New Method for Treating Lattice Point Defects in Covalent Crystals

A new method is proposed for the calculation from first principles of the formation and migration energy of a vacancy or interstitial in covalent crystals. The formation energy of a point imperfection is given by the change in the electrostatic energy of the system of ions arising from the point defect plus the change in energy of the system of valence electrons due to their redistribution associated with the point defect. The redistribution of the valence electrons is determined from a pseudocrystal potential which results from orthogonalizing the valence-electron wave functions to the crystal wave functions of the closed-shell core electrons. The scattering of the valence electrons by the pseudocrystal potential is determined by using the $t$-matrix approximation. The formation energy of an interstitial in diamond, silicon, and germanium turns out to be 1.76, 1.09, and 0.93 eV, respectively; the migration energy of an interstitial is 0.85, 0.51, and 0.44 eV; the formation energy of a vacancy is 3.68, 2.13, and 1.91 eV; and the migration energy of a vacancy turns out to be 1.85, 1.09, and 0.98 eV. The general method presented for treating point defects in covalent crystals can be readily applied to determine the spectrum of localized states at point defects, which is of interest for optical investigations of point defects. Further, the derived expression for the energy of the system of valence electrons can readily be evaluated to show explicitly its dependence on the displacements of the lattice ions resulting from a point defect. Such an expression is needed, for example, for a self-consistent determination of the lattice distortion due to point defects, and furthermore can be used for calculating the elastic constants or lattice force constants from first principles. An approximate expression for the equation of state of the system of valence electrons is given.