GRAPHENE SCIENCE HANDBOOK

GRAPHENE SCIENCE

HANDBOOKNanostructure and

Atomic Arrangement

© 2016 by Taylor & Francis Group, LLC


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E D I T E D B Y

Mahmood Aliofkhazraei • Nasar Ali William I. Milne • Cengiz S. Ozkan

Stanislaw Mitura • Juana L. Gervasoni

GRAPHENE SCIENCE

HANDBOOKNanostructure and

Atomic Arrangement


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CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

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v

ContentsPreface.......................................................................................................................................................................................... ixEditors ..........................................................................................................................................................................................xiContributors ...............................................................................................................................................................................xiii

Section i Atomic Arrangement and Defects

Chapter 1 Graphene Heterostructures ...................................................................................................................................... 3

Zheng Liu and Hong Wang

Chapter 2 Atomic-Scale Defects and Impurities in Graphene .............................................................................................. 21

Rocco Martinazzo

Chapter 3 Atomic Arrangement and Its Effects on Electronic Structures of Graphene from Tight-Binding Description ..........39

Sirichok Jungthawan and Sukit Limpijumnong

Chapter 4 Graphene Plasmonics: Light–Matter Interactions at the Atomic Scale ................................................................ 63

Pai-Yen Chen and Mohamed Farhat

Chapter 5 Graphene/Polymer Nanocomposites: Crystal Structure, Mechanical and Thermal Properties ........................... 77

Fabiola Navarro-Pardo, Ana Laura Martínez-Hernández, and Carlos Velasco-Santos

Chapter 6 Graphene-Like Structures as Cages for Doxorubicin ........................................................................................... 99

Iva Blazkova, Pavel Kopel, Marketa Vaculovicova, Vojtech Adam, and Rene Kizek

Chapter 7 Mathematical Modeling for Hydrogen Storage Inside Graphene-Based Materials .............................................111

Yue Chan

Chapter 8 Morphology of Cylindrical Carbon Nanostructures Grown by Catalytic Chemical Vapor Deposition Method ........................................................................................................................................... 123

S. Ray, M. Jana, and A. Sil

Chapter 9 sp2 to sp3 Phase Transformation in Graphene-Like Nanofilms ............................................................................147

Long Yuan, Zhenyu Li, and Jinlong Yang

Chapter 10 Symmetry and Topology of Graphenes ............................................................................................................... 159

A. R. Ashrafi, F. Koorepazan-Moftakhar, and O. Ori

Section ii Modified Graphene

Chapter 11 N-Doped Graphene for Supercapacitors ............................................................................................................. 167

Dingsheng Yuan and Worong Lin


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vi Contents

Chapter 12 Electrical and Optical Properties and Applications of Doped Graphene Sheets ................................................ 179

Ki Chang Kwon and Soo Young Kim

Chapter 13 Chemical Modifications of Graphene via Covalent Bonding ............................................................................. 207

Liang Cui, Dongjiang Yang, and Jingquan Liu

Chapter 14 Functionalization and Vacancy Effects on Hydrogen Binding in Graphene ...................................................... 221

A. Tapia, C. Cab, and G. Canto

Chapter 15 Modifications of Electronic Properties of Graphene by Boron (B) and Nitrogen (N) Substitution .................... 231

Debnarayan Jana, Palash Nath, and Dirtha Sanyal

Section iii characterization

Chapter 16 Electronic Structure and Topological Disorder in sp2 Phases of Carbon ........................................................... 249

Y. Li and D. A. Drabold

Chapter 17 3D Macroscopic Graphene Assemblies .............................................................................................................. 263

Marcus A. Worsley, Juergen Biener, Michael Stadermann, and Theodore F. Baumann

Chapter 18 3D-AFM-Hyperfine Imaging of Graphene Monolayers Deposit on YBCO-Superconducting Surface ............. 277

Khaled M. Elsabawy

Chapter 19 Phonon Spectrum and Vibrational Thermodynamic Characteristics of Graphene Nanofilms ........................... 289

Alexander Feher, Sergey Feodosyev, Igor Gospodarev, Eugen Syrkin, and Vladimir Grishaev

Chapter 20 Tuning Atomic and Electronic Properties of Graphene by Selective Doping .................................................... 305

Cecile Malardier-Jugroot, Michael N. Groves, and Manish Jugroot

Chapter 21 Scanning Electron Microscopy of Graphene .......................................................................................................319

Yoshikazu Homma, Katsuhiro Takahashi, Yuta Momiuchi, Junro Takahashi, and Hiroki Kato

Chapter 22 Tunneling Current of the Contact of the Curved Graphene Nanoribbon with Metal and Quantum Dots ......... 327

Mikhail B. Belonenko, Natalia N. Konobeeva, Alexander V. Zhukov, and Roland Bouffanais

Chapter 23 Using Few-Layer Graphene Sheets as Ultimate Reference of Quantitative Transmission Electron Microscopy .......................................................................................................................................................... 341

Wang-Feng Ding, Bo Zhao, and Fengqi Song

Section iV Recent Advances

Chapter 24 Computational Modeling of Graphene and Carbon Nanotube Structures in the Terahertz, Near-Infrared, and Optical Regimes ........................................................................................................................................... 359

M. F. Pantoja, D. Mateos Romero, H. Lin, S. G. Garcia, and D. H. Werner


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viiContents

Chapter 25 Design and Properties of Graphene-Based Three Dimensional Architectures .................................................. 375

Chunfang Feng, Ludovic F. Dumée, Li He, Zhifeng Yi, Zheng Peng, and Lingxue Kong

Chapter 26 Electronic Structure of Graphene-Based Materials and Their Carrier Transport Properties ............................. 401

Wen Huang, Argo Nurbawono, Minggang Zeng, Gaurav Gupta, and Gengchiau Liang

Chapter 27 Graphene-Enabled Heterostructures: Role in Future-Generation Carbon Electronics ....................................... 423

Nikhil Jain and Bin Yu

Chapter 28 Recent Progresses and Understanding of Lithium Storage Behavior of Graphene Nanosheet Anode for Lithium Ion Batteries ..................................................................................................................................... 435

Xifei Li and Xueliang Sun

Chapter 29 Study of Transmission, Transport, and Electronic Structure Properties of Periodic and Aperiodic Graphene-Based Structures ................................................................................................................................. 453

Heraclio García-Cervantes, Rogelio Rodríguez-González, José Alberto Briones-Torres, Juan Carlos Martínez-Orozco, Jesús Madrigal-Melchor, and Isaac Rodríguez-Vargas

Chapter 30 Benefits of Few-Layer Graphene Structures for Various Applications ............................................................... 479

I. V. Antonova and V. Ya. Prinz

Chapter 31 Designing Carbon-Based Thin Films from Graphene-Like Nanostructures ...................................................... 497

Cecilia Goyenola and Gueorgui K. Gueorguiev

Chapter 32 Graphene-Based Hybrid Composites ...................................................................................................................517

Antonio F. Ávila, Diego T. L. da Cruz, Hermano Nascimento Jr., and Flávio A. C. Vidal

Chapter 33 Graphene Dispersion by Polymers and Hybridization with Nanoparticles ......................................................... 529

Po-Ta Shih, Kuo-Chuan Ho, and Jiang-Jen Lin

Chapter 34 Magnetocaloric Effect of Graphenes .................................................................................................................. 541

M. S. Reis and L. S. Paixão

Chapter 35 Mode-Locked of Fiber Laser Employing Graphene-Based Saturable Absorber ................................................ 555

Pi Ling Huang, Chao-Yung Yeh, Jiang-Jen Lin, Lain-Jong Li, and Wood-Hi Cheng

Index ......................................................................................................................................................................................... 573


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ix

PrefaceThe theory behind “graphene” was first explored by the physi-cist Philip Wallace in 1947. However, the name “graphene” was not actually coined until 40 years later, where it was used to describe single sheets of graphite. Ultimately, Professor Geim’s group in Manchester (UK) was able to manufacture and see individual atomic layers of graphene in 2004. Since then, much more research has been carried out on the mate-rial, and scientists have found that graphene has unique and extraordinary properties. Some say that it will literally change our lives in the twenty-first century. Not only is graphene the thinnest possible material, but it is also about 200 times stron-ger than steel and conducts electricity better than any other material at room temperature. This material has created huge interest in the electronics industry, and Konstantin Novoselov and Andre Geim were awarded the 2010 Nobel Prize in Physics for their groundbreaking experiments on graphene.

Graphene and its derivatives (such as graphene oxide) have the potential to be produced and used on a commercial scale, and research has shown that corporate interest in the discovery and exploitation of graphene has grown dramati-cally in the leading countries in recent decades. In order to understand how this activity is unfolding in the graphene domain, publication counts have been plotted in Figure P.1. Research and commercialization of graphene are both still at early stages, but policy in the United States as well as in other key countries is trying to foster the concurrent pro-cesses of research and commercialization in the nanotech-nology domain.

Graphene can be produced in a multitude of ways. Initially, Novoselov and Geim employed mechanical exfoliation by using a Scotch tape technique to produce monolayers of the material. Liquid-phase exfoliation has also been utilized. Several bottom-up or synthesis techniques developed for gra-phene include chemical vapor deposition, molecular beam epitaxy, arc discharge, sublimation of silicon carbide, and epi-taxy on silicon carbide.

The first volume of this handbook concerns the fabrica-tion methods of graphene. It is divided into four sections: (1) fabrication methods and strategies, (2) chemical-based meth-ods, (3) nonchemical methods, and (4) advances of fabrication methods.

