*3.1. Maximum Power*

To further analyze the properties of *P*Vmax, we first recall from the seminal work of Whitney [14,15,54] that the power is bounded from above by quantum mechanical constraints. It was shown that the upper bound is reached for a QPC with a sharp step, *γ* → 0, for which, using Equations (2) and (8), the power becomes

$$P\_{\rm sharp} = -\frac{(k\_{\rm B}T\_0)^2}{h} \cdot \frac{eV}{k\_{\rm B}T\_0} \left\{ \frac{eV}{k\_{\rm B}T\_0} - \left(1 + \frac{\Delta T}{T\_0}\right) \ln\left[f\_{\rm R}(E\_0)\right] + \ln\left[f\_{\rm L}(E\_0)\right] \right\}.\tag{21}$$

Maximizing this expression with respect to *eV*/(*k*B*T*0) and *<sup>E</sup>*0/(*k*B*T*0) we find that the maximizing voltage is given by *eV*max = −*ξk*BΔ *T* where *ξ* ≈ 1.14 is the solution of ln(1 + *e*<sup>−</sup>*ξ* ) = −*ξe*<sup>−</sup>*ξ*/(<sup>1</sup> + *e*<sup>−</sup>*ξ* ) [5]. Moreover, the maximizing step energy *E*0,max and temperature difference Δ *T*max are related via [14,28]

$$\frac{E\_{0,\text{max}}}{k\_{\text{B}}T\_0} = \zeta \left( 1 + \frac{\Delta T\_{\text{max}}}{2T\_0} \right). \tag{22}$$

Inserting this expression, together with the relation for the maximizing voltage, into Equation (21) we reach the upper bound for the power established by Whitney [54] and related to the Pendry bound [55],

$$P\_W = -\frac{(k\_\text{B} \Delta T)^2}{h} \zeta \ln\left(1 + e^{\varepsilon}\right) \approx 0.32 \,\frac{(k\_\text{B} \Delta T)^2}{h},\tag{23}$$

which, we emphasize, holds in the linear as well as in the non-linear regime. To relate to this upper bound, in Figure 3a–c we present a set of density plots of *P*Vmax as a function of *<sup>E</sup>*0/(*k*B*T*0) and Δ *T*/*T*0 for different values of step smoothness parameters *γ*.

**Figure 3.** Maximum power with respect to voltage, *P*Vmax, as a function of *<sup>E</sup>*0/(*k*B*T*0) and Δ*T*/*T*0 presented for three different values of the step smoothness *<sup>γ</sup>*/(*k*B*T*0) = 0, 1, 3 (**<sup>a</sup>**–**<sup>c</sup>**). The white dashed lines in (**<sup>a</sup>**,**f**) illustrate Equation (22). (**d**,**<sup>e</sup>**) show close-ups of regions in (**<sup>a</sup>**,**b**), respectively, indicated with yellow dotted rectangles. On the other hand, (**f**) corresponds to an extended parameter regime of (**c**).

From the figure it is clear that for a sharp step, *γ* → 0, there is a broad range of *<sup>E</sup>*0/(*k*B*T*0) and Δ *T*/*T*0 around the dashed line in the (*<sup>E</sup>*0, Δ *<sup>T</sup>*)-space, given by Equation (22), for which *P*Vmax is close to the theoretical maximum value *P*W. For a step smoothness up to *γ* ∼ *k*B*T*0, the situation changes only noticeably for small Δ *T*/*T*0. This is illustrated clearly in the close-ups in Figure 3d,e. Increasing the smoothness further, the region with maximum power close to *P*W shifts to higher values *E*0 and Δ *T*, although still largely centered around Equation (22), as is shown in Figure 3f.

To provide a more quantitative analysis of this behavior, below we investigate two limiting cases for *γ* in further detail.

### 3.1.1. Small Smoothness Parameter *<sup>γ</sup>*/(*k*B*T*0) 1

In the limit, where the value of the smoothness parameter *γ* is small, *<sup>γ</sup>*/(*k*B*T*0) 1, the expression for the transmission probability in Equation (1) can be expanded to leading order in *γ* as [56]

$$D(E) = \theta(E - E\_0) + \gamma^2 \frac{\pi^2}{6} \cdot \frac{d}{dE} \delta(E - E\_0). \tag{24}$$

Inserting this into the expression for the charge current, Equation (2), and performing a partial integration for the delta function derivative, we ge<sup>t</sup> the power

$$P = P\_{\text{sharp}} - \gamma^2 \frac{eV}{h} \cdot \frac{\pi^2}{6} \cdot \frac{d}{dE\_0} \left[ f\_\text{L}(E\_0) - f\_\text{R}(E\_0) \right],\tag{25}$$

with *<sup>P</sup>*sharp given in Equation (21). To estimate how the overall maximum power is modified due to finite smoothness we insert into Equation (25) the values for *eV*/(*k*B*T*0), *<sup>E</sup>*0/(*k*B*T*0) and Δ*T*/*T*0 along the line in the (*<sup>E</sup>*0, <sup>Δ</sup>*T*)-space, see Figure 3, which gives the bounded power for the sharp barrier. We find

$$P(E\_{0, \text{max}}, V\_{\text{max}}) = P\_W \left\{ 1 - 1.06 \left( \frac{\gamma}{k\_B T\_0} \right)^2 \frac{1}{1 + \Delta T / T\_0} \right\},\tag{26}$$

noting that *<sup>E</sup>*0,max and Δ*T* are related via Equation (22). This expression quantifies the effect of the barrier smoothness visible in Figure 3, namely that the maximum power *P*Vmax in the region along the line in the (*<sup>E</sup>*0, Δ*T*) plane defined by Equation (22) is mainly affected for small Δ*T*/*T*0, and approaches *P*W in the strongly non-linear regime, Δ*T*/*T*0 1.
