A comparative study of ${\mathrm{PbTiO}}_3, {\mathrm{PbZrO}}_3$, and the solid solution ${\mathrm{PbZr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_3$ is performed on the soft-mode lattice dynamics within the first Brillouin zone. We consider the six unique B-site orderings for ${\mathrm{PbZr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_3$ representable within the $2\times 2\times 2$ primitive perovskite supercell as well as the virtual crystal approximation (VCA) to extract the phonon dispersion relations of a high-symmetry cubic-constrained form using density functional perturbation theory. We find that the most unstable modes in the rock-salt ordered structure and the VCA, like pure ${\mathrm{PbZrO}}_3$, are antiferrodistortive (AFD) while lower symmetry arrangements are dominated by $\mathrm{\Gamma }$-point ferroelectric (FE) instabilities like pure ${\mathrm{PbTiO}}_3$. Despite similarities in the phonon dispersion relations between the rock-salt ordered supercell and the VCA, the character of modes at high symmetry points are found to be different. In particular, the ${\mathrm{a}}^0{\mathrm{a}}^0{\mathrm{c}}^{-}$ and ${\mathrm{a}}^0{\mathrm{a}}^0{\mathrm{c}}^{+}$ AFD instabilities of the rock-salt ordering are replaced with ${\mathrm{a}}^{-}{\mathrm{b}}^{-}{\mathrm{c}}^{-}$ and ${\mathrm{a}}^{+}{\mathrm{b}}^{+}{\mathrm{c}}^{+}$ instabilities within the VCA. Such a rotation pattern is not seen in any of the supercell-based calculations, thus serving as a quantitative example of the inability of the method to accurately represent local structural distortions. Single modes are found exhibiting dual order parameters. At the zone center, some arrangements show mixed FE and antipolar soft modes (due to Pb motion transverse to the polar axis), and at long wavelengths all arrangements have soft modes of a mixed antipolar and AFD character. These are described with direct analysis of the eigendisplacements.

• Received 4 October 2019
• Revised 27 November 2019

DOI:https://doi.org/10.1103/PhysRevB.100.224305