Doped Mott insulators and high temperature...

Doped Mott insulators and high temperature superconductivity

T. Senthil (MIT) and M. Randeria (OSU)Earlier version: lectures by Patrick Lee and TS at MIT, Sept 09.

Thanks to Patrick Lee for some of the slides.

Wednesday, December 15, 2010

Plan

Lecture 1: The problem of doping a Mott insulatorLecture 2: Cuprate phenomenology + minimal theoryLecture 3: More cuprate phenomenology + minimal theory

Secondary goal: introduction to modern experimental probes of electronic solids


Outline for Lecture 1

1. Cuprates as doped Mott insulators

2. Magnetism and Mott insulators

3. Doping a Mott insulator-(i) some general theoretical questions-(ii) experiments on a few materials.








What is a Mott insulator?

Insulation due to jamming effect of Coulomb repulsion

Coulomb cost of two electronsoccupying same atomic orbital dominant

⇒Electrons can’t move if every possible atomicorbital site is already occupied by another electron.

Odd number of electrons per unit cell: band theory predicts metal.


A useful theoretical model


Complications in many real Mott insulators

1. Orbital degeneracy: More than one atomic orbital may be available for the electron to occupy at each site.

2. Multi-band model may be more appropriate starting point (definitely so if there is orbital degeneracy)

3. Spin-orbit interactions

4. (Obviously) must include long range Coulomb+.............................

In this lecture I will primarily consider situations in which many of these complications (mainly 1-3) are likely unimportant. Fortunately the cuprates likely fall in this class!


When Mott insulator?

Classic Mott insulating materials: transition metal oxides (eg: NiO, MnO, V2O3, La2CuO4, LaTiO3,.......) of 3d series, some sulfides (NiS2), .......3d orbitals close to nucleus: large on-site repulsion compared to inter-site hopping. Will meet some other interesting examples later.

Recent additions: 5d transition metal oxides (eg: Sr2IrO4)Atomic 5d orbitals more extended than 3d, 4d - so why Mott? Mott insulation due to combination of strong spin-orbit + intermediate correlation.

Periodic Table of Elements 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1

Hydrogen 1.00794

H1 Atomic #

Name Atomic Mass

SymbolC Solid

Hg Liquid

H Gas

Rf Unknown

Metals NonmetalsAlkali m

etals

Alkaline

earth metals

Lanthanoids

Transition m

etals

Poor m

etals

Other

nonmetals

Noble gases

Actinoids

2

Helium 4.002602

He2 K

2 3

Lithium 6.941

Li2 1 4

Beryllium 9.012182

Be2 2 5

Boron 10.811

B23 6

Carbon 12.0107

C24 7

Nitrogen 14.0067

N25 8

Oxygen 15.9994

O26 9

Fluorine 18.9984032

F27 10

Neon 20.1797

Ne2 8

KL

3 11

Sodium 22.98976928

Na2 8 1

12

Magnesium 24.3050

Mg2 8 2

13

Aluminium 26.9815386

Al283

14

Silicon 28.0855

Si284

15

Phosphorus 30.973762

P285

16

Sulfur 32.065

S286

17

Chlorine 35.453

Cl287

18

Argon 39.948

Ar2 8 8

KLM

4 19

Potassium 39.0983

K2 8 8 1

20

Calcium 40.078

Ca2 8 8 2

21

Scandium 44.955912

Sc2 8 9 2

22

Titanium 47.867

Ti2 8

10 2

23

Vanadium 50.9415

V2 8

11 2

24

Chromium 51.9961

Cr28

131

25

Manganese 54.938045

Mn28

132

26

Iron 55.845

Fe28

142

27

Cobalt 58.933195

Co28

152

28

Nickel 58.6934

Ni28

162

29

Copper 63.546

Cu2 8

18 1

30

Zinc 65.38

Zn28

182

31

Gallium 69.723

Ga28

183

32

Germanium 72.64

Ge28

184

33

Arsenic 74.92160

As28

185

34

Selenium 78.96

Se28

186

35

Bromine 79.904

Br28

187

36

Krypton 83.798

Kr2 8

18 8

KLMN

5 37

Rubidium 85.4678

Rb2 8

18 8 1

38

Strontium 87.62

Sr2 8

18 8 2

39

Yttrium 88.90585

Y2 8

18 9 2

40

Zirconium 91.224

Zr2 8

18 10

2

41

Niobium 92.90638

Nb2 8

18 12

1

42

Molybdenum95.96

Mo28

1813

1

43

Technetium (97.9072)

