designing electromechanical metamaterial with full nonzero piezoelectric … · for relatively...

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1Department of Materials Science and Engineering, College of Engineering, PekingUniversity, Beijing 100871, China. 2State Key Laboratory of New Ceramics and FineProcessing, School of Materials Science and Engineering, Tsinghua University, Beijing100084, China. 3Beijing Key Laboratory for Magnetoelectric Materials and Devices(BKL-MEMD), Peking University, Beijing 100871, China.*Corresponding author. Email: [email protected]

Yang et al., Sci. Adv. 2019;5 : eaax1782 8 November 2019

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Designing electromechanical metamaterial with fullnonzero piezoelectric coefficientsJikun Yang1, Zhanmiao Li1, Xudong Xin1, Xiangyu Gao1, Xiaoting Yuan1, Zehuan Wang1,Zhonghui Yu1, Xiaohui Wang2, Ji Zhou2, Shuxiang Dong1,3*

Designing topological and geometrical structures with extended unnatural parameters (negative, near-zero,ultrahigh, or tunable) and counterintuitive properties is a big challenge in the field of metamaterials, especiallyfor relatively unexplored materials with multiphysics coupling effects. For natural piezoelectric ceramics, onlyfive nonzero elements in the piezoelectric matrix exist, which has impeded the design and application of pi-ezoelectric devices for decades. Here, we introduce a methodology, inspired by quasi-symmetry breaking, re-alizing artificial anisotropy by metamaterial design to excite all the nonzero elements in contrast to zero valuesin natural materials. By elaborately programming topological structures and geometrical dimensions of the unitelements, we demonstrate, theoretically and experimentally, that tunable nonzero or ultrahigh values of overalleffective piezoelectric coefficients can be obtained. While this work focuses on generating piezoelectric param-eters of ceramics, the design principle should be inspirational to create unnatural apparent properties of othermultiphysics coupling metamaterials.

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INTRODUCTIONMetamaterials are a class of engineered systems that aim at achievingexotic macroscopic physical properties by designedly organized sub-units in the critical dimensions to form, most notably, equivalentparameter values beyond nature (1–8). Inspired by the theoretical pre-diction of left-handed materials from Maxwell’s equations (9), initialresearch studies focused on how to realize materials with negative per-mittivity (10) and permeability (11). Later experimental verification ofnegative refractive index enlightens scientists to craft man-madestructures with subwavelength elements for manipulating wave propa-gation (3, 8, 12), which has been widely applied in designing invisiblecloaks (13,14) andperfect lenses (15), and in transformationoptics (16,17).Compared with photonic crystals, the local physical inhomogeneityin electromagnetic metamaterials, whose unit cell dimensions aremuch smaller than manipulated wavelength, can be disregarded sothat physically equivalent parameters can be defined (1, 2). Besideselectromagnetism, the methodology of achieving pivotal effective indi-ces for targeting metaproperties has been the core concept extended toacoustics (4, 5), mechanics (6, 18), and thermodynamics (7). Whilemetamaterials feature single exotic properties, the strong correlationbetween specific geometries and active ranges restricts tunability andreal-world applications, resulting in tough problems such as narrowbandwidth and high loss in electromagnetic metamaterials (19, 20).To pave the way for practical devices, tunable parameters, and unitelements sensitive to applied external strain, electromagnetic field orother physical fields induced by multiphysics coupling effects areexpected to be introduced to form smart metamaterials (21, 22).

Multifarious complexmethods have been continuously developed tofabricate applicable metamaterials usually requiring topological andgeometrical orderliness on several different scales (23, 24). However,until now, these artificial processing techniques still lack universality,are expensive, and are unable to imitate multiphysical properties

comparable to naturally occurringmaterials. Perfectly manufacturedmetamaterials with both controllable numerous subwavelength unitsand applicable three-dimensional large-volume structures is an eventough challenge (19), while it is necessary for practical macro-sized de-vices (25). To further promote the fabrication feasibility, functional scal-ability, and smart tunability of new metamaterials, designing artificialunit cells with relative easily acquired topological structures, fully usingintrinsic properties of natural materials, and introducing multiphysicscoupling effects should be combined together for thoroughly updatingtraditional solely functional metamaterials and leading to apparentproperties with tunable parameters (21). Guided by this methodology,we propose a systematic strategy to obtain a novel kind of piezoelectricmetamaterials, whose exotic parameters are derived from structural de-signs, while their unit elements are macro-sized piezoelectric ceramicswith naturally occurring electromechanical coupling effects.

