Journal Home > Volume 8 , Issue 2
Journal of Advanced Ceramics 2019, 8(2): 153-173 https://doi.org/10.1007/s40145-018-0311-3
Review | Open Access | Issue | Published: 13 June 2019

Flexoelectric materials and their related applications: A focused review

Show Author's Information Longlong SHUa,b( ),  Renhong LIANGa,  Zhenggang RAOa,b,  Linfeng FEIa,b,  Shanming KEa,b( ),  Yu WANGa,b
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
Keywords:
dielectric constant, flexoelectricity, strain gradient, electric polarization, liquid crystals, sensors and actuators
Cite this article:
SHU L, LIANG R, RAO Z, et al. Flexoelectric materials and their related applications: A focused review. Journal of Advanced Ceramics, 2019, 8(2): 153-173. https://doi.org/10.1007/s40145-018-0311-3
Download citation
EndNote(RIS)
BibTeX

781

Views

65

Downloads

Citations

112

Crossref

N/A

WoS

113

Scopus

0

CSCD

Abstract Full text About this article

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.


menu
Abstract
Full text
Outline
About this article

Flexoelectric materials and their related applications: A focused review

Show Author's information Longlong SHUa,b( ), Renhong LIANGa, Zhenggang RAOa,b, Linfeng FEIa,b, Shanming KEa,b( ), Yu WANGa,b
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

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: dielectric constant, flexoelectricity, strain gradient, electric polarization, liquid crystals, sensors and actuators

References(166)

