Self consistent charge-current in a mesoscopic region attached to superconductor leads

We investigate the behavior of a electric potential profile inside a mesoscopic region attached to a pair of superconducting leads. It turns out that $\mathrm{I}-V$ characteristic curves are strongly modified by this profile. In addition, the electronic population in the mesoscopic region is affected by the profile behavior. We discuss the single particle current and the mesoscopic electronic population within the non-equilibrium Keldysh Green functions technique. The Keldysh technique results are further converted in a self consistent field (SFC) problem by introducing potential profile modifications as proposed by Datta. Evaluation of $\mathrm{I}-V$ characteristics are presented for several values of the model parameters and comparison with current experimental results are discussed.