Abstract
Imposing incommensurable periodicity on the periodic atomic lattice can lead to complex structural phases consisting of locally periodic structure bounded by topological defects1,2,3,4,5,6,7,8. Twisted trilayer graphene (TTG) is an ideal material platform to study the interplay between different atomic periodicities, which can be tuned by twist angles between the layers, leading to moiré-of-moiré lattices9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. Interlayer and intralayer interactions between two interfaces in TTG transform this moiré-of-moiré lattice into an intricate network of ___domain structures at small twist angles, which can harbour exotic electronic behaviours9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. Here we report a complete structural phase diagram of TTG with atomic-scale lattice reconstruction. Using transmission electron microscopy (TEM) combined with a new interatomic potential simulation27,28, we show several large-scale moiré lattices, including triangular, kagome and a corner-shared hexagram-shaped ___domain pattern. Each ___domain is bounded by a 2D network of ___domain-wall lattices. In the limit of small twist angles, two competing structural orders—rhombohedral and Bernal stackings—with a slight energy difference cause unconventional lattice reconstruction with spontaneous symmetry breaking (SSB) and nematic instability, highlighting the importance of long-range interlayer interactions across entire van der Waals layers. The diverse tessellation of distinct domains, whose topological network can be tuned by the adjustment of the twist angles, establishes TTG as a platform for exploring the interplay between emerging quantum properties and controllable nontrivial lattices.
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Data availability
The data that support the findings of this study are presented in the paper, Extended Data and Supplementary Information. Any other relevant data are available from the corresponding authors on request.
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Acknowledgements
D.P., B.K. and H.Y. were supported by the National Research Foundation of Korea (NRF) (grant no. RS-2021-NR061606). H.Y. was supported by the NRF (grant nos. RS-2021-NR060087 and NRF-2022R1A4A1033562) and POSCO Science Fellowship of POSCO TJ Park Foundation. C.P. was supported by the new generation research programme (CG079701) at Korea Institute for Advanced Study (KIAS). E.K. was supported by a KIAS individual grant (CG075002). K.Y. was supported by a KIAS individual grant (CG092501). Y.-W.S. was supported by the KIAS individual grant (CG031509). TEM work was supported by the Center for Materials Analysis at Research Institute of Advanced Materials (RIAM), Seoul National University and the Center for Nano Materials at Sogang University. Computations were supported by the Center for Advanced Computation of KIAS. X.Z., K.D., M.G. and K. Wang were supported by NSF DMREF award 1922165. H.-M.K., S.-G.K. and H.K. acknowledge support from Material Innovation Leading Project through the NRF financed by the Ministry of Science and ICT (2020M3H4A3081879). S.M.Y. was supported by the NRF (grant no. RS-2023-NR076385). P.K. acknowledges the support from DOE (DE-SC0012260) for sample preparation and ARO grant W911NF-21-2-0147 for analysis. K. Watanabe and T.T. acknowledge support from the JSPS KAKENHI (grant no. 21H05233 and 23H02052), the CREST (JPMJCR24A5), JST and World Premier International Research Center Initiative (WPI), MEXT, Japan.
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H.Y. conceived the experiments. D.P., B.K., R.E., X.Z., K.D., M.G., S.H.P. and J.H.L. fabricated samples. D.P., B.K., R.E., H.-M.K., S.-G.K. and H.K. performed TEM experiments. D.P., C.P., E.K., B.K., H.-M.K., S.M.Y., K. Wang, P.K., Y.-W.S. and H.Y. analysed the data. C.P., E.K., K.Y. and Y.-W.S. performed theoretical analysis. K. Watanabet and T.T. grew bulk h-BN crystals. D.P., C.P., Y.-W.S. and H.Y. wrote the manuscript. All authors contributed to the overall scientific interpretation and edited the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Domain contrast image obtained by DF TEM analysis.
