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In published papers, the Gibbs free energy of ferroelectric materials hasusually been quantified by the retention of 6th or 8th order polarizationterms. In this paper, a newly analytical model of Gibbs free energy, thereout,a new model of polarization-electric field hysteresis loops in ferroelectricmaterials has been derived mathematically. As a model validation, four patternsof polarization-electric field hysteresis loops of ferroelectric materials havebeen depicted by using the model. The calculated results indicated that theself-similar model can characterize the various patterns of hysteresis loops inferroelectric materials through adjusting the external excitation or thesynthetically parameter (e.g., electric, temperature, and stress, etc.)employed in the model.

Mathematical modeling of hysteresis loops in ferroelectric materials