Thermodynamics of polydomain heterostructures. I. Effect of macrostresses

Abstract

The thermodynamic principles of the formation of a new class of materials, polydomain heterostructures, are formulated. The polydomain heterostructures can be formed as a result of phase transformations in constrained layers, composed of epitaxial couples or multilayers. Due to the elastic interaction between the layers of a heterostructure, these layers transform into sets of periodically alternating lamellae, or elastic domains. A polydomain layer can consist of either differently oriented domains of the same phase ͑twins͒ or domains of different phases. The goal of this article is to determine the parameters of the polydomain heterostructures and the conditions for their formation, i.e., their dependence on the characteristics of phase transformations, lattice misfits, and the film thickness, as well as external fields. Heterostructures containing ferroelectric and superconductor oxide layers will be considered as examples. In part I the necessary conditions for the formation of polydomain heterostructures have been established on the basis of the analysis of thermodynamic effects of internal and external macrostresses on phase transformations. The thermodynamic criteria are formulated in terms of a few energy parameters: the misfit energies of the phases that form the heterostructure and the energies of interaction between domains. Together with the free energies of the unconstrained phases, these parameters give a quantitative thermodynamic description of polydomain heterostructures provided that the interface effects are negligible. It is shown that polytwin structures are always more stable than single domain structures and therefore, their formation is always possible at some external stresses and temperatures. The stability of heterophase polydomain structures depends on the parameter of incompatibility between the domains, i.e., the ratio of the energy of direct interactions between the domains through their interface to the energy of indirect interaction between domains through the elastic surrounding matrix. If the interdomain interfaces are mobile, their movement may result in superelastic deformation and essential softening of the effective elastic modulus. ͓S0021-8979͑98͒05901-5͔

Strain on ferroelectric thin films

“…These values are the minimum ones preserving the horizontal coherency of the film. This configuration does not fall into the scope of the theory developed by Roytburd et al [85][86][87] and Pertsev et al [123] as the domain periodicity has to be smaller than the thickness of the film. Considering Kittel's law [161], which states that the domain period increases with the square root of the thickness of the film, one sees that Roytburd's model implies to consider thicker film than the one studied by Vlooswijk et al [160].…”

“…With increasing thickness, a transition from a monodomain state (i.e., possibly containing only 180°domains in the tetragonal case) to a polydomain state (with 90°domains in the same symmetry) has been considered theoretically [85][86][87][88]91]. In a monodomain film, the 180°domains reduce the depolarizing field, forming domain stripes as observed on PbTiO 3 thin films [73].…”

“…If an increase of a c with thickness has been reported [79], most the experiments tend to show that a c decreases with thickness [76,80,81]; On the contrary, its evolution with temperature makes a consensus: the decrease of a c with temperature has been observed [74,79,82,83] as well as calculated [84,82]. Roytburd and Alpay, in a series of seminal papers [85][86][87], have calculated domain stability maps that indicate the stable phase as a function of the tetragonality of the film and of the relative difference between the substrate and the film's in-plane lattice constants. These domain stability maps are a generalization of Pompe et al's domain pattern maps developed for twin domains [88] in the case of heterophase polydomain structures.…”

Effect of elastic domains on electromechanical response of epitaxial ferroelectric films with a three-domain architecture

“…As a conclusion, the internal stresses due to lattice misfit can be eliminated in relatively thick films, if all of the three variants of the tetragonal phase are arranged such that the film has the same in-plane size as the substrate. This is based on homogeneous approximation of the film free energy which does not take into account domain architecture [10,29,30].…”

Deformation of Heterogeneous Adaptive Composite

“…Let us assume that initially the active layers are in austenite phase. If there is a misfit between layers characterized by the misfit strain,ε M , then, the elastic energy of internal stress due to interaction between layers is 5) …”