form an infinite three-dimensional lattice.The supercell was composed of eight to nine smaller buildingfragments, which were randomly taken from the C60 and C180

fullerene structures [see Fig. 1(a)]. To obtain a carbon frag-ment, we used a small cuboid with both random dimensionsand spatial orientations to crop the fullerene structures. Foreach fragment, the resulting number of atoms varies from 5 to30, and the shape also varies, as can be seen from Fig. 1(a).Many such random structures were generated, and only thosewith a mass density of ∼ 1.5 g/cm3 and an sp3 fraction lessthan 10% were selected to better model the real GC structure.As a result of the selection, the assembled structure modelswere usually porous due to the relatively low mass density andhigh fraction of sp2. Structural optimization was conductedon the selected structures at a fixed volume to relax most ofthe stresses. Then a further optimization without geometryconfinement completely removed the residual stresses. Therelaxation process converted all one-bond (dangling) andpart of the two-bond (sp hybridization) C atoms to otherhybridization forms.

All the calculations are carried out using density functionaltheory as implemented in the VASP package [20] and adoptingthe projector augmented-wave potentials [21]. The Perdew-Burke-Ernzerhof functional for solids and surfaces (PBEsol[22]) within the gradient generalized approximation was used,which is most suitable for the calculation of GC accordingto the tests carried out by Jiang et al. [18]. The �-point-onlyk-point sampling in the Brillouin zone and a 500 eV energycutoff were used in the calculations. The convergence of energyis tested to be smaller than 1 meV per atom, with the residualstresses and forces in the fully relaxed structures less than0.01 GPa and 0.01 eV/A.

To simulate the shear process, two approaches were used inthe calculations, as illustrated in Fig. 1(b). The first approachis to apply shear strains along a fixed direction, which iswell developed and widely used in previous studies [23–28]on stress-strain relationship and phase transformations. Inthe calculation, we chose the Cartesian coordinate systemso that its z axis is normal to the selected shear crystallineplane, while its x axis is along the chosen shear direction.The quasistatic shear process and relaxed loading path weredetermined by incrementally deforming the lattice vectors ofthe supercell of the GC structure in the direction of the appliedshear strains with an incremental strain step of �εxz = 0.005.At each step, the applied shear strain εxz is fixed, whichdetermines the calculated shear stress σxz, while the otherfive independent components of the strain tensors and all theatoms inside the supercell are simultaneously relaxed until thefollowing conditions are met: (i) all three diagonal componentsof the stress tensor (σxx,σyy,σzz) equal the applied hydro-static pressure, (ii) the other two off-diagonal componentsof the stress tensor (σxy,σyz) are small (<0.01 GPa), and(iii) the force on each atom becomes negligible (<0.01 eV/A).The shape of the (deformed) supercell of the GC structure, thepositions of its atoms, and the relation between the shear stressσxz and shear strain εxz are determined completely at each

shear step by this constrained atomic relaxation, including theeffect of the hydrostatic pressure. The fixed-direction shear isstraightforward, but the shape of the supercell is progressivelydeformed and becomes highly elongated upon continuouslyincreasing shear strain above 100%, which may result ininaccurate simulation due to the limited size of the supercell.

The other approach we adopted is a simple circular shearmodel, which takes advantage of the unlimited number ofshear steps while keeping the cell basically cubic. Also, thisapproach covers more shear directions than the first approachand is thus closer to a real RDAC. It should be noted, however,that the supercell periodicity required by the VASP packageis such a strong constraint that the torque of the sample in areal RDAC cannot be simulated. The circular shear model canbe viewed as an extension of the fixed-direction shear. Theonly difference is that we need to rotate the shear direction(or, equivalently, the coordinate system) around the z axiswhile applying shear deformations. In our calculations, werotated the shear direction M times to accomplish an entirecircle, so there were M directions in total. To keep the strainstep small enough (0.5%), we used N (fixed-direction) shearsubsteps along each of the M directions during the circle, thusgiving M×N steps for a complete strain circle. The rotation ofthe shear direction is accomplished by applying the followingrotation transformation matrix R(θ ) to the lattice vectors of thesupercell of the GC structure and then continuing to shear inthe (new) x-axis direction:

R(θ ) =⎛⎝ cos(θ ) sin(θ ) 0

− sin(θ ) cos(θ ) 00 0 1

⎞⎠, θ = 2π/M.

The circular shear is carried out by first performing the fixed-direction shear in the x direction for N steps, then applying therotation R(θ ) to the sheared lattice vectors of the supercell andcontinuing to shear in the fixed (new) x direction for N steps,and applying the rotation R(θ ) and fixed-direction shear again,until a closed circle is accomplished. In our calculations, we setM = 16 and N = 20. It should be noted that the choice of theseparameters for the circular shear is not unique. But the resultswill not depend on these parameters sensitively, as we willshow below that the results from the fixed-direction shear andcircular shear are close to each other for the GC structure westudy here. For the fixed-direction shear approach, 160 shearsteps were applied, resulting in a total shear strain of 80%. Forthe circular shear approach, where 16×20 = 320 steps form acomplete circle, large and continuous shearing was achievedby applying three strain circles (960 steps) to the structures.

III. RESULTS AND DISCUSSIONS

In Fig. 1(c), we plot the sp3 fraction as a function of pressure(without shear) for three different randomly generated GCstructures, together with the mass density in the inset. The threestructures show the same trend of changes of the sp3 fractionand density under increasing pressure. It is also seen that thedensity increases smoothly with increasing pressure, while thesp3 fraction features a staircaselike increase, which indicates alag of the pressure effect on the local structure transformation.In addition, pressure as large as 40 GPa is required to obtain astructure with a 60% sp3 fraction. Next, we discuss the effect

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UNDERSTANDING SHEAR-INDUCED sp2-TO-sp3 … PHYSICAL REVIEW B 98, 014103 (2018)

FIG. 1. (a) The construction of a random glassy-carbon structure. (b) Schematic illustration of different shear approaches used in thecalculations. (c) Structure transformation of glassy carbon under hydrostatic pressure, as indicated by the change in the sp3 fraction. The insetplots the density against pressure. (d) Structure transformation driven by shear strain with constant hydrostatic pressures. The solid lines aredata from rotational shear, while the dashed lines are from fixed-direction shear. The solid squares indicate four snapshot structures chosen forlater examination, labeled SA-0, -1, -2, and -3 according to the sp3 fraction from small to large.

of shear strain on the transformation, with calculation resultsbased on structure A (SA) of GC.

Utilizing the shear models previously described, we per-formed calculations on SA under various pressures and plottedthe sp3 fraction against the shear strain, as shown in Fig. 1(d),in which each curve is composed of 160 fixed-direction or960 circular-direction shear steps with a step size of 0.5%.The starting point of each curve is the compressed structureof SA at the corresponding pressure in Fig. 1(c). For everysingle shear step, the GC structure either is strained elasticallyor undergoes a local plastic deformation, which is reflectedby the change in the sp3 fraction. It should be noted that thesp fraction is very small, on the order of ∼ 1% for structureswith an sp3 fraction more than 20%; thus the sp2 fractionapproximately equals 100% − sp3 fraction. The two shearapproaches we used gave consistent results in which the sp3

fraction increased remarkably after shear strain was appliedfor P = 5, 20, 40 GPa but not significantly for P = 0. Fromthe circular-direction shearing curves, we observed a trend ofsaturation for the sp3 fraction under larger shear strain. Thehigher the pressure is, the smaller the shear strain neededfor the saturation is. The sp3 fraction keeps fluctuating afterthe saturation, with the saturation value dependent on thepressure. The sp3 fraction of the sample saturated under agiven pressure with shearing is much higher than that underthe same pressure without shearing [see Figs. 1(c) and 1(d)].To verify the independence of the results on the size of thesupercell, we performed the same calculations with a largersupercell containing 249 atoms, which gave nearly identicalresults. In addition, we demonstrated the isotropic properties

of our GC model by applying fixed-direction shear strains inthree perpendicular shear directions on the same GC structure(SA) used in this work, which gave consistent results (the dataare not shown).

To demonstrate the effect of circular shear on the structuraldeformation of GC and its local bond configurations, we plotin Fig. 2 four snapshots at different stages of the circularshear after (constrained) shear structural relaxation at eachstep, under a hydrostatic pressure at 20 GPa. The highlightedtop plane of the supercell rotates in a way consistent withthat shown in Fig. 1(b). The effect of circular shear onlocal bond configurations is complicated. Generally, the sheardeformation will cause bond breaking and rebonding amongseveral neighboring atoms. In Figs. 2(c) and 2(d), we carefullychose the shear deformation steps where one single bondbreaking or rebonding can be seen during the circular sheardeformation, as shown in Figs. 2(e) and 2(f). The first processis the bond breaking between a pair of sp3 atoms at strain step180, induced by the applied shear strain, which transforms intoa pair of sp2 atoms at strain step 181. This happens when thedirection of the instant shear strain �εxz induces elongationof the atomic bond, which helps the pair of atoms overcomethe energy barrier between their atomic configurations. Thesecond process is the bond reforming between a pair of sp2

atoms at strain step 224, boosted by the applied shear strain,which transforms into a pair of sp3 atoms at strain step 225.This happens when the direction of the instant shear strain�εxz moves a pair of sp2 atoms on top of each other so theycan overcome the energy barrier to form a pair of sp3 bondswith the help of the applied external pressure (20 GPa).

014103-3224


FIG. 2. Evolution of the strained structures under rotational shear at 20 GPa. (a)–(d) Structures at strain steps 0, 80, 180, 224, respectively.The top planes of the cell frames are highlighted using thicker red lines. (e) The transformation of a pair of sp3 atoms (strain step 180) to a pairof sp2 atoms (strain step 181), with the instant shear direction �εxz indicated by an arrow. (f) The transformation of a pair of sp2 atoms (strainstep 224) to a pair of sp3 atoms (strain step 225).

For further examination of the stability of the transformedGC structure, we took four structure snapshots (SA-0 to SA-3)in the shear process under different pressures, as indicated bythe solid squares in Fig. 1(d). These structures can be recovered

at ambient conditions (P ≈ 0) after full structural relaxationwith a slight loss of sp3 fraction (on the order of 5%) due tothe structure reconstruction during the stress-releasing process.Three of the recovered structures are shown in Figs. 3(a)–3(c).

FIG. 3. (a)–(c) Recovered structures with different sp3 fractions obtained by full relaxation at ambient conditions (P = 0 GPa) of thesnapshot structures of amorphous carbon under shear [see Fig. 1(d)]. Three colors with different gray scale are used to represent C atoms withsp-, sp2-, and sp3-hybridized bonds. (d) Radial pair distribution function g(r) and (e) bonding angle distribution function f (θ ) of recoveredamorphous carbon structures with different sp3 fractions. The vertical lines are computed g(r) (with r < 4A) and f (θ ) for graphite and cubicdiamond crystal structures.

014103-4225


FIG. 4. Structure transformations driven by shear strains underpressures of 0 and 5 GPa, with a recovered sp3-rich GC structure(SA-3) as the initial structure. The solid lines are data from rotationalshear, while the dashed lines are from fixed-direction shear.

The sp2-rich structure (recovered SA-0) with an 11% sp3

fraction mainly consists of curved fullerene sheets, while thesp3-rich structure (recovered SA-3) with an 88% sp3 fractionconstructs a dense sp3 network, embedded with scattered sp2

C atoms. The radial distribution function (RDF) and bondingangle distribution (BAD) of the recovered SA-0 and SA-3 areplotted in Figs. 3(d) and 3(e), in comparison with those ofdiamond and graphite. The sp2-rich structure has the first RDFpeak at 1.42 A and a BAD peak around 120◦, while the sp3-richstructure has peaks at 1.54 A and 111◦, respectively. These

values of sp2-rich and sp3-rich structures are close to those ofgraphite and diamond crystal.

To understand how shear strains drive the transformation,we applied shear strain on the initially sp3 rich structure(recovered SA-3 with an initial sp3 fraction of 88%) underlow pressure P = 0 and 5 GPa, with data plotted in Fig. 4.For 5 GPa, the sp3 fraction decreases gradually and saturatesat ∼ 60%, which is close to the saturation value plotted inFig. 1(d) under the same pressure and shear strains with alow initial sp3 fraction of ∼ 13%. This finding also appliesto the 0 GPa case, with a quick drop in the sp3 fraction toa saturated value (∼ 25%) slightly higher than that (∼ 20%)in Fig. 1(d) for 0 GPa. The results suggest that the shear stressdoes not always drive the transformation in a specific direction,for example, from sp2 to sp3 hybridization. The saturated sp3

fraction of GC under shear strains is determined by the appliedpressure, independent of its initial fraction. Under 0 GPa, theformation energies of sp2 and sp3 hybridization are close[29,30]. The significant drop in the sp3 fraction from 88%to the saturation of 25% implies that shear strains create localatomic geometry that more readily allows the formation of sp2

hybridization bonds than that of sp3 after bond shear breaking.This can be explained by comparing the bonding structuresof the two types of hybridization. The orbitals of the sp3

hybridization are arranged as a three-dimensional tetrahedron(possibly distorted), which is sensitive to the anisotropic shearstrain, while sp2 are planar orbitals which can form easily inthe plane parallel to the shear plane. As the pressure increasesto 5 GPa, the lower enthalpy, by around 100 meV/atom for thesp3 hybridization [29,30], favors its formation and increases

FIG. 5. (a) Compressive, (b) tensile, and (c) shear stress-strain curves at 0 GPa for the recovered structures with different sp3 fractions:SA-1 (20%), SA-2 (48%), and SA-3 (88%). Maximum stresses are marked with triangles. (d) The calculated electronic DOS spectra of thecorresponding structures before applying compressive, tensile, and shear strains. (e) DOS contributions from a typical sp2 atom and a typicalsp3 atom of the recovered structure SA-3. (f) Total DOS contributions from sp2 atoms and sp3 atoms of the recovered structure SA-3.

014103-5226


its saturation fraction to 60%, although a large percentage ofthe sp2 bonds still exist due to the anisotropic local atomicconfiguration under shear. The competition of the appliedpressure and sheared geometry, with the former favoring sp3

bonds and the latter favoring sp2 bonds, finally determines thedirection of the transformation. For hydrostatic-pressure-onlyexperiments, however, the transformation from the sp2-richphase to the sp3-rich phase at low temperature requires veryhigh pressure due to the large energy barrier between them[31]. To realize such transformations under lower pressure,high temperature is commonly used since thermal fluctuationprovides kinetic energy to reach the global energy minimum.Alternatively, according to our calculation results, shear strainscan also help overcome the energy barrier by introducingcontinuous bond breaking. The shear-induced broken bondscan reform to sp3 hybridization more easily under higherpressure, which correspondingly leads to a higher sp3 fractionsaturation value.

In Fig. 5, we calculate and compare the mechanical strengthand electronic DOS of the recovered structures with differentsp3 fractions at 0 GPa under fixed-direction compression,tension, and shear. By comparing the slopes (related to theelastic constants) and maximum or average stresses (associatedwith the material strengths) of these stress-strain curves, wefind that the structure with a higher sp3 fraction is muchstiffer and harder. Thus the sp3 fraction is an importantmeasure of the mechanical strength of the transformed GCstructures. As pointed out previously [32], the piecewise-linear stress-stain relation in the glassy structure appearsonly locally on an atomic scale. The experimentally observedstress-strain relations for macroscopic glassy samples consistof contributions from different parts of the samples with locallyrandom atomic configurations. The average of these piecewise-linear stress-strain curves over the macroscopic sample gives anonlinear stress-strain relation which starts increasing linearlywith the applied strain when it is small and then approachesthe saturated average stress as the (shear) strain is large [32].The average stress determines the strength of the GC structureunder large applied strains. On the other hand, the electronicDOS of the structures with different sp3 fractions can differ

significantly. For the structure with the larger sp3 fraction, theDOS near the Fermi level is smaller. A gaplike feature wasobserved for the sp3-rich structure, together with a gap statepeaking at the Fermi level. An sp2 carbon atom contributesmuch more in the gap state compared to an sp3 carbon atom, ascan be seen from the site-projected DOS for these two atomsin Fig. 5(e). However, since the sp3 fraction is much largerthan the sp2 fraction, the resulting total contributions [shownin Fig. 5(f)] from these two types of atoms have the same orderof magnitude. In addition, variations exist among the DOSsof the sp3 and sp2 atoms since the bonds in the amorphousnetwork are distorted to a different degree.

IV. SUMMARY

In summary, we performed first-principles calculations onGC and found that shear strain can effectively increase the sp3

fraction of the GC structure under much lower pressure, whichmay ease the industrial production of such sp3-rich structures.The shear stress drives the sp2-sp3 transformation process bybreaking the existing carbon atomic bonds in GC, and then thecarbon atoms will reform into new sp2 or sp3 bonds accordingto the new environment under applied pressure and shearedgeometry, where the former favors sp3 bonds and the latterfavors sp2 bonds, which finally determines the direction of thetransformation. We also found that the sp3-rich structure hasa much higher strength and very different electrical propertiesthan the sp2-rich structures. By changing the pressure, onecan tune the saturated sp3 fraction of the structure with largeshear strains, which makes it versatile for a wide range ofapplications.

ACKNOWLEDGMENTS

This work was supported by the National Natural Sci-ence Foundation of China (Grant No. 11574197) and Min-istry of Science and Technology of China (Grant No.2016YFA0300500). Computation was performed at the Centerfor High Performance Computing, Shanghai Jiao Tong Univer-sity.

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Realization of the high-performance THz GaAs homojunction detector belowthe frequency of Reststrahlen band

Peng Bai,1,2 Y. H. Zhang,1,2,a) X. G. Guo,3 Z. L. Fu,4 J. C. Cao,4 and W. Z. Shen1,2

1Key Laboratory of Artificial Structures and Quantum Control, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China2Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China3School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology,Shanghai 200093, China4Key Laboratory of Terahertz Solid-State Technology, Shanghai Institute of Microsystem and InformationTechnology, Chinese Academy of Sciences, Shanghai 200050, China

(Received 22 September 2018; accepted 23 November 2018; published online 12 December 2018)

High-performance terahertz (THz) detectors are in great need in the applications of security,

medicine, as well as in astronomy. A high responsivity p-GaAs homojunction interfacial workfunc-

tion internal photoemission (HIWIP) detector was demonstrated for a specific frequency (5 THz)

below the frequency of the Reststrahlen band. The experimental results indicate that the optimized

detector shows significant enhancement of the response below the Reststrahlen band in contrast to the

conventional detectors. With the bottom gold layer serving as a perfect reflector, nearly 50% incre-

ment of responsivity and quantum efficiency was obtained further due to the cavity effect. Though

very simple, such reflector design shows a satisfactory effect and is easy to be realized in practical

applications. The resultant peak responsivity of the detector with a bottom reflector could be as high

as 6.8 A/W at 1 V bias. The noise equivalent power is 2:3� 10�12W=Hz1=2. Due to the absorption

ability to normal incident light and high responsivity, the p-GaAs HIWIP detector is promising for

the focal plane array and large-scale pixelless imaging applications. Published by AIP Publishing.https://doi.org/10.1063/1.5061696

Terahertz (THz) detectors have gained more attention and

been intensively explored recently, thanks to their numerous

applications in security, medicine, biology, astronomy, and

non-destructive materials testing.1,2 In view of the availability

of the mature material and processing technology, GaAs-based

detectors are widely used for THz detection. However, strong

absorption by the optical phonons of GaAs causes a dark

region in 32–36 meV (7.8–8.7 THz), which is known as the

Reststrahlen band and limits the photon detection in this

region.3,4 Moreover, the frequency of most of the state-of-the

art THz sources is in the region below the frequency of the

Reststrahlen band of GaAs.5 Therefore, it will be more mean-

ingful to make a study of GaAs-based THz detectors below the

Reststrahlen band for practical applications.

As a natural extension of traditional infrared (IR) quan-

tum well photodetectors (QWPs), THz QWP is a formidable

competitor benefiting from its rapid intrinsic response speed

and flexible wavelength coverage.3,4,6 However, according to

the intersubband transition (ISBT) selection rule, only light

polarized in epitaxial growth direction can lead to the absorp-

tion of n-type QWPs. This feature requires an extra grating

design for the THz QWPs in practical applications, which

makes the device fabrication complicated and increases the

extra cost.7,8 An innovation of the THz QWP was reported in

2015, in which an antenna-coupled microcavity geometry was

adopted.9,10 The responsivity for this complicated structure

QWIP was greatly improved to �5 A/W due to the antenna

gain and the microcavity effect. However, the high cost of the

double-metal cavity structure may be a huge challenge for

large scale applications. Another promising candidate is the

quantum dot-in-well infrared photodetector (DWELL) based

on p-type intersubband transition. This has been recently

demonstrated to exhibit a THz response up to 4.28 THz at

high temperatures (100 K–130 K) with normal incidence.11

Unfortunately, the responsivity of the DWELL in the THz

range is only at the level of �mA/W.

The detector based on the concept of homo-junction inter-

facial workfunction internal photoemission (HIWIP) has been

a competitive detector for THz detection because of its wide

spectrum response coverage, clear physical mechanism and

tailorable cutoff frequency (fc).12 Compared with QWPs, the

greatest advantage of HIWIP is that it allows normal incidence

absorption, which provides significant simplification in the fab-

rication of a large focal plane array.12 In the past 20 years, both

of p-type and n-type GaAs HIWIP detectors have been

achieved.13–17 And, almost all of the detectors showed higher

responsivity (>1 A/W) and detectivity than QWIPs (edge cou-

pled and grating coupled) or DWELL detectors (<0.2 mA/W)

in the THz band. But even so, the performance is still required

to be further improved as possible. High responsivity is helpful

to increase the operation temperature which limits the practical

application for most of the semiconductor THz detectors.

Several theoretical studies show significant improvement of

the quantum efficiency (g) due to the cavity effect, if a pair of

reflectors or a bottom reflector (BR) is applied to the n-GaAs

HIWIP detector.18,19 However, direct experimental support has

not been given till now.

In this paper, we demonstrate a high-performance p-

GaAs HIWIP detector for a peak specific frequency (5 THz)

below the frequency of the Reststrahlen band. First, thea)E-mail: [email protected]

0003-6951/2018/113(24)/241102/5/$30.00 Published by AIP Publishing.113, 241102-1

APPLIED PHYSICS LETTERS 113, 241102 (2018)

229

https://doi.org/10.1063/1.5061696

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http://crossmark.crossref.org/dialog/?doi=10.1063/1.5061696&domain=pdf&date_stamp=2018-12-12

Fresnel matrix method was used to optimize the structural

parameters. Then, the detector was grown and fabricated

accordingly. The optimized detector shows significant enhance-

ment of the response below the Reststrahlen band compared

with the conventional detector. Finally, in order to further

improve the g, extra light propagating to the substrate is uti-

lized with a reflector formed by a gold layer evaporated to the

bottom of the device. Though very simple, such a cavity struc-

ture shows a satisfactory effect and leads to �50% further

enhancement of the responsivity and g, which gives direct

experimental support of the cavity effect in the THz range.

The structural diagram of the p-GaAs HIWIP detector is

shown in Fig. 1(a). From the top to the bottom of the mesa,

the function layers are the top metal electrode, the top con-

tact layer, the top emitter layer, periods of emitter/intrinsic

layers, the bottom intrinsic layer and the bottom contact

layer. A window opens on the top side for front-side illumi-

nation. The basic structure is the multi-period emitter/intrin-

sic layer, where the emitter layers are highly doped. The

detection mechanism of the p-GaAs HIWIP detector is

shown in Fig. 1(b). Free carrier absorption of infrared radia-

tion occurs in the highly doped emitter layers followed by

the internal photoemission of photoexcited carriers. Then,

the photoexcited carriers across the intrinsic barrier are col-

lected under an electric field (see supplementary material).

To obtain a high-performance p-GaAs HIWIP detector,

optimal parameters are needed. The range of the spectral

response for p-GaAs is restricted by the activation energy (D).13

According to the high density (HD) theory,12,14 the calculated

relation of DEv (difference in the valence band edge between

the emitter layer and the intrinsic layer), EF(Fermi level), D, kc

with the doping concentration at Vb ¼ 20 mV; T ¼ 4:2 K is

shown in Fig. 1(c). However, the light–heavy hole transition

effect20 reveals that the HD theory will become invalid at high

doping levels [>2� 1019cm�3, the dashed area in Fig. 1(c)].

Thus, we choose Nd ¼ 1� 1019cm�3, corresponding to the

cutoff frequency of �3.5 THz. The impact ionization theory

gives the period number N � 21 as a limit.17 Therefore, we set

the number of multilayer periods as N ¼ 20.

According to the detection mechanism of the HIWIP

detector,17 the thicknesses of the emitter layer (de) and the

intrinsic layer (di) would affect the internal photoemission

efficiency (gb) and the collection efficiency (gc) of the bar-

rier. In addition, de and di also determine the optical distribu-

tion in the active region. The Fresnel matrix method18,21 was

introduced to calculate the light absorption and propagation

in the multilayer structure. We set the operation temperature

as 4.2 K, the applied bias voltage as Vb ¼ 20 mV and the fre-

quency of incident light as 5 THz (60 lm). The calculated

relationship of de, di and g is shown in Fig. 1(d). The greaches a maximum at de ¼ 17 nm; di ¼ 196 nm.

The wafers were grown according to the optimized

device parameters by MBE technology on a semi-insulating

substrate. Square mesa structures with various areas from

400� 400 lm2 to 1� 1 mm2 were fabricated using optical

lithography and wet-chemical etching. An unoptimized

FIG. 1. (a) The structural diagram of

the p-GaAs HIWIP detector with BR,

(b) the band diagram and the detection

mechanism of the p-GaAs HIWIP

detector, (c) the calculated concentra-

tion dependence of DEv, EF, D, and kc

at Vb ¼ 20 mV, T ¼ 4:2 K, (d) the cal-

culated result of g at different thick-

nesses of the emitter layer (de) and

different thicknesses of the intrinsic

layer (di) of the detector structure

without a bottom reflector.

241102-2 Bai et al. Appl. Phys. Lett. 113, 241102 (2018)

230

ftp://ftp.aip.org/epaps/appl_phys_lett/E-APPLAB-113-024850

p-GaAs HIWIP detector with a similar cutoff frequency was

also fabricated to serve as a reference sample.17 The cutoff

frequency of these two samples is 3.47 THz (optimized) and

3.57 THz (unoptimized), respectively. According to our previ-

ous theoretical studies, the efficiency of the HIWIP detector

would be improved obviously if a resonant cavity structure is

applied.18,19 In order to make use of the extra light propagat-

ing to the substrate and further improve the g, a 100 nm gold

layer was evaporated at the back of the device to serve as a

reflector and evaluate the cavity effect of the device.

The dark current-voltage characteristics at different tem-

peratures are shown in Fig. 2(a). The dark current (id)

increases rapidly with increasing temperature for a given bias

voltage. The discrepancies at low temperatures are due to dif-

ferent field emission currents caused by different intrinsic

layer thicknesses (see supplementary material). The photocur-

rent spectrum of the optimized detector measured at different

bias voltages is shown in Fig. 2(b). The unoptimized detector

is also shown in the inset as a reference for comparison. Both

of the samples show a strong bias dependence. However, the

bias could not be increased infinitely as the dark current

increases with the bias and the response will saturate at a high

bias.16 The deep valley between 270 and 300 cm�1 corre-

sponds to the transverse optical phonon energy in GaAs

(Reststrahlen band). In contrast to the unoptimized detector,

the spectral response of the optimized one showed significant

enhancement below the Reststrahlen band. The apparent peak

frequency (fP) is below the frequency of the Reststrahlen band,

whereas fP in the unoptimized one and nearly all the other

HIWIP-detectors reported before is above the Reststrahlen

band. It should be noted that the photo-response of the detector

is affected by the optical field distribution, the internal photo-

emission efficiency gb and the collection efficiency gc simulta-

neously. The improvement in the low frequency region is

mainly due to the enhanced optical distribution in the active

region. The reduction in the high frequency region is because

of the smaller gb, gc and the relatively low optical field (see

supplementary material). The remarkably reproducible spike

response at 5.1 THz (170 cm�1) is from 1s! 2p transition

from impurity absorption of the Be acceptor.24 This was not

observed in the unoptimized one because the response is

extremely weak below the Reststrahlen band and 5.1 THz is

very close to the cutoff frequency of the detector. Besides, the

unoptimized structure may also cause the poor distribution of

light in the emitter layer.

Figure 3 shows the experimental variation of the respon-

sivity and g with the bias and the wavenumber for the unopti-

mized and optimized detectors, respectively. The mesa depth of

the optimized detector is about twice that of the unoptimized

one. So, the bias voltage applied to the optimized detector

should be twice as that of the unoptimized one to ensure the

same electric field in the active region. The calibrated respon-

sivity (R) [Fig. 3(a)] and the corresponding g [Fig. 3(b)] of the

unoptimized detector show high response above the

Reststrahlen band, which agrees with the previous results.17

The response range above the Reststrahlen band is from

300 cm�1 to 680 cm�1, and shows a peak response at

610 cm�1. In contrast, R [Fig. 3(c)] and g [Fig. 3(d)] of the opti-

mized detector show higher photon response efficiency below

the Reststrahlen band. R below the Reststrahlen band of the

optimized detector is about one order of magnitude higher than

that of the unoptimized one under the same electric field, which

indicates that the optimized structure not only determined the

response frequency of the detector, but also improved the pho-

toresponse efficiency. The remarkable response at 170 cm�1

actually resulted from the combined effect of impurity absorp-

tion of the Be acceptor and free-carrier absorption of the emitter

layer. It should be noted that the impurity absorption of the Be

acceptor would not influence our optimization because the gen-

eral responsivity below the Reststrahlen band itself is higher

than that above the Reststrahlen band even without the impurity

absorption.

