Strongly Correlated Electron Materials: Some DMFT Concepts and Applications Strongly Correlated Electron Materials: Some DMFT Concepts and Applications.

  Strongly Correlated Electron Materials: Some DMFT Strongly Correlated Electron Materials: Some DMFT

Concepts and ApplicationsConcepts and Applications

Gabriel KotliarGabriel Kotliar

and Center for Materials Theory

1111

Colloquium University of Toronto Canada

March 3rd 2011

Colloquium University of Toronto Canada

March 3rd 2011

“Standard Model of Solids “ Band Theory. Fermi Liquid Theory (Landau 1957).

Density Functional Theory (Kohn Sham 1964) energy functional of the density.

2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =Reference Frame for Weakly Correlated Systems.

Starting point for perturbation theory in the screened Coulomb interactions

(Hedin 1965)

Phys. Rev. Lett. 93, 126406 (2004).Phys. Rev. Lett. 93, 126406 (2004).

0( ) *( ) ( )

kjkj kjr r r

er y y

<= å

+ [ - ]KSV10KSG 1G

Many other properties can be computed, Many other properties can be computed, transport, optics, phonons, etc…transport, optics, phonons, etc…

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Cuprate Experimental Phase diagram Cuprate Experimental Phase diagram

Damascelli, Shen, Hussain, RMP 75, 473 (2003)Damascelli, Shen, Hussain, RMP 75, 473 (2003)

Anomalously Anomalously small small

conductivitiesconductivities2 ( )F Fe k k l

h

33

Anomalous resistivitiesAnomalous resistivities

C. Urano et. al. PRL 85, 1052 C. Urano et. al. PRL 85, 1052 (2000) (2000)

Sr2RuO4Sr2RuO4

44

Probing Electronic Structure:Photoemission Probing Electronic Structure:Photoemission

e

Angle integrated spectraAngle integrated spectra

Im ( , ) ( )dk G k A 66

A(k,A(k, A(k,A(k,

Many other spectroscopic tools to “see” Many other spectroscopic tools to “see” correlated electrons !correlated electrons !

b)Strong correlation: fermi liquid parameters b)Strong correlation: fermi liquid parameters can’t be evaluated in perturbation theory can’t be evaluated in perturbation theory

or fermi liquid theory does not work.or fermi liquid theory does not work.

a)a) Weak correlationsWeak correlations

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Shining light on correlated electrons. Shining light on correlated electrons. Optical conductivity. Failure of the StandardOptical conductivity. Failure of the StandardModel: Anomalous Spectral Weight TransferModel: Anomalous Spectral Weight Transfer

Optical Conductivity Optical Conductivity Schlesinger Schlesinger t.al (1993) t.al (1993)

0( )d

= Neff (T, )depends on T = Neff (T, )depends on T

Very Non local transfer of spectral weight in FeSiVery Non local transfer of spectral weight in FeSiD. Van der Marel et.al (2005) [ 1 ev 800 cm-1]D. Van der Marel et.al (2005) [ 1 ev 800 cm-1]

Weight does not recover up to 5 Weight does not recover up to 5 ev. ev.

66Other probes for correlated electrons X-rays, neutrons, electrons, the kitchen sink, Other probes for correlated electrons X-rays, neutrons, electrons, the kitchen sink,

theory ……….theory ……….

How to Make Correlated How to Make Correlated materials ? materials ?

Put open shell in a cagePut open shell in a cage

OxygenOxygen

transition metal iontransition metal ion

Cage : e.g 6 oxygen atoms (octahedra) Cage : e.g 6 oxygen atoms (octahedra)

or other ligands/geometryor other ligands/geometry

Build crystal with this building blockBuild crystal with this building block

or build layers separated by spacers or build layers separated by spacers

Transition metal (open shell )Transition metal (open shell )

Transition metal ionsTransition metal ions

Rare earth ionsRare earth ions

ActinidesActinides

LiLixxCoO2, NaCoO2, NaxxCoO2 CoO2

Battery materialsBattery materials

Thermoelectrics Thermoelectrics

VOVO2 2

Room Room temperature temperature

MIT MIT

LaLa1-x1-xSrxMnO3SrxMnO3

Colossal Colossal MagnetoresistanceMagnetoresistance

LaLa1-x1-xSrSrxxCuO4CuO4

High temperature High temperature superconductor superconductor

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How to find How to find interestinginteresting correlated materials ? correlated materials ?

