The emancipation of flexoelectricity Cite as: J. Appl. Phys. 131, 020401 (2022); https://doi.org/10.1063/5.0079319 Submitted: 19 November 2021 • Accepted: 22 November 2021 • Published Online: 11 January 2022 Irene Arias, Gustau Catalan and Pradeep Sharma COLLECTIONS Paper published as part of the special topic on Trends in Flexoelectricity ARTICLES YOU MAY BE INTERESTED IN Thermal conductivity and management in laser gain materials: A nano/microstructural perspective Journal of Applied Physics 131, 020902 (2022); https://doi.org/10.1063/5.0073507 Soft matter crystallography—Complex, diverse, and new crystal structures in condensed materials on the mesoscale Journal of Applied Physics 131, 020901 (2022); https://doi.org/10.1063/5.0072017 Experiment and modeling of the pulsed lasing in a diode-pumped argon metastable laser Journal of Applied Physics 131, 023104 (2022); https://doi.org/10.1063/5.0079512 J. Appl. Phys. 131, 020401 (2022); https://doi.org/10.1063/5.0079319 131, 020401 © 2022 Author(s). Journal of EDITORIAL scitation.org/journal/jap Applied Physics The emancipation of flexoelectricity Cite as: J. Appl. Phys. 131, 020401 (2022); doi: 10.1063/5.0079319 Submitted: 19 November 2021 · Accepted: 22 November 2021 · View Online Export Citation CrossMark Published Online: 11 January 2022 Irene Arias,1,2 Gustau Catalan,3,4 and Pradeep Sharma5,a) AFFILIATIONS 1 Laboratori de Càlcul Numèric (LaCàN), Universitat Politècnica de Catalunya (UPC), Campus Nord UPC-C2, E-08034 Barcelona, Spain 2 Centre Internacional de Mètodes Numèrics a l’Enginyeria (CIMNE), 08034 Barcelona, Spain 3 ICREA-Institut Català de Recerca i Estudis Avançats, Barcelona, Catalonia, Spain 4 ICN2-Institut Català de Nanociència i Nanotecnologia, Campus UAB, Barcelona, Catalonia, Spain 5 Departments of Mechanical Engineering, Physics, and the Materials Science and Engineering Program, University of Houston, Houston, Texas 77204, USA Note: This paper is part of the Special Topic on Trends in Flexoelectricity. a) Author to whom correspondence should be addressed: psharma@central.uh.edu INTRODUCTION electricity from mechanical pressure thus changes from being a Consider the following: writing (and reading) ferroelectric chemical one, focused on the synthesis of asymmetric materials, to memories without applying voltage;1–3 piezoelectric-like transduc- a structural one, focused on the generation of non-centrosymmetric deformations. tion without piezoelectric materials;4–6 bone remodeling7 and The flexoelectric effect tends to be rather weak for most mate- mammalian hearing;8,9 asymmetric fracture toughness;10 and bulk photovoltaic effects in non-polar materials.11 What do these appar- rials; in bulk ceramics, piezoelectricity wins over flexoelectricity in ently disparate phenomena have in common? Flexoelectricity. terms of the ability to convert mechanical stress into voltage. A Flexoelectricity is defined as a coupling between dielectric further breakthrough, however, came with the realization that at polarization and strain gradient. Although discovered in the late the nanoscale strain, gradients can be much larger than at the mac- 1960s,12–16 the field of flexoelectricity in solid materials lay rela- roscale, and, therefore, the importance of flexoelectricity grows tively dormant and understudied for decades and studies on flexoe- with miniaturization.19–24 With the coming of age of nanotechnol- lectricity