the contacted one. This difference points to a further tensile strain. Refs. suggested that uniaxial tensile strain on 1L-MoS$_2$ induces a E$_{2g}^1$ softening and a splitting in two components: E$_{2g}^{1+}$ and E$_{2g}^{1-}$[@LeeNC2017; @ConlNL2013]. The shift rates for E$_{2g}^{1+}$ and E$_{2g}^{1-}$ are from -0.9 to -1.0cm$^{-1}$/% and from -4.0 to -4.5cm$^{-1}$/%, respectively[@LeeNC2017; @ConlNL2013]. We do not observe splitting, pointing towards a biaxial strain. As for Ref., we calculate a shift rate of E$_{2g}^1$ for biaxial strain from -7.2 to -8.2cm$^{-1}$/%. The amount of tensile strain on the 4L-MoS$_2$ device can thus be estimated. The E$_{2g}^1$ up-shift difference between contacted and as-prepared 4L-MoS$_2$, $\Delta$Pos(E$_{2g}^1$), at 4K is$\sim$-1.0cm$^{-1}$, corresponding to an additional $\sim$0.13% biaxial tensile strain. Thus, assuming a 0.1% strain for the as-prepared 4L-MoS$_2$ due to TEC mismatch with SiO$_2$, we estimate the total strain in the contacted 4L-MoS$_2$ to be$\sim$0.23% at$\sim$4K.

![a) Surface carrier densities required to cross the Q$_1$ sub-band in FL-MoS$_2$ as a function of tensile strain. Theoretical values for 1L (black dots and line) and 3L (red triangle and line) from Ref.; values for 4L (green diamonds and line) are by linear extrapolation. Blue diamond is the present experiment. b) EPC enhancement due to the crossing of the 