Scaling Universality between Band Gap and Exciton Binding Energy of Two-Dimensional Semiconductors

Using first-principles $\text{ }GW$ Bethe-Salpeter equation calculations and the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ theory, we unambiguously show that for two-dimensional (2D) semiconductors, there exists a robust linear scaling law between the quasiparticle band gap (${E}_{g}$) and the exciton binding energy (${E}_{b}$), namely, ${E}_{b}\ensuremath{\approx}{E}_{g}/4$, regardless of their lattice configuration, bonding characteristic, as well as the topological property. Such a parameter-free universality is never observed in their three-dimensional counterparts. By deriving a simple expression for the 2D polarizability merely with respect to ${E}_{g}$, and adopting the screened hydrogen model for ${E}_{b}$, the linear scaling law can be deduced analytically. This work provides an opportunity to better understand the fantastic consequence of the 2D nature for materials, and thus offers valuable guidance for their property modulation and performance control.