Atomic sublattice decomposition of piezoelectric response in tetragonal PbTiO3 , BaTiO3 , and KNbO3

The piezoelectric properties of tetragonal perovskites $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$, $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$, and $\mathrm{K}\mathrm{Nb}{\mathrm{O}}_{3}$ under uniaxial strain along the [001] direction are calculated from first principles. In order to study the piezoelectric response of individual atomic sublattices, the total polarization $P$ and piezoelectric constant ${d}_{33}$ are decomposed into the contributions of $A$-site and $B$-site atoms. These three materials show different behaviors at the atomic scale. In $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$, the Pb and Ti contribute almost equally to the polarization and ${d}_{33}$. In $\mathrm{K}\mathrm{Nb}{\mathrm{O}}_{3}$, most of the polarization and ${d}_{33}$ are contributed by the ferroelectrically active Nb. But in $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$, as the $c$ axis increases, the contribution of Ba to ${d}_{33}$ rises rapidly, reaching 50% of the total ${d}_{33}$ at $c$ = 4.30 \AA{}. This means that the covalent bonding between Ba and O is significant and has an important impact on the piezoelectric response of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$.