vladimir i. anisimovgc.lpi.ru/proceedings/anisimov.pdf · density functional and dynamical...

36
Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov Institute of Metal Physics Ekaterinburg, Russia

Upload: others

Post on 31-Jul-2020

11 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Density Functional and Dynamical

Mean-Field Theory (DFT+DMFT) method

and its application to real materials

Vladimir I. Anisimov

Institute of Metal Physics

Ekaterinburg, Russia

Page 2: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Contents

Method formulation:

Dynamical Mean-Field Theory (DMFT)

Wannier functions as localized orbitals basis

Determination of Hamiltonian parameters

“Constrained DFT” calculations for Coulomb

interaction parameters

DFT+DMFT calculation scheme

Page 3: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Contents

Results of DFT+DMFT calculations:

Strongly correlated metal SrVO3

Metal-insulator transition in V2O3

Heavy fermions in d-system Li2VO4

Charge transfer insulator NiO

Metal-insulator transition with pressure in MnO

Correlated covalent insulators FeSi and FeSb2

Novel superconductors LaOFeAs, BaFe2As2

Jahn-Teller distortions in KCuF3

f-electrons localization in Ce

Page 4: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Dynamical Mean-Field Theory

i

234

5

6

1

Z

• proper time resolved treatment

of local electronic interactions

• includes all many-body correlations !

Hubbard model:

“single-impurity Anderson

model” + self-consistency

Georges and Kotliar (1992)

self-consistency condition:

dynamic mean-field

(hybridization function):

Effective impurity model defined by hybridization function is solved

with an “impurity” solver, e.g., QMC, NRG, ED,3

Metzner,Vollhardt (1989)

Mapping impurity Anderson model on lattice Hubbard model

Page 5: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Including material specific details

DFT+DMFT

Anisimov et al. (1997)

Lichtenstein, Katsnelson (1998)

Kotliar, Vollhardt (2004)

Density Functional Theory:

• material specific: “ab initio”

• fails for strong correlations

Model Hamiltonians:

• input parameters

(t, U, ) unknown

• systematic

many-body approach

• DFT band structure:

• + Coulomb U

• solve by DMFT

Page 6: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

General functionals

(electron density,

spectral density et. ct.)

Model Hamiltonians with

DFT parameters

Problem

“Dream” fully ab-initiomethod

How to define interaction term in

ab-initio but still practical way?Orbitals?

Page 7: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Why Wannier Functions?

Advantages of Wannier function basis set:

<Explicit form of the orbitals

forming complete basis set

< Localized orbitals

< Orbitals are centered on atoms

Wannier functions in real space [1]:

[1] G.H. Wannier, Phys. Rev. 52, 192 (1937)

Bloch functions

like in Hubbard model

Uncertainty of WF definition

for a many-band case:

Unitary matrix

Page 8: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Wannier functions and projection

Eigenvector

element

WF in k-space – projection of the set of trial functions [2] (atomic

orbitals) into Bloch functions subspace :

Bloch functions in DFT basis

(LMTO or plane waves):

coefficients of WF expansion in LMTO-orbitals:

Bloch sums of

LMTO orbitals

[2] D.Vanderbildt et al, Phys. Rev.B 56, 12847 (1997)

Page 9: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Example of WF in real space

WF basis set for V-3d (t2g) subband of SrVO3: XY, XZ, YZ - orbitals

Page 10: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

WF in cuprates

V. V. Mazurenko, et al, Phys. Rev. B 75, 224408 (2007)

Wannier orbitals

centered on

neighboring

copper atoms along

the y axis.

