The flexoelectric effect is the response of electric polarization to a mechanical strain gradient. It can be viewed as a higher-order effect with respect to piezoelectricity, which is the response of polarization to strain itself. However, at the nanoscale, where large strain gradients are expected, the flexoelectric effect becomes appreciable. Besides, in contrast to the piezoelectric effect, flexoelectricity is allowed by symmetry in any material. Due to these qualities flexoelectricity has attracted growing interest during the past decade. Presently, its role in the physics of dielectrics and semiconductors is widely recognized and the effect is viewed as promising for practical applications. On the other hand, the available theoretical and experimental results are rather contradictory, attesting to a limited understanding in the field. This review paper presents a critical analysis of the current knowledge on the flexoelectricity in common solids, excluding organic materials and liquid crystals.

The charged domain wall is an ultrathin (typically nanosized) interface between two domains; it carries bound charge owing to a change of normal component of spontaneous polarization on crossing the wall. In contrast to hetero-interfaces between different materials, charged domain walls (CDWs) can be created, displaced, erased, and recreated again in the bulk of a material. Screening of the bound charge with free carriers is often necessary for stability of CDWs, which can result in giant two-dimensional conductivity along the wall. Usually in nominally insulating ferroelectrics, the concentration of free carriers at the walls can approach metallic values. Thus, CDWs can be viewed as ultrathin reconfigurable strongly conductive sheets embedded into the bulk of an insulating material. This feature is highly attractive for future nanoelectronics. The last decade was marked by a surge of research interest in CDWs. It resulted in numerous breakthroughs in controllable and reproducible fabrication of CDWs in different materials, in investigation of CDW properties and charge compensation mechanisms, in discovery of light-induced effects, and, finally, in detection of giant two-dimensional conductivity. The present review is aiming at a concise presentation of the main physical ideas behind CDWs and a brief overview of the most important theoretical and experimental findings in the field.

Domain walls in ferroic materials have attracted significant interest in recent years, in particular because of the unique properties that can be found in their vicinity. However, to fully harness their potential as nanoscale functional entities, it is essential to achieve reliable and precise control of their nucleation, location, number and velocity. Here, using piezoresponse force microscopy, we show the control and manipulation of domain walls in ferroelectric thin films of Pb(Zr,Ti)O₃ with Pt top electrodes. This high-level control presents an excellent opportunity to demonstrate the versatility and flexibility of ferroelectric domain walls. Their position can be controlled by the tuning of voltage pulses, and multiple domain walls can be nucleated and handled in a reproducible fashion. The system is accurately described by analogy to the classical Stefan problem, which has been used previously to describe many diverse systems and is here applied to electric circuits. This study is a step towards the realization of domain wall nanoelectronics utilizing ferroelectric thin films.

In the pursuit of ferroic-based (nano)electronics, it is essential to minutely control domain patterns and domain switching. The ability to control domain width, orientation and position is a prerequisite for circuitry based on fine domains. Here, we develop the underlying theory towards growth of ultra-fine domain patterns, substantiate the theory by numerical modelling of practical situations and implement the gained understanding using the most widely applied ferroelectric, Pb(Zr,Ti)O 3 , demonstrating controlled stripes of 10 nm wide domains that extend in one direction along tens of micrometres. The observed electrical conductivity along these thin domains embedded in the otherwise insulating film confirms their potential for electronic applications.

Use of ferroelectric domain-walls in future electronics requires that they are stable, rewritable conducting channels. Here we demonstrate nonthermally activated metallic-like conduction in nominally uncharged, bent, rewritable ferroelectric-ferroelastic domain-walls of the ubiquitous ferroelectric Pb(Zr,Ti)O3 using scanning force microscopy down to a temperature of 4 K. New walls created at 4 K by pressure exhibit similar robust and intrinsic conductivity. Atomic resolution electron energy-loss spectroscopy confirms the conductivity confinement at the wall. This work provides a new concept in "domain-wall nanoelectronics".

