Electronic Structure Near the Mott transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

Electronic Structure Near the Mott transition

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline Introduction to the strong correlation

problem and to the Mott transition

Some dynamical mean field ideas

Applications to the Mott transition problem: some insights from studies of model Hamiltonians.

Towards an electronic structure method: applications to materials.


RUTGERS

Momentum Space , bands, k in Brillouin zone is good quantum number.

Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds.

2 ( )F Fe k k l

h

The electron in a solid: wave picture

Maximum metallic resistivity 200 ohm cm


RUTGERS

Standard Model of Solids Qualitative predictions: low temperature dependence of

thermodynamics and transport

~ const ~H constR~VC TOptical response, transitions between bands.

Qualitative predictions. Filled bands-Insulators, Unfilled bands metals. Odd number of electrons metallicity.

Quantitative tools: DFT, LDA, GGA, total energies,good starting point for spectra, GW,and transport


RUTGERS

The electron in a solid: particle picture.

NiO, MnO, …Array of atoms is insulating if a>>aB. Mott: correlations localize the electron

e_ e_ e_ e_

•Think in real space , solid collection of atoms•High T : local moments, Low T spin-orbital order

1

T

•Superexchange


RUTGERS

Mott : Correlations localize the electronLow densities, electron behaves as a particle,use

atomic physics, work in real space.

•One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….)

• Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

•H H H+ H H H motion of H+ forms the lower Hubbard band

•H H H H- H H motion of H_ forms the upper Hubbard band


RUTGERS

Localization vs Delocalization Strong Correlation Problem

• A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem.

•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U or Hartree Fock work well.

•Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands.


RUTGERS

Correlated Materials do big things

Huge resistivity changes V2O3.

Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 .

Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000

(La1-xSrx)MnO3 Colossal Magneto-resistance.


RUTGERS

Strongly Correlated Materials.

Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999).

Huge volume collapses, Ce, Pu……

Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) )


RUTGERS

The Mott transition

Electronically driven MIT. Forces to face directly the localization

delocalization problem. Relevant to many systems, eg V2O3 Techniques applicable to a very broad

range or problems.


RUTGERS

Mott transition in V2O3 under pressure or chemical substitution on V-site


RUTGERS

Universal and non universal features. Top to bottom approach to correlated materials.

Some aspects at high temperatures, depend weakly on the material (and on the model).

Low temperature phase diagram, is very sensitive to details, in experiment (and in the theory).


RUTGERS

Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


RUTGERS

Failure of the Standard Model: NiSe2-xSx

Miyasaka and Takagi (2000)


RUTGERS

Phase Diagrams :V2O3, Ni Se2-x Sx Mc Whan et.

Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976),


RUTGERS

Outline Introduction to the strong correlation problem

and to the Mott transition. DMFT ideas Applications to the Mott transition problem:

some insights from studies of models. Towards an electronic structure method:

applications to materials: NiO, Pu, Fe, Ni, LaSrTiO3, ……….

Outlook


RUTGERS

Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping or chemical potential

Frustration (t’/t)

T temperatureMott transition as a function of doping, pressure temperature etc.


RUTGERS

Limit of large lattice coordination

1~ d ij nearest neighborsijt

d

† 1~i jc c

d

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

~O(1)i i

Un n

Metzner Vollhardt, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89


RUTGERS

Mean-Field : Classical vs Quantum

Classical case Quantum case

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,


t c c c c U n n


RUTGERS

Solving the DMFT equations

G 0 G

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,ED….)Analytical Methods•Extension to ordered states, clusters…….. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .


RUTGERS

DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000).

Identify observable, A. Construct an exact functional of <A>=a, [a] which is stationary at the physical value of a.

Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local

problem, defines a Weiss field. The method is useful when practical and accurate

approximations to the exact functional exist. Example: LDA, GGA, in DFT.


RUTGERS

Example: DMFT for lattice model (e.g. single band Hubbard).Muller Hartman 89, Chitra and Kotliar 99.

Observable: Local Greens function Gii ().

Exact functional [Gii () DMFT Approximation to the functional.

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få


RUTGERS

Extensions of DMFT. Renormalizing the quartic term in the local

impurity action.

EDMFT. Taking several sites (clusters) as local

entity.

CDMFT Combining DMFT with other methods.

LDA+DMFT, GW+EDMFT or “GWU”…..


RUTGERS

Outline

Introduction to the strong correlation problem.

