Flexoelectric materials and their related applications: A focused review

Flexoelectricity refers to the mechanical-electro coupling between strain gradient and electric polarization, and conversely, the electro-mechanical coupling between electric field gradient and mechanical stress. This unique effect shows a promising size effect which is usually large as the material dimension is shrunk down. Moreover, it could break the limitation of centrosymmetry, and has been found in numerous kinds of materials which cover insulators, liquid crystals, biological materials, and semiconductors. In this review, we will give a brief report about the recent discoveries in flexoelectricity, focusing on the flexoelectric materials and their applications. The theoretical developments in this field are also addressed. In the end, the perspective of flexoelectricity and some open questions which still remain unsolved are commented upon.


General concept of flexoelectricity
Materials are usually classified into insulators, semiconductors, and conductors according to their different conduction modes in response to the external electric field. Specifically, the insulators transfer the function of external electric field by inducting rather than conducting. Quite recently a universal physical phenomenon, namely flexoelectricity, has rekindled considerable research interest in insulators due to its great application potential in many fields, e.g., new types of memory, domain engineering, defect tailoring, and non-piezoelectric sensors and actuators [1][2][3][4]. This phenomenon describes the mechanical-electro coupling between the strain gradient and the electric polarization (direct flexoelectric effect), or vice versa, the coupling between the electric field gradient and the stress (converse flexoelectric effect), which can be written as [5]: where P l is the induced polarization, T ij is the induced stress, S ij is the strain, and E k is the electric field, x is the axis of coordinate, µ ijkl is the direct flexoelectric coefficient with respect to strain gradient. The concept of flexoelectricity was firstly originated from liquid crystals [6]. In liquid crystals, flexoelectricity refers to the reorientation of irregularly shaped polarized molecules under strain gradients caused by splay-deformations or bent-deformations, which is different from piezoelectric effect (produced by uniform strain or stress) [7,8]. In some soft materials like hairs and bio-membranes, similar phenomenon that the net charges between the internal and external surface of the material occur in response to the different curvature radiuses was also found [9,10].
As a distinctive mechanical-electro effect, flexoelectricity has two main attractive features. (i) The flexoelectric coefficient is a fourth rank tensor, therefore flexoelectricity could break the limitation of crystal symmetry, and exist in those materials with centrosymmetry, which is quite different from the piezoelectric effect [11][12][13]. (ii) Strain gradient is inversely proportional to the material size, making the nanoscale flexoelectricity extremely large [14][15][16]. Such characteristic is called size effect or scaling effect, and has already been utilized in many nanostructures for sensing and actuating applications [17,18].

