First-principles study of piezoelectricity in tetragonal ${\mathrm{PbTiO}}_3$ and ${\mathrm{PbZr}}_{1/2}{\mathrm{Ti}}_{1/2}{\mathrm{O}}_3$

The local orbital extension of the linearized augmented plane wave (LAPW+LO) method within the general gradient approximation was used to determine structural and ferroelectric properties of tetragonal ${\mathrm{PbTiO}}_3$ (PT), and two chemically ordered ${\mathrm{PbZr}}_{1/2}{\mathrm{Ti}}_{1/2}{\mathrm{O}}_3$ (PZT 50/50) phases, with B-site cations ordered along [001], and [111] directions. Stable ferroelectric ground states were found in all structures. Bulk spontaneous polarization, dynamical charges ${(Z}^{*}),$ and piezoelectric stress tensor elements were determined from ground-state Berry’s phase calculations. Ordering along the polar [001] direction was found to enhance the $e_{33}$ piezoelectric stress modulus in PZT 50/50. While theoretical piezoelectric stress moduli of PT, $e_{15}=3.15\mathrm{}\hspace{0.5em}\mathrm{C}/{\mathrm{m}}^2,$ $e_{31}=-0.93\hspace{0.5em}\mathrm{C}/{\mathrm{m}}^2,$ and $e_{33}=3.23\mathrm{}\hspace{0.5em}\mathrm{C}/{\mathrm{m}}^2,$ agree well with single-crystal experimental data, computed proper moduli of bulk crystalline PZT, $e_{33}\left(P4\mathrm{}\mathrm{mm}\right)=4.81\mathrm{}\hspace{0.5em}\mathrm{C}/{\mathrm{m}}^2$ and $e_{33}\left(I4\mathrm{}\mathrm{mm}\right)=3.60\mathrm{}\hspace{0.5em}\mathrm{C}/{\mathrm{m}}^2,$ differ significantly from low-temperature experimental moduli of polycrystalline samples.