with N$\geq$4[@SpleNL2010]. When an electric field is applied perpendicular to the MoS$_2$ plane, inversion symmetry is broken and the global minimum of the conduction band is shifted to K/K’ in any FL-MoS$_2$[@BrummePRB2015], Figs.\[figure:bands\]b-d. The valleys at K/K’ and at Q/Q’ are characterized by a different electron-phonon coupling (EPC)[@GePRB2013] and, when inversion symmetry is broken, by a different spin-orbit coupling (SOC)[@KadantsevSSC2012]. In particular, both EPC and SOC are larger in the Q/Q’ valleys[@GePRB2013; @KadantsevSSC2012].

![image](Figure_bands_3.eps){width="80.00000%"}

The field-effect transistor (FET) architecture is ideally suited to control the electronic properties of 1L flakes, as it simultaneously provides an electrostatic control of the transverse electric field and the carrier density. In the electric-double-layer (EDL) technique[@FujimotoReview2013], the standard solid gate dielectric is replaced by an ionic medium, such as an ionic liquid or electrolyte. In this configuration, the EDL that forms at the ionic liquid/electrode interfaces supports electric fields in excess of$\sim$10MV/cm[@UenoReview2014], corresponding to surface carrier densities $n_{2d}\gtrsim10^{14}$cm$^{-2}$[@UenoReview2014]. Ionic-liquid gating has been used to tune the Fermi level, $E_F$, in TMDs and explore transport at different carrier concentrations[@SaitoReview2016; @YeScience2012; @BragNL2012; @YuNN2015; @XiPRL2016]. The vibrational properties of TMDs can also be controlled by means of the EDL technique, as suggested by gate-induced softening of Raman-active modes 