Condensate pair fluctuations in a two-dimensional d-wave superconductor and Raman scattering.

Abstract

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