Scattering Theory of Nonlinear Thermoelectricity in Quantum Coherent Conductors

TL;DR

This paper develops a scattering theory for weakly nonlinear thermoelectric transport in quantum conductors, ensuring fundamental conservation laws are maintained, and applies it to analyze heat engine efficiency and thermoelectric refrigerator performance.

Contribution

It extends existing scattering theory to include nonlinear effects in thermoelectric transport, preserving gauge invariance and current conservation, and provides practical calculations for device performance.

Findings

Nonlinear transport coefficients satisfy sum rules for gauge invariance.

Rectification effects can enhance thermoelectric device performance.

The theory accurately predicts efficiency and performance metrics of quantum thermoelectric devices.

Abstract

We construct a scattering theory of weakly nonlinear thermoelectric transport through sub-micron scale conductors. The theory incorporates the leading nonlinear contributions in temperature and voltage biases to the charge and heat currents. Because of the finite capacitances of sub-micron scale conducting circuits, fundamental conservation laws such as gauge invariance and current conservation require special care to be preserved. We do this by extending the approach of Christen and B\"uttiker [Europhys. Lett. 35, 523 (1996)] to coupled charge and heat transport. In this way we write relations connecting nonlinear transport coefficients in a manner similar to Mott's relation between the linear thermopower and the linear conductance. We derive sum rules that nonlinear transport coefficients must satisfy to preserve gauge invariance and current conservation. We illustrate our theory by calculating the efficiency of heat engines and the coefficient of performance of thermoelectric refrigerators based on quantum point contacts and resonant tunneling barriers. We identify in particular rectification effects that increase device performance.