Piezoelectric elements were initially used as the electromechan-ical resonators to form bandgap in locally resonant metamaterials(26, 27). Nowadays, much attention has been focused on piezo-electric metaproperties, in which the piezoelectric response arisesfrom the orderly topological and geometrical structures rather thanthe materials’ chemical composition (18, 28, 29). In the field of piezo-electric metamaterials, exotic engineered piezoelectric constants are thecore targeted parameters to form anomalous properties; for instance,enormous apparent piezoelectric response with an effective piezo-electric coefficient d33 > 3500 pC/N was detected in a lead-free ceramicmetamaterial with designed gradient-generating structures (29). In gen-eral, piezoelectric ceramics are the most widely used electromechanicalmaterials because of their excellent mechanical properties, high Curietemperature, easy preparation process, and economically friendlyperformance (30–34).However, polarized piezoelectric ceramics featurea macroscopic transversely isotropic symmetry similar to 6-mm pointgroup belonging to the hexagonal crystal family (Fig. 1A) (35). Thissymmetry leads to only five naturally occurring nonzero piezoelectricstrain coefficients (which quantifies the ability to transform appliedelectric field into strain), namely, d33, d31 (d32), and d15 (d24), imposingrestriction to the study and application of electromechanical devices fordecades. Crystals with specific point groups have more practical coeffi-cients, like d36, but are still limited (36). Efforts have been devoted to

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achieve better indices, but these few sporadic results did poorly onportability and sustainability because of a lack of clear methodology(30, 31, 37, 38). Meanwhile, the possibility of the remaining zero piezo-electric coefficients has always been ignored for decades, which will firstbe focused and revised in our work.

In this study, full 13 artificial nonzero elements in the piezo-electric strain matrix and the corresponding deformation modes areunprecedentedly excited in their entirety based on 5 natural nonzeroones of ceramics by metamaterial design (see Fig. 1). Some effectivepiezoelectric coefficients exhibit tunable and greatly enhanced valuesby an order ofmagnitude than already existent ones, being even biggerthan the highest natural ones in single crystals (effective d11 of a kindof commercialized PZT-5H ceramics is designed and measured toreach 13,592 pm/V). Composed of above novel metamaterials as unitelements, arrayed electromechanical metamaterials with enhanced20-fold displacement, first-time realized co-fired multilayer shear-mode actuators are designed. These results demonstrate that the piezo-electric metamaterials show great potential in improving traditionalquasi-static or resonant transducers and enlighten brand-new piezo-electric technologies.

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RESULTS AND DISCUSSIONPiezoelectric metamaterial designHere, we show a systematic methodology to excite all nonzero effectivepiezoelectric coefficients in both resonant and quasi-static frequenciesby an electromechanical metamaterial design (Fig. 1B). The piezo-electric matrix of dij in natural ceramics is presented as

½dij�T ¼00d31

00d32

00d33

0d240

d1500

000

24

35T

ði ¼ 1; 2; 3; j ¼ 1; 2; 3; 4; 5; 6Þ

ð1Þ

The longitudinally and transversally extensional modes (d31, d32, andd33) manifest as nonzero normal strain with external electric fieldparallel to polarization direction, while d15 (or d24) is the only nonzeroshearmode with orthogonal orientations. To excite other basic nonzerodij values, there is no choice but to design metamaterials based on thefive natural modes.

To create nonzero piezoelectric strain coefficients dij, our strategystarts by analyzing the requirements of applied electric field (along withidirection) aswell as objective stain type (normal strainwith j=1, 2, and3; shear strain with j = 4, 5, and 6) to determine the adoptive basicnatural modes (selecting d31, d32, d15, d24, or d33) and topologicalstructures (arrangement ways of meta-atoms) based on “center extru-


sion effect” (CEE; which is developed for normal strain; see details be-low) and “diagonal transformation effect” (DTE; which is developed forshear strain). It continues by dividing piezoelectric metamaterials intolattice-like subunits with programmed polarization directions and elec-trodes as locally deformed elements (meta-atoms) to induce anisotropicstrain and, consequently, to excite desired modes. The artificial anisot-ropy actually achieves equivalently reduced symmetry. Finite elementsimulation, theoretical analyses, and experimental measurement jointlyprove the validity of our methodology. The design details fall into twocategories: normal-strain and shear-strain modes.

Normal-strain (j = 1, 2, and 3) metamaterials exhibit contraction orextension deformation. The topological structures for apparent normal-strain modes with effective piezoelectric coefficients d11 (d22), d13 (d23),and d12 (d21) are illustrated in Fig. 2 (A to C), whose detailed designs areexpounded in section S1. Generally speaking, the naturally occurringd15 (or d24) shear mode is stimulated in each meta-atom, functioningas local elements. Head-to-head polarization or electric fields areprogrammed so that these subunits synergistically extrude one anotherto excite an objective orthogonal displacement. Typically, for nonzerod12 (d21) coefficients, stimulation of “2” direction normal strain is quitetough because d15 deformation is totally in perpendicular “13” plane.However, by reusing “extrusion effect,” an extruded strain alongwith2 direction is created with the aid of the positive Poisson’s ratio ofpiezoelectric ceramics.

