Self-Consistent Charge Densities and Surface-State Energies for a One-Dimensional Semi-Infinite Semiconductor Crystal

Publisher Website
Google Scholar
Abstract
Most attempts to understand physical properties of crystal surfaces make the implicit assumption that the charge distribution of the crystal interior is continued up to the surface where it then terminates abruptly. In order to examine this hypothesis, explicit calculations of ionic charge and its position dependence were performed for an essentially one-dimensional-model semiconductor under various conditions. The model is a variation of extended Hückel theory in which Madelung potentials are calculated from ionic charges of the previous iteration and are then incorporated into the Coulomb energies (diagonal Hamiltonian elements) of the subsequent iteration. In all calculations, regardless of the parameters chosen, the self-consistent ionic charges have a greater magnitude at the crystal surface than in the interior. In the less-ionic crystal models this deviation may be as large as 30% of the bulk-charge value. It is shown that the effect is due not to an abnormal Madelung potential at the surface as might be expected, but entirely to the boundary condition which requires wave functions to vanish outside the crystal. The effect of self-consistency upon calculated surface-state energies is qualitatively different in models simulating polar and nonpolar surfaces. By altering parameters associated with the terminal atoms in the chain it is shown that a rather small reconstruction may produce surprisingly large alterations in surface charges and with magnitudes that are not easy to predict in advance. The extension of the conclusions to three-dimensional crystals is discussed. Also, the rather novel numerical methods used in this research are shown to be applicable to more extensive studies of real systems.