a, SAED pattern obtained from TTG. The first-order Bragg peak (\(g=10\bar{1}0\)) is marked with the black dashed circle and the three sets of second-order Bragg peaks, \(g=\bar{1}2\bar{1}0\), \(g=11\bar{2}0\) and \(g=2\bar{1}\bar{1}0\), are marked with red, blue and green dashed circles, respectively. A grey dashed line is drawn to denote the tilt axis. b, A series of DF TEM images obtained as a function of specimen tilt angle in the electron microscope. Four distinct domains are marked as A, B, C and D. c, Electron diffraction intensities experimentally measured from the four different ___domain regions as a function of tilt angle of the specimen. The red, green, black and blue curves correspond to the diffraction intensities obtained from the domains marked as A, B, C and D, respectively. The error bars represent the standard deviations of the intensities obtained from different domains with identical atomic configurations. d, Simulated electron diffraction intensities obtained from different stacking orders in trilayer graphene. The red, green, black and blue curves correspond to the simulated diffraction intensity obtained from ACB, ACA, ABA and ABC stacking orders, respectively.
Extended Data Fig. 2 DF TEM ___domain boundary contrast image obtained by DF TEM analysis.
a–c, DF TEM images obtained from the second-order Bragg peaks. \(g=\bar{1}2\bar{1}0\), \(g=11\bar{2}0\) and \(g=2\bar{1}\bar{1}0\) Bragg peaks are used to obtain the DF images shown in a, b and c, respectively. d, Schematic illustrating the displacement vectors Δi (i = 1, 2, 3) associated with the ___domain boundaries. The displacement vectors Δi are drawn on top of the atomic structure to denote the directions and magnitudes of the displacements. e, Composite-colour DF TEM image obtained from the three sets of DF images shown in Extended Data Fig. 2a–c by summing them after colour contrasting the individual images. As a result, the coloured lines indicate the ___domain walls with the characteristic displacement vectors shown in d. f,g, Distinct ___domain boundary networks formed at the adjacent interfaces. Domain boundary network formed at the bottom interface (f) and at the top interface (g) are drawn as coloured dashed lines. When the bottom interface is twisted with finite angle (θ23 ≠ 0°), a triangular ___domain-wall network appears as a result of the lattice reconstruction as in f. When the top interface exhibits the twist angle θ12 = 0°, a parallel array of ___domain walls appears. The insets in f and g represent the schematic drawings for kagome-like ___domain lattices, in which black and white regions represent the rhombohedral stacking orders and the grey regions indicate the Bernal stacking orders. The ___domain walls are visualized with distinct coloured lines, corresponding to the displacement vectors shown in d with the same colours. In the twisted interface (f), the directions of each ___domain wall are mostly parallel with the displacement vector, indicating that all of the ___domain walls can be characterized with the shear type of displacements. In the untwisted interface (g), one type of ___domain wall (marked by the red lines in g) exhibits the displacement vector that has a non-zero orthogonal component to the ___domain-wall direction, revealing that the uniaxial displacement is incorporated along such ___domain walls.
Extended Data Fig. 3 SSB lattice reconstruction with distinct nematic orders.
a, Low-magnification DF TEM image using the first-order Bragg peak to visualize the ___domain contrast. Inset shows the corresponding SAED pattern. b, \(g=10\bar{1}0\), \(g=2\bar{1}\bar{1}0\), \(g=11\bar{2}0\) and \(g=\bar{1}2\bar{1}0\) DF TEM images obtained from the square region (i) in a. c, \(g=\bar{1}010\), \(g=\bar{2}110\), \(g=\bar{1}\bar{1}20\) and \(g=1\bar{2}10\) DF TEM images obtained from the square region (ii) in a. Figure 2h,i is obtained by summing the four different DF TEM images shown in b and c, respectively, to visualize the ___domain and ___domain-wall contrasts. Note that the nematic boundaries visualized in the \(g=2\bar{1}\bar{1}0\) DF TEM image in b and the \(g=1\bar{2}10\) DF TEM image in c adopt different directions, which are rotated 60° with respect to each other.
