The stochastic analysis of the polarization domain structures, emerging after quenching from a paraelectric to a ferroelectric state, in terms of the polarization correlation functions and their Fourier transforms is a fast and effective tool of the materials structure characterization. In spite of a significant volume of experimental data accumulated over the last three decades for the model uniaxial ferroelectric triglycine sulfate, there were no theoretical tools to comprehend these data until now. This work summarizes the recent progress in understanding of the experiments by means of the original stochastic model of polarization structure formation based on the Landau-Ginzburg-Devonshire theory and the Gauss random field concept assuming the predominance of the quenched polarization disorder over the thermal fluctuations. The system of integrodifferential equations for correlation functions of random polarization and electric field turns out to be analytically solvable. The model provides explanations to a range of experimental results on the polarization formation kinetics including the time-dependent correlation lengths and correlation functions on the macroscopic spatial and time scales. Notably, it predicts the dependence of the ferroelectric coercive field on the initial disordered state characteristics, which can be controlled by quenching parameters like the initial temperature and the cooling rate, thus paving the way for tailoring the functional properties of the material.

Initial conditions after quenching from a high-temperature paraelectric phase to a low-temperature ferroelectric phase have a substantial impact on the temporal development and formation of stable polarization domain structures which eventually determine physical properties and the functionality of ferroelectrics. Based on the recently advanced exactly solvable model of the stochastic domain structure kinetics in a uniaxial ferroelectric [Phys. Rev. B 107, 144109 (2023)], we study the effect of the magnitude of the initial disorder, its initial correlation length and polarization correlation function on the system evolution. For different shapes of the initial correlation function, the time-dependent correlation length and the two-point polarization correlation coefficient are calculated analytically demonstrating universal features and a good agreement with the available experimental data. Particularly, the magnitude of the charge density correlation function reveals a strong reduction of the bound charges at the nominally charged domain walls which was recently observed experimentally in uniaxial ferroelectrics. Consequently, the integrodifferential equations of evolution for the polarization correlation function and the mean polarization are numerically solved for different initial conditions. The temporal dependence of the polarization mean value and variance are evaluated demonstrating the bifurcation behavior depending on the applied electric field. The impact of the initial state properties on the coercive field deciding between the single- and multi-domain final states of the system is disclosed.

Ferroelectrics have a spontaneous electrical polarization that is arranged into domains and can be reversed by an externally-applied field. This high versatility makes them useful in enabling components such as capacitors, sensors, and actuators. Key parameters to tune their dielectric, piezoelectric, and electromechanical performance are the domain structure and the dynamic of the domain walls. In fixed compositions, this is often realized by chemical doping. In addition, structural and microstructural parameters, such as grain size, degree of crystallographic texture and porosity play a key role. An important step forward in the field was the fundamental understanding of the link between the local electric and mechanical driving forces and domain wall motion. Here, the impact of crystal structure and microstructure on these driving forces is reviewed and an engineering toolbox is introduced. An overview of advances in the understanding of domain wall motion on the micro- and nanoscale is provided and discussed in terms of the macroscopic functional performance of polycrystalline ferroelectrics/ferroelastics. In addition, a link to theoretical and computational models is established. The review concludes with a discussion about beyond state-of-the-art characterization techniques, new approaches, and future directions toward non-conventionally ordered ferroelectrics for next-generation nanoelectronic and energy-storage applications.

A self-consistent stochastic model of domain structure formation in a uniaxial ferroelectric, quenched from a high-temperature paraelectric phase to a low-temperature ferroelectric phase, is developed with an account of the applied electric field and the feedback effect via local depolarization fields. Both polarization and field components are considered as Gauss random variables. A system of integrodifferential equations for correlation functions of all involved variables is derived and solved analytically and numerically. Phase diagram in terms of the average value and dispersion of polarization reveals different possible equilibrium states and available final single-domain and multidomain states. The time-dependent evolution of the average polarization and dispersion discloses a bifurcation behavior and the temperature-dependent value of the electric field, deciding between the single-domain and multidomain final states, which can be interpreted as the coercive field. Analytical and numerical results for the time-dependent correlation length and correlation functions exhibit plausible agreement with available experimental data.

Received 5 June 2023DOI:https://doi.org/10.1103/PhysRevB.107.219903©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDomainsFerroelectricityPhase transitionsPhysical SystemsFerroelectricsSingle crystal materialsTechniquesBifurcation analysisLandau-Ginzburg-Devonshire theoryStochastic analysisStochastic analysis methodsStochastic differential equationsCondensed Matter, Materials & Applied PhysicsStatistical Physics

A conceptual problem of the electric-field mediated polarization correlations during a stochastic formation of polarization domain structure in ferroelectrics is addressed by using an exactly solvable stochastic model of polarization development in a uniaxial ferroelectric [Phys. Rev. B 107, 144109 (2023)]. A full set of time-dependent two-point correlation coefficients between all random variables is derived analytically, evaluated numerically and presented graphically in 3D. They are particularly required for the analysis of nonlinear phenomena involving spatial dispersion like optical second harmonic generation and scattering.

