Orbital polarization in correlated electron systems. A. Lichtenstein University of Hamburg

Transcription

1 Orbital polarizatio i correlated electro systems A. Lichtestei Uiversity of Hamburg

2 I collaboratios with: S. Bierma, A. Poteryaev, A. Georges (ENS, Paris) E. Pavarii, O.K. Aderse, (MPI-Stuttgart) M. Katselso (Nijmege) I. Solovyev (Tsukuba) S. Okatov (Hamburg)

3 Outlie Complexity of Correlated Electro Systems Orbital polarizatio startig from atom: - LDA+U ad orbital orderig - LDA+DMFT ad dyamical effects Electroic structure of TM-perovskites Coclusios

4 Map of the Trasitio Metal Oxides ABO 3 metal oxyge rare earth

5 Correlatio drive MIT photoemissio spectra (DOS) A. Fujimori et al. U/W

6 Charge trasfer TMO isulators N(E) d -1 W L p U E F +1 d Zaae-Sawatzky-Alle (ZSA) phase diagram E U W M p-metal V 2 O 5 CuO Charge-Trasfer E g ~ Isulator LaMO 3 V 2 O 3 NiO FeO d-metal Mott-Hubbard E g ~U TiO (W M +W L )/2

7 From Multiplets to Bads Gat(E) -1-1 G(E)=Gat(E)-t TMO Static effects: a) Slater: SP-LDA b) LDA+U 2. Dyamics: a) Hubbard-I b) DMFT

8 Theory: iteractios vs. hoppig Multibad Hubbard model (<im jm 0 >=δ ij δ mm0 ) Coulomb iraatomic iteractio Matrix elemets of electro-electro iteractios: Exact diagoalizatio of atom: t ij =0 gives multiplets! Solutio with hoppigs t ij 0 i solids is ukow!

9 Fuctioals: MFT- DFT- DMFT Weiss Mea-Field Theory (MFT) of classical magets Koh Desity Fuctioal Theory (DFT) of ihomogeeous electro gas i solids Dyamical Mea-Field Theory (DMFT) spectral desity fuctioal of strogly correlated electgro systems

10 Static limit: LDA+U Rotatioally ivariat LDA+U fuctioal Local screed Coulomb correlatios LDA-double coutig term ( σ =Tr( mm0 σ ) ad = " + + ): Occupatio matrix for correlated electros:

11 Slater parametrizatio of U Multipole expasio: Coulomb matrix elemets i Y lm basis: Agular part 3j symbols Slater itegrals:

12 Average iteractio: U ad J Average Coulomb parameter: Average Exchage parameter: For d-electros: Coulomb ad exchage iteractios:

13 Electroic structure of TMO: LDA+U Desity of States (states/ev formula uit) LSDA U= 5e V U= 9e V MO Eergy (e V) 3d MO U= 13eV O 2p DOS 3d LSDA U= 5e V U= 9e V U= 13eV NiO NiO Ee rg y (e V) w(q), mev Spi-wave Spectrum NiO G Z F G L U = 13 LDA exp

14 Spi-Charge ad orbital orderig: LDA+U J(K) Jc Jab Theory Exp KCuF 3 Superexchage iteractio Quadrupolar distortio i KCuF 3

15 Spi ad Orbital momets i CoO I. Solovyev, A. L, ad K. Terakura, PRL 80, 5758 (1998) LDA+U+SO+o-colliear S L L

16 From Atoms to Solids: DMFT Atomic physics (LDA+U) Bads effects (LDA) N(E) N(E) d SL> E F d + 1 E E F E N(E) LDA+DMFT LHB QP UHB E F E

17 DMFT: Self-Cosistet Set of Equatios Σ Σ Σ Gˆ i BZ 1 r Ω ( ω ) = Gˆ ( k, i ) k r ω Σ U Σ ˆ ˆ 1 1 G 0 ( iω ) = G ( iω ) + Σ( iω ) ˆ Σ Σ Σ QMC ED τ U τ Quatum Impurity Solver DMRG FLEX IPT G( τ τ ) Σˆ ew ( ) 1( ) 1 iω = G iω G ( iω ) ˆ 0 ˆ