Carbon is the sixth most abundant element in nature and is an essential element of human life. It has different struc-tures called carbon allotropes. The most common crystal-line forms of carbon are graphite and diamond. Graphite is a three-dimensional allotrope of carbon with a layered struc-ture in which tetravalent atoms of carbon are connected to three other carbon atoms by three covalent bonds and form a hexagonal network structure. Each one of these aforemen-tioned layers is called a graphene layer or sheet. Each sheet is placed in parallel on other sheets. Hence, the fourth valence electron connects the sheets to each other via van der Waals bonding. The covalent bond length is 0.142 nm. The bonds

that are formed by carbon atoms between layers are weak; therefore, the sheets can slide easily over each other. The dis-tance between layers is 0.335 nm. Due to its unique structure and geometry, graphene possesses remarkable physical– chemical properties, including a high Young’s modulus, high fracture strength, excellent electrical and thermal conductiv-ity, high charge carrier mobility, large specific surface area, and biocompatibility.

These properties enable graphene to be considered as an ideal material for a broad range of applications, ranging from quantum physics, nanoelectronics, energy research, catalysis, and engineering of nanocomposites and biomaterials. In this context, graphene and its composites have emerged as a new biomaterial, which provides exciting opportunities for the development of a broad range of applications, such as nano-carriers for drug delivery. The building block of graphene is completely different from other graphite materials and three-dimensional geometric shapes of carbon, such as zero-dimensional spherical fullerenes and one-dimensional carbon nanotubes.

The second volume of this handbook is predominantly about the nanostructure and atomic arrangement of graphene. The chapters in this volume focus on atomic arrangement and defects, modified graphene, characterization of graphene and its nanostructure, and also recent advances in graphene nanostructures. The planar structure of graphene provides an excellent opportunity to immobilize a large number of sub-stances, including biomolecules and metals. Therefore, it is not surprising that graphene has generated great interest for its nanosheets, which nowadays can serve as an excellent plat-form for antibacterial applications, cell culture, tissue engi-neering, and drug delivery.

It is possible to produce composites reinforced with gra-phene on a commercial scale and low cost. In these composites, the existence of graphene leads to an increase in conductivity and strength of various three-dimensional materials. In addi-tion, it is possible to use cheaply manufactured graphene in these composites. For example, exfoliation of graphite is one of the cheapest graphene production techniques. The behavior of many two-dimensional materials and their equivalent three-dimensional forms are completely different. The origin of the aforementioned differences in the behavior of these materials is associated with the weak forces that hold a large number of single layers together to create a bulk material. Graphene can be used in nanocomposites. Currently, researchers have been able to produce several tough and light materials by adding small amounts of graphene to metals, polymers, and ceram-ics. The composite materials usually show better electrical conductivity characteristics compared with pure bulk materi-als, and they are also more resistant against heat.

The third volume describes graphene’s electrical and opti-cal properties and also focuses on nanocomposites and their applications. The fourth volume relates to the mechanical


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x Preface

and chemical properties of graphene and cites recent devel-opments. The fifth volume presents other topics, such as size effects in graphene, characterization, and applications based on size-affected properties. In recent years, scientists have produced advanced composites using graphene, which are excellent from the point of view of mechanical and thermal properties. However, in some of these composites, high elec-trical conductivity only is desirable. For example, the Institute of Metal Research, Chinese Academy of Sciences (IMR, CAS) has created a polymer matrix composite reinforced with graphene, which has a high electrical conductivity. In this composite, a flexible network of graphene has been added to a polydimethylsiloxane matrix (of the silicon family).

Investigation of early corporate trajectories for graphene has led to three major observations. First, the discovery-to-application cycle for graphene seems to be accelerated, for example, compared to fullerene. Even though the discovery of graphene is relatively new, large and small firms have con-tributed to an upsurge in early corporate activities. Second, a rapid globalization has occurred by companies in the United States, Europe, Japan, South Korea, and other developed economies, which were involved in early graphene activities. Chinese companies are currently starting to enter the gra-phene domain, resulting in the expansion of research capabil-ity of nanotechnology. Nevertheless, science alone does not guarantee commercial exploitation. To clarify the issue, the level of corporate patenting in the United Kingdom, which is a pioneer in graphene research, is slightly ahead of Canada and Germany; however, it is dramatically lower than in the United States, Japan, and South Korea. Third, the potential applications of graphene are rapidly expanding. Corporate patenting trends are indicative of their enthusiasm to utilize the features of graphene in various areas, including transis-tors, electronic memory and circuits, capacitors, displays,

solar cells, batteries, coatings, advanced materials, sensors, and biomedical devices. Although graphene was initially proposed as an alternative to silicon, its initial applications have been in electronic inks and additives to resins and coat-ings. We have identified six areas of emerging applications for graphene, including displays/screens, memory chips, biomedical devices, batteries/fuel cells, coatings, inks, and materials. In the investigation of the corporate engagement in graphene, we sought to understand early corporate activity patterns related to broader research and invention trends. In traditional innovation models, a lag between research pub-lication and patenting is consistent with the linear model. However, more recent innovation models are stressing con-current launch, open innovation, and strategic property management.

The sixth volume of this handbook is about the applica-tion and industrialization of graphene, starting with chapters about biomaterials and continues onto nanocomposites, elec-trical/sensor devices, and also new and novel applications.

The editorial team would like to thank all contributors for their excellent chapters contributed to the creation of this handbook and for their hard work and patience during its preparation and production. We sincerely hope that the pub-lication of this handbook will help people, especially those working with graphene, and benefit them from the knowledge contained in the published chapters.

Winter 2016Mahmood Aliofkhazraei

Nasar AliWilliam I. MilneCengiz S. Ozkan

Stanislaw MituraJuana L. Gervasoni

16,000

14,000

12,000

10,000

8000

6000N

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02000 2001 2002 2003 2004 2005 2006 2007

Year2008 2009 2010 2011 2012 2013 2014

FIGURE P.1 Number of documents published around graphene during recent years, extracted fromScopus search engine by searching “graphene” in title + keywords + abstract.


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xi

EditorsMahmood Aliofkhazraei is an assistant professor in the Materials Engineering Department at Tarbiat Modares University. Dr. Aliofkhazraei’s research interests include nan-otechnology and its use in surface and corrosion science. One of his main interests is plasma electrolysis, and he has pub-lished more than 40 papers and a book in this area. Overall he has published more than 12 books and 90 journal articles. He has delivered invited talks, including keynote addresses in several countries. Aliofkhazraei has received numerous awards, including the Khwarizmi award, IMES medal, INIC award, best-thesis award (multiple times), best-book award (multiple times), and the best young nanotechnologist award of Iran (twice). He is on the advisory editorial board of several materials science and nanotechnology journals.

Nasar Ali is a visiting professor at Meliksah University in Turkey. Earlier he held the post of chief scientific officer at CNC Coatings Company based in Rochdale, UK. Prior to this Dr. Ali was a faculty member (assistant professor) at the University of Aveiro in Portugal where he founded and led the Surface Engineering and Nanotechnology group. Dr. Ali has extensive research experience in hard carbon-coating materials, including nanosized diamond coatings and CNTs deposited using CVD methods. He has over 120 interna-tional refereed research publications, including a number of book chapters. Dr. Ali serves on a number of committees for international conferences based on nanomaterials, thin films, and emerging technologies (nanotechnology), and he chairs the highly successful NANOSMAT congress. He served as the fellow of the Institute of Nanotechnology for 2 years on invitation. Dr. Ali has authored and edited several books on surface coatings, thin films, and nanotechnology for lead-ing publishers, and he was also the founder of the Journal of Nano Research. Dr. Ali was the recipient of the Bunshah prize for presenting his work on time-modulated CVD at the ICMCTF-2002 Conference in San Diego, California.

William I. Milne, FREng, FIET, FIMMM, was head of electrical engineering at Cambridge University from 1999 until 2014 and has been director of the Centre for Advanced Photonics and Electronics (CAPE) since 2004. He earned a BSc at St. Andrews University in Scotland in 1970 and later earned a PhD in electronic materials at the Imperial College London. In 2003 he was awarded a DEng (honoris causa) by the University of Waterloo, Canada, and he was elected as Fellow of the Royal Academy of Engineering in 2006. He received the JJ Thomson medal from the Institution of Engineering and Technology in 2008 for achievement in electronics and the NANOSMAT prize in 2010. He is a distinguished visit-ing professor at Tokyo Institute of Technology, Japan, and a distinguished visiting professor at Southeast University in Nanjing, China, and at Shizuoka University, Japan. He is also a distinguished visiting scholar at KyungHee University,

Seoul and a high-end foreign expert for the Changchun University of Science and Technology in China. In 2015, he was elected to an Erskine Fellowship to visit the University of Canterbury, New Zealand. His research interests include large area silicon and carbon-based electronics, thin film materials, and, most recently, MEMS and carbon nanotubes, graphene, and other 1-D and 2-D structures for electronic applications, especially for field emission. He has published/presented approximately 800 papers, of which around 200 were invited/keynote/plenary talks—his “h” index is cur-rently 57 (Web of Science).

Cengiz S. Ozkan has been a professor of mechanical engi-neering and materials science at the University of California, Riverside, since 2009. He was an associate professor from 2006 to 2009 and an assistant professor from 2001 to 2006. Between 2000 and 2001 he was a consulting professor at Stanford University. He earned a PhD in materials science and engineering at Stanford University in 1997. Dr. Ozkan’s areas of expertise include nanomaterials for energy storage; synthesis/processing including graphene, III–V, and II–VI materials; novel battery and supercapacitor architectures; nanoelectronics; biochemical sensors; and nanopatterning for beyond CMOS (complementary metal-oxides semiconduc-tor). He organized and chaired 20 scientific and international conferences. He has written more than 200 technical publica-tions, including journal papers, conference proceedings, and book chapters. He holds over 50 patent disclosures, has given more than 100 presentations worldwide, and is the recipient of more than 30 honors and awards. His important contributions include growth of hierarchical three-dimensional graphene nanostructures; development of a unique high-throughput metrology method for large-area CVD-grown graphene sheets; doping and functionalization of CVD-grown and pristine graphene layers; study of digital data transmission in graphene and InSb materials; memory devices based on inorganic/organic nanocomposites, novel lithium-ion batter-ies based on nano-silicon from beach sand and silicon diox-ide nanotubes; fast-charging lithium-ion batteries based on silicon-decorated three-dimensional nano-carbon architec-tures; and high-performance supercapacitors based on three-dimensional graphene foam architectures.

