Interlayer-coupling-driven stabilization and superconductivity in bilayer CoTe$_2$

Abstract

Interlayer coupling plays a critical role in van der Waals materials by governing lattice stability and emergent quantum phases, yet its impact on few-layer hexagonal CoTe$_2$ remains unclear. Here, using first-principles calculations, we systematically investigate monolayer and bilayer CoTe$_2$ with an emphasis on their electronic structures, lattice dynamics, and electron-phonon coupling, and elucidate the underlying mechanisms driven by interlayer interactions. Our results show that monolayer CoTe$_2$ exhibits pronounced dynamical instability at low temperatures, whereas interlayer coupling stabilizes the bilayer crystal structure and gives rise to phonon-mediated superconductivity with a predicted critical temperature of about $4.7$~K. The stabilization and superconductivity in bilayer CoTe$_2$ are primarily attributed to interlayer-coupling-induced Te-$p_z$ charge redistribution and the associated modification of the Fermi surface and electron-phonon coupling. Finally, we discuss how spin-orbit coupling in bilayer CoTe$_2$ weakens the EPC and suppresses superconductivity. Our work clarifies how interlayer coupling can jointly tune structural stability and superconductivity in few-layer CoTe$_2$, providing insights for engineering quantum phases in layered transition-metal dichalcogenides.

Figures (5)

• Figure 1: (a) Side and top views of the 2L CoTe$_2$ crystal structure. Lattice-dynamical properties of 1L CoTe$_2$ are shown in (b-g): (b) the $\omega_{\boldsymbol{q}\nu}$ calculated using an electron broadening of 0.01 Ry, and projected onto Te$_z$ and Co$_{xy}$ vibrations, where the blue (orange) symbol size is proportional to the Te$_z$ (Co$_{xy}$) contribution; (c) vibration-resolved phonon density of states; (d) harmonic $\omega_{\boldsymbol{q}\nu}$, and anharmonic $\omega_{\boldsymbol{q}\nu}$ obtained from the SSCHA free-energy Hessian at selected temperatures; (e) potential-energy surfaces along the eigen-displacement of the imaginary mode at M (upper panel) and of one imaginary mode at K (lower panel). Insets in (e) display the corresponding phonon polarization vectors in top and side views. (f, g) The Minimum phonon frequency as a function of charge doping and biaxial strain, respectively; the insets show the acoustic-branch dispersions at three representative doping levels and three biaxial strains, respectively.
• Figure 2: Calculated quantities elucidating the mechanism of phonon softening in 1L CoTe$_2$ are summarized in (a--f). (a) The $\omega_{\boldsymbol{q}\nu}$ computed with a relatively large electronic smearing (0.018 Ry); the red symbol size scales with $\lambda_{\boldsymbol{q}\nu}$. (b) Electronic band structure with orbital projections onto Co-$d$ and Te-$p$ states. (c) Fermi surfaces projected onto Co-$d$ and Te-$p$ states; double-headed arrows mark the $\boldsymbol{q}$ vectors where $\chi_{\boldsymbol{q},\nu=1}$ is strongly enhanced (see panel (e)). (d) Map of $\omega_{\boldsymbol{q}\nu=1}$ across an extended BZ. (e) Generalized static electronic susceptibility $\chi_{\boldsymbol{q}\nu=1}$. (f) Static susceptibility within the constant-matrix approximation, $\chi^{\prime}_{\boldsymbol{q}}$.
• Figure 3: Calculated quantities elucidating effect of interlayer coupling in 2L CoTe$_2$ are summarized in (a--h). (a) The $\omega_{\boldsymbol{q}\nu}$ projected onto Te$_z$ and Co$_{xy}$ vibrations; the blue (orange) symbol size scales with the Te$_z$ (Co$_{xy}$) contribution. (b) Vibration-resolved phonon density of states; (c) Potential-energy surfaces for an AA-stacked bilayer constructed from two 1L CoTe$_2$ sheets at different interlayer separations, with atomic displacements along the eigenvectors of the 1L imaginary mode at M (upper) and K (lower) (Fig. 1(e)). (d) Fermi surfaces of the 2L (solid) and 1L (dashed) CoTe$_2$. (e,g) Electronic band structures of 1L and 2L, respectively, projected onto the Te-$p_{xy}$ and Te-$p_{z}$ states. (f) Schematic of the interlayer-coupling-induced band splitting at the $\Gamma$ when going from a 1L to a 2L CoTe$_2$. (h) Planar integral (along $xy$) of charge-density difference for 2L CoTe$_2$ relative to two isolated 1Ls, ($\Delta \rho(z)$), which quantifies net charge accumulation/depletion per $z$ plane along the stacking direction. See Fig. S3 SM for more details.
• Figure 4: (a) Band structures, density of states, and (b) Fermi surfaces of the 2L, calculated with SOC (wSOC) and without SOC (woSOC).
• Figure 5: Calculated quantities elucidating the EPC, the superconducting mechanism, and the SOC effect in the 2L. (a) The $\omega_{\boldsymbol{q}\nu}$ computed using an electron broadening of 0.01 Ry; the red symbol size scales with $\lambda_{\boldsymbol{q}\nu}$. (b) $\alpha^{2}F(\omega)$ and $\lambda(\omega)$ calculated with and without SOC; the shaded region marks where SOC induces the largest deviations. (c) BZ distribution of $\lambda^{\text{SOC}}_{\boldsymbol{q}}(\omega_1,\omega_2)$ with $\omega_1 = 3.50$ meV and $\omega_2 = 8.15$ meV, quantifying the contribution of the phonon modes $(\boldsymbol{q},\nu)$ within the energy window $[\omega_1,\omega_2]$ to $\alpha^{2}F(\omega)$. (d) BZ distribution of $\Delta\lambda_{\boldsymbol{q}}(\omega_1,\omega_2)=\lambda^{\text{woSOC}}_{\boldsymbol{q}}(\omega_1,\omega_2)- \lambda^{\text{SOC}}_{\boldsymbol{q}}(\omega_1,\omega_2)$. The double-headed arrows in (c) and (d) denote representative $\boldsymbol{q}$ vectors where $\lambda^{\mathrm{SOC}}_{\boldsymbol{q}}(\omega_1,\omega_2)$ and $|\Delta\lambda_{\boldsymbol{q}}|$ are particularly pronounced, respectively. (e) Histograms of temperature-dependent superconducting gaps $\Delta_{\boldsymbol{k}}(T)$, for those electronic states, whose Kohn-Sham energies fall within $\pm 0.1$ eV of the Fermi level. (f, g) BZ distribution of $\Delta_{\boldsymbol{k}}(T)$ at $T = 2$ K calculated with and without SOC, respectively; the double-headed arrows are reproduced from panel (d).