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Flexoelectric materials and their related applications: A focused review

Flexoelectric materials and their related applications: A focused review Journal of Advanced Ceramics 2019, 8(2): 153–173 ISSN 2226-4108 https://doi.org/10.1007/s40145-018-0311-3 CN 10-1154/TQ Review a,b,* a a,b Longlong SHU , Renhong LIANG , Zhenggang RAO , a,b a,b,* a,b Linfeng FEI , Shanming KE , Yu WANG School of Materials Science and Engineering, Nanchang University, Nanchang 330031, China Jiangxi Key Laboratory for Two-dimensional Materials and Devices, and Jiangxi Engineering Laboratory for Advanced Functional Thin Films, Nanchang University, Nanchang 330031, China Received: December 5, 2018; Accepted: December 8, 2018 © The Author(s) 2019. Abstract: 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. Keywords: flexoelectricity; strain gradient; electric polarization; dielectric constant; liquid crystals; sensors and actuators phenomenon describes the mechanical-electro coupling 1 General concept of flexoelectricity between the strain gradient and the electric polarization (direct flexoelectric effect), or vice versa, the coupling Materials are usually classified into insulators, between the electric field gradient and the stress semiconductors, and conductors according to their (converse flexoelectric effect), which can be written as different conduction modes in response to the external [5]: electric field. Specifically, the insulators transfer the ∂S ∂E ij function of external electric field by inducting rather PT µµ , (1) l ijkl ij ijkl than conducting. Quite recently a universal physical ∂∂ x x kl phenomenon, namely flexoelectricity, has rekindled where P is the induced polarization, T is the induced l ij considerable research interest in insulators due to its stress, S is the strain, and E is the electric field, x is ij k great application potential in many fields, e.g., new the axis of coordinate, µ is the direct flexoelectric ijkl types of memory, domain engineering, defect tailoring, coefficient with respect to strain gradient. and non-piezoelectric sensors and actuators [1–4]. This The concept of flexoelectricity was firstly originated from liquid crystals [6]. In liquid crystals, flexoelectricity refers to the reorientation of irregularly * Corresponding authors. shaped polarized molecules under strain gradients E-mail: L. Shu, llshu@ncu.edu.cn; S. Ke, ksm@ncu.edu.cn www.springer.com/journal/40145 == 154 J Adv Ceram 2019, 8(2): 153–173 caused by splay-deformations or bent-deformations, piezoelectric coefficient is comparable to 10 nC/m of which is different from piezoelectric effect (produced flexoelectric coefficient. Based on this result, we can by uniform strain or stress) [7,8]. In some soft materials conclude that the flexoelectric coupling in most of the like hairs and bio-membranes, similar phenomenon dielectrics is much weaker than the piezoelectric one. that the net charges between the internal and external (ii) The term “strain gradient” generally leads to surface of the material occur in response to the very sophisticated tensor components of flexoelectric different curvature radiuses was also found [9,10]. coefficient and then makes their measurements very As a distinctive mechanical-electro effect, difficult. According to the early studies by Le Quang flexoelectricity has two main attractive features. (i) The and He [25] and Shu et al. [12], in a material with flexoelectric coefficient is a fourth rank tensor, triclinic symmetry, the possible components of therefore flexoelectricity could break the limitation of flexoelectric coefficients µ could reach to a number ijkl crystal symmetry, and exist in those materials with as high as 54. Therefore, it is almost impossible to centrosymmetry, which is quite different from the precisely extract individual flexoelectric components piezoelectric effect [11–13]. (ii) Strain gradient is in those low- symmetry systems. On the other hand, in inversely proportional to the material size, making the the materials with cubic symmetry (except for the 23 nanoscale flexoelectricity extremely large [14–16]. and m3 point group symmetries), the components of Such characteristic is called size effect or scaling effect, flexoelectric tensor µ could be reduced to only 3, ijkl and has already been utilized in many nanostructures which is generally defined as the transverse flexoelectric for sensing and actuating applications [17,18]. coefficient, longitudinal flexoelectric coefficient, and shear flexoelectric coefficient. During the past few years, three direct measurement 2 Flexoelectricity in materials methods have been developed by using the cantilever beam (Fig. 1(a)), the bottom−up side of truncated 2. 1 Direct and indirect measurement of flexoelectric pyramid (Fig. 1(b)), and the lateral side of truncated coefficients pyramid (Fig. 1(c)) to measure the transverse flexoelectric Although the flexoelectric coupling should be principally coefficient, longitudinal flexoelectric coefficient, and existed in all insulators, studies on exploring the shear flexoelectric coefficient, respectively [5,26]. It is flexoelectric coefficient in specific materials are still worth noting that all the measurement setups in Fig. 1 quite limited. This could be explained by the following are not highly integrated, and therefore the determination two reasons. of all the aforementioned flexoelectric components is (i) In most of the dielectrics, the typical value of the highly constrained. flexoelectric coefficient is only in the range of 0.01–1 In addition to these direct measurements, a few nC/m. This characteristic value is generally regarded as indirect measurements of the flexoelectric coefficients an intrinsic flexoelectric coefficient. In 1986, have been recently developed. Zhou et al. [27] found Tagantsev proposed that the flexoelectric coefficients that the flexoelectric coefficients could be indirectly in general materials are comparable to their e/a value evaluated by using the variation of the stiffness. It is [19,20], where e is the electron charge and a is the because the flexoelectricity will result in a rapid lattice parameter. This theoretical value is in good decline of the stiffness of a dielectric nano-cone- accordance with the recent first principle calculations frustum, especially at small size. The detailed principle developed by Maranganti and Sharma [21] and Hong and structure are described in Fig. 2(a). By studying and Vanderbilt [22,23] in some semiconductors and the nanocompression force vs. the nanocompression ferroelectric perovskites (as shown in Table 1). Note displacement curves of the column and cone-frustum that the unit C/m is quite different with the unit of samples (as shown in Fig. 2(b)), the longitudinal piezoelectric coefficients (C/N). Therefore, it is flexoelectric coefficient could be successfully extracted. inappropriate to make a direct comparison between Hu et al. [28] proposed a special way to generate strain flexoelectricity and piezoelectricity, just by using the gradient by using a shock wave. In their experiment, numerical value of flexoelectric coefficients with that the first-order hydrogen gas gun was employed to fire of the piezoelectric coefficients. Recent studies by a flying plate and hit the non-polarized bulk BaTiO Abdollahi et al. [24] suggested that 1 pC/N of 3 www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 155 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 /∂x and measuring the induced polarization simultaneously. (b) The measuring system of the longitudinal 11 3 flexoelectric coefficients. The measurement mechanism is by applying a mechanical force to the truncated pyramid sample in order to form a strain gradient ∂S /∂x and measuring the induced polarization P simultaneously. (c) The measuring system of 33 3 3 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 /∂x and measuring the induced shear strain S simultaneously. 1 3 13 Fig. 2 (a) Schematic diagram of the nanocompression measurement. In principle, the nanocompression causes homogeneous stress in the constant cross-section pillar, but it will induce stress gradient in the longitudinal direction of the variable cross-section pillar. (b) The nanocompression force as a function of the nanocompression displacement curves of the column and cone-frustum samples with various flexoelectric coupling coefficient. Reproduced with permission from Ref. [27], © AIP Publishing 2016. (c) Schematic view of the one-dimensional shock wave setup for measuring the flexoelectric coefficients. (d) The induced voltage of bulk BaTiO 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. www.springer.com/journal/40145 156 J Adv Ceram 2019, 8(2): 153–173 ceramic (Fig. 2(c)). Through this method, the longitudinal and hence revealed this factor might be dominant in flexoelectric coefficient could be indirectly calculated many other ferroelectrics where the observed flexoelectric by studying the relationship between the induced coefficients were significantly enhanced. voltage and the propagation of the shock waves, as 2. 3 Flexoelectricity in biological materials shown in Fig. 2(d). Recent studies suggest that many biological materials 2. 2 Flexoelectric coefficients in dielectrics and such as bones, hairs, and bio-membranes have ferroelectrics remarkable flexoelectric response [55–59]. The first With the help of those direct and indirect studies of the flexoelectricity in biological materials measurements, a few polymers and dielectrics have could be traced back to 1975 by Williams and Breger been employed for studying their flexoelectric [60]. In this initial study, some of the electro-mechanical coefficients. As shown in Table 1, it is proved that the properties of bones were considered to be likely flexoelectric coefficients in normal dielectrics and originated from “gradient polarization”, but the polymers, represented by TiO ceramic and the 2 mechanism is not clear yet. To clarify the origin of this polyvinylidene fluoride (PVDF) [29–34], are in the electro-mechanical property in bones, Vasquez-Sancho range of 1–10 nC/m, which is in good accordance with et al. [61] compared the flexoelectricity of bone and the intrinsic flexoelectric values. But in some kinds of pure hydroxyapatite (which is the main mineral of bone) ferroelectric ceramics and single crystals, the results by using the traditional cantilever system. As shown in are quite different. Extremely enhanced flexoelectricity Fig. 4(a), the measured flexoelectricity in both bone was found in a series of high permittivity ferroelectrics and pure hydroxyapatite was found to be of the same such as BaTiO (BT) [35,36], (BaSr)TiO (BST) 3 3 order of magnitude, suggesting that the hydroxyapatite [37–39], Ba(TiSn)O (BTS) [40,41], Pb(ZrTi)O (PZT) 3 3 flexoelectricity is the main source of the bending- [42], (KNa)NbO [43], and Pb(Mg Nb )O based 3 1/3 2/3 3 induced polarization in bones. Furthermore, the result ferroelectrics [44–46]. Their measured flexoelectric also revealed that the flexoelectricity in hydroxyapatite coefficients are 3–5 orders of magnitude larger than is helpful to the bone repair and remodeling processes. that in normal dielectrics such as TiO . Intriguingly, It is because that, as shown in Fig. 4(b), a large such enhancement is not a universal phenomenon in ferroelectrics. For the incipient ferroelectric SrTiO and antiferroelectric PbZrO , 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 [5,24,49–54]. The possible mechanisms are summarized in Fig. 3. The state-of- the-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 Fig. 3 Possible mechanisms of enhanced flexoelectricity ceramics, 70% of the enhanced flexoelectricity in BaTiO in dielectrics. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 157 Table 1 Experimental and calculated values of flexoelectric coefficients for some materials μ (nC/m) μ (nC/m) μ (nC/m) 1111 1122 1212 Material Experiment DFT Experiment DFT Experiment DFT GaAs — +0.51 — +0.85 — –0.84 GaP — +0.47 — +0.31 — –0.34 ZnS — –0.31 — –1.5 — –0.61 KCl — +0.40 — –0.12 — –0.23 NaCl — +0.41 — –0.12 — –0.21 TiO — — 2 — — — Polyvinylidene fluoride — — 10 — — — Epoxy resin — — 30 — — — Polyethylene — — 6 — — — Polyethylene terephthalate — — 10 — — — SrTiO 0.2 –0.89 7 +2.3 5.8 –6.6 PbZrO — — 2.5 — — — hydroxyapatite — — 1.2 — — — (KNaLi)(NbSb)O — — 2×10 — — — BT–BZT — — 2.5×10 — — — Pb(ZrTi)O — — 1×10 — — — Pb Sr TiO — — 2×10 — — — 0.3 0.7 3 PMN–PT — — (2–5)×10 — — — PbMg Nb O (3–4)×10 — — — — — 1/3 2/3 3 PIN–PMN–PT — — 1.1×10 — — — 4 4 BaTiO 5×10 –0.36 5×10 +1.6 — –1.5 5 5 (Ba Sr )TiO 1.15×10 10–20 1×10 6–10 — — 1–x x 3 Ba(Ti Sn )O — — 4.5×10 — — — 1–x x 3 flexoelectric electric field will exist in the vicinity of 4(d). This result suggested that flexoelectricity in the cracks of the hydroxyapatite. bio-membrane can drive the activation of tension-gated The bio-membrane flexoelectricity was recently channels within the membrane [67,68]. studied by using the droplet interface bilayer technology 2. 4 Flexoelectricity in liquid crystals (DIB) [62–65]. The principle of DIB in this scenario, as shown in Fig. 4(c) [66], is by using lipids as the The liquid crystal flexoelectricity stems from the 1970s organic surfactant in oil–water emulsion, in order to and has been regarded as one of the fundamental create biomimetic membranes at the intersection of properties of this kind of materials. In general, the neighboring droplets. When two droplets are brought flexoelectric coefficients in liquid crystals are mainly into contact, the lipid monolayers adhere together represented by two independent components, the which act as a capacitor. The contact angle at the component e and the component e , which respond to 1 3 monolayer–bilayer meniscus is determined by the the splay deformation and bend deformation, respectively. balance of the tension between the two interfaces. Up till now, two independent ways have been developed Intriguingly, this technique can be utilized to explore to measure the flexoelectric components e and e in 1 3 the mechano-transduction and interfacial properties of liquid crystals. One is reported by Trabi et al. [69] unsupported liquid biomimetic membranes. Freeman based on the converse flexoelectric effect. As shown in et al. [66] reported that high-frequency membrane Fig. 5(a), to yield a non-uniform electric field, deformation is capable of producing a significant interdigitated electrodes were deposited onto the liquid flexoelectric current, whose value is related to the crystal surface. Due to the coupling of converse curvature of the interfacial membrane, as shown in Fig. flexoelectricity, the liquid crystal was deformed into a www.springer.com/journal/40145 158 J Adv Ceram 2019, 8(2): 153–173 Fig. 4 (a) The flexoelectric coefficient is the constant of proportionality between strain gradient (bending) and bending- induced polarization. (b) Calculated flexoelectric field distribution around a microcrack in bone mineral, the flexoelectric field is biggest at the crack tip and decays progressively away. Reproduced with permission from Ref. [61], © WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2018. (c) The Droplet-Interface Bilayer (DIB) 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 liquid crystals, the typical value of flexoelectric by the distortion of the birefringence pattern. The other components e and e is 1–100 nC/m [71]. Compared 1 3 method is proposed by Harden et al. [70] through the with the ferroelectrics, the flexoelectric coefficients of direct flexoelectric effect, as shown in Fig. 5(b). This liquid crystals are relatively small. However, regardless method is similar to the measurement of flexoelectricity its small coefficients, the flexoelectricity often plays a in dielectrics. The studied nematic liquid crystal (NLC) dominant role in liquid crystals, especially in the widely cells were firstly fixed in a flexible container, and then developed liquid crystal display technology [72–78]. bent by using a loudspeaker. Later, the NLC was Recently, Lee et al. [79] found that the highly distortion oscillated between two fixed posts, and a large of the liquid crystal display could be quantitatively flexoelectric current could be induced. For most of the measured by the variation of the flexoelectric polarization, www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 159 Fig. 5 (a) Schematic view of indirect measurement of flexoelectric coefficient in liquid crystal. An electric field was applied onto the interdigitated electrodes and the flexoelectric coefficients was extracted by the distortion of birefringence images. Reproduced with permission from Ref. [69], © AIP Publishing 2008. (b) Schematic view of the direct measurement of flexoelectric coefficient in liquid crystal. Reproduced with permission from Ref. [70], © American Physical Society 2006. (c) Illustration of the direction of the electric field and the flexoelectric polarization of liquid crystal under the splay and bend deformations. Reproduced with permission from Ref. [79], © Taylor & Francis 2017. (d) Illustration of the field symmetry for positive and negative frames, where the IPS refers to in-plane-switching, FFS refers to fringe-field-switching. Reproduced with permission from Ref. [80], © Optical Society of America 2016. as shown in Fig. 5(c). Moreover, the image flicker is usually orientated in opposite direction so that the problem of liquid crystal mixtures which shows macro piezoelectricity is zero. Once bending this bulk negative dielectric anisotropy can be minimized by the material, a net polarization exists because the bending flexoelectric effect [80,81] (as shown in Fig. 5(d)). induced strain direction of the top and bottom surface is opposite. That is why the surface piezoelectricity can 2. 5 Flexoelectricity in semiconductor couple to the enhanced flexoelectricity in the bulk insulators [86]. Similarly, this mechanism also works Generally speaking, the electro-mechanical coupling is in semiconductor owing to the existence of dead layer. only limited to dielectrics due to the requirement of The dead layer, typically in several micrometers thick, insulation. But more and more studies have disclosed could accumulate the top and bottom surface that semiconductors can be regarded as electro- piezoelectricity, and results in a significant net mechanical materials, e.g., the ZnO semiconductor is a polarization when the semiconductor is bent. As shown piezoelectric material [82–84]. In 2016, Narvaez et al. in Fig. 6(b), the oxygen reduced BaTiO single crystal [85] made a new breakthrough in flexoelectricity and 3 which acts as a good ionic conductor, shows an greatly broadened the concept of flexoelectricity into extremely enhanced flexoelectric-like response. This semiconductors. A dead layer mechanism was proposed and the measured flexoelectricity in some semiconductors phenomenon was also found in Nb-doped TiO can reach a level to 1000 μC/m. semiconductor. Interestingly, the flexoelectricity in As shown in Fig. 6(a), for an arbitrary bulk insulator semiconductor shows a linearized thickness dependence. such as BaTiO single crystal, the surface piezoelectricity The measured effective flexoelectric coefficients were exists due to the symmetry breaking in boundary. At directly proportional with the material thickness, as the top and bottom surfaces, the piezoelectric polarization shown in Fig. 6(c). www.springer.com/journal/40145 160 J Adv Ceram 2019, 8(2): 153–173 Fig. 6 (a) Schematic view of the barrier layer mechanism. For both the bulk insulators and the semiconductors, the surface piezoelectricity will contribute to the flexoelectric coefficients. The barrier layer in semiconductor will block the free charge and have a giant contribution for flexoelectric coefficients. (b) Temperature dependence of the effective transverse flexoelectric coefficients in pure bulk BaTiO single crystal, reduced BaTiO single crystal, and oxidized BaTiO single crystal. (c) Thickness 3 3 3 dependence of the effective transverse flexoelectric coefficients in Nb-doped TiO semiconductor. Reproduced with permission from Ref. [85], © Springer Nature 2016. Inspired by this attractive result, one can expect that in epitaxial ferroelectric thin films [94], the simulated 6 7 –1 –10 m , which may with proper design, the application of semiconductor strain gradient can reach up to 10 make the flexoelectric polarization higher than the flexoelectricity will play an important role in the optimization of the integrated circuit and electro- piezoelectric one. The general theory of flexoelectricity was recently mechanical semiconductor devices. developed. Many of them were focused on the origin of the enhancement of flexoelectricity in some typical 3 Theoretical calculation of flexoelectricity materials [95–97]. Theoretical work on flexoelectricity dates back to the papers by Mashkevich and Tolpygo During the past few years, the studies of flexoelectricity [98], who first proposed the effect, and Kogan [8], who were not solely focused on the experiment level, but formulated the first phenomenological theory. According also led to a lot of theoretical progress. It is worth to the theory proposed by Yudin et al. [99], the flexoelectric noting that the size effect of flexoelectricity was firstly polarization can be accounted for four parts, i.e., proposed by theoretical prediction rather than any surface piezoelectricity, bulk piezoelectricity, surface experiment [14]. Numerous theoretical studies have flexoelectricity, and bulk flexoelectricity [20]. All these demonstrated that the strain gradient becomes large in factors contribute almost equally to the flexoelectric nanoscale, at which the flexoelectricity may even be coefficient. However, a different viewpoint was stated competitive with piezoelectricity [87–93]. Specifically, hereupon by Resta [52] who built the polarization theory www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 161 based on the elemental cubic crystal. His theoretical parameter of each atom. Meanwhile, they