Thermoelectric properties of heavy fermion systems calculated within - PowerPoint PPT Presentation

Thermoelectric properties of heavy fermion systems calculated within a DMFT/NRG-treatment of the periodic Anderson model Gerd Czycholl Institut für Theoretische Physik, Universität Bremen Coworkers: Claas Grenzebach, Frithjof Anders Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.1/9

Contents 1. Typical experimental results 2. Periodic Anderson Model (PAM) 3. Dynamical Mean field theory (DMFT), mapping on single impurity Anderson model (SIAM) 4. Impurity solvers: modified perturbation theory (MPT) and numerical renormalization group (NRG) 5. Results 6. Open problems 7. Discussion and Conclusion Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.2/9

Typical experimental results Temperature dependence of resistivity Jaccard et al. 1997 Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.3/9

Typical experimental results Temperature dependence of thermopower Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.3/9

Typical experimental results Dynamical conductivity Marabelli et al. 1988 Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.3/9

Periodic Anderson Model (PAM) Hamiltonian Rσ + U � � k c † ( E f f † 2 f † Rσ f † R − σ + V ( c † H = ǫ � kσ c � kσ + Rσ f � Rσ f � R − σ f � Rσ f � Rσ + c.c. � � � � � � � kσ Rσ periodic arrangement of localized f-levels, conduction band, hybridization, Coulomb correlation only between localized (f-) electrons Simplifying assumptions: 1. only 2-fold degeneracy of f- and conduction electrons 2. no crystal fields 3. no hybridization dispersion 4. no interaction between conduction and f- electrons k = 2 t � d 5. simple cubic lattice in dimension d : ǫ � l =1 cos( k l a ) Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.4/9

Periodic Anderson Model (PAM) Current operator (consistent with PAM) ∂ǫ � ∂V k � c † ( f † k j x = kσ c � kσ [+ kσ c � kσ + c.c. )] � � ∂k x ∂k x � kσ � �� � =0 , if V k-independent (Czycholl and Leder 1981) in site representation: � ( c † Rσ − c † j x = ita R +∆ x σ c � Rσ c � R +∆ x σ ) � � � R ∆ x σ Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.4/9

Periodic Anderson Model (PAM) Quantities to be calculated one-particle Green function � ∞ dte izt < [ a � kσ ; b † kσ ( t ) , b † G ab kσ ( z ) = ≪ a � kσ ≫ z = − i kσ (0)] + > � � � 0 ( a, bǫ { c, f } ) determines f-electron density of states, for instance: ρ fσ ( E ) = − 1 � ImG ff kσ ( E + i 0) � πN � k Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.4/9

Periodic Anderson Model (PAM) Quantities to be calculated f-electron selfenergy defined by � � � � − 1 kσ ; c † kσ ; f † ≪ c � kσ ≫ z ≪ c � kσ ≫ z z − ǫ � − V � � k = kσ ; c † kσ ; f † ≪ f � kσ ≫ z ≪ f � kσ ≫ z − V z − E f − Σ � kf ( z ) � � Kubo formula for dynamic conductivity: (two-particle Green function) 1 σ xx ( ω ) = − ω + i 0 Imχ j x ,j x ( ω + i 0) � ∞ dte izt < [ j x ( t ) , j x (0)] − > χ j x ,j x ( z ) = − i 0 Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.4/9

