Strain-Enabled Giant Second-Order Susceptibility in Monolayer WSe$_2$

Monolayer WSe$_2$ (ML WSe$_2$) exhibits a high second-harmonic generation (SHG) efficiency under single 1-photon (1-p) or 2-photon (2-p) resonant excitation conditions due to enhanced second-order susceptibility compared with off-resonance excitation states \cite{lin2021narrow,wang2015giant}. Here, we propose a novel strain engineering approach to dramatically boost the in-plane second-order nonlinear susceptibility ($\chi_{yyy}$ ) of ML WSe$_2$ by tuning the biaxial strain to shift two K-valley excitons (the A-exciton and a high-lying exciton (HX)) into double resonance. We first identify the A-exciton and HX from the 2D Mott-Wannier model for pristine ML WSe$_2$ and calculate the $\chi_{yyy}$ under either 1-p or 2-p resonance excitations, and observe a $\sim$ 39-fold $\chi_{yyy}$ enhancement arising from the 2-p HX resonance state compared with the A-exciton case. By applying a small uniform biaxial strain (0.16\%), we observe an exciton double resonance state ($E_{\rm{HX}}$ = 2$E_{\rm{A}}$, $E_{\rm{HX}}$ and $E_{\rm{A}}$ are the exciton absorption energies), which yields up to an additional 52-fold enhancement in $\chi_{yyy}$ compared to the 2-p HX resonance state, indicating an overall $\sim$ 2000-fold enhancement compared to the single 2-p A-exciton resonance state reported in Ref \cite{wang2015giant}. Further exploration of the strain-engineered exciton states (with biaxial strain around 0.16\%) reveals that double resonance also occurs at other wavevectors near the K valley, leading to other enhancement states in $\chi_{yyy}$, confirming that strain engineering is an effective approach for enhancing $\chi_{yyy}$. Our findings suggest new avenues for strain engineering the optical properties of 2D materials for novel nonlinear optoelectronic applications.