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Yuanhao Wei, Yuhao Li, Hanhao Zhang, Shengsheng Lin, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Cun-Fa Gao, [Yan Shi](https://orcid.org/0000-0001-7421-3306)

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[Co-Dominant Piezoelectric and Flexoelectric Effects in Twisted Double Bilayer Graphene](https://mdr.nims.go.jp/datasets/57e7f523-f5ef-4041-b9de-e5b4f3fffd79)

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Co-Dominant Piezoelectric and Flexoelectric Effects in Twisted Double Bilayer GrapheneCitation: Wei, Y.; Li, Y.; Zhang, H.;Lin, S.; Taniguchi, T.; Watanabe, K.;Gao, C.-F.; Shi, Y. Co-DominantPiezoelectric and Flexoelectric Effectsin Twisted Double Bilayer Graphene.Symmetry 2024, 16, 1524. https://doi.org/10.3390/sym16111524Received: 15 October 2024Revised: 8 November 2024Accepted: 11 November 2024Published: 14 November 2024Copyright: © 2024 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).ArticleCo-Dominant Piezoelectric and Flexoelectric Effects in TwistedDouble Bilayer GrapheneYuanhao Wei 1,†, Yuhao Li 2,3,*,†, Hanhao Zhang 2, Shengsheng Lin 2, Takashi Taniguchi 4, Kenji Watanabe 5 ,Cun-Fa Gao 1 and Yan Shi 1,*1 State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronauticsand Astronautics, Nanjing 210016, China2 National Laboratory of Solid-State Microstructures, School of Electronic Science and Engineering andCollaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China3 National Key Laboratory of Spintronics, Nanjing University, Suzhou 215163, China4 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan5 Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan* Correspondence: yuhao@nju.edu.cn (Y.L.); yshi@nuaa.edu.cn (Y.S.)† These authors contributed equally to this work.Abstract: Controlling the balance between piezoelectric and flexoelectric effects is crucial for tailoringthe electromechanical responses of a material. In twisted graphene, it is found that the electromechan-ical response near the domain walls (DWs) is dominated by either the flexoelectric effect as in twistedbilayer graphene (tBLG) or the piezoelectric effect as in twisted monolayer–bilayer graphene (tMBG).The codominance of both effects in a single system is rare. Here, utilizing lateral piezoresponseforce microscopy (LPFM), we show that piezoelectric and flexoelectric effects can coexist and areequally important in twisted double bilayer graphene (tDBG), termed as the piezo-flexoelectric effect.Unlike tBLG and tMBG, distinctive two-step LPFM spatial profiles are captured across the moiréDWs of tDBG. By decomposing the LPFM signal into axisymmetric and antisymmetric components,we find that the angular dependence of both components satisfies sinusoidal relations. Quantitatively,the in-plane piezoelectric coefficient of DWs in tDBG is determined to be 0.15 pm/V by dual ACresonance tracking (DART) LPFM measurement. The conclusion is further supported by continuummechanics simulations. Our results demonstrate that the stacking configuration serves as a powerfultuning knob for modulating the electromechanical responses of twisted van der Waals materials.Keywords: lateral piezoresponse force microscopy; twisted double bilayer graphene; flexoelectriceffect; piezoelectric effect; piezo-flexoelectric effect1. IntroductionThe emergence of electric polarizations and enhanced electrical conductivity at DWsof non-polar materials can admit many new functional devices, such as diodes, memoriesand switches [1–3]. However, so far, the progress in developing these devices has beenslow, partially due to the incomplete understanding of the origins of the polarization andthe lack of an efficient tuning knob for the polarization. For many materials, investiga-tions on polarization are further complicated by the presence of defect states, multiplesources and other order parameters, such as the domain boundary and piezo-flexoelectriceffect [2,4]. Generally, the induced electrical polarization upon mechanical deformation canbe formulated by the constitutive equation [5,6] Pi = eijkε jk + µijkl∂ε jk∂xl, where eijk and µijklare the direct piezoelectric and flexoelectric coefficients; ε jk and∂ε jk∂xlare the strain and straingradient. The first term describes the piezoelectric effect, the second term corresponds tothe flexoelectric effect.Symmetry 2024, 16, 1524. https://doi.org/10.3390/sym16111524 https://www.mdpi.com/journal/symmetryhttps://doi.org/10.3390/sym16111524https://doi.org/10.3390/sym16111524https://creativecommons.org/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/symmetryhttps://www.mdpi.comhttps://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0001-7421-3306https://doi.org/10.3390/sym16111524https://www.mdpi.com/journal/symmetryhttps://www.mdpi.com/article/10.3390/sym16111524?type=check_update&version=1Symmetry 2024, 16, 1524 2 of 8Recently, twisted graphene moiré superlattices are reported to be a flexible and versa-tile platform for observing emergent quantum phenomena, such as superconductivity [7],correlated insulator [8], ferromagnetism [9,10] and even ferroelectricity [11,12]. In thesmall twist angle limit, alternating non-polar stacking domains emerge due to lattice re-construction, and the dislocations are concentrated on the boundaries between adjacentdomains, leading to clean DWs with remarkable strain and strain gradient [13]. The DWwidth is on the order of 10 nm, falling in the mesoscopic regime. This makes twistedgraphene an ideal platform for hosting piezoelectric and flexoelectric polarizations simulta-neously. In practice, previous works suggest the electromechanical responses of DWs aredominated by the flexoelectric effect [14–16] and the piezoelectric effect [16] for tBLG andtMBG, respectively, possibly due to symmetry constraints. Our previous work presentedthe background-signal-free LPFM signal of a tDBG sample [15]. However, the origin ofelectromechanical responses for tDBG are still poorly understood.In this paper, we focus on tDBG samples, revealing distinct LPFM profiles from tBLGand tMBG. Specifically, a two-step LPFM profile was captured with identical conductiveatomic force microscopy (cAFM) current changes. We propose that this two-step profile isattributed to the combination of an axisymmetric flexoelectric effect and an antisymmetricpiezoelectric effect which is consistent with our continuum mechanics simulations, andwe term this as the ‘piezo-flexoelectric effect’. The angular dependence of both compo-nents satisfies a sinusoidal relation, which is a typical behavior for in-plane polarization.Moreover, the in-plane piezoelectric coefficient of tDBG is determined to be 0.15 pm/V byDART LPFM.2. Results and DiscussionHere, two Bernal-stacked graphene bilayers twist at a small in-plane angle θ relative toeach other, where the least energetically favorable configuration is AB-BC stacking (red dot,labeled from top to bottom layers and hyphens for the twisted interface), forming a moirésuperlattice that maximizes the energetically favorable AB-AB (blue dot) and AB-CA (greendot) at the expense of AB-BC through lattice reconstruction [17], as shown in Figure 1a.The DWs are the ‘saddle point’ (SP) stacking (black lines in Figure 1a), also known as strainsoliton [18]. In our experiment, tDBG samples consist of tDBG upon a hexagonal boronnitride (h-BN) substrate. For some samples, large graphite flakes are contacted with tDBGas electrodes, and silver paste is used to connect the large graphite electrode to the externalcircuit for cAFM measurements. More details about sample fabrication can be found inFigure S1 and the Methods section. The principle of LPFM is illustrated in Figure 1b,which demonstrates that LPFM can only capture the component of in-plane polarizationperpendicular to the cantilever axis.Figure 1c,d show the intrinsic LPFM phase and amplitude maps of tDBG01 recoveredfrom the background signal [15,19] (raw LPFM results are presented in Figure S2), whereonly DW regions show nonzero amplitude. In Figure 1e, we show the cAFM current imageof the same area where the current changes at the center of the DWs. The shape of thedomains is also informative. The concave–convex alternant domains can be understoodas the consequence of competition between AB-AB and AB-CA stacking orders resultingin the expansion of AB-AB domains, showing a convex domain for AB-AB and a concavedomain for AB-CA [15,17]. Figure 1f shows the line cuts of LPFM phase, LPFM amplitudeand cAFM current along the arrows in Figure 1c–e with expected 180◦ phase change.However, the amplitude and its spatial profiles are very different from previouslyreported tBLG and tMBG [16]. First, the maximum amplitudes of tDBG DWs are notat the center of DWs, exhibiting two-step profile amplitude, while the amplitude of atMBG DW only gives one peak and tBLG shows two peaks. Meanwhile, the amplitude oftDBG DWs near AB-CA domains side are much higher than that near the AB-AB domainside, as marked by red and blue background in Figure 1f. This phenomenon can beunderstood by the enhancement and weakness of in-plane polarization on two sides of theDW. Second, the LPFM amplitude and cAFM current change at identical positions at theSymmetry 2024, 16, 1524 3 of 8DWs (dashed lines in Figure 1f). The cAFM measurement is highly sensitive to the localstacking of tDBG moiré superlattices [20–23], and an LPFM measurement can capture itsin-plane polarization. Hence, the dashed lines indicate the precise SP locations betweenthe AB-AB and AB-CA stackings, where electromechanical and electric properties changesimultaneously. Specifically, for DW1, the amplitude gives a drop from 75 pm to 50 pm,while the current presents an increase from 10.3 to 15.5 nA, as marked by black arrows. Theamplitudes of three tDBG DWs satisfy a sinusoidal relation (Figure 1g), which matches wellwith previous tMBG and tBLG results [14,16]. From the angle-dependence LPFM resultsof tDBG02 (see Figure S3), Figure 1h gives the in-plane polarization networks of tDBG01,which is consistent with previous tDBG results [15].Symmetry 2024, 16, x FOR PEER REVIEW 3 of 8    Figure 1. LPFM and