Energy-Band Interpolation Scheme Based on a Pseudopotential

Abstract

In this paper an interpolation scheme is developed that depends on only a few parameters. This is done by observing that the effective potential for electrons near the Fermi level can be split into two parts, the part due to the core, and the part due to the other valence electrons. It is assumed that for semiconductors the relative effect of the core is small, so that it can be replaced by an effective repulsive potential. In this way two-parameter pseudopotentials are constructed for diamond and Si that give good agreement with orthogonalized plane wave calculations and experiment at special points of the Brillouin zone, and also yield reasonable results for the bands at other points of the zone. A by-product of the calculations is the discovery of an error in the model for the valence bands of Si near the center of the zone proposed by Dresselhaus. A compromise model is proposed in good agreement with theory and experiment. Good results are obtained for Ge with a three-parameter pseudopotential. Finally, the many experimental facts that have been deduced about the band structures of Si and Ge are augmented by the results of the pseudopotential calculations to yield fairly accurate ($\delta E\lesssim 0.05$ ry) sketches of the energy bands of these crystals along the [100] and [111] directions in the neighborhood of the energy gap.