Ferroelectrics are multifunctional smart materials finding applications in sensor technology, micromechanical actuation, digital information storage, etc. Their most fundamental property is the ability of polarization switching under an applied electric field. In particular, understanding of switching kinetics is essential for digital information storage. In this regard, scaling properties of the temporal polarization response are well-known for 180°-switching processes in ferroelectrics characterized by a unique field-dependent local switching time. Unexpectedly, these properties are now observed in multiaxial polycrystalline ferroelectrics, exhibiting a number of parallel and sequential non-180°-switching processes with distinct switching times. This behavior can be explained by a combination of the multistep stochastic mechanism and the inhomogeneous field mechanism models of polarization reversal. Scaling properties are predicted for polycrystalline ferroelectrics of tetragonal, rhombohedral, and orthorhombic symmetries and are exemplarily demonstrated by the measurements of polarization kinetics in (K,Na)NbO3-based ferroelectric ceramic over a timescale of 7 orders of magnitude. Dynamic scaling properties allow insight into the microscopic switching mechanisms, on the one hand, and into statistical material characteristics, on the other hand, thereby providing the description of temporal polarization with high accuracy. The gained deeper insight into the mechanisms of multistep polarization switching is crucial for future ultrafast and multilevel digital information storage.

Using the Landau-Ginzburg-Devonshire theory, an influence of the misfit strain and surface screening charges, as well as the role of the flexoelectric effect, have been studied by numerical modelling in the case of a rhombohedral lead zirconate-titanate ferroelectric/ferroelastic thin film with an anisotropic misfit produced by a substrate. It was established that the magnitude and sign of the misfit strain influence the domain structure and predominant directions of the polarization vector, providing misfit-dependent phases with different favourable polarization components. Whilst strong enough compressive misfit strains favour a phase with an orthorhombic-like polarization directions, strong tensile misfits only yield in-plane polarization components. The strength of surface screening is seen to condition the existence of closure domain structures and, by increasing, supports the single-domain state depending on the value of the misfit strain. The flexoelectric effect exhibits a weak influence on the phase diagram of multi-domain states when compared with the phase diagram of single-domain states. Its effect, however, becomes significant in the case of skyrmion topological states, which spontaneously form near the film surface when compressive misfit strains are applied. Cooperative influence of the misfit strain, surface screening charges and temperature can set a thin rhombohedral ferroelectric film into a number of different polar and structural states, whereby the role of the flexoelectric effect is pronounced for topologically nontrivial structures.

A stochastic model of electric field-driven polarization reversal in orthorhombic ferroelectrics is advanced, providing a description of their temporal electromechanical response. The theory accounts for all possible parallel and sequential switching events. Application of the model to the simultaneous measurements of polarization and strain kinetics in a lead-free orthorhombic (K,Na)NbO3-based ferroelectric ceramic over a wide timescale of 7 orders of magnitude allowed identification of preferable polarization switching paths, fractions of individual switching processes, and their activation fields. Particularly, the analysis revealed substantial contributions of coherent non-180° switching events, which do not cause macroscopic strain and thus mimic 180° switching processes. * Corresponding author: genenko@mm.tu-darmstadt.de

The review is focused on the phenomenological description of the elastic defects influence on the electrophysical properties of nanosized ferroics. Using the Landau-Ginzburg-Devonshire phenomenological approach we consider the contribution of the elastic defects (e.g. neutral oxygen vacancies) on the phase transitions temperatures, phase diagrams, piezo-resistive, dielectric and polar properties of thin ferroelectric films and nanoparticles. We pay special attention to the role of flexoelectric effect, Vegard strains and stresses, including their synergy, called flexo-chemical coupling, on the defect-driven phenomena in nanoferroics. We explore the influence of mobile charged defects migration (e.g. the drift and diffusion of cations or charged oxygen vacancies) on the domain structure evolution in thin ferroelectric films. Also, we discuss the possibilities of the uncharged elastic defects self-ordering ordering in thin strained oxide films, and show that a thermodynamically stable disordered phase, spatially modulated phases, and long-range ordered phases with defect layers parallel or perpendicular to substrate plane can appear in the phase diagram in dependence on the film-substrate mismatch strain, concentration of elastic defects and Vegard coefficients. Since the long-range ordered ferroic phases can become multiferroic when some elastic dipoles become electric ones, the obtained results open the way to create and control defect-ordered superstructures by the choice of appropriate substrate and defect concentration in thin ferroic films.

