Double Hysteresis Loop of BaTiO3at the Curie Point

It is known that the Curie point $\theta$ of the ferroelectric BaTi${\mathrm{O}}_3$ shifts to higher temperatures when a dc bias field is applied. If the crystal shows a sharp transition, we expect by applying an ac field at the Curie temperature that the crystal would become alternately ferroelectric and nonferroelectric in the cycle of the ac field. This can be seen in the shape of the hysteresis loop at temperatures slightly above $\theta$. In the center of the polarization $P$ versus field $E$ plot, we observe a linear behavior corresponding to the paraelectric state of BaTi${\mathrm{O}}_3$ above $\theta$. At both high voltage ends, however, we observe a hysteresis loop corresponding to the ferroelectric state. A change in temperature causes a change in size and shape of the double hysteresis loops, ranging from a line with curves at the ends (higher temperature) to two overlapping loops (lower temperature). The results obtained allow us to calculate the different constants in the free-energy expression of Devonshire and Slater. One of the results shows that the transition is of the first order since the $P^4$ term turns out to be negative. The properties of the hysteresis loops are discussed, especially the large spontaneous electrical polarization and the low coercive field strength.