Domain formation and strain relaxation in epitaxial ferroelectric heterostructures

The growth of ${\mathrm{PbTiO}}_{3}$ films by a metalorganic chemical-vapor-deposition technique has resulted in three-dimensionally epitaxial heterostructures on various single-crystal substrates. These heterostructures consist of ${\mathrm{PbTiO}}_{3}$ films on the (001) surface of the single crystals: potassium tantalate (${\mathrm{KTaO}}_{3}$), strontium titanate (${\mathrm{SrTiO}}_{3}$), and magnesium oxide (MgO). It was found that the presence of a structural (ferroelectric) phase transition in ${\mathrm{PbTiO}}_{3}$ leads to a ``strain-accommodating'' mechanism in which a domain pattern forms as the system cools through the Curie temperature and limits the extension of interfacial strain in the heterostructure---thus minimizing the total energy of the heterostructure. For ${\mathrm{PbTiO}}_{3}$/${\mathrm{KTaO}}_{3}$(001), the interfacial strain is accommodated by the formation of a periodic domain pattern in the overlayer. In ${\mathrm{PbTiO}}_{3}$/${\mathrm{SrTiO}}_{3}$(001), which exhibits an excellent lattice match between respective a lattice parameters, the film exists as a single c domain. The ${\mathrm{PbTiO}}_{3}$/MgO(001) system, having a poor lattice match for both the a and c axes, appears to find the energy minimum by locking into domains of two-dimensional superlattices with the greatest atomic coincidences. It is found that the nature of the domain pattern depends very strongly on both the film thickness and measuring temperature. A theoretical model of the domain-pattern formation has been developed by using linear-elasticity theory and a Landau-Ginzburg-Devonshire-type phenomenological theory for the substrate and the overlayer, respectively. The theoretical predictions and the experimental measurements were in good agreement in both the thickness and temperature dependence of the relative domain population and the spontaneous strains.