Stanislaw Mitura has been a professor in biomedical engi-neering at Koszalin University of Technology from 2011. He is a visiting professor at the Technical University (TU) of Liberec and was awarded a doctor honoris causa from TU Liberec. He was professor of materials science at Lodz University of Technology from 2001 to 2014. He earned an MSc in physics at the University of Lodz in 1974, a PhD in mechanical engineering at the Lodz University of Technology (1985), and a DSc in materials science at the Warsaw University of Technology in 1993. Professor Mitura’s most prominent


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cognitive achievements comprise the following: from the con-cept of nucleation of diamond powder particles to the syn-thesis of nanocrystalline diamond coatings (NDC); discovery of diamond bioactivity; a concept of the gradient transition from carbide forming metal to diamond film; and technology development of nanocrystalline diamond coatings for medi-cal purposes. Professor Mitura has published over 200 peer-reviewed articles, communications, and proceedings, over 50 invited talks, and contributed to 7 books and proceedings, including Nanotechnology for Materials Science (Pergamon, Elsevier, 2000) and Nanodiam (PWN, 2006). He organized and co-organized several conferences focused on materials science and engineering, especially diamond synthesis under reduced pressure. He is an elected member of the Academy of Engineering in Poland, guest editor in few international jour-nals, including Journal of Nanoscience and Nanotechnology, Journal of Superhards Materials and also a member of the editorial boards of several journals and an elected Fellow of various foreign scientific societies.

Juana L. Gervasoni earned her doctorate in physics at the Instituto Balseiro, Bariloche, Argentina, in 1992. She has been head of the Department of Metal Materials and Nano-structured, Applied Research of Centro Atomico Bariloche (CAB), National Atomic Energy Commission (CNEA), since 2012. She has been member of the Coordinating Committee of the CNEA Controlled Fusion Program since 2013. Her area

of scientific research involves the interactions of atomic par-ticles of matter, electronic excitations in solids, surfaces, and nano-systems, the absorption of hydrogen in metals, and study of new materials under irradiation. Gervasoni is a researcher at the National Atomic Energy Commission of Argentina and the National Council of Scientific and Technological Research (CONICET, Argentina). She teaches at the Instituto Balseiro and is involved in directing graduate students and postdoctorates. She has published over 100 articles in interna-tional journals, some of which have a high impact factor, and she has attended many international conferences. Gervasoni has been a member of the Executive Committee and/or the International Scientific Advisory Board of the International Conference on Surfaces Coatings and Nanostructured Materials (Nanosmat) since 2010, Latin American Conference on Hydrogen and Sustainable Energy Sources (Hyfusen), and the International Conference on Clean Energy (International Conference on Clean Energy, ICCE-2010) and guest editor of the International Journal of Hydrogen Energy (Elsevier). Recently she has focused her research on the study of hydro-gen storage in carbon nanotubes. Along with her academic and research work, Gervasoni is heavily involved in gender issues in the scientific community, especially in Argentina and Latin America. She is a member of the Third World Organization for Women in Science (TWOWS), branch of the Third World Academy of Science (TWAS), Trieste, Italy, since 2010, as well as of Women in Nuclear (WiN) since 2013.


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Contributors

Vojtech AdamDepartment of Chemistry and BiochemistryMendel University in BrnoandCentral European Institute of TechnologyBrno University of TechnologyBrno, Czech Republic

I. V. AntonovaRzhanov Institute of Semiconductor Physics SB RASNovosibirsk, Russia

A. R. AshrafiDepartment of NanocomputingInstitute of Nanoscience and NanotechnologyUniversity of KashanKashan, Iran

Antonio F. ÁvilaDepartment of Mechanical EngineeringUniversidade Federal de Minas GeraisBelo Horizonte, Minas Gerais, Brazil

Theodore F. BaumannPhysical and Life Sciences DirectorateLawrence Livermore National LaboratoryLivermore, California

Mikhail B. BelonenkoVolgograd Institute of BusinessandVolgograd State UniversityVolgograd, Russia

Juergen BienerPhysical and Life Sciences DirectorateLawrence Livermore National LaboratoryLivermore, California

Iva BlazkovaDepartment of Chemistry and BiochemistryMendel University in BrnoBrno, Czech Republic

Roland BouffanaisSingapore University of Technology and DesignSingapore

José Alberto Briones-TorresUnidad Académica de FísicaUniversidad Autónoma de ZacatecasZacatecas, México

C. CabFacultad de IngenieríaUniversidad Autónoma de YucatánYucatán, México

G. CantoCentro de Investigación en CorrosiónUniversidad Autónoma de CampecheCampeche, México

Yue ChanSchool of Mathematical SciencesUniversity of Nottingham, Ningbo ChinaNingbo, China

Pai-Yen ChenDepartment of Electrical and Computer EngineeringWayne State UniversityDetroit, Michigan

Wood-Hi ChengGraduate Institute of Optoelectronic EngineeringNational Chung Hsing UniversityTaichung, Taiwan

Liang CuiChemical and Environmental EngineeringQingdao UniversityQingdao, China

Diego T. L. da CruzGraduate Studies Program on Mechanical EngineeringUniversidade Federal de Minas GeraisBelo Horizonte, Brazil

Wang-Feng DingDepartment of PhysicsHangzhou Normal UniversityHangzhou, China

D. A. DraboldDepartment of Physics and AstronomyOhio UniversityAthens, Ohio

Ludovic F. DuméeInstitute for Frontier MaterialsDeakin Universityand Institute for Sustainability and Innovation Victoria UniversityVictoria, Australia


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xiv Contributors

Khaled M. ElsabawyMaterials Science Unit, Chemistry DepartmentTanta UniversityTanta, Egypt

and

Department of ChemistryTaif UniversityTaif, Saudi Arabia

Mohamed FarhatDivision of Computer, Electrical, and Mathematical

Sciences and EngineeringKing Abdullah University of Science and TechnologyThuwal, Saudi Arabia

Alexander FeherInstitute of PhysicsP. J. Šafárik UniversityKošice, Slovakia

Chunfang FengInstitute for Frontier MaterialsDeakin UniversityVictoria, Australia

Sergey FeodosyevB. I. Verkin Institute for Low Temperature Physics

and Engineering NASUKharkov, Ukraine

S. G. GarciaDepartment of ElectromagneticsUniversity of GranadaGranada, Spain

Heraclio García-CervantesUnidad Académica de FísicaUniversidad Autónoma de ZacatecasZacatecas, México

Igor GospodarevB. I. Verkin Institute for Low Temperature Physics


Cecilia Goyenola Department of Physics, Chemistry, and Biology (IFM)Linköping UniversityLinköping, Sweden

Vladimir GrishaevB. I. Verkin Institute for Low Temperature Physics


Michael N. GrovesDepartment of Chemistry and Chemical EngineeringRoyal Military College of CanadaKingston, Ontario, Canada

Gueorgui K. GueorguievDepartment of Physics, Chemistry, and Biology (IFM)Linköping UniversityLinköping, Sweden

Gaurav GuptaDepartment of Electrical and Computer EngineeringNational University of SingaporeSingapore

Li HeInstitute for Frontier MaterialsDeakin UniversityVictoria, Australia

Kuo-Chuan HoDepartment of Chemical EngineeringNational Taiwan UniversityTaipei, Taiwan

Yoshikazu HommaDepartment of PhysicsTokyo University of ScienceTokyo, Japan

Pi Ling HuangDepartment of PhotonicsNational Sun Yat-sen UniversityKaohsiung, Taiwan

Wen HuangDepartment of Electrical and Computer EngineeringNational University of SingaporeSingapore

Nikhil JainCollege of Nanoscale Science and Engineering State University of New YorkAlbany, New York

Debnarayan JanaDepartment of PhysicsUniversity of CalcuttaWest Bengal, India

M. JanaAdvanced Materials and Process Technology Centre

(AMPTC)Crompton Greaves LimitedGlobal R&D CentreMumbai, Maharashtra, India


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xvContributors

Manish JugrootAdvanced Materials and Process Technology Centre

(AMPTC)Crompton Greaves LimitedGlobal R&D CentreMumbai, Maharashtra, India

Sirichok JungthawanSchool of Physics, Institute of Science and

NANOTEC-SUT Center of Excellence on Advanced Functional Nanomaterials

Suranaree University of TechnologyNakhon Ratchasima, Thailand

Hiroki KatoDepartment of PhysicsTokyo University of ScienceTokyo, Japan

Soo Young KimSchool of Chemical Engineering and

Materials ScienceChung-Ang UniversitySeoul, Republic of Korea

Rene KizekDepartment of Chemistry and BiochemistryMendel University in BrnoandCentral European Institute of TechnologyBrno University of TechnologyBrno, Czech Republic

Lingxue KongInstitute for Frontier MaterialsDeakin UniversityVictoria, Australia

Natalia N. KonobeevaVolgograd State UniversityVolgograd, Russia

F. Koorepazan-MoftakharDepartment of NanocomputingInstitute of Nanoscience and NanotechnologyUniversity of KashanKashan, Iran

Pavel KopelDepartment of Chemistry and BiochemistryMendel University in BrnoandCentral European Institute of TechnologyBrno University of TechnologyBrno, Czech Republic

Ki Chang KwonSchool of Chemical Engineering and Materials ScienceChung-Ang UniversitySeoul, Republic of Korea

Lain-Jong LiInstitute of Atomic and Molecular ScienceTaipei, Taiwan

Xifei Li Department of Mechanical and Materials EngineeringUniversity of Western OntarioLondon, Ontario, Canada

Y. LiDepartment of Physics and AstronomyOhio UniversityAthens, Ohio

Zhenyu LiHefei National Laboratory for Physical Sciences

at MicroscaleUniversity of Science and Technology of ChinaHefei, China

Gengchiau LiangDepartment of Electrical and Computer EngineeringNational University of SingaporeSingapore

