Pair breaking and charge relaxation in superconductors

We present a general formalism based on the quasiclassical Green's function for calculating charge imbalance in nonequilibrium superconductors. Our discussion is sufficiently general that it applies at arbitrary temperatures, and under conditions when the width of quasiparticle states are appreciable due to pair breaking processes, and when strong coupling effects are significant. As a first application we demonstrate in detail how in the limit of smallpair breaking and for a weak coupling superconductor the collision term in the formalism reduces to the one in the quasiparticle Boltzmann equation. We next treat the case of charge imbalance generated by tunnel injection, with pair breaking by phonons and magnetic impurities. Over the range of temperatures investigated exerimentally to date, the calculated charge imbalance is rather close to that evaluated using the Boltzmann equation, even if pair braeking is so strong as almost to destroy superconductivity. Finally we consider charge imbalance generated by the combined influence of a supercurrent and a temperature gradient. We give calculations for a dirty superconductor with scattering by phonons as the pair breaking mechanism, and the results give a reasonable account of the experimental data of Clarke, Fjordboge, and Lindelof. We carry out calculations for the case of impurity scattering along which are valid not only in the clean and dirty limits, but also for intermediate situations. These enable us to see how the large contribution to the charge imbalance found for energies close to the gap edge in the clean case is reduced with increasing impurity scattering