Tutorial notes for the evaluation of thermoelectric quantum bounds in ideal nanostructures

TL;DR

This paper derives explicit quantum bounds on thermoelectric response, power, and efficiency for ideal nanostructures using non-linear scattering theory, providing a tutorial derivation for optimal transmission functions.

Contribution

It offers a detailed, tutorial derivation of quantum bounds on thermoelectric performance in nanostructures using non-linear scattering theory.

Findings

Derived analytic quantum bounds for power and efficiency in thermoelectric devices.

Provided explicit formulas for optimal transmission functions like boxcar and theta shapes.

Validated bounds through theoretical analysis and explicit derivations.

Abstract

The wave-like nature of electrons leads to the existence of upper bounds on the thermoelectric response of nanostructured devices [R. S. Whitney, Phys. Rev. Lett. 112, 130601 (2014); Phys. Rev. B 91, 115425 (2015)]. This fundamental result, not present in classical thermodynamics, was demonstrated exploiting a two-terminal device modelled by non-linear scattering theory. In the present paper, we consider non-linear quantum transport through the same type of device working both as thermal machine and as refrigerator. For both operations, starting from charge and heat current expressions, we provide analytic quantum bounds for power exchanged, thermal currents and device efficiencies. For this purpose, we adopt a transmission function that maximizes the engine efficiency for given power output. For the optimal boxcar- or theta function-transmission shapes, we provide in a tutorial way an explicit deduction of the quantum bound expressions reported in the above cited papers.