also result manifested that the intrinsic flexoelectricity was developed an indirect way for calculating flexoelectric a purely bulk effect, where the surface contribution coefficients which was formulated in such a way that was totally excluded. Later, Yurkov and Tagantsev. the tensor elements can be computed directly in the [100] argued that the direct bulk flexoelectric response context of density-functional calculations, including of a finite sample essentially depends on the surface electronic and lattice contributions [23]. To date, the polarization energy. Recently, Zhang et al. [54] first-principles method could be utilized into the calculated that the surface piezoelectricity which widely calculation of other flexoelectric tensor component and exists in all the unpolarized ceramics and single crystals, much more materials, which is summarized in Table 1 is possibly dominant in the observed flexoelectricity. [102–106]. This statement also well matches their experimental An interesting theoretical progress of flexoelectricity results [54]. The controversy for this issue still exists and we addressed is the application of topology optimization. a comprehensive understood is urgently needed. This method, firstly proposed by Bendsøe and The first-principles calculation of flexoelectric Kikuchi [107] in 1988, has been widely applied to coefficients was firstly developed in several optimize the material property [108,109], especially semiconductors by Maranganti and Sharma [21] until for the design of domains [110,111]. More recently, 2009. Hong et al. [22,23,101] demonstrated the first ab Ghasemi et al. [112,113] presented a new-type initio calculation of the longitudinal flexoelectricity for topology optimization method, which is based on a BaTiO and SrTiO by using a direct approach, where combination of isogeometric analysis (IGA), level set, 3 3 the strain gradient is realized by setting the lattice and point wise density mapping techniques as shown Fig. 7 (a) IGA concept. Each element in the physical space is the image of a corresponding element in the parameter space, and the parameter space is discretized by knot vectors. Control points in IGA are used to discretize the geometry and define the degrees of freedom. Reproduced with permission from Ref. [112], © Elsevier B.V. 2016. (b) The level set function which easily copes with the various numbers of phases, is efficiently satisfied with the multiple constraints. This function intrinsically avoids overlap or vacuum among different phases. Reproduced with permission from Ref. [113], © Elsevier B.V. 2017. (c) Simulated vortex patterns represented by the vorticity calculated from the polarization vector field. A vortex/anti-vortex pair region marked by the dashed line is chosen to illustrate the flexocoupling effects in what follows, and the double-arrow symbol indicates an anti-phase boundary. Typical vortex pair pattern under different flexocoupling of the STO layers. Reproduced with permission from Ref. [114], © The Author(s) 2017. www.springer.com/journal/40145 162 J Adv Ceram 2019, 8(2): 153–173 in Fig. 7(a) and Fig. 7(b), respectively. Through this 8(a)) and the energy conversion efficiency (as shown topology optimization, significant enhanced flexoelectric in Fig. 8(b)) of the flexoelectric energy harvester are coefficients can be obtained in designated materials. much larger than that of the classical piezoelectric Other theoretical calculation methods such as finite energy harvesters. Particularly, when the thickness (h ) element method, phase-filed modeling, and density of the flexoelectric layer is reduced to less than 100 nm, functional theory, are also developed to explore the the power output of the flexoelectric energy harvester direct and converse flexoelectric effect [114–124]. It is is almost 12 times larger than that of the piezoelectric worth noting that Li et al. [114] reported the energy harvester [137]. Choi and Kim [138] fabricated application of flexoelectric phase-field model [125] in a flexoelectric energy harvester device which collects calculating the polar vortices of PbTiO /SrTiO 3 3 energy by a PVDF thin film deposited on the cantilever super-lattices. The results suggested that the induced substrate. The photograph of this device is shown in polarization (represented by the vortex/anti-vortex pair Fig. 8(c). Han et al. [139] prepared a flexoelectric region) is highly related to the given flexoelectric nanogenerator which is consisted of direct-grown coefficients, as shown in Fig. 7(c). 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 4 Applications of flexoelectricity force of 20 N, which is promising for harvesting the mechanical energy. Moreover, Zhu et al. [140] Here we will emphasize on the recent developed designed a flexible flexoelectric fiber which is made of applications of flexoelectricity in several fields, curved piezoelectric composites. As shown in Fig. 8(d), including flexoelectric energy harvester, flexoelectric these fibers possess a stable upward self-poling which sensors and actuators, domain engineering, liquid is induced by flexoelectricity, exhibited a large electric crystal displaying, and some other open applications, output comparable to those of the piezoelectric e.g., flexoelectricity-tuned photovoltaic effect. In the nanogenerators. end, we also list some possible applications where the flexoelectricity might be involved. 4. 2 Actuators 4. 1 Energy harvesters Piezoelectricity has been widely used for sensors and actuators. Similarly, the flexoelectricity could also be Energy harvesters are those devices which can collect utilized for sensing and actuating applications, energy from the external sources like wind energy, especially for nano-sized devices. Compared with the solar power, thermal energy, etc. Specifically, the piezoelectric sensors and actuators, the flexoelectric electromechanical energy harvester can collect energy ones are not limited by the crystal symmetry of the from the mechanical vibrations [126–132]. Recently, materials and the working temperature [141,142]. The the newly designed energy harvester which collect past few years have witnessed many progresses in this energy from the fluctuation of the vibrations (strain aspect. As shown in Fig. 9(a), Zhang et al. [143] gradient) was proposed. Owing to its special size- designed a curved flexoelectric actuator by using dependent effect, the flexoelectric energy harvester is non-polarized PVDF. This actuator exhibited a good suitable to be integrated into small size, where a huge actuating property with a resolution of the displacement strain gradient can be generated. Till now, with the proper design, the mechanical–electrical energy converting reaching up to 1.0 nm and a largest displacement as efficiency of the flexoelectric energy harvester can high as 63.6 nm (as shown in Fig. 9(b)). It is worth reach to 6.6% [133–135]. highlighting that Bhaskar et al. [144] implemented a The potential of flexoelectricity as energy harvester flexoelectric actuator made of micron-sized barium has been predicted by many works. Wang and Wang titanate (as shown in Fig. 9(c)) which is fully compatible [136] developed an analytical model for vibration- for the semiconductor silicon technology. The performance based circular energy harvester that consists of a result (Fig. 9(d)) suggests that this flexoelectric flexoelectric layer and a substrate layer. The result actuator displays comparable performance to the indicated that both the power output (as shown in Fig. actuators use lead-containing piezoelectric material. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 163 Fig. 8 (a) The maximum normalized power output of vibration-based circular energy harvester as a function of the thickness of the flexoelectric layer. (b) The energy converting efficiency of vibration-based circular energy harvester as a function of the thickness of the flexoelectric layer. Reproduced with permission from Ref. [136], © Elsevier Ltd. 2018. (c) The photography of the PVDF-based energy harvester which could collect energy from the excited vibrations. Reproduced with permission from Ref. [138], © IOP Publishing Ltd. 2017. (d) Comparison of the transient output voltage between the unpolarized nanogenerator and polarized nanogenerator. Reproduced with permission from Ref. [140], © Elsevier Ltd. 2018. gradient) is shown in Fig. 10(b). The sensitivity of this 4. 3 Sensors curvature sensor, defined by the slopes of the charge outputs versus curvature, can reach up to 1 pC/m. In The flexoelectric sensors are becoming increasingly addition, Merupo et al. [149] investigated the potential popular because of their small size, absence of de- use of 6.6 µm-thick soft polyurethane films as the large poling and aging problem, and lead-free composition curvature sensor by subjecting flexible aluminum [145]. The state-of-the-art flexoelectric sensor has been foil/PET bi-layered substrate to large deflections. A successfully utilized in many fields such as structural –1 curvature of about 80 m corresponding to a radius of health monitoring, crack detector, and curvature curvature of ~1.2 cm could be effectively sensed under detector [146,147]. Recently, Yan et al. [148] designed a low frequency (3 Hz) bending motion. The pseudo- a BST-based flexoelectric curvature sensor, which can sinusoidal time dependence of the output charge of this transfer the bending deflections directly to the charge sensor is shown in Fig. 10(c). Kwon et al. [150] output. The working principle of this sensor is reported recently that the flexoelectricity can work in illustrated in Fig. 10(a). To precisely detect the bending the microphone. The designed sensitivity of the deflection, two BST micro curvature sensors were attached onto the center side surfaces of an aluminum fabricated barium strontium titanate (Ba Sr TiO ) 0.65 0.35 3 microphone was very high and almost frequency- beam, located symmetrically with respect to its neutral axis. The relationship between flexoelectric charge dependent in wide frequency range, as shown in Fig. outputs of the BST sensors and the curvature (strain 10(d). www.springer.com/journal/40145 164 J Adv Ceram 2019, 8(2): 153–173 Fig. 9 (a) Schematic diagram of the PVDF-based actuator. The symbol θ represents the angle of electrode attachment. To achieve a uniform electric field gradient and application safety, θ in this actuator is set as 120°. (b) Induced displacement as a function of voltage in PVDF-based actuator. Reproduced with permission from Ref. [143], © AIP Publishing 2017. (c) Schematic view of the SrTiO -based actuator. Particularly, the flexoelectric layer is compatible with silicon or any of its gate dielectrics in a completely complementary metal oxide semiconductor-compatible environment. (d) Actuating performance comparison of the SrTiO -based actuator with other actuators. Reproduced with permission from Ref. [144], © Springer Nature 2015. Recently, Gómez et al. [151] prepared an epitaxial 4. 