Tc28

1814

1

44

Ruthenium 101.07

Ru28

1815

1

45

Rhodium 102.90550

Rh28

1816

1

46

Palladium 106.42

Pd28

1818

0

47

Silver 107.8682

Ag2 8

18 18

1

48

Cadmium 112.411

Cd28

1818

2

49

Indium 114.818

In28

1818

3

50

Tin 118.710

Sn28

1818

4

51

Antimony 121.760

Sb28

1818

5

52

Tellurium 127.60

Te28

1818

6

53

Iodine 126.90447

I28

1818

7

54

Xenon 131.293

Xe2 8

18 18

8

KLMNO

6 55

Caesium 132.9054519

Cs2 8

18 18

8 1

56

Barium 137.327

Ba2 8

18 18

8 2

57–7172

Hafnium 178.49

Hf2 8

18 32 10

2

73

Tantalum 180.94788

Ta2 8

18 32 11

2

74

Tungsten 183.84

W28

183212

2

75

Rhenium 186.207

Re28

183213

2

76

Osmium 190.23

Os28

183214

2

77

Iridium 192.217

Ir28

183215

2

78

Platinum 195.084

Pt28

183217

1

79

Gold 196.966569

Au2 8

18 32 18

1

80

Mercury 200.59

Hg28

183218

2

81

Thallium 204.3833

Tl28

183218

3

82

Lead 207.2

Pb28

183218

4

83

Bismuth 208.98040

Bi28

183218

5

84

Polonium (208.9824)

Po28

183218

6

85

Astatine (209.9871)

At28

183218

7

86

Radon (222.0176)

Rn2 8

18 32 18

8

KLMNOP

7 87

Francium (223)

Fr2 8

18 32 18

8 1

88

Radium (226)

Ra2 8

18 32 18

8 2

89–103104

Ruherfordium (261)

Rf2 8

18 32 32 10

2

105

Dubnium (262)

Db2 8

18 32 32 11

2

106

Seaborgium (266)

Sg28

18323212

2

107

Bohrium (264)

Bh28

18323213

2

108

Hassium (277)

Hs28

18323214

2

109

Meitnerium (268)

Mt28

18323215

2

110

Darmstadtium (271)

Ds28

18323217

1

111

Roentgenium (272)

Rg2 8

18 32 32 18

1

112

Ununbium (285)

Uub28

18323218

2

113

Ununtrium (284)

Uut28

18323218

3

114

Ununquadium (289)

Uuq28

18323218

4

115

Ununpentium (288)

Uup28

18323218

5

116

Ununhexium(292)

Uuh28

18323218

6

117

Ununseptium

Uus118

Ununoctium (294)

Uuo2 8

18 32 32 18

8

KLMNOPQ

For elements with no stable isotopes, the mass number of the isotope with the longest half-life is in parentheses.

Periodic Table Design and Interface Copyright © 1997 Michael Dayah. http://www.ptable.com/ Last updated: May 27, 2008

57

Lanthanum 138.90547

La2 8

18 18

9 2

58

Cerium 140.116

Ce2 8

18 19

9 2

59

Praseodymium 140.90765

Pr28

1821

82

60

Neodymium 144.242

Nd28

1822

82

61

Promethium (145)

Pm28

1823

82

62

Samarium 150.36

Sm28

1824

82

63

Europium 151.964

Eu28

1825

82

64

Gadolinium 157.25

Gd2 8

18 25

9 2

65

Terbium 158.92535

Tb28

1827

82

66

Dysprosium 162.500

Dy28

1828

82

67

Holmium 164.93032

Ho28

1829

82

68

Erbium 167.259

Er28

1830

82

69

Thulium 168.93421

Tm28

1831

82

70

Ytterbium 173.054

Yb28

1832

82

71

Lutetium 174.9668

Lu2 8

18 32

9 2

89

Actinium (227)

Ac2 8

18 32 18

9 2

90

Thorium 232.03806

Th2 8

18 32 18 10

2

91

Protactinium 231.03588

Pa28

183220

92

92

Uranium 238.02891

U28

183221

92

93

Neptunium (237)

Np28

183222

92

94

Plutonium (244)

Pu28

183224

82

95

Americium (243)

Am28

183225

82

96

Curium (247)

Cm2 8

18 32 25

9 2

97

Berkelium (247)

Bk28

183227

82

98

Californium (251)

Cf28

183228

82

99

Einsteinium (252)

Es28

183229

82

100

Fermium (257)

Fm28

183230

82

101

Mendelevium (258)

Md28

183231

82

102

Nobelium (259)

No28

183232

82

103

Lawrencium (262)

Lr2 8

18 32 32

9 2

Michael Dayah For a fully interactive experience, visit www.ptable.com. [email protected]


Undoped CuO2 plane: Mott Insulator due to

e- - e- interactionVirtual hopping induces

AF exchange J=4t2/U

Doping a Mott insulator. Holes become mobile and should be a conductor in the absence of strong disorder.

The surprise in cuprates is that it becomes a superconductor!


Undoped CuO2 plane: Mott Insulator due to

e- - e- interactionVirtual hopping induces

AF exchange J=4t2/U

CuO2 plane with doped holes:

La3+ → Sr2+: La2-xSrxCuO4

t

Doping a Mott insulator. Holes become mobile and should be a conductor in the absence of strong disorder.

The surprise in cuprates is that it becomes a superconductor!


