  Two-dimensional (2D) semiconducting transition metal dichalcogenides such as
MoS$_2$ have attracted extensive research interests for potential applications
in optoelectronics, spintronics, photovoltaics, and catalysis. To harness the
potential of these materials for electronic devices requires a better
understanding of how defects control the carrier concentration, character, and
mobility. Utilizing a correction scheme developed by Freysoldt and Neugebauer
to ensure the appropriate electrostatic boundary conditions for charged defects
in 2D materials, we perform density functional theory calculations to compute
formation energies and charge transition levels associated with sulfur
vacancies in monolayer and layered bulk MoS$_2$. We investigate the convergence
of these defect properties with respect to vacuum spacing, in-plane supercell
dimensions, and different levels of theory. We also analyze the electronic
structures of the defects in different charge states to gain insights into the
effect of defects on bonding and magnetism. We predict that both vacancy
structures undergo a Jahn-Teller distortion, which helps stabilize the sulfur
vacancy in the $-1$ charged state.