Flexoelectricity in materials 2. 1 Direct and indirect measurement of flexoelectric coefficients
Although the flexoelectric coupling should be principally existed in all insulators, studies on exploring the flexoelectric coefficient in specific materials are still quite limited. This could be explained by the following two reasons.
(i) In most of the dielectrics, the typical value of the flexoelectric coefficient is only in the range of 0.01-1 nC/m. This characteristic value is generally regarded as an intrinsic flexoelectric coefficient. In 1986, Tagantsev proposed that the flexoelectric coefficients in general materials are comparable to their e/a value [19,20], where e is the electron charge and a is the lattice parameter. This theoretical value is in good accordance with the recent first principle calculations developed by Maranganti and Sharma [21] and Hong and Vanderbilt [22,23] in some semiconductors and ferroelectric perovskites (as shown in Table 1). Note that the unit C/m is quite different with the unit of piezoelectric coefficients (C/N). Therefore, it is inappropriate to make a direct comparison between flexoelectricity and piezoelectricity, just by using the numerical value of flexoelectric coefficients with that of the piezoelectric coefficients. Recent studies by Abdollahi et al. [24] suggested that 1 pC/N of piezoelectric coefficient is comparable to 10 3 nC/m of flexoelectric coefficient. Based on this result, we can conclude that the flexoelectric coupling in most of the dielectrics is much weaker than the piezoelectric one.
(ii) The term "strain gradient" generally leads to very sophisticated tensor components of flexoelectric coefficient and then makes their measurements very difficult. According to the early studies by Le Quang and He [25] and Shu et al. [12], in a material with triclinic symmetry, the possible components of flexoelectric coefficients µ ijkl could reach to a number as high as 54. Therefore, it is almost impossible to precisely extract individual flexoelectric components in those low-symmetry systems. On the other hand, in the materials with cubic symmetry (except for the 23 and m3 point group symmetries), the components of flexoelectric tensor µ ijkl could be reduced to only 3, which is generally defined as the transverse flexoelectric coefficient, longitudinal flexoelectric coefficient, and shear flexoelectric coefficient.
During the past few years, three direct measurement methods have been developed by using the cantilever beam ( Fig. 1(a)), the bottom−up side of truncated pyramid ( Fig. 1(b)), and the lateral side of truncated pyramid ( Fig. 1(c)) to measure the transverse flexoelectric coefficient, longitudinal flexoelectric coefficient, and shear flexoelectric coefficient, respectively [5,26]. It is worth noting that all the measurement setups in Fig. 1 are not highly integrated, and therefore the determination of all the aforementioned flexoelectric components is highly constrained.
In addition to these direct measurements, a few indirect measurements of the flexoelectric coefficients have been recently developed. Zhou et al. [27] found that the flexoelectric coefficients could be indirectly evaluated by using the variation of the stiffness. It is because the flexoelectricity will result in a rapid decline of the stiffness of a dielectric nano-conefrustum, especially at small size. The detailed principle and structure are described in Fig. 2(a). By studying the nanocompression force vs. the nanocompression displacement curves of the column and cone-frustum samples (as shown in Fig. 2(b)), the longitudinal flexoelectric coefficient could be successfully extracted. Hu et al. [28] proposed a special way to generate strain gradient by using a shock wave. In their experiment, the first-order hydrogen gas gun was employed to fire a flying plate and hit the non-polarized bulk BaTiO 3

Fig. 1
Experimental setup for measuring three flexoelectric components. (a) The cantilever beam system which corresponds to the transverse flexoelectric coefficients. The measurement mechanism is by bending the cantilever sample in order to form a strain gradient ∂ S 11 /∂x 3 and measuring the induced polarization simultaneously. (b) The measuring system of the longitudinal flexoelectric coefficients. The measurement mechanism is by applying a mechanical force to the truncated pyramid sample in order to form a strain gradient ∂S 33 /∂x 3 and measuring the induced polarization P 3 simultaneously. (c) The measuring system of the shear flexoelectric coefficients. The measurement mechanism is by applying voltage to the lateral side of the truncated pyramid sample in order to generate the electric field gradient ∂E 1 /∂x 3 and measuring the induced shear strain S 13 simultaneously. The induced voltage of bulk BaTiO 3 samples as a function of shock waves. As the shock wave propagates into the sample, the negative voltage changes to positive voltage, and its value increases rapidly due to the flexoelectric effect which is caused by the strain gradient. Reproduced with permission from Ref. [28], © AIP Publishing 2017. ceramic ( Fig. 2(c)). Through this method, the longitudinal flexoelectric coefficient could be indirectly calculated by studying the relationship between the induced voltage and the propagation of the shock waves, as shown in Fig. 2(d).