Serendipity

An aptitude for making desirable discoveries by accidentAn aptitude for making desirable discoveries by accident

The Edisonian approach to innovation is characterized by trial The Edisonian approach to innovation is characterized by trial and error discovery rather than a systematic theoretical and error discovery rather than a systematic theoretical approach. (e.g. carbon microphone, basis of telephone)approach. (e.g. carbon microphone, basis of telephone)

+ E+ Edisonian approachdisonian approach

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The method works ! Resulted in fascinating The method works ! Resulted in fascinating compounds . Correlated electron materials do compounds . Correlated electron materials do “big things “ . Large volume collapses, ultra “big things “ . Large volume collapses, ultra strong magnets, heavy fermions, ………. , strong magnets, heavy fermions, ………. , high temperature superconductivity ……high temperature superconductivity ……

New phenomenal every few years……..New phenomenal every few years……..

The historical record indicates that Edison's approach was much more complex, that he made The historical record indicates that Edison's approach was much more complex, that he made use of available theories and resorted to trial and error only when no adequate theory existeduse of available theories and resorted to trial and error only when no adequate theory existed

But the serendipity part is is a bit slow…. But the serendipity part is is a bit slow…. ……

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Mean Field Theories Replace a many body problem by a Mean Field Theories Replace a many body problem by a single site problem in an effective medium reference framesingle site problem in an effective medium reference frame

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-

DMFT DMFT

A. Georges and G. Kotliar PRB 45, 6479 (1992).A. Georges and G. Kotliar PRB 45, 6479 (1992).

DMFT self consistency : medium DMFT self consistency : medium to reproduce the exact (best ) local to reproduce the exact (best ) local spectral function of the problemspectral function of the problem. .

Effective medium: quantifieds the Effective medium: quantifieds the notion of “ metallicity” or notion of “ metallicity” or

itineracy itineracy

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

† †

Anderson Imp 0, ,

† † †0 0 0 0 0 0

,

( +c.c).

H c A A A

c c Uc c c

V

c

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Phase diagram :frustrated Hubbard model, integer Phase diagram :frustrated Hubbard model, integer filling filling M. Rozenberg G. Kotliar H. Kajuter G. Thomas PRL75, 105 (1995)

T/W

1010

Quasiparticles Quasiparticles +Hubbard +Hubbard

bandsbands

Transfer Transfer of of

spectralspectralweightweight

Mott transitionMott transition

Coherence Coherence Incoherence Incoherence CrossoverCrossover

Spectral Spectral functionsfunctions

Critical Critical endpoint endpoint

Spinodal Uc2Spinodal Uc2

1111

P. Limelette P. Limelette et.al. et.al. Science Science 302,302,

89 (2003)89 (2003)

T=170T=170

T=300T=300

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf

Phys. Rev. Lett. 75, 105 (1995) Mo, Denlinger, Kim, Park, Allen, Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Saitoh, Muro, Metcalf, Keller, Held,

Eyert, Anisimov, Vollhardt PRL . Eyert, Anisimov, Vollhardt PRL . (2003(2003))

High temperature universality High temperature universality and V2O3and V2O3

CeRhIn5CeRhIn5: : TNTN=3.8 K; =3.8 K; 450 mJ/molK2 450 mJ/molK2 CeCoIn5CeCoIn5: : TcTc=2.3 K; =2.3 K; 1000 mJ/molK2; 1000 mJ/molK2; CeIrIn5CeIrIn5: : TcTc=0.4 K; =0.4 K; 750 mJ/molK2 750 mJ/molK2

4f’s heavy fermions, 115’s, CeMIn5 M=Co, Ir, Rh

Ce

In

Ir

1212

Expts: F. P. Mena et.al, PRB 72, 045119 (2005).K. S. Burch et al., PRB 75, 054523 (2007). E. J. Singley, et, al PRB 65, 161101(R) (2002).

