Towards a Realistic DMFT based Theoretical Transport and Spectroscopy of Correlated Solids

G.Kotliar Physics Department Center for Materials Theory

Rutgers University.

CRISMAT Caen October 30 (2007)

Outline • 1]Introduction to correlated electrons and

DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.

• 2] d’s Doped Titanites. Doping driven Mott transition.[G. Kotliar and G. Palsson PRLPRL 80, (1998), 4775]

• 3] 4f’s 115’s and the tale of multiple hybridization gaps.[K. Haule J. Shim G. Kotliar, Science Nov 1st 2007]

• 4] Conclusions

Correlated Electron Systems Pose Basic Questions in CMT

• FROM ATOMS TO SOLIDS

• How to describe electron from localized to itinerant ?

• How do the physical properties evolve ?

DMFT Spectral Function Photoemission and correlations

• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

e

Angle integrated spectral Angle integrated spectral function function

( , ) ( )dkA k A 88

Georges Kotliar (1992)

DMFT approximate quantum solid as atom in a medium † †

, ,

( )( )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 UcV c c c

1010

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

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

(GW) DFT+DMFT: determine H[k] and density and(GW) DFT+DMFT: determine H[k] and density andself consitently from a functionalself consitently from a functional

and obtain total energies. and obtain total energies. 1212

[ ]*

11

( )( ) (

,)n n

n nk

i ii t k i

V VVa a

aaaa

ew m ww m ww e

-é ùê ú+ - = +Sê ú+ - - S- ë û

å å

1( , )

( ) ( )G k i

i t k i

Spectra=- Im G(k,)

Self consistency for V and

Chitra and Kotliar PRB 62, 12715 (2000) PRB (2001)P.Sun and GK (2005)

Zein et.al. PRL 96, 226403 (2006)). See also Bierman Aryasetiwan and

Georges.

Ir,>=|R, > Gloc=G(R, R’ ’ ) R,R’

1

2

1

1 ( ) Hartreecryst

Coulomb

VG i V

W

r

V P

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

[ , ] [ , , 0, 0]DMFT loc loc nonloc nonlocG W G W G W

Electronic structure problem: compute <r|G|r’> and <r|W|r’> given structure

[ , ] sum all 2PIgraphs= +

+

G W

G

DMFT mapping site or cluster of sites in a self consistent medium. Quantum impurity model, gives and P.Need accurate impurity solvers..

LDA+DMFT­.­V.­Anisimov,­A.­Poteryaev,­M.­Korotin,­A.­Anokhin­and­G.­Kotliar,­J.­Phys.­Cond.­Mat.­35,­7359­(1997)­Review:­G.­Kotliar­S.­Savrasov­­K­Haule­O­Parcollet­V­Oudovenko­C.­Marianetti­RMP­(2006)

Approximate­the­self­energy­of­a­subset­“­uncorrelated­electrons­“­by­dft­Vxc(r)(r,r’)­replace­W()­by­a­static­U­acting­only­on­the­­“correlated­“­set,­which­we­treat­by­DMFT.

“­Local”­­can­mean­a­small­cluster­of­sites­or­multiple­unit­cells.­­Cellular­DMFT­cluster­DMFT.­­

Summary: part 1

• Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004).

• A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996).

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

Spectral function in DMFT analogous to density in DFT

Self consistent Impurity problem, natural language to describe localization/delocalization phenomena. combines atomic physics and band theory

Systematically improvable, cluster DMFT

Recent progress in implementation

Thermoelectric Figure of Merit ZT = T S2 /elLattice

Best Case, Suppose Lattice = 0

ZT = T S2 /el Wiedemann-Franz law L0 = el/T= 2.4 x 10-8 V2/K2

or­­ZT­=­S2/­L0­which­means­that­for­ZT=1,or­S­>­156­V/K­

Basic Scale k/e 86 10-6 V /K

“Best” Thermoelectrics Among Mixed Valence Intermetallics

(Physics Today, March 1997)

0

0.05

0.1

0.15

0.2

0.25

0 50 100 150 200 250 300 350

ZT

T(K)

YbAl3

CePd2.95

0.9CePd2.95

+0.1CePt

3

Fe0.95

Ir0.05

Si

FeSi

G. Mahan B. Sales J. Sharp

“State-of-the-art” Thermoelectric Materials. MRS Talk 2004 B. Sales ONRL

0

0.5

1

1.5

0 200 400 600 800 1000 1200 1400

Fig

ure

of

Mer

it :

ZT

T(K)

Room Temperature

BiSb

Bi2Te

3

PbTe

SiGe

CeFe4-x

CoxSb

12

ORNL Material

Outline • 1]Introduction to correlated electrons and

DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.

• 2] d’s Doped Titanites. Doping driven Mott transition.

• 3] 4f’s 115’s and the tale of multiple hybridization gaps.

• 4] Conclusions

(Tokura et. al. PRL 1993)A doped Mott insulator:La1-ySryO3

La+++ Ti+++ (O3)-- Mott insulator . (3d)1 one electron per site. x holes y electrons.

DMFT calculation U near the Mott transition, M. Rozenberg Zhang and GK PRB (1994)

DMFT black dots.