cAFM mappings in tDBG sample (tDBG01). (a) Schematic of the tDBG moiré superlattice and relative locations of three domains, where red, green and blue dots represent AB-BC, AB-CA and AB-AB domains, respectively. (b) Principle of the LPFM measurement. (c,d) Phase and amplitude images of tDBG01. (e) Current image of the same area in (c,d). (f) Line cuts of the phase, amplitude and current for tDBG01 along black, red and green arrow lines and cross DW1, DW2 and DW3 in (c–e), respectively. (g) Sinusoidal fit of the three amplitudes vs. the cantilever axis angle relative to the DW. (h) Schematic of in-plane polarization networks for tDBG01 sample. Figure 1c,d show the intrinsic LPFM phase and amplitude maps of tDBG01 recovered from the background signal [15,19] (raw LPFM results are presented in Figure S2), where only DW regions show nonzero amplitude. In Figure 1e, we show the cAFM current image of the same area where the current changes at the center of the DWs. The shape of the domains is also informative. The concave–convex alternant domains can be understood as the consequence of competition between AB-AB and AB-CA stacking orders resulting in the expansion of AB-AB domains, showing a convex domain for AB-AB and a concave domain for AB-CA [15,17]. Figure 1f shows the line cuts of LPFM phase, LPFM amplitude and cAFM current along the arrows in Figure 1c–e with expected 180° phase change. However, the amplitude and its spatial profiles are very different from previously reported tBLG and tMBG [16]. First, the maximum amplitudes of tDBG DWs are not at the center of DWs, exhibiting two-step profile amplitude, while the amplitude of a tMBG DW only gives one peak and tBLG shows two peaks. Meanwhile, the amplitude of tDBG DWs near AB-CA domains side are much higher than that near the AB-AB domain side, as marked by red and blue background in Figure 1f. This phenomenon can be understood by the enhancement and weakness of in-plane polarization on two sides of the DW. Sec-ond, the LPFM amplitude and cAFM current change at identical positions at the DWs (dashed lines in Figure 1f). The cAFM measurement is highly sensitive to the local stack-ing of tDBG moiré superlattices [20–23], and an LPFM measurement can capture its in-plane polarization. Hence, the dashed lines indicate the precise SP locations between the AB-AB and AB-CA stackings, where electromechanical and electric properties change simultaneously. Specifically, for DW1, the amplitude gives a drop from 75 pm to 50 pm, while the current presents an increase from 10.3 to 15.5 nA, as marked by black arrows. The amplitudes of three tDBG DWs satisfy a sinusoidal relation (Figure 1g), which matches well with previous tMBG and tBLG results [14,16]. From the angle-dependence LPFM results of tDBG02 (see Figure S3), Figure 1h gives the in-plane polarization net-works of tDBG01, which is consistent with previous tDBG results [15]. This two-step profile phenomenon can be enhanced by adjusting measurement pa-rameters, e.g., drive frequency and cantilever-sample force. More details about the fre-quency-dependent LPFM amplitude of tDBG can be found in Figure S4. Figure 2a,b show Figure 1. LPFM and cAFM mappings in tDBG sample (tDBG01). (a) Schematic of the tDBG moirésuperlattice and relative locations of three domains, where red, green and blue dots represent AB-BC,AB-CA and AB-AB domains, respectively. (b) Principle of the LPFM measurement. (c,d) Phase andamplitude images of tDBG01. (e) Current image of the same area in (c,d). (f) Line cuts of the phase,amplitude and current for tDBG01 along black, red and green arrow lines and cross DW1, DW2 andDW3 in (c–e), respectively. (g) Sinusoidal fit of the three amplitudes vs. the cantilever axis anglerelative to the DW. (h) Schematic of in-plane polarization networks for tDBG01 sample.This two-step profile phenomenon can be enhanced by adjusting measurement param-eters, e.g., drive frequency and cantilever-sample force. More details about the frequency-dependent LPFM amplitude of tDBG can be found in Figure S4. Figure 2a,b show the LPFMresults of tDBG03 (see raw LPFM phase and amplitude images in Figure S5); its amplitudegives a clearer two-step profile than Figure 1d. Figure 2c presents line cuts of phase andamplitude along red and black arrows in Figure 2a,b. Mathematically, this two-step profileLPFM signal (red line) of DW4 can be divided into an axisymmetric vector (green line) anda centrosymmetric vector (blue line), as illustrated in Figure 2d. Hence, the axisymmetricand centrosymmetric components of line cuts in Figure 2c can be calculated, as shown inFigure 2e,f. Those decoupled LPFM results are similar to those of tMBG and tBLG in thefollowing two aspects. First, both the axisymmetric component of tDBG and tMBG showone broad peak, and the centrosymmetric component of tDBG and tBLG give two extremawith 180◦ phase change. Second, the amplitudes of axisymmetric and centrosymmetriccomponents for three DWs (DW4, DW5 and DW6) satisfy sinusoidal relations (Figure 2g),which is a typical feature for in-plane