The topological structures of each normal-strain metamaterial be-have like arm force rod under external stress, and their mechanism isnamedCEE (Fig. 2H, inset). The principle of center extrusion structuresis illustrated in Fig. 2H, and a displacement output along the “3” axisemerges as a joint result of eachmeta-atom.The biggest output acquiredin the metamaterial center is regarded as effective displacement Deff

with values theoretically predicted as

Deff ¼ d15El2t

ð2Þ

where l and t are the length and thickness of a cuboid specimen, re-spectively, and E denotes the applied electric field.

Shear strain (when j = 4, 5, and 6) is an in-plane deformation man-ifesting as transformative rhombic angles. The topological designs forapparent shear modes with effective piezoelectric coefficients d14 (d25),d16 (d26), d34 (d35), and d36 are illustrated in Fig. 2 (D to G) and ex-pounded in detail in section S2. In general, programmed meta-atomsof metamaterials jointly excite anisotropic strain along with twoorthogonal diagonals (extension and contraction, respectively) and re-sult in apparent shear-like deformation. The basic mode of meta-atomsfor coefficient d1j (or d2j) is the d15 (or d24) shear mode, while the d33

Fig. 1. Piezoelectric strainmatrixwith full nonzero elements dijbymetamaterial design in contrastwith only five nonzero ones in natural piezoelectric ceramics.(A) Because of similar 6-mm symmetry of natural piezoelectric ceramics, only five nonzero elements in the piezoelectric strain matrix of natural piezoelectric ceramics exist,namely, d31 (equal to d32), d33, and d15 (equal to d24), with the other 13 elements being totally zero. (B) In this work, the 13 nonzero effective piezoelectric coefficients arecreated by metamaterial design. The method actually achieves macroscopically quasi-symmetry breaking and obtains apparently reduced symmetry.

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mode is adopted for coefficient d34 (d35) and the d31mode is adopted forcoefficient d36.

The mechanism for all shear metamaterials is summarized as DTE,which imitates fork-type structures with rhombic distortion drivenalong diagonals (Fig. 2I, inset). In theory, combination of extensionand contraction along with vertical diagonals is supposed to beequivalent to a pure shear deformation (see detailed analyses inMaterials and Methods). As illustrated in Fig. 2I, for a cube-shapedspecimen, the strain tensor in the prime coordinates (1′2′3′) is

S0 ¼s 0 00 0 00 0 �s

24

35 ð3Þ

Transformed equivalent strain tensor Seff in the original coordinates(123) is presented as

Seff ¼ TS0T�1 ¼

0 0 s0 0 0s 0 0

24

35 ¼ S5 ð4Þ


where T is the transformation matrix. The off-diagonal element S13 isequal to S31, indicating a pure shear strain S5 stimulated equivalently.In consideration of feasibility of polarization and electrode fabrication,only a pair contractive (or extensive) strain is provided along one di-agonal in some metamaterials. However, they still exhibit apparentlyquasi-shear distortion because another diagonal extension can arisefrom the positive Poisson’s ratio of the contraction, just with someextra volume contraction. These metamaterials function in both reso-nant and quasi-static states, with shear strain in resonance frequencybeing actually more ideally pure because of the equipartition of energyin this situation.

With naturally occurring ceramics programmed as unit elements,piezoelectricity is introduced as coupling effects, and a new class ofmetamaterials, piezoelectric metamaterials, is obtained. To clarify thebasic science principles and design methodology, we simplify meta-materials to several meta-atoms. Metamaterials composed of multipleunit elements can also be easily designed (see section S3). However,more meta-atoms usually bring problems such as complex fabricationprocesses and worse device reliability. Our methodology contributes tofull piezoelectric coefficients and practical applications in the situationof only several subunits.

Fig. 2. Schematic designs of piezoelectric metamaterials. With programmed polarization and applied electric field of subunits, the metamaterials realize all effectivenormal or shear-strain modes in both quasi-static and resonant frequencies. (A to G) Schematic metamaterial designs and deformation states by finite elementsimulation in both resonant (with ▲ label) and quasi-static (without ▲ label) states of d11 (d22) mode (A), d13 (d23) mode (B), d12 (d21) mode (C), d14 (d16) mode(D), d16 (d26) mode (E), d34 (d35) mode (F), and d36 mode (G). (H and I) Diagrams of two kinds of fundamental design mechanism learned from natural structureswe established, namely, CEE for effective normal strain (H) and DTE for shear strain (I).

