Cooper-pair-condensate fluctuations and plasmons in layered superconductors

Abstract

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<2Δ 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 Nozières 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 2Δ. The low-frequency (ω<2Δ) plasmon branch is a renormalized version of the AB phonon mode. It thus has quite a different origin than the high frequency (ω>2Δ) 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<2Δ, which may have experimental consequences.

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