Mode propagation in superconductors near T_c

The linearized Eliashberg theory for dirty superconductors in the vicinity of T_c is used to generate the dynamic Ginzburg-Landau (GL) equation for the order parameter and the coupled Boltzmann equation for the quasi-particle distribution function, with and without magnetic impurities and inelastic phonon scattering. The Schmid and Schön (SS) separation of the order parameter into longitudinal (real) and transverse (imaginary) components is shown to be exact to linear order. The former does not couple with the wave vector (k) and frequency (ω) dependent external field, while the latter does. Explicit expressions are exhibited for the current and charge densities to leading order [ Δ T in the gap regime, in which Δτ_s > 1 where τ_s is the conduction electron spin flip lifetime; and Δ² T(ω + iDk²) in the gapless regime, Δτ_s < 1. D is the diffusion constant]. The continuity equation is shown to be obeyed to this order. Use of the Poisson equation in relation to the transverse component leads to a propagating mode, which, in the absence of magnetic impurities, reduces to the SS result. We find that the mode propagates only in the gap regime, whereas it is purely diffusive in the gapless regime and for gapless superconductors. In the presence of magnetic impurities, the solution reduces to the SS limit, but with reduced mode velocity and damping. The charge density appropriate to the propagating mode is shown to be zero to order ω ω_p²τ₁, where τ₁ is the impurity scattering time and θ_p the plasma frequency. The characteristic penetration length of an electric field into a superconductor is calculated, taking into account the phonon inelastic scattering time. The result reproduces the Waldram and SS expressions. Careful examination is made of the conditions under which these results pertain. The characteristic penetration length is also obtained in the gapless regime and for gapless superconductors for which T_cτ_s < 1. Finally, the longitudinal and transverse pair susceptibilities are calculated with and without magnetic impurities and in the gap and gapless regimes.