A stochastic model for the field-driven polarization reversal in rhombohedral ferroelectrics is developed, providing a description of their temporal electromechanical response. Application of the model to simultaneous measurements of polarization and strain kinetics in a rhombohedral Pb(Zr, Ti)O-3 ceramic over a wide time window allows the identification of preferable switching paths, fractions of individual switching processes, and their activation fields. In conjunction, the phenomenological Landau-Ginzburg-Devonshire theory is used to analyze the impact of an external field and stress on switching barriers, showing that residual mechanical stress may promote the fast switching.

Evolution of a stripe array of polarization domains triggered by the oxygen vacancy migration in an acceptor doped ferroelectric is investigated in a self-consistent manner. A comprehensive model based on the Landau-Ginzburg-Devonshire approach includes semiconductor features due to the presence of electrons and holes, and effects of electrostriction and flexoelectricity especially significant near the free surface and domain walls. A domain array spontaneously formed in the absence of an external field is shown to undergo a reconstruction in the course of the gradual oxygen vacancy migration driven by the depolarization fields. The charge defect accumulation near the free ferroelectric surface causes a series of phenomena: (i) symmetry breaking between the positive and negative c-domains, (ii) appearance of an effective dipole layer at the free surface followed by the formation of a surface electrostatic potential, (iii) tilting and recharging of the domain walls, especially pronounced at higher acceptor concentrations. An internal bias field determined by the gain in the free energy of the structure exhibits dependences of its amplitude on time and dopant concentration well comparable with available experimental results on aging in BaTiO3.

Ordering of mobile defects in functional materials can give rise to fundamentally new phases possessing ferroic and multiferroic functionalities. Here we develop the Landau theory for strain induced ordering of defects (e.g. oxygen vacancies) in thin oxide films, considering both the ordering and wavelength of possible instabilities. Using derived analytical expressions for the energies of various defect-ordered states, we calculated and analyzed phase diagrams dependence on the film-substrate mismatch strain, concentration of defects, and Vegard coefficients. Obtained results open possibilities to create and control superstructures of ordered defects in thin oxide films by selecting the appropriate substrate and defect concentration.

A stochastic model for polarization switching in tetragonal ferroelectric ceramics is introduced, which includes sequential 90 degrees- and parallel 180 degrees-switching processes and accounts for the dispersion of characteristic switching times due to a nonuniform spatial distribution of the applied field. It presents merging of the recent multistep stochastic mechanism with the earlier nucleation limited switching and inhomogeneous field mechanism models. The model provides a much better description of simultaneous polarization and strain responses over a wide time window and a deeper insight into the microscopic switching mechanisms, as is exemplarily shown by comparison with measurements on lead zirconate titanate.

Consecutive stochastic 90 degrees polarization switching events, clearly resolved in recent experiments, are described by a nucleation and growth multistep model. It extends the classical Kolmogorov-Avrami-Ishibashi approach and includes possible consecutive 90 degrees- and parallel 180 degrees switching events. The model predicts the results of simultaneous time-resolved macroscopic measurements of polarization and strain, performed on a tetragonal Pb(Zr, Ti)O-3 ceramic in a wide range of electric fields over a time domain of seven orders of magnitude. It allows the determination of the fractions of individual switching processes, their characteristic switching times, activation fields, and respective Avrami indices.

Electric depolarization fields have a great impact on the polarization-switching kinetics in ferroelectrics although they are often neglected in statistical considerations. Analysis of statistical distributions and correlations of polarization and electric field during the field-driven polarization reversal in a bulk polycrystalline ferroelectric by means of the two-dimensional self-consistent mesoscopic switching (SMS) model has revealed that correlations, mediated by electrostatic fields, are mostly isotropic and short range at a typical scale of the mean grain size [Phys. Rev. B 96, 054113 (2017)]. However, the magnitude of emerging depolarization fields remains substantial and strongly influences the switching kinetics. It is known, on the other hand, that the effect of inhomogeneities, such as a granular structure, on the electric field pattern and local field magnitudes is considerably overestimated in two-dimensional simulations. Three-dimensional extension of the SMS model in the current study allows a realistic evaluation of the impact of spatial correlations on the polarization switching in ferroelectric ceramics and opens a possibility to consider materials of different phase symmetries. It is shown that bound charges at grain boundaries due to mismatching grain polarizations as well as the subsequent depolarization fields are essentially dependent on the crystalline symmetry. This explains great differences in statistical field distributions and polarization kinetics observed in ceramics of different phase symmetries. Field correlations are anisotropic, depend on the material symmetry, but remain in all cases short range at the scale of a grain size. This sheds light on the success of models assuming statistically independent switching of different regions. Evolution of the statistical field distributions in the course of polarization reversal is also symmetry dependent but temporal changes in distributions are not substantial which clarifies a good performance of models neglecting the feedback via depolarization fields.

