# Fileset

[Small - 2025 - Ou - Continuous Strain Modulation of Moir  Superlattice Symmetry From Triangle to Rectangle (1).pdf](https://mdr.nims.go.jp/filesets/f55fd02a-dfac-4b40-821a-b4a4896e2493/download)

## Creator

Hao Ou, Koshi Oi, Rei Usami, Takhiko Endo, Keisuke Shinokita, [Ryo Kitaura](https://orcid.org/0000-0001-8108-109X), Kazunari Matsuda, Yasumitsu Miyata, Jiang Pu, Taishi Takenobu

## Rights

[Creative Commons BY-NC Attribution-NonCommercial 4.0 International](https://creativecommons.org/licenses/by-nc/4.0/)

## Other metadata

[Continuous Strain Modulation of Moiré Superlattice Symmetry From Triangle to Rectangle](https://mdr.nims.go.jp/datasets/7d3e2c82-0ca7-4598-b304-fefd3df83c7b)

## Fulltext

Continuous Strain Modulation of Moiré Superlattice Symmetry From Triangle to RectangleRESEARCH ARTICLEwww.small-journal.comContinuous Strain Modulation of Moiré SuperlatticeSymmetry From Triangle to RectangleHao Ou, Koshi Oi, Rei Usami, Takahiko Endo, Keisuke Shinokita, Ryo Kitaura,Kazunari Matsuda, Yasumitsu Miyata, Jiang Pu, and Taishi Takenobu*Moiré superlattices formed in van der Waals (vdW) bilayers of 2D materialsprovide an ideal platform for studying previously undescribed physics,including correlated electronic states and moiré excitons, owing to thewide-range tunability of their lattice constants. However, their crystalsymmetry is fixed by the monolayer structure, and the lack of astraightforward technique for modulating the symmetry of moiré superlatticeshas impeded progress in this field. Herein, a simple, room-temperature,ambient method for controlling superlattice symmetry is reported. Themethod uses vdW heterostructures on a flexible substrate; by bending thesubstrate, a uniaxial strain is introduced. Based on numerical calculations, astrain condition is designed to deform the moiré superlattice from triangularto rectangular, and visualized the continuous deformation of real-space moirésuperlattices using piezoresponse force microscopy. The band calculationsshow that nearly flat moiré minibands remain in the rectangular lattice;therefore, this method provides an additional tuning knob for the Hamiltonianof moiré quantum matter.H. Ou, K. Oi, R. Usami, T. TakenobuDepartment of Applied PhysicsNagoya UniversityNagoya 464-8603, JapanE-mail: takenobu@nagoya-u.jpT. Endo, Y.MiyataDepartment of PhysicsTokyoMetropolitanUniversityTokyo 192-0397, JapanK. Shinokita, K.MatsudaInstitute of AdvancedEnergyKyotoUniversityKyoto 611-0011, JapanR. KitauraResearchCenter forMaterialsNanoarchitectonics (MANA)National Institute forMaterials Science (NIMS)Tsukuba 305-0044, JapanJ. PuDepartment of PhysicsTokyo Institute of TechnologyTokyo 152-8551, JapanThe ORCID identification number(s) for the author(s) of this articlecan be found under https://doi.org/10.1002/smll.202407316© 2025 The Author(s). Small published by Wiley-VCH GmbH. This is anopen access article under the terms of the Creative CommonsAttribution-NonCommercial License, which permits use, distribution andreproduction in any medium, provided the original work is properly citedand is not used for commercial purposes.DOI: 10.1002/smll.2024073161. IntroductionStacking 2D van der Waals (vdW) monolay-ers with dissimilar lattice constants and/orslight rotational misalignment produces amoiré superlattice with a periodicity in-versely related to the magnitude of theinterlayer mismatch.[1,2] The length scaleof the moiré period is typically severaltimes larger than the atomic lattice con-stant (≈0.1 nm).[2] Importantly, the tunabil-ity of the lattice parameters in moiré ma-terials is similar to that of quantum sim-ulators based on ultracold gases, whichare clean models using real physical sys-tems to “simulate” quantum materials.[3,4]A difference in length scales of approx-imately one digit implies a complemen-tary relationship between ultracold gases(≈100 nm) and moiré materials (≈1–100 nm). As such, moiré materials as-sembled from vdW layers are versatileplatforms for simulating and designing the physics of electroniccorrelations and nontrivial band topology.[4–6] The tuning knobsrequired for quantum simulators, as realized in ultracold gases,are lattice constants and crystal symmetry.[7] In moiré materials,however, the controllability of the lattice symmetry is extremelylimited, in stark contrast to the high tunability of lattice constantsin ultracold gases.[4]The limited symmetry of monolayers, which determines thesuperlattice symmetry, mostly limits moiré materials to hon-eycombs and triangles.[4] To obtain nonhoneycombs/triangles,a rectangular superlattice was theoretically proposed based ontwisted bilayer (t-BL) germanium selenide,[8] and recently, it wasdemonstrated by t-BL of WTe2/WSe2.