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