Fractional derivative operators for modeling the dynamic polarization behavior as a function of frequency and electric field amplitude

Polarization phenomena in ferroelectric materials are frequency-dependent, and the present article describes the use of a fractional derivative for the understanding of these phenomena as well as modeling them as functions of frequency and electric field amplitude. The focus was first directed toward the definition and validation of the proposed model through comparisons between simulations and measurements for high electrical field excitation amplitudes on a large frequency bandwidth (major hysteresis loops, measured over 4 decades). Subsequently, the same comparisons were made under ultra-weak as well as weak electric fields. Large frequency bandwidths were tested in each case, and it was shown that the fractional term provided a very accurate modeling of the dynamic behavior of the ferroelectrics. The dielectric permittivity coefficient along the polarization direction ^epsiv33 is a major parameter in ferroelectrics, and the frequency dependence of ^epsiv33 is correctly reproduced by the model. The time-dependence of the polarization reversal/variation was accurately simulated by a fractional derivation (a 0.5 order derivative), however, the use of a first-order derivation term (i.e., viscous losses) was in poor agreement with experimental results. It was found that the model was valid for large excitation field amplitudes as well as for large frequency bandwidths.