First-Principles Determination of the Soft Mode in Cubic ZrO₂

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 ZrO₂ 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 2×2×2 fcc supercell. This size of the supercell gives “exact” phonon frequencies at Γ, 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.