Effect of surface tension and depolarization field on ferroelectric nanomaterial properties

The theory of size effects in nanocrystalline ferroelectric ceramics or nanoparticle powders allowing for surface tension and depolarization field is proposed. Surface tension was included in the free energy functional and extrapolation length was expressed via surface tension coefficient. The latter was shown to be dependent on temperature due to its relation to the dielectric permittivity of the nanoparticles. The depolarization field effect was calculated in the model taking into account the space-charge layer on the surface. The Euler-Lagrange equation for inhomogeneous polarization was solved analytically in both paraelectric and ferroelectric phases. The temperature and size dependence of the spontaneous polarization and dielectric susceptibility of nanomaterials was obtained. The possibility to calculate these and other properties by minimization of the conventional free energy in the form of different power polarization series, but with coefficients that depend on particle size, temperature, contribution of depolarization field and surface tension coefficient, was demonstrated. A comparison with available experimental data is performed.