Effect of correlated hopping on thermoelectric properties: Exact solutions for the Falicov-Kimball model

The effect of correlated hopping on the charge and heat transport is investigated for the Falicov-Kimball model. Exact solutions for the electrical and thermal conductivities and thermoelectric power are obtained within the dynamical mean field theory. The temperature dependences of the transport coefficients are analysed for particular values of correlated hopping which correspond to the significant reconstruction of the density of states and transport function. The cases with strong enhancement of thermoelectric properties are elucidated.


Introduction
Direct transformation of the heat flow into the electric current and verse attracts much attention in science and engineering [1], but its applications are limited by the low thermoelectric figure of merit of the traditional bulk materials -metals and semiconductors -good electric conductors are also good heat conductors. On the other hand, the bad metals or compounds with strong electron correlations, possessing large variability of their band spectrum and density of states depending on the chemical structure and doping, are considered as a possible new candidates for the thermoelectric materials [1,2]. In many cases, the anomalous properties of the strongly correlated materials are ascribed to the on-site Coulomb or spin interactions, which are the leading contributions.
It was already noticed by Hubbard in his seminal article [3] that besides the local Coulomb-type interaction U ini ↑ni ↓ there should be another nonlocal contributions: the intersite Coulomb interaction i j V i jnin j and the so-called correlated hopping i j σ t (2) i j (n iσ +n jσ )c † i σ c j σ and i j σ which reflects the fact that different many-body states can overlap in different amount and, as a result, the value of intersite hopping depends on the occupation of this states. The origin of the correlated hopping can be as a direct intersite interaction as an indirect effective one [4,5]. The effect of local Coulomb interaction is the subject of the famous Hubbard model, which is widely investigated in the theory of strongly correlated electron systems. The correlated hopping is less popular. Even the term used is not well established. Besides the term "correlated hopping" many other are circulated: the "assisted hopping", "bond-charge interaction (repulsion)", "occupation-dependent hopping", "correlated hybridization", etc. Similar contributions in the theory of disordered systems are also known as an "off-diagonal disorder" [6]. Correlated hopping was considered in connection with new mechanisms of high temperature superconductivity [7,8], electronhole asymmetry [9], and enhancement of magnetic properties [10]. Recently, the correlated hopping is examined in application to the quantum dots [11][12][13] and optical lattices [14,15].
In this article we shall consider the effect of correlated hopping on the charge and heat transport for the Falicov-Kimball model [16], which possesses an exact solutions in the dynamical mean field theory (DMFT) [17][18][19][20]. In section 2 we recall the main arguments of the linear response Kubo theory for the charge and heat transport which will be important in further considerations. Section 3 provides the DMFT solutions for the Falicov-Kimball model with correlated hopping and derivations of the charge and energy currents and transport coefficients. In section 4 we consider peculiarities of the charge and heat transport with the change of correlated hopping value and doping and we summarize in section 5.

