Effects of criticality and disorder on piezoelectric properties of ferroelectrics

The piezoelectric response of BaTiO₃ is studied in the vicinity of the cubic to tetragonal phase transition, as a function of temperature and the applied electric field in the polar direction. We also investigate the influence of disorder. In the clean limit we obtain the divergence of the piezoelectric tensor at the critical point. The effect of a small amount of disorder is to translate the critical point in the temperature-electric field phase diagram. For large values of the disorder, the paraelectric to ferroelectric phase transition becomes diffuse but a maximum of the piezoelectric tensor is still obtained even though the divergence of the piezoelectric response is lost. These results are in agreement with experimental observations for the relaxor ferroelectric Pb(Mg_1/3Nb _2/3)O₃-PbTiO₃. We use a Ginzburg-Landau model which explicitly includes the coupling of the polarization to the strain, the electrostatic interaction between polarizations, and a quenched random compressional stress field generated by point defects. The strain field and its associated elastic energy are written in terms of the stress field and the electric polarization by energy minimization subject to elastic compatibility.