Page 11: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

DFT+DMFT calculations scheme

Local Green function:

Dyson equation defines bath Green function:

Self-consistent condition:

Impurity problem defined bath Green

function is solved by QMC

Page 12: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Strongly correlated metal SrVO3

V+4 (d1) ion in cubic

perovskite crystal structureOne electron in partially

filled t2g band

I.Nekrasov et al, Phys. Rev. B 72, 155106 (2005), Phys. Rev. B 73, 155112 (2006)

Page 13: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Strongly correlated metal SrVO3

Strongly correlated metal with pronounced Lower Hubbard band (LHB)

Page 14: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Strongly correlated metal SrVO3

[ ] 1

0 (k)H-)(-Tr Im 1

),(A function Spectral−Σ−= ωω

πωk

Page 15: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Strongly correlated metal SrVO3

2|)(Re

1m

m

mass electron Effective

0

*

≈∂Σ∂

−= =ωωω

)(m

m(k) ~

narrowing Bands

0

1*

kεε−

=

Page 16: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Mott insulator V2O3

Prototypical Mott insulator.

Iso-structural paramagnetic metal to

paramagnetic insulator transition

with small volume change due to

chemical negative pressure.

Page 17: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Mott insulator V2O3

Paramagnetic metal to paramagnetic insulator transition with small change

in corundum crystal structure parameters

Lower Hubbard

Band (LHB)Quasiparticle band

K.Held et al, Phys. Rev. Lett. 86, 5345 (2001), G.Keller et al, Phys. Rev. B 70, 205116 (2004)

Page 18: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Heavy fermions material LiV2O4

Heavy-fermions without f-electrons:

linear specific heat coefficient

g=420 mJ/molK2,

effective electron mass m*/m =25

below TK ~28 K

Cubic spinel crystal structure

with local trigonal symmetry

Page 19: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Heavy fermions material LiV2O4

R.Arita et al, Phys. Rev. Lett. 98, 166402 (2007)

Sharp quasiparticle peak above the Fermi for T=0 limit (PQMC)

Page 20: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Charge transfer insulator NiO

Zaanen-Sawatzky-Allen classification scheme

for transition metal compounds

Mott-Hubbard insulators Charge transfer insulators

Zaanen et al, PRL 55, 418 (1985)

Page 21: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Charge transfer insulator NiO

Charge transfer insulator in paramagnetic phase.

Ni+2 (d8) ion in cubic rock salt crystal structure

J. Kuneš, et al, Phys. Rev. B 75, 165115 (2007)

LDA

DMFT

Page 22: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Charge transfer insulator NiO

Band structure of charge transfer insulator

combines dispersive (itinerant states)

and flat bands (localized states).

O2p bands

Ni3d bands

J. Kunes et al, Phys. Rev. Lett. 99, 156404 (2007)

Page 23: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Metal-insulator transition in MnO

Metal-insulator transition (paramagnetic insulator to paramagnetic metal)

with pressure in MnO accompanied with high-spin to low-spin state transition.

J. Kunes et al, Nature Materials 7, 198 (2008)

Page 24: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Metal-insulator transition in MnO

High-spin state (HS) - t32g e2g configuration

Low-spin state (LS) - t52g e0g configuration

Page 25: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Metal-insulator transition in MnO

collapse. volume withtransition

LS HSin results that

J energy exchange with

competing spliting field

crystal increases pressure

with volumeDecreasing

cf

Page 26: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Correlated covalent insulators FeSi and FeSb2

Transition from non-magnetic semiconductor to paramagnetic metal

with temperature increase in FeSi and FeSb2. Electron doping

in Fe1-xCoxSi results in ferromagnetic metallic state.

Page 27: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Correlated covalent insulators FeSi and FeSb2

J. Kunes et al, Phys.Rev. B 78, 033109 (2008), V. Mazurenko et al, Phys. Rev. B 81, 125131 (2010)

Page 28: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Novel superconductor LaOFeAs

Tc=26K for F content ~11%

Y. Kanamura et al. J. Am. Chem. Soc. 130, 3296 (2008)]

Page 29: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Novel superconductor LaOFeAs

02468

10Total

-5 -4 -3 -2 -1 0 1 2Energy, eV, EF=0

02d-Fe

02

Den

sity

of s

tate

s, (e

V. a

tom

)-1

p-As02p-O

d (x2-y2) Wannier functions (WF) calculated

for all bands (O2p,As4p,Fe3d) and

for Fe3d bands only

All bands WF

constrain DFT

U=3.5 eV

J=0.8 eV

Fe3d band

only WF

constrain DFT

U=0.8 eV

J=0.5 eV

V.Anisimov et al, J. Phys.: Condens. Matter 21, 075602 (2009)