polarization [14,16]. Apart from similarity, the magni-tude of the axisymmetric component of the tDBG LPFM amplitude is only twice as much asthat of the centrosymmetric component, while the amplitude of tMBG is about an order ofmagnitude larger than that of tBLG. These similarities and differences among tBLG, tMBGand tDBG are reminiscent of the in-plane electrical polarization of tDBG generated by theSymmetry 2024, 16, 1524 4 of 8combination of piezoelectric and flexoelectric effects. Noteworthily, the two-step profileis highly sensitive to scan parameters. Figure S6 shows another LPFM scan of the samearea in Figure 2a,b with different drive frequencies, which gives a single-peak profile. Thissensitivity to scan parameters is also shown in the tBLG sample [16].Symmetry 2024, 16, x FOR PEER REVIEW 4 of 8   the LPFM results of tDBG03 (see raw LPFM phase and amplitude images in Figure S5); its amplitude gives a clearer two-step profile than Figure 1d. Figure 2c presents line cuts of phase and amplitude along red and black arrows in Figure 2a,b. Mathematically, this two-step profile LPFM signal (red line) of DW4 can be divided into an axisymmetric vector (green line) and a centrosymmetric vector (blue line), as illustrated in Figure 2d. Hence, the axisymmetric and centrosymmetric components of line cuts in Figure 2c can be calcu-lated, as shown in Figure 2e,f. Those decoupled LPFM results are similar to those of tMBG and tBLG in the following two aspects. First, both the axisymmetric component of tDBG and tMBG show one broad peak, and the centrosymmetric component of tDBG and tBLG give two extrema with 180° phase change. Second, the amplitudes of axisymmetric and centrosymmetric components for three DWs (DW4, DW5 and DW6) satisfy sinusoidal re-lations (Figure 2g), which is a typical feature for in-plane polarization [14,16]. Apart from similarity, the magnitude of the axisymmetric component of the tDBG LPFM amplitude is only twice as much as that of the centrosymmetric component, while the amplitude of tMBG is about an order of magnitude larger than that of tBLG. These similarities and dif-ferences among tBLG, tMBG and tDBG are reminiscent of the in-plane electrical polariza-tion of tDBG generated by the combination of piezoelectric and flexoelectric effects. Note-worthily, the two-step profile is highly sensitive to scan parameters. Figure S6 shows an-other LPFM scan of the same area in Figure 2a,b with different drive frequencies, which gives a single-peak profile. This sensitivity to scan parameters is also shown in the tBLG sample [16].  Figure 2. Decomposition of LPFM signal of tDBG sample (tDBG03). (a,b) LPFM phase and ampli-tude images of tDBG03. Green and blue dots represent the AB-CA and AB-AB domains, respec-tively. (c) Line cuts of the phase and amplitude for tDBG03 along the red and black arrows in (a,b). (d) Schematic of the decomposition process of the DW4 LPFM signal in (c). The red cure is the two-step LPFM signal (A·sin(θ−90°)) of DW4. The green and blue curves represent the decomposed ax-isymmetric and centrosymmetric components, respectively. (e,f) Line cuts of the decomposed phase and amplitude for the axisymmetric and centrosymmetric components, respectively. (g) Sinusoidal fits of the axisymmetric (green) and centrosymmetric (blue) amplitude, respectively. To obtain the in-plane piezoelectric coefficients of DWs in tDBG, we performed DART LPFM measurements on tDBG04, and the typical DART LPFM amplitude and Figure 2. Decomposition of LPFM signal of tDBG sample (tDBG03). (a,b) LPFM phase and ampli-tude images of tDBG03. Green and blue dots represent the AB-CA and AB-AB domains, respec-tively. (c) Line cuts of the phase and amplitude for tDBG03 along the red and black arrows in (a,b).(d) Schematic of the decomposition process of the DW4 LPFM signal in (c). The red cure is thetwo-step LPFM signal (A·sin(θ−90◦)) of DW4. The green and blue curves represent the decomposedaxisymmetric and centrosymmetric components, respectively. (e,f) Line cuts of the decomposed phaseand amplitude for the axisymmetric and centrosymmetric components, respectively. (g) Sinusoidalfits of the axisymmetric (green) and centrosymmetric (blue) amplitude, respectively.To obtain the in-plane piezoelectric coefficients of DWs in tDBG, we performed DARTLPFM measurements on tDBG04, and the typical DART LPFM amplitude and phase (decou-pled from the background) maps of tDBG04 are presented in Figure 3a,b with an expected180◦ phase change and three different amplitudes. One peak profile is captured in the DARTLPFM measurements because the two-step profile is dependent on the scan parameters.Moreover, the centrosymmetric component of the LPFM signal has no contribution to theLPFM amplitude. Zooming in on the white dashed boxed area in Figure 3a, Figure 3cshows selected five DART LPFM amplitude maps of tDBG as the drive voltage increasesfrom 1 V to 4.5 V (see Figure S7 for details), in which the amplitude increases significantly.Figure 3d presents the averaged amplitudes at three DWs of tDBG04 (DW7, DW8 andDW9 are marked