Effects of quasiparticle screening on collective modes. II. Superconductors

We investigate the phase-phase and density-density Green's functions for all temperatures below Tc in clean superconductors. The dispersion relation of the phase-density mode consists of an acoustic branch and a plasma branch, and has striking similarity with those of charge-density-wave systems. Owing to the small electronic effective mass in superconductors, the acoustic branch is found to be important only in the vicinity of Tc. For the plasma branch, we derive a plasma sum rule, showing that the plasma frequency is not affected by the superconducting transition nor the presence of quasiparticles. We also derive microscopic expressions for the phenomenological two-fluid electronic displacements. The normal and superconducting components are found to satisfy the constitutive equations.