The cavity effect of the device with the BR in the THz

region was evaluated by spectral response measurement.

Experimental variation of the responsivity (R) and g with the

bias and the wavenumber for the detector with BR is shown

in Figs. 3(e) and 3(f). The mapping results really show some

improvement of photoresponse for the detector with the BR.

The resultant peak responsivity of the optimized detector

FIG. 2. (a) Experimental dark current of characteristics under different bias

voltages. (b) Infrared photocurrent spectrum of the optimized HIWIP detec-

tor under different bias voltages (relevant electric fields are about 1.37, 1.68,

1.99, and 2.24 kV/cm, respectively) at 3.4 K. The inset shows the photocur-

rent spectrum of the unoptimized HIWIP detector under different bias vol-

tages (relevant electric fields are about 1.39, 1.74, 2.04, and 2.48 kV/cm,

respectively) at 3.4 K. The vertical dashed lines shown in the inset indicate

multiple phonons.22,23


231



with BR could be as high as 6.8 A/W at 1 V bias. The corre-

sponding g also shows some enhancement compared with the

detector without BR. For clarity, R and g at a specific bias

voltage (0.61 V) for detectors with BR (red line) and without

BR (blue line) are presented in Figs. 4(a) and 4(b). Significant

enhancement was achieved for the detector with BR at the

same bias voltage. The light travelling to the substrate could

be effectively used in the structure with BR. The increment of

the g obtained from the experiment is not as obvious as that

predicted by the previous theoretical calculation.19 This may

be caused by the absorption of the thick substrate (�640 lm).

Once lapping and polishing the substrate thinner, the R and gof the detector will be much higher. Furthermore, the bias

dependent peak responsivity (RP) and the corresponding NEP

(NEP ¼ id=RP) for the two detectors (with and without BR)

are presented in Fig. 4(c). These results show a maximum

improvement of 50% for the responsivity at a bias of 0.61 V.

The resultant peak responsivity of the optimized detector with

BR could be as high as 6.8 A/W. And, the minimum NEP

for the two detectors are 2:3� 10�12 W=Hz1=2 (with BR) and

3� 10�12 W=Hz1=2 (without BR), respectively.

The high-performance p-HIWIP detector can be imple-

mented in a hybridized focal plane array, in which the GaAs

substrate is usually completely removed.25 Beyond that,

another potential possible application of the detector is the

large area integrated HIWIP-light emitting diode (LED)

FIG. 3. Variation of (a) and (c) Responsivity and (b) and (d) g (g ¼ h�R=q) with the bias and the wavenumber for the unoptimized and optimized p-GaAs

HIWIP detectors, respectively. Also shown are the (e) responsivity and (f) g of the optimized p-GaAs HIWIP detector with BR.


232

pixelless imaging devices.26 The normal incidence absorp-

tion simplifies the structure of the HIWIP-LED pixelless

imaging devices and makes it more compact. This may have

more advantages over the THz QWP-LED, which needs a

special design for the grating coupler owing to the intersub-

band transition selection rule.27

In conclusion, we demonstrated a high-performance p-

GaAs HIWIP detector for a specific frequency (5 THz) below

the Reststrahlen band of GaAs. First, the Fresnel matrix

method was used to optimize the structural parameters of the

detector. Then, the optimized detector was grown and fabri-

cated accordingly. In contrast to the conventional GaAs

HIWIP detectors, the optimized one shows significant

enhancement of the response below the Reststrahlen band.

The responsivity of optimized GaAs HIWIP detectors could

be as high as 6 A/W, which is also much higher than QWIPs

(edge coupled and grating coupled) and DWELL. Finally, by

a very simple cavity design, which can be easily fabricated

for practical applications, the performance of the p-GaAs

HIWIP detector is further improved. A bottom gold layer is

applied to serve as a BR for the device, which shows nearly

50% enhancement in responsivity compared with the device

without BR. The resultant peak responsivity of the optimized

p-GaAs HIWIP detector with BR could be as high as 6.8 A/

W at 1 V. The NEP of 2:3� 10�12 W=Hz1=2 is demonstrated.

Such a design method could be applied to the HIWIP detec-

tor operating at other peak frequency.

See supplementary material for the detection mecha-

nism, the dark current mechanism, experimental details, and

simulation results of the optical field distribution in the

active region of the devices.

We acknowledge financial support from the Natural

Science Foundation of China (91221201 and 11834011) and the

Shanghai Sailing Program (17YF1429900). Peng Bai is thankful

to Dr. T. M. Wang and Dr. J. Y. Jia for valuable discussions.

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FIG. 4. (a) Responsivity and (b) g of the p-GaAs HIWIP detector with BR (red line) and without BR (blue line). (c) The bias dependent peak responsivity

(with error bar) and the corresponding NEP for the two p-GaAs HIWIP detectors (with and without BR).


233


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Applied Surface Science 439 (2018) 1034–1039


Applied Surface Science

journal homepage: www.elsevier .com/locate /apsusc

Full Length Article

Formation of qualified BaHfO3 doped Y0.5Gd0.5Ba2Cu3O7�d film on CeO2

buffered IBAD-MgO tape by self-seeding pulsed laser deposition

https://doi.org/10.1016/j.apsusc.2018.01.0370169-4332/� 2018 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (Y. Li).

234

Linfei Liu, Wei Wang, Yanjie Yao, Xiang Wu, Saidan Lu, Yijie Li ⇑School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

a r t i c l e i n f o

Article history:Received 13 September 2017Revised 1 December 2017Accepted 4 January 2018Available online 6 January 2018

Keywords:BaHfO3

Y0.5Gd0.5Ba2Cu3O7�d filmPLDSeed layer

a b s t r a c t

Improvement in the in-filed transport properties of REBa2Cu3O7�d (RE = rare earth elements, REBCO)coated conductor is needed to meet the performance requirements for various practical applications,which can be accomplished by introducing artificial pinning centers (APCs), such as second phase dopant.However, with increasing dopant level the critical current density Jc at 77 K in zero applied magnetic fielddecreases. In this paper, in order to improve Jc we propose a seed layer technique. 5 mol% BaHfO3 (BHO)doped Y0.5Gd0.5Ba2Cu3O7�d (YGBCO) epilayer with an inserted seed layer was grown on CeO2 buffered ionbeam assisted deposition MgO (IBAD-MgO) tape by pulsed laser deposition. The effect of the conditionsemployed to prepare the seed layer, including tape moving speed and chemical composition, on the qual-ity of 5 mol% BHO doped YGBCO epilayer was systematically investigated by X-ray diffraction (XRD) mea-surements and scanning electron microscopy (SEM) observations. It was found that all the samples withseed layer have higher Jc (77 K, self-field) than the 5 mol% BHO doped YGBCO film without seed layer. Theseed layer could inhibit deterioration of the Jc at 77 K and self-filed. Especially, the self-seed layer (5 mol%BHO doped YGBCO seed layer) was more effective in improving the crystal quality, surface morphologyand superconducting performance. At 4.2 K, the 5 mol% BHO doped YGBCO film with 4 nm thick self-seedlayer had a very high flux pinning force density Fp of 860 GN/m3 for B//c under a 9 T field, and moreimportantly, the peak of the Fp curve was not observed.

� 2018 Elsevier B.V. All rights reserved.

1. Introduction

REBCO coated conductors are promising conductors for variouskinds of electric power applications owing to their high criticaltemperature (Tc) of over 90 K and large critical current density(Jc) in high magnetic fields [1,2]. Nevertheless, higher Jc in magneticfield is still one unresolved problems for a successful application. Ina magnetic field, the Lorentz force acts on the magnetic flux linesthat have entered the superconductor, and the magnetic flux lineresults in the destruction of the superconducting state. Therefore,it is needed to stop the motion of the magnetic flux line [3].

The introduction of second-phase nanoscale defects into theREBCO matrix has been effective in improving the Jc of REBCOcoated conductors via enhanced flux pinning by the defects.Among these defects, perovskite oxide BaMO3 (BMO, M = Zr, Sn,or Hf) has received particular attention [4–10]. Recently, BHObecame trendy since BHO has been found to give strong pinningwithout significant depression of the Tc of the superconductingmatrix. Tobita et al. first reported a BHO-doped GdBa2Cu3Oy

(GBCO) film deposited by pulsed laser deposition (PLD) [11]. BHOwas also found the most effective APCs to improve the high-fieldperformance of GBCO coated conductors, with the maximum thick-ness dependence and an isotropic angular dependence of criticalcurrent (Ic) values [7].

The SmBa2Cu3Oy (SBCO)/BHO system was extensively studiedin Nagoya University by Y. Yoshida group [12–16]. In early reports,they controlled the nanorod morphology in a SBCO film on aLaAlO3 (LAO) single-crystalline substrate using the seed layer tech-nique at a lower substrate temperature of 720 �C, and fabricated c-axis oriented SBCO film without degradation of superconductingproperties, such as Tc and Jc. They called the seed layer techniqueas the low-temperature growth (LTG) technique [17]. Then theyalso fabricated a BHO-doped SBCO film on a LAO (1 0 0) single crys-talline substrate by PLD using LTG technique and SBCO seed layer.Very recently, it was found that BHO-added SBCO films prepare byLTG method present large values of flux pinning force (Fp) below77 K. Maximum Fp of 105 GN/m3 in 9.0 T at 65 K and 405 GN/m3

in 9.0 T at 40 K have been obtained thanks to the high densityand small size of the nanorods embedded in the superconductingmatrix [12].

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L. Liu et al. / Applied Surface Science 439 (2018) 1034–1039 1035

Increasing the amount of APCs does not increase Jc indefinitely.Instead, it has been shown that the enclosure of too many defectsin the superconducting matrix has a detrimental effect since theresulting current-blocking effect stifles or thwarts the movementof vortices. In our earlier research [18], we investigated the effectof BHO doping concentration in YGBCO thin film prepared byPLD and found that the 5 mol% BHO doping concentration wasthe best in view of the superconducting properties in a magneticfield. However, Jc at 77 K and self-field of the 5 mol% BHO dopedYGBCO film was lower than that of pure YGBCO film.

In this study, we attempted to improve the superconductingproperty of 5 mol% BHO doped YGBCO film using seed layer tech-nique. We fabricated BHO-doped YGBCO films with YGBCO seedlayer and BHO doped YGBCO self-seed layer on metallic substratesby PLD. The effect of the conditions employed to prepare the seedlayer, including thickness and chemical composition, on the qualityof 5 mol% BHO doped YGBCO epilayer was systematically investi-gated. The performances of the films were also compared with 5mol% BHO doped YGBCO film which was fabricated by conven-tional PLD method as reported in [18].

2. Experimental

All the superconducting films were deposited on the bufferedflexible metallic tapes at substrate temperature of 820 �C using reelto reel PLDmethod with a KrF excimer laser (k = 248 nm). The laserbeam was at an angle of 45� with respect to the target surface. Thelaser energy density was 1.0 J/cm2 and the distance between thesubstrate and targets was 40 mm. The oxygen pressure was main-tained at 200 mTorr during the deposition. The films with seedlayer were fabricated via the seed layer technique shown inFig. 1, which consists of two steps. Firstly, YGBCO and 5 mol%BHO doped YGBCO seed layers with different thickness were fabri-cated on buffered flexible metallic tapes at laser repetition rate of10 Hz and different tape moving speed, respectively. Secondly, 5mol% BHO doped YGBCO epilayers were subsequently grown atlaser repetition rate of 160 Hz. We used a mixed target techniqueto add BHO to the YGBCO matrix. 5 mol% BHO powder was mixedwith YGBCO powder to obtain 5 mol% BHO doped YGBCO target. Toshow the effectiveness of the seed layer technique, 5 mol% BHOdoped YGBCO film was fabricated by conventional PLD method.The as-grown films were then ex-situ post annealed at 400 �C forfour hours in pure oxygen at atmospheric pressure. The samplesfabricated without and with seed layer were labelled as S0, S1,S2, S3, S4 and S5, respectively, which were depicted in Table 1.

The buffered flexible metallic tapes were Hastelloy C276 tapes(10 mm in width and 50 lm in thickness) coated with multipleoxide buffer layers, including a biaxially-textured MgO layer fabri-cated by ion beam assisted deposition method (IBAD-MgO). Themultiple oxide buffer layer was PLD-CeO2/IBAD–MgO/Y2O3/Al2O3/

Fig. 1. Schematic diagrams of the cross section of coated conductor architect

235

Hastelloy C 276 (top to bottom). The CeO2 cap layers had a sharp<0 0 l> cube texture. Full-width half-maximum (FWHM) of typicalX-ray x and u scans were about 1� and 4�, respectively. The RootMean Square (RMS) roughness were about 1 nm over 1 lm � 1 lm area and 2 nm over 5 lm � 5 lm area respectively, which indi-cated that CeO2 cap layer had a smooth surface.

The microstructure of the as-grown filmwas evaluated by X-raydiffraction (XRD, D8 discover with GADDS, Bruker Advanced X-raySolution, Inc.) using the h-2h scan mode with Cu-Ka radiation at awavelength of 1.5406 Å. The surface morphology was observed byscanning electron microscopy (SEM, FEI Sirion 200). The thicknessand microstructures of the films were measured by transmissionelectron microscopy (TEM, FEI TECNAI G2) with an accelerationvoltage 200 kV. The Jc values at 77 K and self-field were deter-mined by a standard four-probe method using the 1 lv/cm crite-rion. Transport properties were characterized using a physicalproperties measurement system (PPMS) with a vibrating samplemagnetometer (VSM) probe. The Jc, at an applied field varied from0 to 9 T for B//c-axis of the films, was calculated using the equationof Jc = 20DM/[a(1 � (a/3b))] derived from the extended Bean model[19], where a and b are width and length of the rectangular filmmeasured in centimeter (b > a), respectively, andDM is the magne-tization hysteresis loop width measured in emu/cm3.


3.1. Structure

We first measured the structures of the 5 mol% BHO dopedYGBCO films with seed layers of samples from S1 to S5 by XRD.The typical h-2h XRD patterns of these samples are presented inFig. 2a. It can be observed obviously that only YGBCO (0 0 l) peaksreflecting c-axis oriented YGBCO grains appear. By magnifying theYGBCO (0 0 5) peak (inset in Fig. 2a), a clear peak shift is seen. 5mol% BHO doped YGBCO film deposited on YGBCO seed layer hasa small shift in peak positions of YGBCO (0 0 5) peak from 38.12�to 38.18� with decreasing seed layer tape moving speed from 60m/h to 30 m/h. While for 5 mol% BHO doped YGBCO film depositedon self-seed layer, YGBCO (0 0 5) peak first increases from 38.14� to38.18� and then decreases to 38.1� with the decrease of self-seedlayer tape moving speed from 60 m/h to 15 m/h. The Bragg angleposition of YGBCO (0 0 5) peak in PDF NO.48-0220 is 38.18�. Basedon the calculation method in Ref. [20], e (0 0 5) in S2 and S4 isalmost 0%, which shows no internal strain exists. However, theother samples with seed layers have a very small internal strain.Since the CeO2 cap layers were cut from one long CeO2 bufferedtapes and the thickness of 5 mol% BHO doped YGBCO film wasthe same, the CeO2 (0 0 2) peak intensity should be kept the samefor all the samples. By carefully analyzing the XRD patterns, theratio of (0 0 5) YGBCO (0 0 5) peak intensity to CeO2 (0 0 2) peak

ure based on IBAD-MgO template (a) and a reel-to-reel PLD system (b).

Table 1The main features of all the samples.

Sample S0 S1 S2 S3 S4 S5

Seed layer No YGBCO 5 mol% BHO doped YGBCOSeed layer laser repetition rate (Hz) 10Seed layer tape moving speed (m/h) 60 30 60 30 15Epilayer 5 mol% BHO doped YGBCOEpilayer tape moving speed (m/h) 6Epilayer laser repetition rate (Hz) 160

10 20 30 40 50 60

0

500

1000

1500

2000

2500

3000

3500

4000

38

0

500

1000

1500

2000

(007

)

CeO

2(002

)

(005

)

(006

)

(003

)

S5S4S3

S1

Intensity(cps)

2theta(deg.)

S2

(002

)

(a)

S1 S2 S3 S4 S55

10

15

20

25

30

35

40

45

50

YGBC

O(0

05)/C

eO2(0

02) p

eak

inte

nsity

%

Sample number

(b)

Fig. 2. XRD h-2h patterns of samples S1-S5 (a), inset is magnified XRD pattern of YGBCO (0 0 5) peak, and ratio of YGBCO (0 0 5) peak intensity to CeO2 (0 0 2) peak intensity(b).

1036 L. Liu et al. / Applied Surface Science 439 (2018) 1034–1039

intensity was calculated, which is shown in Fig. 2b. It is evidentthat S4 has the maximum ratio of (0 0 5) YGBCO peak intensityto (0 0 2) CeO2 peak intensity, which shows the best crystallinity.

3.2. Surface morphology

The surface morphologies of the 5 mol% BHO doped YGBCOfilms with seed layers are depicted in Fig. 3. It can be seen first thatthe 5 mol% BHO doped YGBCO films deposited on self-seed layers(Fig. 2c and d) have smoother surface than that deposited onYGBCO seed layer (Fig. 2a and b). Compared with 5 mol% BHOdoped YGBCO film deposited on self-seed layer, 5 mol% BHO dopedYGBCO film deposited on YGBCO seed layer has more and largerparticles. For 5 mol% BHO doped YGBCO film deposited on self-seed layer, the surface is very smooth with a few particles, andbecomes rougher with decreasing seed layer tape moving speed,namely increasing the seed layer thickness. While for that depos-ited on YGBCO seed layer, the same changing trend appears thatthe surface roughness increases with increasing the seed layerthickness.

3.3. Superconducting properties

We now discuss the superconducting properties at 77 K andself-field of all the samples, which are shown in Fig. 4. It can beseen that compared with the 5 mol% BHO doped YGBCO film with-out seed layer, all the samples with seed layer have higher Jc (77 K,self-field). The Jc values of the BHO-doped films deposited on self-seed layers are higher than those of the films deposited on YGBCOseed layers. In the other hand, no matter what seed layer, Jcincreases with firstly increasing seed layer thickness, Jc,S2 > Jc,S1and Jc,S4 > Jc,S3. While further increasing thickness, Jc decreases, Jc,S5 < Jc,S4. S4 with self-seed layer has the highest Jc of 3.76 MA/

23

cm2, which was more than 0.5 times higher than that of the 5mol% BHO doped YGBCO film with the same thick YGBCO seedlayer and 2 times higher than that of the 5 mol% BHO doped YGBCOfilm without seed layer.

In order to investigated the influence of the chemical composi-tion of seed layer on the in-field superconducting property, wemeasured the transport properties of S2 and S4 with same thickseed layer and different seed layer in a magnetic ranging from 0to 9 T applied field parallel to the c-axis (B//c) of the film, at 4.2K, 30 K and 65 K, respectively. The filed dependence of the Jc andflux pinning force density (Fp) estimated from the Jc curves areshown in Fig. 5. From Fig. 5a –c, a decreasing tendency in Jc is seenfor both sample, though the value of Jc under identical magneticfields is much larger in sample 4 than in sample 2. The temperatureis lower, the difference is larger. For example, at 4.2 K, the Jc (258MA/cm2) for the BHO doped film deposited on self-seed layer isapproximately 1.3 times that for the BHO doped film depositedon YGBCO seed layer (196 MA/cm2) at near the self-field, andalmost the same at 65 K. Fig. 5d shows the magnetic fields depen-dence of Fp at 4.2, 30 and 65 K. The value of Fp is much larger insample S4 than in sample S2 at all measured temperature. S4 hasthe maximum Fp value of 860 GN/m3 (4.2 K, 9 T, B//c), 234 GN/m3

(30 K, 9 T, B//c), and 15.8 GN/m3 (65 K, 4 T, B//c). However, wecould not observe the peak of the Fp curve at 4.2 and 30 K. In otherwords, it is possible that both samples can show higher Fp values inhigh magnetic fields over 9 T.

Further investigation was made by TEM. The cross-sectionalTEM micrographs of sample S4 are depicted in Fig. 6. As shownin Fig. 6, the interface between the CeO2 layer and 5% BHO dopedYGBCO superconducting layer is sharp, without any mixture anddiffusion of the phases. No obvious interface between 5% BHOdoped YGBCO seed layer and epilayer is seen, which may be owingto the same chemical composition. The thickness of 5% BHO doped

6

Fig. 3. SEM images of the samples. (a) S1, (b) S2, (c) S3, and (d) S4.

S0 S1 S2 S3 S4 S5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

J c( M

A/c

m2 )

Sample number

at 77 K and self field

Fig. 4. Jc at 77 K and self-field of all the samples.


YGBCO superconducting layer is 350 nm. The film thickness isdirectly proportional to laser repetition rate and tape movingspeed, so we can estimate that the thickness of 5% BHO dopedYGBCO seed layer in sample S4 is 4 nm. In addition, both CeO2

and 5% BHO doped YGBCO layer have good crystallization and c-axis orientation, which are in agreement the XRD result shown inFig. 2. Stacking faults and c-axis aligned defects are observed,which can act as the effective pinning centers and improve thesuperconducting property under magnetic field. We fabricatedthe other three samples at the same deposition conditions as sam-ple S4, and the superconducting layer thickness was measured by

237

TEM. The superconducting layer thickness was 346 nm, 344 nmand 352 nm, respectively. The thickness of 5% BHO doped YGBCOseed layer all were about 4 nm, which can be controlled by fixingthe deposition conditions.

Based on the above results, the superconducting propertieswere improved by using the seed layer deposition method. Andself-seed layer was more effective. These are due to improvementin the structural properties and surface morphologies obtained bythe seed-layer technique. As we all know, there are several factorsaffecting the Jc values, such as crystallinity, internal stress, surfaceroughness, and particles. High crystalline and smooth surface playan important role in increasing the Jc values. On the one hand,structural quality of films can be enhanced by seed layer technique,which was reported by other researchers [21]. Jin-Seo Noh et al.found that it was possible to initiate top-to-bottom crystallinestructures, even at relatively low temperatures just by introducinga thin 800 �C ‘‘seed” layer in a multi-stage pulsed laser depositionprocess. On the other hand, not only crystalline quality but alsosurface morphology can be further improved by self-seeding tech-nique owing to no lattice mismatch between the self-seed layerand the epilayer, which were confirmed by the XRD result inFig. 2 and SEM data in Fig. 3. In comparison with S2 with YGBCOseed layer, S4 with self-seed layer had higher crystallinity andsmoother uniform surface with less and smaller particles. Withincreasing the seed layer thickness, the particles became more,but no significant change in the surface was observed, as shownin Fig. 3. More importantly, the samples with 4 nm thick seed layerhad no internal stress, so they had higher Jc than the samples with2 nm thick seed layer. In addition, BHO doping is good for pining,acts as effective pinning centers, and improves the in-field electri-cal transport properties.

0 20000 40000 60000 8000050

100

150

200

250

J c(MA/

cm2 )

B(oe)

4.2 K, B//C

S2

S4

(a)

0 20000 40000 60000 80000

20

40

60

80

100

120

140

J c(MA/

cm2 )

B(Oe)

30 K, B//C

S2

S4

(b)

0 20000 40000 60000 80000

0

5

10

15

20

25

30

35

65 K, B//C

J c(MA

/cm

2 )

B(oe)

S2

S4

(c)

0 2 4 6 80

500

1000

S4 at 65 K

S2 at 65 K

S4 at 4.2 K

S4 at 30 K

S2 at 4.2 KF p(G

N/m

3 ) at 4

.2 K

and

30

K

B(T)

S2 at 30 K

0

2

4

6

8

10

12

14

16

18

20

22

24

F p(G

N/m

3 ) at 6

5 K

(d)

Fig. 5. In-field Jc at 4.2 K (a), 30 K (b), 65 K (c) and Fp (d) of S2 and S4.

Fig. 6. Cross-sectional TEM image (a) and high magnification image of S4 obtained using the bright field.

1038 L. Liu et al. / Applied Surface Science 439 (2018) 1034–1039

238


4. Conclusions

In summary, pure c-axis oriented 5 mol% BHO doped YGBCOsuperconducting films was fabricated by seeding PLD. Seed layercould inhibit deterioration of the Jc at 77 K and self-filed. Espe-cially, self-seed layer significantly not only improved the crys-tallinity and surface morphology quality of 5 mol% BHO dopedYGBCO superconducting film but also increased self-field and in-field Jc. With increasing self-seed layer thickness, Jc first increasedand then decreased. The optimum thickness of seed layer was 4nm. Compared with the 5 mol% BHO doped YGBCO film withoutseed layer, the Jc (77 K, self-field) of the 5 mol% BHO doped YGBCOfilm with 4 nm thick self-seed layer was increased to three times,which reached above 3.76 MA/cm2. It also had high in-field super-conducting property, the maximum Fp value of 860 GN/m3 (4.2 K,9 T, B//c). However, we can’t observe the peak of the Fp curve at4.2 K, which shows that the sample can show higher Fp values inhigh magnetic fields over 9 T. We believe that this technique notonly has considerable for further improving the superconductingproperties of this process but also provide useful aid for the othermultilayers deposition.

Acknowledgements

This work was supported by National Natural Science Founda-tion of China (Grant numbers 51372150 and 11204174), NationalHigh Technology Research and Development Program of China(Grant number 2014AA032702), and Shanghai Commission ofScience and Technology (Grant numbers 16521108302).

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[8] S. Awaji, Y. Yoshida, T. Suzuki, K. Watanabe, K. Hikawa, Y. Ichino, T. Izumi,High-performance irreversibility field and flux pinning force density inBaHfO3-doped GdBa2Cu3Oy tape prepared by pulsed laser deposition, Appl.Phys. Express 23101 (2015) 3–5.

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PHYSICAL REVIEW B 97, 041113(R) (2018)Rapid Communications

Existence of electron and hole pockets and partial gap openingin the correlated semimetal Ca3Ru2O7

Hui Xing,1 Libin Wen,1 Chenyi Shen,2 Jiaming He,1 Xinxin Cai,3 Jin Peng,4 Shun Wang,1 Mingliang Tian,5,6 Zhu-An Xu,2,6

Wei Ku,1 Zhiqiang Mao,4,* and Ying Liu1,3,6,†1Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), and Shanghai Center for Complex Physics,

School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China2Department of Physics, Zhejiang University, Hangzhou 310027, China

3Department of Physics and Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, USA4Department of Physics and Engineering Physics, Tulane University, New Orleans, Louisiana 70118, USA

5High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China6Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

(Received 8 July 2017; revised manuscript received 19 October 2017; published 19 January 2018)

The electronic band structure of correlated Ca3Ru2O7 featuring an antiferromagnetic (AFM) as well as astructural transition has been determined theoretically at high temperatures, which has led to the understandingof the remarkable properties of Ca3Ru2O7, such as the bulk spin-valve effects. However, its band structure andFermi surface (FS) below the structural transition have not been resolved, even though a FS consisting of electronpockets was found experimentally. Here we report magnetoelectrical transport and thermoelectric measurementswith the electric current and temperature gradient directed along the a and b axes, respectively, of an untwinedsingle crystal of Ca3Ru2O7. The thermopowers obtained along the two crystal axes were found to show oppositesigns at low temperatures, demonstrating the presence of both electron and hole pockets on the FS. In addition,how the FS evolves across T ∗ = 30 K at which a distinct transition from coherent to incoherent behavior occurswas also inferred: the Hall and Nernst coefficient results suggest a temperature- and momentum-dependent partialgap opening in Ca3Ru2O7 below the structural transition with a possible Lifshitz transition occurring at T ∗. Theexperimental demonstration of a correlated semimetal ground state in Ca3Ru2O7 calls for further theoreticalstudies of this remarkable material.

DOI: 10.1103/PhysRevB.97.041113

Layered ruthenates in the Ruddlesden-Popper series family(Sr,Ca)n+1RunO3n+1 [1] have attracted great attention in thecondensed-matter and materials physics community becausethey were found to show a wide range of exciting phenomena,including spin-triplet superconductivity in Sr2RuO4 [2–4],band-dependent Mott metal-insulator transition [5,6], and or-bital ordering [7] in Ca2RuO4, metamagnetism, and correlatedeffects in Sr3Ru2O7 [8–10], making them a canonical complextransition-metal oxide system for the search of new physicalphenomena. The evolution of physics in the Ruddlesden-Popper family (Sr,Ca)n+1RunO3n+1 through the reduction ofcation radius, marked by the change in the system from thequantum magnet Sr3Ru2O7 to the antiferromagnetic metalCa3Ru2O7 [11], as well as the increase in the number ofperovskite RuO2 layers that leads to the transition from a band-dependent Mott insulator Ca2RuO4 to the metallic Ca3Ru2O7

with a k-dependent gap [12,13], is particularly interesting.Ca3Ru2O7 was found to show a paramagnetic metal to AFM

metal transition at TN = 56 K [14]. For 48 K < T < TN , theAFM state is characterized by ferromagnetic bilayers stackedantiferromagnetically along the c axis with the magneticmoments aligned along the a axis. As the temperature is


lowered below Ts = 48 K, the system exhibits a first-orderphase transition, characterized by the switching of magneticmoments from the a to the b axis [15] and multiple otherchanges. Although the orthorhombic crystal symmetry (spacegroup of Bb21m) remains unchanged through the first-ordertransition at Ts , the structural transition is marked by a clearchange in the lattice parameters: The c-axis lattice constant isshortened, whereas those of the a and b axes are enlarged. Suchlattice parameter changes are accompanied by the enhancedrotation and tilting of RuO6 octahedra below Ts [16] asillustrated in Figs. 1(a) and 1(b). Interestingly, the first-orderphase transition at Ts is also accompanied by a sharp increasein the in-plane resistivity ρab [17], followed by a negativedρab/dT , identified previously as a metal-insulator transition.Additionally, a dramatic bulk spin-valve phenomenon wasdiscovered [18] and understood based on the unusual itinerarymagnetic state [19].