The influence of flexoelectric coupling on the internal structure of neutral domain walls in tetragonal phase of perovskite ferroelectrics is studied. The effect is shown to lower the symmetry of 180-degree walls which are oblique with respect to the cubic crystallographic axes, while {100} and {110} walls stay "untouched". Being of the Ising type in the absence of the flexoelectric interaction, the oblique domain walls acquire a new polarization component with a structure qualitatively different from the classical Bloch-wall structure. In contrast to the Bloch-type walls, where the polarization vector draws a helix on passing from one domain to the other, in the flexoeffect-affected wall, the polarization rotates in opposite directions on the two sides of the wall and passes through zero in its center. Since the resulting polarization profile is invariant upon inversion with respect to the wall center it does not brake the wall symmetry in contrast to the classical Bloch-type walls. The flexoelectric coupling lower the domain wall energy and gives rise to its additional anisotropy that is comparable to that conditioned by the elastic anisotropy. The atomic orderof-magnitude estimates shows that the new polarization component P 2 may be comparable with spontaneous polarization P s , thus suggesting that, in general, the flexoelectric coupling should be mandatory included in domain wall simulations in ferroelectrics. Calculations performed for barium titanate yields the maximal value of the P 2 , which is much smaller than that of the spontaneous polarization. This smallness is attributed to an anomalously small value of a component of the "strain-polarization" elecrostictive tensor in this material.2

Flexoelectric effect is the response of electric polarization to the mechanical strain gradient. At the nano-scale, where large strain gradients are expected, the flexoelectric effect becomes appreciable and may substitute piezoelectric effect in centrosymmetric materials. These features make flexoelectricity of growing interest during the last decade. At the same time, the available theoretical and experimental results are rather contradictory. In particular, experimentally measured flexoelectric coefficients in some ferroelectric materials largely exceed theoretically predicted values. Here, we determine the upper limits for the magnitude of the static bulk contribution to the flexoelectric effect in ferroelectrics, the contribution which was customarily considered as the dominating one. The magnitude of the upper bounds obtained suggests that the anomalously high flexoelectric coupling documented for perovskite ceramics can hardly be attributed to a manifestation of the static bulk effect. V C 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4865208] Flexoelectric effect is the response of electric polarization to the mechanical strain gradient. It can be viewed as higher-order effect with respect to piezoelectricity, which is the response of polarization to strain itself. However at the nano-scale, where large strain gradients are expected, the flexoelectric effect becomes appreciable. Besides, in contrast to piezoelectric effect, flexoelectricity is allowed by symmetry in any material. Due to these features flexoelectricity has attracted growing interest during the last decade. On the other hand, the available theoretical and experimental results are rather contradictory, attesting to a limited understanding in the field. In particular, often experimentally measured flexoelectric coefficients largely exceed theoretically predicted values. It is important to distinguish different contributions to the effect: bulk and surface contributions; static and dynamic contributions. The relative magnitude of these contributions is discussed in a recent review article.

Using Ginzburg-Landau theory and two-dimensional (2D) phase field simulations, we analyze the influence of flexoelectric coupling on the domain patterns in ferroelectrics. The phase field simulations predict that a high strength of the flexoelectric coupling leads to formation of a fine structure in domain patterns in ferroelectrics. The fine structure forms when the coupling strength exceeds a critical value and is related to local transition into an incommensurate phase. Depending on the parameters, a structure with stripe patterns with antiparallel polarizations or another one, not seen before, with two-dimensional arrays of alternating vortices is found. Complex domain configurations with coexisting phases and unusual domain walls between them are observed. Although the incommensurate phase does not form for weaker couplings, the influence of flexoelectricity on bulk domain patterns can still be significant. The results of the calculations are rationalized using an analytical model. Directions for the modulation wave vectors in the fine structure are found in the framework of a linear analysis, while the type of the structure-stripes or vortices-is determined by anharmonicity.