[1]
H Lu, C-W Bark, D Esque de los Ojos, et al. Mechanical writing of ferroelectric polarization. Science 2012, 336: 59-61.
DOI Google Scholar
[2]
G Catalan, A Lubk, AHG Vlooswijk, et al. Flexoelectric rotation of polarization in ferroelectric thin films. Nat Mater 2011, 10: 963-967.
DOI Google Scholar
[3]
P Zubko, G Catalan, AK Tagantsev. Flexoelectric effect in solids. Annu Rev Mater Res 2013, 43: 387-421.
DOI Google Scholar
[4]
SV Kalinin, AN Morozovska. Focusing light on flexoelectricity. Nat Nanotechnol 2015, 10: 916-917.
DOI Google Scholar
[5]
LE Cross. Flexoelectric effects: Charge separation in insulating solids subjected to elastic strain gradients. J Mater Sci 2006, 41: 53-63.
DOI Google Scholar
[6]
J Prost, PS Pershan. Flexoelectricity in nematic and smectic-A liquid crystals. J Appl Phys 1976, 47: 2298-2312.
DOI Google Scholar
[7]
VL Indenbom, EB Loginov, MA Osipov. Flexoelectric effect and crystal-structure. Kristallografiya 1981, 26: 1157-1162.
Google Scholar
[8]
SM Kogan. Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Soviet Physics-Solid State 1964, 5: 2069-2070.
Google Scholar
[9]
AG Petrov. Flexoelectricity of model and living membranes. BBA-Biomembranes 2002, 1561: 1-25.
DOI Google Scholar
[10]
KD Breneman, WE Brownell, RD Rabbitt. Hair cell bundles: Flexoelectric motors of the inner ear. PLoS One 2009, 4: e5201.
DOI Google Scholar
[11]
LL Shu, F Li, W Huang, et al. Relationship between direct and converse flexoelectric coefficients. J Appl Phys 2014, 116: 144105.
DOI Google Scholar
[12]
LL Shu, XY Wei, T Pang, et al. Symmetry of flexoelectric coefficients in crystalline medium. J Appl Phys 2011, 110: 104106.
DOI Google Scholar
[13]
PV Yudin, AK Tagantsev. Fundamentals of flexoelectricity in solids. Nanotechnology 2013, 24: 432001.
DOI Google Scholar
[14]
MS Majdoub, P Sharma, T Cagin. Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys Rev B 2008, 77: 125424.
DOI Google Scholar
[15]
TD Nguyen, S Mao, Y-W Yeh, et al. Nanoscale flexoelectricity. Adv Mater 2013, 25: 946-974.
DOI Google Scholar
[16]
K Chu, B-K Jang, JH Sung, et al. Enhancement of the anisotropic photocurrent in ferroelectric oxides by strain gradients. Nat Nanotechnol 2015, 10: 972-979.
DOI Google Scholar
[17]
LL Shu, ZH Yong, XN Jiang, et al. Flexoelectricity in low densification materials and its implication. J Alloys Compd 2017, 695: 1555-1560.
DOI Google Scholar
[18]
J Fousek, LE Cross, DB Litvin. Possible piezoelectric composites based on the flexoelectric effect. Mater Lett 1999, 39: 287-291.
DOI Google Scholar
[19]
AK Tagantsev. Theory of flexoelectric effect in crystals. Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 1985, 88: 2108-2122.
Google Scholar
[20]
AK Tagantsev. Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys Rev B 1986, 34: 5883.
DOI Google Scholar
[21]
R Maranganti, P Sharma. Atomistic determination of flexoelectric properties of crystalline dielectrics. Phys Rev B 2009, 80: 054109.
DOI Google Scholar
[22]
JW Hong, D Vanderbilt. First-principles theory of frozen-ion flexoelectricity. Phys Rev B 2011, 84: 180101.
DOI Google Scholar
[23]
JW Hong, D Vanderbilt. First-principles theory and calculation of flexoelectricity. Phys Rev B 2013, 88: 174107.
DOI Google Scholar
[24]
A Abdollahi, F Vásquez-Sancho, G Catalan. Piezoelectric mimicry of flexoelectricity. Phys Rev Lett 2018, 121: 205502.
DOI Google Scholar
[25]
H Le Quang, QC He. The number and types of all possible rotational symmetries for flexoelectric tensors. P Roy Soc A-Math Phy 2011, 467: 2369-2386.
DOI Google Scholar
[26]
LL Shu, WB Huang, S Ryung Kwon, et al. Converse flexoelectric coefficient f1212 in bulk Ba0.67Sr0.33TiO3. Appl Phys Lett 2014, 104: 232902.
DOI Google Scholar
[27]
H Zhou, YM Pei, JW Hong, et al. Analytical method to determine flexoelectric coupling coefficient at nanoscale. Appl Phys Lett 2016, 108: 101908.
DOI Google Scholar
[28]
TT Hu, Q Deng, X Liang, et al. Measuring the flexoelectric coefficient of bulk barium titanate from a shock wave experiment. J Appl Phys 2017, 122: 055106.
