Effective-surface-energy approach for size effects in ferroics

We present a simple approach enabling analytical treatments of size effects in ferroelectric samples of complicated shapes for the cases where long-range depolarizing effects are not involved. The key element of the approach is the presentation of the energy of the system as the sum of the bulk and effective surface energy (like in the classical nucleation problem), while the latter is expressed as a function of the bulk value of the order parameter. The effective surface energy is calculated in terms of the Kretschmer-Binder framework. The size-driven shift of T-C in the ferroelectric thin films with in-plane polarization and the nanowires with axial polarization is studied using the proposed approach and the results are compared with those exact. In the limit of large extrapolation length, the approach reproduces the exact results (analytical and numerical). For short extrapolation lengths, it can provide a good approximation to the exact results for the case of second-order phase transitions. For ferroelectrics with the first-order phase transition having the maximal correlation length smaller than the extrapolation length (a common situation in perovskites), the approach provides as well an appropriate description of the size effect on the transition temperature. The proposed approach can be used for the description of the size effect not only in ferroelectrics, but in other ferroics as well.