Domain evolution and polarization of continuously graded ferroelectric films

A thermodynamic analysis of graded ferroelectric films demonstrates that, in the equilibrium state, the films are subdivided into alternating single-domain and polydomain bands, the latter consisting of wedge-shape domains. Polarization under an external electrostatic field proceeds through inter-band boundary movements due to the growth or shrinkage of the wedge domains. It is shown how the domain structure and evolution are determined by the principal characteristics of the film: the distribution of the spontaneous polarization and dielectric constant. Graded films exhibit a sharp increase of polarization with the field for weak fields, with a drop of the dielectric constant when the field is increasing. A general approach to finding the dependence of the displacement and the wedge-domain shape on the field as well as analytical solutions for the fourth degree polynomial Landau-Ginzburg-Devonshire and parabolic potentials are presented.