Unrelaxed vacancy formation energies in group-IV elements calculated by the full-potential linear muffin-tin orbital method: Invariance with crystal structure

The unrelaxed vacancy formation energies have been calculated for group-IV elements (Ti, Zr, Hf) in the hexagonal close packed (hcp) and body centered cubic (bcc) structures within the local density approximation to the density functional theory using the full-potential linear muffin-tin orbital method. In hcp-Hf the calculated value of 2.37 eV is in excellent agreement with the experimental value of 2.45+/-0.2 eV. The results found in hcp-Ti and hcp-Zr, i.e., 2.14 eV and 2.07 eV, respectively, can therefore be considered as reliable predictions. In the more open bcc structure, after very conclusive validations of the present procedure in Mo and W by comparison with experiments and other ab initio calculations, vacancy formation energies of 2.2-2.4 eV are obtained in Ti, Zr, and Hf. These energies, which are very similar to those in the hcp structure, are significantly larger than the experimental activation energies for self-diffusion in the bce structure. Assuming that the monovacancy mechanism is dominant in beta-Ti,beta-Zr, and beta-Hf, this demonstrates that structural relaxations with particularly large amplitudes are expected around the vacancy. [S0163-1829(99)00313-6].