**4. Efficiency**

Taking into account the aspect of limited resources, the power output is often not the most significant performance quantifier. A more relevant quantity is then the efficiency of a device. For a heat engine, it is defined as the power output divided by the heat absorbed from the hot bath, Equation (10).

We show the efficiency of the QPC as a steady-state thermoelectric heat engine in Figure 4. Panels (a) to (d) show the efficiency for the sharp barrier as function of voltage *eV*/(*k*BΔ*T*) for different temperature differences Δ*T*/*T*0 and step energies *<sup>E</sup>*0/(*k*B*T*0). For small absolute values of the step energies, see panels (a) and (b) for two examples with *<sup>E</sup>*0/(*k*B*T*0) = −1, 0, the efficiency is rather small

with respect to the Carnot efficiency, *η*/*η*C 0.25 and its overall shape only weakly depends on the temperature difference. This is radically different for larger values of *E*0: panels (c) and (d) of Figure 4 show a strong increase of the efficiency, which for *<sup>E</sup>*0/(*k*B*T*0) = 15 and large temperature differences can reach about 90% of the Carnot efficiency. Also, the stopping voltage *V*s, at which the efficiency is zero and the device stops working as a thermoelectric, is strongly increased, depending on the temperature difference.

**Figure 4.** Efficiency as function of voltage for a sharp barrier (**<sup>a</sup>**–**d**) and for a smooth barrier, *γ* = *k*B*T* (**<sup>e</sup>**–**h**), for selected temperature differences Δ*T* (see different lines) and step energies *E*0 (see different columns).

For large *E*0, see panel (d) of Figure 4, and small temperature differences, where large maximum efficiency values are reached, the efficiency-voltage relation takes a close-to-triangular shape. In this regime, we have that *E* ± *eV*/2 *T*0, *T*0 + Δ*T* for all energies above the step energy *E*0. Therefore, only the *tails* of the Fermi functions contribute in Equations (2) and (3) and the efficiency in linear response in Δ*T* can be approximated as

$$
\eta = \frac{\varepsilon \left| V \right|}{E\_0} \theta \left( \varepsilon V + E\_0 \frac{\Delta T}{T\_0} \right) \ . \tag{29}
$$

This formula describes well the triangular shape of the curves in panel (d), including the stopping voltage at small Δ*T* and large *E*0, given by *eV*s/*k*BΔ*T* ≈ −*E*0/*k*B*T*0, from the argumen<sup>t</sup> of the Heaviside function *θ* in Equation (29). We note that for *V* → *V*s the efficiency *η* → Δ*T*/*T*0 ≈ *η*C, i.e., the efficiency approaches the Carnot limit, see Equation (11). The mechanism for this is the same as described in Ref. [20]; transport effectively takes place in a very narrow energy interval around *E*0, where the distribution functions *f*L(*<sup>E</sup>*0) ≈ *f*R(*<sup>E</sup>*0).

Panels (e) to (h) of Figure 4 show results for the changes in the efficiency for a smooth barrier, *γ* = *k*B*T*0. At temperature differences that are much larger than the smoothness—here the case for *k*BΔ*T*/*γ* = 5, 15—the results for the efficiency are very similar to the case of the sharp barrier. This agrees with the discussion on the power production in the previous section, Section 3. At small temperature differences, however, the efficiency gets strongly reduced by the effect of the smoothness. This is particularly striking for large step energies, see panels (g)–(h) for *E*0/*k*B*T*0 = 5, 15, respectively. Here, efficiencies that were close to Carnot efficiency for a sharp barrier ge<sup>t</sup> reduced by a factor three due to the barrier smoothness. The reason is that increasing smoothness leads to a broadening of the

energy interval where the transport takes place, and hence a breakdown of the mechanism for Carnot efficiency discussed in Ref. [20].
