- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Straintronics and twistronics in bilayer graphene

Federico Escudero1,*, Dong Wang2,*, Pierre A. Pantaleón1, Shengjun Yuan2,3,4, Francisco Guinea1,5, and Zhen Zhan1,†

  • 1IMDEA Nanoscience, Faraday 9, Madrid 28049, Spain
  • 2Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
  • 3School of Artificial Intelligence, Wuhan University, Wuhan 430072, China
  • 4Wuhan Institute of Quantum Technology, Wuhan 430206, China
  • 5Donostia International Physics Center, Paseo Manuel de Lardizábal 4, San Sebastián 20018, Spain

  • *These authors contributed equally to this work.
  • †Contact author: zhenzhanh@gmail.com
PDF

Phys. Rev. Research 8, 023225 – Published 29 May, 2026

DOI: https://doi.org/10.1103/r8sv-6xmp

Abstract

The interplay of twist and strain in bilayer graphene enables the formation of moiré patterns and narrow bands that host correlated and topological phases. While magic-angle twisted bilayer graphene has been widely studied, strain provides an additional and realistic control knob for band engineering. In this work, we first generate a global method to construct commensurate supercells for arbitrary twist and heterostrain. Then, using atomistic tight-binding and strain-extended continuum models to study the commensurate structures, we identify configurations that minimize the bandwidth beyond the magic angle. The results reveal a strong dependence of band narrowing and topology on strain type, magnitude, direction, and lattice relaxation. Particularly, shear strain produces a stronger distortion than uniaxial strain. Including electron-electron interactions through a self-consistent Hartree potential shows that strain broadens the bare bands while reducing electrostatic renormalization. Strain also drives topological transitions as the narrow and remote bands hybridize, establishing twisted and strained bilayer graphene as a tunable platform for flat-band and topological phenomena.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (130)