DOI Google Scholar
[29]
SW Zhang, X Liang, ML Xu, et al. Shear flexoelectric response along 3121 direction in polyvinylidene fluoride. Appl Phys Lett 2015, 107: 142902.
DOI Google Scholar
[30]
SW Zhang, KY Liu, TH Wu, et al. Experimental approach for measuring cylindrical flexoelectric coefficients. J Appl Phys 2017, 122: 144103.
DOI Google Scholar
[31]
SW Zhang, KY Liu, ML Xu, et al. Investigation of the 2312 flexoelectric coefficient component of polyvinylidene fluoride: Deduction, simulation, and mensuration. Sci Rep 2017, 7: 3134.
DOI Google Scholar
[32]
BJ Chu, DR Salem. Flexoelectricity in several thermoplastic and thermosetting polymers. Appl Phys Lett 2012, 101: 103905.
DOI Google Scholar
[33]
BJ Chu, WY Zhu, N Li, et al. Flexure mode flexoelectric piezoelectric composites. J Appl Phys 2009, 106: 104109.
DOI Google Scholar
[34]
Y Zhou, J Liu, XP Hu, et al. Flexoelectric effect in PVDF-based polymers. IEEE Trans Dielect El In 2017, 24: 727-731.
DOI Google Scholar
[35]
WH Ma, LE Cross. Flexoelectricity of barium titanate. Appl Phys Lett 2006, 88: 232902.
DOI Google Scholar
[36]
J Narvaez, S Saremi, JW Hong, et al. Large flexoelectric anisotropy in paraelectric barium titanate. Phys Rev Lett 2015, 115: 037601.
DOI Google Scholar
[37]
WH Ma, LE Cross. Flexoelectric polarization of barium strontium titanate in the paraelectric state. Appl Phys Lett 2002, 81: 3440-3442.
DOI Google Scholar
[38]
LL Shu, T Wang, XN Jiang, et al. Verification of the flexoelectricity in barium strontium titanate through d33 meter. AIP Adv 2016, 6: 125003.
DOI Google Scholar
[39]
Y Li, LL Shu, WB Huang, et al. Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite. Appl Phys Lett 2014, 105: 162906.
Google Scholar
[40]
LL Shu, MQ Wan, ZG Wang, et al. Large flexoelectricity in Al2O3-doped Ba(Ti0.85Sn0.15)O3 ceramics. Appl Phys Lett 2017, 110: 192903.
DOI Google Scholar
[41]
LL Shu, XY Wei, L Jin, et al. Enhanced direct flexoelectricity in paraelectric phase of Ba(Ti0.87Sn0.13)O3 ceramics. Appl Phys Lett 2013, 102: 152904.
DOI Google Scholar
[42]
WH Ma, LE Cross. Strain-gradient-induced electric polarization in lead zirconate titanate ceramics. Appl Phys Lett 2003, 82: 3293-3295.
DOI Google Scholar
[43]
JQ Zhu, TW Chen, LL Shu, et al. Flexoelectric fatigue in (K,Na,Li)(Nb,Sb)O3 ceramics. Appl Phys Lett 2018, 113: 182901.
DOI Google Scholar
[44]
LL Shu, T Li, ZG Wang, et al. Flexoelectric behavior in PIN-PMN-PT single crystals over a wide temperature range. Appl Phys Lett 2017, 111: 162901.
DOI Google Scholar
[45]
WH Ma, LE Cross. Observation of the flexoelectric effect in relaxor Pb(Mg1/3Nb2/3)O3 ceramics. Appl Phys Lett 2001, 78: 2920-2921.
DOI Google Scholar
[46]
J Narvaez, G Catalan. Origin of the enhanced flexoelectricity of relaxor ferroelectrics. Appl Phys Lett 2014, 104: 162903.
DOI Google Scholar
[47]
P Vales-Castro, K Roleder, L Zhao, et al. Flexoelectricity in antiferroelectrics. Appl Phys Lett 2018, 113: 132903.
DOI Google Scholar
[48]
P Zubko, G Catalan, A Buckley, et al. Strain-gradient-induced polarization in SrTiO3 single crystals. Phys Rev Lett 2007, 99: 167601.
DOI Google Scholar
[49]
A Biancoli, CM Fancher, JL Jones, et al. Breaking of macroscopic centric symmetry in paraelectric phases of ferroelectric materials and implications for flexoelectricity. Nat Mater 2015, 14: 224-229.
DOI Google Scholar
[50]
AY Borisevich, EA Eliseev, AN Morozovska, et al. Atomic-scale evolution of modulated phases at the ferroelectric-antiferroelectric morphotropic phase boundary controlled by flexoelectric interaction. Nat Commun 2012, 3: 775.
DOI Google Scholar
[51]
LM Garten, S Trolier-Mckinstry. Enhanced flexoelectricity through residual ferroelectricity in barium strontium titanate. J Appl Phys 2015, 117: 094102.
DOI Google Scholar
[52]
R Resta. Towards a bulk theory of flexoelectricity. Phys Rev Lett 2010, 105: 127601.
DOI Google Scholar
[53]