Essentials of DMFT Applications to the Mott transition problem:


applications to materials Outlook


RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics.

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


RUTGERS

Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)


RUTGERS

X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration. Three peak structure.


RUTGERS

Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


RUTGERS

Insights from DMFT

Three peak structure of the density of states.

In the strongly correlated metallic regime the Hubbard bands are well formed.


RUTGERS

Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure…


RUTGERS

What about experiments?


RUTGERS

Parallel development: Fujimori et.al


RUTGERS

Mott transition in V2O3 under pressure or chemical substitution on V-site


RUTGERS

Anomalous transfer of optical spectral weight V2O3

:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


RUTGERS

Anomalous transfer of spectral weight in v2O3


RUTGERS

Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy


RUTGERS

. ARPES measurements on NiS2-xSex

Matsuura et. al Phys. Rev B 58 (1998) 3690. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998)


RUTGERS

Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]


RUTGERS

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )


RUTGERS

Recent exps. Moo et. al. (2003)Theory Held et. al.


RUTGERS

•Transport in 2d organics. Limlet et. al.


RUTGERS

Strong correlation anomalies

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Transfer of spectral weight which is non local in frequency.

Dramatic failure of DFT based approximations in predicting physical properties.


RUTGERS

Conclusions: generic aspects

Three peak structure, quasiparticles and Hubbard bands.

Non local transfer of spectral weight. Large resistivities.


RUTGERS

Insights from DMFT. Important role of the incoherent part of the

spectral function at finite temperature Physics is governed by the transfer of

spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space pictures are needed as synthesized in DMFT.


RUTGERS

Outline

Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some

insights from studies of models. Towards an electronic structure method:

applications to materials: Pu, Fe, Ni, Ce, LaSrTiO3, NiO,MnO,CrO2,K3C60,2d and quasi-1d organics, magnetic semiconductors,SrRuO4,V2O3………….

Outlook


RUTGERS

Interface DMFT with electronic structure.

Derive model Hamiltonians, solve by DMFT

(or cluster extensions). Full many body aproach, treat light electrons by

GW or screened HF, heavy electrons by DMFT . Treat correlated electrons with DMFT and light

electrons with DFT (LDA, GGA +DMFT)


RUTGERS







RUTGERS

Spectral Density Functional : effective action construction (Chitra and GK).

Introduce local orbitals, R(r-R)orbitals, and local GF G(R,R)(i ) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

Approximate functional using DMFT insights.

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


RUTGERS

Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) phase of Pu: Savrasov GK and Abrahams (Nature 2001) Dai Savrasov GK Migliori Letbetter and Abrahams

(Science 2003) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al

PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al

1997, Nekrasov et.al. 1999, Udovenko et.al 2003) Paramagnetic Mott insulators. NiO MnO, Savrasov and

GK( PRL 2002)……………………


RUTGERS

Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.


RUTGERS

Physics of Pu


RUTGERS

Plutonium Puzzles

o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.

o Many studies (Freeman, Koelling 1972)APW methods

o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give

o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% Is 35% lower than experimentlower than experiment

o This is the largest discrepancy ever known in DFT based calculations.


RUTGERS

DFT Studies LSDA predicts magnetic long range (Solovyev

et.al.)Experimentally Pu is not magnetic. If one treats the f electrons as part of the core

LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that Pu is a weakly correlated system

Alternative approach Wills et. al. (5f)4 core+ 1f(5f)in conduction band.


RUTGERS

Shear anisotropy fcc Pu (GPa)

C’=(C11-C12)/2 = 4.78

C44= 33.59

C44/C’ ~ 8 Largest shear anisotropy in any element!

LDA Calculations (Bouchet et. al.) C’= -48


RUTGERS

Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)


RUTGERS

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003


RUTGERS

Functional approach allows computation of linear response.(S. Savrasov and GK 2002)

Apply to NiO, canonical Mott insulator.

U=8 ev, J=.9ev

Simple Impurity solver Hubbard 1.

Results for NiO: Phonons (Savrasov and Results for NiO: Phonons (Savrasov and Kotliar PRL 2002)Kotliar PRL 2002)Solid circles – theory, open circles – exp. (Roy et.al, 1976)

DMFT


RUTGERS

Phases of Pu


RUTGERS

Dai et. al.


RUTGERS

Epsilon Plutonium.


RUTGERS

Outline

Introduction to the strong correlation problem.

