Time fractional derivative for frequency effect in ferroelectrics

The present article proposes a dynamical model to obtain ferroelectric hysteresis dynamics based on fractional derivatives (Polarization versus Electric field curves). The consideration of a fractional derivative term widely increases the frequency bandwidth of the accuracy of the traditional hysteresis models. For PZT bulk ceramics, the order of the fractional derivative has been found to be 0.5 using experimental data as 10^-3Hz<f<100Hz. For these excitation frequency levels, simulation tests provided good results regarding the comparison of the fractional model and experimental results. With the same set of parameters, it is possible to take into account the nonlinear behavior as f rarr 0: creep phenomenon, ageing ... Next, the model was tested on large frequency bandwidths (>6 decades) and validated with success using the comparison between simulation tests and the only experimental results available in literature obtained in such conditions by Liu and al (J. Phys.:Condens. Matter., 2004, vol.16, pp.1189-1195) for BNT thin film samples. At such frequency levels (>10²Hz), due to power limitations, no ceramic bulk´s experimental results were available.