OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Introduction Point defects, particularly vacancies, determine considerably the nature of various phenomena that occur in metals and alloys. The presence of vacancies is one of the most critical factors that is taken into consideration in the qualitative and quantitative description of diffusion processes accompanying thermal and thermochemical treatment of metals. For instance, recrystallization that develops in plastically deformed materials is based on the phenomenon of self-diffusion, which is closely related to the characteristics of vacancy migration [1, 2]. Polygonization in deformed metals is closely associated with a climb of edge dislocations which is accompanied by the emission or absorption of vacancies [1, 2]. Vacancies have a significant effect on the kinetics of diffusive phase transformations. The coagulation of multiple vacancies is considered as one of the main reasons for the formation of the so-called Kirkendall porosity observed during diffusion welding of some alloys [3–6]. Interstitial defects and vacancies are also essential underway irradiation-induced swelling that is one of the major tasks in nuclear engineering. The key parameter to describe vacancies is the energy of its formation. There are a number of experimental methods to evaluate the vacancy formation energy (VFE) to date. Methods based on precision measurement of heat capacity, electrical resistivity analysis, and positron annihilation spectroscopy (PAS) are among it [7–9]. It should be noted that the experimental determination of the formation energy of point defects is an extremely time-consuming process and is characterized by insufficient accuracy. The appearance and development of effective computational methods, among which the densityfunctional theory (DFT) should especially emphasized, is the result of the intensive development of computational materials science methods used, among other things, for the analysis of defects in the crystal structure. Using DFT, one can easily evaluate the ground state energy for any substance [10] without the introduction of some sort of empirically determined parameters for the calculations. Thus, point defect formation energy can be defined as the difference between the energy values of a supercell containing a vacancy (vacancy supercell, ) vac tot E and a defect-free supercell (bulk supercell, ) bulk tot E . The value obtained by the DFT requires a number of additional corrections to compare with the empirically determined parameters. The features of this approach are described in detail in review publications [11, 12]. One of the stages of DFT computation is associated with the choice of the exchange-correlation (XC) functional. The exact shape of functionals is currently unknown [13] therefore its approximations are used in practice. It should be noted that even if the chosen approximation of XC functional gives the correct result in evaluating some physical property, it may not be appropriate for evaluating another one. There are two widespread approximations among the great number of possible models of XC functionals, namely: the local density approximation (LDA) based on the free electron model [13, 14] and the generalized gradient approximation (GGA) that takes into account not only the electron density, but also its gradient at the considered point in space [15]. Both LDA and GGA functionals are based on a number of simplifications and, for this reason, are characterized by a certain inaccuracy. The choice of a particular XC functional depends on the type of task being solved. For instance, the cohesive energy using GGA-model can be defined more precisely [16] and therefore GGA can be effectively used to calculate the point defects formation energy including vacancies (VFE). However, the inaccuracy of the VFE using the GGA functional turned out to be quite high in practice [17]. In the review paper [11], Freysoldt et al. highlight that using of LDA functional provides a higher accuracy of the VFE evaluating in comparison with computation using GGA. It is associated with the assessment of the inner surface energy contribution arising when one of the atoms is removed from the supercell. The development of new XC functionals and its application for the calculation of various characteristics of materials, including the VFE, make it possible to minimize the deviation of the calculated data from the experimental ones. In particular, the paper [18] reports about efficiency of meta-GGA-functionals. MetaGGA functionals contain the second derivative of the electron density and also take into account the kineticenergy density of electrons, and therefore provide better precision. However, the computations of VFE via the revTPSS functional (one of the most commonly used meta-GGA functionals) did not confirm this
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