Cuprates as doped Mott insulators? Undoped cuprates are (antiferromagnetic) Mott insulators.

Does the Mott insulation play any role in the properties of the doped materials?

Is the doped Mott insulator the right/useful perspective for understanding the doped cuprates?

Despite 20+ years, this remains a contentious issue.

By focusing on phenomenology I will mostly leave it to you to form your own opinion.

However doped Mott insulator perspective leads to interesting conceptual questions, captures a lot of the `zeroth’ order physics, and is important to learn.

So without apology I will (in the theory interludes) consider the problem of doping a Mott insulator.






Briefly discuss general nature of magnetism in Mott insulators


Magnetism and Mott insulators



Spin ladders: A simple example of a quantum paramagnet


Other quantum paramagnets: ``Spin-Peierls”/Valence Bond Solid(VBS) states

• Ordered pattern of valence bonds breaks lattice translation symmetry.

• Ground state smoothly connected to band insulator

• Elementary spinful excitations have S = 1 above spin gap.

(CuGeO3, TiOCl,......)


Most interesting possibility: quantum spin liquids


Brief digression on quantum spin liquids

1. Why are quantum spin liquid Mott insulators interesting?

2. Do quantum spin liquids exist?

3. Can we understand the physics of doped spin liquid Mott insulators?How are they different from physics of doped antiferromagnets?


States of quantum magnetism

19

Ferromagnetism: May be 600 BC

Antiferromagnetism: 1930s

Key concept of broken symmetry.

Prototypical ground state wavefunction: direct product of local degrees of freedom

Short range quantum entanglement.

1930s- present: elaboration of broken symmetry and other states with short range entanglement

| ↑↓↑↓ .........�

| ↑↑↑↑ .........�


Last ≈ 10 years

20

Experimental discovery of quantum spin liquid state*.

Qualitatively new kind of state of matter. Long range quantum entanglement: Prototypical ground state wavefunction Not a direct product of local degrees of freedom.

Many new phenomena - emergence of fractional quantum numbers.

New conceptual and technical theoretical tools to understand.

May be also new kinds of experimental probes will be most useful.

* In d > 1

+

+ .........


Some natural questions

Can quantum spin liquids exist in d > 1?

Do quantum spin liquids exist in d > 1?



Can quantum spin liquids exist in d > 1? Theoretical question

Do quantum spin liquids exist in d > 1?Experimental question



Can quantum spin liquids exist in d > 1? Theoretical question: YES!! (work of many people in last 20 years)

Do quantum spin liquids exist in d > 1?Experimental question: Remarkable new materialspossibly in spin liquid phases


End of digression

Though quantum paramagnets may exist the cuprate Mott insulators are actually antiferromagnetically ordered.

Nevertheless it will be useful to consider the general problem of doping various kinds of Mott insulators, not just antiferromagnets.


How to tell if a Mott insulator has magnetic long range order?

Best option: Neutron diffraction - look for Bragg peaks at the ordering wavevector.

More generally, inelastic neutron scattering measures time dependent spin correlations even if there is no magnetic long range order.

Caveat: Need large single crystals, no strong absorption of neutrons by nuclei,.....

Other methods: NMR, magnetic X-ray diffraction (well suited for iridates)


Neutron scattering:

If there is long range AF order, Bragg peaks appear at G’s.

The direction of the ordered moment can be determined by rotating G.

In the absence of long range order, we can measure equal time correlation function by integrating over ω.


Local moment picture works. Reduced from classical moment of unity due to quantum fluctuations of S=1/2.







The Mott metal-insulator transition

1. Insulator has magnetic order but no Fermi surface

2. With sufficient doping expect to recover Fermi liquid metal which has no magnetism but has a Fermi surface satisfying Luttinger’s theorem (``large Fermi surface”).

Key questions:

How does the Fermi surface die on approaching Mott from metal?

How does magnetism die on approaching metal from Mott?

Difficult old problem in quantum many body physics.


Cuprate example

T

x

Pseudogap

AF Mottinsulator

Non-fermi liquid metal

Fermi liquid


Low doping: AF order. Unit cell is doubled. We have small pockets of total area equal to x times the area of BZ.

Doping x holes in a Mott insulator.

Large doping: no unit cell doubling.

Total Fermi surface area is

Area in the reduced BZ is

?


Evolution from AF to Fermi liquid via superconductor at low T and via pseudo-gap to “strange metal” to fermi liquid at higher T.







How many ways does Nature have to deal with doping a Mott insulator?

Electron doped.

3 Dimension. Brinkman-Rice Fermi liquid.

AF with localized carriers.

Micro phase separation: stripes

Organic ET salts. Metal-insulator transition by tuning U/t.

Possibility of a “spin liquid”.

Doping yields a superconductor.

A second family of HiTc superconductors!


Electron doped side: AF persists to x=0.13 and the doped electrons are localized.

