OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 hypothesis [8]. Thus, the search for XC functionals that make it possible to improve the accuracy of VFE calculations for metals remains an urgent task. The aim of this work was to evaluate the VFE of BCC-, FCC- and HCP-metals widely used in mechanical engineering using DFT and to compare the results obtained by application of various types of XC functionals (GGA and meta-GGA). The results obtained are important for analyzing the effectiveness of DFT computations of point defect formation energy. In addition, the obtained data can be used for reference purposes in the simulation of diffusion processes. Theoretical background The calculations were carried out using the projector-augmented wave (PAW) method with the application of the GPAW code [19, 20] and the atomic simulation environment (ASE) [21], implemented in the Python programming language. The widely used Perdew-Burke-Ernzerhof (PBE) functional of GGA family [22], as well as MGGAC [23] and rMGGAC [24] functionals were used to describe XC potential. The MGGAC functional proposed in [23] is developed for quantum chemistry computations and solid state physics. The authors of this model combined the resulting meta-GGA exchange functional with the GGA correlation one. Using this combination, one can determine the structural and energy properties of solids with high accuracy. The rMGGAC functional proposed by Jana et al. [24] accounts for mismatches in the correlation energy of MGGAC for atoms and ions. The wave functions were described with plane waves. The cut-off energy of 500 eV was used for the plane-wave basis set. Total numbers of k-points generated according to Monkhorst-Pack method was 27 (3 × 3 × 3 along the X, Y and Z axes) for the chosen functionals (PBE, MGGAC and rMGGAC). To improve convergence with respect to Brillouin zone sampling, Marzari-Vanderbilt distribution (cold smearing) with the temperature broadening parameter of 0.2 eV was applied [25]. The energy of vacancy formation was evaluated using supercells approach with a size of 3 × 3 × 3. Detailed information about the parameters used in the computations is given in Appendix A. Calculations were carried out for the following metals: 1) BCC-metals: Li, Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ta, W. 2) FCC-metals: Al, Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au, Pb, Co. 3) HCP-metals: Be, Ti, Zr, Mg, Sc, Zn, Y, Ru, Cd, Hf, Os, Co, Re. To calculate the formation energy of point defect X via DFT, one can use the following formula [11]: [ ] [ ] [ ] f q q tot tot i i F corr i E X E X E bulk n qE E = - - m + + ∑ , (1) where [ ] f q E X is the energy of defect X in charge state q; [ ] q tot E X is the total energy of a supercell containing the defect; [ ] tot E bulk is the total energy of the perfect supercell; ni is a number of atoms of type i that have been added to (in this case it is assumed that ni > 0) or removed from (in this case it is assumed that ni < 0) the supercell to form the defect; μi are the corresponding chemical potentials of the added or removed atoms; f E is the Fermi energy and corr E is a correction term that accounts for finite k-point sampling in the case of shallow impurities (a common term used in the physics of semiconductor). In the case of single vacancy (or monovacancy) formed in a pure metal q = 0; Ecorr = 0; i = 1; n1 = n = –1. Thus, the equation (1) is significantly simplified and takes the following form: [ ] [ ] [ ] , f tot tot E vac E vac E bulk = - + m (2) where m is the chemical potential of the metal analyzed (the chemical potential of a single-element compound is typically used in DFT calculations [26]). This means that the VFE can be derived as the energy difference between a supercell that contains a vacancy and a perfect one (defect-free supercell). However, the total energy is an extensive quantity. In
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