Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly...

23
Correlated Electronic Structure of BaVS 3 Frank Lechermann I. Institut f ¨ ur Theoretische Physik, University of Hamburg, Germany in collaboration with Silke Biermann and Antoine Georges CPHT, ´ Ecole Polytechnique, Palaiseau, France Workshop on “Physics of the low dimensional strongly correlated electronic system : BaVS 3 Orsay, 06.12.07 . – p.1/21

Transcript of Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly...

Page 1: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

Correlated Electronic Structureof BaVS3

Frank LechermannI. Institut fur Theoretische Physik, University of Hamburg, Germany

in collaboration with

Silke Biermann and Antoine GeorgesCPHT,Ecole Polytechnique, Palaiseau, France

Workshop on

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

Orsay, 06.12.07

. – p.1/21

Page 2: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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

Page 3: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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) fromCurie-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 inprimitive cell

- dinterVV ∼ 2dintra

VV

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

Page 4: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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

eeg2

Eg2

Eg1

eg1

Eg

atomic hexagonal orthorhombic

1gA1gA

[Massenetet al., J. Phys.

Chem. Solids.40573]. – p.4/21

Page 5: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

Electronic structure of correlated systems

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

low-energy (coherent)quasiparticleswith 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

Page 6: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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

Page 7: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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−10 −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 andstates originating from atomic-like excitations.

[Georges and Kotliar, PRB45 (1992)]

[Metzner and Vollhardt, PRL62 (1989)]

U

G0(τ−τ ′)

. – p.6/21

Page 8: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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 HDCdd

1

A

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

H =X

T

X

nm

|w0n〉Hnm(T)〈wT

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

Page 9: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

LDA+DMFT for real materials

DMFT loop

DMFT preludeDFT part

update

VKS = Vext + VH + Vxc

[

−∇2

2 + VKS

]

|ψkν〉 = εkν |ψkν〉

from charge density ρ(r) constructupdate

{|χRm

〉} build GKS =[

iωn + µ+ ∇2

2 − VKS

]−1

construct initial G0

impurity solver

Gimpmm′(τ − τ ′) = −〈T dmσ(τ)d†

m′σ′(τ ′)〉Simp

self-consistency condition: construct Gloc

G−10 = G−1

loc + Σimp

Gloc = P(C)R

[

G−1KS −

(

Σimp − Σdc

)]−1

P(C)R

Σimp = G−10 − G−1

imp

ρ

compute new chemical potential µ

ρ(r) = ρKS(r) + ∆ρ(r)

(Appendix A)

. – p.8/21

Page 10: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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) fromCurie-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 inprimitive cell

- dinterVV ∼ 2dintra

VV

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

Page 11: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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

Page 12: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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 2E-E

F (eV)

0

1

2

3

4

5

6

DO

S (1

/eV

)

A1g

Eg

eg

S(3p) -6 -4 -2 0 2 40

2

4

6

. – p.11/21

Page 13: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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 2E-E

F (eV)

0

1

2

3

4

5

6

DO

S (1

/eV

)

A1g

Eg

eg

S(3p) -6 -4 -2 0 2 40

2

4

6

70K<T<240K: orthorhombic Cmc21

-6

-5

-4

-3

-2

-1

0

1

2

3

4

ε−ε F (e

V)

Γ C Y Γ Z E T Z

-1 0 1 2E-E

F (eV)

0

1

2

3

4

5

6D

OS

(1/e

V)

A1g

Eg1

Eg2

eg1

eg2

S(3p)

-6 -4 -2 0 2 40

2

4

6

. – p.11/21

Page 14: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

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

orbital populations ?

nature ofEg states

(Curie-Weiss behavior) ?

