Andreev scattering and cotunneling between two superconductor-normal metal interfaces : the dirty limit

TL;DR

This paper extends the theoretical understanding of crossed Andreev reflections and cotunneling between superconductor-normal interfaces by considering the effects of impurity scattering in dirty superconductors, revealing a longer-range effect and different spatial scaling.

Contribution

It generalizes previous clean superconductor models to dirty superconductors, showing the impact on coherence length and spatial decay of crossed conductance.

Findings

Range of the effect is the coherence length $ ilde{\xi}$, larger than in clean superconductors.

In three dimensions, the algebraic decay scales as 1/r, not 1/r^2.

Calculation involves quasiparticle diffusion probabilities below the superconducting gap.

Abstract

Crossed Andreev reflections and cotunneling occur between two neighbouring superconductor- normal metal or superconducting-ferromagnet interfaces. Previous works assumed a clean BCS superconductor. Here the calculation of the corresponding crossed conductance terms is generalized to a dirty superconductor. The range of the effect is shown to be the coherence length $\tilde{\xi} = \sqrt{\hbar D/\Delta}$, instead of the BCS coherence length $\xi_0$. Moreover, in three dimensions, the algebraic prefactor scales as $1/r$ instead of $1/r^2$. The calculation involves the virtual diffusion probability of quasiparticles below the superconducting gap, in the normal and the anomalous channel.