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Quantifying the Relationship Between Strain and Bandgap in Thin Ga$_2$Se$_2$

We present a rigorous analysis that combines theory, simulation, and experimental measurements to quantify the relationship between strain and bandgap in two dimensional gallium selenide (Ga$_2$Se$_2$). Experimentally, we transfer thin Ga$_2$Se$_2$ flakes onto patterned substrates to deterministically induce multiaxial localized strain. We quantify the local strain using a combination of atomic force microscopy (AFM) measurements and COMSOL Multiphysics simulation. We then experimentally map the strain-induced bandgap shifts using high-resolution hyperspectral PL imaging to generate a robust and statistically significant dataset. We systematically fit this data to extract gauge factors that relate the bandgap shift to the local uniaxial and biaxial strain. We then compute the uniaxial and biaxial strain gauge factors via density functional theory (DFT) and find excellent agreement with the experimentally-determined values. Finally, we show that a simple model that computes bandgap shifts from the local uniaxial and biaxial strain predicts the observed multiaxial bandgap shift with less than 10\% error. The combined results provide a framework for deterministic realization of tailored bandgap profiles induced by controlled strain applied to Ga$_2$Se$_2$, with implications for the future realization of localized quantum emitters for quantum photonic applications.