Doped Mott insulators and high Tc superconductivity Lecture...

Doped Mott insulators and high Tc superconductivity

Lecture 2

T. Senthil and M. Randeria

Wednesday, December 15, 2010

Plan

2

0. Unfinished business from Lecture 1 - Quasi-2d organic materials

1. Cuprate phenomenology


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!


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


EtMe3Sb[Pd(dmit)2]2

Another candidate spin liquid on a triangular lattice

11

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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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

12

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/K

2 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/K

2 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!


End of digression

13

Quantum spin liquids near the Mott transition:

Growing number of experimental candidates - many dramatic phenomena.

A new chapter in condensed matter physics

Back to cuprates...................


High Tc Phase diagram

Plan

1.Overdoped – is it `conventional’?

2.What is strange about the strange metal?

3. Theory interlude


A preliminary look: transport


Overdoped metal

• Does it have a Fermi surface? Size and shape?

Methods to detect – ARPES, deHaas-van Alphen and related quantum oscillations, other….. eg Angle Dependant Magneto-Resistance (ADMR)

• Is it really a Fermi liquid with Landau quasiparticles?


Overdoped metal: Is there a Fermi surface?


deHaas van Alphen, other `quantum oscillations’: classic Fermi surface

determination methods


Remarks on quantum oscillations


Quantum oscillations in Tl-2201

Tc = 10 K, B upto 60 T; oscillations in both M and in c-axis ρ


Thermal conductivity: Wiedemann-Franz law


Is the OD state really a Fermi liquid?



Plan

1.Overdoped – is it `conventional’?

2.What is strange about the strange metal?

3.Theory interlude


What is strange about the strange metal?

25

Mohit Randeria Lec 1

1. Photoemission

Sharp (upto thermal smearing) large Fermi surface but no Landau quasiparticles

2. Transport - linear T resistivity

This lecture - more discussion of transport and other anomalies in strange metal.


The strange metal: electrical transportLinear-T resistivity near optimal doping with nearly zero intercept.

Slope of resistivity/layer roughly the same (1.5 µΩ cm/K) for all materials.

Sheet resistance = ρ/d ~ (h/e^2) T/J

Bi-2201Wednesday, December 15, 2010

Linear resistivity at very low-T

Tied to ``quantum criticality”?

Quantum critical point: second order phase transition at T = 0


Magnetotransport: Hall effect


Optical transport: high frequency tail


Optical transport: low frequency peak


Spin physics: spin susceptibility and NMR relaxation


Dynamic spin correlations: neutron scattering in LSCO


Transition to SC: onset of coherenceARPES results


Onset of coherence in transport


Neutron resonance


Summary on strange metal

Strange metal: Power laws in many physical quantities;

Large Fermi surface but no Landau-like quasiparticles

Slow growth of antiferromagnetic spin correlations

Transition to superconductivity accompanied by appearance of coherent quasiparticles and a sharp spin triplet `resonance’ mode.


Brief theory interlude

Some basic questions

1.How does a metal emerge from a Mott insulator?

2. Why superconductivity?

Simple physical picture (Anderson1987): Superexchange favors formation of singlet valence bonds between localized spins.

Doped Mott insulator: Hole motion in background of valence bonds.


t

Cartoon pictures

Large doping: Hubbard-U not very effective in blocking charge motion

Expect `large Fermi surface’ with area set by 1-x.

What happens as doping is reduced to approach Mott insulator?


Cartoon pictures

t

Low doping: Most of the time most electronsunable to hop to neighboring sites due to Mott-blocking.

If electrons stay localized next to each other long enough, will develop superexchange which will lock their spins into singlets.

Electron configuration changes at long times – conveniently view as motion of holes in sea of singlets.

Resulting state: metallic but with a spin gap due tovalence bond formation => `pseudogap metal”.


Why superconductivity?

Crucial Anderson insight:

Singlet valence bond between localized spins: A localized Cooper pair.

`Pairing’ comes from superexchange due to a repulsive Hubbard interaction.

If spins were truly localized, Cooper pairs do not move => no superconductivity.

Nonzero doping: allow room for motion of valence bonds => superconductivity!

Hole picture: Coherent hole motion in valence bond sea


Fate of collection of valence bonds

Two general possibilities: Valence bonds can crystallize to form a solid (``Valence Bond Solid”) ORStay liquid to form a `Resonating Valence Bond’

Ongoing debates on which one is more relevant but very formation of valence bond crucial ingredient in much thinking about cuprates.

VBS state(with dopinga `bond centered’ stripe)

RVB state = quantum spin liquid


Doped Mott insulators and high Tc superonductivity

T. Senthil and Mohit Randeria

Lecture 3

42


Cartoon understanding of phase diagram

Formation of singlet valence bond

Coherence of hole motion


Does valence bond formation provide a legitimate theoretical route for superconductivity in a repulsive

doped Mott insulator? Many different kinds of studies:

1.1d doped spin ladder: Zero doping – spin gapped insulator due to valence bond formation. Dope – (power law) superconductor .

