Quasiclassical theory of charge transport across mesoscopic normal-metal–superconducting heterostructures with current conservation

We consider the steady-state nonequilibrium behavior of mesoscopic superconducting wires connected to normal-metal reservoirs. Going beyond diffusive limit, we utilize quasiclassical theory and perform a self-consistent calculation that guarantees current conservation through entire system. from ballistic investigate several crucial phenomena such as charge imbalance, momentum-resolved nonequilbrium distributions, current-to-superflow conversion. Connecting earlier results for case, find superconductivity can break down at critical bias voltage $V_\mathrm{c}$. $V_\mathrm{c}$ generally increases interface transparency is reduced, while dependence on mean-free path non-monotonous. discussthe key differences semi-ballistic regimes fully case.