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The elastically driven, self-assembled polydomain nanostructures (elastic domains) in ferroelectric films are both scientifically intriguing and practically important, since the interfaces and long-range elastic and electric fields associated with the periodic nanoscale modulation in structure can endow unique and desirable physical properties to the ferroelectric films, including large piezoelectric and dielectric responses, enhanced pyroelectric, and electrocaloric performances, as well as electro-optical effects (anomalous photovoltaic effect, for example). In this chapter, thermodynamics of the formation of elastic domains in epitaxial ferroelectric films is presented, together with a focused review on the theoretical and experimental studies. In addition, methods of domain engineering aimed at achieving an optimal polydomain structure for certain energy applications are discussed.

A review of theoretical models, phase field modeling and experimental studies of domain structures in epitaxial films is presented. The thermodynamic theory of such domain structures is presented within the macroscopic thermo-mechanical framework. The theory allows for the evaluation of the main parameters of the domain structure using the energy minimization approach applied to the energy of elastic interactions. For homophase (polytwin) films, the thermodynamic theory provides a quantitative tool that can be used to estimate domain fractions in the film and the type of domain structure architecture. For heterophase films, the theory describes (a) the conditions under which two-phase structures can be obtained in epitaxial films, and (b) the phase and domain fractions in these films. The thermodynamic theory can also be used to describe the extrinsic contributions from domain structures to the functional properties of epitaxial ferroelectric films. The review of phase field modeling demonstrates that computational results reproduce the predictions of the thermodynamic theory. It is also shown that the phase field modeling that utilizes the energy minimization procedure for elastic and interfacial energies can be used to predict domain morphology for the films with two-phase structures produced either by phase transformation or through the co-deposition of immiscible phases. The experimental data presented in the review validate predictions of the thermodynamic model and the results of phase field modeling.

The model of the P(E) hysteresis in dielectric—ferroelectric bilayers and weak—strong ferroelectric bilayers is presented with the phase field modeling of the domain structure and P(E) curves in the compositionally graded m BaxSr (1-x) TiO 3 – n BaTiO 3 bilayers on SrTiO 3 substrate. It is shown that the hysteresis loop depends on the film composition and structure and varies from simple rectangular to complex. The coercive field and the single-domain to polydomain transition depend on the film composition and structure. There is a significant difference between the behavior of through thickness domain structures and columnar domain structures localized in one layer.

This study investigates the spinodal mechanism of domain formation and switching through the loss of stability of a single-domain (SD) state with respect to a polydomain (PD) state in thin ferroelectric films with dead layers. The macroscopic thermodynamic model of a homogenized domain structure (DS) is used to analyze the depolarizing field effect on the transitions between SD and PD states and the shape of the P(E) hysteresis loop. Phase field modeling is used to study the inhomogeneities and film thickness effects on DS and P(E) hysteresis. Such modeling of relatively thick films produces P(E) hysteresis loops of a shape similar to that predicted by the macroscopic model. The results of the analysis demonstrate that the SD → PD transition can proceed by the spinodal mechanism in thin films, while the nucleation and growth mechanism should operate in thick films. It is shown that columnar and stripe DS can be produced in thin films.

A thermodynamic analysis of graded ferroelectric films demonstrates that in the equilibrium state the films are subdivided into a single-domain band and a polydomain band which consists of wedge-shape domains. Polarization under an external electrostatic field proceeds through an inter-band boundary movement due to 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 p{sup 4} Landau-Devonshire and parabolic potentials are presented.

A thermodynamic approach was used to describe the formation and magnetoelectric response of composite multiferroic films. Experimental and theoretical results that address the origins of different phase morphologies in epitaxial spinel-perovskite nanostructures grown on differently oriented substrates are presented. A theoretical model of magnetoelectric coupling in multiferroic nanostructures that considers a microscopic mechanism of magnetization in single-domain magnetic nanorods is described. This model explains a discontinuous electromagnetic coupling, as observed experimentally, and predicts a hysteretic behavior of magnetization under external electric fields.

