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Strain fields in twisted bilayer graphene


Van der Waals heteroepitaxy allows deterministic control over lattice mismatch or azimuthal orientation between atomic layers to produce long-wavelength superlattices. The resulting electronic phases depend critically on the superlattice periodicity and localized structural deformations that introduce disorder and strain. In this study we used Bragg interferometry to capture atomic displacement fields in twisted bilayer graphene with twist angles <2°. Nanoscale spatial fluctuations in twist angle and uniaxial heterostrain were statistically evaluated, revealing the prevalence of short-range disorder in moiré heterostructures. By quantitatively mapping strain tensor fields, we uncovered two regimes of structural relaxation and disentangled the electronic contributions of constituent rotation modes. Further, we found that applied heterostrain accumulates anisotropically in saddle-point regions, generating distinctive striped strain phases. Our results establish the reconstruction mechanics underpinning the twist-angle-dependent electronic behaviour of twisted bilayer graphene and provide a framework for directly visualizing structural relaxation, disorder and strain in moiré materials.

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Fig. 1: Four-dimensional STEM Bragg interferometry of TBG.
Fig. 2: Short-range disorder and geometry analysis of TBG.
Fig. 3: Strain mapping of TBG.
Fig. 4: Regimes of reconstruction in TBG.
Fig. 5: Effects of isolated relaxation modes on band structure.
Fig. 6: Visualizing the effect of uniaxial heterostrain on TBG reconstruction.

Data availability

The datasets generated and/or analysed during the current study have been made publicly available from the Zenodo repository. Digital object identifiers (DOIs) to these datasets are provided in refs. 12–30 of the Supplementary Information. Additional information on these datasets and the manuscript figures to which each dataset corresponds are detailed in Supplementary Table 4.

Code availability

The computer code used for the dataset processing and strain analysis has been made publicly available at


  1. 1.

    Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).

    CAS  Article  Google Scholar 

  2. 2.

    Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).

    CAS  Article  Google Scholar 

  3. 3.

    Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    CAS  Article  Google Scholar 

  4. 4.

    Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).

    CAS  Article  Google Scholar 

  5. 5.

    Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    CAS  Article  Google Scholar 

  6. 6.

    Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    CAS  Article  Google Scholar 

  7. 7.

    Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    CAS  Article  Google Scholar 

  8. 8.

    Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    CAS  Article  Google Scholar 

  9. 9.

    Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).

    CAS  Article  Google Scholar 

  10. 10.

    Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).

    CAS  Article  Google Scholar 

  11. 11.

    Jin, C. et al. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature 567, 76–80 (2019).

    CAS  Article  Google Scholar 

  12. 12.

    Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    CAS  Article  Google Scholar 

  13. 13.

    Woods, C. R. et al. Commensurate–incommensurate transition in graphene on hexagonal boron nitride. Nat. Phys. 10, 451–456 (2014).

    CAS  Article  Google Scholar 

  14. 14.

    van Wijk, M. M., Schuring, A., Katsnelson, M. I. & Fasolino, A. Relaxation of moiré patterns for slightly misaligned identical lattices: graphene on graphite. 2D Mater. 2, 034010 (2015).

    Article  CAS  Google Scholar 

  15. 15.

    Dai, S., Xiang, Y. & Srolovitz, D. J. Twisted bilayer graphene: moiré with a twist. Nano Lett. 16, 5923–5927 (2016).

    CAS  Article  Google Scholar 

  16. 16.

    Jain, S. K., Juričić, V. & Barkema, G. T. Structure of twisted and buckled bilayer graphene. 2D Mater. 4, 015018 (2016).

    Article  CAS  Google Scholar 

  17. 17.

    Nam, N. N. T. & Koshino, M. Lattice relaxation and energy band modulation in twisted bilayer graphene. Phys. Rev. B 96, 075311 (2017).

    Article  Google Scholar 

  18. 18.

    Zhang, K. & Tadmor, E. B. Structural and electron diffraction scaling of twisted graphene bilayers. J. Mech. Phys. Solids 112, 225–238 (2018).