Macroscopic and microscopic levels in description of thermoelectric effect
In the linear response Kubo theory the charge j c (r ) and energy (heat) j Q (r ) currents are results of the electro-chemical potential gradient, including electrical field and charge distribution inhomogeneities, and temperature gradient and can be obtained from the following equations [21] j c (r ) = −e L 11 (r , r )∇μ(r )dr − e L 12 (r , r ) ∇T (r ) For the uniform (steady) dc charge and energy currents we can define the dc electric conductivity σ dc = e 2 L 11 , The standard route to introduce macroscopic currents in the microscopic lattice models is the following [21]. First of all one can define the charge polarization byP = i α R i z αni α ,where R i is the lattice site vector andn i α is the particle number operator at site i for particles of kind α with charge z α . Now the charge current can be defined by the continuity equation asĵ = dP dt = 1 i P ,Ĥ .
In a similar way one can define the energy polarization bŷ whereĤ i are the single site (local) contributions in total energyĤ = iĤi , and the energy current bŷ Let us recall how it works for the case of the noninteracting electrons. The Hamiltonian for the noninteracting electrons on lattice can be written aŝ  (2.10) and the energy polarization and current operators now take the form On the other hand, the energy current operator can be rewritten aŝ (2.13) Relation (2.12) is very important. In the cases when it holds, one can immediately write down an expressions for the generalized transport integrals L l m [22], the so-called Boltzmann relations, also known as the Johnson-Mahan theorem in the theory of metals and semiconductors [23,24]. In the simplest form, when the dynamical screening effects can be ignored, they states that the generalized transport integrals for electrons L 11 , L 12 = L 21 , and L 22 can be written in terms of one transport function (relaxation time) I(ω): where f (ω) = 1/(e βω + 1) is the Fermi distribution function. Accounting of the screening effects and inelastic scattering replaces these relations by the generalized one [22]. For the case of noninteracting electrons the transport function has the form where ε k is the band energy (Fourier transform of the hopping integral t i j ), and we get the known rela- (2.16) The main consequences from equations (2.14) are the following. The values of the transport integrals L l m are determined by the features of the transport function I(ω) only within the Fermi window (n = 0) of width ∼ 4T and its "moments" which spread on the larger energy interval (see figure 1). For the metals the transport function I(ω) is strongly reduced and it was shown by Mahan and Sofo [25] that these could be achieved for the δ-function like transport function I (ω) ∼ δ(ω − ε 0 ), when the electronic contribution in thermal conductivity (2.5) vanishes.
In order to get high values of the figure of merit Z T (2.6) for thermoelectric properties in some temperature interval one have to look for the systems with strongly asymmetric transport function I(ω) within the Fermi window with sharp peak at (ω − µ)/T = 1.543404638 or −1.543404638 to enhance the Seebeck coefficient and minimums at (ω − µ)/T = ±2.399357280 to reduce the thermal conductivity. That is why the materials with strong electron correlations as well as disordered systems attract much attention for thermoelectric applications. It is caused by the strong variability of their band structure and density of states in the Fermi window depending on the chemical structure and doping. In particular, it was already shown that doping of Mott insulator can strongly improve its thermoelectric properties [26,27]. In this case, the doping shifts the chemical potential into the lower or upper Hubbard band which produces the strongly asymmetric density of states (d.o.s.) and transport function. Besides the Hubbard type local Coulomb interaction, one can consider the non-local contributions too, which can produce additional enhancements.

Charge and energy current operators and transport coefficients for the Falicov-Kimball model with correlated hopping
A simplest model in the theory of strongly correlated electron systems is the Falicov-Kimball one [16], which considers the local interaction between the itinerant d electrons and localized f electrons. It is a binary alloy type model and it has an exact solution in the dynamical mean field theory (DMFT) [17]. Its enhancement by correlated hopping was also considered and the DMFT solutions with a nonlocal self-energy were obtained [18][19][20].
The connection between the elements of the hopping matrix t i j and initial hopping amplitudes is the Below we shall use the hopping amplitude over empty states as an energy unit: t −− = t 1 = 1. An irreducible part Ξ(ω) as well as the matrix of the λ-fields Λ(ω) are solutions of the system of where an expressions for the Green's function G imp (ω) of the local impurity problem for the Falicov- The previous investigations of the Falicov-Kimball model with correlated hopping [19,20] have elucidated several special cases with a strong reconstruction of the one-particle d.o.s., which in the considered case is equal These crossover points correspond to the special forms of the hopping matrix (3.5): 1. The regular Falicov-Kimball model without correlated hopping corresponds to the case of t 2 = t 3 = 0, when all components of the hopping matrix (3.5) are the same t αβ = t 1 and its determinant is equal to zero det t k = 0 (figure 2a). ?????-5 The dc electric conductivity is connected with the current-current Green's function, which in the considered case has a form where for the D → ∞ hypercubic lattice with unperturbed band energy (3.6) the rigorous replacement An analytic expression for the transport function I (ω) in terms of the solutions of the DMFT impurity problem (3.7) was derived but it is too cumbersome to be presented here.
On the other hand, the energy polarization operator for the Falicov-Kimball model with correlated hopping is equal tô Now, from the continuity equation (2.8) we get an expression for the energy current operator and one could check that it is equal tô That means that the connection (2.12) between the energy and charge current operators holds in the case of correlated hopping and the Boltzmann relations (2.14) can be used in this case too. The numerical results for the charge and heat transport in the systems with correlated hopping are presented in the next section.