Metal–hydride phase transformations in solids commonly proceed with hysteresis. The extrinsic component of hysteresis is the result of the dissipation of energy of internal stress due to plastic deformation and fracture. It can be mitigated on the nanoscale, where plastic deformation and fracture are suppressed and the transformation proceeds through formation and evolution of coherent phases. However, the phase coherency introduces intrinsic thermodynamic hysteresis, preventing reversible transformation. In this paper, it is shown that thermodynamic hysteresis of coherent metal–hydride transformation can be eliminated in epitaxial film due to substrate constraint. Film–substrate interaction leads to formation of heterophase polydomain nanostructure with variable phase fraction which can change reversibly by varying temperature in a closed system or chemical potential in an open system.

The formation and electro-magnetic coupling of self-assembling nanostructures consisting of ferrimagnetic rods embedded in a ferroelectric matrix have been analyzed using a thermodynamic approach. The resulting nanostructure architectures are determined by minimizing the free energy including the elastic energy associated with the epitaxial stresses, which arise during the growth of nanosize constituent phases. The magnetic response to the external electric field has been studied by minimization of the Landau-type free energy for ferrimagnetic and ferroelectric phases with elastic interactions between them. It has been shown that depending on the mechanism of rotation of magnetic moments in single-domain nano-rods the external electric field induces a continuous or discontinuous change of magnetization. The formation of a CoFe2O4--PbTiO3 nanostructure grown on SrTiO3 substrate and its response to an electric field is considered as an example.

Ferroelectric multilayers and superlattices have gained interest for dynamic random access memory (DRAM) applications and as active elements in tunable microwave devices in the telecommunications industry. A number of experimental studies have shown that these materials have many peculiar properties which cannot be described by a simple series connection of the individual layers that make up the heterostructures. A thermodynamic analysis is presented to demonstrate that ferroelectric multilayers interact through internal elastic, electrical, and electromechanical fields and the strength of the coupling can be quantitatively described using Landau theory of phase transformations, theory of elasticity, and principles of electrostatics. The theoretical analysis shows that compositional variations across ferroelectric bilayers result in a broken spatial inversion symmetry that can lead to asymmetric thermodynamic potentials favoring one ferroelectric ground state over the other. Furthermore, the thermodynamic modeling indicates that there is a strong electrostatic coupling between the layers that leads to the suppression of ferroelectricity at a critical paraelectric layer thickness for ferroelectric-paraelectric bilayers. This bilayer is expected to have a gigantic dielectric response similar to the dielectric anomaly near Curie-Weiss temperature in homogeneous ferroelectrics at this critical thickness.

The converse piezoelectric response of a thin film constrained by a substrate is analyzed in different geometries under various boundary conditions. We considerthe effects of elastic deformation of the substrate on the total displacement of thefilm surface induced by the electric field. The change of film thickness and the bending curvature of a film/substrate couple are calculated. For a thin film island clamped on a large thick substrate, the theoretical estimation of the piezoresponse, including a local bending in the vicinity of the island/substrate interface, is in agreement with the finite element calculation.

The theoretical analysis and phase-field modeling of elastic domain structures in constrained layers depending on their misfit and thickness are presented. Domain diagrams that demonstrate how different domain structures can be obtained by manipulating parameters of constraints are constructed for a model system with cubic-tetragonal transformation. It is shown that depending on the misfit, three-domain stress-free structures (hierarchical and cellular) and two-domain uniaxially stressed structures are the equilibrium states for relatively thick films. When the film thickness decreases, the area of a three-domain hierarchical structure shrinks and disappears followed by the disappearing cellular domain structures. In thin films, two-domain structures with asymmetric architectures are stable at negative symmetric misfit (in-plane contraction), while two-domain structures with symmetric domain architectures are stable at positive symmetric misfit (in-plane expansion). The results of the phase-field modeling are discussed in connection with the experimental data available for film/substrate ferroelectric heterostructures.

The evolution of martensitic microstructure and stress in a constrained film coupled with a substrate under cooling and heating is considered. Thermodynamic analysis has done using the special phase diagrams constructed in coordinates temperature-film/ substrate misfit.

Deformation as a result of solid–solid phase transformations, particularly martensitic, is the subject of this paper. The thermodynamic theory of elastic domains is applied to a transformation with a polytwin product phase which consists of two or more different domains (twins) forming plane–parallel alternations. The volume fractions of the phases and different twin components in the product phase are obtained as functions of applied stress (strain) and temperature. It is shown that during stress- or strain-induced transformation a twin fraction in martensite plates changes and a product phase becomes incompatible with an initial phase. This fact leads to the intrinsic instability of a transformational deformation, which appears as a negative Young's modulus under strain-controlled deformation and a thermodynamic hysteresis under stress-controlled deformation. The deformation is irreversible also due to microstructure irreversibility of the transformation from a polytwin state to a single-domain state. Superelastic deformation due to martensitic transformations is discussed as an application of the presented theory.

