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Correlated Electronic Structure of BaVS3

Frank Lechermann I. Institut für Theoretische Physik, University of Hamburg, Germany

in collaboration with

Silke Biermann and Antoine Georges CPHT,École Polytechnique, Palaiseau, France

Workshop on

“Physics of the low dimensional strongly correlated electronic system : BaVS3”

Orsay, 06.12.07

. – p.1/21

Outline

introduction to BaVS3

dynamical mean-field theory in a realistic context:

the LDA+DMFT approach

electronic structure of metallic BaVS3

electronic structure of insulating BaVS3

conclusions

[FL, S. Biermann, and A. Georges, PRB76085101]

[FL, A. Georges, A. Poteryaev, S. Biermann, M. Posternak, A.Yamasaki, and O.K. Andersen,

PRB74, 125120 (2006)]

[FL, S. Biermann, and A. Georges, Progress of Theoretical Physics Supplement160233]

[FL, S. Biermann, and A. Georges, PRL94166402] . – p.2/21

Introduction to BaVS 3

the vanadium sulfide shows three continuous phase transitions:

T∼ 240 K : hexagonal to orthorhombic structural transition

T∼ 70 K : metal-to-insulator transition (MIT) from Curie-Weiss metal to paramagnetic insulator, structural transition to monoclinic phase

T∼ 30 K : incommensurate antiferromagnetic transition

[Sayetatet al.J. Phys. C151627]

orthorhombic (Cmc21) structure at T=100 K: [Ghediraet al., J. Phys. C19 6489]

- zigzag VS3 chains

- two formula units in primitive cell

- dinterVV ∼ 2d intra VV

[Fagotet al., PRL90196401]: large one-dimensional structural fluctuations alongc axis above MIT. – p.3/21

The intriguing physics of BaVS3

Hall coefficient: resistivity/mag. susceptibility: charge density wave below MIT:

[Boothet al., PRB60 14852] [Grafet al., PRB51 2037] [Inamiet al., PRB66 073108]

V(3d1) system

mutually hybridizingA1g orbitals alongc axis

narrowEg bands at

the Fermi level

MIT vanishes at critical

pressure

[Forróet al., PRL851938]

g

3d

e eg2

Eg2

Eg1

eg1

Eg

atomic hexagonal orthorhombic

1gA1gA

[Massenetet al., J. Phys. Chem. Solids.40573]

. – p.4/21

Electronic structure of correlated systems

In strongly correlated solids there are in principle two types of excitations:

low-energy (coherent)quasiparticles with well-defined wave vector

energy shift from the noninteracting eigenvalue

exist on a long but still finite timescale

spectral weightZ

band narrowing byZD ∼ ε⋆ F

will be destroyed at high temperature

high-energy (incoherent)atomic-like excitations

form Hubbard bands around atomic levels

spectral weight1− Z

exist on a short time scale

lower and upper Hubbard band are separated

by an energy scale∆

Mott insulating state for smallt/U

⇒ Theory has to describe the interplay of different energy scales !

ε

ρ

ε

ρ

2ZD

ε

ρ

ε

ρ

. – p.5/21

Dynamical Mean-Field Theory (DMFT)

Hubbard model

at half filling

empty

single

double

occupation

U ≪ t

ideal metal

→

U ∼ t

correlated metal

←

U ≫ t

Mott insulator

U U U U U U

U

U

U U U U U

U U U U U

U U U U U U

U U U U U U

U U U U U U

. – p.6/21

Dynamical Mean-Field Theory (DMFT)

Hubbard model

at half filling

empty

single

double

occupation

U ≪ t

ideal metal

→

U ∼ t

correlated metal

←

U ≫ t

Mott insulator

U U U U U U

U

U

U U U U U

U U U U U

U U U U U U

U U U U U U

U U U U U U

mIntroduce time-dependent “Weiss field” to map lattice problem onto impurity problem by integrating out the effect of all lattice sites but one:

Σimp ≡ G −1 0 −G

−1 imp → DMFT: Gloc = Gimp

⇒ Gloc(iωn) = X

k

1

iωn + µ− εk − Σimp(iωn)

The dynamical mean-fieldG0(τ−τ ′) allows to take care of all local quantum fluctuations within DMFT.

⇒ The theory is designed to treat both quasiparticles and states originating from atomic-like excitations.

