**6. Thermodynamic Uncertainty Relation**

We now turn to the investigation of the TUR, cf. Equation (14), which provides a combined performance quantifier accounting for power output, efficiency and power fluctuations. We first consider the TUR-coefficient *Q*TUR in the linear-response regime. Together with Equation (14), we therefore use the relations for power, power fluctuations, entropy and efficiencies, given in Equations (8)–(13). The linear-response expressions for the charge and heat currents occurring in these relations are given in Equation (16) and we furthermore use *SI* = 2*k*B*T*0*G*. With this, we find

$$Q\_{\rm TUR} = \frac{(GV + L\Delta T)^2}{\Delta T(LT\_0V + K\Delta T) + VT\_0(GV + L\Delta T)} \cdot \frac{T\_0}{2G} \,. \tag{32}$$

Maximizing this expression with respect to voltage we find *V*max → ±<sup>∞</sup>, resulting in *Q*TUR = 1/2, and hence, the inequality becoming an equality. However, this voltage is within a voltage regime where power is dissipated (*P* < 0) and not produced; power production (*P* ≥ 0) would instead require *V*s ≤ *V* ≤ 0. Thus, this is not of practical relevance for the engine performance. Adding the extra condition that *P* ≥ 0 we instead find *V*max → 0. The corresponding value of *Q*TUR on the left-hand

side of Equation (32) then becomes *<sup>L</sup>*2*T*0/(2*GK*). Expressing this in terms of the figure of merit *ZT*, given in Equation (19), we can write the bound on the operationally meaningful TUR-coefficient in the linear-response regime as

$$Q\_{\rm TUR} \le \frac{1}{2} \cdot \frac{ZT}{1 + ZT}.\tag{33}$$

This shows that in the linear response, the parameters of the steady-state thermoelectric heat engine are actually subjected to a tighter bound than given by Equation (32). Please note that this bound is saturated in the limit *V* = 0, where the power production, the power fluctuations, as well as the efficiency all vanish. Also, only for ideal thermoelectrics, with *ZT* → <sup>∞</sup>, does the bound become 1/2. As seen in Figure 9d, this maximal bound is actually reached for large step energies *E*0.

**Figure 9.** Coefficient *Q*TUR 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). We note that we set the *Q*TUR to zero outside the parameter regime where power is produced. The black-dashed line in all panels corresponds to the bound that is given by Equation (33).

The full TUR-coefficient beyond linear response is illustrated in Figures 9 and 10. We find that the inequality *Q*TUR ≤ 1/2 is always respected, even though this is not guaranteed by scattering theory [45]. Interestingly, we find the tighter bound in Equation (33) to be respected for most parameters, even though the inequality is only proven to hold in the linear-response regime and the bound is expressed in terms of linear-response quantities (given by Equation (17)), only. Violations of the bound given in Equation (33) beyond linear response are observed for sufficiently low *E*0 and when the temperature difference is of the order of the magnitude of *E*0 (cf. Figure 9a for a sharp barrier). The regimes where a violation can occur are extended when the barrier is smooth (cf. Figure 9e,f). These violations agree with the general notion that dissipation increases when moving away from the linear response [39]. Furthermore, from Figure 9, we find that in the linear response, as well as for small and negative *E*0, *Q*TUR decreases monotonically as the (negative) voltage is increased. This reflects the behavior of the inverse Fano factor in Figure 7. Importantly, for sharp step energies *E*0, and beyond the linear response, *Q*TUR is a non-monotonic function of the voltage and takes on its maximum at a point where power production is finite. This non-monotonicity is a consequence of the interplay between the monotonically decreasing inverse Fano factor and the strongly increasing efficiency and power (cf. Figures 2 and 4), as the voltage is changed to more negative values.

Figure 10 shows the TUR-coefficient maximized over voltage, *Q*VTUR,max, as a function of the thermal gradient Δ*T* and the step energy *E*0. As for the inverse Fano factor, the maximization only includes the voltage window where power is non-negative. For all values of the barrier smoothness, we find that *<sup>Q</sup><sup>V</sup>*TUR,max generally decreases as a function of Δ*T*, and a closer inspection reveals small non-monotonic features related to the small violations of Equation (33). This contrasts with the maximized inverse Fano factor, which shows the opposite behavior, cf. Figure 8. The decrease of the fluctuations with Δ*T* is thus overcompensated by an increase in dissipation which results in the highest values for *Q*VTUR,max being reached in the linear-response regime. This shows that *<sup>Q</sup><sup>V</sup>*TUR,max is maximal in regimes, where the *<sup>η</sup>V*max is large. Note however that the maximal *Q*TUR is reached at zero voltage, the maximized efficiency *η* is reached close to the stopping voltage *V*S = 0. Furthermore, no features of the line of optimal power production close to *P*W can be identified in the panels of Figure 10.

**Figure 10.** TUR-coefficient maximized over the bias *V*, *Q*VTUR,max, 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>**).
