Plasmons in three-dimensional superconductors

Abstract

We study the plasma branch of an homogeneous three-dimensional electron gas in an $s$-wave superconducting state. Although a sum rule guarantees that the departure of the plasma branch always coincides with the plasma frequency ${\ensuremath{\omega}}_{p}$, the dispersion and lifetime of plasmons is strongly affected by the presence of the pair condensate, especially when ${\ensuremath{\omega}}_{p}$ is close to the pair-breaking threshold $2\mathrm{\ensuremath{\Delta}}$. When ${\ensuremath{\omega}}_{p}$ is between $1.7\mathrm{\ensuremath{\Delta}}$ and $2\mathrm{\ensuremath{\Delta}}$, the level repulsion is strong enough to give the plasma branch an anomalous, downward dispersion and a dispersion minimum strictly lower than ${\ensuremath{\omega}}_{p}$. Then for ${\ensuremath{\omega}}_{p}>2\mathrm{\ensuremath{\Delta}}$, plasmons damp out in pair-breaking excitations, acquiring a small damping rate at zero temperature, which we compute in a non-perturbative way. Finally, the density-density response function displays a resonance near $2\mathrm{\ensuremath{\Delta}}$ (not to be confused with the amplitude mode), which can beat with the main plasma resonance, and subsists for ${\ensuremath{\omega}}_{p}$ large compared to $\mathrm{\ensuremath{\Delta}}$, thereby distinguishing charged from neutral condensates.