18 exp{ τu Multi-bad QMC-scheme Discrete HS-trasformatio (Hirsch, 1983) mm' [ m m' cosh 1 2 ( m + m' )]} = 1 2 S mm ' = ± 1 ( λ ) exp τu mm ' 1 2 = mm ' exp{ λ S mm' mm' ( m Number of Isig fields: N = M( 2M 1), M = { m, σ} Gree Fuctios: 1 Gmm' ( τ, τ ') = Gmm' ( τ, τ ', S) detg Z mm ' ( τ ) G ( τ, τ ', S) = G ( τ, τ ') + V ( τ ) δ δ V σ 1 1 mm' mm ' m mm' ττ ' ( τ ) = λ S ( τ ) σ m mm ' mm ' mm ' m ' mm' = + 1, m < m ' 1, m > m ' H = t c c + U S ij σ + i ij iσ jσ mm' mm' m m' 1 τ m m' U )} mm m τ

19 Metal Isulator trasitio i the Hubbard model i DMFT U = 1 U = 2 U = 2.5 U = 3 U = 4 A.Georges, G.Kotliar, W.Krauth ad M.Rozeberg, Rev. Mod. Phys. 96

20 LDA+DMFT LDA Models approaches Materials-specific (structure, Z, etc.) Fast code packages Fails for strog correlatios Iput parameters ukow Computatioally expesive Systematic may-body scheme BZ 1 r ( ) [ ˆ( ) ˆ ( ) ˆ ( )] 1 ω = I µ + iω H k Σ iω Gˆ i Ω H r k ˆ 0 r r ( k ) = Hˆ LDA( k ) Uˆ dc 0 Multi bad Quatum Mote Carlo

21 DMFT-SCF LDA+DMFT (orthogoal LMTO-TB): ) ( ˆ ), ˆ ( ) ˆ ( ) ˆ ( ), ( ) ( ) ( ) ( ), ( 1 ' ' ' ' 1 ' α ω α ω ω ω ω µ ω ω α + = = Σ + = U i k G U i G i k G i G i k H i i k G IBZ k O BZ k LL LL DMFT LL LDA LL LL h Ee rg y DOS E F sp d dd dd pd d dp pp p d ps s H k +Σ = +Σ H H H H H H H H H s s s s ) ˆ( ) ( ˆ ω r Correlated d-states:

22 Magetism of metals: LDA+DMFT Exchage iteractios i metals Fiite temperature 3d-metal magetism 1,2 1,0 M(T) ad χ(t): LDA+DMFT 1,2 1,0 M(T)/M(0) Global spi flip 0,8 0,6 0,4 M(T) Ni Fe χ(t) 0,8 0,6 0,4 χ -1 M eff 2 /3Tc 0,2 0,2 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 0,0 T/T c A. L., M. Katselso ad G. Kotliar, PRL 87, (2001)

23 E. Pavarii, S. Bierma, A. Poteryaev, A. L, A. Georges, ad O. K. Aderse, PRL (2004) Orthorhombic 3d 1 Perovskites SrVO 3 CaVO 3 Metal m*/m=2.7 Metal m*/m=3.6 LaTi TiO 3 YTiO 3 Isulator Gap=0.2eV Isulator Gap=1.0eV

24 S. Okatov et. al LDA+U results for isulators LaTiO 3 AFM YTiO 3 FM GGA-forces LDA+U

25 LDA results: t 2g DOS all metallic i LDA Crystal-field splittigs w/i t 2g multiplet: (140,200) mev for LaTiO3 ; (200,330) mev for YTiO3

26 LDA+DMFT: DMFT: compariso with experimets SrVO 3 CaVO 3 m*/m=2.2 Exp=2.7 LDA + DMFT m*/m=3.5 Exp=3.6 LaTiO 3 YTiO 3 Exp 0.2 Exp 1.0

27 Orbital orderig i MTiO 3 La Violet: Oxyge Orage: M YTiO3 Occup. LDA DMFT LaTiO Y YTiO E. Pavarii et al. PRL (2004)

28 Coclusios LDA+DMFT is a accurate scheme for realistic descriptio of complex electroic systems Orbital polarizatio i correlated electro system come from the local Coulomb iteractios (LDA+U for spi-orbital ordered isulator ad LDA+DMFT for metals ad paramagetic systems)