Hall Coefficient: expt. Tokura(1993) Theory Kajueter PRB (1996)

LaSrTiO3 photoemission Fujimori et.al.expt. Theory Kajuter and GK

Low T Fermi Liquid

2 2

2

( )( / 3)(1/ )

( ) ( )

( ) ( ) ( )

kk

kk

k

k Z

k

High T Localized “ particle-like” regime

DMFT analysis in limiting cases. Palsson and GK PRL (1998)

Expt. C.C. Hays PRB 90

(1999),10367

Theory : Palsson and Kotliar PRL 80, (1998), 4775

Even more spectacular, electron gas on SrTiO3 interface. Nature (2007)

Theory ? DD Sarma Barman Kajueter Kotliar EPL (1996 )

PRB (2001)

Outline • 1]Introduction to correlated electrons and

DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.

• 2] d’s Doped Titanites. Doping driven Mott transition.

• 3] 4f’s 115’s and the tale of multiple hybridization gaps.

• 4] Conclusions

CeRhIn5: TN=3.8 K; 450 mJ/molK2 CeCoIn5: Tc=2.3 K; 1000 mJ/molK2; CeIrIn5: Tc=0.4 K; 750 mJ/molK2

CeMIn5 M=Co, Ir, Rh

out­of­plane

in-plane

Ce

In

Ir

Ce

In

Ir

CeIn

In

Crystal­structure­of­115’s­­CeMIn5 M=Co, Ir, Rh­

CeIn3­layer

IrIn2­layer

IrIn2­layer

Tetragonal­crystal­structure

4­in­plane­In­neighbors

8­out­of­plane­in­neighbors

3.27au

3.3 au

Very slow crossover!

T*

Slow­crossover­more­consistent­with­NP&F­

T*

coherent­spectral­

weight

T

NP&F:­Nakatsuji,Pines&Fisk,­2004

Buildup­of­coherence­in­single­impurity­case

TK

coherent­spectral­

weight

T

scattering­rate

coherence­peak

Buildup­of­lattice­­coherence

Crossover­around­50K

Angle­integrated­photoemission­

Experimental­resolution­~30meVSurface­sensitivity­at­122­­ev­,­theory­predicts­3meV­broad­band

Expt Fujimori et al., PRB 73, 224517 (2006) P.R B 67, 144507 (2003).

­Theory:­LDA+DMFT,­impurity­solvers­­­SUNCA­and­CTQMC­­Shim­Haule­and­GK­­(2007)

Momentum­resolved­total­spectratrA(,k)

Fujimori,­PRB­

LDA+DMFT­­at­10K ARPES,­HE­I,­15K

LDA­f-bands­[-0.5eV,­0.8eV]­almostdisappear,­only­In-p­bands­remain

Most­of­weight­transferred­intothe­UHB

Very­heavy­qp­at­Ef,hard­to­see­in­total­spectra

Below­-0.5eV:­almost­rigid­downshift

Unlike­in­LDA+U,­no­new­band­at­-2.5eV

Short­lifetime­of­HBs­->­similar­to­LDA(f-core)rather­than­LDA­or­LDA+U

Optical­conductivity

Typical heavy fermion at low T:

Narrow­Drude­peak­(narrow­q.p.­band)

Hybridization­gap

k

Interband­transitions­across­hybridization­gap­->­mid­IR­peak

CeCoIn5

no­visible­Drude­peak

no­sharp­hybridization­gap

F.P.­Mena­&­D.Van­der­Marel,­2005

E.J.­Singley­&­D.N­Basov,­2002

second­mid­IR­peakat­600­cm-1

first­mid-IR­peakat­250­cm-1

•At­300K­very­broad­Drude­peak­(e-e­scattering,­spd­lifetime~0.1eV)­•At­10K:­

•very­narrow­Drude­peak•First­MI­peak­at­0.03eV~250cm-1

•Second­MI­peak­at­0.07eV~600cm-1

Optical­conductivity­in­LDA+DMFT­

Expts:­­F.­P.­Mena,­D.­van­der­Marel,­J.­L.­Sarrao,­PRB 72,­045119­(2005).16.­K.­S.­Burch­et al.,­PRB 75,­054523­(2007).17.­E.­J.­Singley,­D.­N.­Basov,­E.­D.­Bauer,­M.­B.­Maple,­PRB 65,­161101(R)­(2002).

CeIn

In

Multiple­hybridization­gaps

300K

e V

10K

•Larger gap due to hybridization with out of plane In•Smaller gap due to hybridization with in-plane In

non-f­spectra

T=10K T=300Kscattering­rate~100meV

Fingerprint­of­spd’s­due­to­hybridization

Not­much­weight

q.p. bandSO

Momentum­resolved­Ce-4f­spectraAf(,k)

Hybridization­gap

DMFT­qp­bands

LDA­bands LDA­bands DMFT­qp­bands

Quasiparticle­bands

three­bands,­Zj=5/2~1/200

Summary

• 115’s model systems to study the evolution of the f electron as a function of temperature

• Multiple hybridization gaps in optics.

