**3. Power Production**

To characterize the performance of the engine, we first consider the power *P* produced. The power as a function of applied bias *V*, for different values of the step energy *E*0 and temperature difference Δ*T*, is shown in Figure 2 for both sharp (*γ* → 0) and smooth (*γ* = *k*B*T*0) transmission step.

**Figure 2.** Power *P*, normalized by the power bound *P*W, defined in Equation (23), as a function of applied bias *eV*/(*k*BΔ*T*) for a set of step energies *<sup>E</sup>*0/(*k*B*T*0), shown in different columns, and thermal bias values Δ*T*/*T*0, represented by different styles of lines (the same for all panels, see inset in (**g**)). Panels (**<sup>a</sup>**–**d**) [(**<sup>e</sup>**–**h**)] correspond to transmission functions with a step smoothness of *γ* → 0 [*γ* = *k*B*T*0].

As is seen from the figure, a common feature for all P-vs-V curves is that they first increase monotonically from *P* = 0 at *V* = 0 with increasing negative voltage. At some voltage *V*max the power reaches its maximal value, *P*Vmax, and then decreases monotonically to zero, reached at the stopping voltage *V*s. The maximum power with respect to voltage is a function of *<sup>E</sup>*0/(*k*B*T*0), Δ*T*/*T*0 and *<sup>γ</sup>*/(*k*B*T*0), i.e.,

$$P\_{\text{max}}^{\text{V}} = P\_{\text{max}}^{\text{V}} \left( \frac{E\_0}{k\_{\text{B}} T\_0}, \frac{\Delta T}{T\_0}, \frac{\gamma}{k\_{\text{B}} T\_0} \right). \tag{20}$$

In addition, we note that in the linear-response regime we have *P*Vmax = [*L*2/(4*G*)]Δ*T*<sup>2</sup> with *V*max = *V*s/2 = <sup>−</sup>(*L*/[2*<sup>G</sup>*])<sup>Δ</sup>*T*. From Figure 2, it is clear that the power as a function of voltage depends strongly on all parameters *<sup>E</sup>*0/(*k*B*T*0), Δ*T*/*T*0 and *<sup>γ</sup>*/(*k*B*T*0). In particular, going from the linear to the non-linear regime by increasing Δ*T*/*T*0, the maximum power *P*Vmax might increase or decrease depending on the step properties *γ* and *E*0.