4 Domain tailoring and polarization switching growth of columnar porous BaTiO /LaSrMnO /SrTiO / 3 3 3 Si(001) heterostructures by using a complicated recipe Another attractive application of flexoelectricity which both combines the molecule beam epitaxy and should be addressed is the domain tailoring and polymer assisted deposition technology. The illustration polarization switching. As we may know that the of the preparation process is shown in Fig. 11(a). The ferroelectric materials are characterized by their results proved that the ferroelectric polarization of this spontaneous polarization, which can be switched by heterostructure thin film can be reversed by a applying an external electric field. As suggested by Lu mechanical load in epitaxial columnar nanostructures, et al. [1] and Catalan et al. [2], in a nano-sized material, as shown in Fig. 11(b). Similar phenomenon was also e.g., ultrathin ferroelectric film, the spontaneous found in PbTiO thin films [152]. It was experimentally polarization can be switched by mechanical strain 3 proved that, as shown in Fig. 11(c), the mechanical gradient. This powerful function has attracted increasing force shows a comparable ability relative with the interests and hence makes heterostructure thin film a electrically method for switching the domain of the natural scenario for the flexoelectric applications. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 165 Fig. 10 (a) Beam curvature sensing: BST curvature sensor attached to beam. (b) Relationship between charge output and beam curvature-experimental results of BST curvature sensor. Reproduced with permission from Ref. [148], © SPIE 2013. (c) Real time dependence of the output electric charge of the 6.6 µm thick curvature sensor. Reproduced with permission from Ref. [149], © AIP Publishing 2017. (d) The analytical, experimental sensitivity of the flexoelectric microphone at low frequency range (inset: overall sensitivity). Reproduced with permission from Ref. [150], © IOP Publishing Ltd. 2016. Fig. 11 (a) A new thin film preparation approach which combines molecule beam epitaxy and polymer assisted deposition. The complex epitaxial heterostructures were grown in three different stages. (b) Schematic view of the switching of the polarization by an electric field or by a mechanical force across the columnar BTO/LSM/STO/Si(001) heterostructure at room temperature. Reproduced with permission from Ref. [151], © WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2017. (c) Mechanical writing and electrical writing induced variation of the piezoresponse force microscopy phase images of the PbTiO film. The + - tip-induced pressure leads to the switching of the upward polarization (c domain) to the downward polarization (c domain). Reproduced with permission from Ref. [152], © AIP Publishing 2017. www.springer.com/journal/40145 166 J Adv Ceram 2019, 8(2): 153–173 PbTiO thin films. ionic behavior [156] and manipulate the oxygen Moreover, with the help of flexoelectricity, the vacancies [157] (as shown in Fig. 12(d) and Fig. 12(f)). Obviously, the application of strain gradient into self-polarization direction of ferroelectric thin films, the ultra-thin film can result in different vacancy which is of difficulty in control in previous studies, is formation. feasible to be changed by tuning the substrate interfaces and film thicknesses [153,154]. Park et al. 4. 5 Open applications [155] demonstrated that the multiple domain switching pathways in multiaxial ferroelectric materials can be Recent studies showed that the flexoelectricity can be selectively controlled by a newly realized trailing linked to many other important physical behaviors, and flexoelectric field, specifically, by the motion of a hence bringing the flexoelectricity into the totally open mechanically loaded scanning probe microscope tip. applications. For example, the transverse shear surface The illustration of this trailing flexoelectric field is acoustic waves have been found could propagate near shown in Fig. 12(a). Experimental results proved that the flat surfaces of all crystalline dielectrics because of the ferroelectric switching angle of multiferroic the existence of flexoelectric effect [158–160]. Liu et magnetoelectric BiFeO thin film can be stably selected al. [161] proposed that the bended thermoelectric at 71° ferroelastic switching or 109° ferroelectric BiTe film can present bulk photovoltaic effect at switching. The in-plane polarization and out-of-plane infrared wavelengths, which is possibly induced by the polarization in response to the loading forces of the flexoelectric effect, as shown in Fig. 13(a). Yang et al. moving tip shows a significant orientation dependence, [162] proved that the application of flexoelectricity can as shown in Fig. 12(b) and Fig. 12(c), respectively. cause the so-called flexo-photovoltaic effect. In their In addition, with the help of the flexoelectricity, the experiment, the large strain gradient (generated by the microscopic tip can also be used to explore dynamic 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 (black) and out-of-plane polarization P (red) as a function of loading forces with tip motion along x z [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. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 167 Fig. 13 (a) Bulk photovoltaic response of the Bi Te film under different bending distances. The inset gives the photograph of 2 3 the bending setup. Reproduced with permission from Ref. [161], © The Author(s) 2016. (b) Schematic view of the setup for verifying the flexo-photovoltaic effect. (c) The induced photocurrent density as a function of the loading force. (d) Positive photocurrent measured on a TiO (001) face with a 15 µN force applied by the AFM tip. Reproduced with permission from Ref. [162], © The Authors 2018. (e) The diagram illustrates the distribution of carriers and electric dipoles in a strained Bi Te film 2 3 without thermal gradient and with a thermal gradient field. The thermoelectric field has an opposite direction with the flexoelectric field. Reproduced with permission from Ref. [164], © The Authors 2016. (f) Temperature dependence of the Bi TiO -based ceramic plate with large internal field under three-point bending load. The inset shows thermal currents of Na 0.5 0.5 3 the measured thermal currents of this ceramic plate without load. Reproduced with permission from Ref. [165], © American Physical Society 2018. illumination were simultaneously applied onto some mechanical force can result in the sign of the short single crystals, as shown in Fig. 13(b). The result, as circuit current I from negative to positive. Also, this sc shown in Fig. 13(c), suggested that a significant flexo-photovoltaic effect can be improved very photocurrent was induced by the applied mechanical obviously when material dimension is decreased into force. Besides, the generation of such photocurrent was the nanoscale. this study suggest that the application of proved to be only originated from the flexo- flexoelectricity is effective route for improving the photovoltaic effect rather than other factor like Schottky performance of solar cells and optoelectronic devices contact. As shown in Fig. 13(d), the application of [163]. www.springer.com/journal/40145 168 J Adv Ceram 2019, 8(2): 153–173 Recently, a newly reported flexo-caloric effect the non-zero independent component of flexoelectric which defines as the strain gradient induced thermal- coefficients in those low-symmetry crystals? (4) Is current has attracted considerable attentions. As there any way to induce large strain gradient, not just suggested by Liu et al. [164], the flexoelectricity is limited to the size effect? proved to be responsible for the value of Seeback Considering the past few years’ progress in coefficient of some thermoelectric material like BiFe flexoelectric materials and related devices, the authors It is believed that the existence of flexoelectric diploes make the perspective as shown below. can effectively contribute to the thermoelectricity, as 1. Strain gradient is easier to exist in the complex schematically illustrated in Fig. 13(e). Meanwhile, a material and flexible materials, resulting in the liquid remarkable flexo-caloric effect has been found in crystals and bio-materials naturally suitable for Na Bi TiO -based ceramics [165], as shown in Fig. flexoelectricity. Therefore, it is expected that the 0.5 0.5 3 13(f). Combined with the flexoelectric effect and flexoelectricity will be widely used in liquid crystal electrocaloric effect, it is possible to design materials display technology, bio-sensing, bio-medical, and with strong thermo-electric coupling effect for sensing, bio-mimetic materials. thermal energy harvesting, or on-chip solid-state 2. Due to the requirement of miniaturization in cooling applications [166]. electronic devices, the role of flexoelectricity will draw more attentions. In the near future, micro/nano-scale flexoelectric sensing and actuating devices will be 5 Outlook integrated into electronic devices. Among them, it is highly promising to find a lead-free and environmentally In summary, we reviewed the recent progress of friendly flexoelectric material that is compatible for the flexoelectricity, mainly focused on the flexoelectric AlN-based and silicon-based micro-electromechanical materials and their related applications. Flexoelectricity system. is not only limited to the dielectric materials, but also 3. Flexoelectricity is not limited by symmetry, found to exist in the liquid crystals, bio-materials, and giving it more space in material selection. In the future, even semiconductors. The absence of symmetry it is expected to find a kind of natural material that is constraint makes the flexoelectric materials suitable for widely existed and has significantly enhanced flexoelectric most cases where non-uniform electric field distribution properties. and non-uniform strain distribution exist. The recent discoveries utilized the flexoelectricity into many Acknowledgements important application fields such as sensor and actuator, charge transportation, defect formation, domain tailoring, This work was supported by the National Natural Science and some open applications like flexo-photovoltaic Foundation of China under Grant Nos. 11574126 and effect and flexo-caloric effect have been commented. 11604135, and partly by the Natural Science Foundation Although the study of flexoelectricity has an of Jiangxi Province (No. 20161BAB216110), China impressive achievement, the state-of-the-art understanding Postdoctoral Science Foundation (No. 2017M612162), of this field is still in its initial stage. Lots of the and Postdoctoral Science Foundation of Jiangxi Province fundamental problems regarding the flexoelectricity (No. 2017KY02). are unresolved. Herein we can only list parts of them. 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Flexoelectric materials and their related applications: A focused review

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Springer Journals
