Three dimensional phase-field simulations on the frequency dependence of polarization vectors and hysteresis loops in ferroelectric crystals

The phase field approach has been widely used to study the domain structure of ferroelectric crystals in both two and three dimensions (2D and 3D), but in the 3D case, little has been done to address the frequency dependence of ferroelectric characteristics. In this work, we adopt the 3D time-dependent Ginzburg-Landau kinetic equation to calculate the evolution of local polarization vectors and the overall hysteresis loops of ferroelectric crystals under the frequencies from 0.4 kHz to 120 kHz, and then use the fast Fourier transform to analyze the frequency characteristics of the polarizations. It shows the phenomenon of multiple frequencies at low field frequency but not at high one. The distribution and evolution of polarization vectors in x, y, and z directions are obtained, and various forms of electrical hysteresis loops are found from the average of local polarization vectors. The results indicate that, as the frequency increases, the hysteresis loops of P _z versus E _z change from the standard shape to the oval shape, but the loops for P _x and P _y change from the dumbbell shape to an oblique ellipse, and then to figure-eight curve and eventually to the superparaelectric one. The detailed distribution and evolution of the polarization vectors in the crystal are also vividly displayed. Finally, the effects of lattice size, amplitude of the applied field, depolarization energy, and the initial state of polarizations in the crystal are investigated. It shows that the nature of polarization evolution in a 3D crystal is highly complex and that each of these factors can have a significant effect.