Theory of the temperature dependence of the direct gap of germanium

Abstract

A complete pseudopotential calculation of the temperature dependence of the ${\Gamma }_{2^{\prime }}$ conduction-band and the ${\Gamma }_{{25}^{\prime }}$ 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 $c-\mathrm{G}\mathrm{e}$. 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 ${\Gamma }_{{25}^{\prime }}$ with the $L_{3^{\prime }}$ valence bands.