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2019 The Author(s)
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2226-4108
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2227-8508
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10.1007/s40145-018-0311-3
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Abstract

Journal of Advanced Ceramics 2019, 8(2): 153–173 ISSN 2226-4108 https://doi.org/10.1007/s40145-018-0311-3 CN 10-1154/TQ Review a,b,* a a,b Longlong SHU , Renhong LIANG , Zhenggang RAO , a,b a,b,* a,b Linfeng FEI , Shanming KE , Yu WANG School of Materials Science and Engineering, Nanchang University, Nanchang 330031, China Jiangxi Key Laboratory for Two-dimensional Materials and Devices, and Jiangxi Engineering Laboratory for Advanced Functional Thin Films, Nanchang University, Nanchang 330031, China Received: December 5, 2018; Accepted: December 8, 2018 © The Author(s) 2019. Abstract: 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. Keywords: flexoelectricity; strain gradient; electric polarization; dielectric constant; liquid crystals; sensors and actuators phenomenon describes the mechanical-electro coupling 1 General concept of flexoelectricity between the strain gradient and the electric polarization (direct flexoelectric effect), or vice versa, the coupling Materials are usually classified into insulators, between the electric field gradient and the stress semiconductors, and conductors according to their (converse flexoelectric effect), which can be written as different conduction modes in response to the external [5]: electric field. Specifically, the insulators transfer the ∂S ∂E ij function of external electric field by inducting rather PT µµ , (1) l ijkl ij ijkl than conducting. Quite recently a universal physical ∂∂ x x kl phenomenon, namely flexoelectricity, has rekindled where P is the induced polarization, T is the induced l ij considerable research interest in insulators due to its stress, S is the strain, and E is the electric field, x is ij k great application potential in many fields, e.g., new the axis of coordinate, µ is the direct flexoelectric ijkl types of memory, domain engineering, defect tailoring, coefficient with respect to strain gradient. and non-piezoelectric sensors and actuators [1–4]. This The concept of flexoelectricity was firstly originated from liquid crystals [6]. In liquid crystals, flexoelectricity refers to the reorientation of irregularly * Corresponding authors. shaped polarized molecules under strain gradients E-mail: L. Shu, llshu@ncu.edu.cn; S. Ke, ksm@ncu.edu.cn www.springer.com/journal/40145 == 154 J Adv Ceram 2019, 8(2): 153–173 caused by splay-deformations or bent-deformations, piezoelectric coefficient is comparable to 10 nC/m of which is different from piezoelectric effect (produced flexoelectric coefficient. Based on this result, we can by uniform strain or stress) [7,8]. In some soft materials conclude that the flexoelectric coupling in most of the like hairs and bio-membranes, similar phenomenon dielectrics is much weaker than the piezoelectric one. that the net charges between the internal and external (ii) The term “strain gradient” generally leads to surface of the material occur in response to the very sophisticated tensor components of flexoelectric different curvature radiuses was also found [9,10]. coefficient and then makes their measurements very As a distinctive mechanical-electro effect, difficult. According to the early studies by Le Quang flexoelectricity has two main attractive features. (i) The and He [25] and Shu et al. [12], in a material with flexoelectric coefficient is a fourth rank tensor, triclinic symmetry, the possible components of therefore flexoelectricity could break the limitation of flexoelectric coefficients µ could reach to a number ijkl crystal symmetry, and exist in those materials with as high as 54. Therefore, it is almost impossible to centrosymmetry, which is quite different from the precisely extract individual flexoelectric components piezoelectric effect [11–13]. (ii) Strain gradient is in those low- symmetry systems. On the other hand, in inversely proportional to the material size, making the the materials with cubic symmetry (except for the 23 nanoscale flexoelectricity extremely large [14–16]. and m3 point group symmetries), the components of Such characteristic is called size effect or scaling effect, flexoelectric tensor µ could be reduced to only 3, ijkl and has already been utilized in many nanostructures which is generally defined as the transverse flexoelectric for sensing and actuating applications [17,18]. coefficient, longitudinal flexoelectric coefficient, and shear flexoelectric coefficient. During the past few years, three direct measurement 2 Flexoelectricity in materials methods have been developed by using the cantilever beam (Fig. 1(a)), the bottom−up side of truncated 2. 1 Direct and indirect measurement of flexoelectric pyramid (Fig. 1(b)), and the lateral side of truncated coefficients pyramid (Fig. 1(c)) to measure the transverse flexoelectric Although the flexoelectric coupling should be principally coefficient, longitudinal flexoelectric coefficient, and existed in all insulators, studies on exploring the shear flexoelectric coefficient, respectively [5,26]. It is flexoelectric coefficient in specific materials are still worth noting that all the measurement setups in Fig. 1 quite limited. This could be explained by the following are not highly integrated, and therefore the determination two reasons. of all the aforementioned flexoelectric components is (i) In most of the dielectrics, the typical value of the highly constrained. flexoelectric coefficient is only in the range of 0.01–1 In addition to these direct measurements, a few nC/m. This characteristic value is generally regarded as indirect measurements of the flexoelectric coefficients an intrinsic flexoelectric coefficient. In 1986, have been recently developed. Zhou et al. [27] found Tagantsev proposed that the flexoelectric coefficients that the flexoelectric coefficients could be indirectly in general materials are comparable to their e/a value evaluated by using the variation of the stiffness. It is [19,20], where e is the electron charge and a is the because the flexoelectricity will result in a rapid lattice parameter. This theoretical value is in good decline of the stiffness of a dielectric nano-cone- accordance with the recent first principle calculations frustum, especially at small size. The detailed principle developed by Maranganti and Sharma [21] and Hong and structure are described in Fig. 2(a). By studying and Vanderbilt [22,23] in some semiconductors and the nanocompression force vs. the nanocompression ferroelectric perovskites (as shown in Table 1). Note displacement curves of the column and cone-frustum that the unit C/m is quite different with the unit of samples (as shown in Fig. 2(b)), the longitudinal piezoelectric coefficients (C/N). Therefore, it is flexoelectric coefficient could be successfully extracted. inappropriate to make a direct comparison between Hu et al. [28] proposed a special way to generate strain flexoelectricity and piezoelectricity, just by using the gradient by using a shock wave. In their experiment, numerical value of flexoelectric coefficients with that the first-order hydrogen gas gun was employed to fire of the piezoelectric coefficients. Recent studies by a flying plate and hit the non-polarized bulk BaTiO Abdollahi et al. [24] suggested that 1 pC/N of 3 www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 155 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 /∂x and measuring the induced polarization simultaneously. (b) The measuring system of the longitudinal 11 3 flexoelectric coefficients. The measurement mechanism is by applying a mechanical force to the truncated pyramid sample in order to form a strain gradient ∂S /∂x and measuring the induced polarization P simultaneously. (c) The measuring system of 33 3 3 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 /∂x and measuring the induced shear strain S simultaneously. 1 3 13 Fig. 2 (a) Schematic diagram of the nanocompression measurement. In principle, the nanocompression causes homogeneous stress in the constant cross-section pillar, but it will induce stress gradient in the longitudinal direction of the variable cross-section pillar. (b) The nanocompression force as a function of the nanocompression displacement curves of the column and cone-frustum samples with various flexoelectric coupling coefficient. Reproduced with permission from Ref. [27], © AIP Publishing 2016. (c) Schematic view of the one-dimensional shock wave setup for measuring the flexoelectric coefficients. (d) The induced voltage of bulk BaTiO 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. www.springer.com/journal/40145 156 J Adv Ceram 2019, 8(2): 153–173 ceramic (Fig. 2(c)). Through this method, the longitudinal and hence revealed this factor might be dominant in flexoelectric coefficient could be indirectly calculated many other ferroelectrics where the observed flexoelectric by studying the relationship between the induced coefficients were significantly enhanced. voltage and the propagation of the shock waves, as 2. 3 Flexoelectricity in biological materials shown in Fig. 2(d). Recent studies suggest that many biological materials 2. 2 Flexoelectric coefficients in dielectrics and such as bones, hairs, and bio-membranes have ferroelectrics remarkable flexoelectric response [55–59]. The first With the help of those direct and indirect studies of the flexoelectricity in biological materials measurements, a few polymers and dielectrics have could be traced back to 1975 by Williams and Breger been employed for studying their flexoelectric [60]. In this initial study, some of the electro-mechanical coefficients. As shown in Table 1, it is proved that the properties of bones were considered to be likely flexoelectric coefficients in normal dielectrics and originated from “gradient polarization”, but the polymers, represented by TiO ceramic and the 2 mechanism is not clear yet. To clarify the origin of this polyvinylidene fluoride (PVDF) [29–34], are in the electro-mechanical property in bones, Vasquez-Sancho range of 1–10 nC/m, which is in good accordance with et al. [61] compared the flexoelectricity of bone and the intrinsic flexoelectric values. But in some kinds of pure hydroxyapatite (which is the main mineral of bone) ferroelectric ceramics and single crystals, the results by using the traditional cantilever system. As shown in are quite different. Extremely enhanced flexoelectricity Fig. 4(a), the measured flexoelectricity in both bone was found in a series of high permittivity ferroelectrics and pure hydroxyapatite was found to be of the same such as BaTiO (BT) [35,36], (BaSr)TiO (BST) 3 3 order of magnitude, suggesting that the hydroxyapatite [37–39], Ba(TiSn)O (BTS) [40,41], Pb(ZrTi)O (PZT) 3 3 flexoelectricity is the main source of the bending- [42], (KNa)NbO [43], and Pb(Mg Nb )O based 3 1/3 2/3 3 induced polarization in bones. Furthermore, the result ferroelectrics [44–46]. Their measured flexoelectric also revealed that the flexoelectricity in hydroxyapatite coefficients are 3–5 orders of magnitude larger than is helpful to the bone repair and remodeling processes. that in normal dielectrics such as TiO . Intriguingly, It is because that, as shown in Fig. 4(b), a large such enhancement is not a universal phenomenon in ferroelectrics. For the incipient ferroelectric SrTiO and antiferroelectric PbZrO , 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 [5,24,49–54]. The possible mechanisms are summarized in Fig. 3. The state-of- the-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 Fig. 3 Possible mechanisms of enhanced flexoelectricity ceramics, 70% of the enhanced flexoelectricity in BaTiO in dielectrics. