THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into real materials : DMFT at work. From theoretical solid state physics to materials science.

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Insights into real materials : DMFT at work. From theoretical solid state physics to

materials science.


RUTGERS


RUTGERS

Realistic Calculationsof the Electronic Structure of Correlated materials

Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. Inspired by the LDA+U approach (Anisimov, Andersen and Zaanen)

Anisimov Poteryaev Korotin Anhokin and Kotliar (1997). Lichtenstein and Katsenelson (1998)


RUTGERS

Combining LDA and 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 or viewed as parameters


RUTGERS

Spectral Density Functional : 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, 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)]

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


RUTGERS

Spectral Density Functional The exact functional can be built in perturbation

theory in the interaction (well defined diagrammatic rules )The functional can also be constructed expanding around the the atomic limit. No explicit expression exists.

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

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


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

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

Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with = HF

reduces to LDA+U• Removes inconsistencies and shortcomings of this

approach. DMFT retain correlations effects in the absence of orbital ordering.

• 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

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


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% Is 35% lower than experimentlower than experiment

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, using 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 (Savrasov Kotliar and Abrahams Nature 2001)


RUTGERS

Lda vs Exp Spectra


RUTGERS

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


RUTGERS

Case study Fe and Ni

Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures has a

Curie Weiss law with a moment much larger than the ordered moment.

Magnetic anisotropy


RUTGERS

Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK PRL 2001)


RUTGERS

Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

0 3( )q

Meff

T Tc

0 3( )q

Meff

T Tc


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

Outlook

The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR

Strategy: advancing our understanding scale by scale

Generalized cluster methods to capture longer range magnetic correlations

New structures in k space. Cellular DMFT


RUTGERS

Ni low T puzzles

Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis(100) for Nickel .(instead of the correct one (111)

LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)

Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )

I. Yang S. Savrasov and G. Kotliar PRL2001


RUTGERS

Fe and Ni Satellite in minority band at 6 ev, 30 % reduction

of bandwidth, exchange splitting reduction .3 ev Spin wave stiffness controls the effects of spatial

flucuations, it is about twice as large in Ni and in Fe

Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.


RUTGERS


RUTGERS

DMFT: References

Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang

Other work: A. Brandt, W. Nolting, R. Bulla, M. Jarrell, D. Logan, J. Freericks, T. Prushke, W. Metzner, F. Gebhardt, A. Lichtenstein, M. Fleck D. Vollhardt ……………….


RUTGERS

Reviews of DMFT

Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)

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


RUTGERS

Challenges The photoemission in cuprates has a strong

momentum dependence Strong Magnetic Correlations (no orbital

degeneracy) Single Site DMFT does not capture these

effects

2

1~ 0ij i j

j

J S S dd


RUTGERS

Cuprates: Photoemission

–

Transfer of Spectral Weight

with a) temperature and b) doping


RUTGERS

DMFT Spin Orbital Ordered StatesLonger range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al., C-DMFT Kotliar et. al ).


RUTGERS

DMFTFormulation as an electronic structure method (Chitra and Kotliar)Density vs Local Spectral FunctionExtensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)


RUTGERS

Cuprates: Photoemission

–

Transfer of Spectral Weight

with a) temperature and b) doping


RUTGERS

Anomalous Resistivity:LiV2O4

Takagi et.al. PRL 2000


RUTGERS

Anomalous Resistivities: DopedHubbard Model (Prushke and Jarrell 1993)

Title:main.dviCreator:dvips 5.516 Copyright 1986, 1993 Radical Eye SoftwarePreview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


RUTGERS

Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998

Title:

Creator:gnuplotPreview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

Title:

Creator:gnuplotPreview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

IPT

NCA


RUTGERS

Failure of the “Standard Model”: Cuprates

Anomalous Resistivity


RUTGERS

Specific Heat Titanates


RUTGERS

Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455


RUTGERS


RUTGERS

Standard Model

Typical Mott values of the resistivity 200 Ohm-cm

Residual instabilites SDW, CDW, SC

Odd # electrons -> metal

Even # electrons -> insulator

Theoretical foundation: Sommerfeld, Bloch and Landau

Computational tools DFT in LDA

Transport Properties, Boltzman equation , low temperature dependence of transport coefficients2 ( )

MottF Fe k k l

h

Mott


RUTGERS

Failure of the “Standard Model”: Cuprates

Anomalous Resistivity


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,

NRG,ED….)•Analytical Methods

G0 G

Im puritySo lver

S .C .C .


RUTGERS

DMFTFormulation as an electronic structure method (Chitra and Kotliar)Density vs Local Spectral FunctionExtensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)


RUTGERS

DMFT: Methods of Solution


RUTGERS

DMFT Spin Orbital Ordered StatesLonger range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al., C-DMFT Kotliar et. al ).


RUTGERS

Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)


RUTGERS

Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)


RUTGERS

Strongly Correlated Electrons

Competing Interaction

Low T, Several Phases Close in Energy

Complex Phase Diagrams

Extreme Sensitivity to Changes in External Parameters

Need for Quantitative Methods


RUTGERS

Failure of the StandardModel: Anomalous Spectral Weight TransferOptical Conductivity o of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993)

0( )d Neff

0( )d

Neff depends on T


RUTGERS

Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping d or chemical potential

Frustration (t’/t)

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


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

LDA functional

2log[ / 2 ] ( ) ( )

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

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r r

w r

r rr r

- +Ñ - -

+ +-

ò

ò ò

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

Conjugate field, VKS(r)


RUTGERS

Minimize LDA functional

[ ]( )( ) ( ) '

| ' | ( )

LDAxc

KS ext

ErV r V r dr

r r r

d rrdr

= + +-ò

0*2

( ) { )[ / 2 ]

( ) ( ) n

n

ikj kj kj

n KSkj

r f tri V

r r ew

w

r e yw

y +=

+Ñ -=å å

Kohn Sham eigenvalues, auxiliary quantities.


RUTGERS

A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site


RUTGERS

Ising character of the transfer of spectral weight

Ising –like dependence of the photo-emission intensity and the optical spectral weight near the Mott transition endpoint


RUTGERS

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

Spectral Evolution at T=0 half filling full frustration


RUTGERS

Parallel development: Fujimori et.al


RUTGERS


RUTGERS

Pu: Complex Phase Diagram (J. Smith LANL)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into real materials : DMFT at work. From theoretical solid state physics to materials science.

Documents

dmft lda functional

lda dmft static limit

local g slide

ri slide

phase slide

exact density functional

lda o dft

lda u approach anisimov