Modeling of the relaxation time of a ferroïc single crystal

The application of an electrical field E or a mechanical stress σ on a ferroïc single crystal leads to the switching of the dipolar moments inducing the motion of domain walls. According to the approach that we have developed to model the movement of domain walls, we established a relationship between the dielectric (or the piezoelectric) properties and the applied constraints. In this model, we calculate the relaxation time which is the ratio between the restoring force and the friction force. On the other hand the dielectric constant ε and the piezoelectric coefficient d are generally considered as dependent with the relaxation time, so they are functions of the constraint. This result permits us to obtain the ε-E and d-E butterfly loops. This model is potentially applied when the constraint amplitudes E₀ is higher than the coercive constraint E_c. Predictions of the model for single crystals are in well agreement with experimental data presented in literature.