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)Omega(0), where Omega(0) 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%.