Spectral Density Functional: a first principles approach to the electronic structure of correlated...

Spectral Density Functional: a first principles approach to the electronic

structure of correlated solids

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

2001 JRCAT-CERC Workshop on

Phase Control on Correlated Electron Systems

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline Motivation. Some universal

aspects of simple DMFT the Mott transition endpoint in frustrated systems.

Non universal physics requires detailed material modeling. Combining DMFT and band structure a new functional for electronic structure calculations (S. Savrasov and GK)

Results: d electrons Fe and Ni.

(Lichtenstein, Katsenelson and GK, PRL in press)


RUTGERS

Outline

Results: f electrons delta Pu ( S. Savrasov G. K and E. Abrahams,Nature (2001))

Conclusions: further extensions the approach.


RUTGERS

Importance of Mott phenomena

Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation.

The “”in between regime” is ubiquitous central them in strongly correlated systems. Some unorthodox examples

Fe, Ni, Pu.

Solution of this problem and advances in electronic structure theory (LDA +DMFT)


RUTGERS

A time-honored example: Mott transition in V2O3 under pressure

or chemical substitution on V-site


RUTGERS

Phase Diag: Ni Se2-x Sx

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


RUTGERS

Mott transition in layered organic conductors Ito et al. (1986) Kanoda (1987) Lefebvre et al. (2001)


RUTGERS

Theoretical Approach to the Mott

endpoint. DMFT.Mean field approach to quantum

many body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently . Use exact numerical techniques (QMC, ED ) as well as semianalytical (IPT) approaches to solve this simplified problem.

Study simple model Hamiltonians (such as the one band model on simple lattices)

Understand the results physically in terms of a Landau theory :certain high temperature aspects are independent of the details of the model and the approximations used.


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 Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c n nb b b

s st t t t ¯= +òò ò

† †

, ,

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

t c c c c U n n

0

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

10 ( ) ( )n n nG i i iw w m w- = + - D

0

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

Weiss field


RUTGERS

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


RUTGERS

Landau Functional

† †,

2

2

[ , ] ( ) ( ) ( )†

† † † †

0

†

Mettalic Order Para

( )[ ] [ ]

mete

[ ]

[ , ] [ [ ] ]

( )( )

r: ( )

( ) 2 ( )[ ]( )

loc

LG imp

L f f f i i f i

imp

loc f

imp

iF T F

t

F Log df dfe

dL f f f e f Uf f f f d

d

F iT f i f i TG i

i

i

2

2

Spin Model An

[ ] [[ ]2 ]

alogy:

2LG

t

hF h Log ch h

J

G. Kotliar EPJB (1999)


RUTGERS

Functional Approach

The Landau functional offers a direct connection to the atomic energies

Allows us to study states away from the saddle points,

All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional.

Mott transitions and bifurcations of the functional .


RUTGERS

Insights into the Mott phenomenaThe 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

A time-honored example: Mott transition in V2O3 under pressure

or chemical substitution on V-site


RUTGERS

Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar et.al.PRL 84, 5180 (2000))


RUTGERS

Ising character of Mott endpoint

SingularpartoftheWeissfieldisproportionaltoMax{(p-pc)(T-Tc)}1/inmeanfieldand5in3d

couplestoallphysicalquantitieswhichthenexhibitakinkattheMottendpoint.Resistivity,doubleoccupancy,photoemissionintensity,integratedopticalspectralweight,etc.

Divergenceofthethecompressibility,inparticleholeasymmetricsituations,e.g.FurukawaandImada


RUTGERS

Compressibility


RUTGERS

Mott transition endpoint

Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures

Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory

Building of coherence in other strongly correlated electron systems.

condensation of doubly occupied sites and onset of coherence .


RUTGERS

Insights from DMFT: think in term of spectral functions , the density is not changing!

Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit


RUTGERS

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Miyasaka and Tagaki (2000)


RUTGERS

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690


RUTGERS

Two Roads for first principles calculations of

correlated materials using DMFT.

Correlation functions etc..


RUTGERS

Insights from DMFT Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structure (ordered phases)High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT


RUTGERS

LDA+DMFT

The light, SP (or SPD) electrons are extended, well described by LDA

The heavy, D (or F) electrons are localized,treat by DMFT.

LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles of viewed as parameters


RUTGERS

DMFT +LDA : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]

Introduce local orbitals, andf local Greens function by projecting onto the local orbitals.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)]

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

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

( )R r


RUTGERS

LDA+DMFT

The functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit.

DFT is useful because e good approximations to the exact density functional DFT(r)] exist, e.g. LDA….

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.