[9] Therefore, crystal sym-metry is still not a tuning knob formoirématerials, and amethodto control it other than material selection is urgently needed. Inthis study, we focused on the strain effect because the ultimatethinness of the monolayers allows intentional and continuousdeformation by relatively weak forces, such as bending a plas-tic substrate. Although the effect of strain on moiré superlat-tices has already been thoroughly investigated from a theoret-ical viewpoint,[10–12] experimental reports have been limited tounintentional strain effects, and continuous control of the lat-tice constants and symmetry of moiré materials has not yet beenachieved.[12–14] In this study, we demonstrated symmetry manip-ulation of moiré superlattices by continuous uniaxial deforma-tion. We performed a piezoresponse force microscopy (PFM)Small 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (1 of 7)http://www.small-journal.commailto:takenobu@nagoya-u.jphttps://doi.org/10.1002/smll.202407316http://creativecommons.org/licenses/by-nc/4.0/http://creativecommons.org/licenses/by-nc/4.0/http://crossmark.crossref.org/dialog/?doi=10.1002%2Fsmll.202407316&domain=pdf&date_stamp=2025-01-16www.advancedsciencenews.com www.small-journal.comFigure 1. 2D vdW bilayer on plastic substrate. a) Schematic of the sample structure with twisted bilayer WSe2 and hBN transferred on Au/Ni coatedPEN substrate. b) Optical image of PEN substrate with Au/Ni film (left) and magnified microscopic image of the vdW stacking of sample #1 (right).The red dashed-line rectangle indicates the bilayer region. Scale bars: 5 mm (left), 20 μm (right). c) PFM amplitude image of sample #1, which showsclear long-period moiré patterns. Scale bar: 200 nm. Inset: Fast Fourier transform (FFT) image. d) PFM amplitude image of sample #2, which showsclear short-period moiré patterns. Scale bar: 50 nm. Inset: FFT image.technique on a flexible plastic substrate to directly visualize in-tentionally deformed moiré superlattices. As an example of con-tinuous tuning of the crystal symmetry, we realized a lattice sym-metry alternation from triangular to rectangular. The calculatedelectronic band structures of the distorted moiré superlatticespossess nearly flat moiré minibands and follow the continuouslytunable Hubbard model, which is a major step toward the devel-opment of highly tunable quantum simulators based on moirésuperlattices.2. Results and DiscussionThe uniaxial strain was applied to t-BL by simply bending the flex-ible substrate.[15–17] Although this is a well-established method, itis nearly impossible to visualizemoiré superlattices on insulatingplastic substrates using transmission electron microscopy[18] orscanning tunneling microscopy.[18,19] To solve this problem, weused the PFM method, which allows measurements of the insu-lating substrate in air.[9,13,20–22] The PFM technique resolves themoiré superlattice by leveraging the ferroelectricity or flexoelec-tricity inherent in stacked van der Waals heterostructures. In themeasurement process, a conductive AFM cantilever is used toapply a voltage to the sample. The moiré-induced piezoelectricsignal, which results from the strain induced in the material, isthen detected through the deformation of the cantilever. Thanksto the high spatial resolution of AFM, it is possible to effectivelyresolve moiré superlattices with periodicities on the order of afew nanometers or even longer, allowing for the detailed analysisof the moiré pattern at the nanoscale level.[20]Importantly, clear visualization of the moiré superlattice byPFM requires t-BL on the topmost surface of the flat sample.For a flat surface, we realized a strong interaction between the2D vdW materials and plastic substrates using thin Au/Ni filmson plastic substrates (see Figure S1 in the Supporting Infor-mation). For t-BL on the topmost surface, the commonly used“tear-and-stack” method, in which the top surface is hexagonalboron nitride (hBN), cannot be used.[1] Therefore, we adoptedthe dry pick-and-flip assembly technique established by Masub-uchi et al., in which the topmost surface is a t-BL (Figures 1aand S2, Supporting Information).[23] As the target material, weselected single-crystalline triangular WSe2 monolayers grown bythe chemical vapor deposition (CVD) method (Figure S3, Sup-porting Information).[24] We fabricated t-BL WSe2 on hBN andtransferred it to a flexible polyethylene naphthalate (PEN) sub-strate, as shown in Figures 1a,b. To achieve a strong interac-tion between hBN and the substrate, we first deposited a thinAu/Ni film on the PEN substrate, as previous studies have re-ported strong adhesion between deposited metal films and vander Waals materials.