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Investigations on effective values ofpiezoelectric coefficientsThe effective values of artificial coefficients mainly depend on twofactors: (i) topological structures and geometrical sizes of metamaterialsand (ii) the intrinsic electromechanical coupling strength of piezo-electric ceramics. Moreover, the measured or simulated values are alsorelated to electric or mechanical boundary conditions. To verify non-zero piezoelectric coefficients, we adopted finite element method(FEM; by COMSOLMultiphysics) with theoretical analyses and exper-imental results to investigate their magnitudes and variation tendencyalong with geometric dimensions. A kind of relaxor ferroelectrics,PNN-PT (31), and commercialized PZT-5H and PZT-8 are selectedas sample ceramics, and the simulation results are illustrated in Fig. 3.The definition formulas, boundary conditions, and computational de-tails of all artificial coefficients are expatiated inMaterials andMethods,section S4, and table S1. It is obvious thatmetamaterials feature not onlyoverall nonzero but also some tunable and ultrahigh piezoelectric coef-ficients relevant to geometrical dimensions.

Effective piezoelectric coefficients of several normal-strain metama-terials are theoretically predicted as

deffij ¼ d15l2t

ðij ¼ 11; 12; 13; 23Þ ð5Þ

which indicates feasibly tunable values inwide range by changing LTR(length-to-thickness ratio) of specimens (Fig. 3B). Attributed to the


uniform deformation of their topological structures, simulatedvalues match theoretical predictions well. To our best knowledge,the largest piezoelectric strain coefficient of single crystals is reportedto be 7500 pm/V (37), while these electromechanical metamaterialscan be easily tuned and feature much bigger apparent values (above10,000 pm/V) without any additional amplification mechanism. Theirremarkable outputs show great potential for high-performance actuators.

For the d12 (d21) mode, simulation results denote a positive nonlinearcorrelation between the nominal coefficients and LTR (Fig. 3C). Theprecisely analytic solution is hard to be obtained because of the aniso-tropic strain state from extrusion effects. Coefficients of shear modedeffij(ij = 14, 25, 16, 26) increase with the thickness-to-length ratio (TLR)with similar nonlinearity from extrusion deformation (Fig. 3D). Thed34 (d35) shear mode shows steady effective coefficients (Fig. 3E), andd36 has relatively linear increasing values proportional to TLR in a widerange (Fig. 3F). Compared with other shear modes, these unique onesderived by subunits of the d33 or d31mode feature profound advantages.With polarization and applied electric fields being in parallel direction,the electrodes can be one-step fabricated and a relatively higheroperating voltage can be offered without domain switching problem.Further design will demonstrate that these technical improvementsare of great importance in co-fired electromechanical devices.

To experimentally verify the actual performance of piezoelectricmetamaterials, we fabricated and investigated all seven types of piezo-electric metamaterials composed of commercialized PZT-5H ceramics,which are discussed in detail in Materials and Methods and section S5.

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Fig. 3. Magnitudes and variation tendency of effective piezoelectric coefficients of diverse metamaterials by FEM simulation. (A) Geometrical diagram and boundaryconditions of metamaterials for FEM simulation. For normal-strain modes, two side surfaces parallel to objective strain are fixed as zero-displacement boundary. Centerdisplacement along thickness direction behaves as effective output (with “*” label). While in shear-strain modes, a side surface is fixed and metamaterials show apparent sheardeformation. Effective piezoelectric coefficients are calculated based on the motion point (a vertex “*”). (B to F) PNN-PZT, PZT-5H, and PZT-8 chosen as demoed materials; themagnitudes and corresponding variation tendencyof artificial piezoelectric coefficients alongwithgeometry sizes are revealedby finite element simulation,which ared11 (d22, d13,and d23) (B), d12 (d21) (C), d14 (d16) and d16 (d26) (D), d34 (d35) (E), and d36 (F) modes.

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The preparation processes mainly consist of bonding meta-atomswith prepolarization by structural adhesives and coating electrodesby screen printing, and photographs of all metamaterials are shownin fig. S3. Table 1 lists all specimen sizes. Under the condition of steadymechanical boundary, the displacement responses of metamaterials toapplied AC voltages are characterized by a high-precision laserfeedback interferometer and then automatically recorded by a data ac-quisition system (LabVIEW myDAQ) so that the correspondingnominal piezoelectric coefficients can be calculated (see Fig. 4A).The results of two typical metamaterials with the d11 mode (on behalfof normal strain and CEE) and the d36 mode (on behalf of shear strainand DTE) are shown in Fig. 4. Figure 4B presents the measureddisplacement outputs of the d11 mode metamaterial (with LTR = 40)under different driving voltages, where the responses exhibit lownoise, high controllability, and stable repeatability. It is obvious fromFig. 4C that themeasured displacements are comparable to simulationones in the voltage range of 0 to 250 V/mm. The reasonable disparitybetween experimental and analytical data is assigned to technologicalgaps such as full polarization, bonding junction loss, and ideal boundaryconditions.Moreover, when electric field turns stronger than 250 V/mm,the specimen shows unprecedented giant outputs, which are evenhigher than simulation ones and reach around ±4.5 mm under anelectric field of ±400 V/mm. This interesting phenomenon may attri-bute to the additional out-plane shear-bending amplification effects(39), leading to further enhanced displacement. The experimental out-puts below 250 V/mm are fitted, and piezoelectric coefficient d11 iscalculated to be 13,592 pm/V, which is close to the theoretical predic-tion of 16,000 pm/V and nearly 20-fold bigger than natural d33 valuesof PZT-5H (700 pm/V).

