doped mott insulators and high temperature...
TRANSCRIPT
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
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
A useful theoretical model
Wednesday, December 15, 2010
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!
Wednesday, December 15, 2010
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]
Wednesday, December 15, 2010
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!
Wednesday, December 15, 2010
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!
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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.
Briefly discuss general nature of magnetism in Mott insulators
Wednesday, December 15, 2010
Magnetism and Mott insulators
Wednesday, December 15, 2010
Wednesday, December 15, 2010
Spin ladders: A simple example of a quantum paramagnet
Wednesday, December 15, 2010
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,......)
Wednesday, December 15, 2010
Most interesting possibility: quantum spin liquids
Wednesday, December 15, 2010
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?
Wednesday, December 15, 2010
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
| ↑↓↑↓ .........�
| ↑↑↑↑ .........�
Wednesday, December 15, 2010
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
+
+ .........
Wednesday, December 15, 2010
Some natural questions
Can quantum spin liquids exist in d > 1?
Do quantum spin liquids exist in d > 1?
Wednesday, December 15, 2010
Some natural questions
Can quantum spin liquids exist in d > 1? Theoretical question
Do quantum spin liquids exist in d > 1?Experimental question
Wednesday, December 15, 2010
Some natural questions
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
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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)
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
Local moment picture works. Reduced from classical moment of unity due to quantum fluctuations of S=1/2.
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
Cuprate example
T
x
Pseudogap
AF Mottinsulator
Non-fermi liquid metal
Fermi liquid
Wednesday, December 15, 2010
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
?
Wednesday, December 15, 2010
Evolution from AF to Fermi liquid via superconductor at low T and via pseudo-gap to “strange metal” to fermi liquid at higher T.
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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!
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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
Wednesday, December 15, 2010
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).
Wednesday, December 15, 2010
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
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
Metal- insulator transition by tuning U/t.
U/t
x
AF Mott insulator
metal
Cuprate superconductor
Wednesday, December 15, 2010
Metal- insulator transition by tuning U/t.
U/t
x
AF Mott insulator
metal
Cuprate superconductor
Organic superconductor
Wednesday, December 15, 2010
Metal- insulator transition by tuning U/t.
U/t
x
AF Mott insulator
metal
Cuprate superconductor
Organic superconductor
Tc=100K, t=.4eV, Tc/t=1/40.
Tc=12K, t=.05eV, Tc/t=1/40.
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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!
Wednesday, December 15, 2010
No magnetic order
Wednesday, December 15, 2010
A gapless spin liquid
Wednesday, December 15, 2010
Phase diagram
Wednesday, December 15, 2010
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).
Wednesday, December 15, 2010
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!
Wednesday, December 15, 2010
Summary of Lecture 1
1. Cuprates as doped Mott insulators
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
Wednesday, December 15, 2010
Corner sharing octahedrals.
3d
eg
t2g dxy,dyz,dzx
dz2, dx2-y2
Octahedral
field splitting
X2-y2
z2
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Ogata and Fukuyama, Rep. Progress in Physics, 71, 036501 (2008)
Charge transfer insulator.
Electron picture Hole picture
Mott insulator
Wednesday, December 15, 2010
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:
Wednesday, December 15, 2010
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.
Wednesday, December 15, 2010
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