LL Shu, MQ Wan, XN Jiang, et al. Frequency dispersion of flexoelectricity in PMN-PT single crystal. AIP Adv 2017, 7: 015010.
DOI Google Scholar
[54]
XT Zhang, Q Pan, DX Tian, et al. Large flexoelectriclike response from the spontaneously polarized surfaces in ferroelectric ceramics. Phys Rev Lett 2018, 121: 057602.
DOI Google Scholar
[55]
RM Raphael, AS Popel, WE Brownell. A membrane bending model of outer hair cell electromotility. Biophys J 2000, 78: 2844-2862.
DOI Google Scholar
[56]
AG Petrov. Flexoelectricity in lyotropics and in living liquid crystals. In: Flexoelectricity in Liquid Crystals. Imperial College Press, 2012: 177-210.
DOI
[57]
M Abou-Dakka, EE Herrera-Valencia, AD Rey. Linear oscillatory dynamics of flexoelectric membranes embedded in viscoelastic media with applications to outer hair cells. J Non-Newton Fluid 2012, 185-186: 1-17.
DOI Google Scholar
[58]
XN Jiang, WB Huang, SJ Zhang. Flexoelectric nano-generator: Materials, structures and devices. Nano Energy 2013, 2: 1079-1092.
DOI Google Scholar
[59]
HM Kang, ZD Hou, QH Qin. Experimental study of time response of bending deformation of bone cantilevers in an electric field. J Mech Behav Biomed 2018, 77: 192-198.
DOI Google Scholar
[60]
WS Williams, L Breger. Piezoelectricity in tendon and bone. J Biomech 1975, 8: 407-413.
DOI Google Scholar
[61]
F Vasquez-Sancho, A Abdollahi, D Damjanovic, et al. Flexoelectricity in bones. Adv Mater 2018, 30: 1801413.
DOI Google Scholar
[62]
H Bayley, B Cronin, A Heron, et al. Droplet interface bilayers. Mol BioSyst 2008, 4: 1191-1208.
DOI Google Scholar
[63]
MJ Booth, V Restrepo Schild, FG Downs, et al. Functional aqueous droplet networks. Mol BioSyst 2017, 13: 1658-1691.
DOI Google Scholar
[64]
EJ Challita, MM Makhoul-Mansour, EC Freeman. Reconfiguring droplet interface bilayer networks through sacrificial membranes. Biomicrofluidics 2018, 12: 034112.
DOI Google Scholar
[65]
M Makhoul-Mansour, WJ Zhao, N Gay, et al. Ferrofluid-based droplet interface bilayer networks. Langmuir 2017, 33: 13000-13007.
DOI Google Scholar
[66]
EC Freeman, JS Najem, S Sukharev, et al. The mechanoelectrical response of droplet interface bilayer membranes. Soft Matter 2016, 12: 3021-3031.
DOI Google Scholar
[67]
A Kancharala, E Freeman, M Philen. A comprehensive flexoelectric model for droplet interface bilayers acting as sensors and energy harvesters. Smart Mater Struct 2016, 25: 104007.
DOI Google Scholar
[68]
N Tamaddoni, EC Freeman, SA Sarles. Sensitivity and directionality of lipid bilayer mechanotransduction studied using a revised, highly durable membrane-based hair cell sensor. Smart Mater Struct 2015, 24: 065014.
DOI Google Scholar
[69]
CL Trabi, CV Brown, AAT Smith, et al. Interferometric method for determining the sum of the flexoelectric coefficients (e1+e3) in an ionic nematic material. Appl Phys Lett 2008, 92: 223509.
DOI Google Scholar
[70]
J Harden, B Mbanga, N Éber, et al. Giant flexoelectricity of bent-core nematic liquid crystals. Phys Rev Lett 2006, 97: 157802.
DOI Google Scholar
[71]
F Castles, SM Morris, HJ Coles. The limits of flexoelectricity in liquid crystals. AIP Adv 2011, 1: 032120.
DOI Google Scholar
[72]
M Buczkowska, G Derfel. Influence of the surface pretilt angle on spatially periodic deformations in nematic layers. Liq Cryst 2018, 45: 961-964.
DOI Google Scholar
[73]
Y Inoue, M Hattori, H Moritake. Thickness-independent dynamics in cholesteric liquid crystals. Opt Express 2017, 25: 3566-3577.
DOI Google Scholar
[74]
A Poddar, J Dhar, S Chakraborty. Electro-osmosis of nematic liquid crystals under weak anchoring and second-order surface effects. Phys Rev E 2017, 96: 013114.
DOI Google Scholar
[75]
A Varanytsia, LC Chien. Bimesogen-enhanced flexoelectro-optic behavior of polymer stabilized cholesteric liquid crystal. J Appl Phys 2016, 119: 014502.
DOI Google Scholar
[76]
A Varanytsia, LC Chien. Giant flexoelectro-optic effect with liquid crystal dimer CB7CB. Sci Rep 2017, 7: 41333.
DOI Google Scholar
[77]