Figure 4 (D and E) shows the transverse displacement responses of aside vertex of the d36 shear-modemetamaterial. This accordant effectiveoutput by simulation and experiments demonstrates the viability of ourfresh methodology to generate shear-like deformation based onnormal-strainmodes. In practical device applications, the nonlinear ex-trusion effect of d36 or other coefficients may lead to some stress con-centration inmetamaterial bodies especially at interfaces. However, thisphenomenon can be diminished or avoided effectively by creating stressrelease holes [likemultilayer structures in classical piezoelectric technol-ogies (40)], using ductile glues or modifying topological configurations.

The simulated and experimental displacement responses of all otherkinds of metamaterials are shown in fig. S4, and their corresponding


piezoelectric coefficients are listed in Table 1. It is demonstrated that fullnonzero elements of the piezoelectric strain matrix are successfully cre-ated bymetamaterial design.Moreover, some artificial parameters showultrahigh values (like d11 and d13) in comparison with naturally existingones. The experimental results of several coefficients are even muchbigger than the simulated ones (like d12 and d14), which should originatefrom a very strong out-plane (for normal-strainmodes) or in-plane (forshear-strain modes) shear-bending amplification effect of thesetopological structures. The unexpected interesting finding shows thatthe nonlinear extrusion effect may further enhance the performanceof piezoelectric metamaterials in practice.

Potential applications of piezoelectric metamaterialsTo take full advantage of piezoelectric materials for electromechanicalapplications such as actuators, ultrasonic motors, sensors, or energyharvesters, scientists used propermodes to form other diversiform con-figurations. The novel metamaterials bring unprecedented full nonzeroelements to the piezoelectric matrix; moreover, their tunable or ultra-high nature promises to broaden designs of devices, improve their prop-erties, and enlighten brand-new piezoelectric technologies. Here, weshow two potential applications.

Arrayed and enhanced electromechanical metamaterialsUsing metamaterials as unit elements, interesting arrayed electro-mechanical metamaterials can be further constructed through our basicarrangementmethodologies (CEE andDTE) and exhibit enhanced andcompounded extension, contraction, shear, twist, or other programmeddeformation (section S6).

Higher effective piezoelectric constants contribute to bigger appar-ent displacement and lower drive voltage, bringing in benefit to minia-turization and integration ofMEMS (micro-electromechanical systems)driving modules. As illustrated in Fig. 5 (A to C), on the basis of thegiant d11 normal-strain mode (CEE) and arrangement of generalizedDTE, a novel multilayer metamaterial is designed, which exhibits re-markable effective strain (0.4%) and very large apparent displacement(more than 40 mm) under very small applied electric field (200 V/mmfor each layer) (see Materials and Methods and section S6 for FEMcalculation details). These values are 20-fold bigger than traditionald33-derived multilayer piezoelectric actuators of the same geometricalsize and driving voltage. The apparent performance can be mediated asrequired or further optimized by increasing LTR of each layer.

Table 1. Simulated prediction and experimental results of effective piezoelectric coefficients of all seven types of metamaterials based on a kind ofcommercialized PZT-5H ceramics. Several experimental results are even bigger than the simulated ones because of an additional shear-bending amplification effect.

Neotype modes
Specimen sizes
(length × width × thickness)
Simulation values
(pm/V)
Experiment values
(pm/V)

