Finding the quantum thermoelectric with maximal efficiency and minimal entropy production at given power output

Abstract

We investigate the nonlinear scattering theory for quantum systems with strong Seebeck and Peltier effects, and consider their use as heat-engines and refrigerators with finite power outputs. This article gives detailed derivations of the results summarized in Phys. Rev. Lett. 112, 130601 (2014). It shows how to use the scattering theory to find (i) the quantum thermoelectric with maximum possible power output, and (ii) the quantum thermoelectric with maximum efficiency at given power output. The latter corresponds to a minimal entropy production at that power output. These quantities are of quantum origin since they depend on system size over electronic wavelength, and so have no analogue in classical thermodynamics. The maximal efficiency coincides with Carnot efficiency at zero power output, but decreases with increasing power output. This gives a fundamental lower bound on entropy production, which means that reversibility (in the thermodynamic sense) is impossible for finite power output. The suppression of efficiency by (nonlinear) phonon and photon effects is addressed in detail; when these effects are strong, maximum efficiency coincides with maximum power. Finally, we show in particular limits (typically without magnetic fields) that relaxation within the quantum system does not allow the system to exceed the bounds derived for relaxation-free systems, however, a general proof of this remains elusive.

Interfacial thermal transport with strong system-bath coupling: A phonon delocalization effect

“…The intriguing second peak in the transmission for k int 0.03 is a result of Fabry-Pérot-like interference. Such interference patterns have been observed in ordered harmonic systems without an interface where the boundaries to the baths act like partially reflecting surfaces for the phonon modes [38][39][40][41]. For such systems the interference pattern leads to delocalized phonon modes (seen as peaks in the transmission) that resonate with the system-phonon modes (normal modes of the effective force matrix K).…”

Nonlinear thermoelectric efficiency of superlattice-structured nanowires

“…It was shown in Refs. [15,16] that the optimal η at a given P is instead obtained when T (E,V ) at a fixed V has a square shape, allowing all electrons to be transmitted within an energy range determined by the desired efficiency and blocking all electrons outside this energy range. A sharp onset in the transmission function can be achieved by using doping or electrostatic gating to position the lowest 1D subband of a NW close to μ h,c (this idea was originally suggested by Hicks and Dresselhaus [4], although they considered diffusive rather than ballistic transport).…”

“…Recently, Refs. [15,16] addressed, and solved, the problem of finding the transmission function which maximizes the efficiency at a given desired output power, showing that it should have a square shape (i.e., letting all electrons through within a finite energy window and blocking all transport outside this window).…”

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

“…This is the expression of σ when the tight-coupling condition is fulfilled, inserting which into Equation (3) and noting the definition of J 2 and its link with τ, we have…”

“…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].…”