Phase Slip Avalanches in Small Superconductors

We study the effect of phase slips in a quasi 1d superconducting channel along which a current flows and report a new phenomenon where an avalanche of phase slips occurs. This limits the critical current in thin films and wires and drives the system to a topological phase transition at a temperature lower than the bulk critical temperature. We describe the mechanism of such a catastrophic phase slip avalanche and, following Kosterlitz and Thouless, we use group renormalization techniques to derive an exact analytical expression for the critical current as a function of film width and temperature. Our results are in very good agreement with, and reproduce, the available experimental data on superconducting MgB$_2$ thin films. The phenomenon we describe is very general and can be used in the construction of new devices where the superconducting state can coexist with the normal state.