Some recent developments in the calculation of crystal energy bands — new results for the germanium crystal

Abstract

Synopsis Some current methods for determining the electronic structure of crystals are discussed briefly. Among the techniques treated are the cellular method, the orthogonalized plane wave (OPW) method, and the augmented plane wave method. A recently completed calculation of the band structure of the germanium crystal by the OPW method is described. It is found that the band structure obtained depends quite critically upon the assumed crystal potential. The structure of the top of the valence band, which lies at the central point of the reduced zone, is strongly influenced by spin-orbit interaction and by the proximity of the lower conduction bands. The crystal eigensolutions belonging to the two lowest conduction bands converge at different rates and respond in different ways to arbitrary changes in the crystal potential. The latter fact is used as the basis for an interpretation of certain optical properties of germanium-silicon alloys. Our solution indicates that the lowest conduction band had a well-defined minimum at the central point of the reduced zone. If our solution were carried to a higher order approximation, we would expect to find the minimum of this conduction band at positions along the [111] axes rather than at the central point of the zone. The next-to-the-lowest conduction band has six minima along the [100] axes. As we proceed from pure germanium to pure silicon through intermediate compositions of Ge-Si alloys, the states normally occupied by electrons shift from the [111] minima to the [100] minima.