Electronic band structures of correlated metals, such asCa3Ru2O7, which serve as a useful starting point to understandits physical properties, can be calculated if the correlatedeffects are dealt with properly and verified experimentally.Even though the band-structure calculations of Ca3Ru2O7

were attempted [20], no results consistent with experimen-tal results have been reported. On the other hand, a tight-binding argument suggests the presence of hole pockets inaddition to electron ones [21]. Experimentally, angle-resolved




HUI XING et al. PHYSICAL REVIEW B 97, 041113(R) (2018)

0 50 100 150 200 250 300200

300

400

500

B = 0I // b

(µ

cm)

T (K)

0 3 6 9 12 15200

300

400

500

600

700

(µ

cm)

B (T)

T = 2 KB // cI // b

c

a b

b

a

(a)

(b)

(c () e)

(d) (f)

0 10 20 30 40 50 60 70

0.8

1.2

1.6

2.0

2.4 I // a I // b

/70

K

T (K)

T*

Ts

0.06 0.09 0.12 0.15 0.18 0.21

-5

0

5

-10

0

10

20

I // a

T = 2 KB // c

I // b

(µ

cm)

B-1 (T-1)

FIG. 1. (a) Crystal structure of Ca3Ru2O7. The red, green, and blue balls stand for Ca, Ru, and oxygen, respectively. (b) The top view ofthe unit cell with Ca cations neglected. (c) Zero-field resistivity of Ca3Ru2O7 along the b axis as a function of temperature. (d) Temperaturedependence of normalized resistivity along the a and b axes. (e) Field dependence of the b-axis direction resistivity at 2 K for a magnetic fieldalong the c axis. (f) Shubnikov–de Haas oscillations (SdHOs) at 2 K along the a and b axes with the magnetic field along the c axis obtainedfrom ρ(H ) by subtracting the smooth background.

photoemission spectroscopy (ARPES) also revealed the pres-ence of small electron pockets at low temperatures, which willaccount for the observed in-plane metallic behavior . Previousquantum oscillation measurements have yielded partly incon-sistent results in the presence of multiple frequencies [21–23].In addition, ρab(T ) was found to become metallic belowT ∗ = 30 K, raising questions on whether the “insulating” statefor T ∗ < T < Ts is actually metallic possessing a FS and theorigin of change from the incoherent to the coherent behaviorat around T ∗. It also raises an interesting question on the natureof this first-order phase transition at Ts to begin with. In thisregard, the opening of a density wave at Ts was suggested basedon optical spectroscopic studies [24]. No direct evidence forthe presence of a density wave has been found in Ca3Ru2O7

however. Even though a momentum-dependent gap was indeedobserved in ARPES measurements below Ts [22], the FScannot be determined by either the quantum oscillations orthe ARPES measurements at such high temperatures. All thiscalls for alternative methods to determine the FS. Here, usingorientation-dependent magnetoelectrical and thermoelectrictransport measurements, we find experimental evidence for thepresence of both electron and hole pockets and a partial gapopening.

Single crystals of Ca3Ru2O7 were grown by the floating-zone technique. To probe physics related to the in-planeanisotropy, it is critical to use clean twin-free crystals. For this,we performed a systematic screening procedure using x-raydiffraction, Laue diffraction, and superconducting quantuminterference device magnetometry to identify clean twin-free

crystals. Selected crystals were cut along the a and b axes,respectively, with a rectangular shape. Resistivity, Hall, andthermoelectric measurements were performed in a QuantumDesign physical property measurement system with a 14-Tmagnet. A steady-state technique was used in thermoelectricmeasurements. The direction of −∇T relative to directions inthe first Brillouin zone (BZ) is shown schematically by thearrows in the insets of Fig. 2. This allows perturbation of partof the Fermi surfaces with the Fermi velocity parallel to −∇T .For systems with anisotropic electronic states, this method canbe a sensitive probe complementary to well-established probes,such as SdHOs and ARPES measurements.

At high temperatures, Ca3Ru2O7 was found to featuremetallic behavior as shown in the zero-field resistivity dataobtained in a sample prepared by a b-axis crystal in Fig. 1(c).A change in slope appeared at TN , corresponding to the onsetof the AFM transition. Upon further cooling, a sharp jumpin resistivity was found along with a negative slope in ρ(T )as seen previously in the in-plane resistivity measurementswith an unspecified in-plane current direction [16]. At T ∗,resistivity values obtained along both a and b axes were foundto show an incoherent-to-coherent crossover [Fig. 1(d)], wellabove the temperature at which a similar transition was foundin the c-axis resistivity (at T = 8 K). Magnetoresistance at lowtemperatures was found to show SdHOs as seen in the ρ(H )curve in Fig. 1(e). The oscillatory part in ρ(H ), obtained bysubtracting a smooth background, is plotted in Fig. 1(f). Theperiodicity in the �ρ(B−1) for both a- and b-axis resistivitiesgives a frequency in the SdHOs of ∼41 T, suggesting rather

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EXISTENCE OF ELECTRON AND HOLE POCKETS AND … PHYSICAL REVIEW B 97, 041113(R) (2018)

-60

-40

-20

0

20

40

0 10 20 30 40 50 60 70-40

-30

-20

-10

0

T // a T // b

S (

V/K)

B = 0

T* Ts

S =

Sa-S

b (V/

K)

T (K)

(a)

(b)

Sb

Sa

FIG. 2. (a) Zero-field thermopower of Ca3Ru2O7 along the a andb axes as a function of temperature. The inset’s schematics show alow-temperature Fermi-surface schematic adopted from [22] and thedirection of the thermal gradient in the k space. (b) The temperaturedependence of the thermopower anisotropy, defined as �S = Sa − Sb.

tiny FS pockets, ∼1% BZ, a value consistent with those foundpreviously [21,23,25].

Thermopower data measured with the temperature gradientalong the a and b axes, denoted as Sa and Sb, respectively,are shown in Fig. 2. It is seen that both Sa and Sb are positiveand nearly identical at high temperatures. With the decreasingtemperature, the thermopower was found to decrease, but nosignature was found at the magnetic transition around TN .At Ts , a sharp drop was found in both Sa and Sb with thedifference between the two becoming significant. At aroundT ∗ = 30 K, Sb changed sign to positive whereas Sa remainsnegative. The magnitudes of both Sa and Sb were seen todecrease in the low-temperature limit as required for an entropycurrent.

Consider now the implication of the thermopower data.Neglecting correlation effects, thermopower of electrons canbe expressed in terms of conductivity [26,27],

S = αxx

σxx

= −π2k2BT

3|e|∂ ln σ

∂E

∣∣∣∣EF

= −π2k2BT

3|e|[

1

A

∂A

∂E+ 1

l

∂l

∂E

]∣∣∣∣EF

, (1)

where σ denotes the conductivity, α denotes the Peltier con-ductivity, A denotes the FS area, and l denotes the carriermean free path. One can see that thermopower is therefore

a measure of the variation in conductivity with respect tochemical potential. In general, the second term in the squarebrackets in Eq. (1) is much smaller than the first term, thereforethe sign of thermopower is related directly to the carrier typeof the dominating band.

As discussed above, earlier works on quantum oscillation,ARPES, and band-structure calculation have set up a clearboundary condition: tiny FS consisting of possible featuresaround the M and M ′ points (electronlike and holelike pockets)[21,22]. In our measurement, the thermal gradient −∇T wasdirected towards the �-M and �-M ′ directions in k space for Sa

and Sb, respectively. The negative Sa and positive Sb seen at lowtemperatures therefore indicate the existence of dominatingelectron and hole bands in their respective directions: evidencesupporting the presence of both electron and hole pockets inCa3Ru2O7. In a one-band nearly-free-electron approximation[28] S = π2

2kB

eTTF

, the slope of S(T ) at low temperaturesprovides an estimate of the Fermi temperature, leading toT +

F = 350 K for the hole pocket and for the electron pocketT −

F = 425 K. The rather low Fermi temperatures are expectedfor a low carrier-density system. On the other hand, it is alsoimportant to note that, in the presence of subtle FS structures,for instance, a van Hove singularity near the FS [29], orcomplex FS curvatures [30], the sign of thermopower cannotbe linked to the type of carrier directly. However, these specialcases do not seem to occur in our system according to earlierARPES and band-structure calculations as discussed above.

Values of �S = Sa − Sb, a quantitative measure of thethermopower anisotropy, plotted in Fig. 2(b), suggests stronglythe presence of two regimes below Ts = 48 K. The sharpchange also indicates that additional change occurs at around30 K, which cannot be accounted for by the gapped bands at48 K as described in an earlier thermopower measured withan arbitrary in-plane direction [31]. To understand the natureof the electronic state in these two regimes as well as thatof the incoherent-coherent crossover found at T ∗ = 30 K, weinvestigated further the Hall and Nernst effect in Ca3Ru2O7.The Hall resistivity ρH in the inset of Fig. 3(a) is seen todepend on the field linearly at low fields with nonlinearityseen at high fields, which is attributed to multiband effects. TheHall coefficient RH shown in Fig. 3(a) reveals a sign changeat the first-order phase transition at Ts and nonmonotonictemperature dependence at lower temperatures. The sharpincrease in the magnitude of RH suggests a rapid growthin le/ lh and thus a significant reduction in scattering in theelectronlike bands [30]. It is worth noting that similar behaviorhas been found in several two-dimensional charge-density-wave systems featuring saddle points on the FS [32]. Whetherthis applies to Ca3Ru2O7 is yet to be verified.

The Nernst signal ey = Ey/∇T measures the transverseelectric-field Ey generated by a longitudinal temperature gra-dient −∇T in the presence of a magnetic field. Here ey wasmeasured with −∇T along the b axis and was found to dependon the field linearly at low fields. Nonlinearity is seen at highfields [the inset in Fig. 3(b)]. In addition, the temperaturedependence of ey shown in Fig. 3(b) features a sign change at Ts

and slightly below T ∗. A drastic enhancement in its magnitudewas found below T ∗, reaching a value as large as 10 μV/K at15 K. A large ey can arise from several possible sources [28,33].

041113-3242

HUI XING et al. PHYSICAL REVIEW B 97, 041113(R) (2018)

FIG. 3. (a) Temperature dependence of the Hall coefficient inCa3Ru2O7. The inset shows the field dependence of Hall resistivityat various temperatures. (b) Temperature dependence of the Nernstsignal ey in Ca3Ru2O7. The inset shows the field dependence of theNernst signal. The thermal gradient is along the b axis.

In the two-band picture,

ey = S

(α+

xy + α−xy

α+xx + α−

xx

− σ+xy + σ−

xy

σ+xx + σ−

xx

). (2)

where α is the Peltier conductivity tensor with the sign ofcarriers denoted by the superscript “+” or “−” for the holesand the electrons, respectively. We note that from the Hallcoefficient RH (T ), the system is compensated at around Ts ,i.e., σ+

xy = −σ−xy , leading to a vanished second term in Eq. (2).

Therefore, a sizable change in ey around Ts is expected. ey

was also found to change sign at Ts , indicating that the firstterm in ey is comparable to the second term around thistemperature. For the same reason, ey remains relatively small.Below T ∗, RH becomes increasingly negative, whereas ey isnegative and large in magnitude. Therefore the sharp decreasein ey must come from a strong reduction in the first term,i.e., the off-diagonal Peltier coefficient term, which pointsto a change in scattering rate for T < T ∗. This change inthe scattering rate is further demonstrated in Fig. 4. Herewe compare the temperature dependence of the Hall angletan θH = σxy/σxx and the Peltier angle tan θα = αxy/αxx . Theformer corresponds to the carrier mobility and therefore probesthe scattering time whereas the latter is sensitive to the energydependence of the scattering time [33]. It is seen that tan θH

features an increase in its magnitude below Ts but no anomalyaround T ∗. On the other hand, the Peltier angle is seen to showa rapid increase at T ∗. The large Peltier angle, nearly four times

FIG. 4. The temperature dependence of thermal Hall tan θα andHall angle tan θH at 5 T. The inset: a schematic showing a possibleLifshitz transition driven by the shift of chemical potential. Below thestructural transition, the band structure of Ca3Ru2O7 consists of anelectron and hole band at M and M ′ points and another electronlikeband. The chemical potential decreases with lowering temperatureand misses the large electronlike band when cooling below T ∗.

bigger than the Hall angle, suggests a significant change in theenergy dependence of the conductance for T < T ∗.

The above measurements would suggest a momentum-dependent gap opening below Ts , and the electronic state ofCa3Ru2O7 experiences a significant change at around T ∗. ForT ∗ < T < Ts , a limited part of the FS is gapped out, leavingthermopower taken with −∇T along both �-M and �-M ′directions dominated by an electronlike band. As a result, bothSa and Sb are negative, and the anisotropy �S is small. BelowT ∗, however, most of the electronlike band on the FS is gappedout, but the electron and hole pockets near the M and M ′points survive. Furthermore, the temperature-dependent gapopening occurs gradually over a large temperature range as thetemperature is lowered below Ts , which explains the absenceof a clear signature at T ∗ in specific-heat data [17,34].

It is likely that the density-wave formation is responsiblefor the momentum-dependent gap opening, provided that thenesting condition for the density wave varies with the tempera-ture. This will result in a temperature-dependent gap opening.The existing ARPES data appear to support this scenario. Inthis regard, a small jump in the k-dependent gap was foundaround T ∗ in earlier ARPES data (see Fig. 4(c) in Ref. [22]),which not only supports the temperature-dependent nestingcondition picture, but also suggests a Lifshitz transition drivenby a shift in the chemical potential as the temperature is loweredas depicted in the inset of Fig. 4. In this picture, the change in thenesting condition is abrupt at T ∗. Lifshitz transition describesthe change in Fermi-surface topology without breaking anysymmetry of the system. The continuous change in orderparameters, as found in traditional phase transitions, no longerexists. Instead, the topological invariants dictate the transition.Incidentally, an appreciable change in chemical potentialwas indeed found to exist in several semimetals [35,36]. Itis known that a Lifshitz transition will affect the materialproperty significantly due to the reconstructed FS, especiallyin materials with magnetic or charge instabilities. For example,

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EXISTENCE OF ELECTRON AND HOLE POCKETS AND … PHYSICAL REVIEW B 97, 041113(R) (2018)

the nesting condition was found to change significantly at theLifshitz transition in pnictide superconductors [37]. A similarsituation may be encountered in Ca3Ru2O7.

To summarize, we provide experimental evidence for theexistence of both electron and hole pockets in Ca3Ru2O7

at low temperatures through the measurement of anisotropicthermopower. Furthermore, from the measurement of Halland Nernst coefficients, we found evidence for a partialgap opening in an extended temperature range below Ts .These findings help in resolving the standing issue on thelow-temperature Fermi-surface configuration and provide newinsight for further understanding of the intricate behavior ofCa3Ru2O7 at around T ∗.

The authors have benefited from discussions with A.Leggett, Y. Chen, H. Sun, and D. Qian. The work performed atSJTU was supported by MOST (Grant No. 2015CB921104),NSFC (Grants No. 91421304 and No. 11474198), the Funda-mental Research Funds for the Central Universities, at Pennsyl-vania State University by the NSF (Grant No. EFMA1433378),at ZJU was supported by the NSFC under Grants No.U1332209 and No. 11774305, at CAS by the NSFC GrantNo. U1432251, and the CAS/SAFEA international partnershipprogram for creative research teams of China, at Tulanesupported by the U.S. Department of Energy under EPSCoRGrant No. DE-SC0012432 with additional support from theLouisiana Board of Regents.

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Scripta Materialia 150 (2018) 31–35


Scripta Materialia

j ourna l homepage: www.e lsev ie r .com/ locate /scr ip tamat

Regular article

A novel seed/buffer-layer construction for enlarging c-directional growthsector in high performance YBa2Cu3O7−δ bulk

Jun Qian a,1, Ling Tian Ma a,1, Ge Hai Du a, Hui Xiang a, Yan Liu a, Yan Wan a, Si Min Huang a, Xin Yao a,b,⁎,Jie Xiong c, Bo Wang Tao c

a State Key Lab for Metal Matrix Composites, Key Lab of Artificial Structures & Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, 800Dongchuan Road, Shanghai 200240, Chinab Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, Chinac State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China

⁎ Corresponding author.E-mail address: [email protected] (X. Yao).

1 These authors contributed equally to this work.

https://doi.org/10.1016/j.scriptamat.2018.02.0361359-6462/© 2018 Acta Materialia Inc. Published by Elsev

a b s t r a c t
Article history:Received 22 October 2017Received in revised form 21 February 2018Accepted 22 February 2018Available online 20 March 2018

The size enlargement of the c-growth-sector (c-GS) is a matter of great importance in fabricating high-performance YBa2Cu3O7−δ (Y123) bulks. Here, we report a novel construction of film-seed/buffer-layer/main-pellet, in which the 45°-aligned film-seed associated with a would-be [110]-sided buffer-layer was applied.The [110]-sided seed rapidly induced the crystallization of the buffer-layer with the (110) face, which promptedthe second rapid growth on the main-pellet. Consequently, an Y123 bulk with sizable c-GS and high levitationforce was achieved. The new seeding construction with the rapid growth nature of non-equilibrium face is en-couraging for preparing other superconductor bulks in the Y123 family.

© 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords:YBa2Cu3O7−δ superconductor bulkBuffer-layerFilm-seedNon-equilibrium facec growth sector

High-temperature superconductors of REBa2Cu3O7−δ (REBCO,RE123, RE = rare earth elements) bulks with the c-axis orientationhave considerable potential for magnetic applications because of theirexcellent properties, high critical current density and high levitationforce [1–8]. Among them, the Y123 bulk has been preferentially consid-ered for industrial production due to itsmature preparation technology,the top-seeded melt-growth (TSMG) method [9–11]. By that process,the resultant REBCO bulk possesses two types of growth sectors: foura growth sectors (a-GS) and one c growth sector (c-GS), associatedwith grain sector boundaries (a/c-GSB and a/a-GSB), as shown inFig. 1(a). It has been reported that the bulk with a larger size of c-GShas a stronger capacity of levitation force [12,13]. Thus, it is an impor-tant issue to enhance performance of the REBCO superconductor by ex-ploring an effective method to raise the fraction of c-GS in a bulk. In thisregard, two kinds of seeding techniqueswere effectively applied. Firstly,a sizable c-axis oriented seed is used for readily making c-GS initiallylarge [14–17], as shown in Fig. 1(b). Secondly, multi-seeds areemployed for quickly completing the a-GS growth layer, which subse-quently act as a large seed to induce the c-GS growth, as shown inFig. 1(c). In the former case, from a large pre-grown Sm123 singlegrain, Xu gained a 14 × 14 mm2 seed and succeeded in growing the

ier Ltd. All rights reserved.

245

Y123 bulk with the higher levitation force than that induced by asmall seed [14]. Similarly and interestingly, cutting from a huge pre-prepared GdBCO bulk, Nizhelskiy made a distinctively elongated seed(38 × 3 mm2), parallel to the [110] direction at its long edge (so called45° seed), and realized a fast growth in completing the a-b plane [16].More conveniently, exploiting its commercial availability of large-sizeand high thermal stability, Li applied a 9 × 9 mm2 NdBCO film-seed toachieve the Y123 bulk with large c-GS [17]. In the second case,employing multi-seeding melt-growth [18–23], Kim fabricated a seriesof Y123 samples by using 2–5 seeds and found the area fractions ofthe RE123-grain region is in direct proportion to the seed number[19]. Remarkably, Sawamura succeeded in the growth of a large Y123bulk with the diameter of 100 mm by applying 9 crystal-seeds [20].Uniquely, Li exploited two film-seeds and gained the clean (110) grainboundary by placing them with a mutually compatible angle of 90°(termed as (110)/(110) seed arrangement) [21]. Further, Chen ob-served an interesting growth behavior that the precise (110)/(110)seed arrangement formed a (110) plane, possessing a fast growthhabit and leading to a non-equilibrium shape of the Y123 grain, fromwhich the further rapid growth occurred [23]. Obviously, that workbenefited from the film-seed because it was deposited on theMgO sub-strate that is easy to get sliced with the accurate orientation.

On the other hand, the buffer-layer technique, in which a small-pellet is inserted between the seed and the main-pellet, was firstly de-veloped in the TSMG method for inhibiting the diffusion of harmful

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Fig. 1. Schematic illustrations of the cross-sections of Y123 bulks induced by (a) a normal seed (b) a large-sized seed (c) multi-seeds. The a growth sectors and c growth sectors arerepresented as blue and yellow entities, respectively. The schematic diagram of (d) 0° film-seed and (e) 45° film-seed.

32 J. Qian et al. / Scripta Materialia 150 (2018) 31–35

elements from the seed into the bulk [24–26]. Conversely, it was alsoemployed to suppress the diffusion of Ag from the Ag-added SmBCOpellet into the seed, which significantly lowered the melting point ofseed and resulted in the growth failure [27]. Secondly, the buffer-layerwas applied as a larger, secondary seed to enable nucleation and growthof the Y123 bulk [28,29]. Recently, it was found that by tuning its com-ponent, the buffer-layer could play a role in enhancing the thermal sta-bility of the film-seed [30]. In short, the buffer-layer performs a varietyof functions in preventing contamination, acting as a secondly largeseed, and enhancing thermal stability of seed.

In this work, aiming to avoid the complicate crystal-seed prepara-tion and the increased pore density caused by the large seed [17], weproposed a novel seeding construction, consisting of a 45° film-seedand a would-be [110]-sided buffer-layer. Noting that the so-calledwould-be [110]-sided buffer-layer has no specific crystal face at the be-ginning, but it will eventually have [110] sides after the completion ofthe growth. In consequence, the fast growth was realized on bothbuffer-layer and main-pellet, the processing time was reduced and thesizable Y123 bulk with large c-GS and high levitation force wasachieved. More importantly, the new concept of utilizing non-equilibrium (110) face in this work sheds new light on crystal growthof other functional materials.

Y123 bulks were prepared in air by a cold-seeding TSMG method.The initial materials of Y123 and Y2BaCuO5 (Y211) were obtainedthrough a solid-state reaction bymixing Y2O3, BaCO3, and CuO powdersin respective stoichiometric ratios. The fully-mixed raw powders werecalcined at 900 °C for 48 h, and the process was repeated three timesto make sure of the high purity of the initial materials. The precursorpowders were obtained from a mixture of Y123 and Y211 in a molarratio of Y123:Y211 = 10:3 using a mortar and pestle. Additionally,1 wt% of CeO2 was added to the precursor mixture. Then, the precursorpowderswere pressed into pellets of 20mm indiameter, 8mm in thick-ness, and into square-shaped buffer-layers of 10 × 10 × 3 mm3. Twokinds of thin films were employed as seeds to induce the growth ofY123 bulks [31,32]. One was the c-axis oriented Y123 buffered NdBCOfilm with a 0° in-plane alignment (NdBCO[100]film//Y123[100]film//MgO[100]substrate, the 0° film-seed for short), as shown in Fig. 1(d).The other was the c-axis oriented Y123 film-seed with a 45o in-planealignment (Y123[110]film//MgO[100]substrate, the 45° film-seed for

24

short), as shown in Fig. 1(e). The film-seed was placed on the top sur-face of buffer-layer which was located on the top surface of a main-pellet. It should be noting that the normal 0o film-seeds can be readilypurchased from Germany Company of Ceraco ceramic coating GmbH,while the 45° film-seeds are not commercially available. Fortunately,Films Laboratory in University of Electronic Science and Technologycould grow and supply such 45° film-seeds for this work [33].

The pellet was heated up to 1055 °C (themaximum processing tem-perature, Tmax) within 8 h, kept there for 2 h and rapidly cooled within0.5 h to the starting temperature (Tstart) for the seed-induced epitaxygrowth. Then, it was slowly cooled at a rate of 0.5 K/h down to the end-ing temperature (Tend) for completing crystallizationwithin 40 h beforequenching to room temperature.

For levitation force measurement, the buffer-layers of the sampleswere cut off by a commercial cutting machine and the upper surfaceswere polished in order to make sure them flat. Then samples wereannealed in the flowing oxygen from 520 °C to 470 °C for a cooling pe-riod of 300 h. The levitation forces were measured in the zero-field-cooled state at 77 K by using an NdFeB magnet 50 mm in diameterand with a surface magnetic field of 0.5 T for Y123 bulk samples.

To begin with, utilizing a 45° film-seed combining with a would-be[110]-sided buffer-layer, an uncompleted Y123 growth was performedby top-seeded melt-growth, for demonstrating the function of thenew seeding construction. The top morphological characteristic isgiven in Fig. 2(a). Firstly, on the buffer-layer four fast growth regionscan be observed (indicated by the red triangle), initiated rapidly fromthe [110]-sided 45° film-seed (signed by the yellow line) and becamea minimal-sized equilibrium-shape (MSES). Then the grow front ofMSES propagated along both a- and c-axis directions by a normalgrowth habit. On completing its growth in thickness, the buffer-layeracted as a secondary large seed with [110] sides. Analogously, theformation of the second MSES was prompted at the main-pellet by the[110]-sided buffer-layer. It can be seen that there are four fast growthregions with the larger frames on the main-pellet than ones on thebuffer-layer, which certainly helps to generate sizable c-GS in the conse-quent growth in the Y123 bulk.

Subsequently, exploiting the 45° film-seed would-be [110]-sidedbuffer-layer, the completely-grown Y123 bulk was prepared success-fully, labelled as Sample 1. For a comparative study, a normal 0° film-

6

Fig. 2. (a) The top view of an uncompleted Y123 bulk whichwas grown utilizing a 45° film-seed combiningwith a would-be [110]-sided buffer-layer. Top (upper) and side (lower) viewsof c-axis-oriented as-grown Y123 bulks, (b) Sample 1, prepared by using the 45° film-seed with the would-be [110]-sided buffer-layer. (c) Sample 2, prepared by employing the 0° film-seed. (d) The levitation force curve of Sample 1 and Sample 2.

33J. Qian et al. / Scripta Materialia 150 (2018) 31–35

seed was employed to induce the growth of the Y123 bulk, labelled asSample 2. The top and side views of both samples are shown in Fig. 2(b) and (c). The fourfold growth facet lines represent a/a growth sectorboundaries, indicating that two bulks were well grown with four a-GSin the c-axis direction. The triangle region (representing c-GS) [13,16]in the side view of Sample 1 is visibly larger than that of Sample 2, indi-cating the expansion of c-GS in the bulk by the new technique. Such a c-GS enlargement can be attributed partly to two fast growth periods inthe novel construction, and partly to a larger buffer-layer acting as a sec-ondary seed [28,29]. Furthermore, the levitation forces were measuredto be 23.1 and 18.7 N for Sample 1 and 2, respectively as shown in

Fig. 3. The schematic diagram illustrating the sequence from the [110]-sided 45° film-seed to thegrowth of the Y123 bulk.

247

Fig. 2(d), indicating an evident enhancement of superconducting capac-ity by conducting the new idea in this work. In brief, the new seed/buff-ered construction is beneficial for the Y123 bulk to achieve a largervolume fraction of the c growth sector and a better property for engi-neering applications.

On the basis of crystallization theory, high index planes grow fasterthan low ones [34,35]. Thus the (110) face proceeds more rapidly thanthe (100) face, and quickly disappears from the growth front, as ob-served in the Nizhelskiy's work using a [110] direction elongated seedto realize a fast growth on the a-b plane [16]. Fig. 3 illustrates the growthsequence of the sample induced by the film-seed/buffer-layer

[110]-sided buffer-layer to realize double seed-induced rapid growth in top-seededmelt-

Fig. 4. The evolution from initial seedswith a variety of shapes& orientations tominimal-sized equilibrium-shapes (MSES), associatedwith the contribution of the new construction to theamplification of equilibrium shape in our work.

34 J. Qian et al. / Scripta Materialia 150 (2018) 31–35

construction. For a clear understanding, the whole solidification processfrom the seed is divided into three distinct steps. At thefirst step, a rapidgrowth induced by four innate (110) faces of the 45o film-seed occurson the buffer-layer. Because these (110) faces are non-equilibrium,they quickly spread and disappear, leading to an equilibrium shapewith four (100) faces, indicated by the orange frame in Step 1 asshown in Fig. 3. Then at Step 2, following the equilibrium contour, thebuffer-layer migrates its growth front along the [100] and [001] direc-tions at a normal rate. Accordingly, a/c grain sector boundaries (GSBs)extend down until they reach the main-pellet, indicating that the c-axis direction of the buffer-layer is fully grown. As a result, triangleshapes, consisting of a/c-GSBs on each side of the buffer-layer, can beobserved. Noting that the edges of crystallized buffer-layer are parallelto [110] direction,which could be deduced easily according to the orien-tation of film-seed, as indicated by the arrow in Fig. 3. Subsequently, thesecond accelerated growth induced by the completely-grown buffer-layer happens on the main-pellet in Step 3. Similar to the 45° film-seed, the buffer-layer with four (110) faces can also generate four fastgrowth regions on the main-pellet and rapidly form an equilibrium-shape. As it can be observed in Fig. 2(a), four triangle growth regionson the main-pellet are identical with the schematic drawings of Step 3in Fig. 3. In brief, an effect on making c-GS initially large on the main-pellet is realized by twice rapid growth of a-b plane in a short time.Most importantly, the maximal area of the initial a-b plane on themain-pellet can reach to be (

ffiffiffi2

pl)2, when l/h ≤ p, where l and h is the

side length and the thickness of the buffer-layer, p is the value of l/hthat a-b plane and c-axis of buffer-layer are fully grown at the sametime. Such an enlargement of initial a-b plane on the main-pellet leadsto the increase of volume fraction of c-GS in the Y123 bulk.