Charge and heat transport in presence of correlated hopping
It was already mentioned above that for the different values of correlated hopping the different shapes of the one-particle d.o.s. can be realized (figure 2) and the crossover from one regime to another takes place at special forms of the hopping matrix (3.5): either its determinant or some matrix elements are equal to zero. Let us check the behaviour of the charge and heat transport in the vicinity of these crossover points.
First of all we consider the small values of correlated hopping t 2 (below we shall put t 3 = 0). In the absence of correlated hopping t 2 = 0 and for small values of the Coulomb interaction U (figure 3) the shape of the d.o.s. A d (ω) deviates in small amount from the unperturbed one (Gaussian for the D → ∞ hypercubic lattice). The transport function I (ω) in this case slightly vary in the vicinity of the chemical potential value µ d = U /2 and approaches the constant value at large frequencies. The last feature is a consequence of the Gaussian d.o.s. [26], when for the large energy values we still have an exponentially small density of states with a finite relaxation rate, which can not be observed in real systems. Due to this we consider only the small and moderate temperature values when the Fermi window and its moments (see figure 1)    case without correlated hopping t 2 = 0. The temperature dependence of the electric σ dc and thermal κ e conductivities is almost the same and we observe only increase of the Seeback effect at low temperatures similar to the one in the doped Mott insulators [26] (non-zero correlated hopping breaks the electronhole symmetry like doping in the case without correlated hopping). It is known that the temperature behaviour of the thermoelectric properties in Mott insulator in much extent is determined by the temperature dependence of the chemical potential [26,27] and due to the numerical issues we were not able to determine the chemical potential values in the gap at very low temperatures with a precision enough to get the smooth dependences of the Seebeck coefficient.
For the opposite case of the almost independent bands, when t 2 ≈ −1 and off-diagonal elements of the hopping matrix (3.5) are small and change the sign (t +− = t −+ → 0), the d.o.s. and transport function are very smooth in the Fermi window, which results in the metallic behaviour of the electric and thermal conductivity and weak thermoelectric properties ( figure 5). The Coulomb interaction U is less important in this case. In the vicinity of the another crossover point at t 2 = −0.5 (no direct hopping between the sites occupied by f -particles: t ++ = 0) the behaviour is completely different ( figure 6). Exactly at the t 2 = −0.5 value but now the transport function displays the strong enhancement with narrow peak. It should be noted, that in many theoretical simulations the calculation of the two-particle quantities, including the transport function, is problematic and sometimes an approximation I (ω) = πΓA d (ω), replacing by the one-particle quantity, is used instead (see, e.g. [13]). Our results definitely show that such an approximation is not valid and produces a completely different transport functions for many cases.
Different shapes of the transport function in the vicinity of the t 2 = −0.5 value manifest themselves in different transport properties. Exactly at the crossover point t 2 = −0.5 the temperature dependences of the electric σ dc and thermal κ e conductivities as well as of the Seebeck coefficient are similar to the one in the doped small gap Mott insulator. For the t 2 = −0.6 value the metallic behaviour is observed: weak temperature dependence of the electric conductivity and small thermoelectric power. On the other side of the crossover point, an enhancement of all transport coefficients is observed which is more prominent for the electric conductivity and Seebeck coefficient than for the thermal conductivity and results in the increasing of the thermoelectric figure of merit Z T .
For the large values of the Coulomb interaction U = 2 the behaviour is quite different ( figure 7). Now

Conclusions
In this article we have investigated an effect of the correlated hopping on the charge and heat transport and thermoelectric power in correlated material described by the Falicov-Kimball model. Depending on the value of correlated hopping the crossover points which separate regions with different shapes of the d.o.s. and transport function are clarified. The temperature dependences of the electrical and thermal conductivities and thermoelectric power are strongly effected by the presence of singularities and peaks on the transport function and by the temperature evolution of the chemical potential. The largest enhancement of the thermoelectric properties is observed for the values of correlated hopping −t 1 /2 < t 2 < 0, when the direct hopping between the same many body states at different sites is reduced and an indirect one become important. It should be noted that the calculations of the correlated hopping amplitudes for different compounds, see e.g. [11,12], give the absolute values close to this interval but disagree in its sign, which is the main factor according to our results.
Unfortunately, using of the Gaussian d.o.s. produces the non-physical results for the transport function outside the bands, which did not allow to get reasonable values for the chemical potential at very low and very high temperatures and calls for the additional investigations with another unperturbed density of states.