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

Although many processes of nanostructure evolution in solids occur at elevated temperatures, basic data obtained from ground state energetics are used in the modeling of these phenomena. In order to illustrate the effect of phonon and electron excitations on the free binding energy of defect clusters, first-principles calculations are performed for vacancy-solute pairs as well as vacancy and Cu dimers, trimers, and quadromers in bcc Fe. Based on the equilibrium atomic positions determined by the relaxation of the supercell with the defect in the ground state under constant volume (CV) as well as zero pressure (ZP) conditions, the contribution of phonon excitations to the free binding energy is calculated within the framework of the harmonic approximation. The contribution of electron excitations is obtained using the corresponding ground state data for the electronic density of states. Quasi-harmonic corrections to the ZP-based results do not yield significant changes in the temperature range relevant for applications. At 1000 K the maximum decrease/increase of the ZP-based data for the absolute value of the free binding energy with respect to the corresponding ground state value is found for the vacancy-W (43%) / vacancy-Mn (35%) pair. These results clearly demonstrate that contributions of phonon and electron excitation to the free binding energy of the defect clusters are generally not negligible. The general behavior of the free binding energy of vacancy and Cu dimers, trimers and quadromers is similar to that of the vacancy-solute pairs. The results obtained in this work are of general importance for studies on the thermodynamics and kinetics of defect clusters in solids.