Fundamentals of flexoelectricity in solids

Flexoelectricity Modulated Electron Transport of 2D Indium Oxide

The phenomenon of flexoelectricity, wherein mechanical deformation induces alterations in the electron configuration of metal oxides, has emerged as a promising avenue for regulating electron transport. Leveraging this mechanism, stress sensing can be optimized through precise modulation of electron transport. In this study, the electron transport in 2D ultra-smooth In2O3 crystals is modulated via flexoelectricity. By subjecting cubic In2O3 (c-In2O3) crystals to significant strain gradients using an atomic force microscope (AFM) tip, the crystal symmetry is broken, resulting in the separation of positive and negative charge centers. Upon applying nano-scale stress up to 100 nN, the output voltage and power values reach their maximum, e.g. 2.2 mV and 0.2 pW, respectively. The flexoelectric coefficient and flexocoupling coefficient of c-In2O3 are determined as ≈0.49 nC m-1 and 0.4 V, respectively. More importantly, the sensitivity of the nano-stress sensor upon c-In2O3 flexoelectric effect reaches 20 nN, which is four to six orders smaller than that fabricated with other low dimensional materials based on the piezoresistive, capacitive, and piezoelectric effect. Such a deformation-induced polarization modulates the band structure of c-In2O3, significantly reducing the Schottky barrier height (SBH), thereby regulating its electron transport. This finding highlights the potential of flexoelectricity in enabling high-performance nano-stress sensing through precise control of electron transport.

Rayleigh wave propagation in centrosymmetric materials with micro-stiffness, flexoelectric and micro-inertia effects

A theoretical investigation of Rayleigh waves propagation in polarized media has been carried out using a reformulated flexoelectric theory for isotropic dielectrics with micro-inertia effect. Within this non-classical theory, the internal energy density is the functional of the strain tensor, dilatation gradient, deviatoric part of stretch gradient and rotation gradient tensors, polarization vector, and polarization gradient. The obtained system of governing equations additionally contains three material length-scale parameters to account the micro-stiffness effect, one material constant to capture the micro-inertia effect, two flexoelectric constants to describe the flexoelectric effect and three length scale parameters related to the polarization gradient. To solve the coupled governing equations, the method of Lamé-type potentials for mechanical displacement and electric polarization vectors is used. The influences of various factors such as micro-stiffness, flexoelectricity, electric quadrupoles and micro-inertia effects on the phase velocity of the Rayleigh waves in a homogeneous isotropic half-space are studied. It is found that above effects become significant with the increase of the wavenumber. This study can be important for the investigation of high frequency surface acoustic waves in dielectric materials.

Merits and Demerits of Machine Learning of Ferroelectric, Flexoelectric, and Electrolytic Properties of Ceramic Materials

In the present review, the merits and demerits of machine learning (ML) in materials science are discussed, compared with first principles calculations (PDE (partial differential equations) model) and physical or phenomenological ODE (ordinary differential equations) model calculations. ML is basically a fitting procedure of pre-existing (experimental) data as a function of various factors called descriptors. If excellent descriptors can be selected and the training data contain negligible error, the predictive power of a ML model is relatively high. However, it is currently very difficult for a ML model to predict experimental results beyond the parameter space of the training experimental data. For example, it is pointed out that all-dislocation-ceramics, which could be a new type of solid electrolyte filled with appropriate dislocations for high ionic conductivity without dendrite formation, could not be predicted by ML. The merits and demerits of first principles calculations and physical or phenomenological ODE model calculations are also discussed with some examples of the flexoelectric effect, dielectric constant, and ionic conductivity in solid electrolytes.

Revealing Strong Flexoelectricity and Optoelectronic Coupling in 2D Ferroelectric CuInP2S6 Via Large Strain Gradient

The interplay between flexoelectric and optoelectronic characteristics provides a paradigm for studying emerging phenomena in various 2D materials. However, an effective way to induce a large and tunable strain gradient in 2D devices remains to be exploited. Herein, we propose a strategy to induce large flexoelectric effect in 2D ferroelectric CuInP2S6 by constructing a 1D-2D mixed-dimensional heterostructure. The strong flexoelectric effect is induced by enormous strain gradient up to 4.2 × 106 m-1 resulting from the underlying ZnO nanowires, which is further confirmed by the asymmetric coercive field and the red-shift in the absorption edge. The induced flexoelectric polarization efficiently boosts the self-powered photodetection performance. In addition, the improved photoresponse has a good correlation with the induced strain gradient, showing a consistent size-dependent flexoelectric effect. The mechanism of flexoelectric and optoelectronic coupling is proposed based on the Landau-Ginzburg-Devonshire double-well model, supported by density functional theory (DFT) calculations. This work provides a brand-new method to induce a strong flexoelectric effect in 2D materials, which is not restricted to crystal symmetry and thus offers unprecedented opportunities for state-of-the-art 2D devices.

Mechanical Control of Polar Patterns in Wrinkled Thin Films via Flexoelectricity

Intriguing topological polar structures in oxide nanofilms have drawn growing attention owing to their immense potential applications in nanoscale electronic devices. Here, we report a novel route to mechanically manipulate polar structures via flexoelectricity in wrinkled thin films. Our results present a flexoelectric polar transition from a nonpolar state to uniaxial polar stripes, biaxial meronlike or antimeronlike polar structures, and polar labyrinths by varying wrinkle morphologies. The evolution mechanisms and the outstanding mechanical tunability of these flexoelectric polar patterns were investigated theoretically and numerically. This strategy based on flexoelectricity for generating nontrivial polar structures will no longer rely on the superlattice structure and can be widely applicable to all centrosymmetric or noncentrosymmetric materials, providing a broader range of material and structure candidates for polar topologies.

