Size dependence of the ferroelectric transition of small BaTiO3 particles: Effect of depolarization.

TLDR: In this article, the depolarization effect on the ferroelectric transition of small cubic particles with alternating domains separated by 180 degrees of domain walls was investigated. And the results of the calculations showed that the transition temperature of small particles can be substantially lower than that of the bulk transition temperature.

Abstract: A theory has been developed to examine the depolarization effect on the ferroelectric transition of small ${\mathrm{BaTiO}}_{3}$ particles. To reduce the depolarization energy, a crystal would break up into domains of different polarization. In this study, we consider cubic particles with alternating domains separated by 180\ifmmode^\circ\else\textdegree\fi{} domain walls. The depolarization energy and the domain-wall energy were incorporated into the Landau-Ginzburg free-energy density. Assuming a hyperbolic tangent polarization profile across the domain wall, the domain-wall energy \ensuremath{\gamma} and the domain-wall half thickness \ensuremath{\xi} can be obtained by minimizing \ensuremath{\gamma} with respect to \ensuremath{\xi}. To account for ${\mathrm{BaTiO}}_{3}$ not being a perfect insulator, a Schottky space charge layer beneath the particle surface that shields the interior of the crystal from the depolarization field was considered. The equilibrium polarization P and domain width D can be obtained by minimizing the total free-energy density with respect to both P and D. The results of the calculations show that the ferroelectric transition temperature of small particles can be substantially lower than that of the bulk transition temperature as a result of the depolarization effect. Consequently, at a temperature below the bulk transition temperature, the dielectric constant \ensuremath{\epsilon} can peak at a certain cube size L. These results agree with the existing experimental observations. Finally, the theory can also be applied to other ferroelectric materials such as ${\mathrm{KH}}_{2}$${\mathrm{PO}}_{4}$ or ${\mathrm{PbTiO}}_{3}$.