*2.2. Non-Linear-Transport Theory*

The transport properties of the system are described by scattering theory [21]. In the non-lineartransport regime, the scattering properties of the QPC become dependent on the applied voltages *V*L and *V*R of the reservoirs and *V*g of the QPC-defining split gate [50] and possibly also on the temperature bias [5,35,51]. Since the details of this dependence will not be of importance for our analysis, we for simplicity consider a basic model with the QPC-potential capacitively coupled with equal strength, *C*L = *C*R = *C*, to the two terminals L and R. This leads to a modification of the transmission probability as *D*(*E*) → *D*(*E* + *<sup>e</sup>*[(*<sup>V</sup>*R + *<sup>V</sup>*L)*<sup>C</sup>* + *<sup>V</sup>*g*C*g]/[2*<sup>C</sup>* + *<sup>C</sup>*g]), where *C*g is the split gate-QPC capacitance. In the following, we absorb the gate potential dependence into the step energy *E*0 + *eC*g*V*g/(2*<sup>C</sup>* + *<sup>C</sup>*g) → *<sup>E</sup>*0(*Vg*) ≡ *E*0. This modification of the transmission probability guarantees a gauge-invariant formulation of the problem with observable quantities only dependent on the potential differences *V* = *V*L − *V*R, *V*L − *V*g and *V*R − *<sup>V</sup>*g. We here refrain from including the effect of a large temperature difference in the treatment of the transmission probability *<sup>D</sup>*(*E*), which is not required by fundamental principles such as gauge invariance and which has been little addressed so far, and postpone its study to future work.

For the study of the average currents of interest, namely charge current, *I<sup>α</sup>*, and heat current, *J<sup>α</sup>*, we now consider a symmetric biasing *V*L = −*V*<sup>R</sup> = *V*/2. We can then write the average currents that are flowing out of reservoir *α* as

$$I\_a = -\frac{e}{h} \int\_{-\infty}^{\infty} dE \, D(E) \left[ f\_a(E) - f\_{\overline{\mathbb{1}}}(E) \right],\tag{2}$$

and

$$J\_{\mathfrak{a}} = \frac{1}{\hbar} \int\_{-\infty}^{\infty} dE \,(E + \mathfrak{r}\_{\mathfrak{a}} eV/2) D(E) \left[ f\_{\mathfrak{a}}(E) - f\_{\mathfrak{F}}(E) \right] \tag{3}$$

Here, *α* should be understood as follows: L = R and R = L, whereas *τ*L = 1 and *τ*R = −1. In Equations (2) and (3), we have introduced the Fermi distribution functions *fα*(*E*),

$$f\_a(E) = \left[1 + \exp\left(\frac{E + \tau\_a cV/2}{k\_B T\_a}\right)\right]^{-1} \quad \text{for} \quad a = L, \text{R}.\tag{4}$$

While current conservation ensures *I*L = −*I*<sup>R</sup> ≡ *I*, energy conservation results in *J*L = −*J*<sup>R</sup> − *IV*.

To analyze the fluctuations in the system we also need the zero-frequency charge-current noise, given by [52]

$$S\_I = \frac{\epsilon^2}{h} \int\_{-\infty}^{\infty} dE \left\{ D(E) \left[ f\_\mathcal{L}(E) \left( 1 - f\_\mathcal{L}(E) \right) + f\_\mathcal{R}(E) \left( 1 - f\_\mathcal{R}(E) \right) \right] \right. \tag{5}$$

$$+ D(E) \left[ 1 - D(E) \right] \left[ f\_\mathcal{L}(E) - f\_\mathcal{R}(E) \right]^2 \}.$$

In addition to the study of the noise, it is often convenient to analyze the Fano factor

$$F = \frac{S\_I}{|2xI|} \,' \tag{6}$$

being a measure of how much the noise deviates from the one of Poissonian statistics (for which *F* = 1).

### *2.3. Thermodynamic Laws and Performance Quantifiers*

The laws of thermodynamics set very general constraints on the quantities introduced above and on the performance quantifiers, which we are going to study in this paper. We describe these quantities within scattering theory, known to correctly reproduce the laws of thermodynamics [5]. The first law of thermodynamics guarantees energy conservation and can be written as

$$J\_L + J\_R = P.\tag{7}$$

Here, we have introduced the electrical power produced,

$$P = -VI,\tag{8}$$

where −*V I* > 0 if the current flows against the applied bias. Please note that throughout the work, we limit our analysis of performance quantifiers to the relevant regime of positive power production. The second law of thermodynamics states that the entropy production *σ* is non-negative. In our two-terminal geometry, it can be written as

$$
\sigma = -\frac{f\_L}{T\_L} - \frac{f\_R}{T\_R} \ge 0.\tag{9}
$$

This expression determines the direction of energy flows through the system. It equals zero in case that a process is *reversible*.

To determine the performance of the QPC as a heat engine, we now consider three independent quantities and combine them with each other. The first performance quantifier is given by the electrical power, Equation (8), which following the first law, Equation (7), is fully produced from heat.

The second performance quantifier we consider is given by the efficiency

$$
\eta = \frac{P}{J\_{\text{R}}} = -\frac{VI}{J\_{\text{R}}} \,\tag{10}
$$

where *J*R is the heat current that flows out of the hot reservoir. As long as power is positive, the efficiency is bounded by the second law of thermodynamics, Equation (9),

$$0 \le \eta \le \eta\_{\mathbb{C}} \quad \text{with} \quad \eta\_{\mathbb{C}} = 1 - \frac{T\_L}{T\_R} = \frac{\Delta T}{T\_0 + \Delta T},\tag{11}$$

where *η*C denotes the Carnot efficiency. The dissipation arising from an inefficient heat to work conversion is quantified by the entropy, which thereby relates efficiency and produced electrical power to each other

$$
\sigma = \frac{P}{T\_0} \cdot \frac{\eta\_\mathbb{C} - \eta}{\eta}. \tag{12}
$$

It is desirable to have a thermoelectric heat engine which not only produces large power, at high efficiency, but also minimizes fluctuations. The third independent performance quantifier is therefore provided by the low-frequency power fluctuations

$$S\_P = V^2 S\_I.\tag{13}$$

Interestingly, a trade-off between these quantities in the form of a TUR usually exists, as discussed in more detail in Section 6. This trade-off is typically written in the form of [38,39]

$$Q\_{\rm TUR} \equiv \frac{I^2}{S\_I} \cdot \frac{k\_\rm B}{\sigma} \le \frac{1}{2} \,\tag{14}$$

where we have introduced the coefficient *Q*TUR. While this inequality is not always fulfilled for systems well described by scattering theory, see e.g., the discussion in [45,53], we find it to be respected in our system for all parameter values. Importantly this coefficient can be cast into the form [42]

$$Q\_{\rm TUR} = P \frac{\eta}{\eta\_c - \eta} \cdot \frac{k\_{\rm B} T\_0}{S\_P} \, ^\circ \tag{15}$$

where we used Equations (12) and (13). Thus, under the constraint of positive power production and efficiency, we identify *Q*TUR as a convenient combined performance quantifier, accounting for power production, efficiency and power fluctuations together, where 1/2 sets the optimum value.