by red, blue and green dashed polygons in Figure 3c) as a function ofthe drive voltage, respectively, along with their linear fits. The three fitted slopes are thethree projected in-plane piezoelectric coefficients of a DW. Only when the cantilever axis isperpendicular to the DW is the fitted slope equal to the in-plane piezoelectric coefficient ofthe DW, except when setting sample-cantilever angle as 90◦, as the in-plane piezoelectriccoefficient can also be extracted by the sinusoidal fitting of the three slopes, as illustratedin Figure 3e. Consequently, the in-plane piezoelectric coefficient of a DW in tDBG is de-Symmetry 2024, 16, 1524 5 of 8termined to be 0.15 pm/V, which is three times larger than that of tBLG and half that oftMBG [16].Symmetry 2024, 16, x FOR PEER REVIEW 5 of 8   phase (decoupled from the background) maps of tDBG04 are presented in Figure 3a,b with an expected 180° phase change and three different amplitudes. One peak profile is captured in the DART LPFM measurements because the two-step profile is dependent on the scan parameters. Moreover, the centrosymmetric component of the LPFM signal has no contribution to the LPFM amplitude. Zooming in on the white dashed boxed area in Figure 3a, Figure 3c shows selected five DART LPFM amplitude maps of tDBG as the drive voltage increases from 1 V to 4.5 V (see Figure S7 for details), in which the amplitude increases significantly. Figure 3d presents the averaged amplitudes at three DWs of tDBG04 (DW7, DW8 and DW9 are marked by red, blue and green dashed polygons in Figure 3c) as a function of the drive voltage, respectively, along with their linear fits. The three fitted slopes are the three projected in-plane piezoelectric coefficients of a DW. Only when the cantilever axis is perpendicular to the DW is the fitted slope equal to the in-plane piezoelectric coefficient of the DW, except when setting sample-cantilever angle as 90°, as the in-plane piezoelectric coefficient can also be extracted by the sinusoidal fitting of the three slopes, as illustrated in Figure 3e. Consequently, the in-plane piezoelectric coeffi-cient of a DW in tDBG is determined to be 0.15 pm/V, which is three times larger than that of tBLG and half that of tMBG [16].  Figure 3. Determination of in-plane piezoelectric coefficient in tDBG sample (tDBG04). (a,b) Typical DART LPFM phase and amplitude images of tDBG04. (c) DART LPFM amplitude images of the white dashed boxed area in (a) under selected applied voltage from 1.0 V to 4.5 V. DW7, DW8 and DW9 are outlined by red, blue and green dashed polygons, respectively. (d) Averaged DART LPFM amplitude (projected deformation) at three selected DWs (DW7, DW8 and DW9) of tDBG04 as a function of the AC drive voltage, along with the linear fits. The error bars are standard deviations of the amplitude. The slopes of the linear fits, which are measured in-plane piezoelectric coefficients for three DWs in one scan, are 0.15 pm/V, 0.06 pm/V and 0.04 pm/V, respectively. (e) Sinusoidal fit of the three slopes versus the cantilever axis angle relative to the DW. The maximum value of the fit corresponds to the effective in-plane piezoelectric coefficient of tDBG04, which is 0.15 pm/V. We employ continuum mechanics theory to explain the two-step LPFM profile of tDBG. When a conductive tip-induced electric field 𝐸  is applied to the DWs of twisted graphene with an electrical polarization 𝑃 , the induced strain 𝜀  can be expressed as [24] 𝜀 𝑘 𝑄 𝑃 𝐸  , where 𝑘   is the dielectric constant, and 𝑄   is the electro-strictive coefficient. For ideal tDBG moiré superlattices, the moiré DW is shear-type along Figure 3. Determination of in-plane piezoelectric coefficient in tDBG sample (tDBG04). (a,b) TypicalDART LPFM phase and amplitude images of tDBG04. (c) DART LPFM amplitude images of thewhite dashed boxed area in (a) under selected applied voltage from 1.0 V to 4.5 V. DW7, DW8 andDW9 are outlined by red, blue and green dashed polygons, respectively. (d) Averaged DART LPFMamplitude (projected deformation) at three selected DWs (DW7, DW8 and DW9) of tDBG04 as afunction of the AC drive voltage, along with the linear fits. The error bars are standard deviations ofthe amplitude. The slopes of the linear fits, which are measured in-plane piezoelectric coefficients forthree DWs in one scan, are 0.15 pm/V, 0.06 pm/V and 0.04 pm/V, respectively. (e) Sinusoidal fit ofthe three slopes versus the cantilever axis angle relative to the DW. The maximum value of the fitcorresponds to the effective in-plane piezoelectric coefficient of tDBG04, which is 0.15 pm/V.We employ continuum mechanics theory to explain the two-step LPFM profile oftDBG. When a conductive tip-induced electric field Ek is applied to the DWs of twistedgraphene with an electrical polarization Pv, the induced strain εij can be expressed as [24]εij = kimQjkmvPvEk, where kim is the dielectric constant, and Qjkmv is the electrostrictivecoefficient. For ideal tDBG moiré superlattices, the moiré DW is shear-type along the yaxis [23], as shown in Figure 4a, where uy is the only nonzero