2 Flexoelectric coefficients in dielectrics and ferroelectrics
With the help of those direct and indirect measurements, a few polymers and dielectrics have been employed for studying their flexoelectric coefficients. As shown in Table 1, it is proved that the flexoelectric coefficients in normal dielectrics and polymers, represented by TiO 2 ceramic and the polyvinylidene fluoride (PVDF) [29][30][31][32][33][34], are in the range of 1-10 nC/m, which is in good accordance with the intrinsic flexoelectric values. But in some kinds of ferroelectric ceramics and single crystals, the results are quite different. Extremely enhanced flexoelectricity was found in a series of high permittivity ferroelectrics such as BaTiO 3 (BT) [35,36], (BaSr)TiO 3 (BST) [37][38][39], Ba(TiSn)O 3 (BTS) [40,41], Pb(ZrTi)O 3 (PZT) [42], (KNa)NbO 3 [43], and Pb(Mg 1/3 Nb 2/3 )O 3 based ferroelectrics [44][45][46]. Their measured flexoelectric coefficients are 3-5 orders of magnitude larger than that in normal dielectrics such as TiO 2 . Intriguingly, such enhancement is not a universal phenomenon in ferroelectrics. For the incipient ferroelectric SrTiO 3 and antiferroelectric PbZrO 3 , the transverse flexoelectric coefficient is only 1-2 nC/m, which is just comparable with the normal dielectrics [47,48]. In very recent years, the physical mechanism of the enhanced flexoelectricity in some dielectrics was intensively discussed [5,24,[49][50][51][52][53][54]. The possible mechanisms are summarized in Fig. 3. The state-ofthe-art interpretation is not unified and much more work is urgently needed. Herein we want to emphasize that the extremely sensitive unit of the flexoelectric coefficients and their un-integrated electrical measurement may make an interference for the understanding of flexoelectricity. In real finite materials, all the factors, e.g., inner micro strain, polar nanoregions, and surface piezoelectricity, could be easily coupled into the flexoelectric coefficients. Therefore, the observed flexoelectric coefficients in dielectrics should be a competition/combination of many factors. It is noticed that the recent studies by Zhang et al. [54] suggested that the surface piezoelectricity can contribute about 70% of the enhanced flexoelectricity in BaTiO 3 ceramics, and hence revealed this factor might be dominant in many other ferroelectrics where the observed flexoelectric coefficients were significantly enhanced.

3 Flexoelectricity in biological materials
Recent studies suggest that many biological materials such as bones, hairs, and bio-membranes have remarkable flexoelectric response [55][56][57][58][59]. The first studies of the flexoelectricity in biological materials could be traced back to 1975 by Williams and Breger [60]. In this initial study, some of the electro-mechanical properties of bones were considered to be likely originated from "gradient polarization", but the mechanism is not clear yet. To clarify the origin of this electro-mechanical property in bones, Vasquez-Sancho et al. [61] compared the flexoelectricity of bone and pure hydroxyapatite (which is the main mineral of bone) by using the traditional cantilever system. As shown in Fig. 4(a), the measured flexoelectricity in both bone and pure hydroxyapatite was found to be of the same order of magnitude, suggesting that the hydroxyapatite flexoelectricity is the main source of the bendinginduced polarization in bones. Furthermore, the result also revealed that the flexoelectricity in hydroxyapatite is helpful to the bone repair and remodeling processes. It is because that, as shown in Fig. 4(b), a large

Table 1 Experimental and calculated values of flexoelectric coefficients for some materials
flexoelectric electric field will exist in the vicinity of the cracks of the hydroxyapatite. The bio-membrane flexoelectricity was recently studied by using the droplet interface bilayer technology (DIB) [62][63][64][65]. The principle of DIB in this scenario, as shown in Fig. 4(c) [66], is by using lipids as the organic surfactant in oil-water emulsion, in order to create biomimetic membranes at the intersection of neighboring droplets. When two droplets are brought into contact, the lipid monolayers adhere together which act as a capacitor. The contact angle at the monolayer-bilayer meniscus is determined by the balance of the tension between the two interfaces. Intriguingly, this technique can be utilized to explore the mechano-transduction and interfacial properties of unsupported liquid biomimetic membranes. Freeman et al. [66] reported that high-frequency membrane deformation is capable of producing a significant flexoelectric current, whose value is related to the curvature of the interfacial membrane, as shown in Fig.   4(d). This result suggested that flexoelectricity in bio-membrane can drive the activation of tension-gated channels within the membrane [67,68].