Defect Engineering Centrosymmetric 2D Material Flexocatalysts

In this study, the flexoelectric characteristics of 2D TiO2 nanosheets are examined. The theoretical calculations and experimental results reveal an excellent strain-induced flexoelectric potential (flexopotential) by an effective defect engineering strategy, which suppresses the recombination of electron-hole pairs, thus substantially improving the catalytic activity of the TiO2 nanosheets in the degradation of Rhodamine B dye and the hydrogen evolution reaction in a dark environment. The results indicate that strain-induced bandgap reduction enhances the catalytic activity of the TiO2 nanosheets. In addition, the TiO2 nanosheets degraded Rhodamine B, with kobs being ≈1.5 × 10-2  min-1 in dark, while TiO2 nanoparticles show only an adsorption effect. 2D TiO2 nanosheets achieve a hydrogen production rate of 137.9 µmol g-1  h-1 under a dark environment, 197% higher than those of TiO2 nanoparticles (70.1 µmol g-1  h-1 ). The flexopotential of the TiO2 nanosheets is enhanced by increasing the bending moment, with excellent flexopotential along the y-axis. Density functional theory is used to identify the stress-induced bandgap reduction and oxygen vacancy formation, which results in the self-dissociation of H2 O on the surface of the TiO in the dark. The present findings provide novel insights into the role of TiO2 flexocatalysis in electrochemical reactions.

Nanotube ferroelectric tunnel junctions with an ultrahigh tunneling electroresistance ratio

Low-dimensional ferroelectric tunnel junctions are appealing for the realization of nanoscale nonvolatile memory devices due to their inherent advantages of device miniaturization. Those based on current mechanisms have limitations, including low tunneling electroresistance (TER) effects and complex heterostructures. Here, we introduce an entirely new TER mechanism to construct a nanotube ferroelectric tunnel junction with ferroelectric nanotubes as the tunneling region. When rolling a ferroelectric monolayer into a nanotube, due to the coexistence of its intrinsic ferroelectric polarization with the flexoelectric polarization induced by bending, a metal-insulator transition occurs depending on the radiative polarization states. For the pristine monolayer, its out-of-plane polarization is tunable by an in-plane electric field, and the conducting states of the ferroelectric nanotube can thus be tuned between metallic and insulating states via axial electric means. Using α-In2Se3 as an example, our first-principles density functional theory calculations and nonequilibrium Green's function formalism confirm the feasibility of the TER mechanism and indicate an ultrahigh TER ratio that exceeds 9.9 × 1010% of the proposed nanotube ferroelectric tunnel junctions. Our findings provide a promising approach based on simple homogeneous structures for high density ferroelectric microelectric devices with excellent ON/OFF performance.

Advancements and Prospects of Flexoelectricity

The flexoelectric effect, as a novel form of the electromechanical coupling phenomenon, has attracted significant attention in the fields of materials science and electronic devices. It refers to the interaction between strain gradients and electric dipole moments or electric field intensity gradients and strain. In contrast to the traditional piezoelectric effect, the flexoelectric effect is not limited by material symmetry or the Curie temperature and exhibits a stronger effect at the nanoscale. The flexoelectric effect finds widespread applications ranging from energy harvesting to electronic device design. Utilizing the flexoelectric effect, enhanced energy harvesters, sensitive sensors, and high-performance wearable electronic devices can be developed. Additionally, the flexoelectric effect can be utilized to modulate the optoelectronic properties and physical characteristics of materials, holding the potential for significant applications in areas such as optoelectronic devices, energy storage devices, and flexible electronics. This review provides a comprehensive overview of the historical development, measurement of flexoelectric coefficients, enhancement mechanisms, and current research progress of the flexoelectric effect. Additionally, it offers a perspective on future prospects of the flexoelectric effect.

Significant hydrogen generation via photo-mechanical coupling in flexible methylammonium lead iodide nanowires

The flexoelectric effect, which refers to the mechanical-electric coupling between strain gradient and charge polarization, should be considered for use in charge production for catalytically driving chemical reactions. We have previously revealed that halide perovskites can generate orders of higher magnitude flexoelectricity under the illumination of light than in the dark. In this study, we report the catalytic hydrogen production by photo-mechanical coupling involving the photoflexoelectric effect of flexible methylammonium lead iodide (MAPbI3) nanowires (NWs) in hydrogen iodide solution. Upon concurrent light illumination and mechanical vibration, large strain gradients were introduced in flexible MAPbI3 NWs, which subsequently induced significant hydrogen generation (at a rate of 756.5 μmol g-1 h-1, surpassing those values from either photo- or piezocatalysis of MAPbI3 nanoparticles). This photo-mechanical coupling strategy of mechanocatalysis, which enables the simultaneous utilization of multiple energy sources, provides a potentially new mechanism in mechanochemistry for highly efficient hydrogen production.

Curved GaAs cantilever waveguides for the vertical coupling to photonic integrated circuits

We report the nanofabrication and characterization of optical spot-size converter couplers based on curved GaAs cantilever waveguides. Using the stress mismatch between the GaAs substrate and deposited Cr-Ni-Au strips, single-mode waveguides can be bent out-of-plane in a controllable manner. A stable and vertical orientation of the out-coupler is achieved by locking the spot-size converter at a fixed 90 ∘ angle via short-range forces. The optical transmission is characterized as a function of temperature and polarization, resulting in a broad-band chip-to-fiber coupling extending over 150 nm wavelength bandwidth at cryogenic temperatures, with the lower bound for the coupling efficiency into the TE mode being 16±2% in the interval 900-1050 nm. The methods reported here are fully compatible with quantum photonic integrated circuit technology with quantum dot emitters, and open opportunities to design novel photonic devices with enhanced functionality.

Giant Flexoelectricity in Bent Semiconductor Thinfilm

We elucidate the flexoelectricity of semiconductors in the high strain gradient regime, the underlying mechanism of which is less understood. By using the generalized Bloch theorem, we uncover a strong flexoelectric-like effect in bent thinfilms of Si and Ge due to a high-strain-gradient-induced band gap closure. We show that an unusual type-II band alignment is formed between the compressed and elongated sides of the bent film. Therefore, upon the band gap closure, electrons transfer from the compressed side to the elongated side to reach the thermodynamic equilibrium, leading to a pronounced change of polarization along the film thickness dimension. The obtained transverse flexoelectric coefficients are unexpectedly high with a quadratic dependence on the film thickness. This new mechanism is extendable to other semiconductor materials with moderate energy gaps. Our findings have important implications for the future applications of flexoelectricity in semiconductor materials.

