Keldysh field theory approach to direct electric and thermoelectric currents in quantum dots coupled to superconducting leads

TL;DR

This paper applies the Keldysh field theory to analyze electric and thermoelectric currents in quantum dots with superconducting leads, deriving full expressions for thermoelectric effects under various conditions.

Contribution

It introduces a comprehensive Keldysh-based framework to calculate thermoelectric currents in superconductor-quantum dot systems for any temperature difference.

Findings

Derived the full thermoelectric current expression for superconductor-quantum dot-superconductor junctions.

Identified phase-dependent thermoelectric contributions in the linear response regime.

Showed thermoelectric effects dominate at weak coupling, even with particle-hole symmetry.

Abstract

We study the transport properties of a quantum dot contacted to two superconducting reservoirs by means of the Keldysh field theory approach, showing how this technique allows us to straightforwardly recover previous results, resulting extremely effective in dealing with quantum transport problems. In particular, we determine the direct current occurring at equilibrium and the electric and thermoelectric currents triggered when the system is driven out of equilibrium by a voltage or a temperature bias, also for a normal-quantum dot-superconductor junction. The main result of the work is the derivation of the full expression for the thermoelectric current in a superconductor-quantum dot-superconductor junction for any values of the temperature difference between the superconducting leads. We show that in the linear response regime, in addition to the Josephson current, a weakly phase-dependent thermoelectric contribution occurs, provided that electron-hole symmetry is broken. Far from linearity, instead, other contributions arise which lead to thermoelectric effects, dominant at weak coupling, also in the presence of particle-hole symmetry.