4 Flexoelectricity in liquid crystals
The liquid crystal flexoelectricity stems from the 1970s and has been regarded as one of the fundamental properties of this kind of materials. In general, the flexoelectric coefficients in liquid crystals are mainly represented by two independent components, the component e 1 and the component e 3 , which respond to the splay deformation and bend deformation, respectively. Up till now, two independent ways have been developed to measure the flexoelectric components e 1 and e 3 in liquid crystals. One is reported by Trabi et al. [69] based on the converse flexoelectric effect. As shown in technique. This technique uses lipids as an organic surfactant in an oil-water emulsion, creating biomimetic membranes at the intersection of neighboring droplets. When the droplets are brought into contact, the lipid monolayers adhere together in a lipid bilayer, often approximated as a capacitor. The contact angle at the monolayer-bilayer meniscus is determined by the balance of the tension between the two interfaces. (d) At high frequencies of oscillation (50 Hz), which shifts the membrane deformation mechanic from gradual growth and reduction to elastic strain and bending, then producing flexoelectric current. Reproduced with permission from Ref. [66], © Royal Society of Chemistry 2016. splay structure, and this deformation could be quantified by the distortion of the birefringence pattern. The other method is proposed by Harden et al. [70] through the direct flexoelectric effect, as shown in Fig. 5(b). This method is similar to the measurement of flexoelectricity in dielectrics. The studied nematic liquid crystal (NLC) cells were firstly fixed in a flexible container, and then bent by using a loudspeaker. Later, the NLC was oscillated between two fixed posts, and a large flexoelectric current could be induced. For most of the liquid crystals, the typical value of flexoelectric components e 1 and e 3 is 1-100 nC/m [71]. Compared with the ferroelectrics, the flexoelectric coefficients of liquid crystals are relatively small. However, regardless its small coefficients, the flexoelectricity often plays a dominant role in liquid crystals, especially in the widely developed liquid crystal display technology [72][73][74][75][76][77][78]. Recently, Lee et al. [79] found that the highly distortion of the liquid crystal display could be quantitatively measured by the variation of the flexoelectric polarization, as shown in Fig. 5(c). Moreover, the image flicker problem of liquid crystal mixtures which shows negative dielectric anisotropy can be minimized by the flexoelectric effect [80,81] (as shown in Fig. 5(d)).

5 Flexoelectricity in semiconductor
Generally speaking, the electro-mechanical coupling is only limited to dielectrics due to the requirement of insulation. But more and more studies have disclosed that semiconductors can be regarded as electromechanical materials, e.g., the ZnO semiconductor is a piezoelectric material [82][83][84]. In 2016, Narvaez et al. [85] made a new breakthrough in flexoelectricity and greatly broadened the concept of flexoelectricity into semiconductors. A dead layer mechanism was proposed and the measured flexoelectricity in some semiconductors can reach a level to 1000 μC/m.
As shown in Fig. 6(a), for an arbitrary bulk insulator such as BaTiO 3 single crystal, the surface piezoelectricity exists due to the symmetry breaking in boundary. At the top and bottom surfaces, the piezoelectric polarization is usually orientated in opposite direction so that the macro piezoelectricity is zero. Once bending this bulk material, a net polarization exists because the bending induced strain direction of the top and bottom surface is opposite. That is why the surface piezoelectricity can couple to the enhanced flexoelectricity in the bulk insulators [86]. Similarly, this mechanism also works in semiconductor owing to the existence of dead layer. The dead layer, typically in several micrometers thick, could accumulate the top and bottom surface piezoelectricity, and results in a significant net polarization when the semiconductor is bent. As shown in Fig. 6(b), the oxygen reduced BaTiO 3 single crystal which acts as a good ionic conductor, shows an extremely enhanced flexoelectric-like response. This phenomenon was also found in Nb-doped TiO 2 semiconductor. Interestingly, the flexoelectricity in semiconductor shows a linearized thickness dependence. The measured effective flexoelectric coefficients were directly proportional with the material thickness, as shown in Fig. 6(c). Inspired by this attractive result, one can expect that with proper design, the application of semiconductor flexoelectricity will play an important role in the optimization of the integrated circuit and electromechanical semiconductor devices.

