Electron-phonon coupling, critical temperatures and gaps in $\rm{NbSe_2}$/$\rm{MoS_2}$ 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 [Appl. Phys. Lett. 120, 183101 (2022)] 2D van der Waals heterostructure comprising a single layer of MoS$_2$ and few layers of NbSe$_2$. We find strong EPC and a softening of phonon modes in the lowest acoustic branch. While the single MoS$_2$ layer does not actively contribute to the EPC, it significantly elevates the superconducting critical temperature ($T_c$) compared to monolayer NbSe$_2$. This is attributed to the degradation of the charge-density wave (CDW) by the MoS$_2$ layer. Notably, we observe a two-gap superconductivity in $\rm{NbSe_2}$/$\rm{MoS_2}$ and extend our study to three layers of NbSe$_2$. A reduction in $T_c$ with increasing thickness of NbSe$_2$ is observed. We confirm that this trend is consistent with recent experiments, if one goes beyond three layers of NbSe$_2$. We incorporated spin-orbit coupling (SOC) and suggest 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 $T_c$.