We present a microscopic derivation of the effect of current flow on a system near a superconductor-metal quantum critical point. The model studied is a $2d$ itinerant electron system where the electrons interact via an attractive interaction and are coupled to an underlying normal metal substrate which provides a source of dissipation and also provides a source of inelastic scattering that allows us to reach a nonequilibrium steady state. A nonequilibrium Keldysh action for the superconducting fluctuations on the normal side is derived. Current flow, besides its minimal coupling to the order parameter, is found to give rise to two effects. One is a source of noise that acts as an effective temperature $T_{\text{eff}}=eE{v}_F{\tau }_{\text{sc}}$, where $E$ is the external electric field, $v_F$ is the Fermi velocity, and ${\tau }_{\text{sc}}$ is the escape time into the normal metal substrate. Second current flow also produces a drift of the order parameter. Scaling equations for the superconducting gap and the current are derived and are found to be consistent with previous phenomenological treatments as long as a temperature $T\sim T_{\text{eff}}$ is included. The current induced drift is found to produce additional corrections to the scaling which are smaller by a factor of $\mathcal{O}\left(\frac{1}{E_F{\tau }_{\text{sc}}}\right)$, with $E_F$ being the Fermi energy.

• Received 6 August 2008

DOI:https://doi.org/10.1103/PhysRevB.78.214512