Optical signatures of states bound to vacancy defects in monolayer MoS$_2$

We show that pristine MoS$_2$ single layer (SL) exhibits two bandgaps $E_{gparallel}=1.9$ eV and $E_{gperp}=3.2$ eV for the optical in-plane and out-of-plane susceptibilities $chi_parallel$ and $chi_perp$, respectively. In particular, we show that odd states bound to vacancy defects (VDs) lead to resonances in $chi_perp$ inside $E_{gperp}$ in MoS$_2$ SL with VDs. We use density functional theory, the tight-binding model, and the Dirac equation to study MoS$_2$ SL with three types of VDs: (i) Mo-vacancy, (ii) S$_2$-vacancy, and (iii) 3$times$MoS$_2$ quantum antidot. The resulting optical spectra identify and characterize the VDs.