  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.