Sukit LimpijumnongSchool of Physics, Institute of ScienceSuranaree University of TechnologyNakhon Ratchasima, Thailand

and

Thailand Center of Excellence in Physics Commission on Higher Education

Ministry of EducationBangkok, Thailand

H. LinCentral China Normal UniversityWu Han, China

Jiang-Jen LinInstitute of Polymer Science and EngineeringNational Taiwan UniversityTaipei, Taiwan

Worong LinDepartment of Chemistry and Institute of NanochemistryJinan UniversityGuangzhou, China


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xvi Contributors

Jingquan LiuChemical and Environmental EngineeringQingdao UniversityQingdao, China

Zheng LiuCenter for Programmable MaterialsSchool of Materials Science and EngineeringandNOVITASNanoelectronics Centre of ExcellenceSchool of Electrical and Electronic EngineeringandCINTRA CNRS/NTU/THALESNanyang Technological UniversitySingapore

Jesús Madrigal-MelchorUnidad Académica de FísicaUniversidad Autónoma de ZacatecasZacatecas, México

Cecile Malardier-JugrootDepartment of Chemistry and Chemical EngineeringRoyal Military College of CanadaKingston, Ontario, Canada

Rocco MartinazzoDipartimento di ChimicaUniversità degli Studi di MilanoandIstituto di Scienze e Tecnologie MolecolariConsiglio Nazionale delle RicercheMilan, Italy

Ana Laura Martínez-HernándezDivisión de Estudios de Posgrado e InvestigaciónInstituto Tecnológico de QuerétaroandCentro de Física Aplicada y Tecnología AvanzadaUniversidad Nacional Autónoma de MéxicoSantiago de Querétaro, Querétaro, México

Juan Carlos Martínez-OrozcoUnidad Académica de FísicaUniversidad Autónoma de ZacatecasZacatecas, México

D. Mateos RomeroDepartment of ElectromagneticsUniversity of GranadaGranada, Spain

Yuta MomiuchiDepartment of PhysicsTokyo University of ScienceTokyo, Japan

Hermano Nascimento Jr.FIAT Automobile Inc.Betim, Brazil

Palash NathDepartment of PhysicsUniversity of CalcuttaWest Bengal, India

Fabiola Navarro-PardoDivisión de Estudios de Posgrado e InvestigaciónInstituto Tecnológico de QuerétaroandÉnergie Matériaux Télécommunications Institut national de la recherche scientifiqueVarennes, Quebec, Canada

Argo NurbawonoDepartment of Electrical and Computer EngineeringNational University of SingaporeSingapore

O. OriActinium Chemical ResearchRome, Italy

L. S. PaixãoInstituto de FísicaUniversidade Federal FluminenseRio de Janeiro, Brazil

M. F. PantojaDepartment of ElectromagneticsUniversity of GranadaGranada, Spain

Zheng PengInstitute for Frontier MaterialsDeakin UniversityVictoria, AustraliaandAgricultural Product Processing Research InstituteChinese Academy of Tropical Agricultural SciencesGuangdong, China

V. Ya. PrinzRzhanov Institute of Semiconductor Physics SB RAS Novosibirsk, Russia

S. RaySchool of EngineeringIndian Institute of Technology MandiHimachal Pradesh, India

M. S. ReisInstituto de FísicaUniversidade Federal FluminenseRio de Janeiro, Brazil


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xviiContributors

Rogelio Rodríguez-GonzálezUnidad Académica de FísicaUniversidad Autónoma de ZacatecasZacatecas, México

Isaac Rodríguez-VargasUnidad Académica de FísicaUniversidad Autónoma de ZacatecasZacatecas, México

Dirtha SanyalVariable Energy Cyclotron CentreBidhannagarWest Bengal, India

Po-Ta ShihInstitute of Polymer Science and EngineeringNational Taiwan University Taipei, Taiwan

A. SilDepartment of Metallurgical and Materials

EngineeringIndian Institute of Technology RoorkeeUttarakhand, India

Fengqi SongCollege of PhysicsNanjing UniversityNanjing, China

Michael StadermannPhysical and Life Sciences DirectorateLawrence Livermore National LaboratoryLivermore, California

Xueliang SunDepartment of Mechanical and Materials

EngineeringUniversity of Western OntarioLondon, Ontario, Canada

Eugen SyrkinB. I. Verkin Institute for Low Temperature Physics


Junro TakahashiDepartment of PhysicsTokyo University of ScienceTokyo, Japan

Katsuhiro TakahashiDepartment of PhysicsTokyo University of ScienceTokyo, Japan

A. TapiaFacultad de IngenieríaUniversidad Autónoma de YucatánYucatán, México

Marketa VaculovicovaDepartment of Chemistry and BiochemistryMendel University in BrnoandCentral European Institute of TechnologyBrno University of TechnologyBrno, Czech Republic

Carlos Velasco-SantosDivisión de Estudios de Posgrado e InvestigaciónInstituto Tecnológico de QuerétaroandCentro de Física Aplicada y Tecnología

AvanzadaUniversidad Nacional Autónoma de MéxicoSantiago de Querétaro, Querétaro, México

Flávio A. C. VidalFIAT Automobile Inc.Betim, Brazil

Hong WangSchool of Materials Science and

EngineeringNanyang Technological UniversitySingapore

D. H. WernerPennsylvania State UniversityUniversity Park, Pennsylvania

Marcus A. WorsleyPhysical and Life Sciences DirectorateLawrence Livermore National LaboratoryLivermore, California

Dongjiang YangCollege of ChemistryChemical and Environmental EngineeringLaboratory of Fiber Materials and Modern

TextileGrowth Base for State Key LaboratoryQingdao UniversityQingdao, China

Jinlong YangHefei National Laboratory for Physical Sciences

at MicroscaleUniversity of Science and Technology of ChinaHefei, China


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xviii Contributors

Chao-Yung YehMetal Industries Research and Development CenterKaohsiung, Taiwan

Zhifeng YiInstitute for Frontier MaterialsDeakin UniversityVictoria, Australia

Bin YuCollege of Nanoscale Science and EngineeringState University of New YorkAlbany, New York

Dingsheng YuanDepartment of Chemistry and Institute of NanochemistryJinan UniversityGuangzhou, China

Long YuanHefei National Laboratory for Physicial Sciences at

MicroscaleUniversity of Science and Technology of ChinaHefei, China

Minggang ZengDepartment of Electrical and Computer EngineeringNational University of SingaporeSingapore

Bo ZhaoCollege of PhysicsNanjing UniversityNanjing, China

Alexander V. ZhukovSingapore University of Technology and DesignSingapore


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541

34 Magnetocaloric Effect of Graphenes

M. S. Reis and L. S. Paixão

ABSTRACT

Magnetocaloric effect (MCE) is an interesting property of materials, that are able to expel/absorb heat from a thermal reservoir, under a magnetic field change (considering an iso-thermal process), or even increase/decrease their temperature (considering an adiabatic process). These effects are maxi-mum around the critical temperature of the material and therefore ferromagnetic materials are, by far, the most and intensively studied material by the scientific community. For this reason, diamagnetic materials have never been studied in this context, until the present effort, and a number of inter-esting features, with quantum signatures, were discovered.The fundamental model to describe a diamagnetic material is an electron gas, and a huge applied magnetic field promotes degeneracy, named Landau levels. Oscillations on the ther-modynamic quantities are found when the Landau levels cross the Fermi level of the nonperturbed gas at a low-temperature regime. This contribution therefore starts presenting an oscil-latory behavior found in the MCE of diamagnetic materials, which can be tuned as either inverse or normal, depending on the value of the magnetic field change. These results open doors for applications at quite low temperatures. The MCE of non-relativistic diamagnetic materials mentioned above has an oscillatory character and this effect occurs at low tem-perature (~1 K) and high magnetic field (~10 T). A step for-ward was to consider the relativistic properties of graphenes, a two-dimensional massless diamagnetic material, and those

oscillations could be preserved and the effect occurs at a much higher temperature (~100 K) due to the huge Fermi velocity (106 m/s).

34.1 THE MAGNETOCALORIC EFFECT

The magnetocaloric effect (MCE), discovered in 1881 by Warburg [1], is an exciting property, intrinsic to magnetic materials. This effect is induced via coupling of the magnetic sublattice with the magnetic field, which alters the magnetic part of the total entropy due to a corresponding change of the magnetic field. We can see the effect from either an adiabatic or an isothermal process. The material is able to increase/decrease its temperature under an adiabatic process; or even expel/absorb heat from a thermal reservoir under an isother-mal process, as a consequence of changes in the magnetic field. Figure 34.1 illustrates these processes.

From a quantitative point of view, the MCE is measured through the magnetic entropy change ΔS = ΔQ/T, where ΔQ is the amount of heat exchanged between the thermal reser-voir and the magnetic material, when the isothermal process is considered. Analogously, the adiabatic temperature change ΔT characterizes the effect when the adiabatic process is con-sidered. These quantities follow from the entropy when it is expressed as a function of temperature and field, S = S(T, B). Figure 34.2 shows the change in entropy due to a magnetic field, and illustrates how we can obtain the quantities that characterize the MCE, that is, the magnetocaloric potentials.

