of the lattice structure[@GrunerBook2009]).

Since the evolution of the bandstructure with doping is similar in several semiconducting TMDs[@BrummePRB2015; @DasPRB2015; @BrummePRB2016; @WuNatCommun2016; @KangNanoLett2017], this mechanism is likely not restricted to gated MoS$_2$. The $T_c$ increase in correspondence to a Lifshitz transition is reminiscent of a similar behavior observed in CaFe$_2$As$_2$ under pressure[@GonnelliSciRep2016], suggesting this may be a general feature across different classes of materials.

We note that the maximum $T_c\sim11$K is reached at $n_{2d}\simeq 12\cdot 10^{13}$cm$^{-2}$, as reported in Ref.. This is a doping level larger than any doping level which can be associated with the kink. Thus, the Q$_2$ sub-band must be highly populated when the maximum $T_c$ is observed. We address this quantitatively with the Allen-Dynes formula[@AllenDynes], which describes the dependence of $T_c$ by a numerical approximation of the Eliashberg theory accurate for materials with a total $\lambda\lesssim1.5$[@AllenDynes]: $$\label{EqAllenDynes}
T_c(n_{2d})=\frac{\omega_{ln}}{1.2}\mathrm{exp}\left\lbrace\frac{-1.04\left[1+\lambda(n_{2d})\right]}{\lambda(n_{2d})-\mu^*\left[1+0.62\lambda(n_{2d})\right]}\right\rbrace$$ where $\lambda(n_{2d})$ is the total EPC as a function of doping, $\omega_{ln}$ is the representative phonon frequency and $\mu^*$ is the Coulomb pseudo-potential. It is important to evaluate the increase in EPC between the non-superconducting region ($n_{2d}\lesssim6\times10^{13}$cm$^{-2}$) and the superconducting one, i.e. the enhancement in $\lambda$ due to the crossing of the sub-band at Q$_2$. $\Delta\lambda=\lambda(T_c)-\lambda(T_c=0)$ indicates the EPC 