CeCe InIn

InIn

Structure Property Relation: Ce115’s Structure Property Relation: Ce115’s Optics and Multiple hybridization Optics and Multiple hybridization gapsgaps

300K300K

ee VV

10K10K

•Larger gap due to hybridization with Larger gap due to hybridization with out of plane Inout of plane In

•Smaller gap due to hybridization with Smaller gap due to hybridization with in-plane Inin-plane In

non-f spectranon-f spectra

J. Shim K Haule and J. Shim K Haule and GK Science (2007)GK Science (2007)

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Localization Delocalization in ActinidesLocalization Delocalization in Actinides

Mott Transition

Modern understanding of this phenomenaDMFT. Modern understanding of this phenomenaDMFT.

PuPu

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DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)(experiments from Wong et.al, Science, 22 August 2003)1515

DMFT concept: Solids are Made out of Atoms.DMFT concept: Solids are Made out of Atoms.

f shell in a medium . Valence Histogramf shell in a medium . Valence Histogram

| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >®

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Plutonium has an unusual Plutonium has an unusual form of MIXED VALENCEform of MIXED VALENCE

with clear spectral with clear spectral fingerprints. fingerprints.

Shim, Khaule Shim, Khaule Kotliar, Kotliar, Nature, 446, Nature, 446, 513-516 513-516 (2007).(2007).

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PhotoemissioPhotoemissio

nn

Havela et. al. Phys. Rev. Havela et. al. Phys. Rev. B 68, 085101 (2003)B 68, 085101 (2003)

Pu is non magnetic – Cm is magnetic TN ~ 150 K. Pu is non magnetic – Cm is magnetic TN ~ 150 K.

K.Haule J. Shim and GK K.Haule J. Shim and GK Nature 446, 513 Nature 446, 513 (2007)(2007)

Cuprates : fundamental questionsCuprates : fundamental questions

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•Relevant degrees of freedom ?Relevant degrees of freedom ?

•Mechanism of the superconductivity ?Mechanism of the superconductivity ?

•Quasiparticles glued by spin fluctuations, Quasiparticles glued by spin fluctuations,

•or condensation of RVB paired spins . or condensation of RVB paired spins .

[ P. W. Anderson, [ P. W. Anderson, ScienceScience 235235, 1196 (1987), 1196 (1987)

•How to describe the underlying normal state ? How to describe the underlying normal state ?

• Difference among different familiesDifference among different families

K >K > -K>-K>

Slave boson MFT. Slave boson MFT.

• SC order and Tc decrease as x decreases.

• Low doping . pseudogap with D wave symmetry .

• D wave symmetry of the SC OPD wave symmetry of the SC OP

•VVFF is weakly dependent on is weakly dependent on dopingdoping, , . .

• Coherence incoherence crossover Coherence incoherence crossover on the overdoped side. on the overdoped side.

G. Kotliar and J. Liu

PRB 38,5412 (1988)

singlet pairing O

itinerac P

P

y Ob

D

0b ¹

• Related T=0 approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)

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Hubbard model : plaquette in a medium. Hubbard model : plaquette in a medium.

Lichtenstein and Kastnelson PRB (2000)Lichtenstein and Kastnelson PRB (2000)

2121

Link DMFT. Normal state Real Space Picture. Ferrero Link DMFT. Normal state Real Space Picture. Ferrero et. al. (2010) (similar to plaquette Haule and GK) (2006) et. al. (2010) (similar to plaquette Haule and GK) (2006)

• Momentum Space Picture: High T

Singlet formation. S (singlet),T Singlet formation. S (singlet),T (triplet) N=2 singlet, triplet(triplet) N=2 singlet, triplet

E (empty) N=0E (empty) N=0

1+ states with 1 electron 1+ states with 1 electron in + orbin + orb

Underdoped region: arcs shrink as T is reduced. Overdoped Underdoped region: arcs shrink as T is reduced. Overdoped region FS sharpens as T is reducedregion FS sharpens as T is reduced. . 2121

Superexchange Mechanism?Superexchange Mechanism? . . K. Haule and GK K. Haule and GK Phys. Rev. Phys. Rev. B 76, 104509 (2007).B 76, 104509 (2007).Ex= Jij(< Si. Sj >s- < Si . Sj>n)/t

D.J. Scalapino and S.R. White, Phys. Rev. B 58, D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998).8222 (1998).