mostly focused on liquid crystals.17 This situation changed ogy, the development of advanced materials characterization in the early 2000s, with a series of reports from Cross18 showing methods, and commensurate developments in theoretical and com- that the flexoelectric coefficient in materials with high dielectric putational materials science, we have seen an explosion of literature constants could have rather large flexoelectric coefficients — on flexoelectricity. Flexoelectricity has now permeated into sufficiently large to make piezoelectric-like transducers with decent wide-ranging topics: smart material design,25–28 sensors and performance using non-piezoelectric materials.18 actuators,29–31 MEMS/NEMS and memory devices,31,2,32 soft Piezoelectricity, the linear coupling between polarization and robotics,33 energy harvesting,34,11,35–37 2D materials and domain strain, is still the most direct, and generally stronger, form of elec- walls,6,38–40 and understanding of biological phenomena. Much of tromechanical coupling, and entire journals have been dedicated to these developments have been summarized in review articles.24,23,38 their study. Unfortunately, nature has not been very kind to us in Flexoelectricity started out as a proposed replacement of pie- terms of giving us many piezoelectric materials and, even worse, zoelectricity, with disadvantages in terms of magnitude that were the best ones we know tend to contain toxic lead. By contrast, flex- partially compensated by its universality and larger magnitude at oelectricity couples electrical polarization to strain gradients small scales. This “poor man’s piezoelectricity” status, however, is through a fourth order material property tensor. Symmetry tells us being reassessed. Not only can flexoelectricity be an adequate alter- that fourth order tensors are universal and thus all dielectrics native (or, in some cases, the only alternative) to piezoelectricity possess flexoelectricity. This is the key advantage of flexoelectricity. but it can also be a complement to piezoelectricity (the two are not All insulators (and, as recently discovered, also semiconductors), mutually exclusive), and, excitingly, it can generate novel physical will electrically polarize if subjected to non-uniform deformation, responses that would NOT be possible by piezoelectricity alone, and this is true also for biomaterials. The challenge of generating such as asymmetric mechanical responses, mechanical reading and J. Appl. Phys. 131, 020401 (2022); doi: 10.1063/5.0079319 131, 020401-1 Published under an exclusive license by AIP Publishing Journal of EDITORIAL scitation.org/journal/jap Applied Physics writing of ferroelectric domains, or bending-induced bulk photo- REFERENCES voltaic effects. By enabling new physical phenomena, flexoelectric- 1 K. D. N. Cordero-Edwards, A. Abdollahi, J. Sort, and G. Catalan, “Ferroelectrics ity emerges from under the shadow of piezoelectricity as an as smart mechanical materials,” Adv. Mater. 29(37), 1702210 (2017). exciting research subject in its own right; hence the title of this 2 H. Lu, C. W. Bark, D. E. De Los Ojos, J. Alcala, C. B. Eom, G. Catalan, and Editorial. A. Gruverman, “Mechanical writing of ferroelectric polarization,” Science 336(6077), 59–61 (2012). 