Need for a Nonlocal Correlation Potential in Silicon

An attempt is made to fit cyclotron masses and principal energy gaps for silicon using a Heine-Abarenkov-type determination of the core-valence interaction fitted to the atomic spectra of ${\mathrm{Si}}^{3+}$. The valence-valence exchange and correlation potential is approximated by a local potential. The masses and gaps are found to obey a $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}-\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}\text{'}\text{'}$ product relation under variations of the local potential. The theoretical product is 10-25% smaller in absolute value than the experimental product. We conclude that a local approximation to exchange and correlation is inadequate for silicon. If the masses are fitted the gaps are in error by 0.5-0.7 eV. We suggest that screened Hartree-Fock exchange may provide the nonlocality required to overcome these fitting difficulties.