First-principles calculations of piezoelectricity and polarization rotation in $\mathrm{Pb}\left({\mathrm{Zr}}_{0.5}{\mathrm{Ti}}_{0.5}\right){\mathrm{O}}_3$

First-principles local-density approximation calculations of the piezoelectric response of $\mathrm{Pb}\left({\mathrm{Zr}}_{0.5}{\mathrm{Ti}}_{0.5}\right){\mathrm{O}}_3$ (PZT 50/50) are presented for tetragonal and monoclinic phases. We use the linearized augmented plane wave with the local orbital extension method and [001]1:1 ordered PZT 50/50 supercells, constrained with $P4\mathrm{mm}$ tetragonal and $\mathrm{Cm}$ monoclinic symmetry. Calculated internal coordinates of monoclinic PZT 50/50 at the experimental $c/a$ value are in good agreement with measurements of $\mathrm{Pb}\left({\mathrm{Zr}}_{0.52}{\mathrm{Ti}}_{0.48}\right){\mathrm{O}}_3.$ Bulk spontaneous polarization, Born effective charges $Z^{*},$ and piezoelectric coefficients are computed using the Berry’s phase approach. Greatly enhanced piezoelectric coefficients are observed due to polarization rotation as a function of applied strain in the monoclinic phase with fully relaxed internal atomic coordinates. As the polarization rotates within the $\left(11\mathrm{}¯0\right)$ mirror plane between the [001] and $\left[\nu \nu 1\right]$ (pseudocubic) directions, values as large as $e_{33}=12.6\mathrm{}{\mathrm{C}/\mathrm{m}}^2,$ $e_{15}=10.9\mathrm{}$ ${\mathrm{C}/\mathrm{m}}^2,$ $e_{13}=-33{\mathrm{C}/\mathrm{m}}^2,$ and $e_{11}^{\prime }=36\mathrm{}$ ${\mathrm{C}/\mathrm{m}}^2$ are observed at $\nu =1.27\mathrm{}$, where $e_{11}^{\prime }$ is defined as ${0.5(e}_{11}{+e}_{12}).\mathrm{}$ Such large values are consistent with the measured piezoelectric response in ceramic PZT. This supports the polarization rotation mechanism as the origin of the enhanced electromechanical response.