3 C. Stefani, E. Langenberg, K. Cordero-Edwards, D. G. Schlom, G. Catalan, and SUMMARY OF AREAS COVERED N. Domingo, “Mechanical reading of ferroelectric polarization,” J. Appl. Phys. 130(7), 074103 (2021). The articles in the “Trends of Flexoelectricity” Special Topic 4 B. Chu, W. Zhu, N. Li, and L. E. Cross, “Flexoelectric composite—A new pros- Collection in Journal of Applied Physics reflect the diversity and pect for lead-free piezoelectrics,” Funct. Mater. Lett. 3(1), 79–81 (2010). the breadth of the emerging field. The collection covers a wide 5 R. Maranganti, N. D. Sharma, and P. Sharma, “Electromechanical coupling in variety of topics ranging from the application of flexoelectricity in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s func- flexible electronics,41 sensing,42 energy harvesting,43,44 semiconduc- tion solutions and embedded inclusions,” Phys. Rev. B 74(1), 014110 (2006). tors,45,46 to actuators utilized in structural health monitoring.47 6 A. Apte, K. Mozaffari, F. S. Samghabadi, J. A. Hachtel, L. Chang, S. Susarla, and We also see the prospects of using flexoelectricity for mechanical P. M. Ajayan, “2D electrets of ultrathin MoO2 with apparent piezoelectricity,” reading of memory devices3 and topics underpinning the theory of Adv. Mater. 32(24), 2000006 (2020). 7 flexoelectricity.48,49 An extensive and detailed overview of the F. Vasquez-Sancho, A. Abdollahi, D. Damjanovic, and G. Catalan, mathematical and computational modeling of flexoelectricity is “Flexoelectricity in bones,” Adv. Mater. 30(9), 1705316 (2018). presented in Ref. 50. 8 Q. Deng, F. Ahmadpoor, W. E. Brownell, and P. Sharma, “The collusion of flex- oelectricity and Hopf bifurcation in the hearing mechanism,” J. Mech. Phys. Possible approaches of enhancing flexoelectric properties are Solids 130, 245–261 (2019). discussed in polymers,51,52,44,53 ceramics,54–57 ferroelectrics,58–62 9 K. D. Breneman, W. E. Brownell, and A. R. D. Rabbitt, “Hair cell bundles: and two-dimensional (2D) materials.63,62 There is also an intrigu- Flexoelectric motors of the inner ear,” PLoS One 4(4), e5201 (2009). ing proposal regarding flexoelectricity in metals.42 The role of sur- 10 A. Abdollahi, C. Peco, D. Millán, M. Arroyo, G. Catalan, and I. Arias, faces on flexoelectricity is elaborated in Ref. 64 and the origin of “Fracture toughening and toughness asymmetry induced by flexoelectricity,” flexoelectricity from a quantum mechanics point of view is dis- Phys. Rev. B 92(9), 094101 (2015). cussed in Ref. 65. 11 M. M. Yang, D. J. Kim, and M. Alexe, “Flexo-photovoltaic effect,” Science 360(6391), 904–907 (2018). 12 R. B. Meyer, “Piezoelectric effects in liquid crystals,” Phys. Rev. Lett. 22(18), CONCLUDING REMARKS 918 (1969). 13 V. S. Mashkevich and K. B. Tolpygo, “Electrical, optical and elastic properties The “Trends of Flexoelectricity” Special Topic illustrates the of diamond type crystals,” Sov. Phys. JETP 5(3), 435–439 (1957). emancipation of flexoelectricity, with articles covering many of the 14 S. M. Kogan, “Piezoelectric effect during inhomogeneous deformation new possibilities enabled by this phenomenon. Current research and acoustic scattering of carriers in crystals,” Sov. Phys. Solid State 5(10) topics reflect the diversity in this field and range from theoretical (1964). developments, novel devices that exploit flexoelectricity, novel phys- 15 R. D. Mindlin, “Polarization gradient in elastic dielectrics,” Int. J. Solids Struct. 