70K<T<240K: orthorhombic Cmc21

-6

-5

-4

-3

-2

-1

0

1

2

3

4

ε−ε F (e

V)

Γ C Y Γ Z E T Zqc

CDW

-1 0 1 2E-E

F (eV)

0

1

2

3

4

5

6D

OS

(1/e

V)

A1g

Eg1

Eg2

eg1

eg2

S(3p)

-6 -4 -2 0 2 40

2

4

6

. – p.12/21

Page 15: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

Wannier functions for low-energy states

-3

-2

-1

0

1

2

3LDA band structure

-3

-2

-1

0

1

2

3

ε-ε F (

eV)

Γ C Y Γ Z E T Z

-1 0 1 2E-E

F (eV)

0.0

0.5

1.0

1.5

2.0

2.5

DO

S (

1/eV

)

Eg1

Eg2

A1g

Wannier functions in crystal-field basisderived from maximally-localized construction

[Marzari and Vanderbilt, PRB56 12847]

[Souza, Marzari, and Vanderbilt, PRB65 035109]

A1g Eg1 Eg2

Hoppingsin meV

A1g Eg1 Eg2 A1g-Eg1

000 213 0 26 0

0012

-511 44 -12 -146. – p.13/21

Page 16: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

LDA+DMFT in Wannier basisGloc(iωn) =

X

k

[(iωn + µ)1−HLDA(k)−Σ(iωn)]−1

impurity on-site interaction Hamiltonian (U, U ′ = U − 2J, U ′′ = U − 3J):

HU = UX

m

nm↑nm↓ +U ′

2

X

mm′σ

m6=m′

nmσnm′σ +U ′′

2

X

mm′σ

m6=m′

nmσnm′σ

integrated spectral function:

-2 -1 0 1 2 3ω (eV)

0.0

0.5

1.0

1.5

2.0

2.5

DO

S (

1/eV

)

LDA

-1 0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

ρ (1

/eV

)

A1g

Eg1

Eg2

LDA+DMFT

orbital fillings: (ntot = 1)

U ,J (eV) A1g Eg1 Eg2

0.0, 0.0 0.58 0.30 0.12

3.5, 0.7 0.41 0.45 0.14

temperature dependence:

0.0

0.5

1.0

1.5

2.0

-2 -1 0 1 2 3 4ω (eV)

0.0

0.5

1.0

1.5

ρ (1

/eV

)

A1g

Eg1

Eg2

β=10 eV-1

β=30 eV-1

magnetic susceptibility:

0 0.5 1.0 1.5 2.00

1

2

3

4

χ(loc)

0 0.5 1.0 1.5 2.0 2.5T/1000 (K)

A1g

Eg1

Eg2

total

U/J=7 U/J=4

. – p.14/21

Page 17: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

LDA+DMFT Quasiparticle states

Self-energy: Σ(iωn) = ℜΣ(iωn) + ℑΣ(iωn)

analytical continuation and expansion:

ℜΣmm′(ω + i0+) ≈ ℜΣmm′(0) +`

1− [Z−1]mm′

´

ω −O(ω2)

ℑΣmm′(ω + i0+) ≈ −Γmm′ω2 +O(ω3)

det[(ωk1−Z (HLDA(k) + ℜΣ(0)− µ1)] = 0

-0.1

0

0.1

ε-ε F (

eV)

Γ C Y Γ Z E T Z

-0.1

0.0

0.1

ε-ε F (

eV)

Γ M/2 A/2 Z Γ

LDA LDA+DMFT

. – p.15/21

Page 18: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

Recent measurements

Angle-resolved photoemission (ARPES)

[Mitrovic et al., cond-mat/0502144]

[Mo et al., APS March Meeting 2005, unpublished]

Optics

[Kézsmárkiet al., PRL96186402]

00.10.20.30.40.50.60.70.80.91.0

0.01 0.1 1 3 00.10.20.30.40.50.60.70.80.91.0

0 0.5 1.0 1.5 2.0 2.5 3.00

400

800

1200

1600

10K 45K 60K 73K 85K100K115K300K

E c

Ref

lect

ivity

Ec

E c

Ref

lect

ivity

Ec

Eg

A*1g

S(z) E

g

S(3p) V(3d)Eg

A1g

E cE cE cE c

(-1cm

-1)

Energy (eV). – p.16/21

Page 19: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

BaVS3 below MIT: insulating CDW state

Cmc21 structure:

orthorhombic

two equivalent V atoms in

unit cell

d(chain)VV = 5.37 a.u.