  1. E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nat. Rev. Mater. 6, 201 (2021).
  2. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  3. 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).
  4. A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean, A. Rubio, and A. N. Pasupathy, Maximized electron interactions at the magic angle in twisted bilayer graphene, Nature (London) 572, 95 (2019).
  5. M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363, 1059 (2019).
  6. X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al., Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature (London) 574, 653 (2019).
  7. Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young, Independent superconductors and correlated insulators in twisted bilayer graphene, Nat. Phys. 16, 926 (2020).
  8. U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern, et al., Cascade of phase transitions and Dirac revivals in magic-angle graphene, Nature (London) 582, 203 (2020).
  9. 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).
  10. Y. Choi, H. Kim, Y. Peng, A. Thomson, C. Lewandowski, R. Polski, Y. Zhang, H. S. Arora, K. Watanabe, T. Taniguchi, et al., Correlation-driven topological phases in magic-angle twisted bilayer graphene, Nature (London) 589, 536 (2021).
  11. Y. Cao, D. Rodan-Legrain, J. M. Park, N. F. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. Jarillo-Herrero, Nematicity and competing orders in superconducting magic-angle graphene, Science 372, 264 (2021).
  12. J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Graphene bilayer with a twist: Electronic structure, Phys. Rev. Lett. 99, 256802 (2007).
  13. G. Trambly de Laissardière, D. Mayou, and L. Magaud, Localization of Dirac electrons in rotated graphene bilayers, Nano Lett. 10, 804 (2010).
  14. E. Suárez Morell, J. D. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, Flat bands in slightly twisted bilayer graphene: Tight-binding calculations, Phys. Rev. B 82, 121407(R) (2010).
  15. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  16. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  17. P. Moon and M. Koshino, Optical absorption in twisted bilayer graphene, Phys. Rev. B 87, 205404 (2013).
  18. M. Koshino, Interlayer interaction in general incommensurate atomic layers, New J. Phys. 17, 015014 (2015).
  19. P. San-Jose, J. González, and F. Guinea, Non-Abelian gauge potentials in graphene bilayers, Phys. Rev. Lett. 108, 216802 (2012).
  20. Z. Bi, N. F. Q. Yuan, and L. Fu, Designing flat bands by strain, Phys. Rev. B 100, 035448 (2019).
  21. F. Escudero, A. Sinner, Z. Zhan, P. A. Pantaleón, and F. Guinea, Designing moiré patterns by strain, Phys. Rev. Res. 6, 023203 (2024).
  22. G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett. 122, 106405 (2019).
  23. M. Kapfer, B. S. Jessen, M. E. Eisele, M. Fu, D. R. Danielsen, T. P. Darlington, S. L. Moore, N. R. Finney, A. Marchese, V. Hsieh, et al., Programming twist angle and strain profiles in 2D materials, Science 381, 677 (2023).
  24. Y. Hou, J. Zhou, M. Xue, M. Yu, Y. Han, Z. Zhang, and Y. Lu, Strain engineering of twisted bilayer graphene: The rise of strain-twistronics, Small 21, 2311185 (2025).
  25. 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).
  26. C.-C. Lu, Y.-C. Lin, Z. Liu, C.-H. Yeh, K. Suenaga, and P.-W. Chiu, Twisting bilayer graphene superlattices, ACS Nano 7, 2587 (2013).
  27. S. W. Schmucker, C. D. Cress, J. C. Culbertson, J. W. Beeman, O. D. Dubon, and J. T. Robinson, Raman signature of defected twisted bilayer graphene, Carbon 93, 250 (2015).
  28. J. H. Wilson, Y. Fu, S. Das Sarma, and J. H. Pixley, Disorder in twisted bilayer graphene, Phys. Rev. Res. 2, 023325 (2020).
  29. A. Schäpers, J. Sonntag, L. Valerius, B. Pestka, J. Strasdas, K. Watanabe, T. Taniguchi, L. Wirtz, M. Morgenstern, B. Beschoten, et al., Raman imaging of twist angle variations in twisted bilayer graphene at intermediate angles, 2D Mater. 9, 045009 (2022).
  30. T. E. Beechem, T. Ohta, B. Diaconescu, and J. T. Robinson, Rotational disorder in twisted bilayer graphene, ACS Nano 8, 1655 (2014).
  31. A. C. Gadelha, D. A. Ohlberg, F. C. Santana, G. S. Eliel, J. S. Lemos, V. Ornelas, D. Miranda, R. B. Nadas, K. Watanabe, T. Taniguchi, et al., Twisted bilayer graphene: A versatile fabrication method and the detection of variable nanometric strain caused by twist-angle disorder, ACS Appl. Nano Mater. 4, 1858 (2021).