N Vaupotič, S Curk, MA Osipov, et al. Short-range smectic fluctuations and the flexoelectric model of modulated nematic liquid crystals. Phys Rev E 2016, 93: 022704.
DOI Google Scholar
[78]
M Kim, HS Jin, SJ Lee, et al. Liquid crystals for superior electro-optic performance display device with power-saving mode. Adv Opt Mater 2018, 6: 1800022.
DOI Google Scholar
[79]
D-J Lee, J-C Choi, M-K Park, et al. Optical measurement of flexoelectric polarisation change in liquid crystals doped with bent-core molecules using hybrid-aligned structure. Liq Cryst 2017, 44: 1321-1331.
DOI Google Scholar
[80]
MS Kim, PJ Bos, D-W Kim, et al. Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell. Opt Express 2016, 24: 29568-29576.
DOI Google Scholar
[81]
M Kim, HG Ham, HS Choi, et al. Flexoelectric in-plane switching (IPS) mode with ultra-high-transmittance, low-voltage, low-frequency, and a flicker-free image. Opt Express 2017, 25: 5962-5971.
DOI Google Scholar
[82]
D Varshney, S Shriya, S Jain, et al. Mechanically induced stiffening, thermally driven softening, and brittle nature of SiC. J Adv Ceram 2016, 5: 13-34.
DOI Google Scholar
[83]
JH He, CL Hsin, J Liu, et al. Piezoelectric gated diode of a single ZnO nanowire. Adv Mater 2007, 19: 781-784.
DOI Google Scholar
[84]
J Ouyang, W Zhang, SP Alpay, et al. Effect of elastic domains on electromechanical response of epitaxial ferroelectric films with a three-domain architecture. J Adv Ceram 2013, 2: 1-10.
DOI Google Scholar
[85]
J Narvaez, F Vasquez-Sancho, G Catalan. Enhanced flexoelectric-like response in oxide semiconductors. Nature 2016, 538: 219-221.
DOI Google Scholar
[86]
AS Yurkov, AK Tagantsev. Strong surface effect on direct bulk flexoelectric response in solids. Appl Phys Lett 2016, 108: 022904.
DOI Google Scholar
[87]
B He, B Javvaji, XY Zhuang. Size dependent flexoelectric and mechanical properties of barium titanate nanobelt: A molecular dynamics study. Physica B 2018, 545: 527-535.
DOI Google Scholar
[88]
L Qi, SJ Zhou, AQ Li. Size-dependent bending of an electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect. Compos Struct 2016, 135: 167-175.
DOI Google Scholar
[89]
SP Shen, SL Hu. A theory of flexoelectricity with surface effect for elastic dielectrics. J Mech Phys Solids 2010, 58: 665-677.
DOI Google Scholar
[90]
WB Huang, K Kim, SJ Zhang, et al. Scaling effect of flexoelectric (Ba,Sr)TiO3 microcantilevers. Phys Status Solidi RRL 2011, 5: 350-352.
DOI Google Scholar
[91]
G Bai, K Qin, QY Xie, et al. Size dependent flexocaloric effect of paraelectric Ba0.67Sr0.33TiO3 nanostructures. Mater Lett 2017, 186: 146-150.
DOI Google Scholar
[92]
BC Jeon, D Lee, MH Lee, et al. Flexoelectric effect in the reversal of self-polarization and associated changes in the electronic functional properties of BiFeO3 thin films. Adv Mater 2013, 25: 5643-5649.
DOI Google Scholar
[93]
WF Zhou, P Chen, Q Pan, et al. Lead-free metamaterials with enormous apparent piezoelectric response. Adv Mater 2015, 27: 6349-6355.
DOI Google Scholar
[94]
D Lee, A Yoon, SY Jang, et al. Giant flexoelectric effect in ferroelectric epitaxial thin films. Phys Rev Lett 2011, 107: 057602.
DOI Google Scholar
[95]
P Gao, SZ Yang, R Ishikawa, et al. Atomic-scale measurement of flexoelectric polarization at SrTiO3 dislocations. Phys Rev Lett 2018, 120: 267601.
DOI Google Scholar
[96]
AK Tagantsev, K Vaideeswaran, SB Vakhrushev, et al. The origin of antiferroelectricity in PbZrO3. Nat Commun 2013, 4: 2229.
DOI Google Scholar
[97]
M Stengel. Surface control of flexoelectricity. Phys Rev B 2014, 90: 201112.
DOI Google Scholar
[98]
VS Mashkevich, KB Tolpygo. The interaction of vibrations of nonpolar crystals with electric fields. Soviet Physics Doklady 1957, 1: 690.
Google Scholar
[99]
PV Yudin, R Ahluwalia, AK Tagantsev. Upper bounds for flexoelectric coefficients in ferroelectrics. Appl Phys Lett 2014, 104: 082913.
DOI Google Scholar
[100]
AS Yurkov, AK Tagantsev. Strong surface effect on direct bulk flexoelectric response in solids. Appl Phys Lett 2016, 108: 022904.
