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

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. 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. Additionally, quasi-harmonic corrections are applied to the ZP-based data. 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 behaviour 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.