Thermoelectric properties of Mott insulator with correlated hopping at microdoping

An influence of the localization of itinerant electrons induced by correlated hopping on the electronic charge and heat transport is discussed for the lightly doped Mott insulator phase of the Falicov-Kimball model. The case of strongly reduced hopping amplitude between the sites with occupied f-electron levels, when an additional band of localized d-electron states could appear on the DOS in the Mott gap, is considered. Due to the electron-hole asymmetry and anomalous features on the DOS and transport function induced by correlated hopping, a strong enhancement of the Seebeck coefficient is observed at low temperatures, when the flattened dependence is displayed in a wide temperature range.


Introduction
During the last decade, various systems with specific electron properties have attracted a great attention of investigators. They include one-and two-dimensional organic conductors, three-dimensional solids and topological conductors, and right until the fermionic atoms on optical lattices. A great part of their properties can be explained exclusively by a proper treating of the electron dynamics involving the manyelectron effects and electron correlations. One of the promising applications of such systems is the power generation or cooling by means of the thermoelectric effect. Theoretical description of the thermoelectric transport in strongly correlated electron systems is a challenge and requires the development of new approaches, see, e.g., reference [1], and most of the previous investigations were performed for the models with local single-site correlations of the Hubbard or Anderson type.
However, it was originally pointed by Hubbard [2] that the second quantized representation of the inter-electron Coulomb interaction contains, besides the local term U i n i↑ n i↓ , the nonlocal contributions including the inter-site Coulomb interaction i j V i jninj and the so-called correlated hopping i jσ t (2) i j (n iσ +n jσ )c † iσ c jσ , where T is the imaginary-time ordering operator and the angular bracket denotes the quantum statistical averaging with respect to H. Due to the nonlocal character of correlated hopping, it is convenient to treat H t as perturbation and expand around the atomic limit. The corresponding Dyson-type equation can be written in a matrix form as follows: where Ξ i j (ω) is the irreducible cumulant [25,32]. It can be shown that the irreducible cumulant is local in the limit of infinite coordination Z → ∞ [32], Ξ i j (ω) = δ i j Ξ(ω), and can be calculated within the dynamical mean-field theory (DMFT). Now, the formal solution of the Dyson equation (2.9) for lattice Green's function can be written in a matrix form as follows: with the components Here, the band energy is distributed according to the density of states ρ( ), the semi-elliptic one for the Bethe lattice and we introduced two adjugate matrices (2.14) In our case, the scalars C and D are given by (2.15) and An irreducible cumulant Ξ i j (ω) can be found as a solution of the DMFT equations where the local lattice Green's functions are equated with the one of an auxiliary impurity embedded in a self-consistent bath, described by the time-dependent mean field where w 1 = P + = n f , w 0 = P − = 1 − n f , and are the impurity Green's functions of a conduction electron in the presence of a f -state which is either permanently empty or occupied, respectively. After substitution of these expressions in (2.18), one can get the 4th order polynomial equations for g 0 (ω) or g 1 (ω), and details of its solution are given in [28].
For the local single-particle Green's function and, finally, the renormalized DOS of the lattice is expressed in terms of the impurity Green's function The chemical potential for d-electrons µ d is obtained by solving the equation where f (ω) = 1/(e ω/T + 1) is the Fermi function, for a given value of their concentration n d = n d . We now proceed to the calculation of transport properties by linear response theory. The dc charge conductivity σ dc = e 2 L 11 , (2.25) the Seebeck coefficient (thermoelectric power E = S∇T) 26) and the electronic contribution to thermal conductivity are expressed in terms of the transport integrals [33][34][35][36] where I(ω) is the transport function. In the considered case of correlated hopping, the DMFT expression for transport function reads where Φ xx ( ) is the so-called lattice-specific transport DOS [37] and, for the Z = ∞ Bethe lattice with semielliptic DOS, the f -sum rule yields [38] Finally, the transport function reads [28] where E 1 and E 2 are the roots of the denominator in equation (2.11), C(ω) − D(ω) + 2 det t = 0, given by and K(ω) reads (2.34)

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Here, and, for the semielliptic DOS, we find In addition, we calculate the Lorenz number which for the pure metal with degenerate fermions is equal to L 0 = π 2 /3, as it follows from the Wiedemann-Franz law.

