Spectral function of the charge-density-wave (CDW) phonon in an anisotropic CDW superconductor

We investigate the spectral function of the charge-density-wave (CDW) phonon renormalized by electron-phonon interaction in an anisotropic CDW superconductor. The model used for the system is similar to that of Bilbro and McMillan, which shows coexistence of the CDW and superconductivity. In a system with static lattice distortion of wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$, we point out that, in addition to the ordinary electron-phonon interaction, there exists another electron-phonon coupling with zero electron-momentum transfer and creation or annihilation of a single phonon with wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$. This interaction is similar in appearance to that proposed by Balseiro and Falicov for a commensurate CDW and can partly survive the Coulomb screening if the electron-phonon matrix element is $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ dependent. With the parameters pertinent to the situation of $2H$-Nb${\mathrm{Se}}_{2}$ we calculate the spectral function of the CDW phonon as a function of temperature. The predicted frequencies and relative intensities of the CDW and superconducting gap modes are in agreement with the Raman scattering observation by Sooryakumar and Klein in both $A$ and $E$ symmetries.