Cooper-pair-condensate fluctuations and plasmons in layered superconductors

Starting from a given attractive potential, we give a systematic analysis of the spin-singlet s-wave Cooper-pair-condensate fluctuations in a two-dimensional (2D) superconductor. These results are applied to a superlattice of superconducting sheets in which the 2D charge fluctuations are coupled via the Coulomb interaction. Our main interest is how the low-energy Anderson-Bogoliubov (AB) phonon mode in the pair-breaking gap {ω}2{Δ} is modified by the Coulomb interaction. Our formal analysis is valid at arbitrary temperatures. It describes the weakly bound, large-Cooper-pair limit as well as the strongly bound, small-Cooper-pair limit and thus includes both the BCS and Bose-Einstein scenarios (as discussed by Nozi\`eres and Schmitt-Rink as well as Randeira et al.). A comlete normal-mode analysis is given for a charged BCS superconductor, showing how the repulsive (Coulomb) interaction modifies the collective modes of a neutral superconductor. This complements the recent numerical study carried out by Fertig and Das Sarma. We show that the pair-response function shares the same spectrum as the charge-response function, given by the zero of the longitudinal dielectric function {ε}(q,{ω}). In 2D and layered superconductors, there is a low-frequency and high-frequency plasmon branch, separated by a relatively narrow particle-hole continuum at around 2{Δ}. The low-frequency ({ω}2{Δ}) plasmon branch is a renormalized version of the AB phonon mode.It thus has quite a different origin than the high frequency ({ω}>2{Δ}) plasmon, which is analogous to that found in the normal phase. Since the plasmons have low energy at long wavelengths in layered superconductors, they can renormalize the single-particle gap function {Δ}(q,{ω}). One is led to an Eliashberg-type integral equation for the renormalized {Δ}(q,{ω}) in which the dynamically screened Coulomb interaction v(q)/{ε}(q,{ω}) appears. We also point out that the screened longitudinal conductivity σₗ(q,{ω}) of a charged superconductor still contains the AB phonon pole in the region {ω}2{Δ}, which may have experimental consequences.