RUTGERS

LDA+DMFT functional

2

[ , , , ]

log[ / 2 ( , , ')]

( ) ( ) ( ) ( )

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

2 | ' |

[ ] [ ]

n

lda dmft KS

n KS n

KS n n

i

LDAext xc

DC

V G

Tr i V i r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G G

w

r

w w

r w w

r rr r

+G S

=- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò ò

( )01[ ] ( 1) , ( )

2i

DC ab

abi

G Un n n T G i ew

w+

F = - = å

Double counting correction

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


RUTGERS

Spectral density functionalConnection with atomic limit

1[ ] [ ] [ ] logat atG W Tr G Tr G TrG G-F = D - D - +

1 10 atG G

[ ] atSatW Log e

1 10 0[ ] ( ) ( , ') ( ') ( ) ( ) ( ) ( )at a a abcd a b c d

ab

S G c G c U c c c c


RUTGERS

LDA+DMFT Self-Consistency loop

G0 G

Im puritySolver

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


RUTGERS

Realistic DMFT loop

( )k LMTOt H k E® -LMTO

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

ki i Ow w®

10 niG i Ow e- = + - D

0 0

0 HH

é ùê úS =ê úSë û

0 0

0 HH

é ùê úD=ê úDë û

0

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

110

1( ) ( )

( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w

--é ùê ú= +Sê ú- - - Sê úë ûå


RUTGERS

LDA functional

2

[ , , , ]

log[ / 2 ( , , ')]

( ) ( ) ( ) ( )

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

2 | ' |

[ ] [ ]

n

lda dmft KS

n KS n

KS n n

i

LDAext xc

DC

V G

Tr i V i r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G G

w

r

w w

r w w

r rr r

+G S

=- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò ò

( )01[ ] ( 1) , ( )

2i

DC ab

abi

G Un n n T G i ew

w+

F = - = å

Double counting correction


RUTGERS

LDA+DMFT References

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

ALichtensteinandM.KatsenelsonPhys.Rev.B57,6884(1988).

S.SavrasovandG.Kotliar,funcionalformulationforfullselfconsistentimplementation(2001)


RUTGERS

Iron and Nickel: band picture at low T, crossover to real space picture at high T


RUTGERS

Photoemission Spectra and Spin Autocorrelation: Fe(U=2, J=.9ev) (Lichtenstein, Katsenelson,GK prl in press)


RUTGERS

Photoemission and Spin Autocorrelation: Ni (U=3, J=.9 ev)


RUTGERS

Iron and Nickel:mgnetic properties (Lichtenstein, Katsenelson,GK cond-mat 0102297)


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 3.09 (theory) 3.12 (expt)

Ni 1.50 (theory) 1.62 (expt)

Curie Temperature Tc

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


RUTGERS

Fe and Ni

Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe

Classical calculations using measured exchange constants

(Kudrnovski Drachl PRB 2001) Weiss mean field theory gives right Tc for Ni but overestimates

Fe , RPA corrections reduce Tc of Ni by 10% only but reduce Tc of Fe by nearly factor of 2.


RUTGERS

Delocalization-Localization across the actinide series

o f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.

o Pu has a simple cubic fcc structure,the phase which is easily stabilized over a wide region in the T,p phase diagram.

o The phase is non magnetic.o Many LDA , GGA studies ( Soderlind

et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than Is 35% lower than experimentexperiment

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


RUTGERS

Small amounts of Ga stabilize the phase


RUTGERS

Problems with LDA

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% lower than Is 35% lower than experimentexperiment

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


RUTGERS

Problems with LDA LSDA predicts magnetic long range

order which is not observed experimentally (Solovyev et.al.)

If one treats the f electrons as part of the core LDA overestimates the volume by 30%

LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that Pu is a weakly correlated system


RUTGERS

Pu: DMFT total energy vs Volume (S. Savrasov )


RUTGERS

Lda vs Exp SpectraD

OS

, st./

[eV

*cel

l]


RUTGERS

Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)


RUTGERS

Conclusion

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

This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and bands.


RUTGERS

Conclusions Systematic improvements, short

range correlations. Take a cluster of sites, include

the effect of the rest in a G0 (renormalization of the quadratic part of the effective action). What to take for G0:

DCA (M. Jarrell et.al) , CDMFT ( Savrasov and GK )

include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)


RUTGERS

Conclusions

Extensions of DMFT implemented on model systems, carry over to more realistic framework. Better determination of Tcs.

First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT

Improvement in the treatement of multiplet effects in the impurity solvers.

Spectral Density Functional: a first principles approach to the electronic structure of correlated...

Documents

Transcript of Spectral Density Functional: a first principles approach to the electronic structure of correlated...