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 $2.37 \mathrm{eV}$ is in excellent agreement with the experimental value of $2.45\ifmmode\pm\else\textpm\fi{}0.2 \mathrm{eV}.$ The results found in hcp-Ti and hcp-Zr, i.e., $2.14 \mathrm{eV}$ and $2.07 \mathrm{eV},$ 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 $2.2--2.4 \mathrm{eV}$ 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 bcc structure. Assuming that the monovacancy mechanism is dominant in $\ensuremath{\beta}\ensuremath{-}\mathrm{Ti},\ensuremath{\beta}\ensuremath{-}\mathrm{Zr},$ and $\ensuremath{\beta}\ensuremath{-}\mathrm{Hf},$ this demonstrates that structural relaxations with particularly large amplitudes are expected around the vacancy.