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

A theory has been developed to examine the depolarization effect on the ferroelectric transition of small BaTiO3 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°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 and the domain-wall half thickness can be obtained by minimizing with respect to . To account for BaTiO3 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 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 KH2PO4 or PbTiO3.