Most Efficient Quantum Thermoelectric at Finite Power Output

Abstract

Machines are only Carnot efficient if they are reversible, but then their power output is vanishingly small. Here we ask, what is the maximum efficiency of an irreversible device with finite power output? We use a nonlinear scattering theory to answer this question for thermoelectric quantum systems; heat engines or refrigerators consisting of nanostructures or molecules that exhibit a Peltier effect. We find that quantum mechanics places an upper bound on both power output, and on the efficiency at any finite power. The upper bound on efficiency equals Carnot efficiency at zero power output, but decays with increasing power output. It is intrinsically quantum (wavelength dependent), unlike Carnot efficiency. This maximum efficiency occurs when the system lets through all particles in a certain energy window, but none at other energies. A physical implementation of this is discussed, as is the suppression of efficiency by a phonon heat flow. [6,7]. It places fundamental bounds on the efficiency and power output of heat engines and refrigerators made from such systems, such as Carnot's thermodynamic bound on efficiency or Pendry's quantum bound on entropy flow [8].The efficiencies of heat engines, η eng , and refrigerators, η fri , are particularly important (η fri is called the coefficient of performance, COP). These efficiencies are the ratio of power output to power input. For a heat engine, the output is the electrical power, P gen , and the input is the heat flow out of a reservoir (the left (L) reservoir in Fig. 1c), J L . For a refrigerator, it is the inverse. For left (L) and right (R) reservoirs at temperatures T L and T R , Carnot's bounds on these efficiencies are

Implementation of transmission functions for an optimized three-terminal quantum dot heat engine

“…I, the optimal performance of the heat engine is achieved for a transmission function of rectangular shape due to its perfect energy filtering property. 22,23 Quantum dots with a Lorentzian transmission function are clearly suboptimal in this respect. In the following, we investigate alternative realizations based on small arrays of coupled quantum dots, in order to get as close as possible to the optimum.…”

“…of Eq. (22). For strictly one-dimensional systems without loops, one may absorb the sign of the hopping parameters in the definition of the local states of the chain without affecting T (E).…”

“…From the properties of the Landauer integrals (1) and (2) it has been argued 22,23 that the optimal transmission function is rectangular-shaped with T (E) = 1 inside a certain energy window, and zero outside. A second and less obvious question is how this ideal transmission function can be practically implemented using tunnel-coupled quantum dots as building blocks.…”

Thermoelectric energy harvesting with quantum dots

“…An important feature of nonlinear thermoelectrics is that the figure of merit ZT is no longer sufficient to characterize the thermoelectric performance [77]. Instead, one has to rely on quantities such as the maximal efficiency, the efficiency at maximum power [85,86,87] or the maximal efficiency at a given output power [88].…”

Unified Approach to Thermodynamic Optimization of Generic Objective Functions in the Linear Response Regime

“…For example, the power output is often taken as the objective, and the efficiency at maximum power for various kinds of heat engines and heat transfer laws have been investigated following the work of Curzon and Ahlborn [1], in which a bound of efficiency similar to η C was found to be η CA = 1 − √ T c /T h . Recently, investigations on maximum efficiency at a given power and on the controlling protocol for engines to achieve the optimal performance have also attracted much interest [2][3][4][5][6][7]. Besides the power output, there are other suggested objective functions such as (i) the so-called ecological function [8], which is defined as P − T c σ, where P is the power output and σ is the entropy production rate of the two heat reservoirs, and the associated efficiency when the ecological function is optimized is well approximated as (η C + η CA )/2 for endoreversible Carnot engines; (ii) a trade-off function [9], which is defined to be proportional to ηP with η being the thermodynamic efficiency, and for low-dissipation engines the corresponding efficiency at maximum trade-off is in the range [2η C /3, (3 − 9 − 8η C )/2].…”