Modeling of the hyteresys loop of a ferroïc single cristal

We propose a description of the non-linear response of a ferroïc single crystal to the action of an ac constraint, here an electric field, by examining the polarization. For that, we use the model we have develop to describe the movement of the domain walls [1]. We establish the equation of the polarization or of the strain as a function of the time and we deduce the expression of the hysteresis loop. The equation consists of two parts. The first one is related to dynamical processes. It allows both to consider the instantaneous response of the ferroelectric (real part of the dielectric constant) that the delay caused by losses during the switching of the dipoles (imaginary part of the dielectric constant). The frequency and the amplitude dependencies of the slope of loop are included in the description. The second part represents the static contribution. It is described by a nonlinear function which it is the best description of the saturation function. This model allows us to obtain the expressions for the coercive and the remnant values as a function of the frequency and the amplitude of the ac applied constraint. Predictions of the model for single crystals are in well agreement with experimental data.