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The condensate pair fluctuations in the frequency region [omega][approx lt]2[Delta] and the associated Raman-scattering intensity of a two-dimensional (2D) [ital d]-wave weak-coupling BCS superconductor are investigated. Our model includes a dominant [ital d][sub [ital x]][sup 2][minus][ital y][sup 2] ([ital L]=2) and a weaker [ital s]-wave ([ital L]=0) pairing interaction. All response functions involving density and pair operators are evaluated (in the low-[bold q] limit). For neutral [ital d]-wave superconductors (no Coulomb interaction), the expected phononlike phase mode is obtained in the [ital L]=0 channel (which couples to density fluctuations). In contrast, excitonlike modes corresponding to excited Cooper pair states are obtained in the [ital L]=2 channel. We find an amplitude fluctuation mode with frequency [radical]3 [Delta]. For charged [ital d]-wave superconductors, the [ital L]=0 phonon is renormalized into a 2D plasmon, but the [ital L]=2 excitonlike mode remains unaffected by the Coulomb interaction. At [ital T]=0, the latter is shown to be completely washed out in the Raman scattering spectrum ([bold q]=0) due to large [ital p]-[ital h] damping (which arises in the absence of a finite pair-breaking gap in [ital d]-wave superconductors). However, at finite temperatures, we find the energy of the excitonlike mode is drastically lowered [relative tomore » 2[Delta]([ital T])] when the [ital s]-wave attraction is comparable to the [ital d]-wave pairing. This leads to a decrease in the damping and, as a result, the mode shows up as a low-frequency resonance in the Raman cross section. Due to the anisotropy of the [ital d]-wave order parameter, the quasiparticle excitation spectrum and the noninteracting two-particle spectrum are strongly dependent on the direction of [bold q]. We also find that the excitonlike mode frequency becomes anisotropic for wave vectors of the order of [Delta]/[ital v][sub [ital F]].« less

Recent experimental studies by Wolfe, Snoke, and Lin have given strong evidence that paraexcitons in optically excited ${\mathrm{Cu}}_{2}$O crystals form a Bose condensate. In these studies, the observed recombination decay luminescence line shape I(\ensuremath{\omega}) has been analyzed in terms of an ideal Bose-condensed gas. We derive a general expression for I(\ensuremath{\omega}) in terms of the exact single-particle spectral density A(k,\ensuremath{\omega}) of a Bose fluid. This expression is then used to calculate I(\ensuremath{\omega}) for a dilute, weakly interacting Bose gas. Our analytic results, based on the approximate theory due to Popov, are valid in the region just below the Bose-Einstein transition temperature. When formally extended down to T=0, they reproduce the well-known Bogoliubov approximation.

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 \ensuremath{\omega}2\ensuremath{\Delta} 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 \ensuremath{\epsilon}(q,\ensuremath{\omega}). 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\ensuremath{\Delta}. The low-frequency (\ensuremath{\omega}2\ensuremath{\Delta}) plasmon branch is a renormalized version of the AB phonon mode.It thus has quite a different origin than the high frequency (\ensuremath{\omega}g2\ensuremath{\Delta}) 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 \ensuremath{\Delta}(q,\ensuremath{\omega}). One is led to an Eliashberg-type integral equation for the renormalized \ensuremath{\Delta}(q,\ensuremath{\omega}) in which the dynamically screened Coulomb interaction v(q)/\ensuremath{\epsilon}(q,\ensuremath{\omega}) appears. We also point out that the screened longitudinal conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathit{l}}$(q,\ensuremath{\omega}) of a charged superconductor still contains the AB phonon pole in the region \ensuremath{\omega}2\ensuremath{\Delta}, which may have experimental consequences.

There is considerable evidence that the electronic behavior of orthorhombic ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ is dominated by energy bands associated with Cu-O sheets and chains. We discuss the long-wavelength plasmon modes expected in an array of parallel metallic sheets and chains, coupled by the Coulomb interaction and using the random-phase approximation. We show that in the a-b plane, there is a low-energy acoustic plasmon mode, in addition to the usual high-energy bulk plasmon. This new acoustic plasmon does not arise in an array of coupled layers or in an array of parallel chains. Depending as it does on electronic motion in the chains along the b axis, the frequency of the mode is highly anisotropic in the a-b plane.

Recently it is been argued that the plasmon modes expected in the oxide superconductors should be characteristic of a superlattice with a basis of several metallic sheets. We extend this work by including the exchange-correlation effects in a given CuO sheet via local-field corrections (the generalized random-phase approximation). In addition, the effective interaction between electrons in a given CuO sheet is worked out for a superlattice with a basis of several sheets, separating out the effects of the various kinds of plasmons. Numerical results are given for the superlattice plasmon dispersion relations for two and three sheets/unit cell. If the spacing of the sheets is small compared to the superlattice period (as it is in the 2:2:1:2 and 2:2:2:3 superconductors), we find that the low-frequency plasmon branches (/ital n//minus/1 of them, where /ital n/ is the number of sheets/unit cell) are essentially identical to those of an isolated bilayer or trilayer.