CONTENTS

Abstract ..................................................................................................................................................................................... 54134.1 The Magnetocaloric Effect .............................................................................................................................................. 541

34.1.1 Magnetic Entropy Change ................................................................................................................................... 54334.1.2 Adiabatic Temperature Change ........................................................................................................................... 543

34.2 Why Study Magnetocaloric Effect of Graphenes? .......................................................................................................... 54434.3 Magnetic Entropy of Nonrelativistic Diamagnets ........................................................................................................... 544

34.3.1 Grand Potential .................................................................................................................................................... 54434.3.2 Three-Dimensional Nonrelativistic Diamagnets ................................................................................................. 54534.3.3 Two-Dimensional Nonrelativistic Diamagnets .................................................................................................... 546

34.4 Magnetocaloric Potentials of Nonrelativistic Diamagnets .............................................................................................. 54734.4.1 Three-Dimensional Diamagnets .......................................................................................................................... 54734.4.2 Two-Dimensional Diamagnets ............................................................................................................................ 549

34.5 Magnetic Entropy of Relativistic Diamagnets: Graphenes ............................................................................................. 54934.6 Magnetocaloric Potential of Graphene .............................................................................................................................55134.7 Conclusions .......................................................................................................................................................................551Appendix 34A: Density of States of a Two-Dimensional Nonrelativistic Electron Gas .......................................................... 552References ................................................................................................................................................................................. 553


542 Graphene Science Handbook

It is straightforward, the idea, to produce a thermo-magnetic cycle based on the isothermal and/or adiabatic processes (like Brayton and Ericsson cycles). Figure 34.3 presents the Ericsson cycle: it is composed of two isothermal processes and two iso-field processes. Indeed, the idea of magnetic refrigeration began in the late 1920s, when cooling via adiabatic demagnetization

was proposed by Debye [2] and Giauque [3]. The process was later demonstrated by Giauque and MacDougall, in 1933; where they reached 250 mK [4]. The main point to the scien-tific community is: which kind of material optimizes the MCE, to then be used into devices? To find an answer to this question, a brief review on thermodynamics is needed; as given below.

0.0

1.00

1.02

1.04

1.06

1.08

1.10

1.12

0.0

0.95 1.00 1.05 1.10 1.15

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

1.2

T/(Tc)

S (k

B)

S (k

B)

T/(Tc)

j = 1

log(2j + 1)

ΔT

ΔSb0 = 0 b0 = 6.7 × 10−5

FIGURE 34.2 Numerical entropy for a localized ferromagnetic system of angular momenta j = 1. The reduced magnetic field is b0 = μBB/kBTc. We can see the entropy change of an isothermal process as well as the temperature change of an adiabatic process. (Reprinted from Fundamentals of Magnetism, M. S. Reis, Copyright 2013, with permission from Elsevier.)

Adiabatic process

VacuumMagneticmaterial

�ermal reservoirMagnetic material

Isothermal process

FIGURE 34.1 Fundamentals of the MCE. An applied magnetic field either changes the temperature of the magnetic material (consid-ering an adiabatic process), or promotes a heat exchange with a thermal reservoir (considering an isothermal process). (Reprinted from Fundamentals of Magnetism, M. S. Reis, Copyright 2013, with permission from Elsevier.)


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543Magnetocaloric Effect of Graphenes

34.1.1 maGnetic entropy chanGe

We may relate the entropy change and the magnetization of the system using the following Maxwell relation:

∂∂

∂∂

S

B

M

TT B

= .

(34.1)

After integration, the above equation reads as

∆ ∆S T BS

BdB

M

TdB

T BBi

Bf

Bi

Bf

( , ) = =∂∂

∂∂∫∫ .

(34.2)

Thus, one way to obtain the magnetic entropy change is via measurement of magnetization as a function of temperature and magnetic field, that is, M(T, B).

There is another way to obtain the magnetic entropy change: from specific heat, which is defined by the expression

C T

S

TB

B

=∂∂

.

(34.3)

After a simple integration, the above equation leads to

∆ ∆S T BC C

TdTB

T

( , ) = 0

0

−∫ ,

(34.4)

where C0 = CB = 0.From Equation 34.2, we see that the entropy change is a

maximum when ∂M/∂T is maximum, that is, near a ferromag-netic ordering. We can also reach the same conclusion inspect-ing Figure 34.4, which presents the magnetic entropy change for the ferromagnetic material PrNi2Co3 [5]. The referred material

has Tc ≈ 540 K. Therefore, ferromagnetic materials are, by far, the most and intensively studied materials by the scientific com-munity [6] due to the high potential for application into devices. Materials with other kind of ordering, like ferrimagnets and antiferromagnets, have already been largely studied.

34.1.2 adiaBatic temperature chanGe

Entropy is a function of temperature T and magnetic field B

S S T B= ( , ), (34.5)

and then

dS

S

TdT

S

BdB

B T

= ∂∂

+ ∂∂

.

(34.6)

The adiabatic condition means dS = 0, and therefore, con-sidering the definition of specific heat on Equation 34.3 and the Maxwell relation on Equation 34.1, Equation 34.6 reads as

dT

T

C

M

TdB

B B

= − ∂∂

.

(34.7)

Finally, the adiabatic temperature change is

∆ ∆T T BT

C

M

TdB

B BBi

Bf

( , ) .= − ∂∂∫

(34.8)

Thus, in conclusion, to obtain the adiabatic temperature change, we need to measure M(T, B) and C(T, B). However,

1.000.95 1.151.101.051.00

1.02

1.04

1.06

1.08

1.10

1.12

T/(Tc)

S (k

B)

b0 = 6.7 × 10−5

b0 = 0

FIGURE 34.3 Ericsson cycle for the MCE. The reduced magnetic field is b0 = μBB/kBTc. This cycle is one of the proposals for magnetic refrigeration. (Reprinted from Fundamentals of Magnetism, M. S. Reis, Copyright 2013, with permission from Elsevier.)

ΔS (J

/kg-

K)

T (K)

ΔB = 1 TPrNi2Co3

300 400 500 600 700−0.35

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

FIGURE 34.4 Experimental magnetic entropy change for a simple ferromagnetic material: PrNi2Co3. (Reprinted from Fundamentals of Magnetism, M. S. Reis, Copyright 2013, with permission from Elsevier.)


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there is another way to measure this quantity, that is, apply-ing a magnetic field to the sample (adiabatically) and directly measuring the corresponding temperature change.

34.2 WHY STUDY MAGNETOCALORIC EFFECT OF GRAPHENES?

Owing to the large deal of attention driven to materials with long range ordering, MCE on diamagnets have been first stud-ied very recently and it showed interesting properties [7–10]. The simplest model that describes diamagnetic materials is an electron gas; when a magnetic field is applied to this gas, highly degenerated levels (Landau levels) appear. Since the separation of these levels depend on the applied field, the Fermi energy jumps from the nth to the (n − 1)th Landau level. Thus, since thermodynamic quantities depend on the Fermi energy, this mechanism produces the well-known magnetic oscillations. Figure 34.5 illustrates the crossing of Landau levels through the Fermi level. It is known that graphene pres-ents oscillations on electrical conductivity [11] (Shubnikov–de Haas effect), and oscillations on magnetization [12] (de Haas–van Alphen effect), both effects caused by the jumps of the Fermi level. Therefore, it is natural to expect the oscilla-tory behavior to be observed in other quantities, like entropy and, consequently, the MCE.

The exotic electronic properties of graphenes are quite dif-ferent from standard 3D diamagnetic materials. While for a nonrelativistic 3D diamagnetic material the Landau levels are given by a harmonic oscillator [6]:

E i

eB

mi

e

= +

=� ��ω ω1

2with ,

(34.9)

for a 2D Dirac-like system the levels are given by [13]

E j eBvj F= ′ ′ =� � �ω ω with 2 2 ,

(34.10)

where i, j = 0, 1, 2, … and vF = 106 m/s stands for the Fermi velocity (300 times smaller than the speed of light). Note that the Landau levels for a nonrelativistic material are equally spaced, whereas for graphene it is not true. The reason for the difference in the behavior is that in the former case electrons obey the classical parabolic dispersion relation, while in the latter they follow the relativistic linear dispersion relation that comes from Dirac equation.

Therefore, in this chapter, we aim to connect the oscilla-tions found in thermodynamic quantities of graphene with its magnetocaloric properties. We start the analysis with stan-dard nonrelativistic materials and then change to the relativis-tic case to describe graphenes.

34.3 MAGNETIC ENTROPY OF NONRELATIVISTIC DIAMAGNETS

34.3.1 Grand potentiaL

In this section, we briefly review some concepts regarding thermodynamics of electron gas that will be used later on. To discuss the MCE, we must know the entropy of the system

S T B

T B

T( , )

( , ),= − ∂

∂Φ

(34.11)

which can be obtained from the grand potential

Φ( , ) = ( ) 10

T B k T g ze dB− + −∞

∫ ε εβεln( ) ,

(34.12)

where z = eβμ is the fugacity, and g(ε) the one-particle den-sity of states. We considered an electron gas under an applied magnetic field following the conditions of low temperature, εF ≫ kBT, and a magnetic energy μBB from kBT up to εF.

q

q q

q

q

(a) (b) (c)

B = 0 B = 0 B = 0B1 B2 B3B1 < B2 B1 < B3

n + 1

n−1

n = 2

n = 1

n = 1

n = 0

n = 0

n = 0

n

2μBB1

2μBB2

2μBB3

εF0 εF0 εF0εF εF

εF�

FIGURE 34.5 Electron gas under a magnetic field illustrating the crossing of Landau levels through the Fremi level, mechanism that causes oscillations on thermodynamic quantities. (Reprinted from Fundamentals of Magnetism, M. S. Reis, Copyright 2013, with permis-sion from Elsevier.)


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34.3.2 three-dimensionaL nonreLativistic diamaGnets

The density of states is given by [14]

g N Bl

l

BF B

l

l

B

( ) ( )( )

cos/ / //ε ε ε µ π

µε π= + −

−−

=

∞

∑32

14

3 2 1 2 1 2

1

1 2

= +g gB0( ) ( ).ε ε (34.13)

See appendix for details about the procedure to obtain this result. From Equations 34.12 and 34.13, one easily notes that there are two contributions to the grand potential; one that does not depend on the magnetic field B and the other that does, that is,

Φ Φ Φ( , ) ( ) ( , ).T B T T BB= +0 (34.14)

Thus, the total entropy of the electron gas can be written as

S T B S T S T BB( , ) ( ) ( , ).= +0 (34.15)

The first term is the well-known entropy of a free elec-tron gas without applied magnetic field, in which, within the regime of low temperature, kBT ≪ εF, reads as

S T

Nk

t

B

02( )

=2

π,

(34.16)

where t = kBT/εF. Let us now evaluate the second term that corresponds to the magnetic part of the entropy. After integra-tion of Equation 34.12 by parts (twice), we achieve

ΦB D D

D

G zez e

z e

= − +( ) −+

+ ∂∂

−∞

−

∞

−

11

11

1

30

3 10

3 1

βε ε

εε

βεβε( ) ln ( )

( )

G

G ββε ε+

∞

∫ 10

d ,

(34.17)

where

G NB

l

l

BD

B

F l

l

B3

3 2

1

3 2

32

14

( )( )

sin,/

/επ

µε

πµ

ε π=

− −

=

∞

∑ ,,

(34.18)

and

G3 2

3 2

1

5 2

32

14

D BB

F l

l

B

N BB

l

l

B( )

( )cos

/

/επ

µ µε

πµ

ε π= −

− −

=

∞

∑

.

