Soft mode analysis and ferroelectric transition in spherical nanoparticles

We address the dielectric behavior in a free standing spherical particle using the phenomenological Landau theory. Considering the equations of motion and small perturbation against uniform polarization, we obtain the size driven ferroelectricity. The size driven polarization is shown to diverge at a new correlation length which turns out to be the extrapolation length. Mode analysis in such systems is carried out and it is found that the size driven ferroelectricity is associated with a soft mode which takes a minimum value at the new transition temperature similar to that of the bulk system. We suggested that such a mode can be tested in Raman measurement.