Dynamical mean-field theory (DMFT) Large-d limit Idea: For lattice models of correlated and disordered electron systems mathematically a non-trivial limit d → ∞ , t → 0 with dt 2 = const. can be defined (Metzner and Vollhardt 1989). Simplifications: 1. Unperturbed density of states Gaussian (for d-dimensional hypercubic lattice) (U. Wolff 1983; Metzner and Vollhardt 1989) 2. Selfenergy k-independent (site-diagonal, local) (Müller-Hartmann 1989) 3. Current operator vertex corrections vanish (Khurana 1989; Schweitzer und Czycholl 1991) Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Dynamical mean-field theory (DMFT) Mapping on effective single-impurity problem Because of site-diagonality of selfenergy mapping on effective single-impurity problem possible (Brandt and Mielsch 1989; Georges, Kotliar 1993) Single-impurity Anderson model (SIAM) ˜ σ f σ + U V � � � k c † σ f σ f † ( c † ( E f f † 2 f † √ H = ˜ ǫ � kσ c � kσ + − σ f − σ + kσ f σ + c.c. )) � � N σ � � kσ k Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Dynamical mean-field theory (DMFT) Mapping on effective single-impurity problem one-particle f-electron Green function of SIAM 1 G ff SIAM ( z ) = z − E f − Σ f ( z ) − ∆( z ) ˜ V 2 1 � with ∆( z ) = N z − ˜ ǫ � k � k Correlated f-impurity coupled to effective ”bath” (conduction band) described by ∆( z ) Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Dynamical mean-field theory (DMFT) DMFT selfconsistency loop Effective bath Green function ∆( z ) to be determined selfconsistently so that f-electron Green function of PAM and effective SIAM agree: 1 G ff SIAM ( z ) = z − E f − Σ f ( z ) − ∆( z ) = 1 1 � G ff P AM ( z ) = V 2 N z − E f − Σ f ( z ) − z − ǫ � � k k Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Dynamical mean-field theory (DMFT) DMFT selfconsistency loop Initial value: e.g. Σ f ( z ) = 0 G f ( z ) = G ff P AM ; ∆( z ) = z − E f − Σ f ( z ) − G − 1 f ( z ) G ff SIAM ( z ) = G f ( z ) ? New value for Σ f solve SIAM (for E f , U, ∆( z ) ) by suitable ”impurity solver” (approximation or numerical method) Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Dynamical mean-field theory (DMFT) Transport quantities in DMFT Because of vanishing of vertex corrections in DMFT transport quantities can be expressed by single-particle Green functions: dynamical conductivity � dE f ( E ) − f ( E + ω ) σ xx ( ω ) ∼ L ( E, E + ω ) ω 1 � ImG cc R ′ σ ( E + i 0) ImG cc L ( E, E + ω ) = R +∆ x ( E + ω + i 0) R � � R ′ +∆ x � � N R � � R ′ σ ∆ x 1 G cc k ( z ) = � V 2 z − z − E f − Σ f ( z ) − ǫ � k Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Dynamical mean-field theory (DMFT) Transport quantities in DMFT � � � − d f static conductivity: σ xx ( ω = 0) ∼ dE L ( E, E ) dE 1 Resistivity: R ( T ) = σ xx (0) � � � − d f dE ( E − µ ) L ( E, E ) Thermoelectric power (TEP): S ( T ) = 1 dE � � � eT − d f dE L ( E, E ) dE Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.5/9

Impurity solver Possible methods for SIAM 1. Non-crossing approximation (NCA) (Keiter and Kimball 1971) 2. Second order perturbation theory (SOPT) (termed IPT within DMFT-scheme) 3. Exact diagonalization (ED) 4. Quantum Monte Carlo (QMC) (Hirsch and Fye 1983) 5. Numerical renormalization group (NRG) (Krishnamurty, Wilkens, Wilson 1980) 6. modified perturbation theory (MPT) (Martin-Rodero et al. 1982) Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.6/9

Impurity solver Possible methods for SIAM Main properties of impurity solvers: 1. NCA has correct low-T scale (Kondo temperature, but leads to unphysical singularity at Fermi level for low T, not appropriate for low-T transport quantities 2. SOPT fulfills Fermi liquid properties, valid in weak-coupling situation only, violates atomic limit, does not lead to correct Kondo temperature 3. ED leads to discrete spectrum, no static conductivity, not suitable for transport quantities 4. QMC applicable only for high T and not too large U, no low-T transport quantities describable Therefore, here MPT and NRG as impurity solvers. Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.6/9

Impurity solver MPT Ansatz for selfenergy of effective SIAM: α Σ SOP T ( z ) f Σ f ( z ) = Un f − σ + 1 − β Σ SOP T ( z ) f where Σ SOP T ( z ) is the SIAM selfenergy in SOPT relative to the f Hartree-Fock solution. Here the parameters α, β can be determined by the condition that the atomic limit (of vanishing V) and an additional criterion (Fermi liquid sum rule, first four moments) are fulfilled. (Martin-Rodero et al. 1982, Kajueter and Kotliar 1996, Nolting, Meyer et al. 1997-2000) MPT fulfills atomic limit, applicable for all U, T , Fermi liquid properties for low T , but not correct Kondo scale. Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.6/9

Impurity solver NRG … … − Λ −1 − Λ −2 − Λ −3 − Λ −n Λ −n Λ −3 Λ −2 Λ −1 −1 1 Core of the NRG: logarithmic discretisation of the energy axis around the chemical potential µ = 0 with a parameter Λ . Hamiltonian of SIAM: limit of discrete Hamiltonians fulfilling a recursion formula. Solving this semi-infinite chain of discrete models recursively leads to Green functions with discrete poles in spectral representation. To get a continuous Green functions broadening necessary, here broadening by logarithmic Gauss functions with b = 0 . 5 δ ( E − E 0 ) → exp( − b 2 / 4) exp( − (ln E − ln E 0 ) 2 /b 2 ) , bE √ π Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.6/9