Dynamics of order parameters near stationary states in superconductors with a charge-density wave

Abstract

We consider a simple model of a quasi-one-dimensional conductor in which two order parameters (OP) may coexist, i.e., the superconducting OP $\ensuremath{\Delta}$ and the OP $W$ that characterize the amplitude of a charge-density wave (CDW). In the mean field approximation, we present equations for the matrix Green's functions ${G}_{ik}$, where the first subscript $i$ relates to the one of the two Fermi sheets and the other, $k$, operates in the Gor'kov-Nambu space. Using the solutions of these equations, we find stationary states for different values of the parameter describing the curvature of the Fermi surface $\ensuremath{\mu}$, which can be varied, e.g., by doping. It is established, in particular, that in the interval ${\ensuremath{\mu}}_{1}<\ensuremath{\mu}<{\ensuremath{\mu}}_{2}$, the self-consistency equations have a solution for coexisting OPs $\ensuremath{\Delta}$ and $W$. However, this solution corresponds to a saddle point in the energy functional $\ensuremath{\Phi}(\ensuremath{\Delta},W)$, i.e., it is unstable. Stable states are (1) the W state, i.e., the state with the CDW ($W\ensuremath{\ne}0$, $\ensuremath{\Delta}=0$) at $\ensuremath{\mu}<{\ensuremath{\mu}}_{2}$ and (2) the S state, i.e., the purely superconducting state ($\ensuremath{\Delta}\ensuremath{\ne}0$, $W=0$) at ${\ensuremath{\mu}}_{1}<\ensuremath{\mu}$. These states correspond to minima of $\ensuremath{\Phi}$. At $\ensuremath{\mu}<{\ensuremath{\mu}}_{0}=({\ensuremath{\mu}}_{1}+{\ensuremath{\mu}}_{2})/2$, the state (1) corresponds to a global minimum, and at ${\ensuremath{\mu}}_{0}<\ensuremath{\mu}$, the state (2) has a lower energy, i.e., only the superconducting state survives at large $\ensuremath{\mu}$. We study the dynamics of the variations $\ensuremath{\delta}\ensuremath{\Delta}$ and $\ensuremath{\delta}W$ from these states in the collisionless limit. It is characterized by two modes of oscillations, the fast and the slow one. The fast mode is analogous to damped oscillations in conventional superconductors. The frequency of slow modes depends on the curvature $\ensuremath{\mu}$ and is much smaller than $2\ensuremath{\Delta}/\ensuremath{\hbar}$ if the coupling constants for superconductivity and CDW are close to each other. The considered model can be applied to high-${T}_{\text{c}}$ superconductors where the parts of the Fermi surface near the ``hot'' spots may be regarded as the considered two Fermi sheets. We also discuss relation of the considered model to the simplest model for Fe-based pnictides.