[25,26] The lower surface roughness of theAu/Ni film compared to PEN helps enhance the van der Waalsinteraction between hBN and the substrate, resulting in a nearlyflat sample surface.Subsequently, we conducted PFM measurements (see the Ex-perimental Section) and visualized the clear moiré superlat-tices. As illustrated in Figure 1c, the long-period (174 ± 43 nm)Small 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (2 of 7) 16136829, 2025, 25, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202407316 by National Institute For, Wiley Online Library on [18/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.small-journal.comwww.advancedsciencenews.com www.small-journal.comFigure 2. Calculated emergence of rectangular moiré superlattice. a) Schematic representation of the uniaxially strained twisted homobilayer. We as-sumed that the strain was uniaxial and applied it only to the bottom of the two monolayers. The parameter space spanned by the twist angle (𝜃), strainmagnitude (ɛ), and strain angle (𝜑) determines the moiré pattern, which can be characterized by two moiré basis vectors (Lm1 and Lm2) and the angle𝛼 between them. The orange arrow and lines indicate the armchair direction of the bottom layer. The black dashed parallelograms indicate the unit cellof the moiré superlattice. b) Variation of 𝛼 as a function of twist angle and strain magnitude when the strain angle is 0° (armchair direction). The whitelines indicate the emergence of rectangular moiré pattern (𝛼 = 90°), and the blue line shows the 1D pattern (𝛼 = 180°). c) Variation of 𝛼 as a functionof strain angle and strain magnitude when twist angle is 1.35°. The white line corresponds to the rectangular moiré pattern, where 𝛼 = 90°.pattern was observed in a nearly 0°-stacked bilayer WSe2 (sample#1, twist angle of ≈0.1°). The relationship between the period Land the small twist angle 𝜃 is estimated by the equation L ≈ a0/𝜃,where a0 = 0.332 nm represents the lattice constant of monolayerWSe2.[2] Interestingly, themoiré pattern is not uniform due to theunintentional strain and/or inhomogeneous 𝜃, proving possiblelattice distortion by the strain effect from the normal triangularlattice (indicated by the blue triangle) to the deformed rectangularlattice (white triangle).[1,20] In contrast, in the nearly 1°-stackedbilayer WSe2 (sample #2), we observed a more uniform super-lattice (Figure 1d), suggesting that a smaller twist angle is morevulnerable to unexpected inhomogeneity. During the fabricationprocess of heterostructures, unintended random strain may beintroduced. As will be shown later, the smaller the twist angle,the more significantly the moiré superlattice will distort under agiven strain magnitude. Additionally, a smaller twist angle cor-responds to a superlattice with a longer period, making the dis-torting effect more pronounced.[20] The estimated 𝜃 of sample #2was ≈1.35° (L = 14.10 ± 0.07 nm, see the Experimental Section).Because we established a method to visualize the uniformmoiré superlattice on flexible substrates, we introduced a straineffect via substrate bending. To understand the effect of substratebending on the moiré superlattice, we numerically calculated thevariation generated by strain (seeNote S1 in the Supporting Infor-mation). For clarity, we assumed that the strain was uniaxial andapplied it to only one of the two monolayers. As the schematic inFigure 2a shows, we considered the combined effect of 𝜃, strainmagnitude (ɛ), and strain angle (𝜑, where 0° corresponds to theorange line, which is the armchair direction of the strained bot-tom layer) on the moiré superlattice. We described the distor-tion of the superlattice using three parameters of a moiré unitcell (dashed black line parallelogram): lengths of two moiré ba-sis vectors (Lm1 and Lm2), and the moiré angle (𝛼) between them.First, as illustrated in Figure 2b, to argue the possibility of de-formation from triangular to rectangular, we fixed 𝜑 at 0° andclarified the variation of 𝛼, which is a direct indicator of the su-perlattice symmetry, depending on ɛ and 𝜃. The maximal ɛ and𝜃 were set to 2% and 3°, respectively, to meet experimental feasi-bility. The blue line in Figure 2b corresponds to 𝛼 = 180°, whichimplies the existence of 1D moiré superlattices (Figure S4, Sup-porting Information).[13] Moreover, the white lines correspond to𝛼 = 90°, which strongly suggests that deformation into a rect-angle is experimentally possible. The linearity of the white linesalso indicates that a smaller 𝜃 leads to a smaller ɛ required fordeformation.As the next step, to consider the guideline for experiments, wefixed 𝜃 at 1.35° and investigated 𝜑 dependencies because, as ex-perimentally shown in Figures 1c,d, a smaller 𝜃 causes nonuni-formity of moiré superlattice. The 𝜑 was varied from 0° to 30°,that is, from the armchair direction to the zigzag direction ofthe strained monolayer. Again, in Figure 2c, the white line cor-responds to 𝛼 = 90°, indicating that a rectangular lattice can beachieved with a smaller ɛ at 𝜑 = 30° (zigzag direction) than at𝜑 = 0° (armchair direction). Therefore, we chose the combina-tion of parameters (𝜃, ɛ, 𝜑) ≈ (1.35°, 1%, 30°) as a guidelinefor the following experiments. It should be noted that these nu-merical calculations assume that strain is applied to only one ofthe two monolayers, and