Size and doping effects on the coercive field of ferroelectric   nanoparticles
  A microscopic model for describing ferroelectric nanoparticles is proposed which allows us to calculate the polarization as a function of an external electric field, the temperature, the defect concentration and the particle size. The interaction of the constituents of the material, arranged in layers, depends on both the coupling strength at the surface and that of defect shells in addition to the bulk values. The analysis is based on an Ising model in a transverse field, modified in such a manner to study the influence of size and doping effects on the hysteresis loop of the nanoparticles. Using a Green function technique in real space we find the coercive field, the remanent polarization and the critical temperature which differ significantly from the bulk behavior. Depending on the varying coupling strength due to the kind of doping ions and the surface configuration, the coercive field and the remanent polarization can either increase or decrease in comparison to the bulk behavior. The theoretical results are compared with a variety of different experimental data. 
Th. Michael, S. Trimper, J. M. Wesselinowa