Calculation of the Formation Energy of a Schottky Defect in Germanium

The formation energy of a Schottky defect in germanium has been calculated from first principles by using the valence-bond method of Heitler and London. The formation energy of a vacancy is given by the difference between the ground-state energy of the crystal containing the defect and that of the perfect crystal. The latter is derived by using a method based on the general directional theory of valence. The energy of the defect crystal is obtained by using the same general method but taking into account the possible pairing schemes for the vacancy electrons and applying the method of resonance. The tetrahedral valence state of germanium is used as the reference level for the various energies in the calculation. The numerical result obtained depends on the value chosen for the cohesive energy of germanium. Using an average of the three reported values for this quantity, we find for the formation energy at a Schottky defect $E_v=2.21\mathrm{}\pm 0.18$ eV. This is in good agreement with the experimental values.