(34.19)

At low temperature, kBT ≪ εF, the chemical potential approaches the Fermi energy, and therefore z e≈ βεF � 1. With this condition, Equation 34.17 resumes as

Φ Φ ΦB Bno

Bo= + , (34.20)

where

ΦBno

D F D

BB

F l

l

G

N BB

l

= +

= −

−

=

∞

∑3 3

2

3 2

1

5

0 0

3

2 2

1

( ) ( )

( )/

/

ε

πµ µ

ε

G

22 1l

BBF

πµ

ε +

,

(34.21)

has a non-oscillatory character and

ΦBo

BB

F l

l

BFNk T

B

l

l

B=

− −

=

∞

∑32

14

13 2

1

3 2

µε

πµ

ε π/

/

( )cos

ssinh( ),

lx

(34.22)

has an oscillating behavior. In addition,

x

k T

BB

B

= πµ

2 .

(34.23)

Summarizing, the grand potential (Equation 34.14) has two contributions: one that depends on B and the other that does not. The first one also has two contributions (Equation 34.20): one that depends on B as a power law (Equation 34.21), and the other that oscillates depending on the magnetic field (Equation 34.22).

From Equation 34.20, we see that the field-dependent entropy resumes as

S T B S T B

NkB

l

l

B

B Bo

BB

F l

l

B

( , ) ( , )

( )cos

/

/

=

=

−

=

∞

∑32

13 2

1

3 2

µε

πµ

εε πF lx−

4T ( ),

(34.24)

where

T ( )

( )sinh

,xxL x

x=

(34.25)

and L(x) is the Langevin function

L x x

x( ) coth .= − 1

(34.26)

Note that only T ( )x contains information on the tempera-ture; and the period of oscillation depends only on the mag-netic field, and not on the temperature.

The hyperbolic sine in the denominator of the function T ( )x , dampens the entropy. If the thermal energy is large com-pared to magnetic energy kBT ≫ μBB, the entropy becomes very small. On the other hand, if k T BB B� µ , only the term l = 1 contributes notably to the summation. Therefore, we can


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drop out the summation and express the entropy in terms of dimensionless variables; it reads as

S T B

Nk nn xB

B

( , )/

cos( ) ( ),/

= −+

32

11 4

3 2

π T

(34.27)

where

n

BF

B

= −εµ

14

.

(34.28)

In addition, it is convenient to rewrite Equation 34.23 as

x t n= +

π2 14

.

(34.29)

34.3.3 two-dimensionaL nonreLativistic diamaGnets

In two dimensions, the density of states of an electron gas reads (see Appendix 34A for further details)

g g gB( ) ( ),ε ε= +0 (34.30)

where

g

L me0

2

242=

π �,

(34.31)

is the zero-field density of state (with no spin degeneracy), and

g gl

BB

l

l

B

( ) ( ) cos .ε πµ

ε= −

=

∞

∑2 10

1

(34.32)

From the density of states, it is easy again to see that the entropy can be written as a sum of two terms, as in Equation 34.15; one depends only on temperature and the other depends on both, temperature and magnetic field.

Let us now evaluate the field dependent part of the entropy, which we obtain from the grand potential using the density of states (34.32)

ΦB B

l

l

B

T B k T gl

Bze d( , ) ( ) cos ln .= − −

+( )=

∞−

∞

∑∫ 2 1 10

10

πµ

ε εβε

(34.33)

Analogou sly to before, after integration by parts (twice), we achieve

ΦB B

l

lD

BDT B g k T G

B

l( , ) ( ) ( ) | ( ) |= − − +

=

∞∞ ∞∑2 10

1

2 0 2 0ε µ βπ

εG

−

14

2µ βπ

BB

lT BI ( , ) ,

(34.34)

where

G

B

l

l

BzeD

B

B2 1( ) sin ln ,ε µ

ππ

µε βε=

+( )−

(34.35)

G2 1

11

DB

B

B

l

l

B z e( ) cos ,ε µ

ππ

µε βε= −

+−

(34.36)

and

I ( , )

cos

cosh( )

T B

l

B

d

B

F kBT

=

−

→

∞

∫π

µε

β ε µ

ε πε

20

2

4�22

2 2

l

B

l

B

l

BB

BF

B

µ β

πµ

ε

πµ β

cos

sinh

.

(34.37)

Above, the condition εF ≫ kBT implies μ ~ εF, and there-fore z ≫ 1. As a consequence, G2D(∞), G2D(0) and G2 ( )D ∞ are zero; and

G2 0D

BB

l( ) .→ − µ

π (34.38)

Thus, the grand potential reveals two contributions

Φ Φ ΦB Bno

BoT B B T B( , ) ( ) ( , ),= + (34.39)

where

ΦB

noBB g B( ) ( ) ,= 1

60

2µ

(34.40)

has a non-oscillatory character and

ΦBo

B B

l

l

BFT B g k T B

l

l

B lx( , )

( )cos

sinh( ),= −

=

∞

∑21 1

0

1

µ πµ

ε

(34.41)

has an oscillating behavior. Above, x is the same as in Equation 34.23. From Equation 34.40, we can easily see that the non-oscillatory entropy is null. Therefore, the magnetic entropy is

S T B S T B

NkB

l

l

Blx

B Bo

BB

F l

l

BF

( , ) ( , )

( )cos ( )

=

= −

=

∞

∑21

1

µε

πµ

ε T ..

(34.42)

Above

N g d g F

F

= =∫ 0 0

0

ε εε

,

(34.43)


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is the number of electrons that fulfill the gas states up to the Fermi energy in the absence of magnetic field.

Following the same logic employed to 3D electron gases, we can drop the summation of the entropy and write

S T B

N k mm xB

B

( , )cos( ) ( ),= − 2 π T

(34.44)

where

m

BF

B

= εµ

.

(34.45)

Equation 34.23 can be rewritten as

x tm= π2 . (34.46)

34.4 MAGNETOCALORIC POTENTIALS OF NONRELATIVISTIC DIAMAGNETS

34.4.1 three-dimensionaL diamaGnets

From now on, we will investigate the magnetocaloric proper-ties of some nonrelativistic systems, starting with a 3D dia-magnetic material. The standard model for a diamagnetic material is an electron gas; for this reason, this topic was discussed in the last section, and we will apply those results below.

In an isothermal process, we define the entropy change under application of a field B as

∆ ∆S T B S T B S T( , ) ( , ) ( , ).= − 0 (34.47)

As we verified in the last section, the entropy is expressed as a sum of a nonmagnetic term and an oscillating magnetic one. Consequently, the entropy change becomes

∆ ∆S T B S T S T B S T S T

S T B

Bo

Bo

Bo

( , ) [ ( ) ( , )] [ ( ) ( , )]

( , ),

= + − +

=

0 0 0

(34.48)

since S TBo ( , )0 0= . Thus, the magnetic entropy change is

given by Equation 34.27:

∆S

N k nn x

B

= −+

32

11 4

3 2

/

/

cos( ) ( ).π T

(34.49)

Note that the amplitude of the entropy change can be max-imized and minimized by the cosine function. Thus, we can consider the condition

cos( ) ,nπ = ±1 (34.50)

and then an interesting result arises: n even and zero implies in a negative (and minimized) ΔS; n odd implies in a positive (and maximized) ΔS; and, finally, a half-integer makes zero the magnetic entropy change (and then the magnetocaloric

potential of the material). The thermal dependence of the entropy change is ruled by the function T ( )x , and that func-tion peaks at x = 1.6. Therefore, the temperature at which the entropy change is maximum lies at

T K B Tmax[ ] . [ ].= 0 1 (34.51)

Let us consider a piece of gold as an example, whose Fermi energy is 5.51 eV. Consider n = 0 implies, thus, in B = 380 kT, which is absolutely out of the laboratory range; and, analo-gously, n = 104 implies in B = 10 T. Following this example, note that Figure 34.6 presents the magnetic entropy change as a function of the dimensionless temperature t. The scale of t is of the order of 10−5 and then, considering Au, the temperature is of the order of 1 K. Thus, for Au, this effect is comfortably visible at 10 T and 1 K. In addition, as mentioned above, the magnetic entropy change can change its sign depending on the value of n, and this behavior can also be seen in Figure 34.6.

The entropy change as a function of the reciprocal mag-netic field is depicted in Figure 34.7. The effect is maximized around t = 10−5 (see Figure 34.6); for Au it happens around 1 Kelvin as mentioned above. Note that the oscillatory behavior of this quantity is maximized for n odd, zero for n half-inte-ger, and minimized for n even. The modulation presented is given by Equation 34.49 without the cosine.

The MCE is due to a magnetic field change ΔB:0 → B(n). The final value of the magnetic field B(n) can tune the sign of ΔS; and the change of the magnetic field change that is able to do this inversion is (for n ≫ 1)

| ( ) ( ) ( ) .∆ ∆ ∆ ∆B B n B n

nF

B

|= | + − |≈11

2

εµ

(34.52)

Again for Au, note that around 10 T of magnetic field change, an increase of the final value of magnetic field in 10−3 T is enough to invert the magnetic entropy change. In other

ΔS/N

k B (di

men

sionl

ess)

× 1

0−7

t (dimensionless) × 10−5

n = 104

n = 104 + 1/2

n = 104 + 1

−6

−4

−2

0

0 2 4 6 8 10

2

4

6

FIGURE 34.6 Magnetic entropy change for a 3D system as a func-tion of t = kBT/εF. (Reprinted with permission from M. S. Reis, Appl. Phys. Lett., vol. 99, p. 052511. Copyright 2011, American Institute of Physics.)


