Imaging supermoiré relaxation in helical trilayer graphene

Nature Materials (2026)Cite this article In twisted van der Waals materials, local atomic relaxation can alter the underlying electronic structure. Characterizing lattice reconstruction and its susceptibility to strain is essential for understanding emergent electronic states, especially in multilayers in which interference between moiré lattices yields larger supermoiré patterns whose energy is highly sensitive to local stacking. Here we image spatial modulations in the electronic character of helical trilayer graphene, which indicate relaxation into a superstructure of large domains with uniform moiré periodicity. We show that the supermoiré domain size is increased by strain and can be altered in the same device while preserving the local properties within each domain. Finally, we observe a higher conductance at the domain boundaries, consistent with predictions that they host counterpropagating edge modes. Our work provides a real-space visualization of moiré-periodic domains, reveals two independently tunable length scales and demonstrates strain engineering as a route towards designing correlated topological networks at the supermoiré scale.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutThe data that support the findings of this study are available via Zenodo at https://doi.org/10.5281/zenodo.17365682 (ref. 56).Nam, N. N. & Koshino, M. Lattice relaxation and energy band modulation in twisted bilayer graphene. Phys. Rev. B 96, 075311 (2017).Article Google Scholar Carr, S. et al. 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Fractional quantum anomalous Hall effect in multilayer graphene. Nature 626, 759–764 (2024).Article CAS PubMed Google Scholar Hoke, J. Imaging supermoiré relaxation in helical trilayer graphene. Zenodo https://doi.org/10.5281/zenodo.17365682 (2025).Download referencesWe thank A. Sharpe, A. Uri, S. de la Barrera, L.-Q. Xia, A. Young and P. Kim for helpful discussions. This work was supported by the QSQM, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award number DE-SC0021238. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant numbers 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. J.C.H. acknowledges support from the Stanford Q-FARM Quantum Science and Engineering Fellowship. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award number ECCS-2026822.These authors contributed equally: Jesse C. Hoke, Yifan Li, Yuwen Hu.Department of Physics, Stanford University, Stanford, CA, USAJesse C. Hoke, Yifan Li, Yuwen Hu, Julian May-Mann, Trithep Devakul & Benjamin E. FeldmanGeballe Laboratory for Advanced Materials, Stanford, CA, USAJesse C. Hoke, Yifan Li, Yuwen Hu & Benjamin E. FeldmanStanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, USAJesse C. Hoke, Yifan Li, Yuwen Hu & Benjamin E. FeldmanResearch Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, JapanKenji WatanabeResearch Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, JapanTakashi TaniguchiSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarJ.C.H. and Y.L. fabricated the device. J.C.H., Y.L. and Y.H. conducted the SET measurements. J.M.-M., T.D., Y.L., Y.H. and J.C.H. performed the theoretical calculations. T.D. and B.E.F. supervised the project. K.W. and T.T. provided the hBN crystals. All authors contributed to the analysis and writing of the paper.Correspondence to Benjamin E. Feldman.The authors declare no competing interests.Nature Materials thanks Nicolas Leconte and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Schematic of the local stacking configurations within h and \(\bar{h}\) domains and AAA sites. The domains exhibit moiré-periodic order: a honeycomb lattice of alternating AAB and BAA stackings (orange and purple, respectively) with uniform moiré wavelength λM, surrounded by local ABA stacking (white). The h and \(\bar{h}\) domains are related by a C2z transformation. The outermost schematics show the corresponding local real-space alignment of the bottom, middle, and top graphene layers in red, green, and blue, respectively. Here, A and B refer to the sublattices of the monolayer graphene honeycomb lattices.a, Optical micrograph of the sample overlaid with the rasterized trajectory of the SET tip. b, dμ/dn at T = 1.6 K in the vicinity of ν = 4 along the trajectory in a. A single peak is observed throughout. c, Spatial map of θ determined from the location of the ν = 4 peak in b. θ varies by less than 0. 