2. Quasi-1d: Weakly coupled ladders

3. Inhomogenous 2d: Checkerboard Hubbard model

4. Superconductivity in doped VBS Mott insulators (`large-N’ methods): spontaneously generate weakly coupled ladders.

5. Superconductivity in doped spin liquid Mott insulators (i.e insulators with one electron per site)


Superconductivity in doped spin liquids: mean field


Superconductivity in doped spin liquids: variational wavefunctions


Common features of superconductivity in doped (paramagnetic) Mott insulators



Plan

1. Phenomenology of the pseudogap - review, quantum oscillations, and competing orders

2. Some theory

3. Refined basic questions

Pseudogap: loss of ``density of states” in spin, single particle properties


ARPES data fromU. Chatterjee et al (2010)Courtesy: J.C. Campuzano

Bi2212

Some real high Tc phase diagrams


K. Ohishi et al., cond-mat/0412313

Courtesy: J. C. Davis

Text


Summary of ARPES in underdoped pseudogap regime

1. Big antinodal gap – 50 meV or bigger

2. Gapless Fermi arcs near node that shrink as T is reduced;

possibly even extrapolate to 0 at T = 0.

3. Gap is apparently centered on large Fermi surface


New mystery: quantum oscillations in a magnetic field at low T


High field ground state: contrast between under and over-doped


How do all this fit together?


How to fit together?


Arcs versus pockets

Could it be that the arcs are really just one side of a closed pocket near the nodal region?


Competing order and fluctuations

Apart from superconductivity, many other ordered or nearly ordered (i.e short range ordered) states have been reported in the underdoped cuprates.

Some prominent examples:1. Antiferromagnetism/SDW/spin stripes 2. Charge order – charge stripes/CDW/checkerboard3. Nematic order (breaking of lattice rotation symmetry without

breaking translation symmetry). 4. Others - broken T-reversal, circulating currents,......

Implication/importance of these for pseudogap/SC/strange metal not currently understood.


Phase fluctuations above Tc: Nernst/diamagnetism

If Tc controlled by phase stiffness, might expect region with enhanced superconducting phase fluctuations in the `normal’ state above Tc.

Experiment: Microwave conductivity (Corson,…..Orenstein)

Nernst effect and diamagnetism (Wang, Li,……, Ong)(next few slides courtesy of Lu Li)

This fluctuations regime surely exists but does not extend all the way to T*.


Vortices move in a temperature gradientPhase slip generates Josephson voltage

ey = Ey /| T |Nernst signal:

EJ = B x v

Wang et al. PRB(2001)

Vortex Nernst effect

Nernst Region


Magnetization curve of type-II superconductors

Advantage• Clear determination of Hc2 and Hc1 • Area = Condensation energy U

Difficulty•In cuprates, Hc2 ~ 50-150 T• M < 1000 A/m ( ~ 12 G)• HARD to resolve with commercial SQUID magnetometers ( Hmax = 5 T or 7 T)

Diamagnetic signal: need high-resolution magnetometry !

Magnetization study:


Torque magnetometry

Torque on moment: τ = m × B

Deflection of cantilever: τ = k φ

Meff = τ / µ0HVsin(φ)


Torque magnetometry

Torque on moment: τ = m × B

Deflection of cantilever: τ = k φ

B

m×τ

φ

Meff = τ / µ0HVsin(φ)


Meff = τ / µ0HVsin(θ)

Examples of Magnetization curves (I):M vs. T and the onset temperature Tonset

Enhanced diamagnetic signals above TC

Weakly linear orbital background ΔχorbH

Lu Li, ThesisWednesday, December 15, 2010

ConclusionDiamagnetism up to 130 K Non-linear M-H

Bi2212

A universal SC fluctuation onset temperature vs x?


Other order and fluctuations: Antiferromagnetism

AF LRO disappears at very low doping but some soft spin fluctuations persist to high doping. Eg: Neutron resonance in SC state seen in most cuprates.

Resonance frequency decreases with Tc in underdoped.

Soft mode of AF LRO?


`Universal’ spin fluctuation spectrum of superconducting cuprates

Yamada plot for LSCO


Broken translation symmetry I: charge stripes


Broken translation symmetry in STM: methods


12 nm

Science 315, 1380 (2007) , Nature 454, 1072, (2008)

Broken translational symmetry in STM: bond-centered `glass’


4a0

12 nm

Science 315, 1380 (2007) , Nature 454, 1072, (2008)



4a0

12 nm

Cu-O-Cu bond-centered: Trans. symmetry breaking & C4→C2

Science 315, 1380 (2007) , Nature 454, 1072, (2008)



Electronic nematics

Break lattice rotation symmetry without breaking translation symmetry


Correlation with T*


Field induced magnetic ordering at low-T

Magnetic field stabilizes SDW order in favor of superconductivity.


Summary of some important underdoped phenomena


A phenomenological synthesis


Ong `high’ field phase diagram


Key assumption: Electron coherence in a field


Physics across Tc at zero field


Modeling single particle incoherence


Pseudogap and Fermi arcs


Summary of ``synthesis”


Back to basic theory questions

Many different kinds of studies:

1.1d doped spin ladder: Zero doping – spin gapped insulator due to valence bond formation. Dope – (power law) superconductor .

2. Quasi-1d: Weakly coupled ladders

3. Inhomogenous 2d: Checkerboard Hubbard model

4. Superconductivity in doped VBS Mott insulators (`large-N’ methods): spontaneously generate weakly coupled ladders.

5. Superconductivity in doped spin liquid Mott insulators (i.e insulators with one electron per site)


Refined basic theory questions

Is superconductivity with gapless nodal excitations possible in a doped Mott insulator?

Only currently known route is by doping a gapless spin liquid Mott insulator.

Does this force us to a spin liquid based approach to cuprates?


More questions

More generally, large Fermi surface visible (at least at short time scales) already in underdoped.

How should we understand the emergence of the large Fermi surface in a doped Mott insulator?


Even more questions


Last question


Doped Mott insulators and high Tc superconductivity Lecture...

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