*4.2. Power-Efficiency Relations*

The relation between power and efficiency for a sharp barrier, *γ* = 0, was investigated in detail in Refs. [14,15,28,36]. A convenient way to present the efficiency at a given power output, and vice versa, is in the form of lasso diagrams, as shown in Figure 6.

**Figure 6.** So-called lasso diagrams, showing the efficiency at every power output. The parameter that is changed along the lasso-line is the applied voltage *V*. We show results for a sharp barrier (**<sup>a</sup>**–**d**) and for a barrier with smoothness *γ* = *k*B*T*0 (**<sup>e</sup>**–**h**), for selected values of the step energy *<sup>E</sup>*0/(*k*B*T*0) (see different columns) and temperature differences Δ*T*/*T*0 (see different lines), in analogy to Figures 2 and 4.

At small step energies, *<sup>E</sup>*0/(*k*B*T*0) = −1, 0, the maximum power as well as the maximum efficiency are relatively small. However, maximum efficiency and maximum output power basically happen at the same parameter values. This is advantageous for operation of a thermoelectric device, where one typically must decide whether to optimize the engine operation with respect to efficiency or power output.

This trend continues also for larger step energies, see panels (c) and (d) of Figure 6, as long as the temperature difference is larger or of the order of the step energy, *k*BΔ*T E*0 (meaning that *T*0 + Δ*T* > *E*0/*k*B > *T*0). In this case, the power output is close to its maximum value *P* ≈ *P*W, while the efficiency still takes values of up to the order of *η* ≈ 0.6 *η*C, in agreemen<sup>t</sup> with the bounds discussed in Refs. [14,15,28]. These results clearly show the promising opportunities of step-shaped energy-dependent transmissions, as they can possibly be realized in QPCs, for thermoelectric power production.

Please note that the impressively large values for the efficiency at maximum output power do not, however, violate the Curzon–Ahlbohrn [25] bound, *η*CA, which relates to the Carnot efficiency as

$$\eta\_{\rm CA} = \frac{\eta\_{\rm C}}{1 + \left(1 + \Delta T / T\_0\right)^{-1/2}} \ . \tag{30}$$

This predicts a bound on the efficiency at maximum power of *η*CA = 0.5 *η*C in linear response in Δ*T*. That this bound is respected, can for example be verified by noting that the efficiency at maximum power of the grey solid line for Δ*T*/*T*0 = 0.01 in panel (c) is only slightly above 0.4*η*C. Equally, one can check from the green dashed-dotted line in the same panel that the efficiency at maximum power does not exceed the bound for Δ*T*/*T*0 = 5 given by *η*CA = 0.7 *η*C.

For step energies that are large with respect to the temperature of *both* reservoirs, *T*0, *T*0 + Δ*T* < *E*0/*k*B, the power output is reduced, the maximum efficiency, however, increases. In the limit of linear response in the temperature difference, efficiencies close to Carnot efficiency are reached at the expense of close-to-zero power output.

### **5. Power Fluctuations and Inverse Fano Factor**

During recent years it has become clear that in addition to the power and the efficiency as performance indicators of a heat engine, the fluctuations of the power output, *SP*, should also be considered [40]. A reliable operation of the heat engine, i.e., where fluctuations are limited, is desirable. This is particularly relevant for nanoscale devices, where fluctuations are always a sizable effect. To analyze the effect of power fluctuations, we note that the relevant fluctuations in this QPC steady-state thermoelectric heat engine are the charge-current fluctuations, since *SP* = *<sup>V</sup>*<sup>2</sup>*SI*. Therefore, we shift the analysis of power fluctuations to the more straightforward analysis of the Fano factor, see Equation (6).

In Figure 7, we plot the inverse Fano factor 1/*F* as a function of voltage *eV*/(*k*B<sup>Δ</sup> *T*) for different barrier smoothness *γ*, thermal gradients Δ *T*/*T*0, and step energies *<sup>E</sup>*0/(*k*B*T*0). Please note that we set the inverse Fano factor to zero outside the parameter range where power is produced, to be able to use it as a performance quantifier. This performance quantifier 1/*F* is desired to be large, meaning that current fluctuations are small with respect to the average. For all parameters, we find that increasing the (negative) voltage decreases the inverse Fano factor 1/*F* (meaning that the Fano factor *F* increases). This behavior is attributed to the decrease in charge current as the voltage is moved closer to the stopping voltage *V*s, while the total noise is less affected. For small voltages, as well as small and negative step energies, increasing the thermal gradient generally increases the inverse Fano factor. These results can be understood from the linear-response expression for the currents and noise, Equations (16)–(18) and below, giving the inverse Fano factor

$$\frac{1}{F} = \left| \frac{\varepsilon V}{k\_{\rm B} T\_0} + \frac{\varepsilon L}{k\_{\rm B} G} \cdot \frac{\Delta T}{T\_0} \right| \,, \tag{31}$$

where the absolute value can be omitted when focusing on the voltage window in which power is produced. This expression increases with Δ *T* and decreases as *V* goes to more negative values. Increasing Δ *T* thus increases the current without an accompanied increase in fluctuations because *SI* = 2*k*B*T*0*G* is independent of the bias in the linear response. For large step energies *E*0, the inverse Fano factor no longer increases monotonically in Δ *T* but a non-monotonic behavior is observed, indicating a more subtle interplay between the fluctuations and the mean value of the current. We note that for almost all parameter values, the inverse Fano factor is substantially smaller than one which can be attributed to the relatively large thermal noise in the present system.

In Figure 8, the inverse Fano factor maximized over the voltage, (1/*F*) V max, is shown for the same parameters as used in Figures 3 and 5. Please note that the maximization only includes the voltage window where positive electrical power is produced. We find that for all three values of smoothness, the maximum inverse Fano factor increases monotonically with increasing Δ *T*, saturating at values a bit above unity. The Fano factor is thus slightly below unity, a signature of almost uncorrelated, close-to Poissonian, charge transfer (for Poissonian statistics, *F* = 1). At small Δ *T T*0, close to equilibrium, the noise is large even though the average electrical current is small. As noted above, this is purely due to thermal fluctuations, resulting in a small inverse Fano factor.

**Figure 7.** Inverse Fano factor as a function of voltage for sharp barrier (**<sup>a</sup>**–**d**) and for smooth barrier (**<sup>e</sup>**–**h**), for selected gradients Δ*T* (see different lines) and step energies *E*0 (see different columns). Please note that we set the inverse Fano factor to zero outside the parameter regime where power is produced.

**Figure 8.** The inverse Fano factor maximized over all those bias values *V* leading to a non-negative output power, (1/*F*)Vmax, 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>**).
