  Transition metal dichalcogenide field-effect transistors (FETs) have been
actively explored for low-power electronics, light detection, and sensing.
Albeit promising, their performance is strongly limited by low-frequency noise
(LFN). Here, we report on the study of LFN in MoS$_2$ FETs on SiO$_2$
substrates in ambient conditions using photodoping. Using this external
excitation source allows us to access different non-equilibrium steady states
and cross over different noise regimes. We observe a dependence of the noise
power spectrum with the transient decay time window, approaching $1/f$-type
when the system is closer to equilibrium, and identify a dependence of the LFN
with channel thickness. Monolayer/bilayer devices exhibit random telegraph
noise for insulating regimes and $1/f$-type Hooge mobility fluctuations (HMF)
for conductive regimes. Thicker devices exhibit mainly $1/f$-type carrier
number fluctuations (CNF). In the latter, we observe a photodoping-induced
change from a near parabolic to a near linear dependence of the inverse $1/f$
noise amplitude above the threshold gate voltage. This change indicates a
crossover in the LFN mechanism from CNF to HMF. We demonstrate that the study
of conductance and noise under photodoping is an effective tool to identify
dominating carrier noise mechanisms in few-atomic-layer FETs for a wide range
of doping regimes.
