Crystal Potential and Correlation for Energy Bands in Valence Semiconductors

Abstract

A number of forms of the one-electron potential for a semiconductor are evaluated on the basis of orthogonalized-plane-wave (OPW) calculations for Si. Starting with the Hartree-Fock (HF) equations, a method of calculating the exact core exchange is developed and various approximations to the valence exchange are considered. Some of these approximations are considered only to illustrate the effect of certain features on the energy bands. None contain adjustable parameters. The k·p method in conjunction with a priori calculations at k=0 is proposed as a convergent method for obtaining the energy bands throughout the zone and for achieving self-consistency. The procedure is quite simple if the potential is local. A completely self-consistent calculation is carried out using the Slater approximation to the valence exchange. Although agreement with experimenta is reasonable, the differences are such that they cannot be attributed to computational errors. Following the procedure of Kleinman and Phillips the HF valence-exchange matrix elements were estimated from an 8-point sampling of the Brillouin zone. The calculated energy differences between the conduction and valence bands at symmetry points are much larger than the measured values, so that an effective potential including correlations is necessary. Correlations are included in generalized OPW equations. Because the core exchange is essentially unscreened, the core states for orthogonalization are still of the HF type. A nondiagonal self-energy operator is derived on the basis of a Coulomb-hole-plus-screened-exchange approximation suggested by Hedin. Calculations are performed at symmetry points using simplifications of this self-energy in which the nondiagonal components of the dielectric function are neglected. The results are not an improvement over the Slater approximation.