Dynamic hysteresis scaling of first-order phase transition ferroelectrics near the phase transition point in the (η²)³ model

The dynamic hysteresis of the first-order phase transition around the transition point T_c in ferroelectrics is studied by investigating the dynamic response of the Landau-Devonshire (η²)³ model to a time-varying external field of frequency f and amplitude E₀. It is revealed that the single-loop hysteresis as obtained above the upper critical point T⁺ and below the absolute instability point T₀ shows dynamic behaviours very different from the double-loop hysteresis obtained between T_c and T⁺. An extensive calculation reveals a power-law scaling for hysteresis area as a function of E₀, while no reliable power-law scaling for the area as a function of f is available in the low-f regime. The scaling for the double-loop hysteresis seems not to fall into the same class as single-loop hysteresis. Furthermore, the power-law exponents, if any, are somewhat temperature-dependent, but this dependence is very weak in the regime of low E₀ and high f.