Using Landau-Ginzburg-Devonshire theory, we considered the impact of the flexoelectro-chemical coupling on the size effects inpolar properties and phase transitions of thin ferroelectric films with a layer of elastic defects. We investigated a typical case, when defects fill a thin layer below the top film surface with a constant concentration creating an additional gradient of elastic fields. The defective surface of the film is not covered with an electrode, but instead with an ultra-thin layer of ambient screening charges, characterized by a surface screening length. This geometry is typical for the scanning probe piezoelectric force microscopy. Obtained results revealed an unexpectedly strong effect of the joint action of Vegard stresses and flexoelectric effect (shortly flexo-chemical coupling) on the ferroelectric transition temperature, distribution of the spontaneous polarization and elastic fields, domain wall structure and period in thin PbTiO3 films containing a layer of elastic defects. A nontrivial result is the ferroelectricity persisting at film thicknesses below 4 nm, temperatures lower than 350 K and relatively high surface screening length ( 0.1 nm). The origin of this phenomenon is the re-building of the domain structure in the film (namely the cross-over from c-domain stripes to a-type closure domains) when its thickness decreases below 4 nm, conditioned by the flexoelectric coupling and facilitated by negative Vegard effect. For positive Vegard effect, thicker films exhibit the appearance of pronounced maxima on the thickness dependence of the transition temperature, whose position and height can be controlled by the defect type and concentration. The revealed features may have important implications for miniaturization of ferroelectric-based devices.

Using the self-consistent Landau-Ginzburg-Devonshire approach we simulate and analyze the spontaneous formation of the domain structure in thin ferroelectric films covered with the surface screening charge of the specific nature (Bardeen-type surface states). Hence we consider the competition between the screening and the domain formation as alternative ways to reduce the electrostatic energy and reveal unusual peculiarities of distributions of polarization, electric and elastic fields conditioned by the surface screening length and the flexocoupling strength. We have established that the critical thickness of the film and its transition temperature to a paraelectric phase strongly depend on the Bardeen screening length, while the flexocoupling affects the polarization rotation and closure domain structure and induces ribbon-like nano-scale domains in the film depth far from the top open surface. Hence the joint action of the surface screening (originating from e.g. the adsorption of ambient ions or surface states) and flexocoupling may remarkably modify polar and electromechanical properties of thin ferroelectric films.

Measurements of AC losses in a HTS-tape placed in between two bulk magnetic shields of high permeability were performed by applying calorimetric techniques for various asymmetrical shielding arrangements. The experiment was supported by analytical calculations and finite-element simulations of the field and current distributions, based on the Bean model of the critical state. The simulated current and field profiles perfectly reproduce the analytic solutions known for certain shielding geometries. The evaluation of the consequent AC losses exhibits good agreement with measurements for the central position of the tape between the magnets but have increasing discrepancy when the tape is approaching the shields. This can be explained by the increasing contribution of the eddy currents and magnetic hysteresis losses in the conducting shields.

Hybrid structures composed of ferroelectric thin films and functional two-dimensional (2D) materials may exhibit unique characteristics and reveal new phenomena due to the cross-interface coupling between their intrinsic properties. In this report, we demonstrate a symbiotic interplay between spontaneous polarization of the ultrathin BaTiO3 ferroelectric film and conductivity of the adjacent molybdenum disulfide (MoS2) layer, a 2D narrow-bandgap semiconductor. Polarization-induced modulation of the electronic properties of MoS2 results in a giant tunneling electroresistance effect in the hybrid MoS2-BaTiO3-SrRuO3 ferroelectric tunnel junctions (FTJs) with an OFF-to-ON resistance ratio as high as 104, a 50-fold increase in comparison with the same type of FTJs with metal electrodes. The effect stems from the reversible accumulation-depletion of the majority carriers in the MoS2 electrode in response to ferroelectric switching, which alters the barrier at the MoS2-BaTiO3 interface. Continuous tunability of resistive states realized via stable sequential domain structures in BaTiO3 adds memristive functionality to the hybrid FTJs. The use of narrow band 2D semiconductors in conjunction with ferroelectric films provides a novel pathway for development of the electronic devices with enhanced performance.