4(6), 637 (1968). ical phenomena, and investigations in soft matter and biology. It is 16 E. V. Bursian and Z. Oi, “Changes in curvature of a ferroelectric film due to a timely endeavor to collect advances in the finally flourishing field polarization,” Sov. Phys. Solid State USSR 10(5), 2069–2070 (1968). of flexoelectricity and its emergence from the shadows. 17 Flexoelectricity in Liquid Crystals Theory, Experiments and Applications, edited by A. Buka and N. Éber (World Scientific, 2013). 18 L. E. Cross, “Flexoelectric effects: Charge separation in insulating solids sub- ACKNOWLEDGMENTS jected to elastic strain gradients,” J. Mater. Sci. 41, 53 (2006). 19 J. Y. Fu, W. Zhu, N. Li, N. B. Smith, and L. Eric Cross, “Gradient scaling phe- We would like to thank the Journal of Applied Physics edito- nomenon in microsize flexoelectric piezoelectric composites,” Appl. Phys. Lett. rial staff (Brian Solis, Bronte Brecht) and Associate Editor Jiangyu 91(18), 182910 (2007). Li for both the idea of this Special Topic collection as well as their 20 D. Lee, A. Yoon, S. Y. Jang, J. G. Yoon, J. S. Chung, M. Kim, and T. W. Noh, facilitating this task. I.A. acknowledges Generalitat de Catalunya “Giant flexoelectric effect in ferroelectric epitaxial thin films,” Phys. Rev. Lett. (“ICREA Academia” award for excellence in research, and Grant 107(5), 057602 (2011). No. 2017-SGR-1278), the European Research Council (No. 21 G. Catalan, L. J. Sinnamon, and J. M. Gregg, “The effect of flexoelectricity on StG-679451), the Spanish Ministry of Economy and the dielectric properties of inhomogeneously strained ferroelectric thin films,” Competitiveness (No. RTI2018101662-B-I00), and the “Severo J. Phys.: Condens. Matter 16(13), 2253 (2004). Ochoa Programme for Centres of Excellence in R&D” (2019–2023) 22 M. S. Majdoub, P. Sharma, and T. Cagin, “Enhanced size-dependent piezoelec- under Grant No. CEX2018-000797-S funded by MCIN/AEI/ tricity and elasticity in nanostructures due to the flexoelectric effect,” Phys. Rev. B 77(12), 125424 (2008). 10.13039/501100011033. G.C. acknowledges Grant No. 23 S. Krichen and P. Sharma, “Flexoelectricity: A perspective on an unusual elec- PID2019-108573GB-C21 and the “Severo Ochoa Programme for tromechanical coupling,” J. Appl. Mech. 83(3) (2016). Centres of Excellence in R&D” (No. SEV-2017-0706) funded by 24 P. Zubko, G. Catalan, and A. K. Tagantsev, “Flexoelectric effect in solids,” MCIN/AEI/10.13039/501100011033. P.S. is funded by the Annu. Rev. Mater. Res. 43, 387–421 (2013). University of Houston. We would also like to thank Mrs. Kosar 25 J. Yvonnet, X. Chen, and P. Sharma, “Apparent flexoelectricity due to hetero- Mozaffari for help with the bibliography. geneous piezoelectricity,” J. Appl. Mech. 87(11), 111003 (2020). J. Appl. Phys. 131, 020401 (2022); doi: 10.1063/5.0079319 131, 020401-2 Published under an exclusive license by AIP Publishing Journal of EDITORIAL scitation.org/journal/jap Applied Physics 26 Q. Deng, L. Liu, and P. Sharma, “Electrets in soft materials: Nonlinearity, size 46 L. Sun, Z. Zhang, C. Gao, and C. Zhang, “Effect of flexoelectricity on piezo- effects, and giant electromechanical coupling,” Phys. Rev. E 90(1), 012603 (2014). tronic responses of a piezoelectric semiconductor bilayer,” J. Appl. Phys. 27 N. D. Sharma, R. Maranganti, and P. Sharma, “On the possibility of piezoelec- 129(24), 244102 (2021). tric nanocomposites without using piezoelectric materials,” J. Mech. Phys. Solids 47 C. Wei, Z. Wang, and W. Huang, “Performance of a flexoelectric actuator for 55(11), 2328 (2007). lamb wave excitation,” J. Appl. Phys. 129(3), 034902 (2021). 