  Defects can strongly influence the electronic, optical and mechanical
properties of 2D materials, making defect stability under different
thermodynamic conditions crucial for material-property engineering. In this
paper, we present an account of the structural and electronic characteristics
of point defects in monolayer transition metal dichalcogenides $MX_2$ with
$M$=Mo/W and $X$= S/Se, calculated with density-functional theory using the
hybrid HSE06 exchange correlation functional including many-body dispersion
corrections. For the simulation of charged defects, we employ a charge
compensation scheme based on the virtual crystal approximation (VCA). We relate
the stability and the electronic structure of charged vacancy defects in
monolayer MoS$_2$ to an explicit calculation of the S monovacancy in MoS$_2$
supported on Au(111), and find convincing indication that the defect is
negatively charged. Moreover, we show that the finite-temperature vibrational
contributions to the free energy of defect formation can change the stability
transition between adatoms and monovacancies by 300--400 K. Finally, we probe
defect vibrational properties by calculating a tip-enhanced Raman scattering
image of a vibrational mode of a MoS$_2$ cluster with and without an S
monovacancy.