Fig. 4 summarizes the evolution from initial seeds with a variety ofshapes and orientations to their minimal-sized equilibrium-shapes(MSES) in literatures and our work. Normally, the 0°-oriented seedshave been using in the conventional TSSG process, characterized bythe equilibrium shape with the [100] sides. Maintaining its unchangingshape, the 0°-oriented seed induces a steady growth. In the next

24

circumstance, owing to its unstable crystallography faces, an irregularseed quickly changes its shape into an equilibrium one, and its MSESslightly larger than the initial one [15]. Moreover, among 0, 22 and 45°elongated seeds, the largest MSES was achievable by employing a 45°one due to its fast growth nature on the non-equilibrium (110) face[16]. For such a 45° elongated seed, the pre-growth of the large-sizedgrain and precise orientation cutting are required. Eventually, it comesto our unique construction in this work, the 45° film-seed is employed,which has natural [110] edges of Y123 attributed to the intrinsiccleavage-plane of the MgO substrate. More importantly, such a 45°oriented-seed associated with a subsequent-grown buffer-layer with[110]-sides, has a great effect on the enlargement of MSES in themain-pellet, and then raise its volume fraction of c-GS. In other words,the fast growth on the a-b plane can be realized on both buffer-layerand main-pellet, boosted from non-equilibrium (110) faces from boththe 45° film-seed and the [110]-sided buffer-layer. Additionally, thenew construction of thiswork also plays an important role in decreasingthe growth time and increasing the achievable size of Y123 bulks.

In conclusion, a novel construction comprised of the 45° Y123 film-seed and the would-be [110]-sided buffer-layer was developed to fabri-cate Y123 superconductor bulks. As a result, the larger volume fractionof c-GS of sample with the higher levitation forces was achieved. Fur-thermore, the nature of the unique seeding construction is clarifiedthat the (110) faces of 45° film-seed and buffer-layer caused doublerapid growth of a-b plane on the buffer-layer and main-pellet,respectively,making the c-GS initially large. Finally, the scientific under-standing related to non-equilibrium (110) face in this work, is univer-sally worth for growth of other crystal material with excellentperformance.

Acknowledgments

The authors are grateful for financial support from Key Projects forResearch & Development of China (Grant No. 2016YFA0300403), andNational Natural Science Foundation of China (Grant No. 51572171).

8

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1SCIEntIfIC REPORTS | (2018) 8:2192 | DOI:10.1038/s41598-018-20615-7

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Growth and structural characterisation of Sr-doped Bi2Se3 thin filmsMeng Wang1,4,6, Dejiong Zhang 2, Wenxiang Jiang3, Zhuojun Li1,4,6, Chaoqun Han3, Jinfeng Jia3,5, Jixue Li2, Shan Qiao1,4,6, Dong Qian3,5, He Tian2 & Bo Gao1,4,6

We grew Sr-doped Bi2Se3 thin films using molecular beam epitaxy, and their high quality was verified using transmission electron microscopy. The thin films exhibited weak antilocalisation behaviours in magneto-resistance measurements, a typical transport signature of topological insulators, but were not superconducting. In addition, the carrier densities of the non-superconducting thin-film samples were similar to those of their superconducting bulk counterparts. Atom-by-atom energy-dispersive X-ray mapping also revealed similar Sr doping structures in the bulk and thin-film samples. Because no qualitative distinction between non-superconducting thin-film and superconducting bulk samples had been found, we turned to a quantitative statistical analysis, which uncovered a key structural difference between the bulk and thin-film samples. The separation between Bi layers in the same quintuple layer was compressed whereas that between the closest Bi layers in two neighbouring quintuple layers was expanded in the thin-film samples compared with the separations in pristine bulk Bi2Se3. In marked contrast, the corresponding changes in the bulk doped samples showed opposite trends. These differences may provide insight into the absence of superconductivity in doped topological insulator thin films.

Topological insulators (TIs) and superconductors (TScs) are active research fields in condensed matter physics1. The ability of TScs to host gapless Majorana-type collective excitations2,3, which can serve as building blocks for fault-tolerant topological quantum computing4–6, has sparked substantial research interest. One possible route to obtain TScs is to dope TIs. To date, superconductivity has been successfully induced in doped TIs such as Cu-, Sr-, and Nb-doped Bi2Se3

7–9. A few signs of topological superconductivity have been observed in these materials. For example, point contact spectroscopy measurements have revealed a zero-bias conductance peak (ZBCP) in Cu-doped Bi2Se3, suggesting a possible mid-gap state that may be related to the Majorana zero mode10. Signs of odd-parity superconductivity have also been inferred from various measurements including nuclear magnetic resonance experiments and angle-dependent specific heat measurements11,12. However, contradictory results have also been reported. A scanning tunnelling spectroscopy study of Cu-doped Bi2Se3 did not reproduce the finding of ZBCP13. The key problem is the relatively low superconducting volume fraction (~40%)7,14–17 and large super-conducting inhomogeneity in single crystals13,16. To search for smoking-gun type evidence of Majorana zero modes, a number of detection schemas have been proposed, many of which require the preparation of supercon-ducting films from doped TIs18–22. Unfortunately, attempts to grow superconducting doped TI films have not yet been successful23–25. The superconductivity in Cu-doped Bi2Se3 was initially believed to originate from the inter-calation of Cu dopant atoms into van der Waals (vdW) gaps. However, although Cu intercalation was success-fully realised in Bi2Se3 thin films grown by molecular beam epitaxy (MBE), the Cu-doped Bi2Se3 films were not superconducting23. Here, we report our attempt to grow Sr-doped Bi2Se3 thin films using MBE. High-resolution

1CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai, 200050, China. 2Center of Electron Microscopy and State Key Laboratory of Silicon Materials, School of Materials Science and Engineering, Zhejiang University, Hangzhou, 310027, China. 3Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China. 4State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning Road, Shanghai, 200050, China. 5Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China. 6University of Chinese Academy of Sciences, Beijing, 100049, China. Meng Wang, Dejiong Zhang and Wenxiang Jiang contributed equally to this work. Correspondence and requests for materials should be addressed to D.Q. (email: [email protected]) or H.T. (email: [email protected]) or B.G. (email: [email protected])

Received: 18 October 2017

Accepted: 22 January 2018

Published: xx xx xxxx

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http://orcid.org/0000-0002-9009-5992




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high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) examination verified the high quality of the films. In addition, magneto-resistance measurements revealed a weak antilocalisation (WAL) behaviour, which is a typical transport signature of TIs26–30, indicating the well-preserved topological surface state. Similar to Cu-doped Bi2Se3 thin films, the Sr-doped Bi2Se3 thin films were not superconducting, although the carrier densities of the films were similar to those of superconducting bulk SrxBi2Se3 samples8,31. To explore the differences between the non-superconducting thin films and superconducting bulk samples, we per-formed atom-by-atom energy-dispersive X-ray spectroscopy (EDX) mapping. Similar Sr doping structures were observed in both types of samples. The only difference was the opposite trend of expansion/compression of the separation between Bi layers in the Bi2Se3 lattice for the bulk and thin-film doped samples (compared with that in pristine Bi2Se3), which suggests that the emergence of superconductivity in doped Bi2Se3 is possibly related with doping-induced lattice structural change.

ResultsStructural and electrical characterisation of Sr-doped Bi2Se3 thin films. Sr-doped Bi2Se3 thin films were grown using MBE, and HAADF-STEM was used to inspect the film quality. Figure 1a presents a HAADF-STEM image of a thin-film sample, clearly revealing the typical quintuple-layer structure of Bi2Se3. The bright columns correspond to Bi atoms with high atomic number. The Sr dopant atoms are difficult to detect in the HAADF-STEM image because of the low Sr concentration of the sample and the low atomic number of Sr. The inset of Fig. 1a presents a reflection high-energy electron diffraction (RHEED) pattern of an approximately 50-nm-thick film, which confirms the high film quality. To determine whether the thin-film sample was super-conducting similar to bulk Sr-doped Bi2Se3, we performed resistance versus temperature (R–T) measurements, as shown in Fig. 1b. The R–T measurements do not reveal any sign of superconducting behaviour.

The carrier densities in the Sr-doped thin-film samples were determined using Hall measurements. Figure 2a presents the Hall resistivity curves of two thin-film samples. The carrier densities deduced from the linear fit-ting of the curves were 1.11 × 1020 cm−3 and 5.67 × 1019 cm−3 at 5 K, respectively. These numbers are close to the typical carrier density of bulk superconducting Sr-doped Bi2Se3 samples8,31. In addition, a prominent cusp of magneto-conductivity was observed near zero magnetic field at 5 K, as shown in Fig. 2b. The cusp can be attrib-uted to WAL behaviour, which is a typical transport signature of two-dimensional topological surface states and has been observed in many transport measurements of TI thin films26–30. Usually the WAL behaviour can be ana-lysed using the Hikami–Larkin–Nagaoka (HLN) quantum interference model32. We also noticed that Adroguer et al. has proposed a new model to describe the quantum correction to conductivity, which accounts for the Dirac nature of the surface state while HLN model considers a quasi-two-dimensional electron gas with parabolic elec-tron dispersion33,34. Because the carrier density of thin-film samples is in the order of 1019~1020 cm−3, the Fermi level (approximately 320~400 meV above the Dirac Point) crosses with the bulk conduction band according to our previous angle-resolved photoemission spectroscopy (ARPES) measurement results on bulk Sr-doped Bi2Se3 samples with similar carrier density35. In this condition, the HLN model can still be applied to our samples:

σ απ

ψ∆ = − ⋅

+

−

φ φB eh

BB

BB

( ) 12

ln(1)

xx

2

where σ σ σ∆ = −B B( ) ( ) (0)xx xx xx represents the variation of two-dimensional magneto-conductivity, α is the WAL coefficient, ψ is the digamma function, and =φ

φB

el4 2 is the effective magnetic field characterised by the

dephasing length φl . The α value can be 0.5 or 1 depending on the number of topologically protected transport channels. In addition to the cusp near zero magnetic field, the magneto-resistance measurements also revealed a linear magnetic field dependence, as fitted in the inset of Fig. 2b. This linear magneto-resistance behaviour has been observed in many transport measurements of TIs and has been attributed to the scattering of Dirac electrons

Figure 1. Structural and electrical characterisation of Sr-doped Bi2Se3 thin films. (a) Cross-sectional HRTEM image of a film clearly showing the quintuple-layer structure. Inset: Resolved RHEED pattern of a Sr-doped Bi2Se3 thin film. (b) Typical temperature dependence of the resistance of Sr-doped Bi2Se3 thin films.

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in the surface transport channel and the impurity scattering of the electrons in the bulk transport channel27,28,36–38. To achieve a better fit for the WAL behaviour, we subtracted the conductivity deduced from the linear magneto-resistance background from the raw data by extrapolating the linear magneto-resistance towards zero magnetic field27. The fitting yielded values of α = .1 28 and =φl 183 nm, which agrees well with previous meas-urements for Bi2Se3

39–41.To explore the differences between the superconducting bulk and non-superconducting thin-film samples, we

performed atom-by-atom EDX mapping for both types of samples. Figure 3a and b present the EDX mappings for the superconducting Sr0.05Bi2Se3 bulk (Sample A) and non-superconducting thin film Sr0.13Bi2Se3 (Sample B), respectively. No clear differences were observed. Bi(Sr) substitutional and interstitial doping were detected inside the quintuple layers in both types of samples. The slight difference is that Sr dopant atoms intercalated in vdW gaps were frequently visible in the thin-film samples, whereas their presence in the bulk samples was relatively rare. This difference is mainly due to the technical difficulty in imaging the dopant atoms in vdW gaps. Because the dopant atoms are highly mobile in vdW gaps, the bulk samples with low Sr concentration (Sr0.05Bi2Se3) can hardly generate signals in EDX mapping as strong as the thin-film samples with higher Sr concentration (Sr0.13Bi2Se3) can.

Statistical method to determine the separations between adjacent Bi layers. To further explore the differences between the bulk and thin-film samples, we quantitatively evaluated the variation of the Bi2Se3 lat-tice structure along the c-axis. In addition to samples A and B, a pristine Bi2Se3 bulk sample (sample C) was pre-pared for comparison. We performed numerous measurements of the separation between two Bi layers inside the same quintuple layer (d1) and between the closest Bi layers in two neighbouring quintuple layers (d2) for these samples. To reduce any systematic errors, all the high-resolution HAADF-STEM images were obtained under the same experimental settings. The original images were fast Fourier filtered to reduce the noise, which also enhanced the image contrast. Because Bi atoms are much heavier than Se atoms, their intensity was higher. Thus, by setting an appropriate image intensity threshold, all the Bi atoms could be identified along with their atomic locations. The actual position of each individual atom was determined from the two-dimensional Gaussian peak fitting of the image intensity. Using the method described above, the separation between adjacent Bi atoms was

Figure 2. Hall measurements and WAL behaviour of Sr-doped Bi2Se3 thin films. (a) Hall resistivity versus magnetic field of two Sr-doped Bi2Se3 thin films measured at 5 K. (b) WAL behaviour of the thin films at 5 K. The red dashed line in the low-magnetic-field region is a WAL fit to the magneto-conductivity, which was subtracted by the conductivity background deduced from the extrapolating linear magneto-resistance towards zero magnetic field. Inset: The red solid line is a linear fit to the magneto-resistance in the region of magnetic field ≥B 1T, which contributes the conductivity background from the bulk.

Figure 3. EDX mappings of bulk and thin-film samples. (a) EDX mapping for superconducting Sr0.05Bi2Se3 bulk sample and (b) Non-superconducting thin-film Sr0.13Bi2Se3 sample. The white dashed circles denote the Sr dopant atoms located in vdW gaps.

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measured with picometre accuracy. Figure 4 presents a histogram of the separations d1 and d2 for samples A, B, and C. Using pristine Bi2Se3 as a reference, the Gaussian fitting of the histograms indicated that d1 in Sr0.05Bi2Se3 bulk was expanded by 2.1 pm, whereas d2 was compressed by 1.8 pm; in contrast, in the thin-film sample, d1 was compressed by 0.4 pm and d2 was expanded by 1.6 pm. Therefore, the structural changes in the bulk and thin-film Sr-doped Bi2Se3 samples showed opposite trends of compression/expansion, as illustrated in Fig. 5.

DiscussionWhy superconductivity is absent in doped TIs thin films has been elusive for a long time. The answer to this ques-tion is not only necessary to understand the origin of the superconductivity in doped TIs, but will also contribute to the successful growth of superconducting doped TI thin films, which are important for the study of exotic Majorana quasiparticles and the related quantum transport phenomena.

Initially, it was thought that dopants such as Cu atoms intercalated into vdW gaps acted as electron donors and that the superconductivity was generated merely through electron doping. Later experiments performed by Shirasawa et al., however, did not support this speculation23. Although Cu intercalation was confirmed in their thin Bi2Se3 film grown by MBE, the expected superconductivity did not appear. The authors concluded that the electron doping itself could not guarantee the emergence of superconductivity and that other effects such as inho-mogeneity may also be vital for superconductivity. There were some suspicions, for example in Cu doped Bi2Se3, that copper atoms might leak out to the surface, especially in thin-film and nano-flake samples, and the leakage of copper atoms would destroy the superconductivity. For example, Ribak et al. occasionally observed an extremely large band gap in Cu-doped Bi2Se3 superconducting bulk samples using angle-resolved photoemission spectros-copy42. They found through calculations that the application of a uniaxial internal stress along the c-axis could explain the increase of the band gap. They thought that the release of the stress in the layers close to the surface might explain why it was rare to observe such a large band gap in previous ARPES measurements. They claimed that the absence of superconductivity in exfoliated Bi2Se3 was also related with the release of the internal stress42.

In our experiments, we found that Sr doped Bi2Se3 thin-film samples resembled their bulk counterparts in many aspects. TEM inspection revealed good crystallinity of the thin-film samples, and magneto-resistance measurements confirmed the existence of a topological surface state. In addition, the carrier densities of the

Figure 4. Statistics of the separation between Bi layers in the same quintuple layer (d1) and between the closest Bi layers in two neighbouring quintuple layers (d2). (a,b) Sr0.05Bi2Se3 bulk sample; (c,d) pristine Bi2Se3 bulk sample; and (e,f) Sr0.13Bi2Se3 thin-film sample. d2 of the Sr0.13Bi2Se3 thin-film sample was clearly expanded by 3.4 pm compared with that of the Sr0.05Bi2Se3 bulk sample, whereas d1 was compressed by 2.5 pm.

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thin-film samples were similar to those of the bulk Sr-doped Bi2Se3 samples. Even the EDX mappings did not reveal any clear differences in the Sr doping structures between the bulk and thin-film samples. Since our films are quite thick, it is unlikely that the absence of superconductivity is due to the direct tunnelling between the top and bottom surfaces43,44. Because the superconducting properties of doped Bi2Se3 should be mainly affected by its doping structure, the emergence of superconductivity must be highly corelated with the subtle variations of Bi2Se3 lattice structure (One can also argue that the superconductivity is caused by an unknown impurity. But to date, no such impurity has been identified yet).

As the technique used in the growth of thin-film and bulk samples is very different, and the actual Sr concen-tration depends on many experimental details such as the choice of substrate, the environmental temperature during sample growth and the nominal Sr doping level, it will be very hard to get identical actual Sr concentra-tion in thin-film and bulk samples to make a more robust comparison. We think that the best strategy is to turn a non-superconducting film into a superconducting one, then one can make a comparison of lattice constants between a superconducting film and a non-superconducting one grown under similar conditions. We noticed a recent work from Mlack et al. who used voltage pulse/thermal annealing to treat Bi2Se3 nano-flakes (approx-imately 100-nm-thick) capped with Pd electrodes45,46. Although the superconducting behavior observed in the annealed flake was not very clean, this work shows promise that superconducting doped Bi2Se3 thin film is pos-sible to get.

In summary, the emergence of superconductivity in doped Bi2Se3 should be highly corelated with the doping structure. Because no qualitative distinction between superconducting bulk and non-superconducting thin-film samples has been found, one should turn to quantitative structural analysis to understand the emergence of superconductivity in doped Bi2Se3. Future work with better design of experiments is needed to reveal the relation between superconductivity and doping induced lattice structure changes.

MethodsMaterial Synthesis. Sr0.05Bi2Se3 superconducting bulk materials were synthesised by melting a mixture of high-purity Bi, Se, and Sr with a nominal atomic ratio of 2:3:0.05. The mixture was prepared in a nitrogen glove box and then sealed in a quartz ampoule. The ampoule was heated at 850 °C for 24 h and then cooled to 620 °C at a rate of 3 °C/h. The samples were then quenched in ice water. The Sr-doped Bi2Se3 thin-film samples were grown by MBE on insulating SrTiO3(111) substrates. The substrates were heated at approximately 240 °C during film growth. Bi and Se were co-deposited onto the substrate with a flux ratio of ~20:1.

Sample Characterisation. The electrical measurements were performed in a helium-4 cryostat and in a dilution fridge using DC and the lock-in technique.

HRTEM Examination. Cross-sectional samples were prepared for HRTEM examination using a dual-beam microscope (FIB, Quanta 3D, FEG, FEI) with Ga ion milling and a precision ion-polishing system (Gatan 691) with Ar ion milling. The structural defects of the samples were examined with an FEI TITAN Cs-corrected ChemiSTEM operated at an acceleration voltage of 200 kV to avoid knock-on damage. HAADF-STEM analysis was performed using a spherical aberration probe-corrector to achieve a spatial resolution of up to 0.08 nm. The ChemiSTEM EDX provides outstanding sensitivity for determining elements at atomic resolution. To minimize the uncertainty of our measurements, sample drafting was well controlled in the experiments. Any data with an average drafting large than 0.1 pm/unit cell was excluded. To minimize the artificiality caused by different exper-imental conditions, the three samples reported in the text were measured successively in short delay and under nearly the same conditions, including the way of sample preparation, thickness of sample, magnification, camera length, pixel size, probe size, and so on.

Figure 5. Lattice deformation deduced from STEM measurements. Variations of d1 and d2 in bulk and thin-film samples, with pristine Bi2Se3 used as a reference. d1 is the spacing between two Bi layers in the same quintuple layer, and d2 is the spacing between the closest Bi layers in two neighbouring quintuple layers. Inset: Schematic illustration of d1 and d2 in Bi2Se3 lattice.

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AcknowledgementsWe thank W. Li and M.Y for helpful discussions. This work was supported by by National Natural Science Foundation of China under Grant No. 11374321, No. 11474249, No. U1632272, No. U1632266, No. 11574201, No. 11521404 and No. 11227902; by Ministry of Science and Technology of China under Grant No. 2016YFA0301003; by the National 973 Program of China under Grant No. 2015CB654901, by The National Key Research and Development Program of China under Grant 2017YFB0703100. H.T. acknowledges support from Young 1000 Talents Program of China. D.Q. acknowledges support from the Changjiang Scholars Program.

Author ContributionsC. Han and W. Jiang grew the thin-film samples, M. Wang and Z. Li performed structural and electrical measurements, D. Zhang performed the TEM inspections and data analysis. M. Wang, C. Han and D. Zhang contributed equally to this work. J. Jia, J. Li and S. Qiao contributed to the discussions. D. qian, H. Tian and B. Gao designed the experiment and wrote the manuscript.

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Regular Article - Experimental Physics

Cavity optomechanical spectroscopy constraints chameleon darkenergy scenarios

Jian Liu1,2, Ka-Di Zhu1,2,a

1 Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao TongUniversity, 800 DongChuan Road, Shanghai 200240, China

2 Collaborative Innovation Center of Advanced Microstructures, Nanjing, China

Received: 30 September 2017 / Accepted: 12 March 2018 / Published online: 28 March 2018© The Author(s) 2018

Abstract The chameleon scalar field is a matter-coupleddark energy candidate with the screening mechanisms. In thepresent paper,we propose a quantum cavity optomechanicalscheme to detect the possible signature of chameleons via theoptical pump-probe spectroscopy. Compare to the previousexperiment the sensitivity can be improved by the using ofelectrostatic shield and a pump-probe scheme to read theweak frequency splitting. We expect that this work will be auseful addition to the current literature on proposals to detecteffects of dark energy.

1 Introduction

The nature of dark energy is a central mystery in cosmol-ogy, one possibility is that it consists of the scalar fieldswhich may drive the acceleration of the expansion of theuniverse directly. This new light degrees of freedom willcouple to matter fields and leads to long-range fifth forces[1]. But this new forces have not yet been detected on earthor in the solar system [2,3]. One way to alleviate this tensionbetween theory and observation is through the introductionof screening mechanisms. The archetype of this screeningmechanism is the chameleon model [4–6] which proposesthat the coupling to matter depends on the local environ-mental matter density. In dense regions, such as in the lab-oratory, the coupling is very small, and the resulting forcemediated by the chameleon is short ranged, shielding thechameleon interaction from detection. In regions of low den-sity, such as in space, the coupling can be much stronger,and the resulting force mediated by the chameleon is longranged. In a laboratory vacuum, the extremely low densityensures that sufficiently small objects are not screened fromthe scalar field and thus the force arising from the dark energy

a e-mail: [email protected]

scalar could become significantly stronger. There have been anumber of ways for probing the chameleon screening mech-anism in the laboratory proposed or implemented in highvacuum, which have made efforts to derive new limits on thechameleon parameters. These include torsion-balance exper-iments [7,8], gravity resonance spectroscopy [9,10]. Recentsearches using microscopic test masses such as atom, neutron[11–16] and the levitated microspheres [17] often provide thestrongest constraints.

On the other hand, optomechanical coupling betweenthe electromagnetic degrees of freedom and the mechanicalmotion of mesoscopic objects are promising approaches forstudying the transition of a macroscopic degree of freedomfrom the classical to the quantum regime [18]. These sys-tems can also be of considerable technological for improvedmeasurements of displacements [19], forces [20] and masses[21]. Recently, the optical pump-probe technique has becomea popular topic, which affords an effective way to investi-gate the light-matter interaction. Most recently, this opticalpump-probe scheme has also been realized experimentallyin cavity optomechanical systems [22–24]. Several phenom-ena have been demonstrated in different kinds of optome-chanical systems based on the optical pump-probe technol-ogy such as optomechanically induced transparency [25], thelarge change in light velocity [26], optically-tunable delay[27], and light storage [28]. The mechanism underlying theseeffects can be explained as the four-wave mixing (FWM)process. The optical pump-probe technology uses a strongpump laser to stimulate the system to generate coherent opti-cal effect while used the weak laser for probing. This methodis of great interest for applications in nonlinear optical mea-surement within the cavity optomechanical system.

In this paper, we present a novel proposal to probe darkenergy in an all-optical domain based on double levitatedmicroparticles coupled to the cavity. We focus on chameleonscalar field theories, though our methods and results can be

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Fig. 1 Schematic diagram for probing the CIC. The vacuum cavity isdriven by a strong pump pulse and a weak probe pulse. The opticallevitated Silica microspheres with radii of 3 µm are separated by thedistance of 80µm form their mass centers. A shield membrane is placedbetween them to minimize the electromagnetic background forces

generalized to other theories with screening mechanisms.The remainder of the paper is organized as follows. Sec-tion 2 describes the theoretical model. Section 3 gives dis-cussions about the measurement mechanism: (1) once thechameleons appear in the vacuum cavity, the probe spectrumwill present an asymmetric splitting, which strongly revealsthe chameleon induced coupling (CIC), namely, a signalindicative of the dark energy; (2) with optical optomechani-cal system, very tiny splitting in mechanical frequency couldbe readout optically. At last we derive new constraints on thechameleon parameters from existing experiments. Section 4studies the background forces and limits in measurement.We suggest effective methods to eliminate the electrostaticforce background and then check the frequency stability forthermomechanical noise. The last section contains the sum-mary of the main results of the present work. The proposedscheme can be used to rule out large regions of the chameleonparameter space and to detect the fifth force due to the darkenergy field in the laboratory.

2 Theory framework

We consider the two microspheres are placed inside a vacuumcavity consisting of two fixed mirrors. As shown in Fig. 1, theapproach is based on optically trapping a microsphere withlow natural mechanical frequency in the anti-node of an opti-cal standing wave. The dielectric microsphere is attracted tothe anti-node of the field. The resulting gradient in the opticalfield provides a sufficiently deep optical potential well whichallows the particle to be confined in a number of possible trap-ping sites, with precise localization due to the optical standingwave [29–32]. We use the two spheres with the same radiusRi , density ρi , and mass Mi (i = 1, 2). A shield membraneis placed between the sphere A and sphere B to minimize theelectrostatic and Casimir background forces by preventingdirect ac coupling between the masses. We use a silicon die

which is etched into a frame bearing a 1 µm thick membraneof silicon nitride across an area of ∼ 1 mm2. The entire shieldwafer die, including the membrane, is coated with gold onboth sides, Because of the geometry and the tensile stressin the membrane [33], the membrane is expected to be suf-ficiently stiff. Previous experimental works have been suc-cessful at testing gravity at ultrashort distances by use of theFaraday shield [34,35], and similar techniques may work forthe setup proposed here. Thus the experiment’s force noise ispresently induced by the electromagnetic interaction betweenthe shield and the microspheres as we will discuss at last. Theshield separates the left side of the cavity from the right side.In our model, the left cavity modes are strongly driven bythe pump and probe pulses. Since the shielding membraneis apparently fully reflecting, we don’t pump the right cavitywhich only provides the optical trapping of the sphere B.

In what follows we focus on an one-dimensional (1D)model. The Hamiltonian of the left cavity mode is hωca+a,here a is the annihilation operator for the cavity (frequencyωc, damping rate κ). The trapped microspheres are treated asquantum-mechanical harmonic oscillators and their Hamil-tonian can be regarded as hω1o

+A oA and hω2o

+B oB , respec-

tively. Here oA and oB are the annihilation operators of thetwo mechanical modes. We use ωi and γi to denote theirintrinsic frequencies and mechanical damping rates respec-tively. The simultaneous presence of a pump pulse and aprobe pulse generates a radiation pressure force at the beatfrequency, which drives the motion of the oscillator near itsresonance frequency. The Hamiltonian of the interaction ofthe dielectric objects with the cavity fields is provided byoptomechanical coupling hga+a(o+

A + oA) [32], where gis the optomechanical coupling rate. Therefore the Hamilto-nian of the system, in a frame rotating with the input laserfrequency, is then given by

H = h�pua+a + hω1o

+A oA + hω2o

+B oB + hga+a(o+

A + oA)

− i h�p(a − a+) − i h�pr (aeiδt − a+e−iδt ) + Hint ,

(1)

where �pu = ωc − ωp is pump-cavity detuning, δ = ωpr −ωp is the pump-probe detuning. We also introduce the Rabifrequency of the pump field and the probe field inside the cav-ity �p = √

2Ppκ/hωpand �pr = √2Pprκ/hωpr , where Pp

is the pump power, Ppr is the power of the probe field, respec-tively. Hint denotes the chameleon potential energy betweenthe two levitated microspheres as we will discuss below.