    Article  CAS  Google Scholar 

  19. 19.

    Yoo, H. et al. Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene. Nat. Mater. 18, 448–453 (2019).

    CAS  Article  Google Scholar 

  20. 20.

    Rosenberger, M. R. et al. Twist angle-dependent atomic reconstruction and moiré patterns in transition metal dichalcogenide heterostructures. ACS Nano 14, 4550–4558 (2020).

    CAS  Article  Google Scholar 

  21. 21.

    Weston, A. et al. Atomic reconstruction in twisted bilayers of transition metal dichalcogenides. Nat. Nanotechnol. 15, 592–597 (2020).

    CAS  Article  Google Scholar 

  22. 22.

    Alden, J. S. et al. Strain solitons and topological defects in bilayer graphene. Proc. Natl Acad. Sci. USA 110, 11256–11260 (2013).

    CAS  Article  Google Scholar 

  23. 23.

    Uri, A. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581, 47–52 (2020).

    CAS  Article  Google Scholar 

  24. 24.

    Wilson, J. H., Fu, Y., Das Sarma, S. & Pixley, J. H. Disorder in twisted bilayer graphene. Phys. Rev. Res. 2, 023325 (2020).

    CAS  Article  Google Scholar 

  25. 25.

    Kerelsky, A. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

    CAS  Article  Google Scholar 

  26. 26.

    Huder, L. et al. Electronic spectrum of twisted graphene layers under heterostrain. Phys. Rev. Lett. 120, 156405 (2018).

    CAS  Article  Google Scholar 

  27. 27.

    Bi, Z., Yuan, N. F. Q. & Fu, L. Designing flat bands by strain. Phys. Rev. B 100, 035448 (2019).

    CAS  Article  Google Scholar 

  28. 28.

    McGilly, L. J. et al. Visualization of moiré superlattices. Nat. Nanotechnol. 15, 580–584 (2020).

    CAS  Article  Google Scholar 

  29. 29.

    Han, Y. et al. Strain mapping of two-dimensional heterostructures with subpicometer precision. Nano Lett. 18, 3746–3751 (2018).

    CAS  Article  Google Scholar 

  30. 30.

    Yang, H. et al. 4D STEM: high efficiency phase contrast imaging using a fast pixelated detector. J. Phys. Conf. Ser. 644, 012032 (2015).

    Article  CAS  Google Scholar 

  31. 31.

    Jiang, Y. et al. Electron ptychography of 2D materials to deep sub-ångström resolution. Nature 559, 343–349 (2018).

    CAS  Article  Google Scholar 

  32. 32.

    Ophus, C. Four-dimensional scanning transmission electron microscopy (4D-STEM): from scanning nanodiffraction to ptychography and beyond. Microsc. Microanal. 25, 563–582 (2019).

    CAS  Article  Google Scholar 

  33. 33.

    Kim, K. et al. van der Waals heterostructures with high accuracy rotational alignment. Nano Lett. 16, 1989–1995 (2016).

    CAS  Article  Google Scholar 

  34. 34.

    Ozdol, V. B. et al. Strain mapping at nanometer resolution using advanced nano-beam electron diffraction. Appl. Phys. Lett. 106, 253107 (2015).

    Article  CAS  Google Scholar 

  35. 35.

    Kelly, P. Solid Mechanics Lecture Notes (Univ. of Auckland, 2013).

  36. 36.

    McGinty, B. Continuum Mechanics (2012);

  37. 37.

    Butz, B. et al. Dislocations in bilayer graphene. Nature 505, 533–537 (2014).

    CAS  Article  Google Scholar 

  38. 38.

    Fang, S. & Kaxiras, E. Electronic structure theory of weakly interacting bilayers. Phys. Rev. B 93, 235153 (2016).

    Article  CAS  Google Scholar 

  39. 39.

    Carr, S., Fang, S., Zhu, Z. & Kaxiras, E. Exact continuum model for low energy electronic states of twisted bilayer graphene. Phys. Rev. Res. 1, 013001 (2019).