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 157 Table 1 Experimental and calculated values of flexoelectric coefficients for some materials μ (nC/m) μ (nC/m) μ (nC/m) 1111 1122 1212 Material Experiment DFT Experiment DFT Experiment DFT GaAs — +0.51 — +0.85 — –0.84 GaP — +0.47 — +0.31 — –0.34 ZnS — –0.31 — –1.5 — –0.61 KCl — +0.40 — –0.12 — –0.23 NaCl — +0.41 — –0.12 — –0.21 TiO — — 2 — — — Polyvinylidene fluoride — — 10 — — — Epoxy resin — — 30 — — — Polyethylene — — 6 — — — Polyethylene terephthalate — — 10 — — — SrTiO 0.2 –0.89 7 +2.3 5.8 –6.6 PbZrO — — 2.5 — — — hydroxyapatite — — 1.2 — — — (KNaLi)(NbSb)O — — 2×10 — — — BT–BZT — — 2.5×10 — — — Pb(ZrTi)O — — 1×10 — — — Pb Sr TiO — — 2×10 — — — 0.3 0.7 3 PMN–PT — — (2–5)×10 — — — PbMg Nb O (3–4)×10 — — — — — 1/3 2/3 3 PIN–PMN–PT — — 1.1×10 — — — 4 4 BaTiO 5×10 –0.36 5×10 +1.6 — –1.5 5 5 (Ba Sr )TiO 1.15×10 10–20 1×10 6–10 — — 1–x x 3 Ba(Ti Sn )O — — 4.5×10 — — — 1–x x 3 flexoelectric electric field will exist in the vicinity of 4(d). This result suggested that flexoelectricity in the cracks of the hydroxyapatite. bio-membrane can drive the activation of tension-gated The bio-membrane flexoelectricity was recently channels within the membrane [67,68]. studied by using the droplet interface bilayer technology 2. 4 Flexoelectricity in liquid crystals (DIB) [62–65]. The principle of DIB in this scenario, as shown in Fig. 4(c) [66], is by using lipids as the The liquid crystal flexoelectricity stems from the 1970s organic surfactant in oil–water emulsion, in order to and has been regarded as one of the fundamental create biomimetic membranes at the intersection of properties of this kind of materials. In general, the neighboring droplets. When two droplets are brought flexoelectric coefficients in liquid crystals are mainly into contact, the lipid monolayers adhere together represented by two independent components, the which act as a capacitor. The contact angle at the component e and the component e , which respond to 1 3 monolayer–bilayer meniscus is determined by the the splay deformation and bend deformation, respectively. balance of the tension between the two interfaces. Up till now, two independent ways have been developed Intriguingly, this technique can be utilized to explore to measure the flexoelectric components e and e in 1 3 the mechano-transduction and interfacial properties of liquid crystals. One is reported by Trabi et al. [69] unsupported liquid biomimetic membranes. Freeman based on the converse flexoelectric effect. As shown in et al. [66] reported that high-frequency membrane Fig. 5(a), to yield a non-uniform electric field, deformation is capable of producing a significant interdigitated electrodes were deposited onto the liquid flexoelectric current, whose value is related to the crystal surface. Due to the coupling of converse curvature of the interfacial membrane, as shown in Fig. flexoelectricity, the liquid crystal was deformed into a www.springer.com/journal/40145 158 J Adv Ceram 2019, 8(2): 153–173 Fig. 4 (a) The flexoelectric coefficient is the constant of proportionality between strain gradient (bending) and bending- induced polarization. (b) Calculated flexoelectric field distribution around a microcrack in bone mineral, the flexoelectric field is biggest at the crack tip and decays progressively away. Reproduced with permission from Ref. [61], © WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2018. (c) The Droplet-Interface Bilayer (DIB) 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 liquid crystals, the typical value of flexoelectric by the distortion of the birefringence pattern. The other components e and e is 1–100 nC/m [71]. Compared 1 3 method is proposed by Harden et al. [70] through the with the ferroelectrics, the flexoelectric coefficients of direct flexoelectric effect, as shown in Fig. 5(b). This liquid crystals are relatively small. However, regardless method is similar to the measurement of flexoelectricity its small coefficients, the flexoelectricity often plays a in dielectrics. The studied nematic liquid crystal (NLC) dominant role in liquid crystals, especially in the widely cells were firstly fixed in a flexible container, and then developed liquid crystal display technology [72–78]. bent by using a loudspeaker. Later, the NLC was Recently, Lee et al. [79] found that the highly distortion oscillated between two fixed posts, and a large of the liquid crystal display could be quantitatively flexoelectric current could be induced. For most of the measured by the variation of the flexoelectric polarization, www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 159 Fig. 5 (a) Schematic view of indirect measurement of flexoelectric coefficient in liquid crystal. An electric field was applied onto the interdigitated electrodes and the flexoelectric coefficients was extracted by the distortion of birefringence images. Reproduced with permission from Ref. [69], © AIP Publishing 2008. (b) Schematic view of the direct measurement of flexoelectric coefficient in liquid crystal. Reproduced with permission from Ref. [70], © American Physical Society 2006. (c) Illustration of the direction of the electric field and the flexoelectric polarization of liquid crystal under the splay and bend deformations. Reproduced with permission from Ref. [79], © Taylor & Francis 2017. (d) Illustration of the field symmetry for positive and negative frames, where the IPS refers to in-plane-switching, FFS refers to fringe-field-switching. Reproduced with permission from Ref. [80], © Optical Society of America 2016. as shown in Fig. 5(c). Moreover, the image flicker is usually orientated in opposite direction so that the problem of liquid crystal mixtures which shows macro piezoelectricity is zero. Once bending this bulk negative dielectric anisotropy can be minimized by the material, a net polarization exists because the bending flexoelectric effect [80,81] (as shown in Fig. 5(d)). induced strain direction of the top and bottom surface is opposite. That is why the surface piezoelectricity can 2. 5 Flexoelectricity in semiconductor couple to the enhanced flexoelectricity in the bulk insulators [86]. Similarly, this mechanism also works Generally speaking, the electro-mechanical coupling is in semiconductor owing to the existence of dead layer. only limited to dielectrics due to the requirement of The dead layer, typically in several micrometers thick, insulation. But more and more studies have disclosed could accumulate the top and bottom surface that semiconductors can be regarded as electro- piezoelectricity, and results in a significant net mechanical materials, e.g., the ZnO semiconductor is a polarization when the semiconductor is bent. As shown piezoelectric material [82–84]. In 2016, Narvaez et al. in Fig. 6(b), the oxygen reduced BaTiO single crystal [85] made a new breakthrough in flexoelectricity and 3 which acts as a good ionic conductor, shows an greatly broadened the concept of flexoelectricity into extremely enhanced flexoelectric-like response. This semiconductors. A dead layer mechanism was proposed and the measured flexoelectricity in some semiconductors phenomenon was also found in Nb-doped TiO can reach a level to 1000 μC/m. semiconductor. Interestingly, the flexoelectricity in As shown in Fig. 6(a), for an arbitrary bulk insulator semiconductor shows a linearized thickness dependence. such as BaTiO single crystal, the surface piezoelectricity The measured effective flexoelectric coefficients were exists due to the symmetry breaking in boundary. At directly proportional with the material thickness, as the top and bottom surfaces, the piezoelectric polarization shown in Fig. 6(c). www.springer.com/journal/40145 160 J Adv Ceram 2019, 8(2): 153–173 Fig. 6 (a) Schematic view of the barrier layer mechanism. For both the bulk insulators and the semiconductors, the surface piezoelectricity will contribute to the flexoelectric coefficients. The barrier layer in semiconductor will block the free charge and have a giant contribution for flexoelectric coefficients. (b) Temperature dependence of the effective transverse flexoelectric coefficients in pure bulk BaTiO single crystal, reduced BaTiO single crystal, and oxidized BaTiO single crystal. (c) Thickness 3 3 3 dependence of the effective transverse flexoelectric coefficients in Nb-doped TiO semiconductor. Reproduced with permission from Ref. [85], © Springer Nature 2016. Inspired by this attractive result, one can expect that in epitaxial ferroelectric thin films [94], the simulated 6 7 –1 –10 m , which may with proper design, the application of semiconductor strain gradient can reach up to 10 make the flexoelectric polarization higher than the flexoelectricity will play an important role in the optimization of the integrated circuit and electro- piezoelectric one. The general theory of flexoelectricity was recently mechanical semiconductor devices. developed. Many of them were focused on the origin of the enhancement of flexoelectricity in some typical 3 Theoretical calculation of flexoelectricity materials [95–97]. Theoretical work on flexoelectricity dates back to the papers by Mashkevich and Tolpygo During the past few years, the studies of flexoelectricity [98], who first proposed the effect, and Kogan [8], who were not solely focused on the experiment level, but formulated the first phenomenological theory. According also led to a lot of theoretical progress. It is worth to the theory proposed by Yudin et al. [99], the flexoelectric noting that the size effect of flexoelectricity was firstly polarization can be accounted for four parts, i.e., proposed by theoretical prediction rather than any surface piezoelectricity, bulk piezoelectricity, surface experiment [14]. Numerous theoretical studies have flexoelectricity, and bulk flexoelectricity [20]. All these demonstrated that the strain gradient becomes large in factors contribute almost equally to the flexoelectric nanoscale, at which the flexoelectricity may even be coefficient. However, a different viewpoint was stated competitive with piezoelectricity [87–93]. Specifically, hereupon by Resta [52] who built the polarization theory