Temperature dependence of band gaps in Si and Ge

We extend earlier calculations of the temperature dependence of the electronic states at the 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 E₁ and E₂ critical points, as well as the E₀ gap of Si and the E₀^' 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.