The dominant entropy contribution affecting defect concentrations was configurational entropy. Other contributions such as harmonic and anharmonic lattice vibrations were second-order effects and computationally expensive to calculate. Therefore, such contributions were rarely considered in defect investigations. However, to achieve the next accuracy level in defect calculations and thus significantly improve the agreement with experiment, the inclusion of these contributions was critical. The methods needed to compute highly accurate free energies of point defects from first principles were presented here. It was demonstrated how to include all relevant free-energy contributions up to the melting point. The focus was on non-magnetic metals and point defects in the dilute limit. All relevant excitation mechanisms were considered: electronic excitations and ionic vibrations both in the quasiharmonic approximation and explicitly including anharmonicity (i.e., phonon–phonon interaction). Since computing such interactions required sampling large parts of the phase space, straightforward ab initio based simulations (such as molecular dynamics) were in most cases out of reach of supercomputers. To overcome this difficulty, a recently developed hierarchical scheme was presented which permitted so-called coarse-graining of the configuration space and thus efficient calculation of anharmonic contributions to defect formation. The performance and accuracy of the developed methodology was considered for vacancies in aluminium. An important insight was that the entropy of vacancy formation was significantly affected by anharmonicity. It was further shown that the inclusion of all of the aforementioned excitation mechanisms was critical in order to guarantee an accurate description of thermodynamic properties up to the melting point.

Formation Energies of Point Defects at Finite Temperatures. B.Grabowski, T.Hickel, J.Neugebauer: Physica Status Solidi B, 2011, 248[6], 1295–308