Ab initio calculation of the formation energy of charged vacancies in germanium

Density functional theory (DFT) with local density approximation (LDA) has been used to calculate the formation energy (Ef) of the neutral and charged vacancies in germanium single crystal. The standard (four valence electrons) and harder (which treat the semicore 3d states of Ge as valence) projector augmented wave (PAW) potentials were used. Additionally, the effect of including on-site Coulomb interaction, U, for Ge semicore d states within the LDA+U approach was investigated. The LDA+U method improves the LDA band gap which allows investigating the dependence of formation energy of charged vacancies on Fermi level position in the band gap. It was shown that the calculated formation energies of the neutral and charged vacancies are in good agreement with published experimental data.