Theoretical calculation of flexoelectricity
During the past few years, the studies of flexoelectricity were not solely focused on the experiment level, but also led to a lot of theoretical progress. It is worth noting that the size effect of flexoelectricity was firstly proposed by theoretical prediction rather than any experiment [14]. Numerous theoretical studies have demonstrated that the strain gradient becomes large in nanoscale, at which the flexoelectricity may even be competitive with piezoelectricity [87][88][89][90][91][92][93]. Specifically, in epitaxial ferroelectric thin films [94], the simulated strain gradient can reach up to 10 6 -10 7 m -1 , which may make the flexoelectric polarization higher than the piezoelectric one.
The general theory of flexoelectricity was recently developed. Many of them were focused on the origin of the enhancement of flexoelectricity in some typical materials [95][96][97]. Theoretical work on flexoelectricity dates back to the papers by Mashkevich and Tolpygo [98], who first proposed the effect, and Kogan [8], who formulated the first phenomenological theory. According to the theory proposed by Yudin et al. [99], the flexoelectric polarization can be accounted for four parts, i.e., surface piezoelectricity, bulk piezoelectricity, surface flexoelectricity, and bulk flexoelectricity [20]. All these factors contribute almost equally to the flexoelectric coefficient. However, a different viewpoint was stated hereupon by Resta [52] who built the polarization theory www.springer.com/journal/40145 based on the elemental cubic crystal. His theoretical result manifested that the intrinsic flexoelectricity was a purely bulk effect, where the surface contribution was totally excluded. Later, Yurkov and Tagantsev. [100] argued that the direct bulk flexoelectric response of a finite sample essentially depends on the surface polarization energy. Recently, Zhang et al. [54] calculated that the surface piezoelectricity which widely exists in all the unpolarized ceramics and single crystals, is possibly dominant in the observed flexoelectricity. This statement also well matches their experimental results [54]. The controversy for this issue still exists and a comprehensive understood is urgently needed.
The first-principles calculation of flexoelectric coefficients was firstly developed in several semiconductors by Maranganti and Sharma [21] until 2009. Hong et al. [22,23,101] demonstrated the first ab initio calculation of the longitudinal flexoelectricity for BaTiO 3 and SrTiO 3 by using a direct approach, where the strain gradient is realized by setting the lattice parameter of each atom. Meanwhile, they also developed an indirect way for calculating flexoelectric coefficients which was formulated in such a way that the tensor elements can be computed directly in the context of density-functional calculations, including electronic and lattice contributions [23]. To date, the first-principles method could be utilized into the calculation of other flexoelectric tensor component and much more materials, which is summarized in Table 1 [102][103][104][105][106].
An interesting theoretical progress of flexoelectricity we addressed is the application of topology optimization. This method, firstly proposed by Bendsøe and Kikuchi [107] in 1988, has been widely applied to optimize the material property [108,109], especially for the design of domains [110,111]. More recently, Ghasemi et al. [112,113] presented a new-type topology optimization method, which is based on a combination of isogeometric analysis (IGA), level set, and point wise density mapping techniques as shown in Fig. 7(a) and Fig. 7(b), respectively. Through this topology optimization, significant enhanced flexoelectric coefficients can be obtained in designated materials.

Applications of flexoelectricity
Here we will emphasize on the recent developed applications of flexoelectricity in several fields, including flexoelectric energy harvester, flexoelectric sensors and actuators, domain engineering, liquid crystal displaying, and some other open applications, e.g., flexoelectricity-tuned photovoltaic effect. In the end, we also list some possible applications where the flexoelectricity might be involved.