Results and discussion
According to equations (2.25)-(2.28), the transport coefficients are determined by the shape and value of the transport function I(ω) within the so-called Fermi window, defined by function [−d f (ω)/dω], around the Fermi level (chemical potential value) [36]. The largest thermoelectric effect can be observed when transport function is strongly asymmetric at the chemical potential, e.g., it is nonzero on the one side from the chemical potential and zero on the other side. Such case can be achieved in a lightly doped Mott insulators when the chemical potential is stuck at the band edge [30].
In the case of correlated hopping, the Mott insulator phase has some specific features. First of all, the DOS A d (ω) is strongly asymmetric and contains two Hubbard bands, the lower and the upper one, with spectral weights w 0 = 1 − n f and w 1 = n f , respectively. The Mott gap is the largest at t 2 = −0.5 value, see figure 1 (a), when t ++ = 0 and hopping between the sites with occupied f states is prohibited (here and below we put t 1 = 1 and consider the case of t 3 = 0). Besides, at half-filling, n f = 0.5, and exactly at t 2 = −0.5 value, there is a square root singularity at the lower edge of the upper Hubbard band, figure 2, which strongly effects the temperature behaviour of the chemical potential. At high temperatures, the chemical potential µ d is placed closer to the lower Hubbard band, but with the temperature decrease it starts to approach the centre of the Mott gap.
The shape of transport function I(ω) is different and it is strongly affected by the resonant peak [28] placed at the frequency ω = ω res given by Transport function displays a power law frequency dependence at the top of the lower Hubbard band [39], instead of the square root one for DOS, and an anomalous step-like feature at the bottom of the upper Hubbard band. As a result, the temperature dependences of transport coefficients, figure 3, strongly depend on the doping character and on its level. At half filling n d = 0.5 and in the Mott insulator phase, the dc charge and thermal conductivities display typical dependences for the large gap insulators with an exponential decay at β = 1/T → +∞ with divergent Lorenz number L. On the other hand, the Seebeck coefficient is negative and follows the 1/T dependence at small temperatures, which is caused by the above mentioned strong asymmetry of the transport function, power law dependence for the lower Hubbard band and step-like feature for the upper one, see, for comparison, [40]. Now, let us consider the effect of doping. For the hole doping, n d = 0.5 + n c and n c < 0, the chemical potential is placed at T = 0 somewhere at the top of the lower Hubbard band. The temperature dependences of the dc charge and thermal conductivities at high temperatures display the same behaviour as in the Mott insulator case. For low temperatures, when the chemical potential enters the lower Hubbard band, the dc charge conductivity starts to increase, as it should be for bad metal, and the thermal one displays the linear temperature dependence. The strongest doping effect is observed for the thermoelectric transport. Since now the chemical potential approaches the lower Hubbard band with the temperature decreasing faster than in the Mott insulator case, the Seebeck coefficient also increases faster until the chemical potential enters the lower Hubbard band. Then, the transport function becomes smooth within the Fermi window resulting in the lowering of the absolute value of Seebeck coefficient at low temperatures. We can notice a strong deviation of the temperature dependence of thermopower S(T) from the one typical of bad metals [31] and specific for light doping [30]. Now, S(T) exhibits almost a linear temperature dependence below the peak, which is caused by the character of the temperature dependence of the chemical potential due to the sharp features of DOS. Similar behaviour is observed for an electron doping, n c > 0, but now the Seebeck coefficient becomes positive when the chemical potential approaches the upper Hubbard band, and transport coefficients are larger in comparison with the hole doping case because the transport function is larger for the upper Hubbard band in comparison with the lower one. Moreover, the linear segment on S(T) extends and becomes flattened. The results presented above were obtained for the special case of t 2 = −0.5, when the hopping between the sites with occupied f states is zero, t ++ = 0. In figures 4-5, we present the results for the case with small values of t ++ = 0.04. Now, the edge singularity on DOS is smoothed in a narrow peak, whereas the step-like feature on the transport function is replaced by the resonant peak. Nevertheless, the above discussed features are preserved, and we observe only quantitative changes in the temperature dependences of the transport coefficient. The only prominent effect is visible for the electron doping case, n c > 0, when the step-like feature on the transport function is smoothed which leads to the faster decreasing of the Seebeck coefficient at T → 0.