in real experiments, we cannot excludethe simultaneous deformation of the two monolayers. However,the assumption of simultaneous two-monolayer deformation re-quires a much larger ɛ for the deformation to rectangular lattice(Figures S4, S5, and Note S1, Supporting Information), meaningthat (𝜃, ɛ, 𝜑) ≈ (1.35°, 1%, 30°) is the minimum condition.It is important to note that the appearance of a rectangularmoiré superlattice does not necessarily indicate the presence ofa rectangular monolayer lattice. This is because the formation ofa moiré pattern is determined solely by the interference betweenthe two periodic lattices, regardless of the individual lattice ge-ometries. In this case, a strain magnitude of 1% leads to only avery slight deformation of the monolayer lattice, which is negli-gible in terms of its impact on the overall structure. This subtledistortion is not sufficient to induce significant changes in theunderlyingmonolayer lattice, further supporting the idea that themoiré superlattice’s shape arises primarily from the interactionbetween the two lattices, rather than the geometry of each indi-vidual monolayer.We used sample #2 (the estimated 𝜃 is about 1.35°) and,by mounting the substrate on molds, ɛ and 𝜑 were controlledSmall 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (3 of 7) 16136829, 2025, 25, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202407316 by National Institute For, Wiley Online Library on [18/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.small-journal.comwww.advancedsciencenews.com www.small-journal.comFigure 3. Strain-induced triangular-rectangular moiré superlattice variation. a) Schematic of the experimental setup of PFM measurement on bentsubstrate. b–f) PFM phase images sample #2 with substrate mounted on molds with flat surface (b), nominal strain = 0%), radius of 25 mm (c),nominal strain = 0.25%), 16.7 mm (d), nominal strain = 0.38%), 13.9 mm (e), nominal strain = 0.45%), and 12.5 mm (f), nominal strain = 0.5%). Theblue crosses indicate the variation of the moiré angle 𝛼. Scale bars: 50 nm. Insets: FFT images.(Figure 3a).[17,27] For ɛ, we prepared several molds with differentcurvatures and tried to control it from 0% (flat) to 0.5% (radiuscurvature of 12.5 mm) (see the Experimental Section). For 𝜑, weattempted to bend t-BL WSe2 in the zigzag direction (𝜑 = 30°)with respect to the zigzag termination boundary of the startingCVD-grown single-crystal monolayerWSe2.[24,28] As illustrated inFigures 3b–e, we performed PFMmeasurements for the strainedsample #2 and visualized the continuous symmetry modulationof the moiré crystal. Most importantly, as indicated by the anglesin the blue crosses, one can identify a continuous variation ofthe moiré angle from nearly triangular (𝛼 = 120°) to nearly rect-angular (𝛼 = 90°) with increasing the magnitude of strain. FastFourier transform (FFT) images of the superlattice also clearlyshow a continuous symmetrymodulation from triangular to rect-angular. After observing the rectangular lattice, we performed aPFM experiment on the unbent (flat) condition and confirmedthe reversibility of these deformations (Figure S6, Supporting In-formation). Note that, even though the substrate was bent dur-ing the PFM measurements, the surface of the scanned regionremained relatively flat, ruling out the possibility that the distor-tion of the moiré superlattice was caused by changes in surfaceflatness (Figure S7, Supporting Information).To compare the experimental results with the numerical calcu-lations, we directly measured 𝜑 using Raman spectroscopy andsecond-harmonic generationmeasurements (Figures S8, S9, andNote S2, Supporting Information). The obtained 𝜑 = 23°, whichis slightly deviated from 30°. Therefore, the experimental condi-tion for Figure 3f is (𝜃, ɛ, 𝜑) ≈ (1.35°, 0.5%, 23°). As shown inFigure 2 and Figure S4 (Supporting Information), the realizationof a rectangularmoiré under these conditions strongly suggests ahigh likelihood that strain is applied to only one of the twomono-layers. As presented in Figure S4 (Supporting Information), ifboth layers are strained with the same magnitude, the variationin moiré angle 𝛼 is significantly smaller, and achieving a rect-angular superlattice would require an impractically high strainmagnitude. The geometry of the moiré superlattice depends onthe interference between the two monolayer lattices (see Note S1in the Supporting Information). Applying strain to both layers de-forms them simultaneously and hardly changes the interference,preventing the superlattice from distorting significantly. Anotherpossibility is that both layers are strained, but with different mag-nitudes. We calculated this condition by setting the strain mag-nitude in the top layer as a fraction of that in the bottom layer.The results are shown in Figure S5 (Supporting Information).As expected, increasing the fraction leads to higher strain mag-nitudes required for rectangular superlattice formation. Theseconditions require even higher strain magnitudes to distort themoiré superlattice into a nearly rectangular shape. It has alsobeen pointed out that nearly vanishing interlayer friction mightemerge in twisted bilayer structure.