1 Energy harvesters
Energy harvesters are those devices which can collect energy from the external sources like wind energy, solar power, thermal energy, etc. Specifically, the electromechanical energy harvester can collect energy from the mechanical vibrations [126][127][128][129][130][131][132]. Recently, the newly designed energy harvester which collect energy from the fluctuation of the vibrations (strain gradient) was proposed. Owing to its special sizedependent effect, the flexoelectric energy harvester is suitable to be integrated into small size, where a huge strain gradient can be generated. Till now, with the proper design, the mechanical-electrical energy converting efficiency of the flexoelectric energy harvester can reach to 6.6% [133][134][135].
The potential of flexoelectricity as energy harvester has been predicted by many works. Wang and Wang [136] developed an analytical model for vibrationbased circular energy harvester that consists of a flexoelectric layer and a substrate layer. The result indicated that both the power output (as shown in Fig.   8(a)) and the energy conversion efficiency (as shown in Fig. 8(b)) of the flexoelectric energy harvester are much larger than that of the classical piezoelectric energy harvesters. Particularly, when the thickness (h p ) of the flexoelectric layer is reduced to less than 100 nm, the power output of the flexoelectric energy harvester is almost 12 times larger than that of the piezoelectric energy harvester [137]. Choi and Kim [138] fabricated a flexoelectric energy harvester device which collects energy by a PVDF thin film deposited on the cantilever substrate. The photograph of this device is shown in Fig. 8(c). Han et al. [139] prepared a flexoelectric nanogenerator which is consisted of direct-grown piezoelectric PZT on multi-walled carbon nanotubes. This device can repeatedly generate a voltage output of 8.6 V and a current output of 47 nA at a mechanical force of 20 N, which is promising for harvesting the mechanical energy. Moreover, Zhu et al. [140] designed a flexible flexoelectric fiber which is made of curved piezoelectric composites. As shown in Fig. 8(d), these fibers possess a stable upward self-poling which is induced by flexoelectricity, exhibited a large electric output comparable to those of the piezoelectric nanogenerators.

2 Actuators
Piezoelectricity has been widely used for sensors and actuators. Similarly, the flexoelectricity could also be utilized for sensing and actuating applications, especially for nano-sized devices. Compared with the piezoelectric sensors and actuators, the flexoelectric ones are not limited by the crystal symmetry of the materials and the working temperature [141,142]. The past few years have witnessed many progresses in this aspect. As shown in Fig. 9(a), Zhang et al. [143] designed a curved flexoelectric actuator by using non-polarized PVDF. This actuator exhibited a good actuating property with a resolution of the displacement reaching up to 1.0 nm and a largest displacement as high as 63.6 nm (as shown in Fig. 9(b)). It is worth highlighting that Bhaskar et al. [144] implemented a flexoelectric actuator made of micron-sized barium titanate (as shown in Fig. 9(c)) which is fully compatible for the semiconductor silicon technology. The performance result ( Fig. 9(d)) suggests that this flexoelectric actuator displays comparable performance to the actuators use lead-containing piezoelectric material.

3 Sensors
The flexoelectric sensors are becoming increasingly popular because of their small size, absence of depoling and aging problem, and lead-free composition [145]. The state-of-the-art flexoelectric sensor has been successfully utilized in many fields such as structural health monitoring, crack detector, and curvature detector [146,147]. Recently, Yan et al. [148] designed a BST-based flexoelectric curvature sensor, which can transfer the bending deflections directly to the charge output. The working principle of this sensor is illustrated in Fig. 10(a). To precisely detect the bending deflection, two BST micro curvature sensors were attached onto the center side surfaces of an aluminum beam, located symmetrically with respect to its neutral axis. The relationship between flexoelectric charge outputs of the BST sensors and the curvature (strain gradient) is shown in Fig. 10(b). The sensitivity of this curvature sensor, defined by the slopes of the charge outputs versus curvature, can reach up to 1 pC/m. In addition, Merupo et al. [149] investigated the potential use of 6.6 µm-thick soft polyurethane films as the large curvature sensor by subjecting flexible aluminum foil/PET bi-layered substrate to large deflections. A curvature of about 80 m -1 corresponding to a radius of curvature of ~1.2 cm could be effectively sensed under a low frequency (3 Hz) bending motion. The pseudosinusoidal time dependence of the output charge of this sensor is shown in Fig. 10(c). Kwon et al. [150] reported recently that the flexoelectricity can work in the microphone. The designed sensitivity of the fabricated barium strontium titanate (Ba 0.65 Sr 0.35 TiO 3 ) microphone was very high and almost frequencydependent in wide frequency range, as shown in Fig.  10(d).

