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The electron-phonon interaction contribution to the electronic energies isincluded in density functional total energy calculations with ab initiopseudopotentials via the Allen formalism [Phys. Rev. B 18, 5217 (1978)] toobtain temperature dependent electronic structure of diamond and silicon. Thismethod allows us to obtain the thermally-averaged ab initio electronicstructure in a straightforward and computationally inexpensive way. Ourinvestigations on the finite temperature electronic structure of diamond andsilicon lead to a new criterion, temperature transferability, which is requiredin the ab initio pseudopotentials for temperature dependent studies. Thetemperature transferability of the Troullier-Martins pseudopotentials used inthis work is strongly dependent on the cut-off radius and the inclusion of theunbound 3d$^0$ state. The finite temperature indirect band gaps are highlysensitive to the choice of cut-off radius used in the pseudopotentials. Thefinite temperature band structures and density of states show that thermalvibrations affect the electron energies throughout the valence and conductionband. We compare our results on the band gap shifts with that due to theDebye-Waller term in the Allen-Heine theory and discuss the observeddifferences in the zero point and high temperature band gap shifts. Although,the electron energy shifts in the highest occupied valence band and lowestunoccupied conduction band enable to obtain the changes in the indirect anddirect band gaps at finite temperatures, the shifts in other electronic levelswith temperature enable investigations into the finite temperature valencecharge distribution in the bonding region. Thus, we demonstrate that the Allentheory provides a simple and theoretically justified formalism to obtain finitetemperature valence electron charge densities that go beyond the rigidpseudo-atom approximation.

Finite temperature electronic structure of Diamond and Silicon