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words, this effect is sensible to a magnetic field 10,000 times smaller than the applied one.

Owing to its interesting properties this thermal system opens doors for some applications. It is possible to work as a high sensible magnetic field sensor, with a huge magnetic field on the background.

The magnetocaloric potential of the system discussed here is indeed smaller when compared to standard ferromagnetic materials [15]. However, those oscillations can be useful if improved in terms of the magnetocaloric potential; specially for magnetic cooling at quite low temperatures, since the sign of the magnetic entropy change can be ruled by the final value of the applied magnetic field. This effect can indeed be used, if improved, in adiabatic demagnetization refrigerators.

Contrary to what was discussed until now, let us focus on an adiabatic process. In such a process, the system changes its temperature from T0 to TB under application of a magnetic field B. The condition is imposed considering that the entropy of the system at T0 and zero field is the same of the system at TB and under an applied magnetic field B. Thus, from the adiabatic condition

S T S T BB( , ) ( , ),0 0 = (34.53)

we obtain the adiabatic temperature change

∆T T TB= − 0. (34.54)

From Equations 34.16, 34.27, and the adiabatic condition stated above, we obtain

π π π2

0

2 3 2

2 232

11 4

t tn

n xB B= −+

/

cos( ) ( ),/

T

(34.55)

where t0 = t(T0), tB = t(TB), and xB = x(tB). This condition gives a relationship between T0 and TB; and then it is possible to

write the adiabatic temperature change. However, the function T0(TB) from Equation 34.55 is not simple and an approxima-tion is needed. As we are already considering t ≪ 1, we can consider x ≪ 1 (see Equation 34.29). We only need to ensure that n ranges from unity up to the order of 1/t. Following this condition, Equation 34.25 resumes as

T ( ) .x

x≈3

(34.56)

Now we can write our final result from Equation 34.55

∆T Tn

n n=

+ −

0

1 4

cos( )

/ cos( ).

ππ

(34.57)

Note that if n n+ =1 4/ cos( )π , the adiabatic temperature change diverges; and it occurs for n = 1/4, that corresponds to B = 2εF/μB, that no longer fulfill the condition stated above, in which the magnetic field must be up to the order of the Fermi energy. On the other hand, for n ≫ 1, Equation 34.57 resumes as

∆T T

n

n=

0

cos( ).

π

(34.58)

Thus, for n even, the adiabatic temperature change is positive, that is, TB > T0, and the system warms up due to an applied magnetic field. On the other hand, for n odd, TB < T0 and the system cools down due to an applied magnetic field. Of course, for half-integer values of n, the adiabatic tempera-ture change is zero.

Figure 34.8 presents the oscillatory behavior found for the adiabatic temperature change ΔT in Equation 34.57, as well as the approximation for large values of n, Equation 34.58. This

−1.0

−0.5

0.0

0.5

1.0

1 10 100

1.5

2.0

Approximation(large values of n)

Modulation

n (dimensionless)

ΔT/T

0 (di

men

sionl

ess)

FIGURE 34.8 Reduced adiabatic temperature change ΔT/T0 for a 3D system as a function of n (proportional to the reciprocal mag-netic field 1/B). (Reprinted from Solid State Commun., 152, M. S. Reis. 921, Copyright 2012, with permission from Elsevier.)

ΔS/N

k B (di

men

sionl

ess)

×10

−5

n (dimensionless)

Modulation

t = 10−5

0.01 0.1 1 10 100

6

4

2

0

−2

−4

−6

−8

−10

FIGURE 34.7 Magnetic entropy change for a 3D system as a func-tion of n (a function of the inverse magnetic field B). (Reprinted with permission from M. S. Reis, Appl. Phys. Lett., vol. 99, p. 052511. Copyright 2011, American Institute of Physics.)


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figure also contains the modulation of the oscillations found, obtained from Equation 34.57 considering the cosines equal to one.

Let us once again use gold as a real example to illustrate the effect. In this sense, Figure 34.9 presents the prediction of this effect at T0 = 2.56 K and around 8.5 T of magnetic field change. Note, therefore, that this effect is comfortably visible within that range of magnetic field and temperature, reason-able for standard laboratories. The value chosen for the initial temperature T0 corresponds to t = 4 × 10−5, and is a tempera-ture range in which we can work easier.

Considering Figure 34.9, it is possible to see that Gold produces |ΔT(B(n = 11182)) − ΔT(B(n = 11183))| = 50 mK of change of the adiabatic temperature change (around 2.56 K), due to |B(n = 11182) − B(n = 11183)| = 0.8 mT of change of the magnetic field change (with a huge magnetic field on the background). This temperature change is easily measured with a cernox-like sensor, since its accuracy is around 5 mK for this range of temperature.

34.4.2 two-dimensionaL diamaGnets

The realization of a diamagnetic material in two dimensions is a thin film, whose magnetocaloric properties will be dis-cussed below. As we know from the beginning of the present section, the entropy change caused by a field change ΔB:0 → B is simply the magnetic entropy at the final field

∆ ∆S T B S T BBo( , ) ( , ).= (34.59)

Thus, the entropy change, per electron, for a 2D diamag-netic material is given by Equation 34.44

∆S

N k mm x

B

= − 2cos( ) ( ).π T

(34.60)

Let us consider a Gold thin film, with Fermi energy εF = 3.62 eV. From Equation 34.45, it is possible to see that B(m = 1) = 6.2522 × 104 T, and therefore completely out of a laboratory range. On the other hand, B(m = 104) = 6.2522 T, and therefore this is the order of magnitude of m we must consider. Note that the magnetic field change needed to invert the magnetic entropy change (from normal/negative to inverse/positive) is B(m = 104 + 1) − B(m = 104) = 0.7 mT. The temperature dependence of the entropy change lies on the T ( )x function, similarly to the 3D system. This function peaks at x = π2tm = 1.6, and therefore, considering m = 104, the maximum magnetic entropy change occurs at T = 0.7 K.

The dependence of the magnetic entropy change as a func-tion of m is presented in Figure 34.10 for the temperature that maximizes this effect (considering m = 104), that is, 0.7 K. It has an oscillating behavior due to cosine on Equation 34.60, and the envelope of these oscillations comes from the same equation without the cosine function.

34.5 MAGNETIC ENTROPY OF RELATIVISTIC DIAMAGNETS: GRAPHENES

Unlike the systems we studied before, in graphene, electrons behave like relativistic particles with zero rest mass and have an effective “speed of light” equal to Fermi velocity [11]. Thus, dynamics of electrons in graphene is described by Dirac equation. Single particle Dirac Hamiltonian is

HD Fv= ⋅� �α Π, (34.61)

where �α is the Pauli matrix vector and

�Π is the momentum of

ΔT (K

)

B (T)8.500

0.00

−0.01

−0.02

−0.03

0.01

0.02

0.03

8.505 8.510 8.515 8.520

n = 11182T0 = 2.56 Kt0 = 4 × 10−5

n = 11183

FIGURE 34.9 Adiabatic temperature change as function of the applied magnetic field. This example is for 3D Gold (εF = 5.51 eV), under accessible temperature and magnetic field ranges. (Reprinted from Solid State Commun., 152, M. S. Reis. 921, Copyright 2012, with permission from Elsevier.)

ΔS/N

k B (di

men

sionl

ess)

× 1

0−5

10−2 10−1 100 101 102 103 104 105

m,n (dimensionless)

0−2−4−6−8

−10−12−14−16−18

2468

1012

2D Gold

3D Gold

Envelope

T = 0.7 K

FIGURE 34.10 Oscillating magnetic entropy change per electron as a function of the inverse magnetic field B (see Equations 34.28 and 34.45). The oscillatory behavior is evident and has a remarkable difference between 2D (Equation 34.60) and 3D (Equation 34.49) models. For the sake of clearness, above m, n, = 10 only the enve-lope is shown. See text for further details. (Reprinted with permis-sion from M. S. Reis, J. Appl. Phys., 113, 243901. Copyright 2013, American Institute of Physics.)


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the particle. The eigenvalues are

ε j Fj j eBv= sgn( ) | | ,2 �

(34.62)

where j is the Landau level index. Degeneracy of Landau lev-els per graphene area is

�

�g

eB= 2π

,

(34.63)

which is independent of the level. We are already taking into account a factor of 4 corresponding to spin and sublattice degeneracies [16].

To obtain the grand potential, we follow the same pro-cedure of Zhang et al. [17]. They studied the de Haas–van Alphen effect in graphene in the presence of an additional electric field, orthogonal to the magnetic field. The first step is to obtain the total energy of the 2D gas of Dirac electrons, summing the energies of the Landau levels

E g gN

gj

N g

j= +

=∑

0

0

00

[ / ]

mod .

�

� ��

ε µ

(34.64)

Above µ π0 0= �v NF is the zero temperature and field chemical potential, that is, the Fermi energy; and N0 = 1016 m−2 is the density of charge carriers [18]. In addition, the first term counts the energy of the completely occupied Landau levels, and the second one counts the energy of the highest level, par-tially occupied. Here, the notation [z] means integer part of, that is, the largest integer satisfying [z] ≤ z, and mod[z] stands for the fractional part, that is, z = [z] + mod[z]. From the above energy, we write it as a sum of a non-oscillating term plus an oscillating one

E E Eno o= + . (34.65)

where

E

veB

vno F

F

= − +ζπ

µπ

( / )( )

( ),/3 2

2

232

3 2 03

2 2� � (34.66)

and

EeB v

lJ lm lmo F

l

= ′ ′=

∞

∑( )( )cos( ).

