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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