Self-consistent microscopic calculations for non-local transport through nanoscale superconductors

TL;DR

This paper develops a self-consistent microscopic model to analyze non-local transport in nanoscale superconductor hybrid structures under non-equilibrium conditions, revealing how interface transparency and voltage influence transport regimes.

Contribution

It introduces a comprehensive self-consistent approach to study non-local transport in superconductor hybrids, accounting for non-equilibrium effects and arbitrary interface transparencies.

Findings

Crossover from quasi-equilibrium to strong non-equilibrium regimes with increasing transparency and voltage.

Non-local conductance can be described by elastic cotunneling and crossed Andreev transmission in equilibrium.

Additional voltage-dependent contributions are significant in non-equilibrium conditions.

Abstract

We implement self-consistent microscopic calculations in order to describe out-of-equilibrium non-local transport in normal metal-superconductor-normal metal hybrid structures in the presence of a magnetic field and for arbitrary interface transparencies. A four terminal setup simulating usual experimental situations is described by means of a tight-binding model. We present results for the self-consistent order parameter and current profiles within the sample. These profiles illustrate a crossover from a quasi-equilibrium to a strong non-equilibrium situation when increasing the interface transparencies and the applied voltages. We analyze in detail the behavior of the non-local conductance in these two different regimes. While in quasi-equilibrium conditions this can be expressed as the difference between elastic cotunneling and crossed Andreev transmission coefficients, in a general situation additional contributions due to the voltage dependence of the self-consistent order parameter have to be taken into account. The present results provide a first step towards a self-consistent theory of non-local transport including non-equilibrium effects and describe qualitatively a recent experiment [Phys. Rev. Lett. 97, 237003 (2006)].