OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 mately equal to ~11.6, ~13.9 and ~17.0 within the computation of VFE using PBE, MGGAC and rMGGAC respectively. Thus, the results obtained using the widespread PBE XC functional are considerably closer to the experimental data. The scatter in the computed results relative to the experimental data can be evaluated using mean square error (MSE). In this paper, it was calculated according to the following equation: ( ) exp 2 i i calc f f E E MSE n - = ∑ , (4) where i calc f E and expi f E are the calculated and experimental VFE for element of type i, respectively. It should be noted that only experimental values of the VFE, measured using PAS [7], were used in this study. Since the experimental data are presented only for some metals [7], the deviation of the calculated energies remained unknown for the rest, and, for this reason, was not taken into account to calculate the MSE. The MSEs are close for PBE and MGGAC functionals (0.66 and 0.64 eV2, respectively). When using rMGGAC, the MSE is significantly higher (1.11 eV2). Using fig. 2, one can compare the VFE calculated within this study with the experimental results. The comparison of findings was carried out according to the approach proposed by Medasani et al. [8]. From the calculated data, it is clear that the use of the rMGGAC and MGGAC functionals results in VFE overestimation as compared to the experimental values. The VFE computed using the widespread PBE functional quite uniformly distributed relative to the y = x line. In general, the results obtained are characterized by the following trend: Ef PBE < E f MGGAC ≤ E f rMGGAC that is well correlated with the findings of Medasani et al. [8]. a b c Fig. 2. Comparison of experimental and computed values of the vacancy formation energy for the exchange-correlation functionals PBE (a), rMGGAC (b) and MGGAC (c). The dotted line representing the function y = x is shown on the graphs for the convenience of analyzing the obtained data Analyzing the obtained results, it can be noted that patterns defined by computing correspond to the experimental data. The typical dependence of the VFE from the melting temperature was mentioned above. Nevertheless, it is difficult to use the VFE estimated using DFT in subsequent calculations without introducing additional corrections. In particular, the equilibrium concentration of vacancies and the diffusion coefficient depend exponentially on the VFE. It means that these parameters extremely sensitive to the error in determining the latter one. According to T. Mattsson and A. Mattsson [30], to obtain a reasonable value of the defects’ equilibrium concentration at room temperature one need to know the VFE with an accuracy of 0.025 eV. From the presented data it follows that this accuracy is unreachable without additional corrections. One of the approaches used for a posteriori correction of the VFE is to account the energy of the inner surface inside the crystal created by removing one of the atoms [30].
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