Electric-field induced polarization paths in ${\mathrm{P}\mathrm{b}(\mathrm{Z}\mathrm{r}}_{1-x}{\mathrm{Ti}}_x){\mathrm{O}}_3$ alloys

Properties of $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ (PZT) for compositions x near the morphotropic phase boundary and under an electric field are simulated using an ab initio based approach. Applying an electric field of [111] orientation to tetragonal PZT (e.g., $x=0.50)$ leads to the expected sequence of tetragonal, A-type monoclinic, and rhombohedral structures. However, the application of a field of orientation [001] to rhombohedral PZT (e.g., $x=0.47)$ does not simply reverse this sequence. Instead, the system follows a complicated path involving also triclinic and C-type monoclinic structures. These latter phases are found to exhibit huge shear piezoelectric coefficients.