This work studies the temperature-driven ferroelectric phase transition of ferroelectric polarization under the absence of electric field. The modified Heisenberg model in three dimensions was considered and simulated via Monte Carlo simulation. The Metropolis algorithm and the periodic boundary condition were employed. The dependence of electric polarization on temperature was investigated to define the ferroelectric phases and their structural phase transitions. From the results, with well-defined set of relevant temperature parameters, the phase dependent polarization-behavior was found with a sudden change in its behaviors at the transition points. The structure factors were also considered and supported these phase changes. This conclusively pinpointed the important of temperature-dependent parameters in modeling ferroelectric materials.

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Ferroic hysteresis modeling is an approach applicable to study lagging relationship between ferroic properties and external-dynamic-perturbation behaviors. The understanding to the lagging hysteresis behaviors, such as hysteresis area, coercive field, or remnant magnetization/polarization, is essential to the material-industry development. In this review, fundamental analysis as a microscopic study, including spin models, mean-field method and Monte Carlo simulation, and data processing analysis as a linking bridge between macroscopic analysis and the experiment results, such as the artificial neutral network (ANN) and Fourier analysis, were elaborated on how they can be used to model the dynamic ferroic hysteresis. Then, ferroic behaviors were explained on how they depend on the external perturbations, such as the field parameters and temperature. The responses of ferroic behaviors on these factors, in both qualitatively and quantitatively, were also discussed in details.