Temperature dependence of the intrinsic and extrinsic contributions in BiFeO₃-(K_0.5Bi_0.5)TiO₃-PbTiO₃ piezoelectric ceramics

This contribution focuses on the use of modified Rayleigh law as a technique for determining the intrinsic and extrinsic (reversible/irreversible) contributions to the piezoelectric effect up to 150 °C across a broad compositional space, augmenting previous understanding of the BiFeO₃-(K_0.5Bi_0.5)TiO₃-PbTiO₃ system. At room temperature, a mechanistic explanation of the correlation between crystal symmetry, i.e., tetragonal spontaneous strain, x_s, and the Rayleigh relations using Landau theory is provided. The intrinsic response was found to be heavily dependent upon the tetragonal x_s, whereby an optimisation between polarization and permittivity was elucidated, leading to enhanced piezoelectric charge coefficients. A c/a ratio of ∼1.041 was identified at which the room temperature intrinsic and extrinsic effects were at a maximum; a d_init of 183 × 10^-12m/V and Rayleigh coefficient of 59 × 10^-18m²/V² were measured, resulting in the largest piezoelectric charge coefficients. The piezoelectric charge coefficient d₃₃, intrinsic and extrinsic contributions of these materials were all found to increase up to 150 °C while adhering to the Rayleigh model. The reversible extrinsic component of the total reversible response, d_init, was calculated to be relatively minor, 4.9% at room temperature, increasing to 12.1% at 150 °C, signifying its increasing influence to the piezoelectric effect, as domain wall motion is thermally activated. Hence, the phenomenological interpretation provided here may be used as a roadmap to elucidate the origins of the temperature dependence of the piezoelectric effect.