  Strong electron-phonon interaction which limits electronic mobility of
semiconductors can also have significant effects on phonon frequencies. The
latter is the key to the use of Raman spectroscopy for nondestructive
characterization of doping in graphene-based devices. Using in-situ Raman
scattering from single layer MoS$_2$ electrochemically top-gated field effect
transistor (FET), we show softening and broadening of A$_{1g}$ phonon with
electron doping whereas the other Raman active E$_{2g}^{1}$ mode remains
essentially inert. Confirming these results with first-principles density
functional theory based calculations, we use group theoretical arguments to
explain why A$_{1g}$ mode specifically exhibits a strong sensitivity to
electron doping. Our work opens up the use of Raman spectroscopy in probing the
level of doping in single layer MoS$_2$-based FETs, which have a high on-off
ratio and are of enormous technological significance.
