**4. Results**

In this section, we discuss the thermoelectric properties within the single-level model analyzing the role of the electron–electron and electron–vibration interactions between the molecular degrees of freedom. We point out that only the combined effect of these interactions is able to provide a good agreemen<sup>t</sup> between experimental data and theoretical calculations.

The level density *ρ* is shown in the upper left panel of Figure 2, the charge conductance *G* in the upper right panel, the Seebeck coefficient *S* in the lower left panel, and the electronic thermoelectric figure of merit *ZTe<sup>l</sup>* in the lower right panel. All quantities are plotted as a function of level energy  at the temperature *T* = 100 K. For all the quantities, we first analyze the coherent regime (black solid lines in the four panels of Figure 2), which means absence of electron–electron and electron–vibration interactions. Then, we study the effect of a finite electron–vibration coupling *EP* (red dash lines in the four panels of Figure 2) focusing on the intermediate coupling regime. Finally, we consider the combined effect of electron–vibration and electron–electron interactions for all the quantities (blue dash-dot lines in the four panels of Figure 2) analyzing the experimentally relevant regime of a large Coulomb repulsion *U*.

**Figure 2.** Level density *ρ* (**Top Left**); charge conductance *G* (in units of the conductance quantum *G*0) (**Top Right**); Seebeck coefficient *S* (in units of μV/K) (**Bottom Left**); and electronic thermoelectric figure of merit *ZTe<sup>l</sup>* (**Bottom Right**) as a function of level energy  (in units of 0.030 eV) at the temperature *T* = 100 K.

The level density *ρ* per spin reported in the upper left panel of Figure 2 shows the expected decreasing behavior with increasing the level energy . The electron–vibration interaction induces a shift of the curve of about *EP*. In the presence of electron–electron interactions, the behavior is more complex. Actually, in molecular junctions, the strong Coulomb repulsion usually reduces the electronic charge fluctuations and suppresses the double occupation of the electronic levels [1]. For values of  smaller than − *U*, the density is closer to unity, while, for  larger than zero, the density vanishes. For  between − *U* and 0, there is a plateau with a value of the density close to 0.5. Indeed, these phenomena are characteristic of Coulomb blockade effects.

The conductance *G* is shown in upper right panel of Figure 2. At low temperatures, this quantity as a function of the level position  provides essentially the spectral function of the molecular level. Indeed, in the coherent low temperature regime, *G* can be directly related to the transmission with a Lorentzian profile. One of the main effects of an adiabatic oscillator is to shift the conductance peak towards positive energies proportional to the electron–vibration coupling energy *EP*. Apparently, another expected effect is the reduction of the peak amplitude. In fact, electron–vibration couplings on the molecule tend to reduce the charge conduction. As a consequence, electron–vibration couplings induces somewhat longer tails far from the resonance. These features, such as the peak narrowing, are common to other theoretical approaches treating electron–vibration interactions, among which that related to the Franck–Condon blockade [45]. Actually, in a previous paper [39], we successfully compared the results of the adiabatic approximation with those of the Franck–Condon blockade formalism in the low density limit where this latter approach becomes essentially exact [1].

We note that a finite electron–electron interaction not only suppresses the electronic conduction for small values of , but it is also responsible for a second peak centered at  − *U*. In fact, there is a transfer of spectral weight from the main peak to the interaction-induced secondary peak. We stress that these features are compatible with experimental data since conductance gap ascribed to Coulomb blockade effects have been measured in fullerene junctions [1,23].

We investigate the properties of the Seebeck coefficient *S* of the junction in the lower left panel of Figure 2. In analogy with the behavior of the conductance, the main effect of the electron–vibration interaction is to reduce the amplitude of the Seebeck coefficient. Moreover, the shift of the zeroes of *S* is governed by the coupling *EP* as that of the peaks of *G*. Therefore, with varying the level energy , if *G* reduces its amplitude, *S* increases its amplitude in absolute value, and vice versa. This behavior and the values of *S* are in agreemen<sup>t</sup> with experimental data [4,17]. In the Coulomb blockade regime, *S* shows a peculiar oscillatory behavior as a function of the energy , with several positive peaks and negative dips. The energy distance between the peaks (or the dips) is governed by the Hubbard term *U*. Even in this regime, the Seebeck coefficient *S* is negligible for the level energies where the electronic conductance presented the main peaks, that is at  0 and  − *U*. This property turns out to be a result of the strong electron–electron interaction *U* [26]. In any case, close to the resonance (zero values of the level energy ), the conductance looks more sensitive to many-body interactions, while the Seebeck coefficient appears to be more robust.

The electronic conductance *G*, Seebeck coefficient *S*, and electron thermal conductance *Ge<sup>l</sup> K* combine in giving an electronic figure of merit *ZTel*. This latter quantity is shown in lower right panel of Figure 2 at the temperature *T* = 100 K. We stress that, due to the low value of the temperature, the quantity *ZTe<sup>l</sup>* does not show values comparable with unity. However, it is interesting to analyze the effects of many-body interactions on this quantity. A finite value of the electron–vibration coupling *EP* leads to a reduction of the height of the figure of merit peaks. It is worth noting that the position of the peaks in *ZTe<sup>l</sup>* roughly coincides with the position of the peaks and dips of the Seebeck coefficient *S*. Finally, the electron–electron interactions tend to reduce the amplitude and to further shift the peaks of the figure of merit.

