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

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.

💡 Research Summary

In this work, the authors employ first‑principles density‑functional theory (DFT) and density‑functional perturbation theory (DFPT) to investigate the lattice dynamics, electronic structure, and electron‑phonon coupling (EPC) of monolayer (1L) and bilayer (2L) 1T‑CoTe₂. Using the PBE exchange‑correlation functional and three different pseudopotentials (PAW, ONCV, and PseudoDojo), they ensure the robustness of their results. A vacuum spacing of ~15 Å isolates the two‑dimensional slabs, and dense k‑ and q‑meshes are used for accurate Brillouin‑zone sampling. Anharmonic effects at finite temperature are treated with the stochastic self‑consistent harmonic approximation (SSCHA), while machine‑learning potentials accelerate high‑temperature free‑energy calculations.

The monolayer is found to be dynamically unstable at low temperature. Phonon dispersions display imaginary frequencies near the M and K points, originating from out‑of‑plane Te‑p_z and in‑plane Co‑xy vibrations. Real‑space eigenvectors reveal double‑well potential energy surfaces, confirming a structural instability. By recomputing phonon spectra with a larger electronic smearing (0.018 Ry) to eliminate the imaginary modes, the authors expose a pronounced softening of the lowest acoustic branch along the M–K path. The generalized static electronic susceptibility χ_qν and its bare counterpart χ′_q show strong enhancements around the same wave vectors, indicating that the instability is driven primarily by EPC rather than pure Fermi‑surface nesting. Orbital‑projected band structures reveal two hole pockets at the Γ point, composed of hybridized Co‑d and Te‑p states; inter‑pocket nesting vectors coincide with the q‑vectors where χ_qν peaks, further amplifying EPC.

When a second layer is added (AA‑stacked bilayer with an interlayer spacing of 2.66 Å), the situation changes dramatically. The interlayer coupling induces a redistribution of Te‑p_z charge density, which modifies the Fermi surface and reduces the weight of the p‑d hybridized states responsible for strong EPC in the monolayer. Phonon calculations for the bilayer show that, except for a tiny residual imaginary branch near Γ (≈ ‑0.29 meV), all modes are stable. The EPC constant λ drops to ~0.68, and the Eliashberg spectral function α²F(ω) yields an estimated superconducting transition temperature Tc ≈ 4.7 K (using μ* = 0.10 in the McMillan‑Allen‑Dynes formula). This phonon‑mediated superconductivity emerges solely because the interlayer interaction quenches the dynamical instability and weakens the EPC to a regime where Cooper pairing can develop without destroying the lattice.

The authors also explore the effect of spin‑orbit coupling (SOC). Including SOC in the bilayer calculations splits the electronic bands slightly and further reduces the EPC matrix elements, lowering λ to ~0.60 and suppressing Tc to about 3.9 K. Thus SOC acts as a detrimental factor for superconductivity in this system.

Overall, the study demonstrates that interlayer coupling in van‑der‑Waals materials can simultaneously stabilize the crystal structure and enable superconductivity by reshaping the electronic landscape and moderating electron‑phonon interactions. The findings suggest a general route to engineer quantum phases in layered transition‑metal dichalcogenides: by tuning the number of layers (or interlayer distance) one can control charge redistribution, EPC strength, and consequently the emergence of superconductivity or other collective phenomena.