4 Domain tailoring and polarization switching
Another attractive application of flexoelectricity should be addressed is the domain tailoring and polarization switching. As we may know that the ferroelectric materials are characterized by their spontaneous polarization, which can be switched by applying an external electric field. As suggested by Lu et al. [1] and Catalan et al. [2], in a nano-sized material, e.g., ultrathin ferroelectric film, the spontaneous polarization can be switched by mechanical strain gradient. This powerful function has attracted increasing interests and hence makes heterostructure thin film a natural scenario for the flexoelectric applications.
Recently, Gómez et al. [151] prepared an epitaxial growth of columnar porous BaTiO 3 /LaSrMnO 3 /SrTiO 3 / Si(001) heterostructures by using a complicated recipe which both combines the molecule beam epitaxy and polymer assisted deposition technology. The illustration of the preparation process is shown in Fig. 11(a). The results proved that the ferroelectric polarization of this heterostructure thin film can be reversed by a mechanical load in epitaxial columnar nanostructures, as shown in Fig. 11(b). Similar phenomenon was also found in PbTiO 3 thin films [152]. It was experimentally proved that, as shown in Fig. 11(c), the mechanical force shows a comparable ability relative with the electrically method for switching the domain of the  Moreover, with the help of flexoelectricity, the self-polarization direction of ferroelectric thin films, which is of difficulty in control in previous studies, is feasible to be changed by tuning the substrate interfaces and film thicknesses [153,154]. Park et al. [155] demonstrated that the multiple domain switching pathways in multiaxial ferroelectric materials can be selectively controlled by a newly realized trailing flexoelectric field, specifically, by the motion of a mechanically loaded scanning probe microscope tip. The illustration of this trailing flexoelectric field is shown in Fig. 12(a). Experimental results proved that the ferroelectric switching angle of multiferroic magnetoelectric BiFeO 3 thin film can be stably selected at 71° ferroelastic switching or 109° ferroelectric switching. The in-plane polarization and out-of-plane polarization in response to the loading forces of the moving tip shows a significant orientation dependence, as shown in Fig. 12(b) and Fig. 12(c), respectively.
In addition, with the help of the flexoelectricity, the microscopic tip can also be used to explore dynamic ionic behavior [156] and manipulate the oxygen vacancies [157] (as shown in Fig. 12(d) and Fig. 12(f)). Obviously, the application of strain gradient into the ultra-thin film can result in different vacancy formation.