[29] Therefore, we adopted themodel in which only the bottom layer is strained, rather than theone where both layers are strained simultaneously.Still, there is a discrepancy between the theoretical calcula-tion (𝜃, ɛ, 𝜑) ≈ (1.35, 1%, 30°) and the experimental results(1.35°, 0.5%, 23°). The 7° deviation in strain angle is attributed tothe sample fabrication process, as we manually stacked the het-erostructure and transferred it to the PEN substrate. The strainmagnitude also shows a difference, as the experimental strainmagnitude for the rectangular pattern is smaller than the theo-retical prediction.We attribute this difference to two possible rea-sons: First, the experimental strain magnitude was determinedby the curvature of the bent substrate. However, the localizedstrain magnitude in the sample might be higher due to surfaceunevenness, which is difficult to measure. Another possibility isthat the theoretical model cannot fully capture the actual strain-induced deformation. When strain is applied, the twist anglemight change simultaneously,[30] which goes beyond the assump-tions of the model we used. Therefore, to fully understand theSmall 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (4 of 7) 16136829, 2025, 25, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202407316 by National Institute For, Wiley Online Library on [18/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.small-journal.comwww.advancedsciencenews.com www.small-journal.comFigure 4. Valence band structure variation in strained twisted bilayer WSe2. a) Valence band morphology of t-BL WSe2 when heterostrain is applied withmagnitude ɛ of 0% (left), 0.54% (center), and 1.01% (right). The twist angle and strain angle were fixed at 1.35° and 23°, respectively, in correspondencewith the experimental results in Figure 3. b) Band structures of the first two valence bands. c) DOS plots of the first two valence bands. Inset: MagnifiedDOS plot of the first valence band. d) Spatial distribution of the wave function |𝜓 |2 at 1.01% strain magnitude.strain distortion of the moiré superlattice, it is necessary to de-velop a more comprehensive model. Nevertheless, the presentedresults have proven that strain can be intentionally introducedto vdW heterostructures to distort the moiré superlattice in thedesign, which opens the door to studies of combined twist andstrain effects on moiré materials.A previous study showed that the valence band of ≈1° stackedt-BL WSe2 had a bandwidth of ≈1 meV.[31] To confirm whetherthis flat-band nature could be sustained in the strained twistedbilayer WSe2, electronic structure calculations were performed.We adopted an effective continuum model to calculate the bandstructure variation of the moiré superlattice (Note S3, Support-ing Information).[31] The twist angle and strain conditions wereset in correspondence to the experimental results (𝜃, 𝜑) = (1.35°,23°). Note that because the strain magnitudes are different be-tween calculated and experimental results, the parameters wereselected to match the value of 𝛼 among them (Figure S10mSupporting Information). Figure 4a illustrates the variation ofthe first valence band of the mini Brillouin zone (mBZ) whenthe moiré angle 𝛼 is 120°, 106.0°, and 91.5°. The correspond-ing strain magnitudes were 0%, 0.54%, and 1.01%, respec-tively. When strain was absent, the mBZ was a normal hexagon,with energy maxima located at 𝛾 points and minima at k+ (k−)corners.[31] As illustrated in Figures 4b,c, the bandwidth was lessthan 1 meV, indicating a flat band nature. The application of uni-axial strain (0.54%, center of Figure 4a) distorted the shape of themBZ and slightly increased its bandwidth (blue line in Figure 4c).More interestingly, within the first mBZ, the energy maximumsplits into two peaks, which shift in opposite directions from 𝛾due to symmetry breaking.[11] The energy minima also shift to-ward the twom points of the mBZ. This redistribution of energyresults in a saddle-like morphology around 𝛾 .[32,33] As the strainmagnitude was further increased (1.01%, right of Figure 4a), thepeak splitting becamemore distinct and the bandwidthswere stillnarrow (≈1meV at strain of 1.01%),meaning that the anisotropicband structure with a flat-band nature and van Hove singulari-ties (vHSs) can be realized. It can be observed that although themoiré band structure is modified by strain, the band width doesnot show significant changes. We applied a strain with a mag-nitude of no more than 1.01% to the bottom layer, which hardlyaffected its band dispersion.