What is the origin of the p-h asymmetry? Hopping of electron on Cu (d10) is physically different from hopping of a Zhang-Rice singlet located on the oxygen. One possibility is polaron effect is stronger on the electron side.


J=31 meV

X<0.2 commensurate spin order, localized hole. (polaron effect?)

0.2<x diagonal stripe with 1 hole per Ni. (microscopic phase separation into Ni2+ and Ni3+).

Non-metallic until x=0.9

Now ½ hole per linear distance along the stripe (2 Cu sites) : mobile charge.

Period 4a for charge and period 8a for spin.

Smaller J means it is deeper in the Mott phase. Effective hopping is also small and disorder and/or polaron effects favor localized carriers.

Stripes also seen in cuprates.


Tokura et al, PRL 70, 2126 (1993).

X=0 is a band insulator, x=1 is a Mott insulator.

For x=1, Ti is d1 and has S=1/2. Very small optical gap (0.2eV). Surprisingly small TN=150K, (reduced due to orbital degeneracy).

3 dim perovskite structure.

Specific heat = γT


This is an example of “Brinkman-Rice Fermi liquid”.

Diverging mass near the Mott insulator. m*/m=1/xh, z=xh.

σ= e^2nτ/m* is proportional to xh , even though Fermi surface is “large” and has volume x=1-xh as inferred from the Hall effect.

Fermi surface dies not by losing volume (and hence charge carriers) but bydiverging quasiparticle effective mass (and vanishing quasiparticle weight).


X = Cu(NCS)2, Cu[N(CN)2]Br, Cu2(CN)3…..

Q2D organics κ-(ET)2X

anisotropic triangular lattice

dimer model

ET

X

t’ / t = 0.5 ~ 1.1

t’t t

Mott insulator


Pressure tuned superconductivity in the organics

κ-Cu[N(CN)2]Cl t’/t = 0.75

Pressure decreases U/t.

Mott transition is induced by tuning U/t at fixed density of one electron per site.


Metal- insulator transition by tuning U/t.

U/t

x

AF Mott insulator

metal

Cuprate superconductor



U/t

x

AF Mott insulator

metal


Organic superconductor



U/t

x

AF Mott insulator

metal


Organic superconductor

Tc=100K, t=.4eV, Tc/t=1/40.

Tc=12K, t=.05eV, Tc/t=1/40.


Superconductivity in doped ET, (ET)4Hg2.89Br8, was first discovered Lyubovskaya et al in 1987. Pressure data form Taniguchi et al, J. Phys soc Japan, 76, 113709 (2007).

Doping of an organic Mott insulator.


Brief aside: Quantum spin liquids in the organics

Same family of organics also provide fascinating examples of quantum spin liquid Mott insulators.

Many interesting phenomena:

Insulator with specific heat, spin susceptibility like a metal.

Most dramatic: Metallic thermal transport in an insulator!


No magnetic order


A gapless spin liquid


Phase diagram


EtMe3Sb[Pd(dmit)2]2

Another candidate spin liquid on a triangular lattice

47

Highly Mobile Gapless Excitationsin a Two-Dimensional CandidateQuantum Spin LiquidMinoru Yamashita,1* Norihito Nakata,1 Yoshinori Senshu,1 Masaki Nagata,1Hiroshi M. Yamamoto,2,3 Reizo Kato,2 Takasada Shibauchi,1 Yuji Matsuda1*

The nature of quantum spin liquids, a novel state of matter where strong quantum fluctuationsdestroy the long-range magnetic order even at zero temperature, is a long-standing issue inphysics. We measured the low-temperature thermal conductivity of the recently discoveredquantum spin liquid candidate, the organic insulator EtMe3Sb[Pd(dmit)2]2. A sizable lineartemperature dependence term is clearly resolved in the zero-temperature limit, indicating thepresence of gapless excitations with an extremely long mean free path, analogous to excitationsnear the Fermi surface in pure metals. Its magnetic field dependence suggests a concomitantappearance of spin-gap–like excitations at low temperatures. These findings expose a highlyunusual dichotomy that characterizes the low-energy physics of this quantum system.

Spin systems confined to low dimensionsexhibit a rich variety of quantum phenome-na. Particularly intriguing are quantum

spin liquids (QSLs), antiferromagnets with quan-tum fluctuation–driven disordered ground states,which have been attracting tremendous attentionfor decades (1). The notion of QSLs is now firmlyestablished in one-dimensional (1D) spin sys-tems. In dimensions greater than one, it is widelybelieved that QSL ground states emerge when in-teractions among themagnetic degrees of freedomare incompatible with the underlying crystal ge-ometry, leading to a strong enhancement of quan-tum fluctuations. In 2D, typical examples of systemswhere such geometrical frustrations are presentare the triangular and kagomé lattices. Largely trig-gered by the proposal of the resonating-valence-bond theory on a2D triangular lattice and its possibleapplication to high-transition temperature cuprates(2), realizing QSLs in 2D systems has been along-sought goal. However, QSL states are hardto achieve experimentally because the presenceof small but finite 3D magnetic interactionsusually results in some ordered (or frozen) state.Two recently discovered organic insulators,k-[bis(ethylenedithio)-tetrathiafulvalene]2Cu2(CN)3[k-(BEDT-TTF)2Cu2(CN)3] (3) and EtMe3Sb[Pd(dmit)2]2 (4, 5), both featuring 2D spin-1/2Heisenberg triangular lattices, are believed to bepromising candidate materials that are likely tohost QSLs. In both compounds, nuclear magneticresonance (NMR) measurements have shown no

