Electric field induced out-of-plane second-order optical nonlinearity in monolayer transition metal dichalcogenides

Second-order nonlinear optical effects in monolayer transition-metal dichalcogenides (ML TMDCs) have attracted significant attention; these are almost exclusively associated with their in-plane second-order nonlinear susceptibility arising from the intrinsically broken in-plane inversion symmetry. However, a key challenge is the induction and manipulation of out-of-plane symmetry breaking that governs out-of-plane polarized second-order nonlinear processes such as second-harmonic generation in ML TMDCs. Using first-principle density functional theory, we show that applying an electrostatic field perpendicular to the monolayer plane can induce out-of-plane second-order nonlinear susceptibility (${\ensuremath{\chi}}_{zxx}={\ensuremath{\chi}}_{zyy}$) in the visible wavelength range in the four most representative TMDCs (${\mathrm{MoS}}_{2}, {\mathrm{MoSe}}_{2}, {\mathrm{WS}}_{2}, {\mathrm{WSe}}_{2}$), with magnitude comparable to their intrinsic in-plane components (${\ensuremath{\chi}}_{yyy}$). The susceptibility peak values (${\ensuremath{\chi}}_{zxx}^{*}, {\ensuremath{\chi}}_{yyy}^{*}$), with incident energy around half of the C exciton energy in each material, exhibit a linear dependence on the applied field strength $E$. This behavior originates from the joint effects of field-induced asymmetric out-of-plane charge density distribution and structural deformation. Although the asymmetric charge distribution predominantly governs this effect, the structural deformation also contributes to the overall response in all four ML TMDCs. To accurately describe and predict the induced out-of-plane nonlinear susceptibility in ML TMDCs under varied $E$, we introduce a structural deformation descriptor $\ensuremath{\tau}$ which exhibits a linear correlation with ${\ensuremath{\chi}}_{zxx}^{*}$ to measure the magnitude of the electric-field-induced out-of-plane dipole moment. Our study provides an easy-to-implement approach for generating and tuning the out-of-plane second-order optical nonlinearity in ML TMDCs and hence opens a different avenue for investigating active control of their second-order nonlinear optical processes.