Electron-phonon coupling, critical temperatures, and gaps in ${\mathrm{NbSe}}_2/{\mathrm{MoS}}_2$ Ising superconductors

Utilizing Migdal-Eliashberg theory of superconductivity within the first-principles calculations, we work out the role of electron-phonon coupling (EPC) and anisotropic superconducting properties of a recently discovered [Baidya et al., Phys. Rev. B 104, 174510 (2021)] 2D van der Waals heterostructure comprising a single layer of ${\mathrm{MoS}}_2$ and few layers of ${\mathrm{NbSe}}_2$. We find strong EPC and a softening of phonon modes in the lowest acoustic branch. While the single ${\mathrm{MoS}}_2$ layer does not actively contribute to the EPC, it significantly elevates the superconducting critical temperature ($T_c$) compared to monolayer ${\mathrm{NbSe}}_2$. This is attributed to the degradation of the charge-density wave by the ${\mathrm{MoS}}_2$ layer. Notably, we observe a two-gap superconductivity in ${\mathrm{NbSe}}_2/{\mathrm{MoS}}_2$ and extend our study to three layers of ${\mathrm{NbSe}}_2$. A reduction in $T_c$ with increasing thickness of ${\mathrm{NbSe}}_2$ is observed. Incorporation of spin-orbit coupling (SOC) suggests a possible mechanism for Ising superconductivity. We find that SOC reduces EPC while $T_c$ is suppressed concomitantly by about 5K, leading to a closer estimate of the experimental $T_c$.