First-principles energy and stress fields in defectedmaterials

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New methods are presented for microscopically characterizingdefects in materials in terms of local energy and stress fieldscalculated at the first-principles level of theory. Thesefields provide a quantitative measure of the local disturbancecreated by defect-induced electronic and atomic inhomogeneitiesin a solid. The local stress density {σαβ}(r) is computed by explicitly evaluating the strain derivative of a suitably defined energydensity field, ε(r). Althoughε(r) and {σαβ}(r) are defined only up to a gauge transformation, they yield thecorrect total energy and the average macroscopic stress tensor,respectively, when integrated over the entire volume of theunderlying unit cell. In systems with defects, it is shown thatwell-defined averages of ε(r) and{σαβ}(r) can be constructed byrestricting the domain of integration to smaller volumes, whichare integral multiples of the Wigner-Seitz cell for thesupercell containing the defect. These fields can provideimportant insights into the nature of atomic-scale defects. Explicit expressions for ε(r) and{σαβ}(r) are derived within the densityfunctional plane-wave pseudopotential formalism. For the testcases of bulk Al with a vacancy and a Al(001) surface, it isshown that the averaged forms of ε(r) and{σαβ}(r) help characterize the defectsin a physically meaningful manner. Potential applications ofε(r) and {σαβ}(r) tothe characterization of surface relaxations and to multi-scalestudies of materials are suggested.