  32. L. Huder, A. Artaud, T. Le Quang, G. T. De Laissardière, A. G. Jansen, G. Lapertot, C. Chapelier, and V. T. Renard, Electronic spectrum of twisted graphene layers under heterostrain, Phys. Rev. Lett. 120, 156405 (2018).
  33. 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).
  34. F. Mesple, A. Missaoui, T. Cea, L. Huder, F. Guinea, G. Trambly de Laissardière, C. Chapelier, and V. T. Renard, Heterostrain determines flat bands in magic-angle twisted graphene layers, Phys. Rev. Lett. 127, 126405 (2021).
  35. M. Mannaï and S. Haddad, Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides, Phys. Rev. B 103, L201112 (2021).
  36. A. Sinner, P. A. Pantaleón, and F. Guinea, Strain-induced quasi-1D channels in twisted moiré lattices, Phys. Rev. Lett. 131, 166402 (2023).
  37. X. Wang, J. Finney, A. L. Sharpe, L. K. Rodenbach, C. L. Hsueh, K. Watanabe, T. Taniguchi, M. Kastner, O. Vafek, and D. Goldhaber-Gordon, Unusual magnetotransport in twisted bilayer graphene from strain-induced open Fermi surfaces, Proc. Natl. Acad. Sci. USA 120, e2307151120 (2023).
  38. Y. Jiang, J. Mao, J. Duan, X. Lai, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Visualizing strain-induced pseudomagnetic fields in graphene through an hBN magnifying glass, Nano Lett. 17, 2839 (2017).
  39. J. Mao, S. P. Milovanović, M. Anđelković, X. Lai, Y. Cao, K. Watanabe, T. Taniguchi, L. Covaci, F. M. Peeters, A. K. Geim, Y. Jiang, and E. Y. Andrei, Evidence of flat bands and correlated states in buckled graphene superlattices, Nature (London) 584, 215 (2020).
  40. T. Peña, A. Dey, S. A. Chowdhury, A. Azizimanesh, W. Hou, A. Sewaket, C. Watson, H. Askari, and S. M. Wu, Moiré engineering in 2D heterostructures with process-induced strain, Appl. Phys. Lett. 122, 143101 (2023).
  41. M. Kögl, P. Soubelet, M. Brotons-Gisbert, A. V. Stier, B. D. Gerardot, and J. J. Finley, Moiré straintronics: A universal platform for reconfigurable quantum materials, npj 2D Mater. Appl. 7, 32 (2023).
  42. N. Hasan, T. Peña, A. Dey, D. Yoon, Z. Islam, Y. Zhang, M. V. G. Leal, A. M. van der Zande, H. Askari, and S. M. Wu, Strain-induced moiré reconstruction and memorization in two-dimensional materials without twist, arXiv:2510.13699.
  43. X. Gao, H. Sun, D.-H. Kang, C. Wang, Q. J. Wang, and D. Nam, Heterostrain-enabled dynamically tunable moiré superlattice in twisted bilayer graphene, Sci. Rep. 11, 21402 (2021).
  44. I. Sequeira, A. Z. Barabas, A. H. Barajas-Aguilar, M. G. Bacani, N. Nakatsuji, M. Koshino, T. Taniguichi, K. Watanabe, and J. D. Sanchez-Yamagishi, Manipulating moires by controlling heterostrain in van der Waals devices, Nano Lett. 24, 15662 (2024).
  45. Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene, Nature (London) 572, 101 (2019).
  46. J. Yu, G. Jia, Q. Li, Z. Zhan, Y. Wang, K. Xiao, Y. Ju, H. Zhang, Z. Hu, Y. Guo, et al., Strain and twist angle driven electronic structure evolution in twisted bilayer graphene, Rep. Prog. Phys. 89, 048001 (2026).
  47. P. Ouyang, J. Yu, Q. Li, G. Jia, Y. Wang, K. Xiao, H. Zhang, Z. Hu, P. A. Pantaleón, Z. Zhan, et al., Structural and electronic signatures of strain-tunable marginally twisted bilayer graphene, Natl. Sci. Rev. 13, nwaf568 (2026).
  48. R. Carrasco, F. Escudero, Z. Zhan, E. C.-d. Rio, B. Viña-Bausa, Y. Maximenko, P. A. Pantaleon, F. Guinea, and I. Brihuega, Twist and strain in square moiré patterns of stacked graphene layers, Phys. Rev. Lett. 136, 196102 (2026).
  49. J. Herzog-Arbeitman, J. Yu, D. Călugăru, H. Hu, N. Regnault, O. Vafek, J. Kang, and B. A. Bernevig, Topological heavy fermion model as an efficient representation of atomistic strain and relaxation in twisted bilayer graphene, Phys. Rev. B 112, 125128 (2025).
  50. J. Herzog-Arbeitman, D. Călugăru, H. Hu, J. Yu, N. Regnault, J. Kang, B. A. Bernevig, and O. Vafek, Kekulé spiral order from strained topological heavy fermions, Phys. Rev. B 112, 125129 (2025).
  51. A. Zhu, D. Bennett, D. T. Larson, M. M. A. Ezzi, E. Manousakis, and E. Kaxiras, Twisted bilayer graphene from first-principles: Structural and electronic properties, Phys. Rev. B 113, 205125 (2026).
  52. M. Gonçalves, B. Amorim, F. Riche, E. V. Castro, and P. Ribeiro, Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems, Nat. Phys. 20, 1933 (2024).