Tunable Li-ion diffusion properties in MoSSe bilayer anodes by strain gradient

Layered MoSSe nanostructures have been shown as potential candidates for the anode of lithium ion (Li-ion) batteries. The diffusion properties are generally critical to the performance of ionic batteries. The possible migration paths and associated diffusion energy barriers of Li-ions are systematically explored in MoSSe bilayer anodes with different stacking patterns by means of first-principles simulations. It is found that the diffusion properties strongly depend on interfaces and stacking patterns. Furthermore, the simulation results show that the diffusion energy barrier (and thus the diffusion coefficient) can be significantly reduced (enlarged) by applying a positive strain gradient, while increased (reduced) by applying a negative one. For example, the diffusion coefficient is increased roughly by 100 times relative to that of the pristine one when subjected to a strain gradient of 0.02 Å-1. In particular, it is found that less maximum strain is required in the strain-gradient than the uniform strain in order to achieve the same diffusion energy barrier. By careful analysis, the underlying mechanism can be attributed to the flexo-diffusion coupling effect. The coupling strength is characterized by the so-called flexo-diffusion coupling constant which is also calculated for each simulation model. The results of this work may provide valuable insights into the design and optimization of the anodes of ionic batteries.

Flexoelectric Ionic Liquid-Grafted Triblock Copolymers for Energy Harvesting under Flexural Deformation

The goal of the present article is to develop flexoelectric polyelectrolyte elastomers for energy harvesting based on a poly(ethylene glycol)-block-poly(propylene glycol)-block-poly(ethylene glycol) dimethacrylate (PEG-b-PPG-b-PEG-DMA) triblock grafted with an ionic liquid (IL) such as allylmethylimidazolium bis(trifluoromethane sulfonyl) imide (AMIMTFSI). The IL-grafted triblock copolymer network possesses a balance of reasonably good ionic conductivity and high ion polarization during cantilever bending. Of particular importance is the achievement of high flexoelectric coefficients in some flexoelectric polyelectrolyte elastomer (FPE) compositions reaching 1368 μC/m at ambient temperature during mechanical deformation under intermittent square-wave bending mode. With the addition of a 10 wt % lithium bis(trifluoromethane sulfonyl) imide (LiTFSI) salt, the flexoelectric coefficient further improved to 1737 μC/m, which is the highest among all piezoelectric and flexoelectric materials hitherto reported, and thus it opens a new opportunity for clean energy harvesting from a vibrating natural environment.

Stress and Curvature Effects in Layered 2D Ferroelectric CuInP2S6

Nanoscale ferroelectric 2D materials offer the opportunity to investigate curvature and strain effects on materials functionalities. Among these, CuInP2S6 (CIPS) has attracted tremendous research interest in recent years due to combination of room temperature ferroelectricity, scalability to a few layers thickness, and ferrielectric properties due to coexistence of 2 polar sublattices. Here, we explore the local curvature and strain effect on polarization in CIPS via piezoresponse force microscopy and spectroscopy. To explain the observed behaviors and decouple the curvature and strain effects in 2D CIPS, we introduce the finite element Landau-Ginzburg-Devonshire model, revealing strong changes in hysteresis characteristics in regions subjected to tensile and compressive strain. The piezoresponse force microscopy (PFM) results show that bending induces ferrielectric domains in CIPS, and the polarization-voltage hysteresis loops differ in bending and nonbending regions. These studies offer insights into the fabrication of curvature-engineered nanoelectronic devices.

The giant flexoelectric effect in a luffa plant-based sponge for green devices and energy harvesters

Soft materials that can produce electrical energy under mechanical stimulus or deform significantly via moderate electrical fields are important for applications ranging from soft robotics to biomedical science. Piezoelectricity, the property that would ostensibly promise such a realization, is notably absent from typical soft matter. Flexoelectricity is an alternative form of electromechanical coupling that universally exists in all dielectrics and can generate electricity under nonuniform deformation such as flexure and conversely, a deformation under inhomogeneous electrical fields. The flexoelectric coupling effect is, however, rather modest for most materials and thus remains a critical bottleneck. In this work, we argue that a significant emergent flexoelectric response can be obtained by leveraging a hierarchical porous structure found in biological materials. We experimentally illustrate our thesis for a natural dry luffa vegetable-based sponge and demonstrate an extraordinarily large mass- and deformability-specific electromechanical response with the highest-density-specific equivalent piezoelectric coefficient known for any material (50 times that of polyvinylidene fluoride and more than 10 times that of lead zirconate titanate). Finally, we demonstrate the application of the fabricated natural sponge as green, biodegradable flexible smart devices in the context of sensing (e.g., for speech, touch pressure) and electrical energy harvesting.

Lamellar Fluctuations Melt Ferroelectricity

We consider a standard Ginzburg-Landau model of a ferroelectric whose electrical polarization is coupled to gradients of elastic strain. At the harmonic level, such flexoelectric interaction is known to hybridize acoustic and optic phonon modes and lead to phases with modulated lattice structures that precede the state with spontaneously broken inversion symmetry. Here, we use the self-consistent phonon approximation to calculate the effects of thermal and quantum polarization fluctuations on the bare hybridized modes to show that such long-range modulated order is unstable at all temperatures. We discuss the implications for the nearly ferroelectric SrTiO_{3} and KTaO_{3}, and we propose that these systems are melted versions of an underlying modulated state that is dominated by nonzero momentum thermal fluctuations except at the very lowest temperatures.

Flexible BaTiO3 Thin Film-Based Coupled Nanogenerator for Simultaneously Scavenging Light and Vibration Energies

Ferroelectric materials have a variety of properties, such as piezoelectricity, pyroelectricity, and the ferroelectric photovoltaic effect, which enable them to obtain electrical energy from various external stimuli. Here, we report a coupled nanogenerator based on flexible BTO ferroelectric films with a cantilevered beam structure. It combines the photovoltaic and flexoelectric effects in a ferroelectric materials-based coupled nanogenerator for simultaneously scavenging vibration energy and light energy, thus improving energy scavenging performance. As compared with the photovoltaic effect individually, simultaneous vibration and light illumination under a light intensity of 57 mW/cm² at 405 nm can produce a photo-flexoelectric coupling current of 85 nA, where the current peak has been enhanced by 121%. Due to the photo-flexoelectric coupling effect, the device has outstanding charging performance, where a 4.7 μF capacitor can be charged to 60 mV in 150 s. These devices have potential applications in multi-energy scavenging and self-powered sensors.

