Transport theory of multiterminal hybrid structures

We derive a microscopic transport theory of multiterminal hybrid structures in which a superconductor is connected to several spin-polarized electrodes. We discuss the non-perturbative physics of extended contacts, and show that it can be well represented by averaging out the phase of the electronic wave function. The maximal conductance of a two-channel contact is proportional to $(e^2/h) (a_0/D)^2 \exp{[-D/\xi(\omega^*)]}$, where $D$ is the distance between the contacts, $a_0$ the lattice spacing, $\xi(\omega)$ is the superconducting coherence length, and $\omega^*$ is the cross-over frequency between a perturbative regime ($\omega< \omega^*$) and a non perturbative regime ($\omega^* < \omega < \Delta$). The intercontact Andreev reflection and elastic cotunneling conductances are not equal if the electronic phases take a fixed value. However, these two quantities do coincide if one can average out the electronic phase. The equality between the Andreev and cotunneling conductances is also valid in the presence of at least one extended contact in which the phases take deterministic values.