Role of multiple subband renormalization on the thermoelectric - PowerPoint PPT Presentation

Role of multiple subband renormalization on the thermoelectric effect in correlated oxide superlattices Andreas Rüegg, Sebastian Pilgram and Manfred Sigrist Theoretische Physik, ETH Zürich, Switzerland

complex oxide heterostructure [SrTiO 3 ] 5 /[LaTiO 3 ] 1 band Mott insulator insulator annular dark field (STEM), Ohtomo et al., Nature (2002) Hvar, 2008 2

transition metal oxides - interfaces? manganites • bulk transition-metal oxides: - rich and complex phase diagrams cuprates - intrinsic functionalities : � ferroelectricity, superconductivity, CMR, ... - spin , charge and orbital degrees of freedom - strongly correlated electronic systems cf. semi- conductors • artificial nanoscale structures: - novel physics - stabilization of new phases - atomic , electronic and orbital reconstruction - clean doping Tokura, Nagaosa, Science (2000) Dagotto, Science (2005, 2007) 3 Chakhalian et al ., Science (2007) Okamoto, Millis, Nature (2004)

electronic reconstruction 3d-electron charge distribution • general: electronic phase at the interface differs from bulk phase. • important mechanism: transfer of electronic charge across the interface. • “simple” example: metallic interface between insulators. electron energy loss spectra, Ohtomo et al., Nature (2002) SrTiO 3 LaTiO 3 Ohtomo et al. (2002) band/Mott Ohtomo et al. (2004), Thiel et al. (2006), SrTiO 3 LaAlO 3 Reyren et al. (2007) insulators metallic SrTiO 3 LaVO 3 Hotta et al. (2007) SrVO 3 LaVO 3 Sheets et al. (2007) 4 Okamoto and Millis, Nature (2004)

single-particle picture real space: mutual doping (transfer of electronic charge from Mott to band insulator) leads to metallic interfac e. atomic limit: BI MI BI appropriate at high temperatures n 3 d ≈ 1 n 3 d ≈ 0 n 3 d ≈ 0 momentum space: interface states? - quantum confinement (superlattice structure) leads to multiple subbands in a quasi-two dimensional system. ν 3 - familiar from (and successfully applied in) semiconductor physics. 2 1 ! " BI MI BI # k y $ ! # ! " but ! ! ! ! ! " ! # $ # " ! k x how do local correlations modify this picture? 5

semiclassical perspective of transport N M quasi-particle dispersion Fermi surface: BI BI E k ν = E ν ( ε k ) MI MI MI E k ν = 0 in-plane non-interacting � momentum dispersion subband index � E ∇ T (i) multi-subband aspect: � electron-like and hole-like contributions, compensation possible. • e.g. thermopower, Hall constant, ... (ii) correlation effects: need hybridization of itinerant (a) group velocity renormalization � Z ν = ∂ E k ν and (almost) localized degrees � � (relevant for Drude weight) ∂ε k of freedom: interface! � FS (b) renormalization of particle-hole asymmetry ∂ 2 E k ν �� � � � ∂ E k ν � (relevant for thermopower) / α ν = ε k � ∂ε 2 ∂ε k � k FS (c) indirectly through the multi-subband aspect 6

outline • model: - extended single-orbital Hubbard model - effective low-energy theory (slave-boson mean-field theory) • generic “electronic structure” - multiple subbands - quasiparticle renormalization ( group velocity and particle-hole asymmetry ) • parallel transport - thermoelectric effects � Seebeck coefficient (thermopower) - free carrier response � Drude weight, optical conductivity different story: perpendicular • conclusions transport studied by Okamoto and Freericks, Zlatic, Shvaika... transport in multilayered nanostructures J. Freericks 7

modeling Hvar, 2008 8

geometry 3 dimensional model: band insulator Mott insulator band insulator two situations: (SrTiO 3 ; Ti-3 d 0 ) (LaTiO 3 ; Ti-3 d 1 ) (SrTiO 3 ; Ti-3 d 0 ) (i) quantum well BI MI BI BI quantum well BI M ν 3 2 1 electronic sites: cubic lattice BI MI BI (Ti sites) N (ii) superlattice structure counterions (Sr 2+ vs. La 3+ ) x,y BI BI BI BI N z N M M N N 9

electron-ion interaction single-orbital model E C nearest neighbor hopping � V i = − � � � r i − r ion � i j Extended single-orbital Hubbard model: n i + 1 n j + 1 � � � � � � � ˆ c † W ion H = − t ˆ i σ ˆ c j σ + h . c . + U n i ↑ ˆ ˆ n i ↓ + V i ˆ n i W ij ˆ ˆ ij 2 2 � ij � σ i i i � = j i � = j on-site (long range) electron- (long range) ion- repulsion electron interaction ion interaction E C E C W ij = W ion | r i − r j | = ij � � � r ion − r ion � i j parameters at T = 0: BI MI BI studied in the current context by: BI BI M • hopping • Okamoto and Millis t • Lee and MacDonald • on-site U r = U − E C ≥ 0 • Kancharla and Dagotto E C = e 2 • long-range • Freericks et al. � a • geometry: x,y � MI: N � BI: N z M 10

