Strain-induced Giant Second-harmonic Generation in Monolayered $2H$-MoX$_2$ (X=S,Se,Te)

Dynamic second-order nonlinear susceptibilities, $\chi^{(2)}(2\omega,\omega,\omega)\equiv \chi^{(2)}(\omega)$, are calculated here within a fully first-principles scheme for monolayered molybdenum dichalcogenides, $2H$-MoX$_2$ (X=S,Se,Te). The absolute values of $\chi^{(2)}(\omega)$ across the three chalcogens critically depend on the band gap energies upon uniform strain, yielding the highest $\chi^{(2)}(0)\sim$ 140 pm/V for MoTe$_2$ in the static limit. Under this uniform in-plane stress, $2H$-MoX$_2$ can undergo direct-to-indirect transition of band gaps, which in turn substantially affects $\chi^{(2)}(\omega)$. The tunability of $\chi^{(2)}(\omega)$ by either compressive or tensile strain is demonstrated especially for two important experimental wavelengths, 1064 nm and 800 nm, where resonantly enhanced non-linear effects can be exploited: $\chi^{(2)}$ of MoSe$_2$ and MoTe$_2$ approach $\sim$800 pm/V with -2\% strain at 1064 nm.