A high resolution magneto-optical technique with a magnetic indicator film was used to study the effect of a regular array of micrometer-sized mechanical indentations on magnetization dynamics at the surface of a twinned single crystal of YBa2Cu3O7−δ as a function of applied magnetic field, Ha (0 to ±68 mT) at temperatures in the range of 10–70 K. Direct visualization of the magnetic flux interaction with the individual indentations was observed for the first time. During magnetization at low Ha up to 15 mT at 60 K, the first row of indentations provides resistance to the flux front. In the indented area, flux is preferably pinned at the indentation sites. With increasing Ha, the vortex density at the indentations approaches the magnetic flux density in the surrounding material. During demagnetization down to Ha=0 mT, the outgoing flux is trapped by the indentations in areas where the density of the trapped flux is small; flux is also trapped along the (110) twin boundaries behind the indentations. For Ha...

A new class of materials, layer composites containing adaptive polydomain components, is considered. Polydomain structures can be the result of constrained phase transformation. Polydomain structures consist of periodically alternating layers of different stable and metastable phases (heterophase polydomain) or differently oriented domains of product phase (e.g. polytwin). The domain interface movement enables an additional mode of deformation. The theory predicts unusual physical and mechanical properties of composites, particularly, considerable decrease of the effective elastic moduli of composites, nonlinear stress–strain relations, and critical behavior at small thickness of an active layer.

Deformation of an adaptive heterophase crystal with variable microstructure is considered. Under mechanical constraint the phase transformation in single component systems result in an equilibrium two-phase mixture. A typical equilibrium microstructure of a constrained crystal is a polydomain, i.e. an alternation of the plane-parallel layers, or domains, of the parent and product phases with a special crystallographic orientation of interfaces between domains. The relative fractions of the domains are determined by the external conditions. The free energy of a polydomain is a non-convex function of constrained strain. Therefore, the stress–strain relation at displacement controlled deformation of the polydomain is characterized by a negative Young’s modulus. If deformation proceeds under stress control, a hysteretic stress–strain curve on loading and unloading should be observed instead of a negative stress–strain slope. Besides this thermodynamic hysteresis, anelastic hysteresis appears at a constant strain-rate deformation, if the microstructure relaxes more slowly than the strain changes.

The effect of uniaxial external stress fields applied during the cooling down from the deposition temperature on the domain structure of epitaxial ferroelectric or ferroelastic films is investigated using the domain stability map. This map predicts stable domain structures and is constructed in the coordinates of the misfit strain between the cubic substrate and the film, and the tetragonality of the film. It is shown that with the application of a stress field perpendicular to the film-substrate interface, perfectly c-axis oriented films may be obtained.

A map showing regions of stability of possible domain structures and relative fractions of domains in a polytwin structure is developed for epitaxial heterostructures with active layers which undergo a cubic–tetragonal or a tetragonal–orthorhombic transformation. This map, which is also applicable to epitaxial film–substrate systems, shows the dependence of the polytwin structure on misfit strain, lattice parameters of the product phase, and mechanical stress. A uniaxial stress field applied during cooling down from the growth temperature strongly affects the domain structure selection and perfect single-domain structures may be obtained with such fields. For polytwin layers with thicknesses close to the critical thickness for domain formation, microstresses must be taken into consideration and the domain stability map is modified accordingly. Misfit dislocation generation at the deposition temperature is taken into account through a temperature dependent effective substrate lattice parameter. As examples...

