Development of the Local (Site) Symmetry Method in the Supercell Model for a Crystal with an Impurity

Abstract

Symmetry aspects of the periodic model of a crystal with a point defect (a supercell model or a extended unit cell (EUC) model) are considered; splitting of the Wyckoff positions in a primitive crystal cell with the introduction of supercell and the change of the one-electron states classification over k-vector (Brillouin zone folding) are discussed. When considering a point defect in a crystal in the supercell model, it is necessary to take into account the symmetry of the one-electron states of the original crystal at the top of the valence band and at the bottom of the conduction band. The selected supercell should reproduce these states. Each specific selection of a supercell corresponds to a specific splitting of the Wyckoff positions of the original crystal and, as a result, the possibility to place a defect in positions with different point symmetry and to perform a calculation without taking into account the point symmetry of the crystal with a defect at all (site symmetry method). By the results of the calculation of a crystal with a defect in the supercell model, taking into account site symmetry, it is possible to determine the real symmetry of the crystal with a defect, which is essential for the interpretation of experimental data. A copper impurity in a lithium node in a LiCl crystal retains the cubic symmetry of the atom being replaced; the impurity of an iron atom in a titanium site with cubic local symmetry in a SrTiO₃ crystal lowers the symmetry to tetragonal; and the H site in the CsPbI₃ crystal is characterized by the complete removal of point symmetry with the formation of an \({\text{I}}_{2}^{ - }\) ion.