A chameleon scalar field [4] is characterized by an effec-tive potential density Vef f (φ) = V (φ) + A(φ)ρi .The self-interaction V (φ) = 4+nφ−n is characterized by strength ofthe self interaction . The chameleons can drive the cosmicacceleration observed today if is close to the cosmological-constant scale, � 2.4 meV. The coupling function tomatter A(φ) = eφ/M is characterized by an energy scale

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M , which is expected to be below the Planck mass. Thechameleon profile due to an arbitrary static distribution ofmatter can be obtained by solving the non-linear Poissonequation ∇2φ = ∂Vef f /∂φ.

For large global objects, the chameleon field with distancer > Ri from the center is to a good approximation entirelydetermined by the contribution from infinitesimal volumeelements lying within a thin shell of thickness �Ri . In thatcase, the object is said to be screened. For small objects, in thesense that �Ri/Ri > 1, do not have a thin shell. Thus, theirentire volume contributes to the field outside and the objectis said to be unscreened. According Ref. [4], the exteriorsolution for a compact object is given by

φ(r) = − Mi

4πMre−r/Rvac + φvac if

�Ri

Ri> 1, (2)

φ(r) = − Mi

4πMr

3�Ri

Rie−r/Rvac + φvac if

�Ri

Ri� 1. (3)

The shell thickness can be defined as �Ri = Mφvac/ρi Ri .φvac is the background value of φ inside the vacuum cavity,which has the form φvac = ζ [n(n + 1)4+n R2

vac]1/(n+2),where Rvac equals to the radius of the cavity mirrors. The pro-portionality constant ζ is determined by numerically solvingthe equation of motion in the vacuum chamber. In Ref. [16],it was found that ζ is largely sensitive to the assumed cham-ber geometry. Specifically, for n = 1 and the dark energyvalue = 2.4 meV, one finds ζ = 0.68 for a cylinder cav-ity. Dropping the irrelevant constant in the Eqs. (2) and (3),the resulting potential energy is

V (r) = −λ2i

M2i

4πM2

e−r/Rvac

r, (4)

with

λi ={

1 screened

3Mφvac/ρi R2i unscreened.

The separation distance can be regarded as r = r0 + x1 − x2,where x1,2 denote the displacements of mechanical oscil-lators from their equilibrium positions, r0 denotes the dis-tance between the equilibrium positions. In our scheme thetwo levitated microspheres are separated by a distance ofr0 = 80 µm, corresponding to the dark energy scale of = 2.4 meV. Expanding Eq. (4) in the condition of|x1| , |x2| � r0 and working to the lowest order, we canobtain the term of chameleon potential as

V (r) ≈ −λ2i M

2i

4πM2

[1

r0− x1 − x2

r20

+ (x1 − x2)2

r30

− · · ·]

·[

1 − r

Rvac+ · · ·

]. (5)

The first term is constant. The second term represents asteady force does not affect the interactional dynamics. The

term proportional to x1x2 represents the lowest-order cou-pling between the microspheres’ motions. At the distancesof interest here, r/Rvac � 1. In the regime of ωc, ωi � gthe Hamiltonian of gravitational interaction can be obtainedby quantizing mechanical oscillators within rotating waveapproximation [36]

Hint = −λ2i

M2i

2πM2r30

→x1

→x2 ≈ h�(o+

A oB + oAo+B ), (6)

and

� = − λ2i

2π

M2i

M2

1

r30

√M2

i

√ω2i

= − λ2i

2πωi r30

Mi

M2 . (7)

The coefficient � can be defined as the chameleon inducedcoupling (CIC) strength which reveals the chameleonic inter-action between the two coupled oscillators.

Substituting Eqs. (6) and (7) into Eq. (1), we define theoperator sA,B = o+

A,B + oA,B and deal with the meanresponse of the system to the probe field in the presenceof the coupling. Let 〈a〉, 〈a+〉 and 〈sA,B〉 be the expectationvalues of operators a, a+ and sA,B , respectively. Accordingto the Heisenberg equation of motion and the commutationrelations [a, a+] = 1, [o, o+] = 1, the temporal evolutionsof a and sA,B can be obtained and the corresponding equa-tions are given by adding the damping terms

d〈a〉dt

= −(i�pu + κ)〈a〉 + ig〈SA〉〈a〉+ �p + �pr e

−iδt ,

(8)

d2 〈sA〉dt2

+ γ1d 〈sA〉dt

+ (ω21 + �2) 〈sA〉 − �(ω1 + ω2) 〈sB〉

= 2gω1⟨a+⟩ ⟨

a⟩, (9)

d2 〈sB〉dt2

+ γ2d 〈sB〉dt

+ (ω22 + �2) 〈sB〉 − �(ω1 + ω2) 〈sB〉

= −2g�〈a+〉〈a〉. (10)

To solve these equations, we make the ansatz as follows:⟨a(t)

⟩ = a0 + a+e−iδt + a−eiδt , (11)

〈sA(t)〉 = sA0 + sA+e−iδt + sA−eiδt , (12)

〈sB(t)〉 = sB0 + sB+e−iδt + sB−eiδt . (13)

Substituting Eqs. (11)–(13) into Eqs. (8)–(10), respectively,equating terms with the same time dependence, and workingto the lowest order in �pr but to all orders in �p, we canobtain

a0 = �p

i�pu + κ − iδ − igsA0, (14)

a+ = �pr + ia0gsA+i�pu + κ − iδ − igsA0

, (15)

a− = ia0gsA−i�pu + κ + iδ − igsA0

, (16)

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and

(ω21 + �2)sA0 − �(ω1 + ω2)sB0 = 2gω1a

20, (17)

sA+ = �(ω1 + ω2)sB+ + 2gω1(a∗0a+ + a0a∗−)

−δ2 − iγ1δ + ω21 + �2

, (18)

sA− = �(ω1 + ω2)sB− + 2gω1(a∗0a− + a0a∗+)

−δ2 + iγ1δ + ω21 + �2

, (19)

(ω22 + �2)sB0 − �(ω1 + ω2)sA0 = −2g�a2

0, (20)

sB+ = �(ω1 + ω2)sA+ − 2g�(a∗0a+ + a0a∗−)

−δ2 − iγ2δ + ω22 + �2

, (21)

sB− = �(ω1 + ω2)sA− − 2g�(a∗0a− + a0a∗+)

−δ2 + iγ2δ + ω22 + �2

. (22)

Solving Eqs. (14)–(22), we obtain in the steady state,

a+ = �prZ(� − χ) +Uω0

Z(�2 − χ2) + 2χUω0, (23)

and the population ω0 can be resolved by

�2pu = [κ2 + (�pu − gSA0)

2]ω0, (24)

with � = κ−iδ, χ = i�pu−igsA0, Y j = −δ2−iγiδ+ω2i +

�2(i = 1, 2), Z = Y1Y2 −�2(ω1 +ω2)2, U = 2ig2ω1Y2 −

2ig2�2(ω1 + ω2) and sA0, sB0 can be resolved by

(ω21 + �2)sA0 − �(ω1 + ω2)sB0 = 2gω1ω0, (25)

(ω22 + �2)sB0 − �(ω1 + ω2)sA0 = −2g�ω0. (26)

The transmission of the probe beam, defined as the ratio ofthe output and input field amplitudes at the probe frequencyis then given by [18]

t (ωpr ) = �pr/√

2κ − √2κa+

�pr/√

2κ= 1 − 2κa+/�pr . (27)

3 Forecasts and constraints

To illustrate the experimental feasibility of the proposal, wewill choose a set of experimental parameters.

(1) Dielectric object. We assume microspheres fabricatedfrom fused silica with a radius Ri = 3 µm, density ρi =2.3 g/cm3, and a dielectric constant ε = 2. (2) Cavity. Weassume a low-finesse cylinder cavity of length L = 10 cm,the radius of mirrors Rvac = 5 cm, and finesse Fc = 10leading to a cavity decay rate κ = cπ/2FcL = 5 × 108 Hz.Then we obtain the standard optomechanical coupling rateg ≈ 0.25 Hz with waist of the trapping field W = 120 µm[37]. (3) Lasers. The cavity is impinged by a pump pulse ofpower Pp = �2

phωc/2κ = 3×10−5 W with wavelength λ =1064 nm, and we scan the probe frequency across the pumpfrequency in the spectrum. We assume the probe pulse ofpower Ppr = 10−7 W and the pulse frequency ∼ ωi (105 Hz).

Taking a standing wave with wave number σ , to lowestorder near an antinode the potential corresponds to a har-monic oscillator with mechanical frequency [32]

ωi =(

6σ 2 I0ρi c

Reε − 1

ε + 2

)1/2

. (28)

Here I0 is the intracavity field intensity. Then we get ωi =100 kHz with I0 = 10−2 W/µm2. In high vacuum, the domi-nant noise forces acting on the levitated sphere are collisionswith the air molecules, thus the mechanical damping rateone would get for the optical trapping is solely depend onthe properties of the background gas [38],

γi = 3P

Riρiν. (29)

Here ν = √kBT/mgas is the thermal velocity of the gas

molecules, P the gas pressure.The lower pressure limit of sputter-ion pumps is in the

range of 10−11 mbar. Lower pressures in the range of10−12 mbar can only be achieved when the sputter-ion pumpworks in a combination with other pumping methods. Wellestablished are the combinations of a sputter-ion pump with atitanium sublimation pump (TSP) or a non-evaporable gettermodule (NEG). An ultimate pressure of about 2×10−12 Torrwas achieved inside 12 m long chambers made of Al alloy andpumped by a linear non-evaporable getter (NEG) strip and asputter-ion pump of about 50 l s−1 nominal speed [39]. Thispressure could be further reduced to about 5×10−13 Torr byincreasing the pumping speed for Ar and CH4, not pumpedby getters, with the addition of five more sputter-ion pumpsof the same size [40]. Pressures in the low 10−14 Torrrange have been reproducibly achieved in a 3m long vacuumchamber [41]. In our considerations, a conservative valueis taken P = 10−10 Torr, which is usually required forachieving ultrahigh-Q mechanical oscillators and the ultra-sensitive measurements in the levitated optomechanical sys-tem [19,20,32,42,43]. Then we find γi = 10−7.8 Hz in theroom-temperature gas, indicating that ideal oscillators canbe essentially decoupled from their thermal environment.

Compared with the previous scheme to detect chameleonsvia the force-distance properties by the levitated micro-spheres as discussed in Ref. [17], the distinct difference of ourscheme is focused on the quantum optomechanics detectionusing optical nonlinear spectrum. The chameleon inducedcoupling strength � can be obtained from Eq. (7) by settingdifferent energy scales M and . For both the test and sourceobjects are unscreened. The CIC strength can be expressedas

�u(M) = − Mi

2πωi r30 M

2. (30)

In Fig. 2, we have plotted the transmission |t |2 of the probebeam versus detuning at resonance δ − ωi for different

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Fig. 2 The plot of the probe transmission spectrum as a function ofdetuning δ − ωi with different chameleon -matter coupling strength Min the conditions of ri = 3 µm, r0 = 80 µm. The peak splitting caused

by chameleon coupling between objects � can be well recognized inthe spectrum.The other parameters are ωi = 100 kHz, g = 0.25 Hz,κ = 5 × 108 Hz, �p = 1 THz

M . Without the presence of chameleons (M → ∞), wefind a enhanced peak located at the fundamental frequencyωi of the two oscillators as shown in the first picture. Asthe chameleons appear with the strength M = 10−3.6MPl ,the probe transmission spectrum will present an asymmet-ric splitting depicted by the second picture. Here MPl =1/

√8πG is the reduced Planck mass. The two peaks dis-

playing in Fig. 2 for given energy scales M = 10−3.8MPl

and M = 10−4MPl represent the chameleon interactionsbetween the two spheres. The splitting becomes larger withincreasing M . Thus the dark energy signature can be observeddistinctly in the pump-probe spectrum. The splitting distanceD has a simple relationship with the CIC strength � via� = D/2, which provides a straight way to measure theCIC strength on the spectrum. According to the Eq. (30),the minimal detectable field-matter coupling strength Mmin

is mainly determined by the full width at half maximum(FWHM) of the resonance peak on the probe spectrum. Con-sidering Dmin = FWHM ≈ 10−7.8 Hz in Fig. 2, one canobtain �min ≈ 8 × 10−9 Hz, hence Mmin = 10−3.5MPl .

The simultaneous presence of a pump field and a probefield generates a radiation pressure force at the beat fre-quency, which drives the motion of the oscillator near its res-

onance frequency. In the Fig. 3, |N 〉, |n1〉 and |n2〉 denote thenumber states of the cavity photon, and sphere A and sphereB phonons, respectively. |N , n1, n2〉 ↔ |N + 1, n1, n2〉transition changes the cavity field, |N + 1, n1, n2〉 ↔|N , n1 + 1, n2〉 transition is caused by the radiation pres-sure coupling. In a double oscillators system,the couplingbetween two microspheres adds a fourth level. The energy-level can be modified by the chameleonic interaction. TheCIC breaks down the symmetry of the optomechanicalinduced interference, and thereby the single pump-probetransparency window is split into two transparency windows,which yields the chameleon induced optical nonlinearity.

For the both objects become screened, here the CICstrength can be obtained by

�s() = − 6φ2vac

ωi r30ρi Ri

. (31)

Thus in this case, the CIC strength is completely independentof the field-matter interactional strength M , but be changedby the strength of the self interaction . Substituting theminimal detectable CIC strength �min = Dmin/2 ≈ 8 ×10−9 Hz into Eq. (31), we find that the dark energy scale

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Fig. 3 Schematic of the energy-level diagram the energy-level split-ting. |N , n1 + 1, n2〉 ↔ |N , n1, n2 + 1〉 transition is induced by thechameleon coupling

Fig. 4 The expected constraint in the M/MPl ∼ plane forn = 1, theregions in the rectangle (indicated by green) will be excluded. The hori-zontal line marks the range around = 2.4 meV, where the chameleonfield would reproduce the current cosmic acceleration. Recent con-straints from atom interferometry are shown by the blue lines [13,14].The black line indicates limits from neutron interferometry (indicatedby shadows) [11]. The constraint of the levitated microsphere experi-ment [17] are denoted by the red line. Our results are about one or twoorders of magnitude stronger than the ones from atom interferometryat the dark energy range

can be measured down to a precision of the order of ∼10−1 meV.

The Eqs. (30), (31) predict the CIC between the levitatedmicrospheres due to the screening mechanisms in a vacuumchamber. Figure 4 shows our bounds for theories with theexponent n = 1 compared to existing experimental limits onchameleon interactions. The − M area of the green willbe excluded by the quantum optomechanical detection. Thesystem excludes the range M � 10−3.6MPl for the all rangeof . This constraint is about one to two orders of magnitudestronger than the ones from atom interferometry at the rangeof = 2.4 meV [13].

4 Background noise

4.1 Background forces

One can place a stiff metallized shield between the drive andtest masses to minimize the effects of electromagnetic forcesby preventing direct ac coupling between the masses. Thismethod lead to the conclusion that the experiment’s forceresolution is presently limited by an environmental effect,most likely an electrostatic interaction between the shield andthe test mass. Basically, there is a relatively large backgroundforce present at all times between the microspheres and shieldmembrane. Although electrically neutral microspheres areused, they still contain 1014 charges and interact primarily aselectric dipoles.

To study the background forces noise in the system, wefocus on the left cavity where the “sphere A” strongly inter-acts with the shield membrane. We map the system ontothe cavity optomechanics in which the microsphere and theshield are dipole–dipole coupled. The interactional energybetween them can be written as

U = − μiμs

2πε0R30

. (32)

Here μiand μs are the dipole moments of the levitated micro-sphere and the shield membrane. The separation distancecan be regarded as R0 = d0 + x1 − xs , where xs denotethe displacements of shield from their equilibrium positions,d0 denotes the distance between the equilibrium positions.The Hamiltonian of electrostatic interaction can be obtainedby quantizing mechanical oscillators within rotating waveapproximation

HE = − μiμs

πε0d50

→x1

→xs ≈ −h�(o+

A oS + oAo+S ), (33)

and

� = − μiμs

πε0d50

1√MiMs

√ωiωs

, (34)

where o+S (oS) are the bosonic creation(annihilation) opera-

tors for the shield membrane with the oscillation frequency ofωs and the mass Ms . The dipole moments of levitated micro-spheres are measured experimentally. According to the Ref.[17], we assume μi ≈ 102 e · µm and μs ≈ 107 e · µm.The measurement of the membrane’s lowest mechanical res-onance provides the lowest flexural resonance frequencyωs ≈ 130 kHz [44]. If we trap the microsphere near theshield membrane with the distance d0 = 40 µm, we canobtain the coupling � ≈ 3 × 10−3 Hz, which is about 5order of magnitudes higher than the chameleon induced cou-pling rate �.

The Hamiltonian of the system of is then given by

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Fig. 5 The black curve shows the initial resonance of the levitatedmicrosphere A, the red dash curve after considering the dipole forcecoupling between sphere A and the shield membrane which are sepa-rated by a distance of d0 = 40 µm. We assume the shield membrane’slowest mechanical resonance provides the lowest flexural resonancefrequency ωs = 130 kHz. The result shows that the splitting is inde-pendent of the interactions between them. The other parameters are thesame with that in Fig. 2

H = h�pua+a+hωi o

+A oA + hωs o

+S oS + hga+a(o+

A + oA)

− i h�p(a − a+) − i h�pr (aeiδt − a+e−iδt ) + HE .

(35)

The Hamiltonian have the same expression with Eq. (1), thuswe can manipulate it with the same means. In order to studythe interaction between the microsphere and the shield, wedepict the transmission of the probe beam as a function of theprobe-pump detuning without or with the dipole–dipole forcein Fig. 5. Without the presence of electrostatic force (� = 0),we get an enhanced peak which is located at the fundamentalfrequency δ = ωi . By applying the dipole–dipole interactionstrength �, we then depict the probe spectrum by the reddashed line in Fig. 5. Although the interactional strength isvery large(� ∼ 10−3 Hz), interestingly, the probe spectrumwill not be charged, because of the separation of the drivefrequency ωs from the trapping frequency ωi . That is to say,the dipole–dipole coupling can’t induce an optomechanicalsplitting on the spectrum in the condition of ωs �= ωi . Thus itis reasonable to ignore the electrostatic background noise inthe system and the contribution of chameleons will be ableto show itself clearly on the probe spectrum via the splittingas discussed before. We call this phenomenon as an optome-chanical screening and we find any interaction between theshield and the test mass would likely be too small to makethe shield deflect enough to drive the levitated microspherea measurable splitting on spectrum.

Such a shield can screen the chameleon fifth force as well.The change in source mass position will result in a change�φ in the field on the surface of the test mass. The suppres-sion factor can defined as fs = �φ(shield)/�φ(non-shield).According to Ref. [8], fs is approximately equal to fs ≈sech(2mef f ds), where ds is the foil thickness, mef f is the

effective mass of the φ(ρ) fluctuations, it is given by

m2e f f (ρ) ≈ n(n + 1)− 4+n

1+n

( ρi

nM

) n+2n+1

. (36)

Here it is manifest that the effective mass is a function ofthe ambient density, ρi , and that it increases with increasingdensity, exactly as desired. The suppression factor is approx-imately equal to 1 in the case of 2mef f ds < 0.1. In ourscheme, we use a shield membrane of ds = 1 µm. Consid-ering the dimensionless numbers n = 1 and the dark energyscale = 2.4 meV, we can obtain mef f ds ≈ 4 × 10−5 withthe coupling strength Mpl/M = 10−3. Thus mef f ds � 0.1and the force suppression due to shielding membrane can besafely neglected.

Furthermore, the chameleon force depends on the size andgeometry of the vacuum chamber as discussed in details inRef. [16]. Hence we suggest that by comparing the interac-tions of two objects in different chamber geometry, we maybe able to subtract out the common electromagnetic forcebackground. According to Eq. (31) for n = 1, we have

∂D

∂Rvac= 34ζ 2

ωi r30ρi Ri

10/3R1/3vac. (37)

The sensitivity for the radius of the vacuum cavity can bedefined as �Rvac = (∂D/∂Rvac)

−1�D. Consider �D =10−7.8 Hz, when Rvac = 5cm, = 2.4 meV, we get�Rvac = 1.2 µm. Hence we can separate out the signals ofthe chameleons form strong electrostatic force backgroundthrough changing the vacuum cavity radius by microns.

4.2 Thermal noise

The evaluation of the minimum measurable frequency shift,δω, limited by thermomechanical fluctuations of a nanoelec-tromechanical systems (NEMS) resonator can be expressedas [38]

δω =√kBT

Ec

ωi� f

Q. (38)

Here, kB is Boltzmann’s constant, T is the resonator tem-perature, � f is the measurement bandwidth, Q is the qual-ity factor of the microsphere which can be given by Q =ωi/2πγi ≈ 1012. Ec = mω2

i 〈x2c 〉 represents the maximum

drive energy of the microsphere, where we defined 〈xc〉 asthe maximum rms level still consistent with producing a pre-dominantly linear response. For a Gaussian field distribution,the nonlinear coefficients are given by ξ = −2/W 2 [45],Considering the beam waist radius W = 120 µm, for smalldisplacements |xc| � |ξ |−1/2 = 8 × 10−5 m, the nonlin-earity is negligible. In our considerations, xc is taken to be 3orders of magnitude smaller, we choose xc = 8 × 10−8 m,thus the frequency stability will reach δω = 1.5 × 10−8 Hzwith the measurement bandwidth � f = 10−5 Hz at the room

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263


temperature (T = 300 K). Here, � f ≈ 1/2π tm and is depen-dent upon the measurement averaging time tm , thus we gettm ≈ 1.6 × 104 s.

5 Conclusion

We discuss dynamics of the chameleonic coupled quan-tum cavity optomechanical oscillators, then suggest feasiblemethods to detect the chameleon dark energy with high sensi-tivity based the pump-probe technology.Using two levitatedmicrospheres in an ultrahigh-vacuum chamber, we reducedthe screening mechanism. We find that the normal modesplitting can be observed due to the chameleonic inducedcoupling strength. The splitting distance has a simple linearrelationship with the CIC strength, which strongly revealsthe new light degrees of freedom coupled to matter fields.We use a stiff metallized shield to provide the attenuation ofelectrostatic and Casimir forces between two spheres, thendiscuss the background forces between the shield membraneand the levitated sensor. The splitting distance is independentof the interactions between them, leading an effective wayto avoid strong background force noise. Moreover, severalother methods are also proposed to minimize the noises.

Acknowledgements This work was supported by the National NaturalScience Foundation of China (Nos. 11274230 and 11574206), the BasicResearch Program of the Committee of Science and Technology ofShanghai (No. 14JC1491700).

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

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PHYSICAL REVIEW A 97, 033817 (2018)

Multiphoton-resonance-induced fluorescence of a strongly driven two-levelsystem under frequency modulation

Yiying Yan,1,* Zhiguo Lü,2,3,† JunYan Luo,1 and Hang Zheng2,3,‡1Department of Physics, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

2Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China

3Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

(Received 7 January 2018; published 13 March 2018)

We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transitionfrequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming toprovide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory andthe rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- andmultiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridizedrotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-modeFloquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accountingfor the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparentfluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouvilleapproach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA andthe secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse fieldomitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, atthe multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWAspectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of thespectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that thestrict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can beobtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderatelystrong and at the multiphoton resonance with a sufficiently weak transverse field.

DOI: 10.1103/PhysRevA.97.033817

I. INTRODUCTION

In recent years, the study of resonance fluorescence hasbeen renewed in the context of artificial atoms such as quantumdots [1–8], nitrogen-vacancy centers [9], and superconductingqubits [10–12]. In these systems, the standard Mollow tripletas well as the modified Mollow triplet have been observedexperimentally in the case of the monochromatic field. Forinstance, the superconducting qubit allows one to experimen-tally realize resonance fluorescence in a squeezed vacuum [12].More recently, there is increased interest to study the resonancefluorescence from artificial atoms subjected to polychromaticfields as the spectral features turn out to be much richerin a polychromatic field than a monochromatic field [6–8].The studies of resonance fluorescence not only provide abetter understanding of the light-matter interaction but also arerelevant for developing an on-demand single-photon source forapplications in quantum information processing.

Benefiting from the advantages and controllability of arti-ficial atoms, strong and ultrastrong light-matter coupling have

*[email protected]†[email protected]‡[email protected]

become possible in the laboratory, and a number of stimulatingstudies of physics in such coupling regimes beyond the usualrotating-wave approximation (RWA) have been carried out[13–16]. In addition, artificial atoms enable one to realizeexotic driven quantum systems with frequency modulation[9,17–19]. A paradigmatic model of such systems is a two-levelsystem (TLS) that is simultaneously driven by a longitudinalfield and a transverse field, described by the following Hamil-tonian (h = 1):

H (t) = 12 [ω0 + �z cos(ωzt)]σz + �x cos(ωxt)σx, (1)

where ω0 is the static transition frequency of the TLS, σx(y,z)

is the usual Pauli matrix, �z (ωz) is the amplitude (fre-quency) of the longitudinal field that slowly modulates thetwo-level spacing, and �x (ωx) is the amplitude (frequency) ofthe transverse field that near-resonantly drives the transitionbetween the two levels. This bichromatically driven TLSis different from those considered in Refs. [20–24], wherethere is no longitudinal field. The resonance fluorescencespectrum of this bichromatically driven TLS has been studiedrecently in Refs. [25–27] within the framework of the RWA,and the symmetric and asymmetric multipeaked structures offluorescence that are made up of small triplets centered atωx + mωz (m = 0, ± 1, ± 2, . . .) have been found. However,


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YIYING YAN, ZHIGUO LÜ, JUNYAN LUO, AND HANG ZHENG PHYSICAL REVIEW A 97, 033817 (2018)

the origin of the symmetry and asymmetry of the spectrum hasnot been comprehensively studied.

As is known, the bichromatically driven TLS described byEq. (1) has multiple resonance ωx ≈ ω0 − mωz [19,28]. Apartfrom the single-photon resonance frequency ωx ≈ ω0 (m = 0),there are multiphoton-resonance frequencies ωx ≈ ω0 − mωz

(m �= 0). Moreover, it has been illustrated that the multiphotonresonance may have a resonance width comparable with theBloch-Siegert (BS) shift caused by the counter-rotating (CR)terms of the transverse field, and in these cases the RWA forthe transverse field predicts totally different dynamics fromthat without the RWA even though the transverse driving ismoderately strong [28]. Therefore, the validity of the for-malisms with the RWA presented in Refs. [25–27] becomesquestionable in such situations. It is therefore necessary todevelop a theory beyond the RWA to study the multiphoton-resonance-induced fluorescence in the bichromatically drivenTLS. In addition, it is interesting to examine if the BS shiftsignificantly influences the multiphoton-resonance-inducedfluorescence spectrum, which may be relevant to experi-mental test of the effects of the CR terms with the opticalsignals.

In this paper, we extend the counter-rotating-hybridizedrotating-wave (CHRW) method to study the resonance fluo-rescence of the TLS driven simultaneously by the transverseand longitudinal fields. In particular, we are interested inthe multiphoton-resonance-induced fluorescence in the strongdriving regimes, �x ∼ ωz ∼ �z, which is difficult to tacklewith perturbation theory. The CHRW method allows us toderive an effective Hamiltonian periodic in time, from whichwe can derive the two-mode Floquet states and quasienergiesfor the bichromatically driven TLS with Floquet theory otherthan the generalized Floquet theory (GFT). We then solvethe master equation of the driven TLS accounting for thespontaneous decay in the bases of the two-mode Floquet states.Using the solutions derived, we illustrate that the BS shift hasan essential effect on the steady state of the bichromaticallydriven TLS in comparison with the RWA results. We derive aphysically transparent fluorescence spectrum with clear phys-ical significance. Our spectrum is benchmarked against theexact spectrum from the generalized Floquet-Liouville (GFL)formalism, and found to be valid in a wide range of parameterswhere the RWA and secular approximation may break down.It allows us to comprehensively study the effects of the CRterms on the spectrum and analyze in detail the symmetryand asymmetry of the spectrum. In particular, we demonstratea dramatic difference in line shapes of the RWA and non-RWA spectra in the multiphoton-resonance situations wherethe multiphoton resonance has a width comparable with theBS shift.

The rest of paper is organized as follows. In Sec. II, wederive the two-mode Floquet states and quasienergies for thebichromatically driven TLS, and derive the solutions to themaster equation in the bases of the two-mode Floquet states.We illustrate the effect of the BS shift on the steady state. InSec. III, we derive the fluorescence spectrum with the solutionsof the master equation and calculate the line shapes of thefluorescence spectrum in the off- and multiphoton-resonancecases, and benchmark our results and the RWA results withthose from the numerically exact method. We illustrate a

dramatic difference between the RWA and non-RWA theories.Moreover, we analyze the symmetry and asymmetry of thespectrum and figure out the conditions to get a symmetricspectrum, leading to a comprehensive understanding of theorigin of the symmetry and asymmetry of the spectrum. InSec. IV, the conclusions are drawn.

II. TWO-MODE FLOQUET STATES AND THE MASTEREQUATION

To study the fluorescence of the driven TLS, we should takeinto account spontaneous decay. We model the evolution of thedriven TLS with spontaneous emission by the Lindblad masterequation

d

dtρ(t) = −i[H (t),ρ(t)] − κ

2[σ+σ−ρ(t) + ρ(t)σ+σ−

−2σ−ρ(t)σ+], (2)

where ρ(t) is the reduced density matrix of the TLS, κ is theradiative decay rate, and σ± = (σx ± iσy)/2 are the raisingand lowering operators. To derive a physically transparentfluorescence spectrum and go beyond the RWA, we solve themaster equation in the generalized Floquet picture; i.e., wefirst derive the two-mode Floquet states and quasienergies forthe bichromatically driven TLS (without dissipation), and thenwe rewrite the master equation in the bases of the two-modeFloquet states and introduce a partial secular approximation toderive a simplified solution for the master equation.

