Electron–electron and electron-hole interactions in small semiconductor crystallites: The size dependence of the lowest excited electronic state

We model, in an elementary way, the excited electronic states of semiconductor crystallites sufficiently small (∼50 Å diam) that the electronic properties differ from those of bulk materials. In this limit the excited states and ionization processes assume a molecular-like character. However, diffraction of bonding electrons by the periodic lattice potential remains of paramount importance in the crystallite electronic structure. Schrödinger’s equation is solved at the same level of approximation as used in the analysis of bulk crystalline electron-hole states (Wannier excitons). Kinetic energy is treated by the effective mass approximation, and the potential energy is due to high frequency dielectric solvation by atomic core electrons. An approximate formula is given for the lowest excited electronic state energy. This expression is dependent upon bulk electronic properties, and contains no adjustable parameters. The optical f number for absorption and emission is also considered. The same model is applied to the problem of two conduction band electrons in a small crystallite, in order to understand how the redox potential of excess electrons depends upon crystallite size.