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

Abstract

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 temperaturedependent 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 ZPbased 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 D. Murali, M. Posselt, M. Schiwarth: First-principles calculation of defect free energies… MS#BD12759 2 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.

Influence of phonon and electron excitations on the free energy of defect clusters in solids: A first-principles study

“…Moreover, for a given supercell size results of CV and ZP are often very similar. In contrast to the results for the formation energy of a single vacancy and single solutes [17] there is no general trend for the binding energies of the defect clusters given in Table 1: In some cases the CV data are slightly higher than the ZP data, in other cases it is the other way around. For a discussion on the relation between CV and ZP results in dependence on the size of the supercell and on their convergence, the reader is referred to former studies, cf.…”

“…The binding energy of an embedded defect cluster consisting of n species (point defects or foreign atoms) is defined by [17] were used in order to calculate the quantity bind E for the defect clusters.…”

“…the free energy of the defects, must be considered. While some authors have pointed out the importance of considering full DFTbased free formation and migration energies of point defects at nonzero temperatures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] only few studies on the free energy of defect clusters have hitherto been published. The work of Yuge et al [18] is probably the only full DFT-based study which is exclusively related to the free energy of larger embedded defect clusters.…”

“…Refs. [3,4,[15][16][17]20,21]), contributions of phonon and/or electron excitations to the free migration energy were considered. Also in recent DFT studies on the diffusion of foreign atoms or impurities by the interstitial, the vacancy or the dumbbell mechanism (cf.…”

First-principles analysis of solute diffusion in dilute bcc Fe- X alloys

“…However, PBEsol for bcc iron fails at predicting the lattice parameter, giving 2.79Å [85], to be compared with an experimental value of about 2.86Å. At the theoretical lattice parameter (2.79Å), PBEsol gives a rather high vacancy formation enthalpy [46,86], about 2.47 eV, which is beyond the current generally accepted value of about 2.2 eV [46,57,[87][88][89][90]. The results of the PBEsol calculated energies are therefore presented only in Table I and in the Supplemental Material [91].…”