Wave function forms of interlayer excitons in bilayer transition metal dichalcogenides

We numerically solve the electron-hole relative wave function of interlayer excitons in bilayer transition metal dichalcogenides, taking into account the screening effects from both the constituent transition metal dichalcogenides layers and the surrounding dielectric environment. We find that the wave function of the 1s ground state is close to the gaussian form, rather than the well-known exponential decay form of the two-dimensional hydrogen model. Meanwhile, the 2s state has an energy $E_{2s}$ significantly higher than $E_{2p}$ of the 2p state, but becomes close to $E_{3d}$ of the 3d state with $E_{2s}-E_{2p} \approx E_{3d}-E_{2p} \approx E_{2p}-E_{1s}$ under a large interlayer separation and weak environmental screening. Under general conditions, the solved 1s, 2p and 3d wave functions can be fit nearly perfectly by simple analytic forms which smoothly cross from gaussian to exponential decay. These analytic forms can facilitate the accurate evaluation of various exciton quantities for device applications.