Effect of Strain on the Band Gap of Monolayer MoS$_2$

Monolayer molybdenum disulfide ($\mathrm{MoS_2}$) 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). For the unstrained monolayer, we found a standard level of agreement for the bandgap between theory and experiment. For biaxial strain at the experimental Poisson's ratio, we found that 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. The strong disagreement between theory and experiment may reflect an unexpectedly strong effect of the substrate on the strain dependence of the gap. Additionally, we observed a transition from a direct to an indirect bandgap under strain, and (under an equal biaxial strain of 10\%) a semiconductor-to-metal transition, consistent with previous theoretical work.