Enhancement of electrostrictive coefficient Q₃₃ at ferroelectric MPB and its contribution to the maximized piezoelectric coefficient d₃₃

High piezoelectric performance is strongly desired to meet the ever-increasing demands of high-precision actuators and sensors. Ferroelectric morphotropic phase boundary (MPB) is known to exhibit high piezoelectric coefficient and the enhancement of piezoelectricity at MPB is generally ascribed to the maximum extrinsic piezoelectric contributions originating from the easiness of polarization switching. Here we show that in addition to the maximum extrinsic contributions, the highest electrostrictive coefficient Q₃₃ at the MPB which determines how much strain is gained through polarization change, leads to the enhancement of intrinsic piezoelectric coefficient and in turn contributes to the maximized d₃₃ at the MPB of the lead-free Ba(Zr_0.25Ti_0.75)O₃-x(Ba_0.7Ca_0.3)TiO₃ system. The highest Q₃₃ is ascribed to the small 4^th order Landau coefficient at the MPB as found by a revised Landau model. This work helps clarify the origin of high piezoelectric coefficient d₃₃ at ferroelectric MPB. It also suggests a general method to achieve high Q₃₃.