DOI Google Scholar
[101]
JW Hong, G Catalan, JF Scott, et al. The flexoelectricity of barium and strontium titanates from first principles. J Phys: Condens Matter 2010, 22: 112201.
DOI Google Scholar
[102]
AS Banerjee, P Suryanarayana. Cyclic density functional theory: A route to the first principles simulation of bending in nanostructures. J Mech Phys Solids 2016, 96: 605-631.
DOI Google Scholar
[103]
A Kvasov, AK Tagantsev. Dynamic flexoelectric effect in perovskites from first-principles calculations. Phys Rev B 2015, 92: 054104.
DOI Google Scholar
[104]
M Stengel. Unified ab initio formulation of flexoelectricity and strain-gradient elasticity. Phys Rev B 2016, 93: 245107.
DOI Google Scholar
[105]
CE Dreyer, M Stengel, D Vanderbilt. Current-density implementation for calculating flexoelectric coefficients. Phys Rev B 2018, 98: 075153.
DOI Google Scholar
[106]
SH Shi, P Li, F Jin. The mechanical analysis of thermo-magneto-electric laminated composites in nanoscale with the consideration of surface and flexoelectric effects. Smart Mater Struct 2018, 27: 015018.
DOI Google Scholar
[107]
MP Bendsøe, N Kikuchi. Generating optimal topologies in structural design using a homogenization method. Comput Method Appl M 1988, 71: 197-224.
DOI Google Scholar
[108]
MY Wang, XM Wang, DM Guo. A level set method for structural topology optimization. Comput Method Appl M 2003, 192: 227-246.
DOI Google Scholar
[109]
J-H Zhu, W-H Zhang, L Xia. Topology optimization in aircraft and aerospace structures design. Arch Comput Method E 2016, 23: 595-622.
DOI Google Scholar
[110]
JW Hong, G Catalan, DN Fang, et al. Topology of the polarization field in ferroelectric nanowires from first principles. Phys Rev B 2010, 81: 172101.
DOI Google Scholar
[111]
SC Chae, Y Horibe, DY Jeong, et al. Evolution of the domain topology in a ferroelectric. Phys Rev Lett 2013, 110: 167601.
DOI Google Scholar
[112]
H Ghasemi, HS Park, T Rabczuk. A level-set based IGA formulation for topology optimization of flexoelectric materials. Comput Method Appl M 2017, 313: 239-258.
DOI Google Scholar
[113]
H Ghasemi, HS Park, T Rabczuk. A multi-material level set-based topology optimization of flexoelectric composites. Comput Method Appl M 2018, 332: 47-62.
DOI Google Scholar
[114]
Q Li, CT Nelson, SL Hsu, et al. Quantification of flexoelectricity in PbTiO3/SrTiO3 superlattice polar vortices using machine learning and phase-field modeling. Nat Commun 2017, 8: 1468.
DOI Google Scholar
[115]
M Kothari, M-H Cha, K-S Kim. Critical curvature localization in graphene. I. Quantum-flexoelectricity effect. P Roy Soc A-Math Phy 2018, 474: 20180054.
DOI Google Scholar
[116]
YQ Mao, SG Ai, XL Xiang, et al. Theory for dielectrics considering the direct and converse flexoelectric effects and its finite element implementation. Appl Math Model 2016, 40: 7115-7137.
DOI Google Scholar
[117]
Y-J Wang, J Li, Y-L Zhu, et al. Phase-field modeling and electronic structural analysis of flexoelectric effect at 180° domain walls in ferroelectric PbTiO3. J Appl Phys 2017, 122: 224101.
DOI Google Scholar
[118]
A Plymill, HX Xu. Flexoelectricity in ATiO3 (A = Sr, Ba, Pb) perovskite oxide superlattices from density functional theory. J Appl Phys 2018, 123: 144101.
DOI Google Scholar
[119]
SS Nanthakumar, XY Zhuang, HS Park, et al. Topology optimization of flexoelectric structures. J Mech Phys Solids 2017, 105: 217-234.
DOI Google Scholar
[120]
F Deng, Q Deng, SP Shen. A three-dimensional mixed finite element for flexoelectricity. J Appl Mech 2018, 85: 031009.
DOI Google Scholar
[121]
MQ Wan, ZH Yong, WB Huang, et al. Design of a flexure composite with large flexoelectricity. J Mater Sci: Mater El 2017, 28: 6505-6511.
DOI Google Scholar
[122]
L Jiang, X Xu, Y Zhou. Interrelationship between flexoelectricity and strain gradient elasticity in ferroelectric nanofilms: A phase field study. J Appl Phys 2016, 120: 234102.
DOI Google Scholar
[123]
