elementary charge and $V_{th}$ is the threshold voltage required to observe a finite conductivity in the device. We neglect the quantum capacitance $C_q$ of MoS$_2$, since $C_q\gtrsim$100$\mu$F/cm$^2\gg C_{ox}$[@BrummePRB2015]. By using the dielectric constant of SiO$_2\,\epsilon_{ox}$=3.9[@ElKarehBook1995] and an oxide thickness $t_{ox}=90$nm (or $t_{ox}=285$nm, depending on the experiment) we obtain the $n_{2d}$ scale in the top axis of Fig.\[figure:transport\]c, in good agreement with the corresponding values in Fig.\[figure:transport\]a, estimated from the Hall effect measurements in Fig.\[figure:transport\]b.

The bandstructure of field-effect doped NL-MoS$_2$ depends on N[@BrummePRB2015] and strain[@BrummePRB2016]. A fully relaxed N-layer flake, with N$\geq$3, has been predicted to behave as follows[@BrummePRB2015; @BrummePRB2016]: For small doping ($x\lesssim 0.05$e$^-$/unit cell, Figs.\[figure:bands\]b and \[figure:kinks\]a) only the two spin-orbit split sub-bands at K/K’ are populated. At intermediate doping ($0.05\lesssim x \lesssim 0.1$ e$^-$/unit cell, Figs.\[figure:bands\]c and \[figure:kinks\]b), $E_F$ crosses the first spin-orbit split sub-band at Q/Q’ (labeled Q$_1$). For large doping ($x\gtrsim 0.1$ e$^-$/unit cell, Figs.\[figure:bands\]d and \[figure:kinks\]c) $E_F$ crosses the second sub-band (Q$_2$) and both valleys become highly populated[@BrummePRB2015]. Even larger doping ($x\gtrsim 0.35$ e$^-$/unit cell) eventually shifts the K/K’ valleys above $E_F$[@BrummePRB2015].

![image](Figure_kinks_6.eps){width="80.00000%"}

When $E_F$ crosses these high-energy sub-bands at Q/Q’, sharp kinks are expected to appear in the transconductance of gated FL-MoS$_2$[@BrummePRB2016] (see 