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

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 $\tilde{\xi }=\sqrt{\mathrm{\hslash }D\mathrm{/}\mathrm{\Delta }}$, instead of the BCS coherence length ${\xi }_0$. Moreover, in three dimensions, the algebraic prefactor scales as $1\mathrm{/}r$ instead of $1\mathrm{/}{r}^2$. The calculation involves the virtual diffusion probability of quasiparticles below the superconducting gap, in the normal and the anomalous channel.