A. Two-mode Floquet states and quasienergies

According to the GFT, the evolution operator of the bichro-matically driven TLS takes the form [29]

US(t,t ′) =∑α=±

|uα(t)〉〈uα(t ′)|e−iεα (t−t ′), (3)

where |uα(t)〉 is a generalized (two-mode) Floquet state andεα is quasienergy. The subscript α labels physically differentstates. Substituting the ansatz into the Schrödinger equationddt

US(t,t ′) = −iH (t)US(t,t ′), one readily derives the follow-ing equation:

[H (t) − i∂t ]|uα(t)〉 = εα|uα(t)〉. (4)

We below use a routine based on a unitary transformation tosolve the above equation that yields |uα(t)〉 and εα instead ofa standard GFT treatment [29].

We perform a unitary transformation to the two-modeFloquet state [28,30],

|uα(t)〉 = R(t)eS(t)|uα(t)〉, (5)

where R(t) = exp(iωxtσ+σ−) and S(t) = i �x

ωxξ sin(ωxt)σx

with the undetermined parameter ξ ∈ (0,1). When neglectingthe fast-oscillating terms of the transformed Hamiltonian, onegets the equation of motion of the transformed Floquet state[28]:

[HCHRW(t) − i∂t ]|uα(t)〉 = εα|uα(t)〉, (6)

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MULTIPHOTON-RESONANCE-INDUCED FLUORESCENCE OF … PHYSICAL REVIEW A 97, 033817 (2018)

with εα = εα − 12ωx being the transformed quasienergy and

the effective Hamiltonian given by

HCHRW(t) = 1

2{[ω0 + �z cos(ωzt)]J0(Z) − ωx}σz

+1

2[2ω0 + �z cos(ωzt)]J1(Z)σx, (7)

where Z = 2 �x

ωxξ and Jn(Z) is the nth-order Bessel function

of the first kind. The parameter ξ in the effective Hamiltonianis self-consistently determined by �x(1 − ξ ) − ω0J1(Z) = 0.That we use HCHRW(t) as the effective Hamiltonian is the firstapproximation in this paper. In general, the effective Hamil-tonian HCHRW(t) works well in a wide range of parameters[28].

Since HCHRW(t) = HCHRW(t + 2π/ωz), we can solveEq. (6) by the Floquet approach, and the detailed procedurecan be found in Ref. [28], which is more efficient thanthe standard GFT approach. On obtaining the transformedFloquet states |uα(t)〉 and quasienergies εα , we can derivethe two-mode Floquet states by inverting Eq. (5), |uα(t)〉 =e−S(t)R†(t)|uα(t)〉, and quasienergies εα = 1

2ωx + εα . Clearly,e−S(t)R†(t) is periodic in time with periodicity 2π/ωx and|uα(t)〉 = |uα(t + 2π/ωz)〉, leading to the fact that |uα(t)〉includes two basic oscillation frequencies, ωx and ωz. This isalso the reason why we call it the two-mode Floquet state.In addition, it is straightforward to show that |uα,k,l(t)〉 ≡ei(kωx+lωz)t |uα(t)〉 is also a solution to Eq. (4) but with a shiftedquasienergy εα,k,l = εα + kωx + lωz, which forms an infinitetwo-mode Floquet states ladder.

B. Master equation in the generalized Floquet pictureand single-time expectation

With the two-mode Floquet states at hand, we now move tosolve the master equation in the generalized Floquet picture.In the bases of the two-mode Floquet states, we can rewrite themaster equation as

d

dtραβ(t) = −i αβραβ(t) +

∑α′,β ′

Lαβ,α′β ′ (t)ρα′β ′(t), (8)

where αβ = εα − εβ , and ραβ(t) = 〈uα(t)|ρ(t)|uβ(t)〉 is theelement of the density matrix in the generalized Floquetpicture. We present the derivation of the above equation inAppendix A. The time-dependent decay rates in the aboveequation read

Lαβ,α′β ′(t) =∑k,l

L(k,l)αβ,α′β ′e

i(kωx+lωz)t , (9)

where

L(k,l)αβ,α′β ′ = κ

2

∑n,m

[2Y

(n,m)αα′ X

(k−n,l−m)β ′β

−∑

λ

(δα,α′X

(n,m)β ′λ Y

(k−n,l−m)λβ

+δβ,β ′X(n,m)αλ Y

(k−n,l−m)λα′

)]. (10)

Here, X(k,l)αβ (Y (k,l)

αβ ) denotes the two-mode Fourier compo-nent of 〈uα(t)|σ+(−)|uβ(t)〉 and its explicit form is given inAppendix A. In the generalized Floquet picture, one findsthat the dissipation terms depend on time. In spite of theircomplicated forms, we can invoke reasonable approximationto derive a solution with a satisfactory accuracy.

The solution to Eq. (8) can be formally divided into twoparts, i.e.,

ρ(t) = δρ(t) + ρ(st)(t), (11)

where δρ(t) denotes the solution to the homogeneous part ofEq. (8) while ρ(st)(t) denotes the steady state in the long-timelimit. Below we separately calculate these two parts.

We first derive the explicit form for δρ(t). Based on thesame spirit of the secular approximation, we omit the fast-oscillating terms in the homogeneous part of Eq. (8), yieldingthe following equations:

d

dtδρ++(t) = (L(0,0)

++,++ − L(0,0)++,−−)δρ++(t),

(12)d

dtδρ+−(t) = (L(0,0)

+−,+− − i +−)δρ+−(t).

This is the second approximation introduced in the presentpaper. We call this treatment partial secular approximation todistinguish it from the usual secular approximation invoked tothe total master equation. Consequently, we have the formalanalytical solutions for the homogeneous part:

δρ++(t) = δρ++(0)e−�relt , (13)

δρ+−(t) = δρ∗−+(t) = δρ+−(0)e−(i +−+�deph)t , (14)

where

�rel = L(0,0)++,−− − L(0,0)

++,++

= κ∑k,l

(|X(k,l)+− |2 + |X(k,l)

−+ |2), (15)

�deph = −L(0,0)+−,+−

= κ

2

∑k,l

(|X(k,l)−+ |2 + |X(k,l)

+− |2 + 4|X(k,l)++ |2) (16)

are time-averaged relaxation and dephasing rates for the two-mode Floquet states, respectively.

The steady state can be calculated by recalling that it canbe expanded as

ρ(st)αβ (t) =

∑k,l

ρ(k,l)αβ ei(kωx+lωz)t , (17)

where ρ(k,l)αβ is a time-independent two-mode Fourier com-

ponent to be determined. By substituting the expansion intoEq. (8), one readily finds the following equations:

i( αβ + kωx + lωz)ρ(k,l)αβ =

∑α′,β ′,n,m

L(k−n,l−m)αβ,α′β ′ ρ

(n,m)α′β ′ . (18)

In general, Eq. (18) can be numerically solved by introducingan appropriate truncation and yields the knowledge of thesteady state of the driven TLS. Clearly, the accuracy of theobtained steady state depends on the accuracy of the two-mode

033817-3268


Floquet states. In other words, the accuracy of the steady stateis limited by the accuracy of the CHRW Hamiltonian in ourformalism. In principle, the exact steady state can also beobtained as long as two-mode Floquet states are exact.

We now show how to calculate the physical quantities bythe solution derived above. For the population difference of theTLS, we have

〈σz(t)〉 = Tr[σzρ(t)]

=∑α,β

〈uα(t)|σz|uβ(t)〉ρβα(t)

=∑

α,β,k,l

Z(k,l)αβ ρβα(t)ei(kωx+lωz)t , (19)

where the two-mode Fourier expansion for 〈uα(t)|σz|uβ(t)〉has been used and Z

(k,l)αβ is given in Eq. (A12). In the long-time

limit, the time-averaged population difference can be obtained:

〈σz(t → ∞)〉 =∑

α,β,k,l

Z(k,l)αβ ρ

(−k,−l)βα , (20)

where the overline indicates the average over time. The time-averaged excited-state population can be obtained accordingly:

ρ++ = 12 [1 + 〈σz(t → ∞)〉], (21)

which influences the emission processes. Obviously, otherphysical quantities of interest can be evaluated similarly.

To end this section, we briefly discuss the validity of thepresent method, which is referred to as the CHRW methodhereafter. Obviously, its validity depends on the validity ofHCHRW(t) and Eqs. (13) and (14). Roughly speaking, HCHRW(t)works well for �x/ωx < 1 (0.2) and �z/ω0 < 0.2 (1) as longas ωz < ωx [28]. In the valid regime of HCHRW(t), Eqs. (13)and (14) turn out to be justified if | +−| > κ because of thepartial secular approximation.

C. Dynamics of population difference and time-averagedexcited-state population

To exemplify the performance of the CHRW method aswell as the effects of the CR terms of the transverse field, wefirst calculate the dynamics of the population difference andcompare our results with those of the numerically exact methodand the RWA method. The exact results can be obtained byusing the GFT to solve the master equation (2) directly. This isdifferent from the above formalism. We state in Appendix B thedetailed procedure to calculate the time evolution of the TLSdescribed by the master equation (2) by using the GFT directly,which is referred to as the GFL formalism [31]. The RWAresults are obtained by the method presented in Ref. [25], wherethe CR terms of the transverse field have been neglected andthe master equation with the RWA Hamiltonian is numericallysolved in a rotating frame by the Floquet theory without anyother approximation (its validity fully depends on the RWA),which is referred to as the RWA method. In addition, to carryout numerical calculation, we choose the parameters ω0 = 1meV, ωz = 0.18ω0, and κ = 1010 s−1 [26]. Moreover we set�z = 0.15ωz and �x ∼ ωz. Clearly, the CHRW method isapplicable to other driving regimes as long as the parametersare in the range of validity.

FIG. 1. Dynamics of population difference for ωx = 0.7588ω0,ωz = 0.18ω0, �z = 0.15ω0, and κ = 0.00658ω0. The transversefield strengths are (a) �x = 0.09ω0 and (b) �x = 0.24ω0. Insets:Dynamics at the time interval (900,1000)ω−1

0 .

Figures 1(a) and 1(b) display the dynamics of the populationdifference of the TLS calculated from the three methods:CHRW, GFL, and RWA, for ωx = 0.7588ω0 and two differentstrengths �x = 0.09ω0 and �x = 0.24ω0, respectively. Forboth the weak and strong transverse fields, one finds thatboth short-time and long-time behaviors predicted by theCHRW method are in excellent agreement with those fromthe numerically exact GFL method. Additionally, the insetsof Figs. 1(a) and 1(b) show that the CHRW method capturesfast-oscillating behaviors caused by the CR terms of thetransverse field. In contrast, the RWA method is valid for weakdriving but becomes invalid for strong driving compared to theGFL method. More specifically, the fast-oscillating behaviorsare averaged out due to the RWA (see the dynamics in theinsets) and there is a large difference in the long-time behaviorsbetween the RWA and non-RWA results in Fig. 1(b), implyingthat the CR terms have essential contribution to the steady statein the strong driving regime. In addition, for other driving pa-rameters, it is straightforward to verify that the CHRW methodcan yield sufficiently accurate results compared to the exactGFL results provided the parameters are in the valid regime ofHCHRW(t) [28] and | +−| > κ . The latter requirement is dueto the partial secular approximation we used.

We continue to analyze in detail the difference betweenthe RWA and non-RWA theories in the long-time limit. In

033817-4269


FIG. 2. Time-averaged excited-state population as a function ofωx for ωz = 0.18ω0, �z = 0.15ω0, and κ = 0.00658ω0. Transversefield strengths are (a) �x = 0.09ω0 and (b) �x = 0.24ω0. Inset in (a):Zoom of the resonance peaks as indicated by the arrow.

Figs. 2(a) and 2(b), we calculate the time-averaged excited-state population as a function of the transverse frequency ωx

for �x = 0.09ω0 and �x = 0.24ω0, respectively. For �x =0.09ω0, the CHRW and RWA results are found to almostcoincide with each other except for very small shifts betweenthem. The inset in Fig. 1(a) shows the shift between the twopeaks at ωx ≈ 1.36ω0. This type of shift is actually the BSshift caused by the CR terms of the transverse field. Obviously,except the peaks near ω0 ± 2ωz, the BS shifts are far smallerthan the widths of resonance peaks and thus can be safelyneglected. For �x = 0.24ω0, Fig. 2(b) shows that the CHRWcurve (solid line) is apparently shifted from the RWA one(dashed line). Interestingly, the peaks at ωx ≈ 0.75ω0 andωx ≈ 1.25ω0 have widths comparable with the BS shifts. Insuch cases, the resonance positions of the non-RWA Hamil-tonian become “far” off-resonance positions for the RWAHamiltonian, and one can expect that the RWA and non-RWAsteady-state behaviors can be quite different. This is actuallythe reason that the large difference appears in Fig. 1(b), whereωx = 0.7588ω0 used in Fig. 1(b) corresponds to the resonancefrequency (the abscissa of the maximum) of the non-RWAmultiphoton-resonance peak near 0.75ω0 in Fig. 2(b).

We now analyze quantitatively the BS shifts in Fig. 2.In general, the values of the BS shifts can be given by the

FIG. 3. Time-averaged excited-state population vs ωx and �x

calculated by the CHRW method for ωz = 0.18ω0, �z = 0.15ω0,and κ = 0.00658ω0. The dashed line and dotted line show theresonance positions at ωx ≈ ω0 − mωz (m = 0,2) determined by thetime-averaged transition probability P obtained from the CHRWHamiltonian.

difference in the abscissas of the maxima of the non-RWAand RWA peaks representing the resonance ω0 ≈ ωx + mωz,namely, the BS shift is quantified by δω

(m)BS = ω(m)

res − ω(m,RWA)res ,

where ω(m)res (ω(m,RWA)

res ) denotes the abscissa of the maximum ofresonance peak at ωx ≈ ω0 − mωz for the non-RWA (RWA)Hamiltonian. Alternatively, ω(m)

res can be efficiently determinedfrom the time-averaged transition probability [28]

P = |〈+|US(t,t ′)|−〉|2 = 12 (1 − |Z(0,0)

++ |2) (22)

with |∓〉 being the bare ground and excited states of the TLS, bysearching the transverse frequencies that satisfy P = 1

2 . In theweak damping regime, resonance frequencies ω(m)

res calculatedfrom P and ρ++ are consistent with each other. In Fig. 3, weshow the behavior of the time-averaged excited-state popu-lation versus ωx and �x for ωz = 0.18ω0 and �z = 0.15ω0,which is obtained by the CHRW method. We also calculate theresonance frequencies ωx ≈ ω0 − mωz (m = 0,2) by using P ,shown by dashed and dotted lines. Indeed, one finds that theresonance positions given by P and ρ++ are consistent witheach other. Additionally, this result displays the shifts of theresonance peaks as the variation of �x . In Tables I and II,we present the resonance frequencies calculated from P = 1

2by three different methods: GFT, CHRW, and RWA. Here theGFT results are obtained by calculating |Z(0,0)

++ | with a standardGFT approach (see the details in Refs. [28,29] and references

TABLE I. Resonance frequencies (ωx ≈ ω0 − mωz) calculatedfrom P and BS shifts δω

(m)BS quantified by the difference between

the GFT and RWA results in units of ω0 for �x = 0.09ω0.

m GFT CHRW RWA δω(m)BS

−2 1.349265 1.349262 1.347528 0.001737−1 1.163870 1.163864 1.161986 0.0018840 1.002029 1.002024 1 0.0020291 0.840214 0.840205 0.838014 0.0022002 0.654905 0.654896 0.652472 0.002433

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TABLE II. Resonance frequencies (ωx ≈ ω0 − mωz) calculatedfrom P and BS shifts δω

(m)BS quantified by the difference between the

GFT and RWA results in units of ω0 for �x = 0.24ω0.

m GFT CHRW RWA δω(m)BS

−2 1.269542 1.269514 1.256016 0.0135260 1.014855 1.014349 1 0.0148552 0.758869 0.758839 0.743984 0.014885

therein). The CHRW and RWA results are obtained by usingthe CHRW and RWA Hamiltonians. Comparing the CHRWresults with GFT results, we find that the CHRW method indeedeffectively takes the BS shifts into account. It allows us to figureout sufficiently accurate multiphoton-resonance positions andthus study the multiphoton-resonance-induced fluorescence.

To end this section, let us discuss the influence of the usualsecular approximation on the excited-state population. If weuse the secular approximation for the total master equation(8) (not just for its homogeneous part), we obtain the CHRW-SA results in Fig. 2. Specifically, ρ++ is obtained with thesimplified population difference

〈σz(t → ∞)〉 =∑

α

Z(0,0)αα ρ(0,0)

αα , (23)

where

ρ(0,0)++ = L(0,0)

++,−−�rel

=∑

k,l |X(k,l)−+ |2∑

k,l(|X(k,l)+− |2 + |X(k,l)

−+ |2),

ρ(0,0)−− = 1 − ρ

(0,0)++ =

∑k,l |X(k,l)

+− |2∑k,l(|X(k,l)

+− |2 + |X(k,l)−+ |2)

(24)

are the average populations of the two-mode Floquet states. Itis evident that the CHRW-SA results overestimate the excited-state population at the narrow multiphoton-resonance peaks,indicating inadequacy of the secular approximation in suchsituations. In fact, such narrow multiphoton resonancemay beencountered over a wide range of the parameters. In the fol-lowing, we shall see the influence of the secular approximationon the fluorescence spectrum.

III. FLUORESCENCE SPECTRUM

A. Derivation of the fluorescence spectrum

The steady fluorescence spectrum is given by the Fouriertransform of the two-time correlation function [32,33]

S(ω) ∝ Re∫ ∞

0lim

t ′→∞g(1)(τ ; t ′)e−iωτ dτ, (25)

with

g(1)(τ ; t ′) = TrS+R[U †(t ′ + τ )σ+U (t ′ + τ )

×U †(t ′)σ−U (t ′)ρ(0) ⊗ ρR] (26)

being the two-time correlation function, where U (t) representsthe time evolution operator of the composite system of theTLS and radiative reservoir. ρR is the reference state of thereservoir. Making use of the quantum regression theory [34],we obtain the two-time correlation function from the single-time expectation 〈σ+(t)〉 by just replacing the initial conditionρ(0) with σ−ρ(t ′) and thus derive the following spectrumfunction:

S(ω) = Scoh(ω) + Sinc(ω), (27)

Scoh(ω) ∝∑k,l

∑α,β,n,p

X(n,p)αβ ρ

(k−n,l−p)βα

∑α′,β ′,m,q

Y(m,q)α′β ′ ρ

(k−m,l−q)β ′α′ δ(ω − kωx − lωz), (28)

Sinc(ω) ∝∑k,l

{2X

(k,l)++

P(−k,−l)++ �rel + Q

(−k,−l)++ (ω − kωx − lωz)

�2rel + (ω − kωx − lωz)2

+ X(k,l)+−

P(−k,−l)−+ �deph + Q

(−k,−l)−+ (ω − kωx − lωz − +−)

�2deph + (ω − kωx − lωz − +−)2

+X(k,l)−+

P(−k,−l)+− �deph + Q

(−k,−l)+− (ω − kωx − lωz + +−)

�2deph + (ω − kωx − lωz + +−)2

}, (29)

where P(k,l)αβ = ReF (k,l)

αβ , Q(k,l)αβ = ImF

(k,l)αβ , and

F(k,l)αβ =

∑λ,n,m

Y(n,m)αλ ρ

(k−n,l−m)λβ −

∑m,q

ρ(m,q)αβ

∑α′,β ′,n,p

Y(n,p)α′β ′ ρ

(k−n−m,l−p−q)β ′α′ . (30)

The detailed derivation is presented in Appendix C. We stress that we introduced the partial secular approximation to the masterequation (8) when deriving the above spectral function.

In this paper, we focus on the spectral features of the incoherent part of the spectrum, Sinc(ω). The leading-order terms of theincoherent part can be extracted from Eq. (29) by retaining ρ

(k,l)αβ if k = l = 0 and α = β but setting ρ

(k,l)αβ = 0 otherwise. We

obtain a more simplified and physically transparent expression:

Sinc(ω) ∝∑k,l

{|X(k,l)

++ |2[1 − (ρ(0,0)++ − ρ

(0,0)−− )2]

�rel

�2rel + (ω − kωx − lωz)2

+ |X(k,l)+− |2ρ(0,0)

++�deph

�2deph + (ω − kωx − lωz − +−)2

+ |X(k,l)−+ |2ρ(0,0)

−−�deph

�2deph + (ω − kωx − lωz + +−)2

}. (31)

033817-6271


If the average populations of the Floquet states given in Eq. (24)are used in the above expression, this result corresponds to thespectrum derived with the secular approximation as usual forthe total master equation (8). Obviously, Eq. (31) indicatesthat the spectrum consists of a series of emission lines theweights, line widths, and positions of which are related to thepopulations, the decay rates, and the quasienergy spacing ofthe two-mode Floquet states.

In general, the validities of Eqs. (29) and (31) are different.Roughly speaking, provided that the driving parameters arein the valid regime of the CHRW Hamiltonian, the formerholds if | +−| > κ while the latter holds if | +−| � κ . Thedifference in the valid conditions for Eqs. (29) and (31) resultsfrom the fact that the secular approximation was used for thehomogeneous part of the master equation to derive Eq. (29)but for the total master equation to arrive at Eq. (31).

The present formalism allows an interpretation of fluores-cence in the context of the transitions between the two-modeFloquet states, through which the spectral features can beunderstood. Let us explore the physical significance of X

(k,l)αβ as

the characteristics of the spectrum strongly depend on X(k,l)αβ .

Recalling its definition in Eq. (A3), we can rewrite it in analternative form:

X(k,l)αβ = lim

T →∞1

T

∫ T

0〈uα,N+k,M+l(t)|σ+|uβ,N,M (t)〉dt, (32)

where N and M are arbitrary integers. From this expression,we note that X

(k,l)αβ is the transition matrix element between

〈uα,N+k,M+l(t)| and |uβ,N,M (t)〉, which can be used to indi-cate the transition between |uα,N+k,M+l(t)〉 and |uβ,N,M (t)〉.Basically, a transition from |uα,N+k,M+l(t)〉 to |uβ,N,M (t)〉generates a photon with the frequency given by the differenceof the quasienergies of the two-mode Floquet states, i.e.,ω = εα,N+k,M+l − εβ,N,M = αβ + kωx + lωz, which are thepositions of the emission lines in Eqs. (29) and (31). Conse-quently, several such different transitions are possible to causeseveral individual emission lines, yielding the multipeakedspectrum. In general, only specific transitions contribute tothe spectrum, which can be figured out by analyzing theproperties of X

(k,l)αβ . Specifically, for the bichromatically driven

TLS under study, it follows from Eq. (A3) that X(k,l)αβ = 0

with k being even numbers and l being any integer, butX

(k,l)αβ becomes nonvanishing with k being odd numbers and

certain values of l depending on the longitudinal modulation.Moreover, provided that �x/ωx < 1, X

(k,l)αβ decreases rapidly

with the increase of k (this is due to the properties of thehigher-order Bessel functions, and physically this means thatthe contributions of the multiphoton processes of the transversefield are negligible). Therefore, it is sufficient to consider X

(k,l)αβ

with k = 1. All in all, the present formalism allows us tocomprehensively understand the spectral properties illustratedbelow.

B. Off-resonance and multiphoton-resonance-inducedfluorescence

In this section, we study fluorescence under the off- andmultiphoton-resonance conditions. We first benchmark theresults of Eq. (29) (CHRW), Eq. (31) (CHRW-SA), and the

FIG. 4. Incoherent fluorescence spectrum for �x = 0.09ω0, ωx =0.8737ω0, ωz = 0.18ω0, �z = 0.15ω0, and κ = 0.00658ω0. TheCHRW curve shows the result from Eq. (29) while the CHRW-SAcurve shows the result from Eq. (31).

RWA method [25] against those of the exact GFL methodto further confirm the performance of various approximatemethods. In Appendix B, we present the detailed procedure tocalculate the spectrum with the GFL formalism. In Figs. 4–8,one notes that the CHRW results are in good agreement withthe GFL results in both the off-resonance and multiphoton-resonance cases as well as in relatively weak and strong drivingregimes, whereas the RWA and CHRW-SA results may differfrom the exact results under certain conditions. Below we focuson the spectral features of the fluorescence and the CR-inducedmodifications to the spectrum.

To begin with, we show that the emission line at the drivingfrequency ωx may be eliminated from the spectrum. Figure 4shows the line shapes of the spectrum in the off-resonancecase, i.e., �x = 0.09ω0 and ωx = 0.8737ω0. It is evident thatthere is no emission line at ωx . This is actually due to thefact that |X(1,0)

++ | = 0 with the given parameters, correspondingto the fact that the specific transitions that yield the emissionline at ωx are physically forbidden under certain conditions.Since |X(1,0)

++ | has multiple zeros (see Fig. 10), the phenomenonof the missing emission line at ωx can be predicted in otherparameter regimes. Furthermore, in the following we will seethat this phenomenon can be influenced by the CR terms,leading to a discrepancy between the RWA and non-RWAresults. On the other hand, one notes that the spectrum is foundto be asymmetric when ωx �= ω0. Asymmetric spectra are alsofound in Ref. [27] in the perturbative regime (ωz ∼ �x � �z)but cannot be given by the analytical result in Ref. [26],which predicts that the spectrum is always symmetric. Theartefact symmetry of the analytical result is because of the usedapproximation and is further discussed in the next section.

Previously, we have shown that the spectrum is symmetric inthe RWA case when ωx = ω0 [25]. Here we examine whetherthe spectrum remains symmetric in the non-RWA case whenωx = ω0 and the driving strength is moderately strong, e.g.,�x = 0.24ω0. Figure 5(a) displays the line shapes of thespectrum for ωx = ω0. It is found that the RWA spectrum issymmetric while the CHRW and GFL spectra are not symmet-ric. Since the RWA theory predicts symmetric spectra at the

033817-7272


FIG. 5. Incoherent fluorescence spectrum for �x = 0.24ω0, ωz =0.18ω0, �z = 0.15ω0, and κ = 0.00658ω0. The transverse frequen-cies used are (a) ωx = ω0 and (b) ωx = 1.0144ω0.

RWA single-photon resonance, it is then intuitive to imaginethat the non-RWA theory may predict symmetric spectra atthe shifted single-photon resonance. Figure 5(b) shows thespectrum for the transverse frequency ωx = ω0 + �2

x/(4ω0),which takes the second-order BS shift [30,35,36] into account.One notes that the non-RWA spectrum is almost symmetric [bycalculation one finds that Sinc(ωx − ωz) − Sinc(ωx + ωz) �= 0is not vanishingly small, i.e., the non-RWA spectrum is notstrictly symmetric]. However, the RWA spectrum becomesobviously asymmetric due to the detuning. The present resultssuggest that the CR terms of the transverse field can break thesymmetry of the spectrum at ωx = ω0 but cause the almostsymmetric spectra at the shifted single-photon resonance,which will be verified further in the next section.

Next, we examine the line shapes of the fluorescence spec-trum at the multiphoton-resonance positions in Fig. 2(a), e.g.,ωx = 0.8402ω0 and 0.6549ω0 for �x = 0.09ω0. Figure 6 dis-plays the multiphoton-resonance-induced fluorescence spec-tra. In Fig. 6(a), the spectra are apparently asymmetric andseem to consist of three separated small triplets centered at ωx ,ωx + ωz, and ωx + 2ωz. Clearly, the triplet centered at ωx + ωz

has the greatest intensity. Additionally, the blue sideband ofthe small triplet centered at ωx nearly vanishes, which canbe attributed to |X(1,0)

+− | � 0. In Fig. 6(b), the spectra are alsosomewhat asymmetric and are made up of single-peaked emis-sion lines at ωx + mωz (m = 1,2,3) and a double-peaked line

FIG. 6. Incoherent part of the multiphoton-resonance-inducedfluorescence spectrum for �x = 0.09ω0, ωz = 0.18ω0, �z = 0.15ω0,and κ = 0.00658ω0. The transverse frequencies are (a) ωx =0.8402ω0 and (b) ωx = 0.6549ω0. Insets: Zoom of the spectralcomponents as indicated by the arrows.

centered atωx . This time one finds that the most intense spectralcomponent appears at ω ≈ ωx + 2ωz. These results suggestthat for the multiphoton-resonance conditions (ωx + mωz ≈ω0) the spectrum is generally asymmetric and the most intensespectral component may occur at ω ≈ ωx + mωz. Interest-ingly, in some senses, the line shapes shown in Figs. 6(a) and6(b) resemble those obtained in the monochromatic field underthe three- and five-photon resonance conditions [31] but differfrom those predicted with the same bichromatically driven TLSin the perturbative regime [27]. In addition, Fig. 6(b) shows thatthe CHRW-SA and RWA results are not consistent with theGFL and the CHRW results, indicating the breakdown of thesecular approximation and RWA. The breakdown of the formercan be simply attributed to the fact that it overestimates theexcited-state population at the narrow multiphoton resonance(see Fig. 2). The breakdown of the RWA is because it missesthe effects of the CR terms.