displacement. Thus, only∂uy∂x and ∂2uy∂x2 need to be taken into consideration. In Figure 4b(i), we sketch the profile ofthe displacement across the DW, where the profile of the displacement was adopted fromRef. [25]. For a linear constitutive relation, the in-plane piezoelectric (Figure 4b(ii)) andflexoelectric (Figure 4b(iii)) polarizations show axisymmetric and centrosymmetric profiles,respectively. Surprisingly, the linear combination of piezoelectricity and flexoelectricitygives a two-step profile (Figure 4b(iv)), which is highly consistent with the LPFM signalof tDBG; we call it ‘piezo-flexoelectricity’. Based on the threefold rotational symmetry oftDBG, a two-dimensional map of the in-plane piezo-flexoelectric polarization is presentedin Figure 4c. The AB-CA domains surround the in-plane polarization clockwise, and AB-ABdomains enclose the in-plane polarization anticlockwise, while the magnitude of in-planepolarization near the AB-CA domain is larger than that of the AB-AB domain.Symmetry 2024, 16, 1524 6 of 8Symmetry 2024, 16, x FOR PEER REVIEW 6 of 8   the y axis [23], as shown in Figure 4a, where 𝑢  is the only nonzero displacement. Thus, only  and  need to be taken into consideration. In Figure 4b (ⅰ), we sketch the pro-file of the displacement across the DW, where the profile of the displacement was adopted from Ref. [25]. For a linear constitutive relation, the in-plane piezoelectric (Figure 4b (ⅱ)) and flexoelectric (Figure 4b (ⅲ)) polarizations show axisymmetric and centrosymmetric profiles, respectively. Surprisingly, the linear combination of piezoelectricity and flexo-electricity gives a two-step profile (Figure 4b (ⅳ)), which is highly consistent with the LPFM signal of tDBG; we call it ‘piezo-flexoelectricity’. Based on the threefold rotational symmetry of tDBG, a two-dimensional map of the in-plane piezo-flexoelectric polariza-tion is presented in Figure 4c. The AB-CA domains surround the in-plane polarization clockwise, and AB-AB domains enclose the in-plane polarization anticlockwise, while the magnitude of in-plane polarization near the AB-CA domain is larger than that of the AB-AB domain.  Figure 4. Continue mechanics analysis of piezoelectric and flexoelectric effects in tDBG. (a) Sche-matic of a perfect tDBG moiré DW. The red and blue hexagonal grids represent the upper and lower bilayer graphene layers, respectively. (b) (ⅰ) The displacement (𝑢 ) field across the shear DW in (a). (ⅱ–ⅳ) Spatial profiles of in-plane piezoelectric, flexoelectric and piezo-flexoelectric polarizations are proportional to the first derivative of displacement (∂𝑢 /𝜕𝑥 ), second derivative of displacement (∂ 𝑢 /𝜕𝑥 ) and linear combination of the first and second derivatives of displacement (𝐴 ∙ ∂𝑢 /𝜕𝑥𝐵 ∙ ∂ 𝑢 /𝜕𝑥  ), respectively. (c) Two-dimensional in-plane piezo-flexoelectric polarization image. The color denotes the direction of polarization in the inset. Green and blue dots represent AB-CA and AB-AB domains, respectively. 3. Conclusions In conclusion, tDBG moiré superlattices represent a pristine platform for investigat-ing electromechanical couplings at the mesoscopic scale. Our findings reveal the coexist-ence of piezoelectricity and flexoelectricity in the DWs of tDBG and offer new ways to manipulate electromechanical couplings in moiré superlattices. The interplay of electronic state transition and polarization competition in a moiré system, such as opposite Chern numbers in twisted transition metal dichalcogenide [26,27], can be derived from polariza-tion competition among ferroelectricity, piezoelectricity and even flexoelectricity [28], which may not only advance our understanding of the electronic phenomena but also show promise for the development of innovative technologies with tailored electronic and mechanical functionalities.   Figure 4. Continue mechanics analysis of piezoelectric and flexoelectric effects in tDBG. (a) Schematicof a perfect tDBG moiré DW. The red and blue hexagonal grids represent the upper and lowerbilayer graphene layers, respectively. (b) (i) The displacement (uy) field across the shear DW in (a).(ii–iv) Spatial profiles of in-plane piezoelectric, flexoelectric and piezo-flexoelectric polarizationsare proportional to the first derivative of displacement (∂uy/∂x), second derivative of displacement(∂2uy/∂x2) and linear combination of the first and second derivatives of displacement (A·∂uy/∂x +B·∂2uy/∂x2), respectively. (c) Two-dimensional in-plane piezo-flexoelectric polarization image. Thecolor denotes the direction of polarization in the inset. Green and blue dots represent AB-CA andAB-AB domains, respectively.3. ConclusionsIn conclusion, tDBG moiré superlattices represent a pristine platform for investigatingelectromechanical couplings at the mesoscopic scale. Our findings reveal the coexistenceof piezoelectricity and flexoelectricity in the DWs of tDBG and offer new ways to ma-nipulate electromechanical couplings in moiré superlattices. The interplay of electronicstate transition and polarization competition in a moiré system, such as opposite