After the analysis of the effects of many-body interactions on the charge conductance and Seebeck coefficient, we can make a comparison with the experimental data available in [17] and shown in Figure 1. These data are plotted again in Figure 3 together with the fit discussed in Figure 1. We recall

that for fullerene junctions the one-level model discussed in the previous section is characterized by the following parameters: *E*0 − *μ* = 0.057 eV, Γ = 0.032 eV, and *α* = 0.006 eV/V. We remark that the level energy  used in the previous discussion is related to the energy *E*0 and the gate voltage *VG* through Equation (19). Therefore, once the value of *E*0 is fixed, one can switch from the energy  to the gate voltage *VG*. Before introducing many-body effects, we consider a slight shift of the level position considering the case *E*0 − *μ* = 0.065 eV reported in Figure 3. This energy shift is introduced to counteract the shifts of the peaks (conductance) or zeroes (Seebeck) introduced by many-body interactions which, in addition, reduce the amplitudes of response functions. The aim of this paper is to provide an optimal description for both charge conductance *G* and Seebeck coefficient *S*.

**Figure 3.** Charge conductance *G* (in units of the conductance quantum *G*0) (**Left**); and Seebeck coefficient *S* (in units of μV/K) (**Right**) as a function of gate voltage *VG* (in units V) at the temperature *T* = 100 K: experimental data (black circles), data fit (red solid line) corresponding to one-level model with energy *E*0 − *μ* = 0.057 eV, coherent results (blue dash line) corresponding to one-level model with energy *E*0 − *μ* = 0.065 eV, effect of the only electron–vibration coupling *EP* = 0.018 eV (magenta dash-dot line), and effect of additional electron–electron interaction *U* = 0.3 eV (orange double dash-dot line).

Starting from the level energy *E*0 − *μ* = 0.065 eV, in Figure 3, we analyze the effect of the electron–vibration coupling in the intermediate regime *EP* = 0.018 eV. The shift induced in the zero of the Seebeck coefficient is still compatible with experimental data. Moreover, the electron–vibration interaction shifts and reduces the peak of the charge conductance in an important way. However, this is still not sufficient to ge<sup>t</sup> an accurate description of the charge conductance. One could increase the value of the coupling energy *EP*, but, this way, the shift of the conductance peak becomes too large with a not marked reduction of the spectral weight.

Another ingredient is necessary to improve the description of both conductance *G* and Seebeck coefficient *S*. In our model, the additional Coulomb repulsion plays a concomitant role. Its effects poorly shift the zero of the Seebeck coefficient and slightly modifies the curve far from the zero. Therefore, the description of the Seebeck coefficient remains quite accurate as a function of the gate voltage. On the other hand, it provides a sensible reduction of the conductance amplitude with a not large shift of the peak. Hence, the effects of Hubbard term are able to improve the theoretical interpretation of the experimental data for the conductance *G* and the Seebeck coefficient *S* close to the resonance. Far from the resonance, in a wider window of gate voltages, theory predicts the existence of a secondary peak of the conductance and a complex behavior of the Seebeck coefficient due to Coulomb blockade effects. The features are not negligible as a function of the gate voltage.

As far as we know, experimental measurements of the electronic thermal conductance *Ge<sup>l</sup> K* have become only very recently available [21]. Indeed, it is important to characterize this quantity since it allows determining the thermoelectric figure of merit. Therefore, in Figure 4, we provide the theoretical prediction of the electronic thermal conductance *Ge<sup>l</sup> K* as a function of *VG* starting from the optimized

values of the one-level parameters used to describe both charge conductance and Seebeck coefficient in an accurate way. We stress that the plotted thermal conductance is expressed in terms of the thermal conductance quantum *g*0(*T*) = *<sup>π</sup>*2*k*2*BT*/(3*h*) [46]. The main point is that, in the units chosen in Figure 4, the thermal conductance *Ge<sup>l</sup> K* shows a strong resemblance with the behavior of the charge conductance *G* in units of the conductance quantum *G*0 as a function of the gate voltage *VG*. We remark that, at *T* = 100 K, *g*0(*T*) 9.456 × 10−<sup>11</sup> (W/K) 100 pW/K. The values of the thermal conductance *Ge<sup>l</sup> K* shown in Figure 4 are fractions of *g*0(*T*), therefore they are fully compatible with those estimated experimentally in hydrocarbon molecules [19] (50 pW/K).

**Figure 4.** Electronic thermal conductance *Ge<sup>l</sup> K* in units of thermal conductance quantum *g*0(*T*) (*g*0(*T*) = *<sup>π</sup>*2*k*2*BT*/(3*h*)) as a function the voltage gate *VG* in units of Volt at the temperature *T* = 100 K: coherent results (black solid line) corresponding to one-level model with energy *E*0 − *μ* = 0.065 eV, effect of the only electron–vibration coupling *EP* = 0.018 eV (red dash line), and effect of additional electron–electron interaction *U* = 0.3 eV (blue dash-dot line).