[34] Since the dispersion of the moiréband is determined by both themonolayer band structure and theinterlayer coupling-induced moiré potential, the band width re-mained largely unaffected. However, due to the distortion of themoiré superlattice, the geometry of the moiré potential is tuned,breaking the initial symmetry of the strain-free case and intro-ducing anisotropy into the band structure.With magnified electron–electron interactions, vHSs are re-lated to many exotic phenomena and new phases of matter be-cause of strengthened instability, such as correlation-inducedgap opening and the emergence of superconductivity.[35,36] Weshow that strain barely affects the bandwidth of the valenceband, and the robust vHS in the strained sample allows theobservation of correlation-related phenomena. Furthermore, thereal-space superlattice exhibited continuous and observable sym-metry change, manifesting as high tunability in shape whileSmall 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (5 of 7) 16136829, 2025, 25, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202407316 by National Institute For, Wiley Online Library on [18/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.small-journal.comwww.advancedsciencenews.com www.small-journal.commaintaining a narrow bandwidth to ensure sufficient correla-tion. When strain magnitude was 1.01%, the moiré superlat-tice exhibited a near-rectangular periodicity with highly localizedhole wavefunction distribution (Figure 4d). Simultaneously, thestrain-induced distortion of the wavefunction was observed, i.e.,the anisotropy emerged within the flat-band regime. Thus, weobtained a platform for studying the anisotropic Hubbard modelin rectangular superlattices.[37,38] Combined with shape tunabil-ity (Figure S11, Supporting Information), the rectangular moirésuperlattice in strained t-BL WSe2 shows tunable anisotropy inboth the shape and intercell hopping processes, whichmakes thematerial system an ideal fine-tuning quantum simulator for in-vestigating correlation-related phenomena.In addition, the distorted superlattice symmetry and bandstructure lead to modifications in optical properties, particularlyin the behavior of excitons trapped by the moiré potential, aspreviously reported.[13] With the established method for apply-ing controllable strain, it is now possible to systematically studythe detailed relationship between strain and excitonic behaviorthrough optical measurements. Given the nanometer-scale sizeof the moiré unit cell, high-resolution techniques, such as tip-enhanced Raman spectroscopy, present a promising direction forfuture studies.[39] This remains an intriguing aspect and will bea focus of future research.3. ConclusionWe demonstrated the application of intentional strain to t-BLWSe2 by fabricating the sample on a flexible substrate. The clearmoiré superlattice resolved by PFM proves the feasibility of theproposed structure, in which we successfully realized continu-ous distortion of the superlattice from triangular to rectangular.We further studied the strain effect on the band structure and re-vealed an anisotropic band structure with a flat-band nature andvHS. The methods and results of this study pave the way towardthe broader utility of 2D vdW heterostructures as highly tunablequantum simulators.4. Experimental SectionSample Fabrication: WSe2 monolayers were grown on SiO2/Si sub-strates by CVD process.[24] hBN crystals were purchased from 2D semi-conductors and mechanically exfoliated into flakes on a SiO2 (300 nm)/Sisubstrate; its thickness was estimated by optical microscopy and deter-mined by atomic force microscopy. The vdW stacks were fabricated us-ing a polymer-assisted transfer method. Two types of polymers were pre-pared: one was Elvacite 2552C dissolved in anisole (EA. weight ratio 1:1.5),and the other was a mixture of Elvacite with ionic liquid ([EMIM][TFSI])dissolved in anisole (EIA. weight ratio Elvacite:[EMIM][TFSI]: anisole =1:0.4:1.5). Prior to sample collection, small drops of both types of poly-mers were picked and transferred onto glass slides using a toothpick. Theslide glasses were then annealed at 180 °C in air for 1 h. EA polymer wasfirst used to pick up hBN flakes at 80 °C. It should be noted that the thick-ness of the hBN flakes is between 30 and 40 nm, which minimizes theeffect of the surface roughness of the Au/Ni film. The hBN was then usedto pick monolayer WSe2. hBN was first put into contact with the mono-layer at 90 °C, and the transfer stage was heated to 130 °C to increasethe contact area. Subsequently, the monolayer was picked up at 80 °C.The second monolayer was picked afterward, with twist angle control byrotating the transfer stage. The whole stack on EA polymer was then trans-ferred to EIA polymer at 40 °C. At this stage, the bilayer was encapsulatedby hBN and the EIA polymer. After transferring the EIA polymer (with thehBN-bilayer stack) onto the substrate, the bilayer was placed on the top fordirect PFM