Modeling of the properties of a ferroïc single crystal based on the motion of domain walls

The ferroïc materials exhibit a nonlinear behavior largely due to the motion of domain walls. To well understand the resulting properties, it is necessary to develop a model describing these movements. In this work, we establish the motion equation of one domain wall induced by application of an electric field E or of a mechanical stress σ. The effects of a magnetic field could also be envisaged. The domain wall is viewed as an equivalent rigid plane in interaction with the crystalline imperfections, these latter being represented in terms of viscous damping force and of restoring force. The model can be applied to single crystals at low constraint amplitudes because the walls density is a constant as well as at high amplitudes if this density remains constant. The polarization P, the strain S, the dielectric constant ε and the piezoelectric coefficient d are obtained from the model. In particular, the frequency dependence of ε and d (real and imaginary parts) is calculated and the corresponding expressions of the relaxation times are obtained. We determine also the variation of ε and d with the amplitude of the ac applied constraint (E or σ) in the limit of low intensities which results in the hyperbolic law (this latter corresponds to the full expression of the Rayleigh law). The temperature dependence is also included. The expressions of various cycles (P, S, ε and d as a function of E or σ) are also achieved. Predictions of the model for single crystals are in well agreement with experimental data presented in literature.