Impurity and fluctuation effects in charge-density-wave superconductors

We formulate a theory of a charge-density-wave (CDW) distorted superconductor, in the presence of impurities and CDW fluctuations. This theory is expected to be applicable to the layered compound $2H$-Nb${\mathrm{Se}}_2$. Our primary aim is to compute the superconducting density of states $N_s\left(\omega \right)$. We also obtain the temperature dependence of the order parameter $\Delta$. We show, when all other parameters are fixed, that nonmagnetic impurities will raise the superconducting transition temperature and $\Delta$. Since the CDW gap $W_Q$ is expected to decrease with concentration, this effect should be even more pronounced. For nonmagnetic impurities, $N_s\left(\omega \right)$ is found to be "BCS-like" with a gap at $\Delta \ne {\Delta }^{\mathrm{BCS}}$, with subsidiary structure at $\omega ={({\Delta }^2+W_Q^2)}^{\frac{1}{2}}$. As the nonmagnetic impurity concentration is further increased this structure rounds out and $N_s\left(\omega \right)$ approaches the strict BCS limit. CDW fluctuation effects are expected to lead to fine structure in $N_s\left(\omega \right)$ at the characteristic phonon frequency. In principle, an analysis of (future) tunneling data will give detailed information about the CDW excitations, some of which has been obtained from Raman spectroscopy.