d11 (LTR = 40)
20 mm × 7 mm × 0.5 mm 16,000 13,592
d13 (LTR = 28)
14 mm × 7 mm × 0.5 mm 11,200 6,984
d12 (LTR = 14)
14 mm × 14 mm × 1 mm 200 2,205
d14 (TLR = 0.1)
10 mm × 10 mm × 1 mm 28 64
d16 (TLR = 0.1)
10 mm × 10 mm × 1 mm 22 37
d34 (TLR = 0.1)
10 mm × 10 mm × 1 mm 531 714
d36 (TLR = 1/14)
14 mm × 14 mm × 1 mm 372 259
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Co-firing multilayer shear-mode actuatorsThe co-fired multilayer structure is an efficient method to manufactureintegrated andmultifunctional devices such as actuators andmultilayerceramics capacitors. Since tape casting and co-fired technologies wereinvented decades ago, shear-modemultilayer actuators have never beensuccessfully fabricated using these methods because the polarizationand applied electric field are in orthogonal directions, and electrodesrequire polishing and repreparation after polarization.However, our pi-ezoelectric metamaterials promise to first solve the puzzling problemsby d34 (d35) or d36 normal-strain–derived shearmodes. Figure 5 (D to F)and section S7 demonstrate multilayer d36 shear-mode ceramic actua-


tors. Surfaces of every other layer are partitioned into four squares, withtheir electrodes coated; the remaining surfaces function as groundingelectrodes, forming interdigital structures. After squares in each layerpolarized along the same direction, one diagonal positive and anotherdiagonal negative driving voltages will stimulate in-plane d36 shear de-formation and will eventually obtain shear-mode multilayer actuators.Conventional d36 face-shear actuators (35) usually require an expensivecrystal resonator, while the designed novel co-firing ones based on ce-ramic metamaterials feature low operating voltage, higher mechanicalstrength, lower cost, and still comparable precise positioning, promisingto motivate research in brand-newminiaturized actuating devices such

Fig. 4. Experimental verification of metamaterials with effective piezoelectric coefficients d11 (on behalf of normal strain and CEE) and d36 (on behalf of shearstrain and DTE). (A) Schematic diagram of test system for measuring the displacement performance of metamaterials. The system consists of a driving signal moduleand a high-precision displacement measuring module. (B and C) Real-time displacement responses (B) and amplitudes (C) of d11 mode metamaterial (with LTR = 40)under sinusoidal driving signals with diverse voltages. (D and E) Real-time displacement responses (D) and amplitudes (E) of d36 mode metamaterial (with TLR = 1/14).Photo credit: Jikun Yang, Department of Materials Science and Engineering, College of Engineering, Peking University, Beijing 100871, China.

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as nanoshear actuators, piezoelectrically walked step motors, and ultra-sonic motors in the microelectronics industry.

Besides inspiring brand-new piezoelectric inventions, electro-mechanical metamaterials with full nonzero coefficients will hopefullyrenovate and enhance traditional piezoelectric devices. Previouslynondispersive shear horizontal waves used in damage detection aregenerated by the d15 mode, while novel artificial modes with mildervoltage may be more competitive (35). A kind of shear-bending actua-tors reported by us before can also be classified into the applicationexamples of d11 mode metamaterials (39). Serving as addictive tunableelements with abundant deformation modes, electromechanical meta-materials are able to introduce piezoelectric tunability to traditionalsmart metadevices (25).

CONCLUSION AND DISCUSSIONWe implement a metamaterial design to piezoelectric ceramics andreport a novel class of electromechanical metamaterials with fully un-natural nonzero piezoelectric coefficients, which promises to pro-foundly promote the fundamental research in piezoelectric materialsand technologies. Themethodology of CEE andDTE arises frombasicconcepts of condensed matter physics, including symmetry principle,finite element idea, and multiphysics coupling effects.

The broken symmetry, an extensive mechanism in physics, leads tomany charming phenomena such as antiferromagnetism and super-conductivity. Usually, novel broken symmetry is pretty tough to con-trollably build in the atomic level or micro-level, while our resultsdemonstrate that some equivalent effects may be architected throughmacroscopic metamaterial design, in which the finite meta-atomsfunction as locally equivalent elements to introduce artificial anisotropy.


Naturally nonexistent effective physical parameters may be induced byprogrammed quasi-symmetry breaking not only in polycrystallinematerials but also in amorphous materials or crystals. Like amorphousmetallic glass, a kind of magnetostriction material of 6-mm symmetry,overall nonzero 18 magnetostriction coefficients are supposed to be de-signed and generated, despite the hard control of the space-confinedmagnetic field.

Previousmetamaterials aim at single exotic function, while the com-bined effects derived from topological structures and intrinsic nature ofmetamolecules have been paid little attention, such as the novelelectromechanical metamaterial in this work. Further explorationshould aim at introducing multiphysical coupling effects to form smartmetamaterials, which hopefully feature properties such as tunability byexternal physical field or self-adaptability to external environment (forexample, pyroelectric metamaterials with controllable current and Kerreffect with tunable oriented refractive index).