Research on Output Characteristics of Microscale BST Laminate Structure Based on Mixed Finite Element Method

The flexoelectric effect, which is sensitive to size, refers to the phenomenon of coupling between the strain gradient and electrical polarization and involves higher-order derivatives of physical quantities such as displacement, and the analytical process is complicated and difficult. Therefore, in this paper, a mixed finite element method is developed considering the effects of size effect and flexoelectric effect on the electromechanical coupling behavior of microscale flexoelectric materials. Based on the theoretical model of enthalpy density and the modified couple stress theory, the theoretical model and finite element model of microscale flexoelectric effect are established, and the Lagrange multiplier is used to coordinate the higher-order derivative relationship between the displacement field and its gradient, and the C¹ continuous quadrilateral 8-node (displacement and potential) and 4-node (displacement gradient and Lagrange multipliers) flexoelectric mixed element. By comparing the numerical calculation results and analytical solutions of the electrical output characteristics of the microscale BST/PDMS laminated cantilever structure, it is proved that the mixed finite element method designed in this paper is an effective tool for studying the electromechanical coupling behavior of flexoelectric materials.

Ionic Liquids Exhibit the Piezoelectric Effect

The piezoelectric effect was discovered over a century ago, and it has found wide application since that time. The direct piezoelectric effect is the production of charge upon application of force to a material, and the converse piezoelectric effect is a change in the material dimension(s) upon the application of a potential. To date, piezoelectric effects have been observed only in solid-phase materials. We report here the observation of the direct piezoelectric effect in room-temperature ionic liquids (RTILs). The RTILs 1-butyl-3-methyl imidazolium bis(trifluoromethyl-sulfonyl)imide (BMIM⁺TFSI^-) and 1-hexyl-3-methyl imidazolium bis(trifluoromethylsulfonyl) imide (HMIM⁺TFSI^-) produce a potential upon the application of force when confined in a cell, with the magnitude of the potential being directly proportional to the force applied. The effect is one order of magnitude smaller than that seen in quartz. This is the first report to our knowledge of the direct piezoelectric effect in a neat liquid. Its discovery has fundamental implications about the organization and dynamics in ionic liquids and invites theoretical treatment.

Critical Roles of Impurities and Imperfections in Various Phases of Materials

In many materials, impurities and imperfections play a critical role on the physical and chemical properties. In the present review, some examples of such materials are discussed. A bulk nanobubble (an ultrafine bubble) is stabilized against dissolution by hydrophobic impurities attached to the bubble surface. An acoustic cavitation threshold in various liquids decreases significantly by the presence of impurities such as solid particles, etc. The strength of brittle ceramics is determined by the size and number of pre-existing microcracks (imperfections) in the specimen. The size effect of a BaTiO₃ nanocrystal is influenced by the amount and species of adsorbates (impurities) on its surface as adsorbate-induced charge-screening changes the free energy. The dielectric constant of an assembly of BaTiO₃ nanocubes is influenced by a small tilt angle (imperfection) between two attached nanocubes, which induces strain inside a nanocube, and is also influenced by the spatial strain-relaxation due to defects and dislocations (imperfections), resulting in flexoelectric polarization.

Impact of 3D Curvature on the Polarization Orientation in Non-Ising Domain Walls

Ferroelectric domain boundaries are quasi-two-dimensional functional interfaces with high prospects for nanoelectronic applications. Despite their reduced dimensionality, they can exhibit complex non-Ising polarization configurations and unexpected physical properties. Here, the impact of the three-dimensional (3D) curvature on the polarization profile of nominally uncharged 180° domain walls in LiNbO₃ is studied using second-harmonic generation microscopy and 3D polarimetry analysis. Correlations between the domain-wall curvature and the variation of its internal polarization unfold in the form of modulations of the Néel-like character, which we attribute to the flexoelectric effect. While the Néel-like character originates mainly from the tilting of the domain wall, the internal polarization adjusts its orientation due to the synergetic upshot of dipolar and monopolar bound charges and their variation with the 3D curvature. Our results show that curved interfaces in solid crystals may offer a rich playground for tailoring nanoscale polar states.

Configurable Crack Wall Conduction in a Complex Oxide

Mobile defects in solid-state materials play a significant role in memristive switching and energy-efficient neuromorphic computation. Techniques for confining and manipulating point defects may have great promise for low-dimensional memories. Here, we report the spontaneous gathering of oxygen vacancies at strain-relaxed crack walls in SrTiO₃ thin films grown on DyScO₃ substrates as a result of flexoelectricity. We found that electronic conductance at the crack walls was enhanced compared to the crack-free region, by a factor of 10⁴. A switchable asymmetric diode-like feature was also observed, and the mechanism is discussed, based on the electrical migration of oxygen vacancy donors in the background of Sr-deficient acceptors forming n⁺-n or n-n⁺ junctions. By tracing the temporal relaxations of surface potential and lattice expansion of a formed region, we determine the diffusivity of mobile defects in crack walls to be 1.4 × 10^-16 cm²/s, which is consistent with oxygen vacancy kinetics.

