Virtual-crystal approximation that works: Locating a compositional phase boundary in $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$

We present a method for modeling disordered solid solutions, based on the virtual crystal approximation (VCA). The VCA is a tractable way of studying configurationally disordered systems; traditionally, the potentials which represent atoms of two or more elements are averaged into a composite atomic potential. We have overcome significant shortcomings of the standard VCA by developing a potential which yields averaged atomic properties. We perform the VCA on a ferroelectric oxide, determining the energy differences between the high-temperature rhombohedral, low-temperature rhombohedral, and tetragonal phases of $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ at $x=0.5\mathrm{}$ and comparing these results to superlattice calculations and experiment. We then use our method to determine the preferred structural phase at $x=\mathrm{0.4.}\mathrm{}$ We find that the low-temperature rhombohedral phase becomes the ground state at $x=0.4,$ in agreement with experimental findings.