Herein above we have considered the case of half-filling of f particle states, n f = 0.5. However, in our previous investigations [28], it was found that for the case of f particle doping, n f > 0.5, the third narrow band appears [figure 1 (c) and (d)] when the hopping amplitude between the sites occupied by f particle becomes very small, |t ++ | t 1 . Now, the DOS contains three bands: the lower and the upper Hubbard bands with equal spectral weights w 0 = 1 − n f and the narrow middle band with spectral 13703-8 Thermoelectric properties of Mott insulator with correlated hopping at microdoping , this middle band shrinks to a level with a δ-peak on DOS, whereas it gives no contribution to the transport function ( figure 6). Hence, the gaps on the DOS and transport function are of different width. In the cases of a pure large gap insulator, n d = 1 − n f + n c with n c = 0, at very low temperatures, T → 0, the chemical potential is placed in the centre of large gap, i.e., between the top of lower Hubbard band and δ-peak from localized states on DOS. On the other hand, these localized states do not contribute to transport coefficients, and the transport function I(ω) "feels" the larger gap between the top of the lower Hubbard The switching on of the hopping between the sites occupied by f electrons, t ++ > 0 (t 2 < −0.5), leads to the spreading of the δ-peak from localized states on DOS into the narrow band, and the narrow resonant peak appears on the transport function, figure 8. Now, the gaps on the DOS and transport function are of the same width, which restores the typical behaviour for the Mott insulator case. The transport properties, figure 9, of the hole doped Mott insulator, n d = 1 − n f + n c with n c < 0, are similar to the one discussed in the previous case of t ++ = 0, but one can notice a strong enhancement of the dc charge conductivity for the electron doping, n c > 0, when chemical potential enters the resonant peak on the transport function. Moreover, for electron doping, the Seebeck coefficient displays almost flat temperature dependence in a wide temperature range and it starts to decrease in an expected manner at T → 0.
For larger values of the hopping amplitude t ++ between the sites occupied by f electrons, the middle band of localized states joins with the upper Hubbard band. Now, we observe a huge enhancement at the bottom of the upper Hubbard band both on DOS and on the transport function due to resonant peak, figure 10. The temperature dependences of the dc charge and thermal conductivities are similar to the above discussed but the thermopower displays an anomaly for the case of Mott insulator, figure 11. Now, the Seebeck coefficient S(T) is positive at high temperatures and increases with its decreasing up to some temperature value. For lower temperatures it starts to decrease, changes its sign and rapidly increases at low temperatures.

Conclusions
In this article we have discussed the peculiarities of the charge and thermal transport in the Falicov-Kimball model with correlated hopping at a light doping of the Mott insulator phase. We consider the cases of the strongly reduced hopping amplitude t ++ between the sites occupied by the f electrons, when the DOS and transport function display anomalous features, including edge singularity, resonant peak, and additional band of localized states at n f > 0.5. At half-filling n f = 0.5 and in the Mott insulator phase, n c = 0, the dc charge and thermal conductivities display a typical behaviour for the large gap insulators with asymmetric DOS, whereas the light hole and electron doping restore the bad metallic conductivity with an enhanced thermopower for the electron doping case. Outside the half-filling case, when n f > 0.5, and for the completely reduced hopping between the sites occupied by f -electrons, the gaps on the DOS and transport function do not coincide, which causes an anomalous thermoelectric transport at low temperatures featuring a strong reduction of the Lorenz number and a huge enhancement of thermopower for the electron doping case.