Intrinsic Piezoelectric Response in Perovskite Alloys: PMN-PT versus PZT

First-principles supercell calculations and the modern theory of polarization are used to compute the $e_{33}$ piezoelectric coefficients of $\mathrm{Pb}\left({\mathrm{Zr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_3\right)$ (PZT) and $0.60\mathrm{Pb}\left({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}\right){\mathrm{O}}_3+0.40\mathrm{}{\mathrm{PbTiO}}_3$ (PMN-PT) alloys. A drastic enhancement by a factor of 2.7 is found for $e_{33}$ when going from PZT to PMN-PT. The huge value of $e_{33}$ in PMN-PT comes from the large response of the internal coordinates of Pb, Ti, Nb, and O atoms to a macroscopic strain. On the other hand, the Mg atoms contribute little to the piezoelectricity for dielectric and elastic reasons.