KR-slave-bosons and heterostructures effective 1D Schrödinger equation: � z 2 � � l ε k + λ l ψ k ν ( l ) − t z l z l + γ ψ k ν ( l + γ ) = E k ν ψ k ν ( l ) . γ = ± 1 hopping renormalization subband index Lagrange multiplier non-interacting in- layer index in-plane momentum plane dispersion KR-slave-boson mean-field theory provides a way to find the local (low-energy) self-energy i) layer-resolved hopping renormalization: � � (1 − n l + d 2 l )( n l − 2 d 2 n l − 2 d 2 l ) + d l l z l = z l ( n l , d l ) = � n l (1 − n l / 2) (amplitude of) electronic density double occupancy ii) free energy density: 2 2 2 + 1 2 + 1 � � 1 + e − β E k ν ( n,d, λ ) � 1 + e − β E k ν ( n,d, λ ) � � � � � � � � � f ( n, d, λ ) = − f ( n, d, λ ) = − log log + U r + U r ( λ l − V l ) n l ( λ l − V l ) n l d l d l n l W ll ′ n l ′ − n l W ll ′ n l ′ − 2 2 β N || β N || k ν k ν l l ll ′ ll ′ l l iii) saddle-point equations: maximize with respect to λ λ minimize with respect to n, d n, d Gutzwiller, PRL (1963) Kotliar, Ruckenstein, PRL (1986) 11 Brinkmann, Rice, PRB (1970) AR, Pilgram, Sigrist, PRB (2007)

typical solution N = M = 10 , U r = 22 t, E c = 0 . 8 t ( T = 0) 1 BI BI BI charge density n l 0.5 � � � t E C /t λ TF ∝ a MI MI MI MI E C 0 20 40 60 80 10 M N single-particle 5 λ l potential 0 20 40 60 80 0.2 double occupancy 0.1 d l (amplitude) 0 U r /t 20 40 60 80 l 12

(coherent) charge density Mott regime layer-resolved spectral density: ) *+,- U r = 23 t ! A ( ω ) *+,- n .,-/ n U = 16 t .,-/ "$( metallic interfaces "$' n coh 0 ω n, A ( ω ) "$& n coh "$% n coh " 0 ω ) ! !" ! # " # !" (two-site) DMFT data courteously l taken from Okamoto and Millis (2004) Okamoto and Millis, PRB (2004) 13 AR, Pilgram, Sigrist, PRB (2007)

“electronic structure” Hvar, 2008 14

quantum well subband dispersion ν 3 � z 2 � � l ε k + λ l ψ k ν ( l ) − t z l z l + γ ψ k ν ( l + γ ) = E k ν ψ k ν ( l ) 2 1 γ = ± 1 BI MI BI partially filled subbands % N = 8 $ U r = 24 t ! E C = 0 . 8 t E k ν /t " # ! " ν = 1 , 2 ! ! ! $ ! ! ! " # " ! ε k /t * Ueda, Rice, PRL (1985) 15 AR, Pilgram, Sigrist, PRB (2007)

envelope wave functions � z 2 � � l ε k + λ l ψ k ν ( l ) − t z l z l + γ ψ k ν ( l + γ ) = E k ν ψ k ν ( l ) γ = ± 1 effective potential square of envelope wave function $% % &% double well ε k /t = - 4 "% ! &% l - 1) ε k ] /t ! $% '% [ λ l + ( z 2 $% single quantum well &% "% N = 10 U r = 14 t ! &% ε k /t = 4 ! $% % ! !" ! # " # !" l

quantum well subband dispersion ν 3 � z 2 � � l ε k + λ l ψ k ν ( l ) − t z l z l + γ ψ k ν ( l + γ ) = E k ν ψ k ν ( l ) 2 1 γ = ± 1 BI MI BI partially filled subbands % N = 8 $ “band insulator” hybridization of almost U r = 24 t ! localized (correlated) and itinerant degrees E C = 0 . 8 t E k ν /t interface " of freedom # reminiscent of heavy- “Mott insulator” ! " fermion systems * ν = 1 , 2 ! ! ! $ high thermopower? ! ! ! " # " ! ε k /t * Ueda, Rice, PRL (1985) 17 AR, Pilgram, Sigrist, PRB (2007)

quantum well renormalization ν 3 � z 2 � � l ε k + λ l ψ k ν ( l ) − t z l z l + γ ψ k ν ( l + γ ) = E k ν ψ k ν ( l ) . 2 1 γ = ± 1 BI MI BI ! quasi-particle weight: N = 8 “band insulator” U r = 24 t � Z ν = ∂ E ν k � � ∂ε k Z ν &'# � FS “Mott insulator” suppressed in “Mott insulator” & ! " # $ % !! !" ν particle-hole asymmetry: ) interface ∂ 2 E ν k �� � � � ∂ E ν k � / α ν = ε k ( � ∂ε 2 ∂ε k α ν � k FS & enhanced at interface! ! ( ! " # $ % !! !" ν subbands 18

thermoelectric effects Hvar, 2008 19