Crystal Potential and Energy Bands of Semiconductors. III. Self-Consistent Calculations for Silicon

Abstract

An approximately self-consistent crystal potential is constructed for Si from a superposition of free-atom core and a sampling of crystal valence band charge densities. Valence-core exchange is calculated directly from core wave functions while valence-valence exchange is included using momentum-independent and momentum-dependent approximations taken from the results for a free-electron gas. The resulting crystal potential is surprisingly similar to one previously obtained by Woodruff from a superposition of free-atom charge densities. The calculated valence wave functions in the core region differ substantially from those of Woodruff because of the variational method used by him to calculate wave functions in that region. As a result the calculated energy gap is changed from Woodruff's value of 4 ev to about 1.5 ev, in substantially better agreement with the experimental value (1.1 ev). The various uncertainties in the calculation are listed; it is concluded that the relative position of levels near the band gap should be correct to within about 1 ev. Effective masses are also calculated and compared with experiment; the agreement is quite good.