**1. Introduction**

Nanoscale thermodynamics has attracted considerable attention during the last three decades. Key motivations are the prospect of on-chip cooling and power production as well as an enhanced thermoelectric performance arising from unique properties of nanoscale systems, such as quantum size effects and strongly energy-dependent transport properties [1–9]. Among various nanoscale systems, quantum point contacts (QPC) [10] are arguably the simplest devices which show a thermoelectric response [11]. A requirement of such a response is an energy-dependent transmission probability [12,13], which breaks the electron-hole symmetry. Within non-interacting scattering theory, the transmission probability fully determines the thermoelectric response of a two-terminal device. The QPC and similar devices provide a particularly interesting thermoelectric platform as their transmission probability may approximate a step function, maximizing the power generation [14,15]. This feature is in contrast to the case of a quantum dot, where the transmission probability may approximate a Dirac delta distribution, maximizing the efficiency of heat-to-power conversion [16–20].

Most previous studies on the thermoelectric properties of QPCs focused on the linear response regime [11,12,21–24]. In this regime, the optimal performance of thermodynamic devices was extensively investigated, especially the efficiency at maximum power which is limited by the Curzon-Ahlborn efficiency [25–27]. There are however several works considering various aspects of the thermoelectric response in the non-linear regime [14,15,28–37]. This includes a Landauer–Büttiker scattering approach to the weakly non-linear regime [35,37], detailed investigations of the relation between power and efficiency when operating the QPC as a heat engine or refrigerator [14,15,36,37] as well as the full statistics of efficiency fluctuations [28].

Here, we review the thermoelectric effect of a QPC acting as a steady-state thermoelectric heat engine. We focus on the non-linear-response regime and analyze the output power and the efficiency for different parameter regimes, varying the smoothness of the step in the transmission probability of the QPC. In addition to a high efficiency and power production, it is desirable to have as little fluctuations as possible in the output of a heat engine. However, these three quantities, which we will analyze as three independent performance quantifiers, are often restricted by a thermodynamic uncertainty relation (TUR), preventing the design of an efficient and powerful heat engine with little fluctuations [38–44]. In this paper, we use a TUR-related coefficient as an additional combined performance quantifier, accounting for power output, efficiency, and fluctuations together. While TURs have been rigorously proven for time-homogeneous Markov jump processes with local detailed balance [39,41], they are not necessarily fulfilled in systems well described by scattering theory [45]. Nevertheless, we find the TUR to be valid in a temperature- and voltage-biased QPC. We note that recently, it has been shown that a weaker, generalized TUR applies whenever a fluctuation theorem holds [46,47]. Here, we show further constraints on the TUR under the restriction that the thermoelectric element *produces* power, necessary to define a useful performance quantifier. Interestingly, in linear response, this constraint can be related to the figure of merit, *ZT*.

This paper is structured as follows. In Section 2, we introduce the model of a QPC with smooth energy-dependent transmission, as well as the transport quantities and resulting performance quantifiers of interest. The latter are then analyzed for the QPC with different degrees of smoothness of the transmission function, namely the output power in Section 3, the efficiency in Section 4, the (power) fluctuations in Section 5, and the combined performance quantifier deduced from the TUR in Section 6.

### **2. Model System and Transport Theory**

We consider the two-terminal setup shown in Figure 1, with a single-mode QPC connected to a left (L) and a right (R) electronic reservoir, characterized by electrochemical potentials *μ*L = *μ*0 − *eV*L and *μ*R = *μ*0 − *eV*R, and kept at temperatures *T*L = *T*0 (cold reservoir) and *T*R = *T*0 + Δ *T* (hot reservoir), respectively. Here, *V*L and *V*R are externally applied voltages, *μ*0 denotes the electrochemical potential in the absence of voltage bias, *T*0 corresponds to the background temperature and Δ *T* ≥ 0 stands for the temperature difference due to heating of the right reservoir. In the following, we always set *μ*0 as the reference energy.
