Thermal phase slips in superconducting films near the critical current at arbitrary temperatures

Abstract

We develop a theory of thermal phase slips in disordered superconducting films biased near the critical current $I_c(T)$. Generalizing recent results obtained close to $T_c$, we show that the optimal fluctuation governing the phase-slip barrier in two dimensions satisfies the exactly integrable Boussinesq equation for arbitrary temperatures $T<T_c$. Since both the transverse and longitudinal sizes of the optimal nucleus diverge as $I\to I_c(T)$, the Usadel equation for quasiparticles in the presence of a slowly varying order parameter can be solved perturbatively using a gradient expansion. The resulting field theory for a complex order parameter is further reduced to the Boussinesq free energy for a single real field, with the coefficients expressed as Matsubara sums over the solutions of the uniform Usadel equation at $I_c(T)$. The activation barrier near $I_c$ has the asymptotic form $\Delta F(T,I\to I_c) = E(T) (1-I/I_c)^\alpha$. We calculate $E(T)$ over the full temperature range for both two-dimensional films ($\alpha=3/4$) and one-dimensional wires ($\alpha=5/4$). For films, the theory is valid within a narrow $10\%$ window below $I_c(T)$, where the saddle-point configuration remains vortex-free. For wires, $\Delta F(T,I\to I_c)$ provides a good approximation to the activation barrier for all temperatures and currents.