  Monolayer WSe$_2$ (ML WSe$_2$) exhibits high second harmonic generation (SHG)
efficiency under single 1-photon (1-p) or 2-photon (2-p) resonance conditions
due to enhanced second-order susceptibility compared with the off-resonance
state \cite{lin2021narrow,wang2015giant}. Here, we propose a novel strain
engineering approach to dramatically boost the in-plane second-order
susceptibility ($\chi_{yyy}$) 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 and compare the enhanced $\chi_{yyy}$ at single 1-p and 2-p resonance
states. The strongest enhancement arises from the 2-p resonance state of HX. By
applying a small uniform biaxial strain (0.16%), we observe an excitonic double
resonance state ($E_{\rm{HX}}$ = 2$E_{\rm{A}}$, $E_{\rm{HX}}$ and $E_{\rm{A}}$
are the exciton absorption energies). Strain-tuning can yield up to a $50$X
enhancement in $\chi_{yyy}$ compared to the single 2-p resonance state of HX,
resulting in a further three orders of magnitude enhancement in SHG efficiency.
Further exploration of strain states near 0.16\% reveals double resonance also
occurs at other wavevectors near the K valley, leading to a further increase in
$\chi_{yyy}$, confirming that strain engineering is an effective methodology
for enhancing $\chi_{yyy}$. Our findings suggest new avenues for strain
engineering the optical properties of 2D materials for novel nonlinear
optoelectronics applications.