MATERIALS AND METHODSMetamaterial design methodologyTowholly create nonzero piezoelectric coefficients dij, we developed twobasic topological deformation or arrangement methods, including CEE(for normal strain) and DTE (for shear strain), which contributes toartificial anisotropy and changes the apparent symmetry of 6-mmpointgroup. The normal-strain modes can be excited through extrusion ofmeta-atoms, while for shear-strain modes, artificial anisotropic strainalong different diagonals is the basic requirement for shear-like de-formation. For DTE, the combination of extension and contractionstrains along with diagonals can be demonstrated to be equivalent to apure shear strain in theory. As illustrated in Fig. 2I, for a square-shaped

Fig. 5. Designs of arrayed electromechanical metamaterials and brand-new co-firing shear-mode actuators. (A to C) Arrayed normal-strain metamaterial (length ×width × thickness: 2 cm × 2 cm × 1 cm; 20 layers, PNN-PZT) based on novel d11 mode elements and fork-type arrangement ways shows a very large apparent displacement(over 40mm). (D toF) The puzzling problemsof co-fired shear-modemultilayer structure are expected to be solved. On the basis of new fundamentald36 normal-strain–derivedshear-mode metamaterials and specific interdigital electrodes (D and E), multilayer shear-mode co-fired structures are designed, exciting perfect shear deformation (F).

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specimen, tensile strain s and compressive strain −s are applied simul-taneously along the (101) and (�101) orientations, respectively.Whenwetake 1′ and 3′ axes as the prime coordinates, the strain tensor is

S0 ¼

s 0 00 0 00 0 �s

24

35 ð6Þ

Using the transformation matrix (where parameter q = 45°)

T ¼cosq 0 sinq0 1 0

�sinq 0 cosq

24

35 ð7Þ

The transformed equivalent strain tensor Seff in the original coor-dinates (123) can be converted into

Seff ¼ TS0T�1 ¼

0 0 s0 0 0s 0 0

24

35 ¼ S5 ð8Þ

The off-diagonal components S13 and S31 have the same values s,which means that a pure shear strain S5 is generated equivalently.

In the situation where only compressive strain 3′ is produced insome metamaterial modes, the transformed equivalent strain tensorSeff is

Seff ¼ TS0T�1 ¼ T

0 0 00 0 00 0 �s

24

35T�1 ¼ 1

2

s 0 s0 0 0s 0 s

24

35

¼ � 12

0 0 s0 0 0s 0 0

24

35� 1

2

s 0 00 0 00 0 s

24

35 ð9Þ

The strain tensor in the original coordinates (123) is composed oftwo items: The first item is equivalent shear strain, and the second itemdenotes some extra volume contraction, which will not influence theapplication ofmetamaterials. The results prove that shear-like deforma-tion was successfully excited. To satisfy the preparation feasibility andconvenient real-world application, we usually adopted the design of on-ly contraction (or extension) along with one diagonal, and the other di-agonal deformation was automatically generated because of the positivePoisson’s effect.

Finite element simulationBasic electromechanical metamaterialsThe deformation of the basic 13 electromechanicalmetamaterials or thespecific values and variation tendency of nonzero piezoelectric coeffi-cients in Fig. 3 were simulated by the COMSOL code using the piezo-electric device module. The geometrical models of all modes were builtby parameterized coordinates and divided into unit elements accordingto the topological designs as shown in Fig. 2 and table S1. The ceramicmaterials PZT-5H and PZT-8 are the default materials in the materiallibraries, and PNN-PZT is a user-defined new material, the full set ofparameters of which is derived from our previous work (31). The dif-


ferent unit cells were polarized alongwith head-to-head or back-to-backdirections by establishing rotated coordinate systems and applied volt-ages to electrode surfaces by 200 V/mm (note that for some shearmodes, the electric field is average values; see section S4). With bound-ary conditions of partially fixed constraints and very detailedmesh gen-eration controlled by physical fields, the displacements and effectivecoefficients were simulated in the steady state. To calculate the effectivevalues of piezoelectric coefficients, centers of normal-strain modes andvertices of shear-strain modes were regarded as motion points. Thelength of shear-strain specimens for calculating effective piezoelectriccoefficients was assumed to be the original value without regard to con-traction. Parameterization function was used to investigate their varia-tion tendency with TLR or LTR. As a comparison to experimentalresults as shown in Fig. 4 and fig. S4, the simulated results were basedon default PZT-5H in the material libraries of COMSOL, while its d33,d31, and d15 values were altered to 700, −320, and 800 pm/V accordingto the commercialized PZT-5H we used.Arrayed and enhanced electromechanical metamaterialsThe arrayed and enhanced electromechanical metamaterials are themultilayer structures of the d11 mode. The geometrical model contains20 PNN-PZT ceramic layers, and each layer was connected by poly-methylmethacrylate (with thickness and length being both 1/10 of thick-ness of each PNN-PZT layer) in the material libraries. With sidesurfaces of the bottom layer fixed, the displacement output wascalculated.Co-firing multilayer shear-mode actuatorsThe simulation for shear-type multilayer actuators was similar with ar-rayed normal-strain metamaterials. Notably, the interdigital electrodesand ceramic layers with various polarization directions should be care-fully set up.With one side surface fixed, the shear-type deformation andstress distribution were simulated.