We now illustrate a marked difference between the RWAand non-RWA spectra at the multiphoton-resonance position inFig. 2(b), e.g., �x = 0.24ω0 and ωx = 0.7588ω0. Figure 7(a)shows that the RWA and the non-RWA spectra have a dra-matic difference in the intensities of the emission lines. It isevident that the components of the RWA and non-RWA spectracentered at ωx + 2ωz dramatically differ from each other. In

033817-8273


FIG. 7. Incoherent part of the multiphoton-resonance-inducedfluorescence spectrum for �x = 0.24ω0, ωx = 0.7588ω0, ωz =0.18ω0, κ = 0.00658ω0, and two values of �z.

addition, the emission line of the non-RWA spectrum at ωx

disappears while that of the RWA spectrum still exists. This isattributed to the fact that the zeros of |X(1,0)

++ | of the non-RWAtheory are shifted from those of the RWA theory due to the CRterms of the transverse field. Figure 7(b) shows that when �z =0 the difference between the RWA and non-RWA theoriesin the monochromatic field is not as obvious as that in thebichromatic field. Moreover, the intensity of fluorescence for�z = 0 is much weaker than that for �z = 0.15ω0, indicatingthat the longitudinal modulation may be used to enhance theemission under certain conditions. The present result indicatesthat the CR terms have non-negligible contributions to thespectrum at certain multiphoton resonance. In addition, amarked difference between the RWA and non-RWA spectracan be predicted at other resonance positions of the resonanceband ωx ≈ ω0 ± 2ωz (see Fig. 3).

Apart from the strong driving regime �x ∼ ωz ∼ �z,the marked difference between the RWA and the non-RWAtheories can also be found in other regimes. We considerωx = 0.8694ω0 and ωz = 0.18ω0 and weaker strengths �x =0.12ω0 and �z = 0.015ω0. Figure 8(a) shows that the differ-ence between the RWA and the non-RWA spectra is similarto that shown in Fig. 6(b). This time the marked differencebetween the RWA and the non-RWA spectra occurs at ωx + ωz.However, Fig. 8(b) shows that when �z = 0, such difference

FIG. 8. Incoherent part of the multiphoton-resonance-inducedfluorescence spectrum for �x = 0.12ω0, ωx = 0.8694ω0, ωz =0.18ω0, κ = 0.00658ω0, and two values of �z.

between the RWA and non-RWA theories vanishes. The com-parison between the cases of �z = 0 and �z �= 0 in Figs. 7 and8 indicates that the effects of the CR terms on the spectrummay be much more easily detected in the bichromatic casethan in the monochromatic case. Below we discuss in detailthe breakdown of the RWA.

The difference between the RWA and non-RWA theoriesillustrated can be attributed to the combined effect of the BSshifts and narrow resonance widths. To understand how thedifference arises, it is instructive to recall that ωx = 0.6549ω0

[Fig. 6(b)] and 0.7588ω0 [Fig. 7(a)] are the multiphoton-resonance frequencies (ωx + 2ωz ≈ ω0) of the CHRW Hamil-tonian for the different driving strengths. From Tables I and II,one finds that there are BS shifts given by δω

(2)BS between the

RWA and non-RWA Hamiltonians. Moreover, Fig. 2 shows thatthe corresponding resonance peaks are very narrow, the widthsof which may be comparable with the BS shifts. Therefore, theresonance frequencies of the non-RWA Hamiltonian are actu-ally “far” off-resonance frequencies for the RWA Hamiltonianin spite of the small detuning δω

(2)BS. It follows that a consid-

erable decrease in the steady excited-state population can beexpected for the RWA Hamiltonian at the resonance positionsof the non-RWA Hamiltonian, leading to the fact that the in-tensity of the RWA spectrum differs from that of the non-RWAspectrum. In contrast, if the resonance width is much larger than

033817-9274


the BS shift, the latter may be safely neglected, which leadsto the consistency of the RWA and non-RWA theories [seeFig. 6(a)]. All in all, in the multiphoton-resonance situationswhere the resonance width is comparable with the BS shift,the RWA and non-RWA spectra become markedly different.

To end this section, we give some remarks on the abovefindings. First, most importantly, it is found that the CR termis essential to the multiphoton-resonance-induce fluorescenceif the resonance has a width comparable with the BS shift.At such a resonance, the RWA and non-RWA spectra canbe quite different. Second, the emission line at the drivingfrequency ωx can be eliminated from the spectrum at thezeros of |X(1,0)

++ |. In general, the zero of |X(1,0)++ | depends on the

driving parameters. Similar to the resonance positions, theremay exist a non-negligible shift between the zeros predictedwith and without the CR terms, leading to the fact that theemission line at ωx appears in the RWA spectrum but vanishesin the non-RWA spectrum [see Fig. 7(a)] and vice versa.Third, the CR terms can break the symmetry of the spec-trum in the vanishing detuning case. The spectrum generallyhas asymmetric multipeaked structures in the off-resonancecases as well as multiphoton-resonance cases, similar to theproperties of the fluorescence from a bichromatically drivenTLS with transverse fields different from the present model[20,21,23,24]. In the following, we explore how the symmetryoccurs and how it is broken by the CR terms.

C. Symmetry and asymmetry of the spectrum

The origin of the symmetry and asymmetry of the spectrumhas not been analyzed in detail even within the framework ofthe RWA. In Ref. [27], the symmetry is simply attributed to thefact that at ωx = ω0 saturation is achieved for arbitrary longi-tudinal modulation strength while the asymmetry is attributedto the unequal redistribution of populations among the doublydressed states. However, it is easy to verify that the RWA spec-trum is symmetric as long as ωx = ω0 even if saturation is notachieved (in the presence of a weak enough transverse field);when the populations of the two-mode Floquet states (doublydressed states) are equal, i.e., ρ

(0,0)++ = ρ

(0,0)−− , the spectrum

can be asymmetric, e.g., the multiphoton-resonance-inducedfluorescence. In the following we first explore the origin ofthe symmetry and asymmetry with the RWA Hamiltonian andthen study how the CR terms of the transverse field influencethe symmetry of the spectrum.

We use Eq. (31) to analyze the symmetry of the spectrumas it is not only adequate to give accurate description in mostcases but also the leading-order contribution to the spectrum. Ingeneral, if the spectrum is symmetric, Eq. (31) indicates that thesymmetry axis may be ω = ωx + mωz (m = 0, ± 1, ± 2, . . .).For the symmetric spectrum, we must have the same weightsof the emission lines the positions of which are symmetric withrespect to the possible symmetry axis:

|X(1,m+l)++ | = |X(1,m−l)

++ | (l = 1,2,3, . . .), (33)

and

|X(1,m+l)+− |2ρ(0,0)

++ = |X(1,m−l)−+ |2ρ(0,0)

−− (l = 0,1,2, . . .). (34)

In principle, for a given m, i.e., a certain symmetry axis,these equalities strongly constrain the values of the driving

FIG. 9. |X(k,l)++ | with k = 1 and l = ±1, ± 2 vs ωx calculated

from the RWA Hamiltonian for ωz = 0.18ω0, �z = 0.15ω0, andvarious �x .

parameters to obtain the symmetric spectrum. On the otherhand, for any given m, i.e., every possible symmetry axis,if the equalities (33) and/or (34) cannot hold simultaneously,one concludes that the spectrum is asymmetric. In practice,we find that the conditions to get the symmetric spectrumare very limited, and the spectrum is generally asymmetric.However, if the transition matrix element X

(k,l)αβ is analytically

calculated in a limiting case as given in Appendix D, theequalities (33) and (34) can always be satisfied, i.e., thespectrum is always symmetric. This suggests that improperapproximations invoked in calculating the transition matrixelement would cause artefact symmetry of the spectrum.

We first consider the RWA Hamiltonian and m = 0. InFig. 9, we display the matrix elements |X(1,±l)

++ | numericallycalculated as a function of ωx with l = 1,2 for ωz = 0.18ω0,�z = 0.15ω0, and various �x . Interestingly, the numerical re-sults show that when l = 1,2, the curves of |X(1,l)

++ | and |X(1,−l)++ |

are mirror images of each other about ωx = ω0. By furthernumerical calculation, we find that this mirror symmetry existsbetween |X(1,l)

αβ | and |X(1,−l)βα | for l = 0,1,2,3, . . .. Moreover,

Fig. 9 also indicates that the mirror symmetry is independentof the driving strength �x . As a result, we have the equalities|X(1,l)

αβ | = |X(1,−l)βα | as long as ωx = ω0. Thus, the equalities (33)

hold under the same condition. On the other hand, for the RWAHamiltonian, we find |X(k,l)

αβ | = 0 if k �= 1 from Eq. (A3) in thelimit of Z → 0. By this observation and using Eq. (24), wehave ρ

(0,0)++ = ρ

(0,0)−− at ωx = ω0 because of |X(1,l)

+− | = |X(1,−l)−+ |.

Therefore, the equalities (34) also hold at ωx = ω0. It turnsout that the basic equalities |X(1,l)

αβ | = |X(1,−l)βα | at ωx = ω0

guarantee that the spectrum is symmetric when ωx = ω0.Next, we examine if the spectrum may be symmetric

about ω = ωx + mωz for the RWA Hamiltonian and m �= 0.In Fig. 10, we show the behaviors of |X(1,m+l)

++ | and |X(1,m−l)++ |

with m = 1,2 and l = 1,2 as a function of ωx for two values of�x . For m = 1 and �x = 0.009ω0, Fig. 10(a) shows that thepart of the curves of |X(1,2)

++ | (solid line) and |X(1,0)++ | (dashed

line) nearly overlap with each other around ωx = 0.82ω0, andso do those of |X(1,3)

++ | (dot-dashed line) and |X(1,−1)++ | (dotted

033817-10275


FIG. 10. |X(k,l)++ | with k = 1 and various l vs ωx calculated from

the RWA Hamiltonian for ωz = 0.18ω0, �z = 0.15ω0, and two valuesof �x .

line), indicating that |X(1,1+l)++ | may equal to |X(1,1−l)

++ | closeto the multiphoton-resonance frequency ωx = ω0 − ωz. Byfurther numerical calculation carried out to find the valuesof ωx close to 0.82ω0 that satisfy the equalities |X(1,1+l)

++ | =|X(1,1−l)

++ |, we find that the values vary slightly with l, i.e., theequalities |X(1,1+l)

++ | = |X(1,1−l)++ | cannot simultaneously hold at

a nontrivial fixed ωx . Nevertheless, we find that the differencebetween |X(1,1+l)

++ | and |X(1,1−l)++ | at ωx = 0.82ω0 is so small

that one has |X(1,1+l)++ | � |X(1,1−l)

++ | for l = 1,2,3, . . .. However,when �x = 0.09ω0, Fig. 10(b) shows that one cannot evenfind such a nontrivial ωx that the difference between |X(1,1+l)

++ |and |X(1,1−l)

++ | is vanishingly small. Moreover, by comparingFigs. 10(a) and 10(b), it is intuitive to imagine that the dif-ference between |X(1,1+l)

++ | and |X(1,1−l)++ | becomes smaller and

smaller at ωx = 0.82ω0 as �x → 0. For m = 2, Figs. 10(c) and10(d) display a similar phenomenon as shown in Figs. 10(a) and10(b). Therefore, the present results suggest that, when �x �ωz, |X(1,m+l)

++ | � |X(1,m−l)++ | for l = 1,2,3, . . . occurs at the

multiphoton-resonance frequencies ωx = ω0 − mωz (m �= 0).

On the other hand, one can verify that |X(1,m+l)+− | � |X(1,m−l)

−+ |for l = 0,1,2,3, . . . under the same condition, leading to|X(1,m+l)

+− |2ρ(0,0)++ � |X(1,m−l)

−+ |2ρ(0,0)−− . To summarize, one cannot

get a strictly symmetric spectrum in the multiphoton-resonancecases but only gets an almost symmetric fluorescence spec-trum because of |X(1,m+l)

αβ | � |X(1,m−l)βα | when �x � ωz and

at multiphoton-resonance position ωx = ω0 − mωz (m �= 0).This is further confirmed by calculating the line shapes withthe RWA method (the results are not shown here). Despite thefact that Eq. (31) cannot predict quantitative accurate results(the secular approximation becomes invalid) when �x ∼ κ ,the equalities (33) and (34) can still be used to indicate thesymmetry condition, indicating that X

(k,l)αβ plays a fundamental

role in the symmetry of the spectrum.The above analysis leads us to draw the conclusion that for

the RWA Hamiltonian the spectrum is strictly symmetric whenωx = ω0, i.e., at the RWA single-photon resonance frequency.When �x � ωz, the spectrum is almost symmetric at themultiphoton-resonance frequency ωx = ω0 − mωz (m �= 0).Otherwise, the spectrum is asymmetric. Moreover, our analysisindicates that the symmetry of the spectrum can be attributedto the behaviors of X

(k,l)αβ under certain conditions. In the

following, we study how the symmetry condition is modifiedby the CR terms of the transverse field.

We explore how the CR terms influence the symmetry inthe moderately strong driving regime and in the case of m = 0.In Fig. 11(a), we calculate |X(1,±l)

++ | with l = 1,2 as a functionof ωx for �x = 0.24ω0. In comparison with Fig. 9(c), we notethat the central crossing of |X(1,l)

++ | and |X(1,−l)++ | is shifted to

the right side of ωx = ω0, i.e., the curves obtained from theCHRW Hamiltonian have been shifted due to the CR terms. Itis then intuitive to ask whether the shifts of the central crossingof |X(1,l)

++ | and |X(1,−l)++ | are equal for different l. To answer this

question, we numerically search the solutions, denoted as ω(l)X ,

to the equations |X(1,l)++ | − |X(1,−l)

++ | = 0 (l = 1,2) for variableωx near ω0. The behaviors of ω

(l)X as a function of �x are

shown in Fig. 11(b). We provide not only the numerical resultsfrom the CHRW Hamiltonian but also those from the originalHamiltonian with the aid of the GFT approach [28]. It is foundthat the curves for different l do not coincide, implying that theshifts of the central crossing positions of |X(1,±l)

++ | for l = 1,2are different. Consequently, the equalities |X(1,l)

++ | = |X(1,−l)++ |

for l = 1,2,3, . . . cannot simultaneously hold at a fixed ωx ,namely, the symmetry is broken by the CR terms. Neverthe-less, when �x < 0.3ω0, Fig. 11(b) shows that the differencebetween the curves is very small and the behaviors of ω

(l)X can

be approximated by the shifted single-photon resonance fre-quency that accounts for the second-order BS shift. In general,

we find |X(1,l)αβ | � |X(1,−l)

βα | at ωx = ω0 + �2x

4ω0. This is actually

the reason that we observe an almost symmetric non-RWAspectrum in Fig. 5(b) but an asymmetric non-RWA spectrumin Fig. 5(a). The present analysis leads us to conclude thatfor the non-RWA Hamiltonian we can only obtain an almostsymmetric spectrum at the shifted single-photon resonancefrequency when the driving strength is moderately strong.

In the multiphoton-resonance cases (m �= 0), the condi-tions to obtain almost symmetric spectra for the non-RWA

033817-11276


FIG. 11. (a) |X(k,l)++ | with k = 1 and l = ±1, ± 2 vs ωx calculated

from the CHRW Hamiltonian for �x = 0.24ω0, ωz = 0.18ω0, and�z = 0.15ω0. (b) Central crossing position ω

(l)X as a function of �x for

ωz = 0.18ω0, and �z = 0.15ω0. Inset: Zoom of curves in the interval(0.240,0.245)ω0.

Hamiltonian are the same as those for the RWA Hamiltoniansince the CR terms become insignificant if �x � ωz < ω0.In other cases, the non-RWA spectrum can be markedlyasymmetric.

IV. CONCLUSIONS

In summary, we have studied the fluorescence spectrum ofthe strongly driven TLS under frequency modulation in theoff-resonance and multiphoton-resonance cases by using theCHRW method. By the combination of unitary transformationand Floquet theory, the CHRW method allows us to efficientlycalculate the two-mode Floquet states and quasienergies fromthe effective Hamiltonian instead of the original Hamiltonianof the bichromatically driven TLS, which is more efficient thanthe usual GFT approach. We solved the master equation of thedriven TLS accounting for spontaneous emission in the basesof the two-mode Floquet states and got the time evolutionand the steady state of the dissipative bichromatically drivenTLS, and derived the fluorescence spectrum. It is foundthat the fluorescence spectrum of the present formalism isnot only physically transparent but also adequately accuratecompared to the numerically exact results over a wide rangeof the driving parameters where the RWA and the secularapproximation may break down, which are frequently used inthe analytical calculation.

In contrast with the previous spectra that are derived withinthe framework of the RWA and/or perturbation theory [25–27],the validity of which is limited, the present spectrum enablesus to include the driving much more accurately as long as thetwo-mode Floquet states are calculated accurately, which canbe given by the semianalytical CHRW method or the standardGFT approach (but with a lower calculation efficiency). Thus,it is applicable to the regime where the RWA and perturbationtheories are inapplicable. Additionally, it is compatible withnot only numerical implement but also analytical calculation.In particular, the spectrum in Ref. [26] can be easily reproducedfrom Eq. (31) as long as we derive the two-mode Floquet statesin a limiting case, i.e., �x � ωz.

We illustrated several features of the fluorescence spec-trum, which may be strongly modified by the CR termsof the transverse field. First, we demonstrated that in themultiphoton-resonance situations where the resonance widthsare comparable with the BS shifts, the line shapes of the RWAand non-RWA spectra markedly differ from each other, whichis relevant to the effect of the BS shifts on the steady state.Second, the emission line at ωx may vanish due to the zeros of|X(1,0)

++ |. The zeros can be shifted by the CR terms, leading to theinconsistency of the RWA and non-RWA spectra when drivingis moderately strong. Third, the symmetry of the spectrumcan be attributed to the behaviors of the transition matrixelements, X(k,l)

αβ . It turns out that for the RWA Hamiltonian thesymmetry of the spectrum can be expected as long as ωx = ω0.However, when the transverse field is moderately strong, theCR terms can break the symmetry of the spectrum in the caseof ωx = ω0 but cause almost symmetric spectra at the shiftedsingle-photon resonance that takes the BS shift into account. Inaddition, at the multiphoton resonance, the almost symmetricspectra occur when the transverse field is sufficiently weak. Ourresults suggest that the strongly driven TLS under frequencymodulation may be a promising candidate for studying themultiphoton-resonance-induced fluorescence and the effectsof the CR terms in the moderately strong driving regime(where the effects of the CR terms become insignificant inthe monochromatically driven TLS).

ACKNOWLEDGMENTS

This work was supported by the National Natural ScienceFoundation of China (Grants No. 11647082, No. 11474200,No. 11774311, and No. 11774226).

APPENDIX A: DERIVATION OF THE MASTER EQUATIONIN THE GENERALIZED FLOQUET PICTURE

It is straightforward to rewrite the master equation in thebases of the two-mode Floquet states. Differentiating ραβ(t) =〈uα(t)|ρ(t)|uβ(t)〉 with respect to time t , one finds

d

dtραβ (t) = d〈uα(t)|

dtρ(t)|uβ(t)〉 + 〈uα(t)|ρ(t)

d|uβ(t)〉dt

+〈uα(t)|dρ(t)

dt|uβ(t)〉

= −i(εα − εβ)ραβ(t) − κ

2〈uα(t)|[σ+σ−ρ(t)

+ ρ(t)σ+σ− − 2σ−ρ(t)σ+]|uβ(t)〉, (A1)

033817-12277


where we have used the relation given in Eq. (4). Insertingthe completeness relation of the generalized Floquet states∑

λ |uλ(t)〉〈uλ(t)| = I , where I represents an identity matrix,between the operators of the dissipation terms, one gets,e.g.,

〈uα(t)|σ+σ−ρ(t)|uβ(t)〉=∑α′,λ

〈uα(t)|σ+|uλ(t)〉〈uλ(t)|σ−|uα′(t)〉〈uα′(t)|ρ(t)|uβ(t)〉

=∑α′,λ

〈uα(t)|σ+|uλ(t)〉〈uλ(t)|σ−|uα′(t)〉ρα′β(t)

=∑

α′,β ′,λ

〈uα(t)|σ+|uλ(t)〉〈uλ(t)|σ−|uα′ (t)〉δβ,β ′ρα′β ′ (t).

(A2)

To proceed, we derive the two-mode Fourier expansion of〈uα(t)|σj |uβ(t)〉 (j = ±,z) with the two-mode Floquet statesobtained in Sec. II A as follows:

〈uα(t)|σ+|uβ(t)〉= 〈uα(t)|R(t)eS(t)σ+e−S(t)R†(t)|uβ(t)〉

= 1

2{1 + cos[Z sin(ωxt)]}eiωx t 〈uα(t)|σ+|uβ(t)〉

+1

2{1 − cos[Z sin(ωxt)]}e−iωx t 〈uα(t)|σ−|uβ(t)〉

−1

2i sin[Z sin(ωxt)]〈uα(t)|σz|uβ(t)〉

=∑k,l

1

2[f +

k−1x(+)αβ,l + f −

k+1x(−)αβ,l − f z

k x(z)αβ,l]e

i(kωx+lωz)t

≡∑k,l

X(k,l)αβ ei(kωx+lωz)t . (A3)

Here, we have used the following Fourier expansions:

1 ± cos[Z sin(ωxt)] =∑

k

f ±k eikωx t , (A4)

i sin[Z sin(ωxt)] =∑

k

f zk eikωx t , (A5)

〈uα(t)|σj |uβ(t)〉 =∑

l

x(j )αβ,le

ilωzt , (A6)

with their Fourier components given by

f ±k = ωz

2π

∫ 2π/ωz

0{1 ± cos[Z sin(ωxt)]}e−ikωx t dt

= [1 ± J0(Z)]δk,0 ±∞∑

n=1

J2n(Z)(δ2n,k + δ−2n,k), (A7)

f zk = ωz

2π

∫ 2π/ωz

0i sin[Z sin(ωxt)]e

−ikωx tdt

=∞∑

n=1

J2n−1(Z)(δ2n−1,k − δ1−2n,k), (A8)

x(j )αβ,l = ωz

2π

∫ 2π/ωz

0〈uα(t)|σj |uβ(t)〉e−ilωzt dt. (A9)

In the same way, we have

〈uα(t)|σ−|uβ(t)〉 =∑k,l

Y(k,l)αβ ei(kωx+lωz)t , (A10)

〈uα(t)|σz|uβ(t)〉 =∑k,l

Z(k,l)αβ ei(kωx+lωz)t , (A11)

where Y(k,l)αβ = X

(−k,−l)∗βα and

Z(k,l)αβ = (f +

k − δk,0)x(z)αβ,l + f z

k+1x(−)αβ,l − f z

k−1x(+)αβ,l . (A12)

Substituting Eqs. (A3) and (A10) into Eq. (A2), we have

〈uα(t)|σ+σ−ρ(t)|uβ(t)〉 =∑

α′,β ′,k,l

(∑λ,n,m

X(n,m)αλ Y

(k−n,l−m)λα′

)

× ei(kωx+lωz)t δβ,β ′ρα′β ′ (t). (A13)

Obviously, similar expressions for 〈uα(t)|ρ(t)σ+σ−|uβ(t)〉 and〈uα(t)|σ−ρ(t)σ+|uβ(t)〉 can be obtained. Thus, we get themaster equation in the generalized Floquet picture.

APPENDIX B: GENERALIZED FLOQUET-LIOUVILLEFORMALISM

It has been shown that the GFT can be applied to thequantum system driven by the polychromatic transverse fieldin the presence of the dissipation [31]. Obeying the samerules, we use the GFT to solve the master equation (2). Wefirst rewrite the master equation in terms of the Bloch vector〈�σ (t)〉 = (〈σ+(t)〉,〈σ−(t)〉,〈σz(t)〉)T, which reads

id

dt〈�σ (t)〉 = M(t)〈�σ (t)〉 + �b, (B1)

where

M(t) =⎛⎝−ω0 − �z cos(ωzt) − i κ

2 0 �x cos(ωxt)0 ω0 + �z cos(ωzt) − i κ

2 −�x cos(ωxt)2�x cos(ωxt) −2�x cos(ωxt) −iκ

⎞⎠, (B2)

and

�b = i(0,0, − κ)T. (B3)

For the homogeneous part of the Bloch equation (B1), weassume that it possesses the following ansatz according to the

GFT:〈�σ (t)〉 = �(t,t ′)〈�σ (t ′)〉, (B4)

with the evolution operator being

�(t,t ′) =∑

j=±,z

|φj (t)〉〈ϕj (t ′)|e−iεj (t−t ′), (B5)

033817-13278


where |φj (t)〉 and 〈ϕj (t ′)| are 3 × 1 and 1 × 3 vectors, re-spectively, and εj is called the characteristic exponent (todistinguish it from the quasienergies aforementioned).

We now calculate |φj (t)〉. Differentiating �(t,t ′) withrespect to t , one finds

[M(t) − i∂t ]|φj (t)〉 = εj |φj (t)〉, (B6)

which is similar to Eq. (4). One readily derives the two-mode Fourier expansion for the coefficient matrix M(t) =∑

k,l M(k,l)ei(kωx+lωz)t with

M (0,0) = diag(−ω0 − i

κ

2,ω0 − i

κ

2, − iκ

), (B7)

M (0,±1) = diag

(−�z

2,�z

2,0

), (B8)

M (±1,0) =

⎛⎜⎝

0 0 �x

2

0 0 −�x

2

�x −�x 0

⎞⎟⎠. (B9)

We assume the two-mode Fourier expansion for |φj (t)〉:|φj (t)〉 =

∑k,l

∣∣φ(k,l)j

⟩ei(kωx+lωz)t , (B10)

where |φ(k,l)j 〉 = (φ(k,l)

+,j ,φ(k,l)−,j ,φ

(k,l)z,j )T is a 3 × 1 time-

independent vector, each component of which is to bedetermined below. Substituting the two-mode Fourierexpansions of M(t) and |φj (t)〉 into Eq. (B6), we find atime-independent algebra equation:∑

k,l

[M (n−k,m−l) + (kωx + lωz)δk,nδl,m]∣∣φ(k,l)

j

⟩ = εj

∣∣φ(n,m)j

⟩.

(B11)

This equation represents a matrix equation M|φj 〉〉 = εj |φj 〉〉,where the “element” of M is given by

Mnm,kl = M (n−k,m−l) + (kωx + lωz)δk,nδl,m, (B12)

and the vector |φj 〉〉 is composed of |φ(k,l)j 〉 arranged in order.

In a similar way, we can evaluate 〈ϕj (t ′)| as well. Specifi-cally, we differentiate �(t,t ′) with respect to t ′ and get

〈ϕj (t ′)|[M(t ′) + i←−∂t ′ ] = εj 〈ϕj (t ′)|. (B13)

We use the two-mode Fourier expansion

〈ϕj (t ′)| =∑k,l

⟨ϕ

(k,l)j

∣∣e−i(kωx+lωz)t ′ , (B14)

where 〈ϕ(k,l)j | = (ϕ(k,l)

j,+ ,ϕ(k,l)j,− ,ϕ

(k,l)j,z ) is a 1 × 3 time-independent

vector. From Eq. (B13), we have∑k,l

⟨ϕ

(k,l)j

∣∣Mkl,nm = εj

⟨ϕ

(n,m)j

∣∣, (B15)

which corresponds to 〈〈ϕj |M = εj 〈〈ϕj | with the row vectorconsisting of 〈ϕ(k,l)

j |. Equations (B11) and (B15) indicate that|φj 〉〉, 〈〈ϕj |, and εj are the right eigenvector, left eigenvector,and eigenvalue of M, respectively. Thus, the evolution opera-tor �(t,t ′) can be obtained whenM is diagonalized. Formally,the solutions to the Bloch equations with an inhomogeneouspart can be found as

〈�σ (t)〉 = �(t,t ′)〈�σ (t ′)〉 − i

∫ t

t ′�(t,s)�bds. (B16)

The explicit form for each component of the Bloch vector canbe expressed in terms of the two-mode Fourier componentsand characteristic exponents as follows:

〈σj (t)〉 =∑

j1,j2=±,z

∑k,l,n,m

φ(k,l)j,j1

ei(kωx+lωz)t

{ϕ

(n,m)j1,j2

〈σj2 (t ′)〉e−i(nωx+mωz)t ′e−iεj1 (t−t ′) − κϕ(n,m)j1,z

1 − ei(nωx+mωz−εj1 )(t−t ′)

i(εj1 − nωx − mωz)e−i(nωx+mωz)t

}.

