Use of site symmetry in supercell models of defective crystals: polarons in CeO2

Abstract

In supercell calculations of defective crystals, it is common to place a point defect or vacancy in the atomic position with the highest possible point symmetry. Then, the initial atomic structure is often arbitrary distorted before its optimization, which searches for the total energy minimum. In this paper, we suggest an alternative approach to the application of supercell models and show that it is necessary to preliminarily analyze the site symmetry of the split Wyckoff positions of the perfect crystal supercell atoms (which will be substituted or removed in defective crystals) and then perform supercell calculations with point defects for different possible site symmetries, to find the energetically most favorable defect configuration, which does not necessarily correspond to the highest site symmetry. Using CeO₂ as an example, it is demonstrated that this use of the site symmetry of the removed oxygen atoms in the supercells with vacancies allows us to obtain all the possible atomic and magnetic polaron configurations, and predict which vacancy positions correspond to the lowest formation energies associated with small polarons. We give a simple symmetry based explanation for the existence of controversies in the literature on the nature of the oxygen vacancies in CeO₂. In particular, the experimentally observed small polaron formation could arise for oxygen vacancies with the lowest C_s site symmetry, which exist in 3 × 3 × 3 and larger supercells. The results of first principles calculations using a linear combination of atomic orbitals and hybrid exchange–correlation functionals are compared with those from previous studies, obtained using a widely used DFT+U approach.