Theory of Raman scattering of superconducting amplitude modes in charge-density-wave superconductors

The theory of Raman scattering is developed in those materials where superconductivity and the charge-density wave (CDW) coexist, such as in ${\mathrm{NbSe}}_2$. In the case of the absence of the CDW, the ‘‘effective densities’’ characterizing the Raman scattering and the superconducting phase modes are coupled and contribute to the Raman spectra, while the superconducting amplitude mode is unobservable by light scattering. Littlewood and Varma pointed out that in the presence of the CDW the CDW phonon is also coupled to the amplitude mode and a new type of bound state occurs below twice the superconducting gap 2Δ. In the present theory symmetry arguments are explored, and thus different scattering channels (e.g., crystal harmonics and spherical functions) are introduced. In the channel L=0, the real charge density occurs and is screened by the long-range Coulomb force, and only the coupled amplitude-phonon mode with a bound state as predicted by Littlewood and Varma appears. For L≠0, the effective density enters and the bound state shows an effective-density character except for very large coupling of the CDW phonon to the amplitude mode, which must be also associated with an instability of the s-type superconductivity. It is shown that the L=0 channel cannot be observed separately because it is always mixed with other channels L≠0 obeyed by crystal symmetry. In this mixed channel, bound states with amplitude-mode and effective-charge character may occur, but it is argued that the effective-charge bound state is the dominant one except for the case of anomalously large coupling between the CDW phonon and the amplitude mode.