Discommensuration-driven superconductivity in the charge density wave phases of transition-metal dichalcogenides

We introduce a McMillan-Ginzburg-Landau theory to describe the cooperative coexistence of charge density and superconducting order in two-dimensional crystals. With a free energy that explicitly accounts for the competition between commensurate and incommensurate ground states, we are able to map the transition between these phases and monitor the development of discommensurations in the near-commensurate regime. Attributing the enhancement of superconducting order to density wave fluctuations, we propose a coupling scheme that yields a phase diagram in qualitative agreement with experiments in conducting transition-metal dichalcogenides. The model predicts the development of nonuniform superconductivity similar to that arising from a pair density wave, with a spatial texture driven by the underlying charge density wave fluctuations.

Figure 1

Phase diagram obtained by minimizing ${\mathcal{F}}_{\text{CDW}}$. Labels C, NC and IC stand for commensurate, near-commensurate and homogeneously incommensurate CDW phases, respectively. When ${\mathcal{F}}_{\text{CDW}}<0$, the system is in a CDW state and the C phase corresponds to $\eta =0$. The green line represents the C-IC boundary $E_c\left(t\right)$. The red line indicates the boundary of the SC phase including the linear $E$ dependence in the CDW-SC coupling $a_s$ of Eq. (3) ($a_1=500E$); it becomes the gray line if $a_s$ is $E$ independent ($a_1=500\times 2.1$). The inset shows the equilibrium $\eta$ at $t_c$ (first-order transition) and at low temperature.

Figure 2

(a) Real-space plot of the density profile $\delta \rho \left(\mathbf{r}\right)$ at $E=2.2, t=-1.7$ (in units of $\sqrt{3}a/2\pi$, with $a$ the lattice constant). The yellow-dashed lines mark the places where the phase of each CDW order parameter, ${\psi }_j\left(\mathbf{r}\right)$, jumps by $\pi$. (b) and (c) respectively show the phase and amplitude of ${\psi }_j\left(\mathbf{r}\right)$ along the white vertical cut marked in (a). (d) The SC order parameter $\mathrm{\Phi }\left(\mathbf{r}\right)$ in the same region as (a). (e) $\mathrm{\Phi }\left(\mathbf{r}\right)$ along the vertical cut marked in (a).

Figure 3

(a) Schematic of the distinct nonuniform SC regimes spatially correlated with the DC network: nucleation and expansion of the SC order parameter ($T_{\text{SC}}^{\text{1D}}<T\le T_{\text{SC}}^{\text{0D}}$), percolation ($T_{\text{DC}}^{\text{2D}}<T\le T_{\text{SC}}^{\text{1D}}$), and finite everywhere. See Supplemental Fig. S4 for actually calculated textures. (b) Illustration of how the connectivity in the percolation regime constrains the vortex structure, with impact in the magnetic response.