long-range magnetic order down to a temperaturecorresponding to J/12,000, where J (~250 K forboth compounds) is the nearest-neighbor spininteraction energy (exchange coupling) (3, 5). In atriangular lattice antiferromagnet, the frustrationbrought on by the nearest-neighbor Heisenberg

interaction is known to be insufficient to destroythe long-range ordered ground state (6). This hasled to the proposals of numerous scenarios whichmight stabilize a QSL state: spinon Fermi surface(7, 8), algebraic spin liquid (9), spin Bose metal(10), ring-exchange model (11), Z2 spin liquidstate (12), chiral spin liquid (13), Hubbard modelwith a moderate onsite repulsion (14, 15), andone-dimensionalization (16, 17). Nevertheless,the origin of the QSL in the organic compoundsremains an open question.

To understand the nature of QSLs, knowledgeof the detailed structure of the low-lying elemen-tary excitations in the zero-temperature limit, par-ticularly the presence or absence of an excitationgap, is of primary importance (18). Such infor-mation bears immediate implications on the spincorrelations of the ground state, as well as thecorrelation length scale of the QSL. For example,in 1D spin-1/2 Heisenberg chains, the elementaryexcitations are gapless spinons (chargeless spin-1/2 quasiparticles) characterized by a linear en-ergy dispersion and a power-law decay of the spincorrelation (19), whereas in the integer spin casesuch excitations are gapped (20). In the organiccompound k-(BEDT-TTF)2Cu2(CN)3, where thefirst putative QSL state was reported (3), the pres-ence of the spin excitation gap is controversial(18, 21). In this compound, the stretched, non-

REPORTS

1Department of Physics, Graduate School of Science, KyotoUniversity, Kyoto 606-8502, Japan. 2RIKEN, Wako-shi, Saitama351-0198, Japan. 3Japan Science and Technology Agency,Precursory Research for Embryonic Science and Technology(JST-PRESTO), Kawaguchi, Saitama 332-0012, Japan.

*To whom correspondence should be addressed. E-mail:[email protected] (M.Y.); [email protected] (Y.M.)

t

Non-magnetic layer(EtMe3Sb, Et2Me2Sb)

Pd(dmit)2 moleculeA

B C

Fig. 1. The crystal structure of EtMe3Sb[Pd(dmit)2]2 and Et2Me2Sb[Pd(dmit)2]2. (A) A view parallelto the 2D magnetic Pd(dmit)2 layer, separated by layers of a nonmagnetic cation. (B) The spinstructure of the 2D planes of EtMe3Sb[Pd(dmit)2]2 (dmit-131), where Et = C2H5, Me = CH3, anddmit = 1,3-dithiole-2-thione-4,5-dithiolate. Pd(dmit)2 are strongly dimerized (table S1), formingspin-1/2 units [Pd(dmit)2]2

– (blue arrows). The antiferromagnetic frustration gives rise to a state inwhich none of the spins are frozen down to 19.4 mK (4). (C) The spin structure of the 2D planes ofEt2Me2Sb[Pd(dmit)2]2 (dmit-221). A charge order transition occurs at 70 K, and the units areseparated as neutral [Pd(dmit)2]2

0 and divalent dimers [Pd(dmit)2]22–. The divalent dimers form

intradimer valence bonds, showing a nonmagnetic spin singlet (blue arrows) ground state with avery large excitation gap (24).

4 JUNE 2010 VOL 328 SCIENCE www.sciencemag.org1246

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Highly Mobile Gapless Excitationsin a Two-Dimensional CandidateQuantum Spin LiquidMinoru Yamashita,1* Norihito Nakata,1 Yoshinori Senshu,1 Masaki Nagata,1Hiroshi M. Yamamoto,2,3 Reizo Kato,2 Takasada Shibauchi,1 Yuji Matsuda1*

The nature of quantum spin liquids, a novel state of matter where strong quantum fluctuationsdestroy the long-range magnetic order even at zero temperature, is a long-standing issue inphysics. We measured the low-temperature thermal conductivity of the recently discoveredquantum spin liquid candidate, the organic insulator EtMe3Sb[Pd(dmit)2]2. A sizable lineartemperature dependence term is clearly resolved in the zero-temperature limit, indicating thepresence of gapless excitations with an extremely long mean free path, analogous to excitationsnear the Fermi surface in pure metals. Its magnetic field dependence suggests a concomitantappearance of spin-gap–like excitations at low temperatures. These findings expose a highlyunusual dichotomy that characterizes the low-energy physics of this quantum system.