Domain Switching in BaTiO3 Films Induced by an Ultralow Mechanical Force

Low-energy switching of ferroelectrics has been intensively studied for energy-efficient nanoelectronics. Mechanical force is considered as a low-energy consumption technique for switching the polarization of ferroelectric films due to the flexoelectric effect. Reduced threshold force is always desirable for the considerations of energy saving, easy domain manipulation, and sample surface protection. In this work, the mechanical switching behaviors of BaTiO₃/SrRuO₃ epitaxial heterostructure grown on Nb:SrTiO₃ (001) substrate are reported. Domain switching is found to be induced by an extremely low tip force of 320 nN (estimated pressure ∼0.09 GPa), which is the lowest value ever reported. This low mechanical threshold is attributed to the small compressive strain, the low oxygen vacancy concentration in BaTiO₃ film, and the high conductivity of the SrRuO₃ electrode. The flexoelectricity under both perpendicular mechanical load (point measurement) and sliding load (scanning measurement) are investigated. The sliding mode shows a much stronger flexoelectric field for its strong trailing field. The mechanical written domains show several advantages in comparison with the electrically written ones: low charge injection, low energy consumption, high density, and improved stability. The ultralow-pressure switching in this work presents opportunities for next-generation low-energy and high-density memory electronics.

Merits and Demerits of ODE Modeling of Physicochemical Systems for Numerical Simulations

In comparison with the first-principles calculations mostly using partial differential equations (PDEs), numerical simulations with modeling by ordinary differential equations (ODEs) are sometimes superior in that they are computationally more economical and that important factors are more easily traced. However, a demerit of ODE modeling is the need of model validation through comparison with experimental data or results of the first-principles calculations. In the present review, examples of ODE modeling are reviewed such as sonochemical reactions inside a cavitation bubble, oriented attachment of nanocrystals, dynamic response of flexoelectric polarization, ultrasound-assisted sintering, and dynamics of a gas parcel in a thermoacoustic engine.

Controllable Piezo-flexoelectric Effect in Ferroelectric Ba0.7Sr0.3TiO3 Materials for Harvesting Vibration Energy

The rapid development of the automotive and aerospace industries has led to an increasingly urgent need for electromechanical coupling materials and devices. Here, we have demonstrated the tunable piezo-flexoelectric effect in ferroelectric Ba_0.7Sr_0.3TiO₃ materials for scavenging vibration energy. The positive peak output current of an ITO/Ba_0.7Sr_0.3TiO₃/Ag cantilever device based on the flexoelectric effect is only 45 nA at room temperature, which is promoted to 90 nA by the piezo-flexoelectric effect. In addition, the piezo-flexoelectric current of the device can be further boosted to 270 nA by increasing the working temperature to 41.0 °C with a corresponding enhancement ratio of 348.28%. The significantly improved piezo-flexoelectric current is ascribed to the ultrahigh dielectric constant, which is related to the tetragonal-cubic phase transition of the Ba_0.7Sr_0.3TiO₃ materials. This work reveals the temperature-modulated piezo-flexoelectric effect in ferroelectric Ba_0.7Sr_0.3TiO₃ materials, providing a convenient route for scavenging and sensing of vibration energy.

Highly heterogeneous epitaxy of flexoelectric BaTiO3-δ membrane on Ge

The integration of complex oxides with a wide spectrum of functionalities on Si, Ge and flexible substrates is highly demanded for functional devices in information technology. We demonstrate the remote epitaxy of BaTiO₃ (BTO) on Ge using a graphene intermediate layer, which forms a prototype of highly heterogeneous epitaxial systems. The Ge surface orientation dictates the outcome of remote epitaxy. Single crystalline epitaxial BTO_3-δ films were grown on graphene/Ge (011), whereas graphene/Ge (001) led to textured films. The graphene plays an important role in surface passivation. The remote epitaxial deposition of BTO_3-δ follows the Volmer-Weber growth mode, with the strain being partially relaxed at the very beginning of the growth. Such BTO_3-δ films can be easily exfoliated and transferred to arbitrary substrates like Si and flexible polyimide. The transferred BTO_3-δ films possess enhanced flexoelectric properties with a gauge factor of as high as 1127. These results not only expand the understanding of heteroepitaxy, but also open a pathway for the applications of devices based on complex oxides.

The integration of epitaxial complex oxides on semiconductor and flexible substrates is required but challenging. Here, the authors report the highly heterogeneous epitaxy of transferrable BaTiO_3-δ membrane with enhanced flexoelectricity on Ge (011).

Effect of Pre-Polarization Process on the Apparent Piezoelectric Response Measured by Point-Ring Method in Ferroelectric Perovskite Oxide Ceramics

Both flexoelectricity and piezoelectricity belong to the electromechanical coupling effect. While piezoelectricity only exists in materials whose crystal structure is noncentrosymmetric and a pre-polarization process is necessary for ferroelectric piezoelectric ceramics, flexoelectricity theoretically exists in all dielectric materials and does not require pre-polarization. However, this does not mean that flexoelectricity would not be affected by the pre-polarization process, considering that flexoelectricity is a polarization phenomenon. In this work, we prepared ferroelectric perovskite oxide ceramics Ba1−xCaxTiO3 and revealed a strong effect of the pre-polarization process on the flexoelectric response of the ceramics, characterized by the apparent piezoelectric response measured by the point-ring method. The effective piezoelectric coefficient was separated into the one contributed by the flexoelectric(-like) response and the piezoelectric(-like) response by quasi-static d33 measurement and a two-step point-ring testing method. The effective piezoelectric coefficient contributed by the flexoelectric(-like) response of the ceramics could be largely enhanced to be over 350 pC/N after a 900 V polarization, larger than the standard piezoelectric response. The pre-polarization process was suggested to alter the polarization state and defect distributions, which would further change the overall flexoelectric response (both intrinsic and extrinsic parts) of the samples. Our work indicates a facile method to enhance the apparent piezoelectric response of flexoelectric materials under a bending mode.

Computation of Flexoelectric Coefficients of a MoS2 monolayer with a Model of Self-consistently Distributed Effective Charges and Dipoles

Flexoelectricity is an electromechanical coupling phenomenon, that can generate noticeable electric polarization in dielectric materials for nanoscale strain gradients. It is gaining an increasing attention because of its potential applications, and the fact that experimental results were initially an order of magnitude higher than initial theoretical predictions. This stimulated intense experimental and theoretical researches to investigate flexoelectric coefficients in dielectric materials such as two-dimensional materials. In this work, we concentrate on the calculation of the flexoelectric coefficients of 2D-MoS2 thanks to a model using self-consistently determined charges and dipoles on the atoms. More specifically, we study the importance of two contributions which were neglected/omitted in previous papers using this model, namely the charge term in the total polarization and the conservation of electric charge through a Lagrange multiplier. Our calculations demonstrate that the results for flexoelectric coefficient computed with this improved definition of polarization agree better with experimental measurements, provided consistent definitions for signs are used. Additionally, we show how two physical contributions with opposite signs compete to give net values of flexoelectric coefficients that can be either positive of negative depending on their relative importance, and give net values for the case of MoS2.

