  Monolayer $\mathrm{MoS_2}$ under strain has many interesting properties and
possible applications in technology. A recent experimental study examined the
effect of strain on the bandgap of monolayer $\mathrm{MoS_2}$ on a mildly
curved graphite surface, reporting that under biaxial strain with a Poisson's
ratio of 0.44, the bandgap decreases at a rate of 400 meV/% strain. In this
work, we performed density functional theory (DFT) calculations for a
free-standing $\mathrm{MoS_2}$ monolayer, using the generalized gradient
approximation (GGA) PBE, the hybrid functional HSE06, and many-body
perturbation theory with the GW approximation using PBE wavefunctions
(G0W0@PBE). We found that under biaxial strain with the experimental Poisson's
ratio, the bandgap decreases at rates of 63 meV/% strain (PBE), 73 meV/% strain
(HSE06), and 43 meV/% strain (G0W0@PBE), which are significantly smaller than
the experimental rate. We also found that PBE predicts a similarly smaller rate
(90 meV/% strain) for a different Poisson's ratio of 0.25. Spin-orbit
correction (SOC) has little effect on the gap or its strain dependence.
Additionally, we observed a semiconductor-to-metal transition under an equal
tensile biaxial strain of 10% and a transition from a direct to an indirect
bandgap, consistent with previous theoretical work.