Spin systems confined to low dimensionsexhibit a rich variety of quantum phenome-na. Particularly intriguing are quantum

spin liquids (QSLs), antiferromagnets with quan-tum fluctuation–driven disordered ground states,which have been attracting tremendous attentionfor decades (1). The notion of QSLs is now firmlyestablished in one-dimensional (1D) spin sys-tems. In dimensions greater than one, it is widelybelieved that QSL ground states emerge when in-teractions among themagnetic degrees of freedomare incompatible with the underlying crystal ge-ometry, leading to a strong enhancement of quan-tum fluctuations. In 2D, typical examples of systemswhere such geometrical frustrations are presentare the triangular and kagomé lattices. Largely trig-gered by the proposal of the resonating-valence-bond theory on a2D triangular lattice and its possibleapplication to high-transition temperature cuprates(2), realizing QSLs in 2D systems has been along-sought goal. However, QSL states are hardto achieve experimentally because the presenceof small but finite 3D magnetic interactionsusually results in some ordered (or frozen) state.Two recently discovered organic insulators,k-[bis(ethylenedithio)-tetrathiafulvalene]2Cu2(CN)3[k-(BEDT-TTF)2Cu2(CN)3] (3) and EtMe3Sb[Pd(dmit)2]2 (4, 5), both featuring 2D spin-1/2Heisenberg triangular lattices, are believed to bepromising candidate materials that are likely tohost QSLs. In both compounds, nuclear magneticresonance (NMR) measurements have shown no

long-range magnetic order down to a temperaturecorresponding to J/12,000, where J (~250 K forboth compounds) is the nearest-neighbor spininteraction energy (exchange coupling) (3, 5). In atriangular lattice antiferromagnet, the frustrationbrought on by the nearest-neighbor Heisenberg

interaction is known to be insufficient to destroythe long-range ordered ground state (6). This hasled to the proposals of numerous scenarios whichmight stabilize a QSL state: spinon Fermi surface(7, 8), algebraic spin liquid (9), spin Bose metal(10), ring-exchange model (11), Z2 spin liquidstate (12), chiral spin liquid (13), Hubbard modelwith a moderate onsite repulsion (14, 15), andone-dimensionalization (16, 17). Nevertheless,the origin of the QSL in the organic compoundsremains an open question.

To understand the nature of QSLs, knowledgeof the detailed structure of the low-lying elemen-tary excitations in the zero-temperature limit, par-ticularly the presence or absence of an excitationgap, is of primary importance (18). Such infor-mation bears immediate implications on the spincorrelations of the ground state, as well as thecorrelation length scale of the QSL. For example,in 1D spin-1/2 Heisenberg chains, the elementaryexcitations are gapless spinons (chargeless spin-1/2 quasiparticles) characterized by a linear en-ergy dispersion and a power-law decay of the spincorrelation (19), whereas in the integer spin casesuch excitations are gapped (20). In the organiccompound k-(BEDT-TTF)2Cu2(CN)3, where thefirst putative QSL state was reported (3), the pres-ence of the spin excitation gap is controversial(18, 21). In this compound, the stretched, non-

REPORTS

1Department of Physics, Graduate School of Science, KyotoUniversity, Kyoto 606-8502, Japan. 2RIKEN, Wako-shi, Saitama351-0198, Japan. 3Japan Science and Technology Agency,Precursory Research for Embryonic Science and Technology(JST-PRESTO), Kawaguchi, Saitama 332-0012, Japan.

*To whom correspondence should be addressed. E-mail:[email protected] (M.Y.); [email protected] (Y.M.)

t

Non-magnetic layer(EtMe3Sb, Et2Me2Sb)

Pd(dmit)2 moleculeA

B C

Fig. 1. The crystal structure of EtMe3Sb[Pd(dmit)2]2 and Et2Me2Sb[Pd(dmit)2]2. (A) A view parallelto the 2D magnetic Pd(dmit)2 layer, separated by layers of a nonmagnetic cation. (B) The spinstructure of the 2D planes of EtMe3Sb[Pd(dmit)2]2 (dmit-131), where Et = C2H5, Me = CH3, anddmit = 1,3-dithiole-2-thione-4,5-dithiolate. Pd(dmit)2 are strongly dimerized (table S1), formingspin-1/2 units [Pd(dmit)2]2

– (blue arrows). The antiferromagnetic frustration gives rise to a state inwhich none of the spins are frozen down to 19.4 mK (4). (C) The spin structure of the 2D planes ofEt2Me2Sb[Pd(dmit)2]2 (dmit-221). A charge order transition occurs at 70 K, and the units areseparated as neutral [Pd(dmit)2]2

0 and divalent dimers [Pd(dmit)2]22–. The divalent dimers form

intradimer valence bonds, showing a nonmagnetic spin singlet (blue arrows) ground state with avery large excitation gap (24).