www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 161 based on the elemental cubic crystal. His theoretical parameter of each atom. Meanwhile, they also result manifested that the intrinsic flexoelectricity was developed an indirect way for calculating flexoelectric a purely bulk effect, where the surface contribution coefficients which was formulated in such a way that was totally excluded. Later, Yurkov and Tagantsev. the tensor elements can be computed directly in the [100] argued that the direct bulk flexoelectric response context of density-functional calculations, including of a finite sample essentially depends on the surface electronic and lattice contributions [23]. To date, the polarization energy. Recently, Zhang et al. [54] first-principles method could be utilized into the calculated that the surface piezoelectricity which widely calculation of other flexoelectric tensor component and exists in all the unpolarized ceramics and single crystals, much more materials, which is summarized in Table 1 is possibly dominant in the observed flexoelectricity. [102–106]. This statement also well matches their experimental An interesting theoretical progress of flexoelectricity results [54]. The controversy for this issue still exists and we addressed is the application of topology optimization. a comprehensive understood is urgently needed. This method, firstly proposed by Bendsøe and The first-principles calculation of flexoelectric Kikuchi [107] in 1988, has been widely applied to coefficients was firstly developed in several optimize the material property [108,109], especially semiconductors by Maranganti and Sharma [21] until for the design of domains [110,111]. More recently, 2009. Hong et al. [22,23,101] demonstrated the first ab Ghasemi et al. [112,113] presented a new-type initio calculation of the longitudinal flexoelectricity for topology optimization method, which is based on a BaTiO and SrTiO by using a direct approach, where combination of isogeometric analysis (IGA), level set, 3 3 the strain gradient is realized by setting the lattice and point wise density mapping techniques as shown Fig. 7 (a) IGA concept. Each element in the physical space is the image of a corresponding element in the parameter space, and the parameter space is discretized by knot vectors. Control points in IGA are used to discretize the geometry and define the degrees of freedom. Reproduced with permission from Ref. [112], © Elsevier B.V. 2016. (b) The level set function which easily copes with the various numbers of phases, is efficiently satisfied with the multiple constraints. This function intrinsically avoids overlap or vacuum among different phases. Reproduced with permission from Ref. [113], © Elsevier B.V. 2017. (c) Simulated vortex patterns represented by the vorticity calculated from the polarization vector field. A vortex/anti-vortex pair region marked by the dashed line is chosen to illustrate the flexocoupling effects in what follows, and the double-arrow symbol indicates an anti-phase boundary. Typical vortex pair pattern under different flexocoupling of the STO layers. Reproduced with permission from Ref. [114], © The Author(s) 2017. www.springer.com/journal/40145 162 J Adv Ceram 2019, 8(2): 153–173 in Fig. 7(a) and Fig. 7(b), respectively. Through this 8(a)) and the energy conversion efficiency (as shown topology optimization, significant enhanced flexoelectric in Fig. 8(b)) of the flexoelectric energy harvester are coefficients can be obtained in designated materials. much larger than that of the classical piezoelectric Other theoretical calculation methods such as finite energy harvesters. Particularly, when the thickness (h ) element method, phase-filed modeling, and density of the flexoelectric layer is reduced to less than 100 nm, functional theory, are also developed to explore the the power output of the flexoelectric energy harvester direct and converse flexoelectric effect [114–124]. It is is almost 12 times larger than that of the piezoelectric worth noting that Li et al. [114] reported the energy harvester [137]. Choi and Kim [138] fabricated application of flexoelectric phase-field model [125] in a flexoelectric energy harvester device which collects calculating the polar vortices of PbTiO /SrTiO 3 3 energy by a PVDF thin film deposited on the cantilever super-lattices. The results suggested that the induced substrate. The photograph of this device is shown in polarization (represented by the vortex/anti-vortex pair Fig. 8(c). Han et al. [139] prepared a flexoelectric region) is highly related to the given flexoelectric nanogenerator which is consisted of direct-grown coefficients, as shown in Fig. 7(c). 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 4 Applications of flexoelectricity force of 20 N, which is promising for harvesting the mechanical energy. Moreover, Zhu et al. [140] Here we will emphasize on the recent developed designed a flexible flexoelectric fiber which is made of applications of flexoelectricity in several fields, curved piezoelectric composites. As shown in Fig. 8(d), including flexoelectric energy harvester, flexoelectric these fibers possess a stable upward self-poling which sensors and actuators, domain engineering, liquid is induced by flexoelectricity, exhibited a large electric crystal displaying, and some other open applications, output comparable to those of the piezoelectric e.g., flexoelectricity-tuned photovoltaic effect. In the nanogenerators. end, we also list some possible applications where the flexoelectricity might be involved. 4. 2 Actuators 4. 1 Energy harvesters Piezoelectricity has been widely used for sensors and actuators. Similarly, the flexoelectricity could also be Energy harvesters are those devices which can collect utilized for sensing and actuating applications, energy from the external sources like wind energy, especially for nano-sized devices. Compared with the solar power, thermal energy, etc. Specifically, the piezoelectric sensors and actuators, the flexoelectric electromechanical energy harvester can collect energy ones are not limited by the crystal symmetry of the from the mechanical vibrations [126–132]. Recently, materials and the working temperature [141,142]. The the newly designed energy harvester which collect past few years have witnessed many progresses in this energy from the fluctuation of the vibrations (strain aspect. As shown in Fig. 9(a), Zhang et al. [143] gradient) was proposed. Owing to its special size- designed a curved flexoelectric actuator by using dependent effect, the flexoelectric energy harvester is non-polarized PVDF. This actuator exhibited a good suitable to be integrated into small size, where a huge actuating property with a resolution of the displacement strain gradient can be generated. Till now, with the proper design, the mechanical–electrical energy converting reaching up to 1.0 nm and a largest displacement as efficiency of the flexoelectric energy harvester can high as 63.6 nm (as shown in Fig. 9(b)). It is worth reach to 6.6% [133–135]. highlighting that Bhaskar et al. [144] implemented a The potential of flexoelectricity as energy harvester flexoelectric actuator made of micron-sized barium has been predicted by many works. Wang and Wang titanate (as shown in Fig. 9(c)) which is fully compatible [136] developed an analytical model for vibration- for the semiconductor silicon technology. The performance based circular energy harvester that consists of a result (Fig. 9(d)) suggests that this flexoelectric flexoelectric layer and a substrate layer. The result actuator displays comparable performance to the indicated that both the power output (as shown in Fig. actuators use lead-containing piezoelectric material. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 163 Fig. 8 (a) The maximum normalized power output of vibration-based circular energy harvester as a function of the thickness of the flexoelectric layer. (b) The energy converting efficiency of vibration-based circular energy harvester as a function of the thickness of the flexoelectric layer. Reproduced with permission from Ref. [136], © Elsevier Ltd. 2018. (c) The photography of the PVDF-based energy harvester which could collect energy from the excited vibrations. Reproduced with permission from Ref. [138], © IOP Publishing Ltd. 2017. (d) Comparison of the transient output voltage between the unpolarized nanogenerator and polarized nanogenerator. Reproduced with permission from Ref. [140], © Elsevier Ltd. 2018. gradient) is shown in Fig. 10(b). The sensitivity of this 4. 3 Sensors curvature sensor, defined by the slopes of the charge outputs versus curvature, can reach up to 1 pC/m. In The flexoelectric sensors are becoming increasingly addition, Merupo et al. [149] investigated the potential popular because of their small size, absence of de- use of 6.6 µm-thick soft polyurethane films as the large poling and aging problem, and lead-free composition curvature sensor by subjecting flexible aluminum [145]. The state-of-the-art flexoelectric sensor has been foil/PET bi-layered substrate to large deflections. A successfully utilized in many fields such as structural –1 curvature of about 80 m corresponding to a radius of health monitoring, crack detector, and curvature curvature of ~1.2 cm could be effectively sensed under detector [146,147]. Recently, Yan et al. [148] designed a low frequency (3 Hz) bending motion. The pseudo- a BST-based flexoelectric curvature sensor, which can sinusoidal time dependence of the output charge of this transfer the bending deflections directly to the charge sensor is shown in Fig. 10(c). Kwon et al. [150] output. The working principle of this sensor is reported recently that the flexoelectricity can work in illustrated in Fig. 10(a). To precisely detect the bending the microphone. The designed sensitivity of the deflection, two BST micro curvature sensors were attached onto the center side surfaces of an aluminum fabricated barium strontium titanate (Ba Sr TiO ) 0.65 0.35 3 microphone was very high and almost frequency- beam, located symmetrically with respect to its neutral axis. The relationship between flexoelectric charge dependent in wide frequency range, as shown in Fig. outputs of the BST sensors and the curvature (strain 10(d). www.springer.com/journal/40145 164 J Adv Ceram 2019, 8(2): 153–173 Fig. 9 (a) Schematic diagram of the PVDF-based actuator. The symbol θ represents the angle of electrode attachment. To achieve a uniform electric field gradient and application safety, θ in this actuator is set as 120°. (b) Induced displacement as a function of voltage in PVDF-based actuator. Reproduced with permission from Ref. [143], © AIP Publishing 2017. (c) Schematic view of the SrTiO -based actuator. Particularly, the flexoelectric layer is compatible with silicon or any of its gate dielectrics in a completely complementary metal oxide semiconductor-compatible environment. (d) Actuating performance comparison of the SrTiO -based actuator with other actuators. Reproduced with permission from Ref. [144], © Springer Nature 2015. Recently, Gómez et al. [151] prepared an epitaxial 4. 