  53. V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, Tight-binding approach to uniaxial strain in graphene, Phys. Rev. B 80, 045401 (2009).
  54. M. A. Vozmediano, M. Katsnelson, and F. Guinea, Gauge fields in graphene, Phys. Rep. 496, 109 (2010).
  55. B. Amorim, A. Cortijo, F. De Juan, A. G. Grushin, F. Guinea, A. Gutiérrez-Rubio, H. Ochoa, V. Parente, R. Roldán, P. San-Jose, et al., Novel effects of strains in graphene and other two dimensional materials, Phys. Rep. 617, 1 (2016).
  56. G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, and H. Terrones, Electronic and optical properties of strained graphene and other strained 2D materials: A review, Rep. Prog. Phys. 80, 096501 (2017).
  57. H. Suzuura and T. Ando, Phonons and electron-phonon scattering in carbon nanotubes, Phys. Rev. B 65, 235412 (2002).
  58. J. L. Manes, Symmetry-based approach to electron-phonon interactions in graphene, Phys. Rev. B 76, 045430 (2007).
  59. F. Guinea and N. R. Walet, Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers, Proc. Natl. Acad. Sci. USA 115, 13174 (2018).
  60. T. Cea, N. R. Walet, and F. Guinea, Electronic band structure and pinning of Fermi energy to van Hove singularities in twisted bilayer graphene: A self-consistent approach, Phys. Rev. B 100, 205113 (2019).
  61. Z. A. Goodwin, V. Vitale, X. Liang, A. A. Mostofi, and J. Lischner, Hartree theory calculations of quasiparticle properties in twisted bilayer graphene, Electron. Struct. 2, 034001 (2020).
  62. T. Cea and F. Guinea, Coulomb interaction, phonons, and superconductivity in twisted bilayer graphene, Proc. Natl. Acad. Sci. USA 118, e2107874118 (2021).
  63. V. T. Phong, P. A. Pantaleón, T. Cea, and F. Guinea, Band structure and superconductivity in twisted trilayer graphene, Phys. Rev. B 104, L121116 (2021).
  64. D. E. Parker, T. Soejima, J. Hauschild, M. P. Zaletel, and N. Bultinck, Strain-induced quantum phase transitions in magic-angle graphene, Phys. Rev. Lett. 127, 027601 (2021).
  65. G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Global phase diagram of the normal state of twisted bilayer graphene, Phys. Rev. Lett. 128, 156401 (2022).
  66. H. Hu, G. Rai, L. Crippa, J. Herzog-Arbeitman, D. Călugăru, T. Wehling, G. Sangiovanni, R. Valentí, A. M. Tsvelik, and B. A. Bernevig, Symmetric Kondo lattice states in doped strained twisted bilayer graphene, Phys. Rev. Lett. 131, 166501 (2023).
  67. E. J. Mele, Commensuration and interlayer coherence in twisted bilayer graphene, Phys. Rev. B 81, 161405(R) (2010).
  68. G. Trambly de Laissardière, D. Mayou, and L. Magaud, Numerical studies of confined states in rotated bilayers of graphene, Phys. Rev. B 86, 125413 (2012).
  69. L. D. Landau, L. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity (Elsevier, Amsterdam, Netherlands, 2012), Vol. 7.
  70. M. E. Gurtin, E. Fried, and L. Anand, The Mechanics and Thermodynamics of Continua (Cambridge University Press, Cambridge (UK) and New York (USA), 2010).
  71. F. Escudero, F. Guinea, and Z. Zhan, Geometrical properties of strained and twisted moiré heterostructures, J. Phys.: Condens. Matter 38, 103001 (2026).
  72. F. Escudero, Z. Zhan, P. A. Pantaleón, and F. Guinea, Moiré-driven equilibrium, arXiv:2601.00948.
  73. A. Artaud, L. Magaud, T. Le Quang, V. Guisset, P. David, C. Chapelier, and J. Coraux, Universal classification of twisted, strained and sheared graphene moiré superlattices, Sci. Rep. 6, 25670 (2016).
  74. X. Lai, G. Li, A. M. Coe, J. H. Pixley, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Moiré periodic and quasiperiodic crystals in heterostructures of twisted bilayer graphene on hexagonal boron nitride, Nat. Mater. 24, 1019 (2025).
  75. See Supplemental Material at http://link.aps.org/supplemental/10.1103/r8sv-6xmp for details on commensurate structures with twist and strain, the tight-binding model, relaxation effects, Berry curvature and topology, and extended numerical results.
  76. J. H. Los, L. M. Ghiringhelli, E. J. Meijer, and A. Fasolino, Improved long-range reactive bond-order potential for carbon. I. Construction, Phys. Rev. B 72, 214102 (2005).
  77. A. N. Kolmogorov and V. H. Crespi, Registry-dependent interlayer potential for graphitic systems, Phys. Rev. B 71, 235415 (2005).
  78. A. Ramires and J. L. Lado, Electrically tunable gauge fields in tiny-angle twisted bilayer graphene, Phys. Rev. