Theory of the temperature dependence of the direct gap of germanium
A complete pseudopotential calculation of the temperature dependence of the ${\ensuremath{\Gamma}}_{{2}^{\ensuremath{'}}}$ conduction-band and the ${\ensuremath{\Gamma}}_{{25}^{\ensuremath{'}}}$ valence-band states is performed. The calculation uses the lattice dynamics of Weber's bond-charge model and a local pseudopotential with a basis of 59 plane waves. Debye-Waller (DW) terms and also self-energy terms are included. While the DW terms alone are only slightly larger than experimental results, the addition of the self-energy correction results in a total shift of the gap with increasing temperature which is nearly three times as large as that observed. These results are compared with calculations and experiments for the second-order Raman scattering of $c\ensuremath{-}\mathrm{G}\mathrm{e}$. We conclude that the pseudopotential-rigid-ion calculation overestimates the electron-2-TA-phonon coupling while it underestimates the coupling of the electrons with two TO phonons. The self-energy effects are particularly large for optical phonons connecting the ${\ensuremath{\Gamma}}_{{25}^{\ensuremath{'}}}$ with the ${L}_{{3}^{\ensuremath{'}}}$ valence bands.