Engineering of atomic-scale flexoelectricity at grain boundaries

Flexoelectricity is a type of ubiquitous and prominent electromechanical coupling, pertaining to the electrical polarization response to mechanical strain gradients that is not restricted by the symmetry of materials. However, large elastic deformation is usually difficult to achieve in most solids, and the strain gradient at minuscule is challenging to control. Here, we exploit the exotic structural inhomogeneity of grain boundary to achieve a huge strain gradient (~1.2 nm-1) within 3-4-unit cells, and thus obtain atomic-scale flexoelectric polarization of up to ~38 μC cm-2 at a 24° LaAlO3 grain boundary. Accompanied by the generation of the nanoscale flexoelectricity, the electronic structures of grain boundaries also become different. Hence, the flexoelectric effect at grain boundaries is essential to understand the electrical activities of oxide ceramics. We further demonstrate that for different materials, altering the misorientation angles of grain boundaries enables tunable strain gradients at the atomic scale. The engineering of grain boundaries thus provides a general and feasible pathway to achieve tunable flexoelectricity.

Coexistence of Flexo- and Ferro-Electric Effects in an Ordered Assembly of BaTiO3 Nanocubes

It has been reported that the flexoelectric effect could be dominant in the nanoscale. The discrepancy between theory and experiments on the frequency dependence of the dielectric constant of an ordered assembly of BaTiO₃ nanocubes is nearly resolved by assuming the coexistence of flexo- and ferro-electric effects. Although flexoelectric polarizations perpendicular to the applied alternating electric field contribute to the dielectric constant, those parallel to the electric field do not contribute because the magnitude of the flexoelectric polarization does not change due to the mismatch of strain at the interface of the nanocubes. On the other hand, some dielectric response is possible for the ferroelectric component of the polarization parallel to the electric field.

Direct and Converse Flexoelectricity in Two-Dimensional Materials

Building on recent developments in electronic-structure methods, we define and calculate the flexoelectric response of two-dimensional (2D) materials fully from first principles. In particular, we show that the open-circuit voltage response to a flexural deformation is a fundamental linear-response property of the crystal that can be calculated within the primitive unit cell of the flat configuration. Applications to graphene, silicene, phosphorene, boron nitride, and transition-metal dichalcogenide monolayers reveal that two distinct contributions exist, respectively of purely electronic and lattice-mediated nature. Within the former, we identify a key metric term, consisting in the quadrupolar moment of the unperturbed charge density. We propose a simple continuum model to connect our findings with the available experimental measurements of the converse flexoelectric effect.

Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity

The Timoshenko beam model is applied to the analysis of the flexoelectric effect for a cantilever beam under large deformations. The geometric nonlinearity with von Kármán strains is considered. The nonlinear system of ordinary differential equations (ODE) for beam deflection and rotation are derived. Moreover, this nonlinear system is linearized for each load increment, where it is solved iteratively. For the vanishing flexoelectric coefficient, the governing equations lead to the classical Timoshenko beam model. Furthermore, the influence of the flexoelectricity coefficient and the microstructural length-scale parameter on the beam deflection and the induced electric intensity is investigated.

Flexoelectric and Piezoelectric Coupling in a Bended MoS2 Monolayer

Low-dimensional (LD) transition metal dichalcogenides (TMDs) in the form of nanoflakes, which consist of one or several layers, are the subject of intensive fundamental and applied research. The tuning of the electronic properties of the LD-TMDs are commonly related with applied strains and strain gradients, which can strongly affect their polar properties via piezoelectric and flexoelectric couplings. Using the density functional theory and phenomenological Landau approach, we studied the bended 2H-MoS2 monolayer and analyzed its flexoelectric and piezoelectric properties. The dependences of the dipole moment, strain, and strain gradient on the coordinate along the layer were calculated. From these dependences, the components of the flexoelectric and piezoelectric tensors have been determined and analyzed. Our results revealed that the contribution of the flexoelectric effect dominates over the piezoelectric effect in both in-plane and out-of-plane directions of the monolayer. In accordance with our calculations, a realistic strain gradient of about 1 nm−1 can induce an order of magnitude higher than the flexoelectric response in comparison with the piezoelectric reaction. The value of the dilatational flexoelectric coefficient is almost two times smaller than the shear component. It appeared that the components of effective flexoelectric and piezoelectric couplings can be described by parabolic dependences of the corrugation. Obtained results are useful for applications of LD-TMDs in strain engineering and flexible electronics.

Analysis of the Piezoelectric Properties of Aligned Multi-Walled Carbon Nanotubes

Recent studies reveal that carbon nanostructures show anomalous piezoelectric properties when the central symmetry of their structure is violated. Particular focus is given to carbon nanotubes (CNTs) with initial significant curvature of the graphene sheet surface, which leads to an asymmetric redistribution of the electron density. This paper presents the results of studies on the piezoelectric properties of aligned multi-walled CNTs. An original technique for evaluating the effective piezoelectric coefficient of CNTs is presented. For the first time, in this study, we investigate the influence of the growth temperature and thickness of the catalytic Ni layer on the value of the piezoelectric coefficient of CNTs. We establish the relationship between the effective piezoelectric coefficient of CNTs and their defectiveness and diameter, which determines the curvature of the graphene sheet surface. The calculated values of the effective piezoelectric coefficient of CNTs are shown to be between 0.019 and 0.413 C/m², depending on the degree of their defectiveness and diameter.

