We extend earlier calculations of the temperature dependence of the electronic states at the Γ point of Si and Ge to other points of the Brillouin zone. Thus we are able to calculate the temperature dependence of gaps and critical-point energies: the indirect gap and the $E_1$ and $E_2$ critical points, as well as the $E_0$ gap of Si and the $E_0^{\mathcal{’}}$ gap of Ge. Both the Fan self-energy and the Debye-Waller terms of the electron-phonon interaction are included. The theoretical results, corrected for the contribution of thermal expansion to the temperature shifts, show satisfactory agreement with experimental data.

• Received 17 September 1984

DOI:https://doi.org/10.1103/PhysRevB.31.2163