  We investigate an excitonic peak appearing in low-temperature
photoluminescence of monolayer transition metal dichalcogenides (TMDCs), which
is commonly associated with defects and disorder. First, to uncover the
intrinsic origin of defect-related excitons, we study their dependence on gate
voltage, excitation power, and temperature in a prototypical TMDC monolayer,
$MoS_2$. We show that the entire range of behaviors of defect-related excitons
can be understood in terms of a simple model, where neutral excitons are bound
to ionized donor levels, likely related to sulphur vacancies, with a density of
$7\cdot10^{11} cm^{-2}$. Second, to study the extrinsic origin of
defect-related excitons, we controllably deposit oxygen molecules in-situ onto
the surface of $MoS_2$ kept at cryogenic temperature. We find that in addition
to trivial p-doping of $3\cdot10^{12} cm^{-2}$, oxygen affects the formation of
defect-related excitons by functionalizing the vacancy. Combined, our results
uncover the origin of defect-related excitons, suggest a simple and conclusive
approach to track the functionalization of TMDCs, benchmark device quality, and
pave the way towards exciton engineering in hybrid organic-inorganic TMDC
devices.
