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 defects such as the vacancy and foreign atoms. Since most processes occur at elevated temperatures not only the energetics of defects in the ground state but also their temperature-dependent free energies must be known. The first-principles calculation of contributions of phonon and electron excitations to free formation, binding, and migration energies of defects is illustrated in the case of bcc-Fe. First of all, the ground state properties of the vacancy, the foreign atoms Cu, Y, Ti, Cr, Mn, Ni, V, Mo, Si, Al, Co, O, and the O-vacancy pair 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, the phonon contribution to defect free energies is calculated within the harmonic approximation using the equilibrium atomic positions determined in the ground state under CV and ZP conditions. In most cases the ZP-based free formation energy decreases monotonously with temperature, whereas for CV-based data both an increase and a decrease were found. The application of a quasi-harmonic correction to the ZP-based data does not modify this picture significantly. However, the corrected data are valid under zero-pressure conditions at higher temperatures than in the framework of the purely harmonic approach. The difference between CV- and ZP-based data is mainly due to the volume change of the supercell since the relative arrangement of atoms in the environment of the defects is nearly identical in the two cases. A simple transformation similar to the quasi-harmonic approach is found between the CV- and ZP-based frequencies. Therefore, it is not necessary to calculate these quantities and the corresponding defect free energies separately. In contrast to ground state energetics the CV- and ZP-based defect free energies do not become equal with increasing supercell size. Third, it was found that the contribution of electron excitations to the defect free energy can lead to an additional deviation of the total free energy from the ground state value or can compensate the deviation caused by the phonon contribution. 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. The consideration of the temperature dependence of the magnetization yields self-diffusion data in very good agreement with experiments.