Chernnumbers in twisted transition metal dichalcogenide [26,27], can be derived from polar-ization competition among ferroelectricity, piezoelectricity and even flexoelectricity [28],which may not only advance our understanding of the electronic phenomena but alsoshow promise for the development of innovative technologies with tailored electronic andmechanical functionalities.4. MethodsSample Fabrication. All devices were fabricated using a modified ‘tear-and-stack’method (see Figure S1a for details), similar to Ref. [29]. Typical optical images of tDBGdevices are shown in Figure S1b.Scanning Probe Microscopy Measurements. LPFM, cAFM and DART LPFM modeswere adopted in this work. Those measurements are carried out on an Oxford InstrumentsAsylum Research MFP-3D Origin atomic force microscope. ASYELEC-01-R2 probes coatedwith 5-nm Ti and 20-nm Ir were used in all modes with a spring constant of 2.8 N/m. Thetypical free resonance frequency is ~75 kHz, and LPFM contact resonance frequency is~780 kHz.Supplementary Materials: The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/sym16111524/s1. Correspondence and requests for materialsshould be addressed to Y.L. and Y.S.https://www.mdpi.com/article/10.3390/sym16111524/s1https://www.mdpi.com/article/10.3390/sym16111524/s1Symmetry 2024, 16, 1524 7 of 8Author Contributions: Y.L. and Y.S. conceived the experiment. Y.W. and Y.L. performed the AFMmeasurements. H.Z. and Y.L. fabricated the devices with the help of S.L. T.T. and K.W. provided thebulk BN crystals. Y.L., Y.S., H.Z. and Y.W. analyzed the data. Y.L., Y.S., C.-F.G., H.Z. and Y.W. wrotethe paper with input from all authors. All authors have read and agreed to the published version ofthe manuscript.Funding: This work was supported by the National Key Research and Development Programof China (2021YFA0715600), National Natural Science Foundation of China (12072150, 12274222and 12404217), National Science Foundation of Jiangsu Province (BK20220756), Natural ScienceFoundation of Guangdong Province (2022A1515011773), and National Natural Science Foundation ofChina for Creative Research Groups (51921003). K.W. and T.T. acknowledge support from the JSPSKAKENHI (21H05233 and 23H02052) and World Premier International Research Center Initiative(WPI), MEXT, Japan.Data Availability Statement: Data are contained within the article and Supplementary Materials.Conflicts of Interest: The authors declare no competing interests.References1. Catalan, G.; Seidel, J.; Ramesh, R.; Scott, J.F. Domain wall nanoelectronics. Rev. Mod. Phys. 2012, 84, 119–156. [CrossRef]2. Nataf, G.F.; Guennou, M.; Gregg, J.M.; Meier, D.; Hlinka, J.; Salje, E.K.H.; Kreisel, J. Domain-wall engineering and topologicaldefects in ferroelectric and ferroelastic materials. Nat. Rev. Phys. 2020, 2, 634–648. [CrossRef]3. Meier, D.; Selbach, S.M. Ferroelectric domain walls for nanotechnology. Nat. Rev. Mater. 2022, 7, 157–173. [CrossRef]4. Xia, Y.; Ji, Y.; Liu, Y.; Wu, L.; Yang, Y. Controllable Piezo-flexoelectric Effect in Ferroelectric Ba0.7Sr0.3TiO3 Materials for HarvestingVibration Energy. ACS Appl. Mater. Interfaces 2022, 14, 36763–36770. [CrossRef]5. Zubko, P.; Catalan, G.; Tagantsev, A.K. Flexoelectric effect in solids. Annu. Rev. Mater. Res. 2013, 43, 387–421. [CrossRef]6. Wang, B.; Gu, Y.; Zhang, S.; Chen, L.-Q. Flexoelectricity in solids: Progress, challenges, and perspectives. Prog. Mater. Sci. 2019,106, 100570. [CrossRef]7. Cao, Y.; Fatemi, V.; Fang, S.; Watanabe, K.; Taniguchi, T.; Kaxiras, E.; Jarillo-Herrero, P. Unconventional superconductivity inmagic-angle graphene superlattices. Nature 2018, 556, 43–50. [CrossRef]8. Cao, Y.; Fatemi, V.; Demir, A.; Fang, S.; Tomarken, S.L.; Luo, J.Y.; Sanchez-Yamagishi, J.D.; Watanabe, K.; Taniguchi, T.; Kaxiras, E.;et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 2018, 556, 80–84. [CrossRef]9. Sharpe, A.L.; Fox, E.J.; Barnard, A.W.; Finney, J.; Watanabe, K.; Taniguchi, T.; Kastner, M.A.; Goldhaber-Gordon, D. Emergentferromagnetism near three-quarters filling in twisted bilayer graphene. Science 2019, 365, 605–608. [CrossRef]10. Serlin, M.; Tschirhart, C.L.; Polshyn, H.; Zhang, Y.; Zhu, J.; Watanabe, K.; Taniguchi, T.; Balents, L.; Young, A.F. Intrinsic quantizedanomalous Hall effect in a moiré heterostructure. Science 2020, 367, 900–903. [CrossRef]11. Zheng, Z.; Ma, Q.; Bi, Z.; de la Barrera, S.; Liu, M.H.; Mao, N.; Zhang, Y.; Kiper, N.; Watanabe, K.; Taniguchi, T.; et al.Unconventional ferroelectricity in moire heterostructures. Nature 2020, 588, 71–76. [CrossRef] [PubMed]12. Klein, D.R.; Xia, L.-Q.; MacNeill, D.; Watanabe, K.; Taniguchi, T.; Jarillo-Herrero, P. Electrical switching of a bistable moirésuperconductor. Nat. Nanotechnol. 2023, 18, 331–335. [CrossRef] [PubMed]13. Kazmierczak, N.P.; Van Winkle, M.; Ophus, C.; Bustillo, K.C.; Carr, S.; Brown, H.G.; Ciston, J.; Taniguchi, T.; Watanabe, K.;Bediako, D.K. Strain fields in twisted bilayer graphene. Nat. Mater. 2021, 20, 956–963. [CrossRef] [PubMed]14. McGilly, L.J.; Kerelsky, A.; Finney, N.R.; Shapovalov, K.; Shih, E.M.; Ghiotto, A.; Zeng, Y.; Moore, S.L.; Wu, W.; Bai, Y.; et al.Visualization of moire superlattices. Nat. Nanotechnol. 