Thermodynamic theory of intrinsic finite-size effects in PbTiO₃ nanocrystals. I. Nanoparticle size-dependent tetragonal phase stability

We propose a phenomenological intrinsic finite-size effect model for single domain, mechanically free, and surface charge compensated PbTiO₃ (PT) nanocrystals with no depolarization fields, undergoing a first-order tetragonal → cubic ferrodistortive phase transition. We then derive a Landau-Devonshire type free-energy functional, which is commensurate with the solution of the free-energy equation for the polar point group 4mm, wherein the nanoparticle size (ξ) is a variable. By using experimental particle size-dependent spontaneous polarization (P→_s) data for PbTiO₃, we compute the Landau coefficients up to the sixth order in the range <150 nm as a function of ξ. This thermodynamic potential takes into account the size dependence of the Landau coefficients in a consistent manner, and is able to predict the size-induced phase transition as well as the metastable tetragonal phase in the cubic phase field. We then construct a free-energy density surface in ΔG-P→_s-ξ space, which describes the decrease in tetragonal phase stability with decreasing ξ rigorously.