Landau theory of the size-driven phase transition in ferroelectrics

The influence of size on the dielectric behaviour of ferroelectrics is discussed using phenomenological Landau theory. Three types of free-standing geometry are used in our calculations-film, cylinder and sphere. For films, there is a size-driven phase transition (i.e. a transition from a ferroelectric state to a paraelectric state as the thickness of the film is decreased) so long as the surface ferroelectricity is weaker than that of bulk. The polarization becomes zero below a critical size at which the susceptibility has a maximum. Otherwise the susceptibility decreases as the film thickness decreases and no size-driven phase transition exists. However, for cylinders and spheres there is always a size-driven phase transition, and so the dielectric susceptibility is always enhanced at small size. The sphere geometry has the largest critical size amongst the three geometries. In order to fit the experimental measurements on fine-grained samples, we renormalize our calculations for the sphere geometry using a Gaussian distribution function to represent the variation of particle size. The renormalization rounds the peak in the susceptibility and noticeably shifts its position to smaller size if the standard deviation of the size distribution is comparable with its mean. The critical size judged from dielectric measurements could therefore be smaller than that of an isolated sphere. Our calculations are in qualitative agreement with experimental measurements on the susceptibility of barium titanate and lead titanate.