The influence of size on the dielectric behaviour of ferroelectrics is discussed using phenomenological Landau theory. Three types of free-standing geometry are used in our calculations-film cylinder and sphere. For films, there is a size-driven phase transition (i.e. a transition from a ferroelectric state to a paraelectric state as the thickness of the film is decreased) so long as the surface ferroelecvicity is weaker than that of bulk. The polariaion becomes zero below a critical size at which the susceptibility has a maximum. Otherwise the susdeptibility decreases~as the film thickmess decreases and no size-driven phase tmnsition exists However. for cylinders and spheres there is always a size-driven phase transition. "d so the dielectric susceptibility is always enhanced at small size. The sphere geometty has the l a e s t critical size amongst the three geometries. In order to fit the e x p r i m e d measurements on fine-grained samples, we renormalize our calculations for the sphere geometry using 3 Gaussian distribution function to represent the variation of paaicle size. The renormalization rounds the peak in the susceptibility and noticeably shifts its position to smaller size if the standard deviation of the size distribution is comparable with its mean. The critical size judged from dielectric measurements could therefore be smaller than that of an isolated sphere. Our calculations are in qualitative agreement with experimental measurements on the Susceptibility of barium t i t " and lead titanate.

The Curie temperature of ferroelectric films described by the transverse Ising model was studied under the mean-field theory. The film layer number, the surface interaction and the surface layer number dependence of the Curie temperature were obtained. There is a critical surface interaction strength, which is JSC=1.25 J for single-surface-layer films, and JSC=1.078 J for multiple-surface-layer films. If the surface interaction strength exceeds the critical value, there exists an optimum film thickness which possesses the maximum Curie temperature; then the surface interaction strength is weaker than the critical value, the Curie temperature decreases monotonically with increasing film thickness, and there exist critical sizes or critical thicknesses at which the ferroelectricity will disappear if the surface interaction is weak enough.

We have performed first-principles calculations on the (001) surface of cubic SrHfO(3) and SrTiO(3) with SrO and BO(2) (B = Ti or Hf) terminations. Surface structure, partial density of states, band structure, and surface energy have been obtained. For the BO(2)-terminated surface, the largest relaxation appears on the second-layer atoms but not on the first-layer ones. The analysis of the structure relaxation parameters reveals that the rumpling of the (001) surface for SrHfO(3) with SrO termination is stronger than that for SrTiO(3). For the HfO(2)-terminated surface of SrHfO(3), the surface state appears near the M point of its band structure.

A one-dimensional lattice model with non-ferroelectric boundary sites has been used to study the effect of size on the polarization-reversal process in ferroelectric films. We assume non-ferroelectric imperfections at the boundary sites which act as nuclei for the polarization reversal, and include them explicitly in the free energy. The dependences of the switching time on film thickness and on electric field strength have been obtained, and fitted with empirical formulae. The time evolution of the polarization under a step field has also been simulated, from which we can clearly see domain wall formation and motion. The polarization and current under a bipolar pulse signal are also obtained to demonstrate the different memory state responses, in which a back-switching phenomena can be observed clearly.