[Georges and Kotliar, PRB45 (1992)] [Metzner and Vollhardt, PRL62 (1989)]

U

G0(τ−τ ′)

. – p.6/21

LDA+DMFT for real materials

so far only single-band, model-type hopping: Gloc(iωn) = X

k

1

iωn + µ− εk − Σ(iωn)

for real materials: εk → HLDA(k)

⇒ Gloc(iωn) = X

k

[(iωn + µ)1−HLDA(k)−HDC −Σ(iωn)] −1

HLDA(k) =

0

@

Hsp(k) Hsp,d(k)

Hd,sp(k) Hdd(k)

1

A , Σimp =

0

@

0 0

0 Σdd

1

A , HDC =

0

@

0 0

0 HDC dd

1

A

HamiltonianHLDA(k) to be written in Wannier(-like) basis{wn(r−T)}

Ĥ = X

T

X

nm

|w0n〉Hnm(T)〈w T m| , Hnm(k) ∼

X

T

eik·THnm(T)

Hamiltonian may include only strongly correlated orbitals, but also weakly correlated orbitals

double-counting correctionHDC has to take care of correlations already included in LDA

in most cases no charge-density update in the self-consistency cycle, i.e.,

DMFT works “post processing” to an LDA calculation

bottleneck: quantum impurity solver ! . – p.7/21

LDA+DMFT for real materials

DMFT loop

DMFT preludeDFT part

update

V̂KS = V̂ext + V̂H + V̂xc

[

−∇ 2

2 + V̂KS

]

|ψkν〉 = εkν |ψkν〉

from charge density ρ(r) construct update

{|χ Rm

〉} build ĜKS = [

iωn + µ+ ∇2

2 − V̂KS ]−1

construct initial Ĝ0

impurity solver

G imp mm′

(τ − τ ′) = −〈T̂ d̂mσ(τ)d̂ † m′σ′

(τ ′)〉Simp

self-consistency condition: construct Ĝloc

Ĝ−10 = Ĝ −1 loc + Σ̂imp

Ĝloc = P̂ (C) R

[

Ĝ−1KS − (

Σ̂imp − Σ̂dc )]−1

P̂ (C) R

Σ̂imp = Ĝ −1 0 − Ĝ

−1 imp

ρ

compute new chemical potential µ

ρ(r) = ρKS(r) + ∆ρ(r)

(Appendix A)

. – p.8/21

Back to BaVS3 ...

the vanadium sulfide shows three continuous phase transitions:

T∼ 240 K : hexagonal to orthorhombic structural transition

T∼ 70 K : metal-to-insulator transition (MIT) from Curie-Weiss metal to paramagnetic insulator, structural transition to monoclinic phase

T∼ 30 K : incommensurate antiferromagnetic transition

[Sayetatet al.J. Phys. C151627]

orthorhombic (Cmc21) structure at T=100 K: [Ghediraet al., J. Phys. C19 6489]

- zigzag VS3 chains

- two formula units in primitive cell

- dinterVV ∼ 2d intra VV

[Fagotet al., PRL90196401]: large one-dimensional structural fluctuations alongc axis above MIT. – p.9/21

LDA calculations for metallic BaVS3

previous calculations:

[ M. Nakamura, A. Sekiyama, H. Namatame, A. Fujimori, H. Yoshihara, T. Ohtani, A. Misu,

and M. Takano, PRB4916191 (1994)]

[L.F. Mattheiss, Solid State Commun.93791 (1995)]

[M.H. Whangbo, H.J. Koo, D. Dai, and A. Villesuzanne, J. Solid State Chem.165345 (2001)]

[X. Jiang and G. Y. Guo, PRB70 035110 (2004)]

[A. Sanna, C. Franchini, S. Massidda, and A. Gauzzi, PRB70235102 (2004)]

our calculations:

mixed-basis pseudopotential code

[B. Meyer, C. Elsässer, FL and M. Fähnle,FORTRAN 90 Program for Mixed-Basis-Pseudopotential

Calculations for Crystals, MPI für Metallforschung, Stuttgart]

. – p.10/21

LDA results for metallic BaVS3

T>240K: hexagonalP63/mmc

-6

-5

-4

-3

-2

-1

0

1

2

3

4

ε- ε F

(e V

)

Γ K M Γ A H L A

-1 0 1 2 E-E

F (eV)

0

1

2

3

4

5

6

D O

S (1

/e V

)

A 1g

E g

e g

S(3p) -6 -4 -2 0 2 4 0

2

4

6

. – p.11/21

LDA results for metallic BaVS3

T>240K: hexagonalP63/mmc

-6

-5

-4

-3

-2

-1

0

1

2

3

4

ε- ε F

(e V

)

Γ K M Γ A H L A

-1 0 1 2 E-E

F (eV)

0

1

2

3

4

5

6

D O

S (1

/e V

)

A 1g

E g

e g

S(3p) -6 -4 -2 0 2 4 0

2

4

6

70K

LDA results for metallic BaVS3

narrow (0.7 eV)Eg bands

at the Fermi level

broader (2.7 eV) foldedA1g band

2kF alongΓ-Z: 0.94c∗

experimental2kF: qCDWc =0.5c ∗

Fermi surface not flattened