Experimental verificationMetamaterial preparationA kind of commercialized PZT-5H (with d33 coefficients being 650to 700 pm/V, and d31 and d15 coefficients being around −320 and800 pm/V, respectively; Suzhou PANT Piezoelectric TechnologyCorporation) plate was cut into meta-atoms with desired geometricsizes using a precision cuttingmachine (CNC-400,MTICorporation).After silver paste electrodes were screen-printed onto surfaces with apostfire at 650°C for 0.5 hour, these meta-atoms were polarized insilicone oil bath under a DC electric field of 10 kV cm−1 in 120°C for15 min. Later on, the initial electrodes were removed (not necessary forall modes), and meta-atoms were arranged and bonded together bystructural adhesives according to the specific topological structure de-sign. Newly designed electrodes were printed to apply the drive voltage,and wires were bonded onto the electrodes by low-temperature silverpaste without postfire procedures for further measurement.Parameter measurementThe piezoelectric strain coefficients of all seven kinds of metamaterialspecimens were measured using electric field–induced displacementmethod. The specimens were installed onto the three-dimensionalprinted fixture according to the desired boundary conditions. A sinus-oidal driving voltage was produced using a signal generator (TektronixAFG3022B) and amplified using a high-voltage amplifier (PINTEKHA-405 or Kepco BOP-1000M). The driving voltages were monitoredusing an oscilloscope (Keysight MSOX4024A), and the metamaterialdisplacement responses were measured using a high-precision laserfeedback interferometer (LeiCe LY1000) with nanometer resolution.

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SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/11/eaax1782/DC1Section S1. Designing piezoelectric metamaterials with normal-strain modeSection S2. Designing piezoelectric metamaterials with shear-strain modeSection S3. Piezoelectric metamaterials composed of multiple meta-atomsSection S4. FEM simulation detailsSection S5. Experimental verification detailsSection S6. Arrayed metamaterials with enhanced performanceSection S7. Design of neotype co-firing multilayer shear-mode actuatorsFig. S1. d34 mode piezoelectric metamaterials composed of various numbers of meta-atoms.Fig. S2. Specimen photographs of all seven kinds of piezoelectric metamaterials.Fig. S3. Test system for measuring effective displacement outputs of metamaterials.Fig. S4. Displacement responses to electric field of other five kinds of metamaterials.Fig. S5. Arrayed shear-mode metamaterials with in-plane arrangement way.Fig. S6. Preparation processes of neotype co-firing shear-mode multilayer actuator.Table S1. Schematic diagrams and definition formulas for calculating all effective piezoelectriccoefficients.

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Acknowledgments: We appreciate the initial suggestion on metamaterial research fromL. Li. Funding: This work was supported by the National Natural Science Foundationof China (grant nos. 51772005 and 51072003) and the Beijing Key Laboratory forMagnetoeletric Materials and Devices. Author contributions: J.Y. designed and fabricatedpiezoelectric metamaterials, performed the experiments and FEM simulations, and analyzedthe data under the guidance of S.D. J.Y. and X.X prepared specimens. J.Y., Z.L., and X.G.conducted the experimental verification. Z.W., X.Y., and Z.Y. helped to optimize deviceapplications with J.Y. X.W. and J.Z. provided the necessary guidance for experiments. Themanuscript was written by J.Y. and finally modified by S.D. S.D. provided guidance during allstages. Competing interests: The authors declare that they have no competing interests.Data and materials availability: All data needed to evaluate the conclusions in the paperare presented in the paper and/or the Supplementary Materials. Additional data related to thispaper may be requested from the authors.

Submitted 1 March 2019Accepted 17 September 2019Published 8 November 201910.1126/sciadv.aax1782

Citation: J. Yang, Z. Li, X. Xin, X. Gao, X. Yuan, Z. Wang, Z. Yu, X. Wang, J. Zhou, S. Dong,Designing electromechanical metamaterial with full nonzero piezoelectric coefficients. Sci.Adv. 5, eaax1782 (2019).

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Designing electromechanical metamaterial with full nonzero piezoelectric coefficients

Shuxiang DongJikun Yang, Zhanmiao Li, Xudong Xin, Xiangyu Gao, Xiaoting Yuan, Zehuan Wang, Zhonghui Yu, Xiaohui Wang, Ji Zhou and

DOI: 10.1126/sciadv.aax1782 (11), eaax1782.5Sci Adv

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