BEM evaluation of surface octahedral strains and internal strain gradients in 3D-printed scaffolds used for bone tissue regeneration

Most of the mechnoregulatory computational models appearing so far in tissue engineering for bone healing predictions, utilize as regulators for cell differentiation mainly the octahedral volume strains and the interstitial fluid velocity calculated at any point of the fractured bone area and controlled by empirical constants concerning these two parameters. Other stimuli like the electrical and chemical signaling of bone constituents are covered by those two regulatory fields. It is apparent that the application of the same mechnoregulatory computational models for bone healing predictions in scaffold-aided regeneration is questionable since the material of a scaffold disturbs the signaling pathways developed in the environment of bone fracture. Thus, the goal of the present work is to evaluate numerically two fields developed in the body of two different compressed scaffolds, which seem to be proper for facilitating cell sensing and improving cell viability and cell seeding efficiency. These two fields concern the surface octahedral strains that the cells attached to the scaffold can experience and the internal strain gradients that create electrical pathways due to flexoelectric phenomenon. Both fields are evaluated with the aid of the Boundary Element Method (BEM), which is ideal for evaluating with high accuracy surface strains and stresses as well as strain gradients appearing throughout the analyzed elastic domain.

Emergence of non-trivial polar topologies hidden in singular stress field in SrTiO3: topological strain-field engineering

Discovery of non-trivial topological structures in condensed matters holds promise in novel technological paradigms. In contrast to ferromagnetics, where a variety of topological structures such as vortex, meron, and skyrmion have been discovered, only few topological structures can exist in ferroelectrics due to the lack of non-collinear interaction like the Dzyaloshinskii-Moriya interaction in ferromagnetics. Here, we demonstrate that polarization structures with a wide range of topological numbers (winding numbernfrom -3 to +1) can be mechanically excited and designed by the mode-I singular stress field formed near the crack-tip in incipient ferroelectric SrTiO₃. Our phase-field simulations based on Ginzburg-Landau theory successfully reveals that the near-tip polar topology is driven by the flexoelectric coupling with intense strain gradient at the tip, while a variety of the far-field topological structures is triggered by a collaboration between the electrostrictive and flexoelectric effects. The strain (gradient) field analysis further shows that the unexpected topological characters are implied in the singular stress field, which develops a variety of polar topologies near the crack tip. Therefore, our work provides a novel insight into the unusual interplay between mechanical- and ferroelectric-topologies, i.e. 'topological strain-field engineering', which paves the way to the mechanical design of functional topologies in the matter.

A Framework for Low-Intensity Low-Frequency Ultrasound Neuromodulation Sonication Parameter Identification from Micromechanical Flexoelectricity Modelling

Low-intensity, low-frequency ultrasound (LILFU) has recently emerged as a promising technique to modulate non-invasively nerve activities at lower cost than other traditional and more-invasive neuromodulation methods. However, there is currently no consensus on the optimum sonication parameters to be used in LILFU applications, and most of the accepted ranges have arisen from trial-and-error approaches. Here we utilise a recently proposed micromechanics model of membrane flexoelectricity, a potential candidate for neuromodulation, and simulate action potentials/membrane polarisation triggered by acoustic pulses of different pulse frequencies, pulse magnitudes and duty cycles. Results reveal that, at constant duty cycles, increasing the transmit frequency increases the thresholds of both the pulse magnitude and the elastic energy rate density required to mechanically trigger an action potential, whereas at constant frequencies, increasing the duty cycle reduces both. The influence of transmit frequency is weakened at lower duty cycles. Our simulation results offer some guidance on the selections of sonication parameters used in LILFU for neurologic disorder treatments in the context of the flexoelectricity hypothesis.

Ferroelectric Self-Polarization Controlled Magnetic Stratification and Magnetic Coupling in Ultrathin La0.67Sr0.33MnO3 Films

Multiferroic oxide heterostructures consisting of ferromagnetic and ferroelectric components hold the promise for nonvolatile magnetic control via ferroelectric polarization, advantageous for the low-dissipation spintronics. Modern understanding of the magnetoelectric coupling in these systems involves structural, orbital, and magnetic reconstructions at interfaces. Previous works have long proposed polarization-dependent interfacial magnetic structures; however, direct evidence is still missing, which requires advanced characterization tools with near-atomic-scale spatial resolutions. Here, extensive polarized neutron reflectometry (PNR) studies have determined the magnetic depth profiles of PbZr_0.2Ti_0.8O₃/La_0.67Sr_0.33MnO₃ (PZT/LSMO) bilayers with opposite self-polarizations. When the LSMO is 2-3 nm thick, the bilayers show two magnetic transitions on cooling. However, temperature-dependent magnetization is different below the lower-temperature transition for opposite polarizations. PNR finds that the LSMO splits into two magnetic sublayers, but the inter-sublayer magnetic couplings are of opposite signs for the two polarizations. Near-edge X-ray absorption spectroscopy further shows contrasts in both the Mn valences and the Mn-O bond anisotropy between the two polarizations. This work completes the puzzle for the magnetoelectric coupling model at the PZT/LSMO interface, showing a synergic interplay among multiple degrees of freedom toward emergent functionalities at complex oxide interfaces.

Flexophotovoltaic Effect in Potassium Sodium Niobate/Poly(Vinylidene Fluoride-Trifluoroethylene) Nanocomposite

Flexoelectricity is an electromechanical coupling effect in which electric polarization is generated by a strain gradient. In this investigation, a potassium sodium niobite/poly(vinylidene fluoride-trifluoroethylene) (KNN/PVDF-TrFE)-based nanocomposite is fabricated, and the flexoelectric effect is used to enhance the photovoltaic current (I pv) in the nanocomposite. It is found that both a pyroelectric current and photovoltaic current can be generated simultaneously in a light illumination process. However, the photovoltaic current (I pv) in this process contributes ≈85% of the total current. When assessing the effect of flexoelectricity with a curvature of 1/20, the I pv of the curved KNN/PVDF-TrFE (20%) (K/P-20) composite increased by ≈13.9% compared to that of the flat K/P-20 nanocomposite. Similarly, at a curvature of 1/20, the I pv of the K/P-20 nanocomposite is 71.6% higher than that of the PVDF-TrFE film. However, the photovoltaic effect induced by flexoelectricity is much higher than the increased polarization from flexoelectricity, so this effect is called as the flexophotovoltaic effect. Furthermore, the calculated energy conversion efficiency of the K/P-20 film is 0.017%, which is comparable to the previous research result. This investigation shows great promise for PVDF-based nanocomposites in ferroelectric memory device applications.