28 A. Mocci, J. Barceló-Mercader, D. Codony, and I. Arias, “Geometrically polar- 48 S. Sharma, D. Singh, R. Vaish, R. Kumar, and V. S. Chauhan, “Performance ized architected dielectrics with apparent piezoelectricity,” J. Mech. Phys. Solids indexes for flexoelectricity in transverse and longitudinal modes,” J. Appl. Phys. 157, 104643 (2021). 129(14), 145105 (2021). 29 A. Abdollahi and I. Arias, “Constructive and destructive interplay between pie- 49 H. Le Quang and Q.-C. He, “Compact explicit matrix representations of the zoelectricity and flexoelectricity in flexural sensors and actuators,” J. Appl. Mech. flexoelectric tensor and a graphic method for identifying all of its rotation and 82(12), 121003 (2015). reflection symmetries,” J. Appl. Phys. 129(24), 244103 (2021). 30 Z. Wang, X. X. Zhang, X. Wang, W. Yue, J. Li, J. Miao, and W. Zhu, “Giant 50 D. Codony, A. Mocci, J. Barcelo-Mercader, and I. Arias, “Mathematical and flexoelectric polarization in a micromachined ferroelectric diaphragm,” Adv. computational modeling of flexoelectricity,” J. Appl. Phys. (online 2021). Funct. Mater. 23(1), 124 (2013). 51 M. Saadeh, Y. Aceta, P. Frère, and B. Guiffard, “Enhancing flexoelectricity in 31 U. K. Bhaskar, N. Banerjee, A. Abdollahi, E. Solanas, G. Rijnders, and PEDOT:PSS polymer films with soft treatments,” J. Appl. Phys. 130(1), 014103 G. Catalan, “Flexoelectric MEMS: Towards an electromechanical strain diode,” (2021). Nanoscale 8(3), 1293–1298 (2016). 52 E. Lorenzett, K. S. Moreira, Y. A. S. d. Campo, D. Mehler, A. L. Devens, 32 U. K. Bhaskar, N. Banerjee, A. Abdollahi, Z. Wang, D. G. Schlom, G. Rijnders, M. A. Noras, and T. A. L. Burgo, “Flexoelectric characterization of dielectrics and G. Catalan, “A flexoelectric microelectromechanical system on silicon,” Nat. under tensile, compressive, and flexural loads by non-contact kelvin probe mea- Nanotechnol. 11(3), 263 (2016). surements,” J. Appl. Phys. 129(20), 204502 (2021). 33 M. Grasinger, K. Mozaffari, and P. Sharma, “Flexoelectricity in soft elastomers 53 X. Chen, J. Yvonnet, H. S. Park, and S. Yao, “Enhanced converse flexoelectric- and the molecular mechanisms underpinning the design and emergence of giant ity in piezoelectric composites by coupling topology optimization with homoge- flexoelectricity,” Proc. Natl. Acad. Sci. U.S.A. 118(21) (2021). nization,” J. Appl. Phys. 129(24), 245104 (2021). 34 L. Shu, S. Ke, L. Fei, W. Huang, Z. Wang, J. Gong, and G. Catalan, 54 M. Hahn, S. Trolier-McKinstry, and R. J. M, Jr., “Flexoelectric barium stron- “Photoflexoelectric effect in halide perovskites,” Nat. Mater. 19(6), 605–609 tium titanate (BST) hydrophones,” J. Appl. Phys. 129(6), 064504 (2021). (2020). 55 Z. Yu, Z. Wang, S. Shu, T. Tian, W. Huang, C. Li, S. Ke, and L. Shu, “Local 35 B. Wang, S. Yang, and P. Sharma, “Flexoelectricity as a universal mechanism structural heterogeneity induced large flexoelectricity in Sm-doped PMN-PT for energy harvesting from crumpling of thin sheets,” Phys. Rev. B 100(3), ceramics,” J. Appl. Phys. 129(17), 174103 (2021). 035438 (2019). 56 Y. L. Tang, Y. L. Zhu, M. J. Zou, Y. J. Wang, and X. L. Ma, “Coexisting mor- 36 Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, “Nanoscale flexoelectric photropic phase boundary and giant strain gradient in BiFeO3 films,” J. Appl. energy harvesting,” Int. J. Solids Struct. 