Dynamical Mean Field Theory Approach to the Electronic Structure Problem of Solids

Gabriel Kotliar Physics Department and Center for Materials

Theory Rutgers University

Theories of Excited States in Molecules and Nanostructures Workshop sponsored by DOE Basic Energy Sciences June 13-14, 2010 Marriott Inner Harbor, Baltimore, MD

Work in collaboration with K Haule (Rutgers)

DMFT• Designed to treat strongly correlated electron

materials [ for example Mott transition problem]• Designed to compute one electron spectral

functions, photoemission and BIS • Designed to treat finite electronic temperature• Combines ideas of physics (bands ) and

chemistry (local CI)• It is a relatively new method. Still rapidly

developing.

DMFT : the middle way

• More expensive than density functional theory ( because it targets spectral properties)

• Less expensive than direct application of QMC or CI (because it only uses these tools locally )

• Utilizes advances in electronic structure [ DMFT can be built on top of LDA, hybrid-DFT, GW ] and techniques such as QMC or CI, and its various levels of approx to solve the impurity problem.

• Greens function method, based on a judicious use of the local approximation. Solved the Mott transition problem in the context of the model Hamiltonians. Goal, combine those ideas with technology from electronic structure methods to understand and predict properties of correlated materials.

• Testing methods: “simple” models, experiments, predictive power ?

CeIn

In

Cerium 115Multiple hybridization gaps

300K

eV

10K

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

non-f spectra

C. Yee G. Kotliar K. Haule Phys. Rev. B 81, 035105 (2010)

Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).

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atomic levels

Quantifying the degree oflocalization/delocalization

( ),

( ),

( )[ ]n

imp n

A

G i

i

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w

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®

Impurity Solver

Machine for summing all local diagrams in PT in U to all orders.

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

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| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >® Determine energy and and S self consistently from extremizing a functional : the spectral density functional . Chitra and Kotliar (2001) . Savrasov and Kotliar (2001) Full self consistent implementation . Review: Kotliar et.al. RMP (2006)

12

1( , )

( ) ( )G k i

i t k i

Spectra=- Im G(k,w)

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++

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,[ ] [ , ]dft lda dmf loct G Ur r+G ¾¾®G

abcdU U®

Diagrams: PT in W and G. 1 1 1 1

0

1 1[ , ] [ ] [ ] [ , ]

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Introduce projector Gloc Wloc

Main steps in DMFT

• 1) Solve for atomic shell in a medium, Gloc Ploc Sloc and Wloc [Impurity Solver]

• 2) Embed Sloc Ploc to obtain the solid greens functions. [Embedding]

• 3) Project the full greens function to get the local greens function of the relevant shell.

[Projection or Truncation]• 4) Recompute the medium in which the atom is

embedded. [ Weiss fields]• Postprocessing: evaluate total energies, A(k,omega)

sigma(omega) ………

ImpuritySolver

General impurity problem

Diagrammatic expansion in terms of hybridization D+Metropolis sampling over the diagrams

•Exact method: samples all diagrams!•Allows correct treatment of multiplets

P. Werner et. al. PRL (2006) K.H.aule Phys. Rev. B 75, 155113 (2007)

An exact impurity solver, continuous time QMC - expansion in terms of hybridization

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,

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locloc f spd loc ff

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W Ww

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Recent Work on determining U, and Edc using SC GW.

12

LDA+DMFT as an approximation to the general scheme

, ,

dmft ,

0 0 [ ] [ ]

0 [ ] [ ]spd sps spd f

ff f spd ff

Vxc k Vxc k

Edc Vxc k Vxc k

æ ö æ ö÷ ÷ç ç÷ ÷S +ç ç÷ ÷ç ç÷ ÷ç çS -è ø è ø®

Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule and GK EPL 84, 57009 (2008)

U is parametrized in terms of Slater integrals F0 F2 F4 ….

Photoemission

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

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

alpa->delta volume collapse transition

Curium has large magnetic moment and orders antiferromagnetically Pu does is

non magnetic.

F0=4,F2=6.1

F0=4.5,F2=7.15

F0=4.5,F2=8.11

DMFT Phonons in fcc d-Pu

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

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

-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 Beyond GW

Coordination Sphere Coordination Sphere

GW+DMFT Why it should work ? GW+DMFT proposed and fully implmented in the context of a a one orbital lattice model.P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Test various levels of self consistency in Gnonloc Pinonloc P.Sun and GK PRL (2004). See also GW+dc+U Biermann, F.Aryasetiawan. and A. Georges, PRL 90, 86402 (2003)

Test notion of locality in LMTO basis set in various materials. N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, (2006). Include higher order graphs, first implementation of GW+DMFT (with a perturbative impurity solver).

N Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)

Cutoff Radius R

Challenges

• Optimal choice of projectors.• Basis sets [LMTO, LAPW, plane

waves+PAW’s…..]• Optimal description of the “weakly correlated

sector” [ dft , GW, hybrids ]• Cluster DMFT • Determination of the screened F0, F2, F4• Nanostructures and interfaces

Nanostructures-interfaces-impurities

Local self energy approximation