Self-consistent electronic structure of a vacancy in aluminum

The electronic structure of a vacancy in Al has been studied using a self-consistent pseudopotential scheme, in which the environment of the vacancy was simulated by a supercell containing 27 atomic sites. The charge-density perturbation due to the vacancy was found to be quite short ranged, indicating that the supercell used was large enough to effectively isolate individual vacancy potentials. At the center of the vacancy, the charge density was 0.27 electrons per atom; this value is compared with the results of previous calculations. The local densities of states at a vacancy and its first- and third-nearest-neighbor shells were investigated, and evidence for a vacancy-associated resonance state was found at ∼0.3 Ry below the Fermi energy. Finally, the vacancy-formation energy was also calculated in the ideal (unrelaxed) structure using the $\overrightarrow{\mathrm{G}}$-space formalism for calculating total energies. A value of 1.9 eV was obtained; this is 2.9 times the experimental value. The effects of lattice relaxation, the supercell method, and defect potentials on the electronic structure and the resulting formation energy are discussed.