Page 30: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Novel superconductor LaOFeAs

0

0,5

Spe

ctra

l fun

ctio

n, e

V-1

xy

0

0,5 yz, zx

0

0,5 3z2-r

2

-3 -2 -1 0 1 2 3Energy, eV, EF=0

0

0,5 x2-y

2

DMFT results for Hamiltonian

and Coulomb interaction

parameters calculated

with Wannier functions

for Fe3d bands only

U=0.8 eV

J=0.5 eV

Page 31: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Novel superconductor LaOFeAs

DMFT results for Hamiltonian

and Coulomb interaction

parameters calculated

with Wannier functions

for all bands (O2p,As4p,Fe3d)

U=3.5 eV

J=0.8 eV

Moderately correlated

regime with significant

renormalization for

electronic states on the

Fermi level (effective

mass m*~2) but

no Hubbard band.

Page 32: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

- Good agreement with PES and

ARPES data

- DMFT bands εDMFT(k) are

very well represented by scaling

εDMFT(k)=εLDA(k)/(m*/m)

S. de Jong et al. (2009)

Chang Liu et al. (2008) This work

BaFe2As2: DMFT results vs ARPES experiment

Page 33: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Correlations and lattice distortion: KCuF3

d-level scheme:

3d9

free ion cubic

eg

t2g

Crystal structure and

Orbital order (OO):

• pseudo cubic perovskite I4/mcm

• cooperative JT distortion

below 1000 K

• Neel temperature ~38 K

• dx2 - y2 hole antiferroorbital ordering

1 2 3

Energy (eV)

0

2

4

6

8

Den

sity

of s

tate

s

x2 - y

2

3z2 - r

2

t2g orbitals

GGA (Cu 3d) density of state:

metallic solution -> inconsistent with exp

tetragonal

(c/a < 1)

KCuF3: a prototype eg (3d9) Jahn-Teller system

Page 34: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Correlations and lattice distortion: KCuF3

KCuF3: GGA+DMFT results

0 1 2 3 4 5 6 7

JT distortion, δJT (% of a)

0

0.05

0.1

0.15

0.2

0.25

0.3

Ene

rgy

(eV

per

fu)

GGA

GGA+DMFT (U=7 eV, J=0.9 eV)

Total energy:

U = 7.0 eV, J = 0.9 eV

GGA:

GGA+DMFT:

• metallic solution

• total energy almost const for

JT distortion < 4 %

• no JT distortion (orbital order)

for T > 100 K !

inconsistent with experiment

• paramagnetic insulator

• energy gain of ~ 175 meV

• antiferro-orbital order

• optimal JT distortion at 4.2 %

• JT distortion persists up

to 1000 K (melting tem-re)

in good agreement with exp

structural relaxation due to

electronic correlations !

17

5 m

eV

Leonov et al., Phys. Rev. Lett. 101, 096405 (2008)

Page 35: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

f-electrons localization in Ce

-2 -1 0 1 2 3Energy (eV)

0

10

20

30

40

50

DO

S (S

tate

s/eV

/ato

m)

TotalCe-f

DFT

Ce a - g transition with 15% volume change.

Kondo temperature TK 1000K (a) , 30K (g)

M.B. Zoelfl et al, Phys. Rev. Lett. 87, 276403 (2001)

Page 36: Vladimir I. Anisimovgc.lpi.ru/proceedings/anisimov.pdf · Density Functional and Dynamical Mean-Field Theory (DFT+DMFT) method and its application to real materials Vladimir I. Anisimov

Conclusions

• Dynamical mean-field theory (DMFT) is a powerful tool

to study correlation effects

• Ab-initio definition of correlated orbitals and

interaction strength (U) between them based on

Wannier functions formalism results in “first-

principles” DFT+DMFT calculations scheme

• DFT+DMFT method was successful in describing

paramagnetic Mott insulators, correlated metals, charge

transfer insulators, metal-insulator transitions with

pressure and temperature, cooperative Jahn-Teller

lattice distortions