A phase field study of frequency dependence and grain-size effects in nanocrystalline ferroelectric polycrystals

In this study we take the view that frequency dependence of ferroelectric hysteresis is a result of direct competition between the speed of polarization evolution and the speed of external loading. We used the Ginzburg-Landau kinetic equation to evaluate the evolution of polarization vectors. We also devised a polycrystal model with a core-shell grain configuration to reflect the effect of the grain-boundary (GB) affected zone. The phase-field results showed that the coercive field tended to increase with frequency, but remnant polarization increased only slightly while the dielectric constant and piezoelectric constant d₃₃, tended to decrease. We also found that, while both hysteresis and butterfly loops exhibited the familiar sharp tails at low frequencies, the tails disappeared and the loops became elliptic- and kidney-shaped, respectively, at high frequencies. The calculated low-frequency phenomena are widely supported by experiments, but the high-frequency ones are not commonly found in the literature. We substantiated both types of findings with details of the underlying domain dynamics. They clearly showed a complete 180° polarization reversal at low frequencies, but stopped mostly at 90° at high frequencies. We also examined the influence of the kinetic coefficient and the loading amplitude, and found that, as either increases, the elliptic and kidney shapes of the loops would occur at a higher frequency. The calculated grain-size effects indicated that the remnant polarization, dielectric constant, and d₃₃ all decreased with decreasing grain size. This is again widely supported by experiments. But we also found that the grain-size effect of coercive field is more complicated. It may increase or decrease, and it is the magnitude of spontaneous polarization of the GB affected zone that determines its outcome.