A tight-binding model for illustrating exciton confinement in semiconductor nanocrystals

The Brus equation describes the relation between the lowest energy of an electron-hole pair and the size of a semiconductor crystallite. However, taking the strong confinement regime as a starting point, the equation does not cover the transition from weak to strong confinement, the accompanying phenomenon of charge-carrier delocalization, or the change in the transition dipole moment of the electron-hole pair state. Here, we use a one-dimensional, two-particle Hubbard model for interacting electron-hole pairs that extends the well-known tight-binding approach through a point-like electron-hole interaction. On infinite chains, the resulting exciton states exhibit the known relation between the Bohr radius, the exciton binding energy, and the effective mass of the charge carriers. Moreover, by introducing infinite-well boundary conditions, the model enables the transition of the exciton states from weak to strong confinement to be tracked, while straightforward adaptations provide insights into the relation between defects, exciton localization, and confinement. In addition, by introducing the dipole operator, the variation of the transition dipole moment can be mapped when shifting from electron-hole pairs in strong confinement to delocalized and localized excitons in weak confinement. The proposed model system can be readily implemented and extended to different multi-carrier states, thus providing researchers a tool for exploring, understanding, and teaching confinement effects in semiconductor nanocrystals under different conditions.