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Utilizing Migdal-Eliashberg theory of superconductivity within thefirst-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 heterostructurecomprising a single layer of MoS$_2$ and few layers of NbSe$_2$. We find strongEPC and a softening of phonon modes in the lowest acoustic branch. While thesingle MoS$_2$ layer does not actively contribute to the EPC, it significantlyelevates the superconducting critical temperature ($T_c$) compared to monolayerNbSe$_2$. 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$_2$. Areduction in $T_c$ with increasing thickness of NbSe$_2$ is observed. Weconfirm that this trend is consistent with recent experiments, if one goesbeyond three layers of NbSe$_2$. We incorporated spin-orbit coupling (SOC) andsuggest a possible mechanism for Ising superconductivity. We find that SOCreduces EPC while $T_c$ is suppressed concomitantly by about 5K, leading to acloser estimate of the experimental $T_c$.

Electron-phonon coupling, critical temperatures and gaps in $\rm{NbSe_2}$/$\rm{MoS_2}$ Ising Superconductors