Power-law temperature dependence of the inelastic-scattering rate in disordered superconductors

We present a theory of the quasiparticle inelastic lifetime ${\mathrm{\tau }}_{\mathrm{in}}$ in disordered superconducting films. We find that both the Coulomb and the electron-phonon contribution to ${\mathrm{\tau }}_{\mathrm{in}}^{\mathrm{-}1}$ are enhanced by disorder, and that for reasonably strong electron-phonon coupling the latter is dominant. In contrast to clean superconductors, the scattering rate is larger than the recombination rate at all temperatures. This leads to a power-law temperature dependence of ${\mathrm{\tau }}_{\mathrm{in}}^{\mathrm{-}1}$, in agreement with experimental observations. The theory quantitatively accounts for the magnitude, disorder dependence, and temperature dependence of ${\mathrm{\tau }}_{\mathrm{in}}$ measured in recent experiments.