• Very different Ce-In hybridizations with In

out of plane being larger.

J. Shim K Haule and G.K Science Express November 1st (2007).

Outline • 1]Introduction to correlated electrons and

DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.

• 2] d’s Doped Titanites. Doping driven Mott transition.

• 3] 4f’s 115’s and the tale of multiple hybridization gaps.

• 4] Conclusions

Conclusion

• Strongly Correlated electrons, still fertile ground for discovery of new thermoelectrics.

• Theory has improved, DMFT! can it play now some role in assisting and guiding experimental discoveries ?

Na0.7CoO2 (Terasaki). Good oxide thermoelectric

K. Fujita et al.Jpn. J. Appl. Phys.40 (2001) 4644

DMFT study of Nax CoO2

Foo et.al. PRL 247001

CoO2NaCoO2

Theoretical Issues: Na-Induced Correlations in NaxCoO2

C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007)

• What is the minimal model of the cobaltes ? • t2g orbitals + binary potential a see which

results of the Li /Na vacancy .• Why are correlations stronger near a band

insulator than near a Mott insulator ?• U < Uc2 , hole moves in a restricted space

(where potential is low) and is strongly correlated.

• DMFT calculations account for the Curie Weiss phase and the Fermi liquid phase

Assume Na patterns of Zandbergen et. al.PRB 70 024101

C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007) . A

DMFT calculations with and without disorder U=3 ev.

C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007)

x=.33 QP dispersion LDA+DMFTC. Marianetti K. Haule and O Parcollet to

appear in PRL

FeSb2 Bentien et. al.

Marcasite Structure FeSb2

Silicon. Treat all electrons as correlated. First order PT as impurity solver. [Cluster version of GW] LMTO basis setF. Aryasetiawan and O. Gunnarson, Phys. Rev. B 49, 16 214 (1994).

Zein Savrasov and Kotliar PRL 96, 226403 (2006)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2

Sigma s GW

Sigma p GW

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 1 2

D Sigma s

D Sigma p

GW self energy for Si Self energy corrections beyond GW

Coordination Sphere Coordination Sphere

Locality of correlations Zein Savrasov and Kotliar PRL 96, 226403 (2006))

Conclusions

• DMFT as a technique, makes contact with experiments, total energies, phonons, photoemission,ARPES,optics,…

• Concepts “ cell “in a quantum medium, spectral function, temperature dependent electronic structure, transfer of spectral weight.

• Method under development, but already gives some exciting results.

• Ultimate goal, is to be able to focus on deviations from DMFT.

Conclusions

• Correlations in sp electrons (worse case ) require 3 coordination spheres.

• 4f’s single site works reasonably well for the Ir 115. Quantum critical point : 2 site DMFT ?

• 5f’s Pu as a mixed valent metal. Cm RKKY metal.

• 3d’s. High Tc. Nodal antinodal dichotomy, novel type of Mott transition. Two gap scenario in SC state ?

Thanks!!

Realistic DMFT: past succeses> and future perspectives for

modelling electric> and thermal transport.

Download the thesis of gunnar.

Download the papers of petrovic and bentien

Dowload paper by indranil paul.

Download paper by gunnar.

Download the stuff on latio3-in particular goodenough

Paper.

The future

• Clear theoretical problems.

• The techniques used for titanites should apply to cobaltites and misfit cobaltates.

• Disorder. Electron eelctron interactions. Importance of detailed modelling.

• The techniques used for 115’s should be useful for SbF. Substittutions. Decrease in thermal conductivity. Tricks ?

Summary part 3

• What is the minimal model of the cobaltes ? • t2g orbitals + binary potential a see which

results of the Li /Na vacancy .• Why are correlations stronger near a band

insulator than near a Mott insulator ?• U < Uc2 , hole moves in a restricted space

(where potential is low) and is strongly correlated.

• DMFT calculations account for the Curie Weiss phase and the Fermi liquid phase

References: part 3

•   C. Marianetti, G. Kotliar, and G. Ceder, Nature Materials 3, 627 - 631 (2004).

• C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98,176405 (2007)

• C. Marianetti, K. Haule and O Parcollet cond-mat (2007)

Alternative theory : low spin to high spin Khaliullin Phys. Rev. Lett. 96, 216404 (2006)

• Actinides, phonons, role of multiplets, spectral signatures, Pu as mixed valent metal.

• Cobaltates, key role of inhomogeneities bringing correlations near a (correlated) insulator. DMFT treatment of an alloy.

Conclusions :chemistry brings out different aspects of localization delocalization physics.

•115’s delocalization transition as a function of T. Spectral function as a coherence order parameters. Multiple hybridization gaps.

after G. Lander, Science (2003)and Lashley et. al. PRB (2006).

Mott Transition

PuPu

Mott transition across the actinides. B. Johansson Phil Mag. 30,469 (1974)]

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

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

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

Double well structure and Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

F(T,V)=Fphonons+Finvar

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

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

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

C. C. Hays,* J.-S. Zhou, J. T. Markert, and J. B. Goodenough

REVIEW B VOLUME 60, NUMBER 14 1 OCTOBER 1999-II