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Dynamical Mean-Field Theory description of the voltage induced transition in a non-equilibrium superconductor

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.