First-Principles Determination of the Soft Mode in Cubic ${\mathrm{ZrO}}_2$

A direct approach to calculate the phonon dispersion using an ab initio force constant method is introduced. The phonon dispersion and structural instability of cubic ${\mathrm{ZrO}}_2$ are found using a supercell method in the local-density approximation. The force constants are determined from the Hellmann-Feynman forces induced by the displacement of an atom in the $2\times 2\times 2$ fcc supercell. This size of the supercell gives “exact” phonon frequencies at $\gamma ,X,L,W$ Brillouin zone points. The phonon dispersion curves show a pronounced soft mode at the $X$ point, in agreement with the experimentally observed cubic to tetragonal phase transition.