The thermodynamic effects of microstresses on the formation of polydomain heterostructures are considered. The microstresses arise due to the periodic deviation of the actual misfit from the average misfit on the interfaces between a polydomain layer and neighboring layers. The thermodynamic analysis is restricted to heterostructures with D/h

The discovery of polydomain phases has been one of the fundamental results of the experimental and theoretical studies of solid phase transformations. These phases consist of the alternation of domains: twins of a product phase or layers of different product phases. Elastic interactions between domains lead to the formation of the equilibrium polydomain structure corresponding to the free energy minimum. Thus these domains have been named “elastic domains”. The elastic domains form the polydomain phase whose average properties are dependent on its domain structure. Its basic element is a polytwin: a plane-parallel plate consisting of alternating plane-parallel domains or twins. The crystallographic orientation of the polytwin interfaces and its internal structure are equilibrium characteristics of the phase transition. Thus the parameters of the domain structure are the additional degrees of freedom of a polydomain phase. This fact considerably determines the properties of the polydomain phase, a...

The domain structure of ferroelectric or ferroelastic materials, arising as a result of the cubic-tetragonal transition inside an epitaxial layer, are considered. A thermodynamic theory of the size and arrangement of the domains has been developed. The dependence of the equilibrium domain structure on the epitaxial layer thickness has been shown.

In solid state phase transformations a single crystal or a grain of a polycrystal are often transformed into a regular system of structural domains which are differently oriented variants of the transformed phase. These domains appear as a result of the reduction in crystal symmetry caused by the transformation and are related to each other in a twin-like fashion. An assembly of plane-parallel twins forms a polysynthetic twin or polytwin.[1] Polytwins, in turn, may be parts of more complex hierarchical structures. "Polysandwich" structures of this kind are characteristic of many types of structural phase transli~rmations in solids, martensitic, ferroelastic, ordering and decomposition.[2] These polydomain structures arise in both first and second order phase transformations. If the polydomain region is embedded in an elastic matrix, elastic interactions between domains mediated by the matrix lead to the formation of an equilibrium polydomain structure corresponding to a minimum of the tree energy. Therefi~l'e, these domains have been termed "elastic domains". If the characteristic length of the domain structure is small in comparison to the characteristic size of the solid, this homoor heterophase polydomain structure can be treated as a special type of polydomain phase with domain structure parameters as additional internal thermodynamical variables. These variables control the properties of the polydomain phase as well as the thermodynamics and kinetics of phase transformations to a considerable extent.

Phase field simulations were used to study domain structures in model systems containing homogeneous or bi-layer ferroelectric films with different chemical free energy density profiles and different relative strength of dipole-dipole and elastic interactions.

Magnetic force microscopy of amorphous compressed Terfenol‐D films of micron thickness on both crystalline Si and glassy silica shows that their ferromagnetic domain structure is one dimensional and periodic. On Si the structure is aligned with respect to the easy elastic direction of the substrate. The square of the domain period is proportional to the thickness of the film. All facts indicate that the domain morphology in these films is determined by a substantial elastic contribution to the domain wall and film/substrate interaction energies.

The time-dependent relaxation of remanant polarization in epitaxial lead zirconate titanate $[{\mathrm{P}\mathrm{b}(\mathrm{Z}\mathrm{r}}_{0.2}{\mathrm{Ti}}_{0.8}{)\mathrm{O}}_{3},\mathrm{P}\mathrm{Z}\mathrm{T}]$ ferroelectric thin films, containing a uniform two-dimensional grid of 90\ifmmode^\circ\else\textdegree\fi{} domains (c axis in the plane of the film), is examined using voltage-modulated scanning force microscopy. 90\ifmmode^\circ\else\textdegree\fi{} domain walls preferentially nucleate 180\ifmmode^\circ\else\textdegree\fi{} reverse domains during relaxation, which grow and coalesce as a function of relaxation time. Relaxation is seen to saturate at different levels depending on the write voltage. Late (saturation) stages of relaxation are accompanied by pinning and faceting of the domain walls (drastically reducing the wall mobility), which is direct evidence of the role of defect sites and crystallographic features on polarization relaxation. The kinetics of relaxation is modeled through the nucleation and growth Johnson-Mehl-Avrami-Kolmogorov theory with a decreasing driving force.

Layer composites containing adaptive polydomain components are considered. Polydomain structures can be result of constrained phase transformation. Polydomain structures consist of periodically alternating layers of different phases or differently oriented domains of product phase. The domain interface movement enables an additional mode of deformation. The theory predicts unusual physic and mechanical properties of composites, particularly, considerable decrease of the effective elastic moduli of composites, nonlinear stress-strain relations, critical behavior at decreasing thickness of an active layer. For ferroelectric or ferromagnetic polydomain components superelasticity can be governed and stiffed by the bias electric or magnetic field.