(B17)

We now calculate the fluorescence spectrum in the steady-state limit. According to the quantum regression theory [34], wecan derive the two-time correlation function from the single-time expectation 〈σ+(t)〉 by replacing the initial condition ρ(t ′) withσ−ρ(t ′) (see the details in the next section). Denoting t = t ′ + τ , we have the two-time correlation function

g(1)(τ ; t ′) =∑

j1=±,z

∑k,l,n,m

φ(k,l)+,j1

ei(kωx+lωz)t

{[ϕ

(n,m)j1,+

1 + 〈σz(t ′)〉2

− ϕ(n,m)j1,z

⟨σ−(t ′)

⟩]

× e−iεj1 τ e−i(nωx+mωz)t ′ − κϕ(n,m)j1,z

e−i(nωx+mωz)t 1 − ei(nωx+mωz−εj1 )τ

i(εj1 − nωx − mωz)〈σ−(t ′)〉

}. (B18)

In the long-time limit, t ′ → ∞, we omit the oscillatory terms explicitly depending on t ′ and get the following two-time correlationfunction:

g(1)(τ ) =∑

j1,j2=±,z

∑k,l,n

m,p,q

κ2φ(n,m)+,j1

ϕ(n−k,m−l)j1,z

φ(p−k,q−l)−,j2

ϕ(p,q)j2,z

[εj1 − (n − k)ωx − (m − l)ωz](pωx + qωz − εj2 )ei(kωx+lωz)τ

+∑

j1=±,z

∑k,l

⎧⎨⎩1

2φ

(k,l)+,j1

ϕ(k,l)j1,+ −

∑j2=±,z

∑n,m,p,q

φ(k,l)+,j1

[1

2ϕ

(n,m)j1,+ φ

(n+p−k,m+q−l)z,j2

− ϕ(n,m)j1,z

×φ(n+p−k,m+q−l)−,j2

(1 − κ

i(εj1 − nωx − mωz)

)]ϕ

(p,q)j2,z

i(εj2 − pωx − qωz)

}ei(kωx+lωz)τ−iεj1 τ . (B19)

033817-14279


On performing integration given in Eq. (25), we obtain the fluorescence spectrum

S(ω) =∑

j1,j2=±,z

∑k,l,n

m,p,q

πκ2φ(n,m)+,j1

ϕ(n−k,m−l)j1,z

φ(p−k,q−l)−,j2

ϕ(p,q)j2,z

[εj1 − (n − k)ωx − (m − l)ωz](pωx + qωz − εj2 )δ(ω − kωx − lωz)

+ Re∑

j1=±,z

∑k,l

1

i(ω + εj1 − kωx − lωz)

⎧⎪⎨⎪⎩

1

2φ

(k,l)+,j1

ϕ(k,l)j1,+ −

∑j2=±,z

∑n,m,p,q

φ(k,l)+,j1

[1

2ϕ

(n,m)j1,+ φ

(n+p−k,m+q−l)z,j2

−ϕ(n,m)j1,z

φ(n+p−k,m+q−l)−,j2

(1 − κ

i(εj1 − nωx − mωz)

)]κϕ

(p,q)j2,z

i(εj2 − pωx − qωz)

⎫⎪⎬⎪⎭. (B20)

The formal spectral function can be used to numerically calculate the line shape of the fluorescence.

APPENDIX C: DERIVATION OF THE FLUORESCENCE SPECTRUM IN THE GENERALIZED FLOQUET PICTURE

In the Markovian approximation, we have

g(1)(τ ; t ′) = TrS{σ+TrR[U (t ′ + τ,t ′)σ−ρ(t ′) ⊗ ρRU †(t ′ + τ,t ′)]}, (C1)

which means that we can obtain the two-time correlation function from the single-time expectation 〈σ+(t ′ + τ )〉 by using theinitial condition σ−ρ(t ′) when solving the master equation. In the generalized Floquet picture, the single-time expectation can beformally obtained as

〈σ+(t)〉 =∑

α,β,k,l

X(k,l)αβ ei(kωx+lωz)t ρβα(t),

=∑k,l

ei(kωx+lωz)t

{2X

(k,l)++ [ρ(st)

++(t) − 1

2+ δρ++(t ′)e−�rel(t−t ′)] + X

(k,l)+− [δρ−+(t ′)e−�deph(t−t ′) + ρ

(st)−+(t)]

+X(k,l)−+ [δρ+−(t ′)e−�deph(t−t ′) + ρ

(st)+−(t)]

}, (C2)

where we used Eqs. (13) and (14) and relation X(k,l)++ = −X

(k,l)−− . To calculate g(1)(τ ; t ′) in the steady-state limit, we use the initial

condition σ−ρ(st)(t ′) in the above equation, i.e., we use the following replacements:

δραβ(t ′) → 〈uα(t ′)|[σ− − 〈σ−(t ′)〉]ρ(st)(t ′)|uβ(t ′)〉, (C3)

ρ(st)αβ (t) → 〈σ−(t ′)〉ρ(st)

αβ (t ′ + τ ), (C4)

12 → 1

2 〈σ−(t ′)〉, (C5)

where 〈σ−(t ′)〉 appears because of the property of trace preserving of the master equation. Consequently, we arrive at the followingexpression:

g(1)(τ ; t ′) =∑k,l

ei(kωx+lωz)(t ′+τ )

⎧⎪⎪⎪⎨⎪⎪⎪⎩∑α,β

n,p

X(n,p)αβ ρ

(k−n,l−p)βα

∑r,q

ei(rωx+qωz)t ′∑α′,β ′h,m

Y(h,m)α′β ′ ρ

(r−h,q−m)β ′α′

+∑n,m

[2X(k,l)++ F

(n,m)++ e−�relτ + X

(k,l)+− F

(n,m)−+ e(i +−−�deph)τ + X

(k,l)−+ F

(n,m)+− e(−i +−−�deph)τ ]ei(nωx+mωz)t ′

⎫⎪⎪⎪⎬⎪⎪⎪⎭

, (C6)

where we used the two-mode Fourier expansions of 〈σ−(t ′)〉, ρ(st)(t ′ + τ ), and 〈uα(t ′)|[σ− − 〈σ−(t ′)〉]ρ(st)(t ′)|uβ(t ′)〉 =∑k,l F

(k,l)αβ ei(kωx+lωz)t ′ with F

(k,l)αβ given in Eq. (30). As t ′ → ∞, we omit the oscillatory terms explicitly depending on t ′ and

033817-15280


thus obtain the two-time correlation function with τ dependence only:

g(1)(τ ) = limt ′→∞

g(1)(τ ; t ′) =∑k,l

ei(kωx+lωz)τ∑

α,β,n,p

X(n,p)αβ ρ

(k−n,l−p)βα

∑α′,β ′,h,m

Y(h,m)α′β ′ ρ

(−k−h,−l−m)β ′α′

+∑k,l

ei(kωx+lωz)τ [2e−�relτX(k,l)++ F

(−k,−l)++ + e(i +−−�deph)τX

(k,l)+− F

(−k,−l)−+ + e(−i +−−�deph)τX

(k,l)−+ F

(−k,−l)+− ]. (C7)

With Eq. (C7) at hand, we can evaluate the integral in Eq. (25) and derive the fluorescence spectrum.

APPENDIX D: ANALYTICAL CALCULATION OF THEFLUORESCENCE SPECTRUM

We show that the present formalism also allows us toanalytically calculate the spectrum, e.g., the result given inRef. [26]. To this end, we derive the two-mode Floquet statesanalytically in a limiting case. We first transform the originalHamiltonian with a new generator

S(t) = i

2

[ω0t + �z

ωz

sin(ωzt)

]σz, (D1)

and then retain the near resonance terms but neglect the off-resonance terms. In doing so, we find the resulting Hamiltonian

H ′(t) = eS(t)[H (t) − i∂t ]e−S(t)

= �x

2(e−iωx t + eiωx t )

[σ+e

iω0t+i�zωz

sin(ωzt) + H.c.]

= �x

2J−m

(�z

ωz

)[σ+ei(ω0−ωx−mωz)t + H.c.], (D2)

where the last line only includes the near resonant termsω0 − ωx − mωz ≈ 0. As is known, this treatment is expectedto be justified when ωz � �x [37]. Otherwise, it breaks down,leading to the fact that the derived spectrum with this method isinsufficient to capture the spectral features given in this paper.This Hamiltonian can be further simplified by the rotatingoperation with

R(t) = exp[−i(ω0 − mωz − ωx)σ+σ−t], (D3)

which leads to

H = R(t)[H ′(t) − i∂t ]R†(t) = δm

2σ+σ− + Fm

2σx, (D4)

where δm = ω0 − ωx − mωz and Fm = �xJ−m( �z

ωz). The

eigenstates of H are given by

|u±〉 = ±√

�m ∓ δm

2�m

|−〉 +√

�m ± δm

2�m

|+〉, (D5)

with the corresponding eigenenergies E±m = 1

2 (δm ± �m),where �m = √δ2

m + F 2m. Thus, we have the two-mode

Floquet states

|u±(t)〉 = ei 12 ω0t−S(t)R†(t)|u±〉 (D6)

and quasienergies ε± = E±m − 1

2ω0. The transition matrix ele-

ments X(k,l)αβ can thus be found from the following expression:

〈uα(t)|σ+|uβ(t)〉 = 〈uα|R(t)eS(t)σ+e−S(t)R†(t)|uβ〉= 〈uα|σ+|uβ〉ei

�zωz

sin(ωzt)+i(mωz+ωx )t

=∑

l

〈uα|σ+|uβ〉Jl

(�z

ωz

)

× ei(l+m)ωzt+iωx t , (D7)

which indicates that

X(k,l)αβ = δk,1Jl−m

(�z

ωz

)〈uα|σ+|uβ〉. (D8)

With the analytical form of X(k,l)αβ and quasienergies at hand,

we can now derive the spectrum; i.e., we just need to calculatethe ingredients in the formal spectrum (31). It is easy to verifythat the analytical spectrum derived is exactly the same as thatgiven in Ref. [26] (apart from the difference in the notation andconstants) and is inapplicable to the strong driving regime, e.g.,�x ∼ ωz ∼ �z � κ .

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https://doi.org/10.1088/1361-6633/aa5170

https://doi.org/10.1088/1361-6633/aa5170

https://doi.org/10.1088/1361-6633/aa5170

https://doi.org/10.1088/1361-6633/aa5170

















https://doi.org/10.1103/PhysRevA.60.R4245




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https://doi.org/10.1103/PhysRev.138.B979








Annals of Physics 396 (2018) 245–253


Annals of Physics

journal homepage: www.elsevier.com/locate/aop

Exotic odd–even parity effects in transmissionphase, (Andreev) conductance, and shot noise ofa dimer atomic chain by topologyBing Dong a,b,*, X.L. Lei a,ba Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department ofPhysics and Astronomy, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai 200240, Chinab Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China


Article history:Received 2 November 2017Accepted 8 July 2018Available online 27 July 2018

Keywords:ConductanceShot noiseDimer chainTopologyEdge state

a b s t r a c t

We investigate the transport properties through a finite dimerchain connected to two normal leads or one normal and one su-perconductor (SC) leads. The dimer chain is described by the Su–Schrieffer–Hegger model and can be tuned into a topologicallynontrivial phase with a pair of zero-energy edge states (ZEESs). Wefind that if the dimer chain is of nontrivial topology, (1) it will showopposite odd–even parity of the number of sites, in comparisonwith the topologically trivial and plain chains, in the (Andreev)transmission probability at the Fermi energy, the noise Fano factorin the zero bias limit, and even the transmission phase due tothe coupled ZEESs; (2) the ZEES can determine appearance of theAndreev bound states at the site connected to the SC lead, andthereby induces a nonzero-bias-anomaly in the Andreev differen-tial conductance of the hybrid junction; (3) the transmission phaseof the normal junction has a unique 2π continuous phase variationat the zero-energy resonant peak that is also different from theusual phase shift in resonant point in usual systems.

© 2018 Elsevier Inc. All rights reserved.

* Corresponding author at: Key Laboratory of Artificial Structures andQuantumControl (Ministry of Education), Departmentof Physics and Astronomy, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai 200240, China.

E-mail address: [email protected] (B. Dong).

https://doi.org/10.1016/j.aop.2018.07.0120003-4916/© 2018 Elsevier Inc. All rights reserved.

283


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246 B. Dong, X.L. Lei / Annals of Physics 396 (2018) 245–253

1. Introduction

Much intention has been paid, in recent years, to the physics of topological state in solid-statephysics [1]. The experimental realization of the topology in one-dimensional (1D) systems [2,3], bymeasuring the quantized Zak phase [4], has stimulated over again extensive investigation on the Su–Schrieffer–Hegger (SSH) model [5]. It has been already verified that some physical systems, e.g., thetwo-dimensional graphene ribbon [6] and p-orbital optical lattice system [7], can be mapped to theSSH model. The main feature of this simple SSH model is that it has two topologically differentphases, which can be distinguished via the presence or absence, controlled by tuning the dimerizationstrength, of twofold degenerate zero-energy edge states (ZEESs) under the open boundary condition(OBC) [6–13]. It is therefore an intriguing issue to provide experimental confirmation for the existenceof the ZEESs in the topological dimer chain. Transport measurement is, out of question, one ofthe most important means to signature the edge states, when a finite dimer chain, for instance,the graphene ribbon, is connected to two normal leads. Recently, some works have already beendone to discuss incoherent tunneling of the finite dimer chain subject to a large bias voltage [14]and/or a high frequency ac electric field [15,16]. In this paper, we examine the linear and nonlinear(Andreev) tunneling in the coherent regime for both a normal junction and a hybrid junction involvingsuperconductor (SC).

2. Model of dimer chain

We consider a 1D dimer lattice of N sites, which contains two sublattices a and b in each unitcell with alternatingly modulated nearest-neighbor hopping amplitudes between them. It is just thecelebrated SSH Hamiltonian,

H1D =

∑jσ

(t1d

†aσ ,jdbσ ,j + t2d

†aσ ,j+1dbσ ,j + H.c.

), (1)

where d†aσ ,j (daσ ,j) and d†

bσ ,j (dbσ ,j) are the fermion creation (annihilation) operators of electrons on thesublattices a and b of the jth unit cell with spin-σ , respectively. t1(2) = t0 ∓ δt denote the hoppingamplitudes in the unit cell and between two adjacent unit cells, δt being the dimerization strength.Notice that the dimer chain showing topologically either trivial or nontrivial property can be easilycontrolled by simply tuning the relative strength of the intracell-to-intercell couplings, δt [8,6,12,13].Throughout we will measure all energies in units of t0 and use units e = h = kB = 1.

3. Normal junction

In this paper, we propose two kinds of transport measurements for detecting the zero-mode byconnecting the left and right ends of the dimer chain respectively to two reservoirs. We consider thefirst setup that two leads are both normal metals (NCN junction), Hη =

∑kσ (εηkσ − µη)c

†ηkσ cηkσ ,

where c†ηkσ (cηkσ ) is the creation (annihilation) operator of an electron withmomentum k and spin-σ ,

energy εηk, and chemical potential µη in the lead η (η = {L, R}). The tunneling Hamiltonian for thecoupling between the chain and the leads are

HT =

∑kσ

(γLc†Lkσdaσ ,1 + γRc

†Rkσda(b)σ ,N + H.c.). (2)

The right end site of the chain could be sublattice either a or b depending on that the number N of thesites is even or odd. Here γη describes the tunnel-coupling matrix element between the QD and leadη and the corresponding coupling strength is defined as Γη = 2π

∑k|γη|

2δ(ω − εηk). Without lossof generality, we use the wide band limit and assumed that ΓL = ΓR = Γ is independent of energyto avoid undesirable effects from the conduction band edge. In addition, in order not to disturb thezero-energy mode of the dimer chain as far as possible, we set the tunnel-coupling Γ = 0.1 in ourfollowing calculations to guarantee that it is much weaker than hopping amplitude, Γ ≪ t0.

284

B. Dong, X.L. Lei / Annals of Physics 396 (2018) 245–253 247

Applying the nonequilibrium Green function (NGF) method, we can evaluate the current from theleft lead to the chain and its shot noise as [17]

I =

∫dωπ

T (ω) (fR − fL) , (3)

S = 2∫

dωπ

{T (ω) [fR(1 − fL) + fL(1 − fR)] − T (ω)2[fR − fL]2

}. (4)

with the Fermi distribution fη = [e(ω−µη)/T − 1]−1 at the temperature T and the transmissionprobability T (ω) = ΓLΓR|Gr

1N (ω)|2. Here, Gr

1N (ω) is the retarded GF between the first and last sitesof the chain.

It is seen that the transport properties are completely determined by the transmission coefficient.Therefore we first examine how the transmission probability evolves when the atomic chain changesfrom metallic to topological. In Fig. 1(a), we plot the transmission spectrum T (ω) as functions of thenumber of sites for the dimer chains with δt = 0.2. It is observed that the transmission probabilityat ω = 0 shows (1) a nearly perfect transmission even for the even number of sites, N = 16, but(2) a rapid decrease with increasing length of the chain, and (3) on the contrary, a nearly perfectreflection for the case of odd-site dimer chain. These behaviors are clearly different from those ofthe plain chain (δt = 0). In fact, we can obtain the exact analytical expressions for T (0) [18,19].For the plain chain, the transmission probability has the well-known odd–even parity dependence:T (0) = 16ΓLΓRt20/(ΓLΓR + 4t20 )

2≃ 0 for an even-site chain and T (0) = 4ΓLΓR/(ΓL + ΓR)2 = 1 for an

odd-site chain. In contrast, for the case of dimer chain, we have

T (0) =

⎧⎪⎪⎨⎪⎪⎩(Γ /t2)2(t2/t1)N

[(Γ /2t2)2(t2/t1)N + 1]2, N is even;

4(t2/t1)(N−1)

[(t2/t1)(N−1) + 1]2, N is odd,

(5)

showing an opposite odd–even parity. For instance, the dimer chainwith δt = 0.2 has a nearly perfecttransmission at ω = 0 only if N = 16, whereas it has a perfect reflection if N is odd.

These peculiar behaviors can be ascribed to the appearance of the ZEESs of the dimer chain. Asshown in Fig. 1(b) for N = 18, the dimer chain has a topologically nontrivial phase in the regime ofδt > 0 characterized by the presence of zero-energy states under the OBC, whereas no edge statesexist in the regime of δt < 0. From the distribution probability of the zero-energy states along thechain in the case of topological regime, δt = 0.2 [Fig. 1(d, e)], we find that these states are indeedmostprobably occupied at the two endpoints of the chain. In addition, we find that the overlap integral ofthe wave function between the two coupled ZEESs cannot be negligible for the short chain, N = 18[Fig. 1(d)], which induces a nonzero coupling between the two edge states. As a result, electron can betransferred coherently from one end to the other via the two ZEESs when the energy of the incidentelectron is zero, rather than via the scattering states as the odd-site plain chain does. Nevertheless,the overlap integral is quite sensitive to the chain length. It decays rapidly and becomes infinitesimalfor the long chain, for instance, N = 28 [Fig. 1(e)]. In this case, the two ZEESs become localized atthe two endpoints respectively, and the transport channel closes. Moreover, for the odd-site chain,one ZEES exists at the whole regime of δt [Fig. 1(c)]. But this state is always localized at one end ofthe chain [Fig. 1(f)] (the left end for δt > 0 whereas the right end for δt < 0), except for the plainchain (δt = 0). Of course, the single localized edge state cannot support electron tunneling, indicatingstrong localization in the odd-site dimer atomic wire.

The transmission probability at the Fermi energy can be detected in experiments by measuringthe linear conductance, G = (2e2/h)T (0), at zero temperature. We then calculate G and plot them,in Fig. 2, as functions of chain length for various dimerization strengths. It is evident that the dimerchain displays the opposite odd–even parity to the plain chain, when the number of sites of the chainis sufficient large depending on the value of the dimerization strength. Moreover, the conductanceof the dimer chain shows an exponential decay with increasing chain length, scaled properly asexp(−2Nδt/t0), while the plain chain does not. This fact confirms that the long dimer chain is an idealAnderson insulator. In addition, the shot noise Eq. (4) can be approximated as S = (4e2/h)T (0)[1 −

285


Fig. 1. (Color online) (a) Dependence of the transmission probability on the number of atoms for the dimer chainswith δt = 0.2.(b, c) Energy spectrum of the dimer chains with even number N = 18 and odd number N = 17 of sites, respectively. (d, e, f)Norm of wave functions of the zero-energy modes at each site for the even-site short chain N = 18, long chain N = 28, andodd-site chain N = 17.

Fig. 2. The zero-temperature conductance as a function of the chain length for various δt = 0.15 (black line), 0.2 (redline), and 0.3 (blue line). Inset: The conductance of the topological chain shows a perfect exponential decay with a decaycoefficient proportional to 2δt/t0 . While for the plain chain, no exponential decay is found (purple line). (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this article.)

T (0)]eV in the limit of small bias voltage, V = µL −µR, at zero temperature. Therefore, it is expectedthat the Fano factor F = S/2I of the dimer chain will also display the different odd–even behaviorfrom the plain chain, as shown in Fig. 3.

We now analyze the transmission phase since it provides a complementary information to depicttransmission coefficient, t(ω) =

√ΓLΓRGr

1N (ω) [20,21]. For the plain chain, irrespective of the numberN of sites, the transmission resonant peaks are all out of phase, i.e. the phase increase continuously butrapidly by π from one resonant peak to the next, as seen by the dashed thin lines in Fig. 4(a) [20–22].

286


Fig. 3. (Color online) The Fano factor S/2I as a function of the bias voltage V for different chain lengths with δt = 0.2 (thicklines) and δt = 0 (thin lines) at zero temperature.

Fig. 4. (a) Transmission phase as functions of the electron energy for the even- and odd-site chains with δt = 0.2 (thick lines)and 0 (thin lines). (b) Trajectory of the transmission amplitude for theN = 16 dimer chain. The black line depicts the path of theelectron energy range from ω = 0 to 0.55, the red line for ω = 0.55–0.86, and the blue line for ω = 0.86–1.0. (c) Trajectory ofthe transmission amplitude for the N = 17 dimer chain for ω = 0–0.45 (black line), ω = 0.45–0.7 (red line), and ω = 0.7–1.0(blue line). The arrows indicate the starting point of the two paths. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

The odd-site dimer chain has the similar phase variation behavior for both the bulk states and the ZEES,even though the transmission amplitude is nearly zero atω = 0. This behavior is illustrated in Fig. 4(c),showing that the path of t(ω) as a function of energy encircles the origin of the complex plane butalways stay away from it. As the energy of incident electron is close to the bulk states, the circle radiiare nearly equal to 1 [red and blue lines in Fig. 4(c)], which are the typical trajectory of transmissionamplitude for a single-channel symmetric double-barrier structure [21,22]. Nevertheless, the radiusof the trajectory becomes very small near the edge state [black line in Fig. 4(c)]. While for the case ofeven-site dimer chain, transmission amplitude starts from the positive real axis with |t| = 1 and thenevolves along the black line as seen in Fig. 4(b) to approach the origin. This indicates a continuousphase variation from 0 to 2π when the energy of incident electron sweeps through the zero-energyresonance, and manifests that the two sides of the zero-energy resonance are in phase, which is justopposite to the regular resonances in the plain chain and even the resonances via the bulk states inthe dimer chain. It can thereby be regarded as a unique signal for the appearance of the ZEESs.

4. Hybrid junction

We turn to the second transport setup at which the right lead is replaced with a SC (NCS junction),HR =

∑kσ εRkc

†Rkσ cRkσ +

∑k(∆c†

Rk↓c†R−k↑

+ H.c.), where ∆ is the SC gap. Since we are interested

287


only in the Andreev tunneling in the subgap region, ∆ can be set as the biggest energy scale andthereby the role of the SC lead is solely to induce s-wave pairing for the site connected to the SClead, i.e. the right end site in the NCS junction. It is of course possible that superconductivity isinduced in a longer region of the dimer chain. But this is a next order effect and we do not considerit in the present paper. To account for the proximity effect, we make use of the site⊗Nambu space,ψ = (da↑,1, . . . , da(b)↑,N , d

†a↓,1, . . . , d

†a(b)↓,N )

T , to rewrite the effective Hamiltonian for the SSH modelin matrix form as H1D =

12ψ

†Kψ , where the 2N × 2N matrix K is given by

K =

(K0 KSKS −K0

), (6)

in terms of the two N × N matrices,

K0 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 t1 0 · · · · · · 0t1 0 t2 0 · · · 0

0 t2 0. . .

. . ....

.... . .

. . .. . .

. . ....

.... . .

. . .. . .

. . . t1(2)0 · · · · · · · · · t1(2) 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (7)

and KS having only one nonzero element for the right end site of the chain, KS,NN = ΓS/2 (Here weuse ΓS instead of ΓR to signify the SC hybrid effect). Without loss of generality, we set ΓS = 3Γ in thecalculations such that a pair of the Andreev bound states (ABSs) is established unambiguously. Theretarded GF Gr (ω) of the chain, defined as Gr (t, t ′) = −iθ (t − t ′)⟨[ψ(t), ψ†(t ′)]⟩, can be formally givenby the Dyson equation

Gr (ω) =

[ωI − K −

(Σ r 00 Σ r

)]−1

, (8)

where I is a 2N × 2N unit matrix and the N × N retarded self-energyΣ r is due to the coupling to theleft normal lead. It has only one nonzero diagonal element,Σ r

11 = −iΓL/2.Employing NGF method, we can derive the Andreev current for the subgap tunneling as [19,23]

IA = −

∫dωπ

TA(ω) (1 − fL − f−L) , (9)

with the hole distribution function f−L = [e(−ω−µL)/T − 1]−1, and the Andreev reflection probabilityTA(ω) = Γ 2

L |Gr1,N+1(ω)|

2. In this kind of hybrid transport device, the chemical potential of the SC leadis usually set as a reference, and the external bias voltage is applied to the normal lead, µL = V . Thenoise of the Andreev current can therefore be calculated at zero temperature as [23,24]

SA = 4∫ V

−V

dωπ

TA(ω)[1 − TA(ω)]. (10)

We then analyze the topological effect on theAndreev reflection spectrum TA(ω) of the dimer chain.We have the exact expressions for TA(0) [19]. The odd–even parity is also evident in the Andreevconductance of the plain chain as in the NCN junction: TA(0) = (ΓLΓS t20/2)

2/[(ΓLΓS)2 + t40 ]2

≃ 0 if Nis even, while T (0) = 4Γ 2

L Γ2S /(Γ

2L + Γ 2

S )2

= 0.36 if N is odd. For the dimer chain, we obtain [19]

TA(0) =

⎧⎪⎪⎨⎪⎪⎩(ΓLΓS/2t22 )

2(t2/t1)4N

[(ΓLΓS/4t22 )2(t2/t1)4N + 1]2, N is even;

Γ 2L Γ

2S (t2/t1)

4N

4[(ΓL/2)2(t2/t1)4N + (ΓS/2)2]2, N is odd.

(11)

An opposite odd–even parity is also found, for instance, TA(0) = 0.29 for the short even-site chain(N = 16), while TA(0) ≃ 0 for the N = 17 chain with δt = 0.2. More interestingly, we find

288


Fig. 5. (Color online) (a) Dependence of the Andreev reflection spectrum on the even number of sites for the dimer chains withδt = 0.2 and ΓS = 3Γ . (b) The Andreev reflection spectrum of the odd site N = 17 with various dimerization strengths. (c, d)Energy spectrum of the dimer chains with the even site N = 18 and the odd site N = 17, respectively.

from Fig. 5(a) that TA(ω) exhibits triple peaks, say one zero-energy peak and two sharp peaks locatedequally at the left and right sides of the central peak, for the even-site dimer chain in the topologicallynontrivial phase. In addition, the height of the two side peaks is nearly unity for short chains anddecreases with increase of the chain length also but relativelymuch slowly comparedwith the centralpeak. The side peaks can be ascribed to the emergence of a pair of ABSs adhered to the ZEES due toSC proximity effect, as illustrated in Fig. 5(c). While in the topologically trivial case (δt < 0), no ZEESmeans no ABS, and thus no side peaks. Another intriguing result is that for the odd-site dimer chain,TA(ω) shows either a single central peak if δt > 0 or two side peaks if otherwise [Fig. 5(b)]. This isbecause the isolated ZEES is located at the left end of the odd-site chain in the case of δt > 0, therebyno electronic state exists at the right end to support the ABS; on the contrary, the degenerate singleZEES at the right end is splitting owing to the SC proximity effect and becomes a pair of ABSs in thecase of δt < 0 [Fig. 5(d)].

The ZEES supported ABS can be detected through two quantities in the nonlinear regime. The firstquantity, for example, is the nonzero-bias-anomaly in the differential Andreev conductance. Indeed,from the inset figure in Fig. 6(a), it is observed a upward jump in the Andreev current IA-V curvesstemming from the sharp peak of the ABS for the short even-site dimer chains in the topologicallynontrivial phase. Correspondingly, the ABS induces a sharp downward jump in the current noise,i.e. the Fano factor F = SA/2IA. Likewise, a rapid downward jump to Poissonian, i.e. F = 1, is alsofound for the noise of the odd-site chain, N = 17, in the case of δt < 0. While in the case of δt > 0,the noise remains superPoissonian, i.e. F ≈ 2, until the bulk state begins to contribute [Fig. 6(b)].

289


Fig. 6. (Color online) The Fano factor of the Andreev current, SA/2IA at zero temperature, as a function of the bias voltage V for(a) different chain lengths with δt = 0.2 at the topologically nontrivial case (the thin line denotes the result of δt = 0), and (b)various δt for the odd-site chain (N = 17). Inset in (a) plots the corresponding IA-V curves.

5. Summary

Using the SSH tight-binding model, we have analyzed the topological effect on the transportproperties of a 1D dimer chain when the chain is connected to two normal leads or to one normaland one SC leads by means of NGF method. It has been found that the topologically nontrivial chainpossesses an opposite odd–even parity dependences of the (Andreev) conductance and the noise Fanofactor in the zero-bias limit with respect to the number of sites, compared with results of the plainchain. Moreover, we have predicted a nonzero-bias-anomaly in the Andreev differential conductanceof theNCS junction, and ascribed it to the emergence of ABSs due to the joint effects of the ZEES and theSC proximity effect. Besides, we have also analyzed the transmission phase of the NCN junction andfound a unique 2π continuous phase variation at the zero-energy resonant peak only when the dimerchain is in the topologically nontrivial phase. Finallywewould like tomention thatwe propose, in thispaper, two simple transport experiments to detect the ZEES, which could provide useful informationto identify the topological quantum phase transition in the 1D atomic wire.

Acknowledgments

This work was supported by Projects of the National Basic Research Program of China (973Program) under Grant No. 2011CB925603, and the National Science Foundation of China under GrantNo. 11674223.

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