4 Domain tailoring and polarization switching growth of columnar porous BaTiO /LaSrMnO /SrTiO / 3 3 3 Si(001) heterostructures by using a complicated recipe Another attractive application of flexoelectricity which both combines the molecule beam epitaxy and should be addressed is the domain tailoring and polymer assisted deposition technology. The illustration polarization switching. As we may know that the of the preparation process is shown in Fig. 11(a). The ferroelectric materials are characterized by their results proved that the ferroelectric polarization of this spontaneous polarization, which can be switched by heterostructure thin film can be reversed by a applying an external electric field. As suggested by Lu mechanical load in epitaxial columnar nanostructures, et al. [1] and Catalan et al. [2], in a nano-sized material, as shown in Fig. 11(b). Similar phenomenon was also e.g., ultrathin ferroelectric film, the spontaneous found in PbTiO thin films [152]. It was experimentally polarization can be switched by mechanical strain 3 proved that, as shown in Fig. 11(c), the mechanical gradient. This powerful function has attracted increasing force shows a comparable ability relative with the interests and hence makes heterostructure thin film a electrically method for switching the domain of the natural scenario for the flexoelectric applications. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 165 Fig. 10 (a) Beam curvature sensing: BST curvature sensor attached to beam. (b) Relationship between charge output and beam curvature-experimental results of BST curvature sensor. Reproduced with permission from Ref. [148], © SPIE 2013. (c) Real time dependence of the output electric charge of the 6.6 µm thick curvature sensor. Reproduced with permission from Ref. [149], © AIP Publishing 2017. (d) The analytical, experimental sensitivity of the flexoelectric microphone at low frequency range (inset: overall sensitivity). Reproduced with permission from Ref. [150], © IOP Publishing Ltd. 2016. Fig. 11 (a) A new thin film preparation approach which combines molecule beam epitaxy and polymer assisted deposition. The complex epitaxial heterostructures were grown in three different stages. (b) Schematic view of the switching of the polarization by an electric field or by a mechanical force across the columnar BTO/LSM/STO/Si(001) heterostructure at room temperature. Reproduced with permission from Ref. [151], © WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2017. (c) Mechanical writing and electrical writing induced variation of the piezoresponse force microscopy phase images of the PbTiO film. The + - tip-induced pressure leads to the switching of the upward polarization (c domain) to the downward polarization (c domain). Reproduced with permission from Ref. [152], © AIP Publishing 2017. www.springer.com/journal/40145 166 J Adv Ceram 2019, 8(2): 153–173 PbTiO thin films. ionic behavior [156] and manipulate the oxygen Moreover, with the help of flexoelectricity, the vacancies [157] (as shown in Fig. 12(d) and Fig. 12(f)). Obviously, the application of strain gradient into self-polarization direction of ferroelectric thin films, the ultra-thin film can result in different vacancy which is of difficulty in control in previous studies, is formation. feasible to be changed by tuning the substrate interfaces and film thicknesses [153,154]. Park et al. 4. 5 Open applications [155] demonstrated that the multiple domain switching pathways in multiaxial ferroelectric materials can be Recent studies showed that the flexoelectricity can be selectively controlled by a newly realized trailing linked to many other important physical behaviors, and flexoelectric field, specifically, by the motion of a hence bringing the flexoelectricity into the totally open mechanically loaded scanning probe microscope tip. applications. For example, the transverse shear surface The illustration of this trailing flexoelectric field is acoustic waves have been found could propagate near shown in Fig. 12(a). Experimental results proved that the flat surfaces of all crystalline dielectrics because of the ferroelectric switching angle of multiferroic the existence of flexoelectric effect [158–160]. Liu et magnetoelectric BiFeO thin film can be stably selected al. [161] proposed that the bended thermoelectric at 71° ferroelastic switching or 109° ferroelectric BiTe film can present bulk photovoltaic effect at switching. The in-plane polarization and out-of-plane infrared wavelengths, which is possibly induced by the polarization in response to the loading forces of the flexoelectric effect, as shown in Fig. 13(a). Yang et al. moving tip shows a significant orientation dependence, [162] proved that the application of flexoelectricity can as shown in Fig. 12(b) and Fig. 12(c), respectively. cause the so-called flexo-photovoltaic effect. In their In addition, with the help of the flexoelectricity, the experiment, the large strain gradient (generated by the microscopic tip can also be used to explore dynamic 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 (black) and out-of-plane polarization P (red) as a function of loading forces with tip motion along x z [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. www.springer.com/journal/40145 J Adv Ceram 2019, 8(2): 153–173 167 Fig. 13 (a) Bulk photovoltaic response of the Bi Te film under different bending distances. The inset gives the photograph of 2 3 the bending setup. Reproduced with permission from Ref. [161], © The Author(s) 2016. (b) Schematic view of the setup for verifying the flexo-photovoltaic effect. (c) The induced photocurrent density as a function of the loading force. (d) Positive photocurrent measured on a TiO (001) face with a 15 µN force applied by the AFM tip. Reproduced with permission from Ref. [162], © The Authors 2018. (e) The diagram illustrates the distribution of carriers and electric dipoles in a strained Bi Te film 2 3 without thermal gradient and with a thermal gradient field. The thermoelectric field has an opposite direction with the flexoelectric field. Reproduced with permission from Ref. [164], © The Authors 2016. (f) Temperature dependence of the Bi TiO -based ceramic plate with large internal field under three-point bending load. The inset shows thermal currents of Na 0.5 0.5 3 the measured thermal currents of this ceramic plate without load. Reproduced with permission from Ref. [165], © American Physical Society 2018. illumination were simultaneously applied onto some mechanical force can result in the sign of the short single crystals, as shown in Fig. 13(b). The result, as circuit current I from negative to positive. Also, this sc shown in Fig. 13(c), suggested that a significant flexo-photovoltaic effect can be improved very photocurrent was induced by the applied mechanical obviously when material dimension is decreased into force. Besides, the generation of such photocurrent was the nanoscale. this study suggest that the application of proved to be only originated from the flexo- flexoelectricity is effective route for improving the photovoltaic effect rather than other factor like Schottky performance of solar cells and optoelectronic devices contact. As shown in Fig. 13(d), the application of [163]. www.springer.com/journal/40145 168 J Adv Ceram 2019, 8(2): 153–173 Recently, a newly reported flexo-caloric effect the non-zero independent component of flexoelectric which defines as the strain gradient induced thermal- coefficients in those low-symmetry crystals? (4) Is current has attracted considerable attentions. As there any way to induce large strain gradient, not just suggested by Liu et al. [164], the flexoelectricity is limited to the size effect? proved to be responsible for the value of Seeback Considering the past few years’ progress in coefficient of some thermoelectric material like BiFe flexoelectric materials and related devices, the authors It is believed that the existence of flexoelectric diploes make the perspective as shown below. can effectively contribute to the thermoelectricity, as 1. Strain gradient is easier to exist in the complex schematically illustrated in Fig. 13(e). Meanwhile, a material and flexible materials, resulting in the liquid remarkable flexo-caloric effect has been found in crystals and bio-materials naturally suitable for Na Bi TiO -based ceramics [165], as shown in Fig. flexoelectricity. Therefore, it is expected that the 0.5 0.5 3 13(f). Combined with the flexoelectric effect and flexoelectricity will be widely used in liquid crystal electrocaloric effect, it is possible to design materials display technology, bio-sensing, bio-medical, and with strong thermo-electric coupling effect for sensing, bio-mimetic materials. thermal energy harvesting, or on-chip solid-state 2. Due to the requirement of miniaturization in cooling applications [166]. electronic devices, the role of flexoelectricity will draw more attentions. In the near future, micro/nano-scale flexoelectric sensing and actuating devices will be 5 Outlook integrated into electronic devices. Among them, it is highly promising to find a lead-free and environmentally In summary, we reviewed the recent progress of friendly flexoelectric material that is compatible for the flexoelectricity, mainly focused on the flexoelectric AlN-based and silicon-based micro-electromechanical materials and their related applications. Flexoelectricity system. is not only limited to the dielectric materials, but also 3. Flexoelectricity is not limited by symmetry, found to exist in the liquid crystals, bio-materials, and giving it more space in material selection. In the future, even semiconductors. The absence of symmetry it is expected to find a kind of natural material that is constraint makes the flexoelectric materials suitable for widely existed and has significantly enhanced flexoelectric most cases where non-uniform electric field distribution properties. and non-uniform strain distribution exist. The recent discoveries utilized the flexoelectricity into many Acknowledgements important application fields such as sensor and actuator, charge transportation, defect formation, domain tailoring, This work was supported by the National Natural Science and some open applications like flexo-photovoltaic Foundation of China under Grant Nos. 11574126 and effect and flexo-caloric effect have been commented. 11604135, and partly by the Natural Science Foundation Although the study of flexoelectricity has an of Jiangxi Province (No. 20161BAB216110), China impressive achievement, the state-of-the-art understanding Postdoctoral Science Foundation (No. 2017M612162), of this field is still in its initial stage. Lots of the and Postdoctoral Science Foundation of Jiangxi Province fundamental problems regarding the flexoelectricity (No. 2017KY02). are unresolved. Herein we can only list parts of them. 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