Essentials of DMFT Applications to the Mott transition problem:


applications to materials: Outlook


RUTGERS

Outlook

Local approach to strongly correlated electrons.

Many extensions, make the approach suitable for getting insights and quantitative results in correlated materials.


RUTGERS

Conclusion

The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.

This has lead to extensions to more realistic models, and a beginning of a first principles approach to the electronic structure of correlated materials.


RUTGERS

Outlook Systematic improvements, short range

correlations, cluster methods, improved mean fields.

Improved interfaces with electronic structure.

Exploration of complex strongly correlated materials. Correlation effects on surfaces,

large molecules, systems out of equilibrium, illumination, finite currents, aeging.


RUTGERS

Acknowledgements: Development of DMFT

Collaborators: V. Anisimov,G. Biroli, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, K. Haujle, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, O. Parcollet , G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang

Support: NSF DMR 0096462

Support: Instrumentation. NSF DMR-0116068

Work on Fe and Ni: ONR4-2650

Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02


RUTGERS


RUTGERS


RUTGERS

Expts’ Wong et. al.


RUTGERS


RUTGERS


RUTGERS

E-DMFT references

H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph.D thesis (1995)).

Q. Si and J L Smith PRL 77 (1996)3391 . R. Chitra and G.Kotliar Phys. Rev. Lett

84, 3678-3681 (2000 ) Y. Motome and G. Kotliar. PRB 62, 12800 (2000) R. Chitra and G. Kotliar

Phys. Rev. B 63, 115110 (2001) S. Pankov and G. Kotliar PRB 66, 045117 (2002)


RUTGERS

1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o o o oc Go c n n U n nb b

s st t t t d t t ¯ ¯+òò

† †

, ,


t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

( , ')Do t t+


RUTGERS

Cluster extensions of DMFT

Two impurity method. [A. Georges and G. Kotliar (1995 unpublished ) and RMP 68,13 (1996) , A. Schiller PRL75, 113 (1995)]

M. Jarrell et al Dynamical Cluster Approximation [Phys. Rev. B 7475 1998]

Periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000).

G. Kotliar S. Savrasov G. Palsson and G. Biroli Cellular DMFT [PRL87, 186401 2001]


RUTGERS

C-DMFT

C:DMFT The lattice self energy is inferred from the cluster self energy.

0 0cG G ab¾¾® c

abS ¾¾®Sij ijt tab¾¾®

Alternative approaches DCA (Jarrell et.al.) Periodic clusters (Lichtenstein and Katsnelson)


RUTGERS

C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1


RUTGERS

DMFT plus other methods.

DMFT+ LDA , V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988).

S. Savrasov and G.Kotliar, funcional formulation for full self consistent implementation. Savasov Kotliar and Abrahams . Application to delta Pu Nature (2001)

Combining EDMFT with GW. Ping Sun and Phys. Rev. B 66, 085120 (2002). G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241


RUTGERS

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)


RUTGERS

LDA+DMFT Self-Consistency loop

G0 G

Im puritySo lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


RUTGERS

QP in V2O3 was recently found Mo et.al


RUTGERS

Ni and Fe: theory vs exp ( T=.9 Tc)/ ordered moment

Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (expt)

eff high T moment

Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt)

Curie Temperature Tc

Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)


RUTGERS


G0 G

Im puritySo lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


RUTGERS

LDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

[ ]

R R

n

n KS

KS n n

i

LDAext xc

DC

R

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r


RUTGERS

Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

AppreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy


RUTGERS

Comments on LDA+DMFT

• Static limit of the LDA+DMFT functional , with = HF reduces to LDA+U

• Removes inconsistencies of this approach,• Only in the orbitally ordered Hartree Fock

limit, the Greens function of the heavy electrons is fully coherent

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.


RUTGERS

LSDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

[ ]

R R

n

n KS

KS n nKS i

LDAext xc

DC

R

Tr i V r r

V r r dr B r m r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- - S +

+ + +-

F - F

åò ò

ò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

KS KS ab [ ( ) ( ) G V ( ) ( ) ]LDA DMFT a br m r r B r


RUTGERS






It is useful to introduce a Lagrange multiplier conjugate to a, [a,

It gives as a byproduct a additional lattice information.


RUTGERS

Solving the DMFT equations

G 0 G

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,ED….)Analytical Methods•Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .


RUTGERS


G0 G

Im puritySolver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


DMFT

U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

Documents

Electronic Structure Near the Mott transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University