Lifetime of a Quasi-Particle in a Superconductor at Finite Temperatures and Application to the Problem of Thermal Conductivity

Abstract

The theory of the lifetime of a quasi-particle at finite temperatures is developed by using a generalization of the Green's function formalism of Nambu. The explicit results for the lifetime, as a function of the excitation energy and the temperature, are applied to the problem of electronic thermal conductivity when interaction with the phonons is the limiting mechanism for the heat flux. The lifetime appears in a general expression for the thermal conductivity which is derived from the corresponding Boltzmann equation by making a relaxation-time approximation. The resulting curve for the ratio of the thermal conductivity in the superconducting state to that in the normal state versus the reduced temperature is found to decrease monotonically and to have a slope of about 2 at the transition temperature.