5 Open applications
Recent studies showed that the flexoelectricity can be linked to many other important physical behaviors, and hence bringing the flexoelectricity into the totally open applications. For example, the transverse shear surface acoustic waves have been found could propagate near the flat surfaces of all crystalline dielectrics because of the existence of flexoelectric effect [158][159][160]. Liu et al. [161] proposed that the bended thermoelectric BiTe 3 film can present bulk photovoltaic effect at infrared wavelengths, which is possibly induced by the flexoelectric effect, as shown in Fig. 13(a). Yang et al. [162] proved that the application of flexoelectricity can cause the so-called flexo-photovoltaic effect. In their experiment, the large strain gradient (generated by the atomic force microscopy tip) and 405 nm laser Fig. 12 (a) Schematic of polarization switching due to the trailing flexoelectric field tracing the SPM tip motion. The in-of-plane polarization P x (black) and out-of-plane polarization P z (red) as a function of loading forces with tip motion along [110] orientation (b) and [110] orientation (c). Reproduced with permission from Ref. [155], © Springer Nature 2018. (d) and (e) The normalized vacancy concentration maps after mechanical scanning, where (d) and (e) were performed using a sharp and blunt tip with a contact force of 9.5 μN, respectively. Reproduced with permission from Ref. [157], © The Author(s) 2017. illumination were simultaneously applied onto some single crystals, as shown in Fig. 13(b). The result, as shown in Fig. 13(c), suggested that a significant photocurrent was induced by the applied mechanical force. Besides, the generation of such photocurrent was proved to be only originated from the flexophotovoltaic effect rather than other factor like Schottky contact. As shown in Fig. 13(d), the application of mechanical force can result in the sign of the short circuit current I sc from negative to positive. Also, this flexo-photovoltaic effect can be improved very obviously when material dimension is decreased into the nanoscale. this study suggest that the application of flexoelectricity is effective route for improving the performance of solar cells and optoelectronic devices [163].
Recently, a newly reported flexo-caloric effect which defines as the strain gradient induced thermalcurrent has attracted considerable attentions. As suggested by Liu et al. [164], the flexoelectricity is proved to be responsible for the value of Seeback coefficient of some thermoelectric material like BiFe 3 . It is believed that the existence of flexoelectric diploes can effectively contribute to the thermoelectricity, as schematically illustrated in Fig. 13(e). Meanwhile, a remarkable flexo-caloric effect has been found in Na 0.5 Bi 0.5 TiO 3 -based ceramics [165], as shown in Fig.  13(f). Combined with the flexoelectric effect and electrocaloric effect, it is possible to design materials with strong thermo-electric coupling effect for sensing, thermal energy harvesting, or on-chip solid-state cooling applications [166].

Outlook
In summary, we reviewed the recent progress of flexoelectricity, mainly focused on the flexoelectric materials and their related applications. Flexoelectricity is not only limited to the dielectric materials, but also found to exist in the liquid crystals, bio-materials, and even semiconductors. The absence of symmetry constraint makes the flexoelectric materials suitable for most cases where non-uniform electric field distribution and non-uniform strain distribution exist. The recent discoveries utilized the flexoelectricity into many important application fields such as sensor and actuator, charge transportation, defect formation, domain tailoring, and some open applications like flexo-photovoltaic effect and flexo-caloric effect have been commented.
Although the study of flexoelectricity has an impressive achievement, the state-of-the-art understanding of this field is still in its initial stage. Lots of the fundamental problems regarding the flexoelectricity are unresolved. Herein we can only list parts of them.
(1) Should the intrinsic flexoelectric coefficients of some kind of materials be beyond the theoretical value? Once it exists, the replacement of conventional electromechanical device by flexoelectric device may come true in the future. (2) The physical origin of the observed flexoelectricity in many perovskites is still unclear. Is there any universal rule to predict the flexoelectric coefficients in all kinds of the flexoelectric materials? (3) How to utilize the flexoelectricity into the low-symmetry crystals, especially for measuring the non-zero independent component of flexoelectric coefficients in those low-symmetry crystals? (4) Is there any way to induce large strain gradient, not just limited to the size effect?
Considering the past few years' progress in flexoelectric materials and related devices, the authors make the perspective as shown below.
1. Strain gradient is easier to exist in the complex material and flexible materials, resulting in the liquid crystals and bio-materials naturally suitable for flexoelectricity. Therefore, it is expected that the flexoelectricity will be widely used in liquid crystal display technology, bio-sensing, bio-medical, and bio-mimetic materials.
2. Due to the requirement of miniaturization in electronic devices, the role of flexoelectricity will draw more attentions. In the near future, micro/nano-scale flexoelectric sensing and actuating devices will be integrated into electronic devices. Among them, it is highly promising to find a lead-free and environmentally friendly flexoelectric material that is compatible for the AlN-based and silicon-based micro-electromechanical system.
3. Flexoelectricity is not limited by symmetry, giving it more space in material selection. In the future, it is expected to find a kind of natural material that is widely existed and has significantly enhanced flexoelectric properties.