Stress/electrical scaling in ferroelectrics

The present paper introduces a scaling law correlating three factors of ferroelectric materials in quite simple functions: the polarization (P), the electrical field (E), and the stress (T). The law is based on the physical symmetries of the problem and renders it possible to express the mechanical stress (T) as an electric field equivalent { E_eq =Th [P (E=0,T)] } and, as a consequence, also the relationship between strain (S) and polarization (P). Three materials with various phase structure (tetragonal, rhombohedral, and morphotropic) were used for the verification. It was found that such an approach permitted the prediction of the maximal stress using only purely electrical measurements [i.e., measurements of S (E) and P (E)]. Once this law was validated for the compressive stresses, the mapping could be extended in order to predict the polarization behavior in the tensile stress zone. It was shown that the polarization behaved differently as a function of the compressive and tensile stresses. The scaling law could also predict the piezoelectric constant (d₃₃) under stress using only purely electrical measurements. Reciprocally, the dielectric constant (ε₃₃) under an electrical field can be predicted using only purely mechanical measurements. Comparisons of hysteresis loops were made and a good agreement was found for the polarization versus electric field, compressive stresses, and, albeit to a lesser extent, for the coefficients (ε₃₃) and d₃₃.