Self-consistent microscopic calculations for nonlocal transport through nanoscale superconductors

Abstract

We implement self-consistent microscopic calculations in order to describe out-of-equilibrium nonlocal 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 quasiequilibrium to a strong nonequilibrium situation when increasing the interface transparencies and the applied voltages. We analyze in detail the behavior of the nonlocal conductance in these two different regimes. While in quasiequilibrium 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 toward a self-consistent theory of nonlocal transport including nonequilibrium effects and describe qualitatively a recent experiment [P. Cadden-Zimansky and V. Chandrasekhar, Phys. Rev. Lett. 97, 237003 (2006)].