T < TMIT: Im structure[Fagotet al., Solid State Sci.7 718]

monoclinic

doubling of unit cell

four inequivalent V atoms

tetramerization (trimerization)

dominant2kF distortion

∆d(chain)VV =−0.07 a.u.

∆d(chain)VV =−0.17 a.u.

∆d(chain)VV =+0.19 a.u.

∆d(chain)VV =+0.10 a.u.

V(4) (∆dVS=−0.020 a.u.)

V(3) (∆dVS=+0.001 a.u.)

V(2) (∆dVS=+0.034 a.u.)

V(1) (∆dVS=+0.016 a.u.). – p.17/21

Page 20: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

LDA+DMFT for insulating BaVS 3

12 Wannier functions from LDA

-2

-1

0

1

2

ε-ε F (

eV)

V Z Γ A M L

LDA + cluster-DMFT with 4-site impurity

Eg1 majority occupation on V(1)/V(2)

mixedA1g /Eg1 occupation on V(3)/V(4)

substantial intersiteΣ(ω) between V(3)/V(4)

Im (40 K) Wannier-DOS

0123 A

1g

Eg2

Eg1

0123

DO

S (

1/eV

)

0123

-2 -1 0 1 2 3 4E-E

F (eV)

0123

V(4)

V(3)

V(2)

V(1)

U = 3.5 eV, J = 0.7 eV

0

1

0

1

ρ (1

/eV

)

0

1

-2 -1 0 1 2 3 4ω (eV)

0

1

V(4)

V(3)

V(2)

V(1)

. – p.18/21

Page 21: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

LDA+DMFT for insulating BaVS 3

orbital occupations:

V1 V2 V3 V4 〈V〉

LDA DMFT LDA DMFT LDA DMFT LDA DMFT LDA DMFT

A1g 0.49 0.12 0.40 0.11 0.62 0.47 0.61 0.34 0.53 0.26

Eg1 0.46 0.89 0.44 0.85 0.28 0.46 0.37 0.62 0.39 0.70

Eg2 0.05 0.03 0.07 0.07 0.11 0.03 0.10 0.02 0.08 0.03

sum 1.00 1.04 0.91 1.03 1.01 0.96 1.08 0.98

self energyΣ(iωn)

-1

0

1

0 1ω

n (eV)

-1

0

1Σ (e

V)

Re ΣIm Σ

1

V1 V2

V3 V4

-0.2

0.0

0.2A

1g - A

1g

A1g

- Eg1

Eg1

- A1g

Eg1

- Eg1

0.0 0.5 1.0ω

n (eV)

-0.2

0.0

0.2

Σ (e

V)

0.5 1.0

Re ΣIm Σ

V1-V2 V2-V3

V3-V4 V4-V1

onsite correlations intersite correlations. – p.19/21

Page 22: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

Conclusions

BaVS3 poses interesting test case in strongly correlated physics

Exhibits competing itinerant and localized states

DFT-LDA not sufficient to treat the compound adequately

LDA+DMFT capable of revealing basic mechanisms

Quasi-localizedEg electrons (Curie-Weiss behavior)

Fermi-surface deformation results in CDW instability for

A1g electrons

. – p.20/21

Page 23: Correlated Electronic Structure of BaVS3 · Electronic structure of correlated systems In strongly correlated solids there are in principle two types of excitations: low-energy (coherent)quasiparticles

Test case: MLWFs for SrVO3

SrVO3 is a3d1 transition-metal oxide with full cubic symmetry:

-8

-6

-4

-2

0

2

4

6

8

ε−ε F (

eV)

R Γ X M Γ -8 -6 -4 -2 0 2 4 6 8E-E

F (eV)

0

2

4

6

8

DO

S (

1/eV

)

totalV(t

2g)

V(eg)

O(2p)

0 1 2 3 4 5 6 7distance along [110] (a.u.)

0

1

2

3

4

5

Ψ (a

.u.-3

/2)

0 1 2 3 4 5distance along [a,1,0] (a.u.)

0.0

0.5

1.0

Ψ (a

.u.-3

/2)

. – p.21/21