/3 2

1

11

π ππ π

�

(34.67)

Above J1 is the integral

J pe

t tdt

e

t t idt

tp tp

1 2

00

( ) =( 1)

=( )

− −∞∞

+−

+∫∫ π πIm ,

(34.68)

and

′ =m N

B0

0φ,

(34.69)

where φ π015 22 06 10= = × −�/ Tme . is the magnetic flux

quantum.The next step is to evaluate the grand potential from the

total energy above. According to the procedure reported by Sharapov et al. [12], the grand potential of electrons in gra-phene can be obtained evaluating

Φ( , ) = ( ) ( )0 0 0T B P E dT µ µ µ µ−−∞

∞

∫ ,

(34.70)

where the distribution function PT(z) takes into account the effect of temperature

P zk T

z

k T

T

BB

( )cosh

.= 1

42

2

(34.71)

At low temperatures, we obtain for the grand potential a non-oscillatory term that depends only on the field and an oscillating term dependent on temperature and field. The non-oscillatory part of the grand potential is

ΦB

no F

F

Bv

eBv

( )( )

( )( )

./= − +ζπ

µπ

3 2

2

232

3 2 03

2

/

� � (34.72)

To obtain the oscillatory part of the grand potential, it is convenient to use the complex form of Equation 34.68 and perform the integration in the complex plane. After that we find

ΦBo F

l

T BeB v lm

l

ly

ly( , )

( ) cos( )( ) sinh( )

,= ′

=

∞

∑2 2

0 1

2πµπ

π

(34.73)

where

y tm= ′2 2π . (34.74)

Since the non-oscillatory term does not depend on the tem-perature, it follows that the non-oscillatory entropy is null. Therefore, the magnetic entropy reads as

S T B N km

lm

llyB

oB

l

( , )cos( )

( ).=′

′

=

∞

∑21

0

1

πT

(34.75)

Note that the temperature dependence of the oscillat-ing contribution to the magnetic entropy, which is given by


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Equation 34.25, is found to be the same for both 3D and 2D nonrelativistic systems. As we argued before, the hyperbolic sine in the denominator of the function T ( )y dampens all terms of the summation for l ≥ 2. Thus, the magnetic entropy per electron reads as

S T B

N k mm yB

o

B

( , )cos( ) ( ).

0

2=′

′π T

(34.76)

34.6 MAGNETOCALORIC POTENTIAL OF GRAPHENE

From the result of the previous section, it is easy to

see that S TBo ( , )0 0= . Therefore, the entropy change,

∆ ∆S T B S T BBo( , ) ( , )= , is

∆S

N k mm y

B0

2=′

′cos( ) ( ).π T

(34.77)

The oscillating MCE for graphenes is presented as a function of m′ in Figure 34.11. The change of magnetic field change required to invert the MCE (from normal to inverse) for a graphene (ca. 3.4 T) is much higher than the 3D nonrela-tivistic case (ca. 1 mT); therefore, it would be easier to verify the effect. In addition, the oscillation is rapidly smashed by the hyperbolic sine in the denominator of the function T ( )y in Equation 34.77.

Figure 34.12 presents the entropy change as a function of temperature for some values of m′, that is, some values of magnetic field. Note that odd (even) values of m′ minimize (maximize) the magnetic entropy change. The temperature of the maximum entropy change, given by the function T ( )y , is

T K B Tmax[ ] . [ ].= 5 3 (34.78)

If we consider m′ = 3, the corresponding magnetic field is B = 6.9 T. Therefore, the temperature of the maximum entropy change is 36.6 K, quite comfortable to work with. For m′ = 1, the temperature of maximum entropy change is even higher, T = 109.3 K; the temperature at which Figure 34.11 was plotted.

34.7 CONCLUSIONS

The community studying the MCE sums efforts for applica-tions considering only materials with cooperative orderings, like ferro-, antiferro-, and even ferrimagnetic materials. An interesting system with a high possibility of applications was recently presented, considering 3D [7,8] and 2D [10] stan-dard nonrelativistic diamagnetic materials, like Gold. Their oscillating MCE is shown to work at low temperatures (ca. 1 K) considering feasible values of magnetic field change. To increase interest on this effect, the oscillating MCE of a 2D massless Dirac-like system, graphene, was explored [9]. It was shown that the temperature in which this effect appears is quite comfortable (ca. 37 K), due to the relativistic proper-ties of the electrons in this material. In addition, the change of magnetic field change required to invert the MCE from normal to inverse is ca. 1 mT for 3D nonrelativistic material (useful for application in high sensible magnetic field sensor at low tem-peratures), and ca. 3.4 T for graphene. Table 34.1 summarizes some aspects of MCE of 3D and 2D diamagnets and compare with graphene. We see the applied magnetic field and the tem-perature of maximum entropy change for that field. Data were extracted from figures and examples discussed.

Equations 34.27, 34.44, and 34.76 suggest that the mag-netic entropy has a general form that scales with the dimen-sion d of the system as follows:

B B T Bd / cos( ) ( ),2 1/ /T (34.79)

ΔS/N

0kB (

dim

ensio

nles

s)

m′ = N0ϕ0/B (dimensionless)0

0.0

−0.3

−0.6

−0.9

0.3

0.6

0.9

1.2

1 2 3 4

B = 20.6 T B = 10.3 T

Envelope

T = 109.3 K

B = 6.9 T

FIGURE 34.11 Oscillating magnetic entropy change as a func-tion of m′, which is inversely proportional to the magnetic field B. (Reprinted with permission from M. S. Reis, Appl. Phys. Lett., 101, 222405. Copyright 2012, American Institute of Physics.)

ΔS/N

0kB (

dim

ensio

nles

s)

T = 54.6 K

T = 36.6 K

T = 109.3 K

m′ = 3

m′ = 2

m′ = 1

t (dimensionless)0.00

0.0−0.1

−0.2−0.3

−0.4−0.5

−0.6−0.7

0.1

0.20.3

0.05 0.10 0.15 0.20 0.25 0.30

FIGURE 34.12 Oscillating magnetic entropy change as a function of temperature. We can see several curves for different values of magnetic field. (Reprinted with permission from M. S. Reis, Appl. Phys. Lett., 101, 222405. Copyright 2012, American Institute of Physics.)


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irrespective of being a relativistic material by nature (gra-phene) or not. This general magnetic dependence of the entropy is responsible for the unique effects found on mag-netocaloric properties of 2D and 3D diamagnetic materials. However, the order of magnitude of the entropy change is dif-ferent for each kind of material because of the proportionality constants involved. In the case of graphene, the proportion-ality involves the Fermi velocity, which is quite huge. This makes graphene special because MCE can be observed in comfortable values of temperature and field.

APPENDIX 34A: DENSITY OF STATES OF A TWO-DIMENSIONAL NONRELATIVISTIC ELECTRON GAS

We show below the procedure to obtain the density of states; we use as a simple example a 2D nonrelativistic electron gas. The one-particle density of states follows easier from the Laplace transformation of the canonical partition function

Z B1 ( )β in the Boltzmann limit (kBT ≫ εF)

gi

e Z dB

i

i

( ) ( ) ,επ

β ββ ε

β

β

= ′ ′′

− ∞

+ ∞

∫12

1

(34A.1)

where, for noninteracting systems, the following relationship between canonical and grand canonical partition functions holds

ln ( , ) ( ).Z T B zZ B= 1 β (34A.2)

Thus, at this point, we need to obtain the grand partition function in the Boltzmann limit

ln ( , ) ,Z T B zen

n= ∑ −βε

(34A.3)

but before, the energy spectrum εn for this model.The Hamiltonian of the present model, that is, a 2D nonrel-

ativistic electron gas with a transversal magnetic field �B Bk= ˆ,

can be written as

H = + +

12

2 2

mp p eBx

ex y( ) ,

(34A.4)

and then the Schrödinger equation reads as

− + +

=�2 2

22

02

212m

d

dxm x x x x

ee n n nω φ ε φ( ) ( ) ( ),

(34A.5)

where

x

k

eB

eB

mny

en0

12

= = = +

��, , .ω ε ωand

(34A.6)

Above, ky = 2πny/L, where ny = 0, 1, 2, …, is related to the translational symmetry along the y axis; and the energy spec-trum represents the Landau levels, where n is the Landau level index. Note that the harmonic oscillator of Equation 34A.5 has several centers x0 that depend on ny and B. Since these centers must be within the considered plane, that is, 0 ≤ x0 ≤ L

then 0 ≤ ≤n gy � , where �g L eB h= 2 / is the degeneracy of the Landau level n (note that for each n, there are ny possibilities).

Thus, Equation 34A.3 can be rewritten as

ln ( , ) exp

sinh( ),

Z T B zg n

zL eB

h y

n

= − +

=

=

∞

∑0

2

12

21

� �β ω

(34A.7)

where y = μBBβ. Thus, a simple comparison of Equations 34A.2 and 34A.7 leads to

Z

L eB

h yB1

2

21

( )sinh( )

.β =

(34A.8)

From the above and Equation 34A.1, it is possible to write the density of states we are looking for

gL eB

h i

e

Bd

Bi

i

( )sinh( )

,επ µ β

ββ ε

β

β

=′

′

′

− ∞

+ ∞

∫2

21

2

(34A.9)

that reads as

g g gB( ) ( ),ε ε= +0 (34A.10)

where

g

L me0

2

242=

π �,

(34A.11)

is the density of state (with no spin degeneracy), of the non-perturbed 2D electron gas; and

g gl

BB

l

l

B

( ) ( ) cos .ε πµ

ε= −

=

∞

∑2 10

1 (34A.12)

TABLE 34.1Summary of Maximum Entropy Change for a Feasible Applied Magnetic Field

3D 2D GrapheneB (T) 10 6 10Tmax (K) 1 0.7 53ΔS/NkB 4.6 × 10−7 6.2 × 10−5 0.3

Note: Comparison between data shown in figures and examples.


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REFERENCES

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