  Using dynamical mean-field theory (DMFT) we study a simplified model for
heterostructures involving superconductors. The system is driven
out-of-equilibrium by a voltage bias, imposed as an imbalance of chemical
potential at the interface. We solve the self-consistent DMFT equations using
iterative second-order perturbation theory in the Nambu-Keldysh formalism. We
show that the superconducting state is destabilized by voltage biases of the
order of the energy gap. We demonstrate that the transition to the normal state
occurs through an intermediate bad superconducting state, which is
characterized by a smaller value of the order parameter and incoherent
excitations. We discuss the energetic balance behind the stabilization of such
exotic superconducting state.
