Temperature dependence of band gaps in Si and Ge
We extend earlier calculations of the temperature dependence of the electronic states at the \ensuremath{\Gamma} 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.