Effects of material thickness and surrounding dielectric medium on Coulomb interactions and two-dimensional excitons

We examine the impact of quantum confinement on the interaction potential between two charges in two-dimensional semiconductor nanosheets in solution. The resulting effective potential depends on two length scales, namely, the thickness $d$ and an emergent length scale ${d}^*\ensuremath{\equiv}\ensuremath{\epsilon}d/{\ensuremath{\epsilon}}_{\text{sol}}$, where $\ensuremath{\epsilon}$ is the permittivity of the nanosheet and ${\ensuremath{\epsilon}}_{\text{sol}}$ is the permittivity of the solvent. In particular, quantum confinement, and not electrostatics, is responsible for the logarithmic behavior of the effective potential for separations smaller than $d$, instead of the one-over-distance bulk Coulomb interaction. Finally, we corroborate that the exciton binding energy also depends on the two-dimensional Bohr radius ${a}₀$ in addition to the length scales $d$ and ${d}^*$ and analyze the consequences of this dependence.