Size dependence of the ferroelectric transition of smallBaTiO3particles: Effect of depolarization

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 ${\mathrm{PbTiO}}_{3}$.