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A theory is developed for intermediate monoclinic (FE m ) phases near morphotropic phase boundaries in ferroelectrics of complex oxides. It is based on the conformal miniaturization of stress-accommodating tetragonal domains under the condition of low domain-wall energy density. The microdomainaveraged lattice parameters are determined and attributed to the parameters of an adaptive monoclinic phase. The theory is applied to the temperature, electric field, and compositional dependent FE m lattice parameters. The predictions of the theory are rigidly obeyed over the entire FE m stability range. [1][2][3][4]. An explanation of these phases in terms of the conventional Ginzburg-Landau-Devonshire (GLD) theory of homogeneous ferroelectric phases is quite difficult. The homogeneous FE m phase can be described only if eight terms in the free energy expansion are included and thus eight fitting parameters are introduced [5]. However, the GLD theory cannot explain the observed special relations between the crystal lattice parameters of the FE m phase discussed in this Letter. Basic-principles calculations (based on an assumption of microscopic homogeneity of the ferroelectric order) have shown that the intermediate phases could be attributed to rotational polarization instabilities [6 -8]. These instabilities, in fact, demonstrate that the crystallographic anisotropy of the polarization direction vanishes (or at least is drastically reduced) near the MPB. Accordingly, the domainwall energy will also be dramatically decreased.The purpose of this Letter is to demonstrate that a homogeneous ferroelectric state is unstable under these special conditions. For very low values of domain-wall energies, the system transforms into a mixed (or adaptive) state [9,10]. This adaptive state is inhomogeneous on the nanoscale and homogeneous on the macroscale. The nanoscale microstructure of this adaptive state is a miniaturized microdomain structure determined by the accommodation of the misfit-generated stress and electric field. We show that changes in the microdomain topology result in gradual average symmetry adaptations. Previous experimental studies have shown microdomains on the scale of 10 nm, which gradually change with temperature and composition [11], consistent with the theory.The conventional ferroelastic microstructure consists of polydomain plates [12,13]. Each plate is formed by alternating layers of twin-related domains, as shown in Fig. 1 [14]. The relative thicknesses of the domain layers are adjusted to establish the macroscopic invariance of the habit plane [12,13] and in so doing eliminate longrange stress fields generated by crystal lattice misfits FIG. 1. Dark-field TEM image of a stress-accommodating polydomain structure in a CuAu alloy [14]. The structure of the adaptive phase has the same morphology but is conformally miniaturized to reach nano-or subnanoscale. Then white and black stripes become microdomains that are ''invisible'' to the usual diffraction measurement and the macroplates become macrodomains ...

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Ferroelectric and ferroelastic phases with very low domain wall energies have been shown to form miniaturized microdomain structures. A theory of an adaptive ferroelectric phase has been developed to predict the microdomain-averaged crystal lattice parameters of this structurally inhomogeneous state. The theory is an extension of conventional martensite theory, applied to ferroelectric systems with very low domain wall energies. The case of ferroelectric microdomains of tetragonal symmetry is considered. It is shown for such a case that a nanoscale coherent mixture of microdomains can be interpreted as an adaptive ferroelectric phase, whose microdomain-averaged crystal lattice is monoclinic. The crystal lattice parameters of this monoclinic phase are self-adjusting parameters, which minimize the transformation stress. Self-adjustment is achieved by application of the invariant plane strain to the parent cubic lattice, and the value of the self-adjusted parameters is a linear superposition of the lattice constants of the parent and product phases. Experimental investigations of Pb(Mg1/3Nb2/3)O3–PbTiO3 and Pb(Zn1/3Nb2/3)O3–PbTiO3 single crystals confirm many of the predictions of this theory.

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