**4. Thermoelectric Transport**

Until now we showed results for the charge transport driven by an electric bias of the leads due to different chemical potentials. The GME formalism allows also, in a straightforward way, the presence of a temperature bias. Instead of different chemical potentials in the left and right leads, *μ<sup>L</sup>*,*R*, one can easily consider different temperatures, *TL*,*R*, and calculate the resulting currents after switching on the contacts between the leads on the central system. Notice that, like in the case of an electric bias, there is no requirement that the temperature bias is small, such that the nonlinear thermoelectric regime is directly accessible [69]. In addition, since the Coulomb interaction between electrons in the central system is already incorporated via the Fock space, the GME allows the inclusion of Coulomb blocking and other electron correlation effects in the thermoelectric transport [70,71].

The thermoelectric transport at nanoscale is a reach and active topic within the context of the modern quantum thermodynamics, partly motivated by novel ideas on the conversion of wasted heat into electricity, and partly by the characterization of nanoscale system by methods complementary to pure electric transport [72]. For example, an effect specific to nanosystems is the sign change of the thermoelectric current or voltage when the electronic energy spectrum consists of discrete levels. This effect was predicted in the early 90' [73] and detected experimentally for quantum dots [74–76] and molecules [77]. This means that thermoelectric current in a nanoelectronic system may flow from the hotter contact to the colder one, but also from the colder to the hotter, although the second possibility might appear counter-intuitive. A simple explanation of this sign change of the current is that in a nanoscale system with discrete resonances the current can be seen as having two components, one carried by populated states above the Fermi energy, and another one carried by depopulated states below it. By analogy with a semiconductor, the former states correspond to electrons in the conduction band and the later states to holes in the valence band. Whereas an electric bias drives the electric currents due to particles and holes in the same direction, such that they always add up, a thermal bias drives them in opposite directions, such that the net current is their difference, which can be positive, negative, or zero.

We can describe this effect with the GME, first assuming a simple model with unidimensional and discretized leads, and just a single site in between them as central system. By using the Markov approximation one can show analytically that the current in the leads, in the steady state, are obtained as [71]

$$f\_{L,R} = \frac{1}{\tau^2} \frac{V\_L^2 V\_R^2}{V\_L^2 + V\_R^2} \left[ f\_L(E) - f\_R(E) \right] \,, \tag{51}$$

where*VL*,*<sup>R</sup>* are the coupling parameters of the leads with the central site, *τ* is the hopping energy on the leads, and *E* is the energy of the central site. We see that the sign of the current depends on the difference between the Fermi energies in the leads at the resonance energy,

$$f\_l(E) = \frac{1}{e^{(E-\mu\_l)/k\_B T\_l} + 1}, \ l = L, R\ . \tag{52}$$

Thus, in the presence of a thermal bias, say *TL* > *TR*, but in the absence of an electric bias, i.e., *μL* = *μ<sup>R</sup>*, the current is zero and changes sign around *μl* = *E*. In addition, the current may also vanish if the chemical potential in the leads is sufficiently far from the resonance such that the two Fermi functions are both close to zero or one. Which means that if the central system has more resonant energies the current may also change sign when *μl* is somewhere between two of them.

In Figure 7 we show an example of thermoelectric currents calculated with the GME, using the same model as in Section 3. The lowest single-particle levels having energies *ε*1 = 0.375 meV and *ε*2 = 3.37 meV are followed by the two-particle singlet state with *Es* = 5.39 meV and triplet with *Et* = 5.62 meV, and then by another excited two-body state with zero spin with energy *Ex* = 10.5 meV. We consider temperatures *kBTL* = 0.5 meV and *kBTR* = 0.05 meV in the left and right lead, respectively (or *TL* = 5.8 K and *TR* = 0.58 K), and equal chemical potentials. In Figure 7a one can see the time dependence of the currents in the leads after they are coupled to the central system, for two values of the chemical potentials, 4.8 meV and 5.4 meV, selected on each side of the singlet state. Compared to the results shown in Section 3 here we increased the coupling parameters between the leads and the central system 1.4 times, such that the steady state is reached sooner.

**Figure 7.** (**a**) The time evolution of the currents in the left and right leads, *JL*,*R*, driven by a temperature bias where *TL* = 5.8 K and *TR* = 0.58 K. With red color the results for the chemical potential *μL* = *μR* = 48 meV, and with blue color for *μL* = *μR* = 54 meV. In the steady state the currents have opposite sign; (**b**) The current in the steady state for two different temperatures of the left lead, *TL* = 5.8 K (red) and *TL* = 11.5 K (blue), for variable chemical potentials *μL* = *μR*.

As predicted by Equation (51), the currents in the steady state have opposite sign. But in fact, as shown by the red curve of Figure 7b, here we do not resolve the energy interval between the singlet and triplet states with *kBTL* > *Et* − *Es* = 0.23 meV, such that we obtain one single (common) sign change for these two levels (or "resonances"). Next, by increasing the chemical potential within the larger gap between *Es* and *Ex* the current in the steady state approaches zero and changes sign again, for *μl* ≈ 7.0 meV, and for *μl* ≈ 7.8 meV when the temperature of the hot lead is doubled, *TL* = 11.5 K.

By varying the chemical potential below the singlet energy *Es* we obtain a similar decreasing trend of the current, except that now there is no sign change close to the energy *ε*2 = 3.37 meV, but only a succession of minima and maxima. The reason is the level broadening due to the coupling of the central system with the leads [71]. Still, from such data one can observe experimentally the charging energy, as the interval between consecutive maxima, or minima, or mid points between them [77].

In the present review we show only the thermoelectric current, which corresponds to a short-circuit experimental setup, i.e., a circuit without a load. To obtain a voltage with the GME method one has to simulate a load by considering also a chemical potential bias. Thus, one can obtain the open-circuit voltage, which corresponds to that electric bias *μR* − *μL* which totally suppresses the thermoelectric current, or the complete I-V characteristic of the "thermoelectric device". Interestingly, the sign change of the thermoelectric current or voltage can also be obtained by increasing the temperature of the hot lead, while keeping the other lead as cold as possible [70,78–80].

A novel example of sign reversal of the thermoelectric current has been recently predicted in tubular nanowires, either with a core-shell structure or made of a topological insulator material, in the presence of a transversal magnetic field [81]. In this case the energy spectra are continuous, but organized in subbands which are nonmonotonic functions of the wavevector along the nanowire, yielding a transmission function nonmonotonic with the energy, and the reversal of the thermoelectric current, even in the presence of moderate perturbations [82,83].
