Effects of Two-Dimensional Material Thickness and Surrounding Dielectric Medium on Coulomb Interactions and 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 $d^* \equiv \epsilon d / \epsilon_{\text{sol}}$, where $\epsilon$ is the permittivity of the nanosheet and $\epsilon_{\text{sol}}$ 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 exciton Bohr radius $a_0$ in addition to the length scales $d$ and $d^*$ and analyze the consequences of this dependence.