Grain-boundary effect on the Curie-Weiss law of ferroelectric ceramics and polycrystalline thin films:: Calculation by the method of effective medium

The influence of grain boundaries on the dielectric properties of ferroelectric ceramics and polycrystalline thin films is described theoretically by the method of effective medium. Grain boundaries are modeled by low-permittivity (dead) layers, which do not exhibit ferroelectric instability. The effective permittivity of a polycrystalline material is calculated in the paraelectric regime above the transition temperature. The calculations are based on the solution of electrostatic problem for a spherical dielectric inclusion separated from the surrounding dissimilar matrix by a low-permittivity interface layer. For isotropic bulk ceramics, an analytic expression is derived for the effective permittivity as a function of the grain size, dead-layer thickness, and its permittivity. Temperature dependence of the aggregate dielectric response is calculated for BaTiO3 (BT) ceramics of different grain sizes and found to be in good agreement with measurements. It is shown that grain boundaries not only renormalize the Curie-Weiss temperature and constant, but may also cause deviations from the Curie-Weiss law. For BT polycrystalline thin films grown on dissimilar substrates, numerical calculations of the effective dielectric constants are performed, taking into account both the grain-boundary and substrate effects on the film anisotropic dielectric response. Theoretical predictions are compared with the grain size dependence of the permittivity of BT films grown on Pt-coated Si.