The possibility of the formation of periodic arrangements of 90°-domains in ferroelectric epitaxial films is discussed. Domains can construct a simple alternation of two domains or more complex hierarchical polydomain structures. The physical reason for their formation is a relaxation of the elastic energy of misfit stresses in an epitaxial system. Therefore, they are called elastic domains. The thermodynamic theory of elastic domains predicts architectures of polydomain structures, domain fractions in them, their scales and effect of external mechanical and electrical fields on these characteristics. The theory is supposed to provide a guidance for modeling and engineering polydomain structures in ferroelectric epitaxial heterostructures. Current theoretical and experimental studies on polydomain heterostructures are discussed.

Thermodynamic analysis of a graded ferroelectric epitaxial film demonstrates that in the equilibrium state the films subdivide into an alternation of polydomain bands consisting of wedge- or lens-shape domains and single-domain bands. Polarization under an external electrical field proceeds through an interband boundary movement due to the growth/shrinkage of the wedge domains. The domain structure and its evolution are determined by the distribution of the spontaneous polarization and dielectric constant. Through film grading, it is possible to design a controlled domain structure with desirable tunability. Graded films exhibit a sharp increase of the dielectric constant as electrical field approaches zero.

New phenomenon in epitaxial systems - the formation of strain controlled modulated domain and heterophase structures in epitaxial layers - is under theoretical investigation. These structures should form due to the elastic interaction of an epitaxial layer with a substrate (or another epitaxial layer). Modulated structures consist of periodically alternating layers of differently oriented domains of one phase (e.g. twins) or layers of different stable and metastable phases. It can also include stabilized virtual phases i.e. metastable phases which cannot be formed in bulk systems.

The domain structure in bilayer ferroelectric films was studied by using an analytical thermodynamic theory and computer simulations utilizing the phase field model. It is demonstrated that in a low applied field, a self-poled state can be produced in a bilayer film with one layer in a polydomain state and the other layer in a single-domain state. The presence of the layer with a polydomain structure results in a higher dielectric constant and lower coercive field as compared to a bilayer in a single-domain state. The increase in the applied field results in the transition to a single-domain state in the whole bilayer. The thermodynamic potentials of the layers and the energies of electrostatic and elastic interactions determine the critical fields which can control the transitions between single-domain and polydomain states as well as polarization switching of bilayers. Different thermodynamic hysteresis loops are obtained depending on the relationship between these critical fields and the amplitude of the applied ac field. The phase field model of the domain structure produces results that confirm main predictions of the analytical theory.

The equilibrium domain structure and its evolution under an electric field in ferroelectric bilayers and graded multilayers are considered. The equilibrium bilayer is self-poled and contains a single-domain and a polydomain (with 180 domains) layers. The polarization of a graded multilayer proceeds by movement of wedge-like domains as a result of progressive transformation of polydomain layers to a single-domain state. The theory provides the principal explanation of peculiarities of dielectric behavior of graded ferroelectric films and can be applied to graded ferromagnetics and ferroelastics., Comment: 4 pages, 3 figures

Landau–Devonshire-type phenomenological thermodynamic theory is employed to explain the electric field dependence of piezoelectric properties of tetragonal single domain PbZrxTi1−xO3 (PZT). The strong nonlinearity of the converse piezoelectric coefficient under a large external electric field is proved to be intrinsic both in bulk crystal and epitaxial tetragonal PZT thin films. The tunability of piezoelectric responses by an external electric field and its dependence on the film/substrate misfit and elastic compliance of thin films are characterized quantitatively. The theoretical predictions are in good agreement with the experimental results of piezoresponse scanning microscopy. Due to the large tunability of piezoresponse in it, PZT 50/50 (x=0.5) epitaxial film is a promising candidate for use in future tunable devices.

The domain structure of oxide superconductors arises as a result of twinning associated with the tetragonal-orthorhombic transition inside a restricted volume (a grain in a polycrystal, an epitaxial layer, or a part of an initial single crystal). A thermodynamic theory of the size and arrangement of the domains has been developed. There are two types of interfaces in equilibrium domain structures: coherent twin boundaries and boundaries between polytwin layers with microstresses. Elastic vortex-interface interactions have been estimated.