XF Xu, LM Jiang, YC Zhou. Reduction of leakage currents in ferroelectric thin films by flexoelectricity: A phase field study. Smart Mater Struct 2017, 26: 115024.
DOI Google Scholar
[124]
KM Hamdia, H Ghasemi, XY Zhuang, et al. Sensitivity and uncertainty analysis for flexoelectric nanostructures. Comput Method Appl M 2018, 337: 95-109.
DOI Google Scholar
[125]
AK Yadav, CT Nelson, SL Hsu, et al. Observation of polar vortices in oxide superlattices. Nature 2016, 530: 198-201.
DOI Google Scholar
[126]
C Mo, J Davidson, WW Clark. Energy harvesting with piezoelectric circular membrane under pressure loading. Smart Mater Struct 2014, 23: 045005.
DOI Google Scholar
[127]
S Azizi, A Ghodsi, H Jafari, et al. A conceptual study on the dynamics of a piezoelectric MEMS (Micro Electro Mechanical System) energy harvester. Energy 2016, 96: 495-506.
DOI Google Scholar
[128]
A Jasim, H Wang, G Yesner, et al. Optimized design of layered bridge transducer for piezoelectric energy harvesting from roadway. Energy 2017, 141: 1133-1145.
DOI Google Scholar
[129]
S Ravi, A Zilian. Monolithic modeling and finite element analysis of piezoelectric energy harvesters. Acta Mech 2017, 228: 2251-2267.
DOI Google Scholar
[130]
VK Wong, JH Ho, AB Chai. Performance of a piezoelectric energy harvester in actual rain. Energy 2017, 124: 364-371.
DOI Google Scholar
[131]
ZB Yang, YQ Wang, L Zuo, et al. Introducing arc-shaped piezoelectric elements into energy harvesters. Energy Convers Manag 2017, 148: 260-266.
DOI Google Scholar
[132]
ZY Zhou, WY Qin, P Zhu. Harvesting acoustic energy by coherence resonance of a bi-stable piezoelectric harvester. Energy 2017, 126: 527-534.
DOI Google Scholar
[133]
L Qi, SJ Huang, GY Fu, et al. On the mechanics of curved flexoelectric microbeams. Int J Eng Sci 2018, 124: 1-15.
DOI Google Scholar
[134]
Z Yan. Modeling of a piezoelectric/piezomagnetic nano energy harvester based on two dimensional theory. Smart Mater Struct 2018, 27: 015016.
DOI Google Scholar
[135]
X Liang, SL Hu, SP Shen. Nanoscale mechanical energy harvesting using piezoelectricity and flexoelectricity. Smart Mater Struct 2017, 26: 035050.
DOI Google Scholar
[136]
KF Wang, BL Wang. Energy gathering performance of micro/nanoscale circular energy harvesters based on flexoelectric effect. Energy 2018, 149: 597-606.
DOI Google Scholar
[137]
KF Wang, BL Wang. An analytical model for nanoscale unimorph piezoelectric energy harvesters with flexoelectric effect. Compos Struct 2016, 153: 253-261.
DOI Google Scholar
[138]
S-B Choi, G-W Kim. Measurement of flexoelectric response in polyvinylidene fluoride films for piezoelectric vibration energy harvesters. J Phys D: Appl Phys 2017, 50: 075502.
DOI Google Scholar
[139]
JK Han, DH Jeon, SY Cho, et al. Nanogenerators consisting of direct-grown piezoelectrics on multi-walled carbon nanotubes using flexoelectric effects. Sci Rep 2016, 6: 29562.
DOI Google Scholar
[140]
RJ Zhu, ZM Wang, H Ma, et al. Poling-free energy harvesters based on robust self-poled ferroelectric fibers. Nano Energy 2018, 50: 97-105.
DOI Google Scholar
[141]
MC Ray. Analysis of smart nanobeams integrated with a flexoelectric nano actuator layer. Smart Mater Struct 2016, 25: 055011.
DOI Google Scholar
[142]
MC Ray. Mesh free model of nanobeam integrated with a flexoelectric actuator layer. Compos Struct 2017, 159: 63-71.
DOI Google Scholar
[143]
SW Zhang, KY Liu, ML Xu, et al. A curved resonant flexoelectric actuator. Appl Phys Lett 2017, 111: 082904.
DOI Google Scholar
[144]
UK Bhaskar, N Banerjee, A Abdollahi, et al. A flexoelectric microelectromechanical system on silicon. Nat Nanotechnol 2016, 11: 263-266.
DOI Google Scholar
[145]
SR Kwon, WB Huang, SJ Zhang, et al. Flexoelectric sensing using a multilayered barium strontium titanate structure. Smart Mater Struct 2013, 22: 115017.
DOI Google Scholar
[146]
WB Huang, X Yan, SR Kwon, et al. Flexoelectric strain gradient detection using Ba0.64Sr0.36TiO3 for sensing. Appl Phys Lett 2012, 101: 252903.
