Generalized scaling law for exciton binding energy in two-dimensional materials

Binding energy calculation in two-dimensional (2D) materials is crucial in determining their electronic and optical properties pertaining to enhanced Coulomb interactions between charge carriers due to quantum confinement and reduced dielectric screening. Based on full solutions of the Schr\"odinger equation in screened hydrogen model with a modified Coulomb potential ($1/r^{\beta-2}$), we present a generalized and analytical scaling law for exciton binding energy, $E_{\beta} = E_{0}\times \big (\,a\beta^{b}+c\big )\, (\mu/\epsilon^{2})$, where $\beta$ is a fractional-dimension parameter accounted for the reduced dielectric screening. The model is able to provide accurate binding energies, benchmarked with the reported Bethe-Salpeter Equation (BSE) and experimental data, for 58 mono-layer 2D and 8 bulk materials respectively through $\beta$. For a given material, $\beta$ is varied from $\beta$ = 3 for bulk 3D materials to a value lying in the range 2.55$-$2.7 for 2D mono-layer materials. With $\beta_{\text{mean}}$ = 2.625, our model improves the average relative mean square error by 3 times in comparison to existing models. The results can be used for Coulomb engineering of exciton binding energies in the optimal design of 2D materials.