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Phenomenology broadly similar to kappa-ET spin liquid.

Weak Mott insulator - close to pressure driven Mott transition.

No magnetic ordering to T << Jbut gapless spin excitations (NMR, specific heat).


Metallic thermal transport in a Mott insulator

48

exponential decay of the NMR relaxation indicatesinhomogeneous distributions of spin excitations(22), which may obscure the intrinsic propertiesof the QSL. A phase transition possibly associatedwith the charge degree of freedom at ~6 K furthercomplicates the situation (23). Meanwhile, inEtMe3Sb[Pd(dmit)2]2 (dmit-131) such a transi-tion is likely to be absent, and a muchmore homo-geneous QSL state is attained at low temperatures(4, 5). As a further merit, dmit-131 (Fig. 1B) hasa cousinmaterial Et2Me2Sb[Pd(dmit)2]2 (dmit-221)with a similar crystal structure (Fig. 1C), whichexhibits a nonmagnetic charge-ordered state witha large excitation gap below 70 K (24). A com-parison between these two related materials willtherefore offer us the opportunity to single outgenuine features of the QSL state believed to berealized in dmit-131.

Measuring thermal transport is highly advan-tageous for probing the low-lying elementaryexcitations in QSLs, because it is free from thenuclear Schottky contribution that plagues theheat capacity measurements at low temperatures(21). Moreover, it is sensitive exclusively to itin-erant spin excitations that carry entropy, whichprovides important information on the nature of the

spin correlation and spin-mediated heat transport.Indeed, highly unusual transport properties includ-ing the ballistic energy propagation have been re-ported in a 1D spin-1/2 Heisenberg system (25).

The temperature dependence of the thermalconductivity kxx divided by Tof a dmit-131 singlecrystal displays a steep increase followed by arapid decrease after showing a pronounced maxi-mum at Tg ~ 1 K (Fig. 2A). The heat is carriedprimarily by phonons (kxx

ph) and spin-mediatedcontributions (kxx

spin). The phonon contributioncan be estimated from the data of the nonmagneticstate in a dmit-221 crystal with similar dimensions,which should have a negligibly small kxx

spin. Indmit-221, kxx

ph/T exhibits a broad peak at around1 K, which appears when the phonon conductiongrows rapidly and is limited by the sample bound-aries. On the other hand, kxx/Tof dmit-131, whichwell exceeds kxx

ph/T of dmit-221, indicates a sub-stantial contribution of spin-mediated heat con-duction below 10K. This observation is reinforcedby the large magnetic field dependence of kxx ofdmit-131, as discussed below (Fig. 3A). Figure2B shows a peak in the kxx versus T plot for dmit-131, which is absent in dmit-221. We thereforeconclude that kxx

spin and kxxspin/T in dmit-131 have

a peak structure at Tg ~ 1 K, which characterizesthe excitation spectrum.

The low-energy excitation spectrum can beinferred from the thermal conductivity in the low-temperature regime. In dmit-131, kxx/T at lowtemperatures is well fitted by kxx/T= k00/T + bT2

(Fig. 2C), where b is a constant. The presence of aresidual value in kxx/T at T!0 K, k00/T, is clearlyresolved. The distinct presence of a nonzero k00/Tterm is also confirmed by plotting kxx/T versus T(Fig. 2D). In sharp contrast, in dmit-221, a corre-sponding residual k00/T is absent and only a pho-non contribution is observed (26). The residualthermal conductivity in the zero-temperature limitimmediately implies that the excitation from theground state is gapless, and the associated correla-tion function has a long-range algebraic (power-law)dependence. We note that the temperature depen-dence of kxx/T in dmit-131 is markedly differentfrom that in k-(BEDT-TTF)2Cu2(CN)3, in whichthe exponential behavior of kxx/Tassociated withthe formation of excitation gap is observed (18).

Key information on the nature of elementaryexcitations is further provided by the field depen-dence of kxx. Because it is expected that kxx

ph ishardly influenced by the magnetic field, particu-larly at very low temperatures, the field depen-dence is governed by kxx

spin(H) (26). The obtainedH-dependence, kxx(H), at low temperatures isquite unusual (Fig. 3A). At the lowest temperature,kxx(H) at low fields is insensitive toH but displaysa steep increase above a characteristic magneticfieldHg ~ 2 T. At higher temperatures close to Tg,this behavior is less pronounced, and at 1K kxx(H)increases with H nearly linearly. The observedfield dependence implies that some spin-gap–likeexcitations are also present at low temperatures,along with the gapless excitations inferred fromthe residual k00/T. The energy scale of the gap ischaracterized by mBHg, which is comparable tokBTg. Thus, it is natural to associate the observedzero-field peak in kxx(T)/Tat Tgwith the excitationgap formation.