2020, 15, 580–584. [CrossRef]15. Li, Y.; Wang, X.; Tang, D.; Wang, X.; Watanabe, K.; Taniguchi, T.; Gamelin, D.R.; Cobden, D.H.; Yankowitz, M.; Xu, X.; et al.Unraveling Strain Gradient Induced Electromechanical Coupling in Twisted Double Bilayer Graphene Moiré Superlattices.Adv. Mater. 2021, 33, 2105879. [CrossRef]16. Zhang, H.; Wei, Y.; Li, Y.; Lin, S.; Wang, J.; Taniguchi, T.; Watanabe, K.; Li, J.; Shi, Y.; Wang, X.; et al. Layer-DependentElectromechanical Response in Twisted Graphene Moiré Superlattices. ACS Nano 2024, 18, 17570–17577.17. Kerelsky, A.; Rubio-Verdú, C.; Xian, L.; Kennes, D.M.; Halbertal, D.; Finney, N.; Song, L.; Turkel, S.; Wang, L.; Watanabe, K.; et al.Moiréless correlations in ABCA graphene. Proc. Natl. Acad. Sci. USA 2021, 118, e2017366118.18. Alden, J.S.; Tsen, A.W.; Huang, P.Y.; Hovden, R.; Brown, L.; Park, J.; Muller, D.A.; McEuen, P.L. Strain solitons and topologicaldefects in bilayer graphene. Proc. Natl. Acad. Sci. USA 2013, 110, 11256. [CrossRef]19. Jungk, T.; Hoffmann, Á.; Soergel, E. Quantitative analysis of ferroelectric domain imaging with piezoresponse force microscopy.Appl. Phys. Lett. 2006, 89, 163507. [CrossRef]20. Zhang, S.; Song, A.; Chen, L.; Jiang, C.; Chen, C.; Gao, L.; Hou, Y.; Liu, L.; Ma, T.; Wang, H.; et al. Abnormal conductivity inlow-angle twisted bilayer graphene. Sci. Adv. 2020, 6, eabc5555. [CrossRef]21. Rosenberger, M.R.; Chuang, H.-J.; Phillips, M.; Oleshko, V.P.; McCreary, K.M.; Sivaram, S.V.; Hellberg, C.S.; Jonker, B.T. TwistAngle-Dependent Atomic Reconstruction and Moiré Patterns in Transition Metal Dichalcogenide Heterostructures. ACS Nano2020, 14, 4550–4558. [CrossRef]https://doi.org/10.1103/RevModPhys.84.119https://doi.org/10.1038/s42254-020-0235-zhttps://doi.org/10.1038/s41578-021-00375-zhttps://doi.org/10.1021/acsami.2c09767https://doi.org/10.1146/annurev-matsci-071312-121634https://doi.org/10.1016/j.pmatsci.2019.05.003https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26154https://doi.org/10.1126/science.aaw3780https://doi.org/10.1126/science.aay5533https://doi.org/10.1038/s41586-020-2970-9https://www.ncbi.nlm.nih.gov/pubmed/33230334https://doi.org/10.1038/s41565-022-01314-xhttps://www.ncbi.nlm.nih.gov/pubmed/36717710https://doi.org/10.1038/s41563-021-00973-whttps://www.ncbi.nlm.nih.gov/pubmed/33859383https://doi.org/10.1038/s41565-020-0708-3https://doi.org/10.1002/adma.202105879https://doi.org/10.1073/pnas.1309394110https://doi.org/10.1063/1.2362984https://doi.org/10.1126/sciadv.abc5555https://doi.org/10.1021/acsnano.0c00088Symmetry 2024, 16, 1524 8 of 822. Li, Y.; Xue, M.; Fan, H.; Gao, C.-F.; Shi, Y.; Liu, Y.; Watanabe, K.; Tanguchi, T.; Zhao, Y.; Wu, F.; et al. Symmetry Breaking andAnomalous Conductivity in a Double-Moiré Superlattice. Nano Lett. 2022, 22, 6215–6222. [CrossRef]23. Zhang, S.; Xu, Q.; Hou, Y.; Song, A.; Ma, Y.; Gao, L.; Zhu, M.; Ma, T.; Liu, L.; Feng, X.-Q.; et al. Domino-like stacking orderswitching in twisted monolayer–multilayer graphene. Nat. Mater. 2022, 21, 621–626. [CrossRef] [PubMed]24. Gruverman, A.; Kalinin, S.V. Piezoresponse force microscopy and recent advances in nanoscale studies of ferroelectrics. J. Mater.Sci. 2006, 41, 107–116. [CrossRef]25. Lebedeva, I.V.; Lebedev, A.V.; Popov, A.M.; Knizhnik, A.A. Dislocations in stacking and commensurate-incommensurate phasetransition in bilayer graphene and hexagonal boron nitride. Phys. Rev. B 2016, 93, 235414. [CrossRef]26. Park, H.; Cai, J.; Anderson, E.; Zhang, Y.; Zhu, J.; Liu, X.; Wang, C.; Holtzmann, W.; Hu, C.; Liu, Z.; et al. Observation offractionally quantized anomalous Hall effect. Nature 2023, 622, 74–79. [CrossRef] [PubMed]27. Foutty, B.A.; Kometter, C.R.; Devakul, T.; Reddy, A.P.; Watanabe, K.; Taniguchi, T.; Fu, L.; Feldman, B.E. Mapping twist-tunedmultiband topology in bilayer WSe2. Science 2024, 384, 343–347. [CrossRef]28. Zhang, X.-W.; Wang, C.; Liu, X.; Fan, Y.; Cao, T.; Xiao, D. Polarization-driven band topology evolution in twisted MoTe2 andWSe2. Nat. Commun. 2024, 15, 4223. [CrossRef]29. Wong, D.; Nuckolls, K.P.; Oh, M.; Lian, B.; Xie, Y.; Jeon, S.; Watanabe, K.; Taniguchi, T.; Bernevig, B.A.; Yazdani, A. Cascade ofelectronic transitions in magic-angle twisted bilayer graphene. Nature 2020, 582, 198–202. [CrossRef]Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individualauthor(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury topeople or property resulting from any ideas, methods, instructions or products referred to in the content.https://doi.org/10.1021/acs.nanolett.2c01710https://doi.org/10.1038/s41563-022-01232-2https://www.ncbi.nlm.nih.gov/pubmed/35449221https://doi.org/10.1007/s10853-005-5946-0https://doi.org/10.1103/PhysRevB.93.235414https://doi.org/10.1038/s41586-023-06536-0https://www.ncbi.nlm.nih.gov/pubmed/37591304https://doi.org/10.1126/science.adi4728https://doi.org/10.1038/s41467-024-48511-xhttps://doi.org/10.1038/s41586-020-2339-0 Introduction  Results and Discussion  Conclusions  Methods  References