*4.1. Maximum Efficiency*

We now focus our study on the maximum value of the efficiency that can be reached over the whole range of voltages, *η*Vmax, a function of *<sup>E</sup>*0/(*k*B*T*0), Δ*T*/*T*0, *<sup>γ</sup>*/(*k*B*T*0). The results of this maximization procedure are shown in Figure 5, where panel (a) corresponds to a sharp barrier (*γ* = 0) while smooth barriers with *γ* = *k*B*T*0 and *γ* = 3*k*B*T*0 are presented in panels (b) and (c), respectively.

**Figure 5.** Density plots of the maximum efficiency *η*Vmax (that is, maximized over the voltage *V*) as a function of temperature difference Δ*T* and step energy *E*0, for three different values of barrier smoothness, *γ*/*k*B*T*0 = 0, 1, 3 (**<sup>a</sup>**–**<sup>c</sup>**). Please note that in panel (**c**) the axes ranges are strongly enlarged.

Two important results can be immediately seen from these density plots of the efficiency *η*Vmax as a function of temperature difference Δ*T*/*T*0 and step energy *<sup>E</sup>*0/(*k*B*T*0). First, we confirm the observations about the response to small temperature differences made from Figure 4. While for a sharp barrier, efficiencies close to Carnot efficiency are reached in the linear response (close to the stopping voltage, as we know from Figure 4), for even only slightly smoothed barriers this is not the case anymore. For *γ*/*k*B*T*0 = 3, the maximum efficiency in the linear response is even suppressed down towards zero. This clearly shows that whenever the barrier step is not truly sharp, *non-linear response* is required to ge<sup>t</sup> a thermoelectric response with large power output and with high efficiency. Second, panels (b) and (c) of Figure 5 show that for temperature differences much larger than the smoothness—or, in other words, with one of the reservoir temperatures being much larger than the smoothness—almost the same (large) efficiency as in the sharp-barrier case is found, as long as the step energy is sufficiently large. Note, however, that these large-efficiency regions are *far* from those regions, which were previously identified as the ones of large power output, and are furthermore limited to regions with very large temperature differences and step energies.