Next we examined a dynamical aspect of thespin-mediated heat transport. An important ques-tion is whether the observed energy transfer viaelementary excitations is diffusive or ballistic. Inthe 1D spin-1/2 Heisenberg system, the ballisticenergy propagation occurs as a result of the con-servation of energy current (25). Assuming thekinetic approximation, the thermal conductivityis written as kxx

spin = Csvs‘s /3, where Cs is the spe-cific heat, vs is the velocity, and ‘s is themean freepath of the quasiparticles responsible for the ele-mentary excitations. We tried to estimate ‘s sim-ply by assuming that the linear term in the thermalconductivity arises from the fermionic excitations,in analogy with excitations near the Fermi surfacein metals. The residual term is written as k00/T ~(kB

2/da!)‘s, where d (~3 nm) and a (~1 nm) areinterlayer and nearest-neighbor spin distance. Weassumed the linear energy dispersion e(k)= !vsk,a 2D density of states and a Fermi energy com-parable to J (26). From the observed k00/T, wefind that ‘s reaches as long as ~1 mm, indicating

1.0

0.8

0.6

0.4

0.2

0.0

xx/T

(W

/K2 m

)

0.100.080.060.040.020.00

dmit-131

dmit-221

!-(BEDT-TTF)2Cu2(CN)3

("2)

0.8

0.6

0.4

0.2

0.00.30.0 T (K)

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

xx/T

(W

/K2 m

)

1086420T (K) T 2 (K2)

Tg

dmit-131 (spin liquid) dmit-221 (non-magnetic)

1.6

1.2

0.8

0.4

0.0

xx (W

/K m

)

1086420T (K)

A B CD

Fig. 2. The temperature dependence of kxx(T)/T (A) and kxx(T) (B) of dmit-131 (pink) and dmit-221(green) below 10 K in zero field [kxx(T) is the thermal conductivity]. A clear peak in kxx/T is observed indmit-131 at Tg ~ 1 K, which is also seen as a hump in kxx. Lower temperature plot of kxx(T)/T as a functionof T2 (C) and T (D) of dmit-131, dmit-221, and k-(BEDT-TTF)2Cu2(CN)3 (black) (18). A clear residual ofkxx(T)/T is resolved in dmit-131 in the zero-temperature limit.

Fig. 3. (A) Field dependence ofthermal conductivity normalizedby the zero field value, [kxx(H) –kxx(0)]/kxx(0) of dmit-131 at lowtemperatures. (Inset) The heat cur-rent Q was applied within the 2Dplane, and the magnetic field H wasperpendicular to the plane. kxx andkxy were determined by diagonaland off-diagonal temperature gra-dients, DTx and DTy, respectively.(B) Thermal-Hall angle tanq(H) =kxy/(kxx – kxxph)as a function ofH at0.23 K (blue), 0.70 K (green), and1.0 K (red).

0.3

0.2

0.1

0.0

-0.1

{ xx

(H) -

xx

(0) }

/ xx

(0)

121086420

0H (T)

0.23 K 0.70 K 1.0 K

Hg

-0.1

0.0

0.1

tan

(H)

1210864200H (T)

A

B

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dmit quantum spin liquid

Gapless excitations are mobile in dmit spin liquid!


Summary of Lecture 1


2. Magnetism and Mott insulators- apart from antiferromagnetism, possibility of quantum paramagnets; quantum spin liquid most dramatic. Useful theoretically to think about fate of doping all kinds of Mott insulators, not just antiferromagnets.

3. Doping a Mott insulator-(i) some general theoretical questionsHow does Fermi surface die? How does magnetism evolve?

-(ii) experiments on a few materials. Cuprates most spectacular - superconductivity, strange metal, and pseudogap between fermi liquid and Mott insulator


Corner sharing octahedrals.

3d

eg

t2g dxy,dyz,dzx

dz2, dx2-y2

Octahedral

field splitting

X2-y2

z2


Ogata and Fukuyama, Rep. Progress in Physics, 71, 036501 (2008)

Charge transfer insulator.

Electron picture Hole picture

Mott insulator


Doping a charge transfer insulator: The “Zhang-Rice singlet”

Symmetric orbital

centered on Cu.Anti-symmetric orbital

Due to AF exchange between Cu and O, the singlet symmetric orbital gains a large energy, of order 6 eV. This singlet orbital can hop with effective hopping t given by:


Also from Raman scattering.

Largest J known among transition metal oxide, except for the Cu-O chain compound where J=220meV.

By fitting the spin wave dispersion measured by neutron scattering. (also needs a small ring exchange term.)

Spin flip breaks 6 bonds, costs 3J.


What is unique about the cuprates?

Pure CuO2 plane Single band Hubbard model, or its strong coupling limit, the t-J model.

⇒

Dopeholes t

J t ≈ 3 J1) low dimension

2) H = J Σ Si · Sjnnlarge J = 135 meV

Competition:

t favors delocalization of electrons

J favors ordering of localized spins3) quantum spin S =1/2 (NNN hopping t’ may explain asymmetry

Between electron and hole doping )Wednesday, December 15, 2010

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