*doping range* where the sub-band crossing occurs (between $R_s$ minimum and maximum, i.e. the *lower* and *upper* bounds of each kink sets the resolution of this approach). Each sub-band crossing starts after the $R_s$ minimum at lower doping, then develops in correspondence of the inflection point, and is complete once the $R_s$ maximum is reached.

We show evidence for this behavior in Fig.\[figure:kinks\], where we plot $T_c$ (panel d) and $R_s^0$ (panel e) as a function of $n_{2d}$. The electric field is applied both in liquid-top-gate (filled dots and dashed line) and dual-gate (solid red line) configurations. For comparable values of $n_{2d}$, the liquid-gate geometry features larger $R_s^0$ than back-gated. This difference is due to increased disorder introduced when $n_{2d}$ is modulated via ionic gating[@Gonnelli2dMater2017; @PiattiAppSS2017; @PiattiAPL2017; @GallagherNatCommun2015; @OvchinnikovNatCommun2016]. Two kinks appear in the $n_{2d}$ dependence of $R_s$: a low-doping one for $1.5\cdot 10^{13} \lesssim n_{2d} \lesssim 2\cdot 10^{13}$cm$^{-2}$, and a high-doping one for $7\cdot 10^{13} \lesssim n_{2d} \lesssim 9\cdot 10^{13}$cm$^{-2}$. The plot of the SC dome of gated MoS$_2$ on the same $n_{2d}$ scale shows that the low-doping kink appears well before the SC onset, while the second appears immediately after, before the maximum $T_c$ is reached.

These results 