Extended Data Fig. 4 Domain boundary contrast images obtained from TTG.
a–f, SAED (panels (i)) and DF TEM images (panels (ii)–(iv)) using the three sets of second-order Bragg peaks obtained from TTG with various twist-angle combinations. Each of the Bragg peaks used to obtain DF TEM images shown in panels (ii)–(iv) are marked with red, green and blue arrows in the SAED images. SAED and DF TEM images are obtained for the experimentally observed phases shown in the main text (Fig. 3): simple triangular ___domain lattice (a), simple triangular ___domain lattice bound by moiré-of-moiré boundaries (b), double-coloured triangular ___domain lattice (c), kagome-like ___domain lattice (d), coloured triangular ___domain lattice (e) and hexagram ___domain lattice (f).
Extended Data Fig. 5 Structural transition between different ___domain lattices.
a,b, Domain contrast image (\(g=10\bar{1}0\) DF TEM image (a)) and the ___domain boundary contrast image (summation of \(g=2\bar{1}\bar{1}0\), \(g=11\bar{2}0\) and \(g=\bar{1}2\bar{1}0\) DF TEM images (b)) obtained from the same region of the TTG specimen. Transition from the simple triangular ___domain lattice bound by moiré-of-moiré boundaries (left) to the double-coloured triangular ___domain lattice near the commensurate condition of 2θ12 + θ23 = 0 (centre) to another locally commensurate coloured triangular ___domain lattice (right). c,d, Domain contrast image (\(g=10\bar{1}0\) DF TEM image (c)) and the ___domain boundary contrast image (summation of \(g=2\bar{1}\bar{1}0\), \(g=11\bar{2}0\) and \(g=\,\bar{1}2\bar{1}0\) DF TEM images (d)) obtained from another region of the TTG specimen. Transition from the simple triangular ___domain lattice (left) to the kagome-like ___domain lattice (right) is shown to exhibit complex local minimum ___domain configuration in the middle.
Extended Data Fig. 6 Modified KC potential.
Spurious out-of-plane displacement of Bernal-stacked (a) and rhombohedral-stacked (b) trilayer graphene. Red arrows indicate tiny but finite displacements around 1.4 × 10−3 Angstrom. c, Gaussian-type next-nearest-neighbour interlayer interaction potential that depends only on the plane-projected distance r∥ between carbon atoms on the top and bottom layers. d, The energy difference between Bernal (EB) and rhombohedral (ER) trilayer graphene as a function of control parameter V. Without V, EB is larger than ER with KC potential. With V > 0.15 meV, the Bernal stacking becomes more stable than the rhombohedral stacking.
Extended Data Fig. 7 Comparison of relaxed structures with different total energy configurations.
a, Simulated TEM images for the alternate stacking case of θ12 + θ23 = 0°, corresponding to panels (i) in Fig. 3. Grey colour corresponds to Bernal stacking and black and white colours to rhombohedral stacking. The left panel is a moiré ___domain pattern in the case of Bernal stacking in the ground state and the right panel shows the fully relaxed ___domain pattern in the case that rhombohedral stacking has a lower total energy than Bernal stacking. b, Simulated TEM images for the helical stacking case of θ12 − θ23 = 0°, corresponding to the panels (vi). The left and right panels are obtained with same conditions used in a and used same colour scheme. Scale bars, 100 nm.
Extended Data Fig. 8 Structures and energies of the ___domain wall in TTG.
Stacking ___domain structures of TTG for θ12 = −θ23 = θ (a) and θ12 = 0, θ23 = θ (b). The height variation dz along the blue (Bernal–Bernal boundary) and red (Bernal–rhombohedral boundary) arrows are drawn in c and d, respectively. In c and d, from the top to bottom are line profiles of dz for the top, middle and bottom layers, respectively. e, Energies (E) of TTG as a function of θ for the cases in a and b are plotted with blue and red circles, respectively. From the slopes of E(θ), the energies of ___domain walls are estimated to be 137 and 84 meV Å−1, respectively.
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Supplementary Information
Supplementary sections 1–13, including Supplementary Figs. 1–18 and Supplementary Table 1.
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Park, D., Park, C., Yananose, K. et al. Unconventional ___domain tessellations in moiré-of-moiré lattices. Nature 641, 896–903 (2025). https://doi.org/10.1038/s41586-025-08932-0
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DOI: https://doi.org/10.1038/s41586-025-08932-0