1∘ across the entire area, highlighting the high degree of uniformity in the device.a-c, Spatial dependence of dμ/dn in a 1 × 1 μm2 area of the sample at ν = 4 (a), ν = − 4 (b) and ν = 0 (c). Scale bar: 250 nm. Similar supermoiré modulations are visible for each incompressible state. This is consistent with the theoretical expectations that valley-contrasting domain boundary modes are present for each of these states. Data measured at T = 1.6 K for ∣ν∣ = 4 and at T = 330 mK for ν = 0.Spatial dependence of dμ/dn at T = 330 mK in a 1 × 1 μm2 area of the sample for different incompressible states, respectively labeled by their Chern number C and zero-field intercept (C, ν). Scale bar: 250 nm. Similar supermoiré modulations occur in all cases. This is again consistent with theoretical expectations that the valley-contrasting domain boundary modes are also stable in the presence of a magnetic field.a, A representative plot of dμ/dn near ν = 4. Δn denotes the full width half maximum (FWHM) of the dμ/dn peak. Blue dots are data and the dark blue curve is a Gaussian fit. b, Map of the spatial distribution of Δn. Scale bar: 500 nm. c, The corresponding local twist angle disorder Δθ = θ(nmax + Δn/2) − θ(nmax − Δn/2), where nmax is the density where the Gaussian peak is maximized and θ(n) is given by the equation in the “Twist angle determination" subsection in the Methods. From the data, we estimate the median Δθ across the sample to be Δθ = 0.012∘. This is (up to factors of order unity) an effective bound on the overall twist angle variability on length scales smaller than the ~ 100 nm spatial resolution of the SET probe. Notably, Δθ exhibits a periodic spatial dependence at the supermoiré scale. It is enhanced along domain boundaries and maximized at the AAA sites, while it is minimized within the domains. This observation is consistent with theoretical calculations of the relaxed HTG lattice structure: within the domains, the system is expected to be moiré-periodic and can be characterized by a single (relaxed) twist angle, whereas at the domain walls and AAA sites, the lattice structure is predicted to be moiré-aperiodic. The aperiodicity is expected to be larger at the AAA sites than at the domain walls. The black dashed lines indicate the domain walls, while black dots at their intersections correspond to the AAA sites.a-b, Stitched images of dμ/dn at ν = 4 during the first (a) and second (b) rounds of measurement. Scale bar: 1 μm. All data measured at T = 1.6 K.Hofstadter spectrum of the h domain. Colors indicate separate contributions from the K (red) and \({K}^{{\prime} }\) (blue) valleys, whose spectra are distinct due to C2z symmetry breaking. The Hofstadter spectrum of \(\bar{h}\) domains is identical, except that K and \({K}^{{\prime} }\) are exchanged.a, Spatial map of θ, reproduced from Extended Data Fig.2. b, Peak dμ/dn at ν = 4 within the black box in a. The data are identical to that in Fig. 2b. c, dμ/dn at T = 330 mK as a function of n and B in nine select locations (colored dots in a and b). Each Landau fan shows qualitatively similar behavior despite spanning different locations across several microns and representing a range of twist angles, differing apparent strain profiles, and different high symmetry locations within the supermoiré lattice.a-b, dμ/dn at T = 330 mK as function of n and B, respectively measured in the center of a domain during the first (a) and second (b) round of measurements. The data in a are identical to Fig. 4a and are reproduced here for ease of comparison.a, Optical image of the graphene flakes used in the HTG device after being cut into four pieces with an atomic force microscope. b, The final stack immediately after deposition onto a Si/SiO2 substrate. A small overlap between the sacrificial layer (black) and the top graphene layer (blue) was used to prevent relative sliding between the hBN and top graphene layer. The relative angles between the bottom (red), middle (green), and top (blue) graphene flakes match closely to the target twist angles during stacking. c, Image of the final device after etching and metallization.Supplementary Sections 1–6 and Figs. 1–19.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsHoke, J.C., Li, Y., Hu, Y. et al. Imaging supermoiré relaxation in helical trilayer graphene. Nat. Mater. (2026). https://doi.org/10.1038/s41563-025-02423-3Download citationReceived: 27 November 2024Accepted: 28 October 2025Published: 05 January 2026Version of record: 05 January 2026DOI: https://doi.org/10.1038/s41563-025-02423-3Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative