Theory of charge-density-wave superconductors

It is shown theoretically that the superconducting state coexisting with charge-density wave (CDW, its wave vector $\overrightarrow{\mathrm{Q}}$) is characterized by the spatially varying periodic order parameter $\Delta \mathrm{}\left(Q\right)\mathrm{}$ in addition to the ordinary uniform order parameter $\Delta \mathrm{}\left(0\right)\mathrm{}$. Perturbational calculation based on the linearized gap equation yields the depression of the transition temperature of this state. The upper critical field $H_{c2}\mathrm{}\left(T\right)\mathrm{}$ is derived from the Ginzburg-Landau (GL) equation modified so as to include the effect of the CDW, and shown to exhibit a positive curvature in $H_{c2}\mathrm{}\left(T\right)\mathrm{}$ vs $T$ curve when applied magnetic field is perpendicular to $\overrightarrow{\mathrm{Q}}$. The parallel $H_{c2}\mathrm{}\left(T\right)\mathrm{}$ follows the ordinary GL theory. Physical implication of this result is discussed in connection with highly anisotropic superconducting materials.