Theory of Size-Driven Transitions in Displacive and Order-Disorder Ferroelectrics

We present a simple theory for structural transitions in displacive ferroelectrics of the perovskite type. In our theory, the competition between the elastic energy cost for the displacement of the homopolar ion from the centrosymmetric position, and the energy gain due to a ferroelectric ordering of the dipoles formed by the ionic displacements, leads naturally to a first-order transition from a paraelectric to a ferroelectric phase. This transition takes place at a certain temperature $T_c(L)$ as the temperature is decreased and, at a certain size $L_c (T)$ as the size of the system is increased. The transition temperature $T_c (L)$ is suppressed as the sample size is reduced, and vanishes for samples below a certain size. For order-disorder ferroelectrics, our theory shows that the suppression of $T_c$ by a reduction in system size is not appreciable, a result that is borne out by experiments.