Topological defects in systems with two competing order parameters: Application to superconductors with charge- and spin-density waves

On the basis of coupled Ginzburg-Landau equations we study nonhomogeneous states in systems with two order parameters (OPs). Superconductors with a superconducting OP $\Delta$ and a charge- or spin-density wave with amplitude $W$ are examples of such systems. When one OP, say $\Delta$, has a form of a topological defect, like, e.g., a vortex or domain wall between the domains with the phases 0 and $\pi$, the other OP $W$ is determined by the Gross-Pitaevskii equation and is localized at the center of the defect. We consider in detail the domain-wall defect for $\Delta$ and show that the shape of the associated solution for $W$ depends on temperature and doping (or on the curvature of the Fermi surface) $\mu$. It turns out that, provided the temperature or doping level is close to some discrete values $T_n$ and ${\mu }_n$, the spatial dependence of the function $W\left(x\right)$ is determined by the form of the eigenfunctions of the linearized Gross-Pitaevskii equation. The spatial dependence of $W_0$ corresponding to the ground state has the form of a soliton, while other possible solutions $W_n\left(x\right)$ have nodes. The inverse situation when $W\left(x\right)$ has the form of a topological defect and $\Delta \left(x\right)$ is localized at the center of this defect is also possible. In particular, we predict a surface or interfacial superconductivity in a system where a superconductor is in contact with a material that suppresses $W$. This superconductivity should have rather unusual temperature dependence existing only in certain intervals of temperature. Possible experimental realizations of such nonhomogeneous states of OPs are discussed.

Figure 1

Coordinate dependence of $W_0$ (red), $W_1$ (black), $W_2$ (green), and $W_3$ (blue) near the corresponding “energy” levels ${\mathcal{E}}_N$ for the case when the superconducting state is favored far from the defect at $x=0$. Exactly at ${\mathcal{E}}_N$, as follows from Eqs. (14) and (15), $W=0$. Note that in the opposite case when the CDW or the SDW state is more favorable at $x\to \infty$ one needs to make the exchange $\Delta \leftrightarrow W$ and, correspondingly, ${\mathcal{U}}_w\leftrightarrow {\mathcal{U}}_s$ and ${\kappa }_s\leftrightarrow {\kappa }_w$, and the shown curves will describe the dependence $\Delta \left(x\right)$ while $W=W_{\infty }tanh\left({\kappa }_{w}x\right)$.

Figure 2

The coefficient $C$ in Eq. (17) on ${\mu }_c$ for $r=5.0$. The critical doping ${\mu }_c$ is calculated from Eq. (3) and represents a line in the $({\mu }_{\phi },{\mu }_0)$ plane, which is shown in the upper part of the figure. In the lower part, the value of $C$ along the obtained ${\mu }_c$ line is presented as a function of ${\mu }_{\phi }$ (inserting the corresponding value of ${\mu }_0$). The coefficient $C$ is negative; thus, as $\eta >0$ due to conditions of realization of the superconducting state far from the topological defect, from Eq. (17) it is seen that $\delta \left[{\mu }^2\right]<0$.