First-principles energy and stress fields in defected materials

Abstract New methods are presented for microscopically characterizing defects in materials in terms of local energy and stress fields calculated at the firstprinciples level of theory. These fields provide a quantitative measure of the local disturbance created by defect-induced electronic and atomic inhomogeneities in a solid. The local stress density σαβ(r) is computed by explicitly evaluating the strain derivative of a suitably defined energy density field, ε(r). Although ε(r) and σαβ(r) are defined only up to a gauge transformation, they yield the correct total energy and the average macroscopic stress tensor, respectively, when integrated over the entire volume of the underlying unit cell. In systems with defects, it is shown that well-defined averages of ε(r) and σαβ(r) can be constructed by restricting the domain of integration to smaller volumes, which are integral multiples of the Wigner–Seitz cell for the supercell containing the defect. These fields can provide important insights into the nature of atomic-scale defects. Explicit expressions for ε(r) andσαβ(r) are derived within the density functional plane-wave pseudopotential formalism. For the test cases of bulk Al with a vacancy and a Al(001) surface, it is shown that the averaged forms of ε(r) and σαβ(r) help characterize the defects in a physically meaningful manner. Potential applications of ε(r) and σαβ(r) to the characterization of surface relaxations and to multi-scale studies of materials are suggested.