51(18), 3218 (2014). Phys. 129(18), 184101 (2021). 37 R. Mbarki, N. Baccam, K. Dayal, and P. Sharma, “Piezoelectricity above the 57 D. Tian, Y. Hou, W. Zhou, and B. Chu, “Flexoelectric response of ferroelectric curie temperature? Combining flexoelectricity and functional grading to enable ceramics with reduced surface layer effect,” J. Appl. Phys. 129(19), 194103 (2021). high-temperature electromechanical coupling,” Appl. Phys. Lett. 104(12), 122904 58 X. Wen, G. Yang, Q. Ma, Y. Tian, X. Liu, D. Xue, Q. Deng, and S. Shen, (2014). “Flexoelectricity in compositionally graded Ba1–xSrxTiO3 ceramics,” J. Appl. 38 F. Ahmadpoor and P. Sharma, “Flexoelectricity in two-dimensional crystalline Phys. 130(7), 074102 (2021). and biological membranes,” Nanoscale 7(40), 16555–16570 (2015). 59 G. Lu, S. Li, X. Ding, J. Sun, and E. K. H. Salje, “Tip-induced flexoelectricity, 39 T. Dumitrică, C. M. Landis, and B. I. Yakobson, “Curvature-induced polariza- polar vortices, and magnetic moments in ferroelastic materials,” J. Appl. Phys. tion in carbon nanoshells,” Chem. Phys. Lett. 360, 182 (2002). 129(8), 084104 (2021). 40 60 K. Chu, B. K. Jang, J. H. Sung, Y. A. Shin, E. S. Lee, K. Song, and C. H. Yang, M. J. Zou, Y. L. Tang, Y. P. Feng, W. R. Geng, X. L. Ma, and Y. L. Zhu, “Enhancement of the anisotropic photocurrent in ferroelectric oxides by strain “Influence of flexoelectric effects on domain switching in ferroelectric films,” gradients,” Nat. Nanotechnol. 10(11), 972–979 (2015). J. Appl. Phys. 129(18), 184103 (2021). 41 61 Y. X. Liu, Y. Cai, Y. S. Zhang, X. Deng, N. Zhong, P. H. Xiang, and Y. Liu, R.-M. Niu, S. D. Moss, P. Finkel, X.-Z. Liao, and J. M. Cairney, C. G. Duan, “Van der Waals epitaxy for high-quality flexible VO2 film on mica “Atomic coordinates and polarization map around a pair of 12[011¯] dislocation substrate,” J. Appl. Phys. 130(2), 025301 (2021). cores produced by plastic deformation in relaxor ferroelectric PIN-PMN-PT,” 42 A. S. Yurkov and P. Yudin, “Flexoelectricity in metals,” J. Appl. Phys. 129(19), J. Appl. Phys. 129(23), 234101 (2021). 195108 (2021). 62 J. Li, W. Xiong, X. Huang, W. Chen, and Y. Zheng, “Phase field study on the 43 B. Javvaji, R. Zhang, X. Zhuang, and H. S. Park, “Flexoelectric electricity gen- effect of substrate elasticity on tip-force-induced domain switching in ferroelec- eration by crumpling graphene,” J. Appl. Phys. 129(22), 225107 (2021). tric thin films,” J. Appl. Phys. 129(24), 244105 (2021). 44 K. S. Moreira, E. Lorenzett, A. L. Devens, Y. A. S. d. Campo, D. Mehler, and 63 W. Hao, Z. Wu, X. Li, and Y. Pu, “Edge effect on flexoelectronic properties of janus T. A. L. Burgo, “Low-cost elastomer-based flexoelectric devices,” J. Appl. Phys. MoSSe nanoribbons: A first-principles study,” J. Appl. Phys. 129(18), 185101 (2021). 129(23), 234502 (2021). 64 C. A. Mizzi and L. D. Marks, “The role of surfaces in flexoelectricity,” J. Appl. 45 Y. Qu, F. Jin, and J. Yang, “Magnetically induced charge redistribution in the Phys. 129(22), 224102 (2021). bending of a composite beam with flexoelectric semiconductor and piezomag- 65 F. Zypman, “Quantum flexoelectric nanobending,” J. Appl. Phys. 129(19), netic dielectric layers,” J. Appl. Phys. 129(6), 064503 (2021). 194305 (2021). J. Appl. Phys. 131, 020401 (2022); doi: 10.1063/5.0079319 131, 020401-3 Published under an exclusive license by AIP Publishing