measurements. A flexible PEN film (thickness = 125 μm) wasfirst deposited with Ni (2 nm) and Au (10 nm) via vacuum evaporation.The EIA polymer was brought into contact with the substrate at 150 °C, atwhich the polymer would melt. By removing the substrate from the glassslide, part of the polymer was left on the substrate, with the hBN-bilayerstack buried underneath. The residual polymer was then washed away bysequentially immersing the substrate in chloroform (2 min), acetone (10min), and IPA (10 min).PFM Measurements: The moiré pattern of the near-0° bilayer WSe2sample #1 wasmeasured using the vertical PFMmode. Annealing was notperformed. The horizontal PFM mode was adopted for the near-1° bilayerWSe2 sample #2. To conduct PFM measurements on the bent substratemounted onmold, we first sandwiched the PEN substrate between the capand mold and heated the whole structure at 160 °C for 3 h to fix the shapeof the substrate. In order to conduct the PFMmeasurements, the cap wasremoved, leaving space for contact between the AFM cantilever and sur-face of the substrate. To mount the substrate on the mold without a cap,we additionally used the PDMS thin film (thickness = 165 μm) as an adhe-sion layer. PFM measurements were performed using a Veeco Multimode8 atomic force microscope in an ambient environment. We used the SCM-PIT-V2 cantilever, which has a force constant of ≈3 N m−1, for all PFMmeasurements. For the vertical PFM mode measurements, we selected acontact resonance frequency of ≈280 kHz; for the horizontal PFM modemeasurements, the contact resonance frequency was set to≈700 kHz. Theapplied a.c. bias was between 800 and 2000 mV, and the scan speed of thecantilever was set between 1.5 and 3 Hz for optimal resolution of the ac-quired images. For all measurements, the contact force was less than 60nN. The collected data was processed using the Gwyddion software. Nobackground subtraction process was adopted, and the scale range of eachimage was limited to within 3 × RMS for better visibility of the moiré pat-tern. The period L in long-period moiré structures with a stacking angleof approximately zero degrees (sample #1) was estimated from the PFMimage. It contains a very large error (±43 nm) due to the nonuniform dis-tortion in this sample. On the other hand, the moíre period in sample #2was estimated by FFT images.[9] To improve the signal-to-noise ratio ofthe obtained moíre pattern, the raw PFM images were Fourier filtered witha threshold of 70% of themaximum intensity. We further fit the fast Fouriertransform satellite peak intensity to a Gaussian function.Optical Measurements: Raman and photoluminescence spectra wererecorded using a JASCO NRS-5100 instrument. The wavelength of the ex-citation laser was 532 nm through a 100x lens, with a power of 130 μW.SHG measurements were performed using a home-built optical sys-tem with a linearly polarized femtosecond laser (80 MHz) as the excita-tion source. The wavelength of the laser was fixed at 900 nm, with a powerof 730 μW. The second harmonic signal was collected at a wavelength of450 nm. The angular dependence of the signal was measured by rotatingthe angle of the excitation laser using a 𝜆/2 waveplate.Supporting InformationSupporting Information is available from the Wiley Online Library or fromthe author.AcknowledgementsThe authors gratefully acknowledge M.S., Y.T., X.T.W., and X.Y.W. forfruitful discussions on the experimental design. This work was financiallysupported by JSPS KAKENHI Grant Numbers JP20H05867, JP20H05664,JP21H05232, JP21H05234, JP21H05235, JP21H05236, JP22K18986,JP22H00280, JP22H04957, JP22KJ1555, JP22H01899, JP23H05469 andJST, CREST Grant Numbers JPMJCR23A4 and JPMJCR23O3, Japan. H.O.is sponsored by the JSPS Research Fellowship.Small 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (6 of 7) 16136829, 2025, 25, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202407316 by National Institute For, Wiley Online Library on [18/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.small-journal.comwww.advancedsciencenews.com www.small-journal.comConflict of InterestThe authors declare no conflict of interest.Data Availability StatementThe data that support the findings of this study are available from the cor-responding author upon reasonable request.KeywordsMoiré superlattice, strain engineering, Transition metal dichalcogenides,van der Waals heterostructuresReceived: August 20, 2024Revised: December 25, 2024Published online: January 16, 2025[1] C. N. Lau, M. W. Bockrath, K. F. Mak, F. Zhang,Nature 2022, 602, 41.[2] K. F. Mak, J. Shan, Nat. Nanotechnol. 2022, 17, 686.[3] I. Bloch, J. Dalibard, S. Nascimbene, Nat. Phys. 2012, 8, 267.[4] M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D.N. Basov, A. N. Pasupathy, A. Rubio, Nat. Phys. 2021, 17, 155.[5] B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M.Noack, H. Shi, S. R. White, S. Zhang, G. K.-L. Chan, Science 2017,358, 1155.[6] E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F.Mak, T. Senthil, E. Tutuc, A. Yazdani, A. F. Young, Nat. Rev. Mater.2021, 6, 201.[7] F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, Y. Takahashi, Nat. Rev.Phys. 2020, 2, 411.[8] D. M. Kennes, L. Xian, M. Claassen, A. Rubio, Nat. Commun. 2020,11, 1124.[9] K. Kang, W. Zhao, Y. Zeng, K. Watanabe, T. Taniguchi, J. Shan, K. F.Mak, Nat. Nanotechnol. 2023, 18, 861.[10] L. Huder, A. Artaud, T. Le Quang, G. T. de Laissardière, A. G. M.Jansen, G. Lapertot, C. Chapelier, V. T. Renard, Phys. Rev. Lett. 2018,120, 156405.[11] Z. Bi, N. F. Q. Yuan, L. Fu, Phys. Rev. B 2019, 100, 035448.[12] J.-B. Qiao, L.-J. Yin, L. He, Phys. Rev. B 2018, 98, 235402.[13] Y. Bai, L. Zhou, J. Wang, W. Wu, L. J. McGilly, D. Halbertal, C. F. B. Lo,F. Liu, J. Ardelean, P. Rivera, N. R. Finney, X.-C. Yang, D. N. Basov, W.Yao, X. Xu, J. Hone, A. N. Pasupathy, X.-Y. Zhu, Nat. Mater. 2020, 19,1068.[14] M. Huang, Z. Wu, J. Hu, X. Cai, E. Li, L. An, X. Feng, Z. Ye, N. Lin, K.T. Law, N. Wang, Natl. Sci. Rev. 2023, 10, nwac232.[15] J. Pu, Y. Yomogida, K.-K. Liu, L.-J. Li, Y. Iwasa, T. Takenobu,Nano Lett.2012, 12, 4013.[16] J. Pu, K. Funahashi, C.-H. Chen, M.-Y. Li, L.-J. Li, T. Takenobu, Adv.Mater. 2016, 28, 4111.[17] J. Pu, W. Zhang, H. Matsuoka, Y. Kobayashi, Y. Takaguchi, Y. Miyata,K. Matsuda, Y. Miyauchi, T. Takenobu, Adv. Mater. 2021, 33, 2100601.[18] C. Zhang, C.-P. Chuu, X. Ren, M.-Y. Li, L.-J. Li, C. Jin, M.-Y. Chou, C.-K.Shih, Sci. Adv. 2017, 3, 1601459.[19] Z. Zhang, Y. Wang, K. Watanabe, T. Taniguchi, K. Ueno, E. Tutuc, B. J.LeRoy, Nat. Phys. 2020, 16, 1093.[20] L. J. McGilly, A. Kerelsky, N. R. Finney, K. Shapovalov, E.-M. Shih, A.Ghiotto, Y. Zeng, S. L.Moore,W.Wu, Y. Bai, K.Watanabe, T. Taniguchi,M. Stengel, L. Zhou, J. Hone, X. Zhu, D. N. Basov, C. Dean, C. E.Dreyer, A. N. Pasupathy, Nat. Nanotechnol. 2020, 15, 580.[21] K. Yasuda, X. Wang, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, Sci-ence 2021, 372, 1458.[22] X. Wang, K. Yasuda, Y. Zhang, S. Liu, K. Watanabe, T. Taniguchi, J.Hone, L. Fu, P. Jarillo-Herrero, Nat. Nanotechnol. 2022, 17, 367.[23] S. Masubuchi, M. Sakano, Y. Tanaka, Y. Wakafuji, T. Yamamoto, S.Okazaki, K. Watanabe, T. Taniguchi, J. Li, H. Ejima, T. Sasagawa, K.Ishizaka, T. Machida, Sci. Rep. 2022, 12, 10936.[24] N. Wada, J. Pu, Y. Takaguchi, W. Zhang, Z. Liu, T. Endo, T. Irisawa,K. Matsuda, Y. Miyauchi, T. Takenobu, Y. Miyata, Adv. Funct. Mater.2022, 32, 2203602.[25] S. B. Desai, S. R. Madhvapathy, M. Amani, D. Kiriya, M. Hettick, M.Tosun, Y. Zhou, M. Dubey, J. W. Ager III, D. Chrzan, A. Javey, Adv.Mater. 2016, 28, 4053.[26] J. Shim, S.-H. Bae, W. Kong, D. Lee, K. Qiao, D. Nezich, Y. J. Park,R. Zhao, S. Sundaram, X. Li, H. Yeon, C. Choi, H. Kum, R. Yue, G.Zhou, Y. Ou, K. Lee, J. Moodera, X. Zhao, J.-H. Ahn, C. Hinkle, A.Ougazzaden, J. Kim, Science 2018, 362, 665.[27] H. Ito, Y. Edagawa, J. Pu, H. Akutsu, M. Suda, H. M. Yamamoto, Y.Kawasugi, R. Haruki, R. Kumai, T. Takenobu, Phys. Status Solidi 2019,13, 1900162.[28] Y. Kobayashi, S. Yoshida, M. Maruyama, H. Mogi, K. Murase, Y.Maniwa, O. Takeuchi, S. Okada, H. Shigekawa, Y. Miyata, ACS Nano2019, 13, 7527.[29] G. Ru, W. Qi, K. Tang, Y. Wei, T. Xue, Tribol. Int. 2020, 151, 106483.[30] S. Zhu, P. Pochet, H. T. Johnson, ACS Nano 2019, 13, 6925.[31] T. Devakul, V. Crépel, Y. Zhang, L. Fu, Nat. Commun. 2021, 12, 6730.[32] A. Ziletti, S. M. Huang, D. F. Coker, H. Lin, Phys. Rev. B 2015, 92,085423.[33] Y. Hu, X. Wu, Y. Yang, S. Gao, N. C. Plumb, A. P. Schnyder, W. Xie, J.Ma, M. Shi, Sci. Adv. 2022, 8, eadd2024.[34] N. Jena, Dimple, R. A., A. Rawat, M. K. Mohanta, A. De Sarkar, Phys.Rev. B 2019, 100, 165413.[35] F, Gebhard, The Mott Metal-Insulator Transition: Models and Methods,Springer, Berlin, Heidelberg, 1997.[36] R. S. Markiewicz, Int. J. Mod. Phys. B 1991, 05, 2037.[37] M. Laubach, R. Thomale, C. Platt, W. Hanke, G. Li, Phys. Rev. B 2015,91, 245125.[38] A. Szasz, J. Motruk, Phys. Rev. B 2021, 103, 235132.[39] A. C. Gadelha, D. A. A. Ohlberg, C. Rabelo, E. G. S. Neto, T. L.Vasconcelos, J. L. Campos, J. S. Lemos, V. Ornelas, D. Miranda, R.Nadas, F. C. Santana, K. Watanabe, T. Taniguchi, B. Van Troeye, M.Lamparski, V. Meunier, V.-H. Nguyen, D. Paszko, J.-C. Charlier, L. C.Campos, L. G. Cançado, G. Medeiros-Ribeiro, A. Jorio, Nature 2021,590, 405.Small 2025, 21, 2407316 © 2025 The Author(s). Small published by Wiley-VCH GmbH2407316 (7 of 7) 16136829, 2025, 25, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/smll.202407316 by National Institute For, Wiley Online Library on [18/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.small-journal.com Continuous Strain Modulation of Moir80é Superlattice Symmetry From Triangle to Rectangle 1. Introduction 2. Results and Discussion 3. Conclusion 4. Experimental Section Supporting Information Acknowledgements Conflict of Interest Data Availability Statement Keywords