Metal-ferroelectric supercrystals with periodically curved metallic layers

Simultaneous manipulation of multiple boundary conditions in nanoscale heterostructures offers a versatile route to stabilizing unusual structures and emergent phases. Here, we show that a stable supercrystal phase comprising a three-dimensional ordering of nanoscale domains with tailored periodicities can be engineered in PbTiO₃-SrRuO₃ ferroelectric-metal superlattices. A combination of laboratory and synchrotron X-ray diffraction, piezoresponse force microscopy, scanning transmission electron microscopy and phase-field simulations reveals a complex hierarchical domain structure that forms to minimize the elastic and electrostatic energy. Large local deformations of the ferroelectric lattice are accommodated by periodic lattice modulations of the metallic SrRuO₃ layers with curvatures up to 10⁷ m^-1. Our results show that multidomain ferroelectric systems can be exploited as versatile templates to induce large curvatures in correlated materials, and present a route for engineering correlated materials with modulated structural and electronic properties that can be controlled using electric fields.

Switching a Polar Metal via Strain Gradients

Although rare, spontaneous breakdown of inversion symmetry sometimes occurs in a material which is metallic: these are commonly known as polar metals or ferroelectric metals. Their polarization, however, is difficult to switch via an electric field, which limits the experimental control over band topology. Here we investigate, via first-principles theory, flexoelectricity as a possible way around this obstacle with the well-known polar metal LiOsO_{3}. The flexocoupling coefficients are computed for this metal with high accuracy with an approach based on real-space sums of the interatomic force constants. A Landau-Ginzburg-Devonshire-type first-principles Hamiltonian is built and a critical bending radius to switch the material is estimated, whose order of magnitude is comparable to that of BaTiO_{3}.

Strain Gradient Elasticity in SrTiO3 Membranes: Bending versus Stretching

Young's modulus determines the mechanical loads required to elastically stretch a material and also the loads required to bend it, given that bending stretches one surface while compressing the opposite one. Flexoelectric materials have the additional property of becoming electrically polarized when bent. The associated energy cost can additionally contribute to elasticity via strain gradients, particularly at small length scales where they are geometrically enhanced. Here, we present nanomechanical measurements of freely suspended SrTiO₃ crystalline membrane drumheads. We observe an unexpected nonmonotonic thickness dependence of Young's modulus upon small deflections. Furthermore, the modulus inferred from a predominantly bending deformation is three times larger than that of a predominantly stretching deformation for membranes thinner than 20 nm. In this regime we extract a strain gradient elastic coupling of ∼2.2 μN, which could be used in new operational regimes of nanoelectro-mechanics.

Flexoelectricity in atomic monolayers from first principles

We study the flexoelectric effect in fifty-four select atomic monolayers using ab initio Density Functional Theory (DFT). Specifically, considering representative materials from each of the Group III monochalcogenides, transition metal dichalcogenides (TMDs), Groups IV, III-V, and V monolayers, Group IV dichalcogenides, Group IV monochalcogenides, transition metal trichalcogenides (TMTs), and Group V chalcogenides, we perform symmetry-adapted DFT simulations to calculate transversal flexoelectric coefficients along the principal directions at practically relevant bending curvatures. We find that the materials demonstrate linear behavior and have similar coefficients along both principal directions, with values for TMTs being up to a factor of five larger than those of graphene. In addition, we find electronic origins for the flexoelectric effect, which increases with monolayer thickness, elastic modulus along the bending direction, and sum of polarizability of constituent atoms.

Mechanics of free-standing inorganic and molecular 2D materials

The discovery of graphene has triggered a great interest in inorganic as well as molecular two-dimensional (2D) materials. In this review, we summarize recent progress in the mechanical characterization of free-standing 2D materials, such as graphene, hexagonal boron nitride (hBN), transition metal-dichalcogenides, MXenes, black phosphor, carbon nanomembranes (CNMs), 2D polymers, 2D metal organic frameworks (MOFs) and covalent organic frameworks (COFs). Elastic, fracture, bending and interfacial properties of these materials have been determined using a variety of experimental techniques including atomic force microscopy based nanoindentation, in situ tensile/fracture testing, bulge testing, Raman spectroscopy, Brillouin light scattering and buckling-based metrology. Additionally, we address recent advances of 2D materials in a variety of mechanical applications, including resonators, microphones and nanoelectromechanical sensors. With the emphasis on progress and challenges in the mechanical characterization of inorganic and molecular 2D materials, we expect a continuous growth of interest and more systematic experimental work on the mechanics of such ultrathin nanomaterials.

A Physical Perspective to the Inductive Function of Myelin-A Missing Piece of Neuroscience

Starting from the inductance in neurons, two physical origins are discussed, which are the coil inductance of myelin and the piezoelectric effect of the cell membrane. The direct evidence of the coil inductance of myelin is the opposite spiraling phenomenon between adjacent myelin sheaths confirmed by previous studies. As for the piezoelectric effect of the cell membrane, which has been well-known in physics, the direct evidence is the mechanical wave accompany with action potential. Therefore, a more complete physical nature of neural signals is provided. In conventional neuroscience, the neural signal is a pure electrical signal. In our new theory, the neural signal is an energy pulse containing electrical, magnetic, and mechanical components. Such a physical understanding of the neural signal and neural systems significantly improve the knowledge of the neurons. On the one hand, we achieve a corrected neural circuit of an inductor-capacitor-capacitor (LCC) form, whose frequency response and electrical characteristics have been validated by previous studies and the modeling fitting of artifacts in our experiments. On the other hand, a number of phenomena observed in neural experiments are explained. In particular, they are the mechanism of magnetic nerve stimulations and ultrasound nerve stimulations, the MRI image contrast issue and Anode Break Excitation. At last, the biological function of myelin is summarized. It is to provide inductance in the process of neural signal, which can enhance the signal speed in peripheral nervous systems and provide frequency modulation function in central nervous systems.