First-principles calculation of defect free energies: General aspects illustrated in the case of bcc-Fe

Modeling of nanostructure evolution in solids requires comprehensive data on the properties of intrinsic point defects, foreign atoms and defect clusters. Since most processes occur at elevated temperatures not only the energetics of these species in the ground state but also their temperature-dependent free energies must be known. These data can be used to obtain improved, temperature-dependent input parameters for atomistic or object kinetic Monte Carlo simulations and rate theory.
The first-principles calculation of contributions of phonon and electron excitations to free formation, binding, and migration energies is illustrated in the case of bcc-Fe. First of all, the ground state properties of the defects are determined under constant volume (CV) as well as zero pressure (ZP) conditions, and relations between the results of both kinds of calculations are discussed. Second, vibrational and electronic contributions to defect free energies are calculated using the equilibrium atomic positions determined in the ground state for the CV and the ZP case. Additionally, the quasi-harmonic approach is applied to ZP-based data in order to obtain results closest to the experimental conditions at elevated temperatures. However, in most cases considered this leads only to minor modifications. In contrast to ground state energetics the CV- and ZP-based defect free energies do not become equal with increasing supercell size. A simple transformation is found between the CV- and ZP-based frequencies and between the corresponding free energies. Finally, self-diffusion via the vacancy mechanism is investigated. The ratio of the respective CV- and ZP-based results for the vacancy diffusivity is nearly equal to the reciprocal of that for the equilibrium concentration. This behavior leads to almost identical CV- and ZP-based values for the self-diffusion coefficient. Obviously, this agreement is accidental and cannot be generalized to other cases.The consideration of the temperature dependence of the magnetization yields self-diffusion data in very good agreement with experiments