First-principles formation energies of monovacancies in bcc transition metals

Monovacancies for seven bcc d-transition metals V, Cr, Fe, Nb, Mo, Ta, and W have been studied in detail from first-principles calculations. A full-potential, linear muffin-tin-orbital (FP-LMTO) method has been used in conjunction with both the local-density approximation (LDA) and the generalized-gradient approximation (GGA) to calculate volume-relaxed vacancy formation energies in all seven metals. A complementary ab initio pseudopotential (PP) method has been used to calculate both volume- and structure-relaxed LDA formation energies and formation volumes in V, Nb, Mo, Ta, and W. Fully relaxed PP geometries have also been applied to FP-LMTO LDA and GGA calculations. From these results, the following clear trends and conclusions emerge: (i) for the same fully relaxed geometry, FP-LMTO-LDA and PP-LDA formation energies are nearly identical; (ii) the lowest calculated formation energies are within or close to experimental error bars for all bcc metals except Cr, and the overall agreement with experiment is better for the $4d$ and $5d$ metals than the $3d$ metals; (iii) GGA and LDA formation energies are very similar for the $4d$ and $5d$ metals but for the $3d$ metals, and especially Fe, GGA performs better; (iv) volume- and structural-relaxation contributions lower the calculated formation energy by 0.1--0.5 eV, and improve agreement with experiment; (v) fully relaxed LDA formation volumes are in the narrow range (0.45--0.62)Ω₀, where Ω₀ is the equilibrium atomic volume; and (vi) the dominant structural effects are an approximate 5% inward relaxation of the first near-neighbor shell for group-V metals and a corresponding 1% inward relaxation for group-VI metals, with the exception of Mo, for which the second-shell atoms also relax inward by about 1%.