Static and dynamic properties of ferroelectric thin film multilayers

A thermodynamic theory of ferroelectric thin film multilayers is developed. The free energy function is written down using a multilayer model in which c-domain layers of one ferroelectric material alternate with a-domain layers of a second ferroelectric material. We assume that the interfaces are perfectly sharp and that the polarization at these boundaries is zero. The renormalization of the free energy coefficients due to the stresses in the films and to the depolarizing field was taken into account, as well as the renormalization of the coefficients of the polarization gradients. The equilibrium inhomogeneous polarization temperature and its thickness dependence were determined from the solutions of the Euler-Lagrange equations resulting from the minimization of the free energy functional. A thickness induced ferroelectric phase transition is shown to exist and its transition temperature and critical layer thickness depend on the domain orientation. The criteria for ''a/c'', ''c/c'' and ''a/a'' domain multilayering are calculated and expressed via coefficients of the free energy density and the layer thickness. The calculated multilayer susceptibility diverges at the transition temperature of the thickness induced ferroelectric phase. This divergence is shown to be the origin of the giant dielectric response observed in some multilayers. The theory gives an excellent fit to the temperature dependence of the giant susceptibility observed recently in multilayers of PbTiO$_{3}$-Pb$_{1-x}$La$_{x}$TiO$_{3}$ ($x=0.28$).