DOI Google Scholar
[147]
W Huang, SR Kwon, FG Yuan, et al. A flexoelectric micro-accelerometer. ASME 2012, 9: 597-603.
DOI Google Scholar
[148]
X Yan, WB Huang, SR Kwon, et al. Design of a curvature sensor using a flexoelectric material. In: Proceedings of the SPIE 8692, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, 2013: 86920N.
DOI
[149]
VI Merupo, B Guiffard, R Seveno, et al. Flexoelectric response in soft polyurethane films and their use for large curvature sensing. J Appl Phys 2017, 122: 144101.
DOI Google Scholar
[150]
SR Kwon, WB Huang, SJ Zhang, et al. Study on a flexoelectric microphone using barium strontium titanate. J Micromech Microeng 2016, 26: 045001.
DOI Google Scholar
[151]
A Gómez, JM Vila-Fungueiriño, R Moalla, et al. Electric and mechanical switching of ferroelectric and resistive states in semiconducting BaTiO3-δ films on silicon. Small 2017, 13: 1701614.
DOI Google Scholar
[152]
HD Lu, S Liu, ZY Ye, et al. Asymmetry in mechanical polarization switching. Appl Phys Lett 2017, 110: 222903.
DOI Google Scholar
[153]
R Guo, L Shen, H Wang, et al. Tailoring self-polarization of BaTiO3 thin films by interface engineering and flexoelectric effect. Adv Mater Interfaces 2016, 3: 1600737.
DOI Google Scholar
[154]
IS Vorotiahin, AN Morozovska, EA Eliseev, et al. Flexocoupling impact on the kinetics of polarization reversal. Phys Rev B 2017, 95: 014104.
DOI Google Scholar
[155]
SM Park, B Wang, S Das, et al. Selective control of multiple ferroelectric switching pathways using a trailing flexoelectric field. Nat Nanotechnol 2018, 13: 366-370.
DOI Google Scholar
[156]
D Seol, SM Yang, S Jesse, et al. Dynamic mechanical control of local vacancies in NiO thin films. Nanotechnology 2018, 29: 275709.
DOI Google Scholar
[157]
S Das, B Wang, Y Cao, et al. Controlled manipulation of oxygen vacancies using nanoscale flexoelectricity. Nat Commun 2017, 8: 615.
DOI Google Scholar
[158]
EA Eliseev, AN Morozovska, MD Glinchuk, et al. Lost surface waves in nonpiezoelectric solids. Phys Rev B 2017, 96: 045411.
DOI Google Scholar
[159]
WJ Yang, Q Deng, X Liang, et al. Lamb wave propagation with flexoelectricity and strain gradient elasticity considered. Smart Mater Struct 2018, 27: 085003.
DOI Google Scholar
[160]
AN Morozovska, MD Glinchuk, EA Eliseev, et al. Flexocoupling-induced soft acoustic modes and the spatially modulated phases in ferroelectrics. Phys Rev B 2017, 96: 094111.
DOI Google Scholar
[161]
YC Liu, JD Chen, C Wang, et al. Bulk photovoltaic effect at infrared wavelength in strained Bi2Te3 films. Apl Mater 2016, 4: 126104.
DOI Google Scholar
[162]
MM Yang, DJ Kim, M Alexe. Flexo-photovoltaic effect. Science 2018, 360: 904-907.
DOI Google Scholar
[163]
W Lee, D Kim, J Lim, et al. Light-induced electrical switch via photo-responsive nanocomposite film. Sensor Actuat B Chem 2018, 266: 724-729.
DOI Google Scholar
[164]
YC Liu, JD Chen, HY Deng, et al. Anomalous thermoelectricity in strained Bi2Te3 films. Sci Rep 2016, 6: 32661.
DOI Google Scholar
[165]
P Chen, HF Zhang, BJ Chu. Strain gradient induced thermal-electrical response in paraelectric Na0.5Bi0.5TiO3- based ceramics. Phys Rev Materials 2018, 2: 034401.
DOI Google Scholar
[166]
S Patel, A Chauhan, NA Madhar, et al. Flexoelectric induced caloric effect in truncated Pyramid shaped Ba0.67Sr0.33TiO3 ferroelectric material. J Electron Mater 2017, 46: 4166-4171.
DOI Google Scholar
Publication history
Copyright
Acknowledgements
Rights and permissions

Publication history

Received: 05 December 2018
Accepted: 08 December 2018
Published: 13 June 2019
Issue date: June 2019

Copyright

© The author(s) 2019

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11574126 and 11604135, and partly by the Natural Science Foundation of Jiangxi Province (No. 20161BAB216110), China Postdoctoral Science Foundation (No. 2017M612162), and Postdoctoral Science Foundation of Jiangxi Province (No. 2017KY02).

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Return