    CAS  Article  Google Scholar 

  40. 40.

    Guinea, F. & Walet, N. R. Continuum models for twisted bilayer graphene: effect of lattice deformation and hopping parameters. Phys. Rev. B 99, 205134 (2019).

    CAS  Article  Google Scholar 

  41. 41.

    Latychevskaia, T. et al. Holographic reconstruction of interlayer distance of bilayer two-dimensional crystal samples from their convergent beam electron diffraction patterns. Ultramicroscopy 219, 113021 (2020).

    Article  CAS  Google Scholar 

  42. 42.

    Cao, Y. et al. Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016).

    CAS  Article  Google Scholar 

  43. 43.

    Boresi, A. P. & Schmidt, R. J. in Advanced Mechanics of Materials 55–72 (Wiley, 2003).

  44. 44.

    Metcalf, T. R. Resolving the 180-degree ambiguity in vector magnetic field measurements: the ‘minimum’ energy solution. Sol. Phys. 155, 235–242 (1994).

    Article  Google Scholar 

  45. 45.

    Lu, W. et al. Implementation of higher-order variational models made easy for image processing. Math. Methods Appl. Sci. 39, 4208–4233 (2016).

    Article  Google Scholar 

  46. 46.

    Duan, J. et al. An edge-weighted second order variational model for image decomposition. Digit. Signal Process. 49, 162–181 (2016).

    Article  Google Scholar 

  47. 47.

    Savitzky, B. et al. py4DSTEM: a software package for multimodal analysis of four-dimensional scanning transmission electron microscopy datasets. Preprint at (2020).

  48. 48.

    Ophus, C., Ciston, J. & Nelson, C. T. Correcting nonlinear drift distortion of scanning probe and scanning transmission electron microscopies from image pairs with orthogonal scan directions. Ultramicroscopy 162, 1–9 (2016).

    CAS  Article  Google Scholar 

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We thank P. Kim for discussions. The major experimental work was supported by the Office of Naval Research Young Investigator Program under award no. N00014-19-1-2199 (D.K.B.). M.V.W. acknowledges support from an NSF GRFP award and a UC Berkeley Chancellor’s Fellowship. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under contract no. DE-AC02-05CH1123. C.O. acknowledges the support of the Department of Energy Early Career Research Award programme. S.C. acknowledges support from the NSF through grant no. OIA-1921199. J.C. and H.G.B. acknowledge support from the Presidential Early Career Award for Scientists and Engineers (PECASE) through the US Department of Energy. D.K.B. acknowledges support from the Rose Hills Foundation through the Rose Hills Innovator Program. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant no. JPMXP0112101001, JSPS KAKENHI grant no. JP20H00354 and the CREST (JPMJCR15F3), JST.

Author information




N.P.K., M.V.W., C.O., K.C.B., H.G.B. and D.K.B. conceived the study. M.V.W. designed and fabricated the samples. M.V.W., K.C.B. and J.C. acquired the 4D-STEM data. N.P.K., C.O. and H.G.B. created the data analysis code. S.C. carried out the band structure calculations and finite-element modelling. T.T. and K.W. provided the bulk hBN crystals. N.P.K. and M.V.W. processed the data. N.P.K., M.V.W. and D.K.B. analysed the data and wrote the manuscript. All the authors contributed to the overall scientific interpretation and edited the manuscript.

Corresponding author

Correspondence to D. Kwabena Bediako.

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The authors declare no competing interests.

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Peer review information Nature Materials thanks Jeil Jung, Jannik Meyer 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.

Supplementary information

Supplementary Information

Supplementary Sections 1–12, Figs. 1–28 and Tables 1–3.

Supplementary Table 4

Details of all relevant 4D-STEM datasets (uploaded to Zenodo repository) corresponding to refs. 12–30 of the Supplementary Informaton.

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Kazmierczak, N.P., Van Winkle, M., Ophus, C. et al. Strain fields in twisted bilayer graphene. Nat. Mater. (2021).

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