How is the energy distributed How is the energy distributed in q and w ?in q and w ?

Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988)

Expts; Dai et.al. Expts; Dai et.al. 2222

Building phase diagram Building phase diagram magnetization at T=0 vs magnetization at T=0 vs ..

Single siteSingle site

Two siteTwo site

2020

Origin of magnetism :Comparing the AF Origin of magnetism :Comparing the AF

and the “underlying PM state and the “underlying PM state ““<EK>sdw -<EK>pm<EK>sdw -<EK>pm

LSCO gains kinetic energy when it LSCO gains kinetic energy when it magnetizes. [Mott ] NCCO pays kinetic magnetizes. [Mott ] NCCO pays kinetic

energy [Slater ] energy [Slater ]

NCCO magnetizes to lower its NCCO magnetizes to lower its double occupancy ! Slater. double occupancy ! Slater.

pmsdwUn n Un n

Can be traced to the structure: absence of Can be traced to the structure: absence of apical oxygens reduces the charge transfer apical oxygens reduces the charge transfer

energy energy 2424

Weber Haule and GK Nature Weber Haule and GK Nature Physics 10, 1038 (2010).

Cuprates SuperconductorsCuprates Superconductors

• Plaquette DMFT reasonable reference frame to think about the qualitative physics of cuprates, starting from high temperatures.

• High Tc materials. are near the single site DMFT Mott boundary. LSCO more correlated than NCCO, role of apical oxygens.

• High temperature superconductivity occurs in the region where neither wave/itinerant nor localized/ particle picture fully applies. [ Alterantive viewpoint to spin fluctuation theory ] , i.e. where perturbation theory fails more catastrophically ( Murphy’s law).

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Realistic DMFT as a conceptual tool and a computational tool Realistic DMFT as a conceptual tool and a computational tool

DMFT (simple yet accurate ? ) reference frame to think about DMFT (simple yet accurate ? ) reference frame to think about electrons in solids and compute their properties. electrons in solids and compute their properties.

Compare different “states” of the system for the same value of Compare different “states” of the system for the same value of parameters. parameters. Understand Mechanism for ordering , magnetic, Understand Mechanism for ordering , magnetic, superconducting, exotic, ………. superconducting, exotic, ……….

Bridge between atomic information and physical and Bridge between atomic information and physical and spectroscopical properties. [Structure-Property relation spectroscopical properties. [Structure-Property relation Learning --> Design ? ] Learning --> Design ? ]

Qualitative and quantitative system specific results gives us Qualitative and quantitative system specific results gives us confidence in the method. Many examples (sp, 3d,4d, 5d, 4f, confidence in the method. Many examples (sp, 3d,4d, 5d, 4f,

5f…) 5f…)

New arenas Interfaces, junctions heterostructures, artificial New arenas Interfaces, junctions heterostructures, artificial materials containing correlated electrons materials containing correlated electrons

32322626

““Matthias’s Rules” for High TcMatthias’s Rules” for High Tc

• Metals. Must have d electrons (not just s s-p, nor f). Stay away from oxides.

• High symmetry is good, cubic is best. Nb3Sn

• Certain electron concentrations are favored (look for peak in density of states at Fermi level)

• Stay away from theorists

• “ Do not follow my rules “

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Thanks!! for your attention!

$upport : NSF -DMR , DOE-Basic Energy Sciences, DOE-CMSN, AOSR - MURI, NSF-materials world network.

K. Haule K. Haule S. Savrasov S. Savrasov C. WeberC. WeberC MarianettiC Marianetti J. Shim J. Shim

Reference: Reference: G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Parcollet, and C. Marianetti, Rev. Mod. Phys. 78, 000865 (2006)Rev. Mod. Phys. 78, 000865 (2006)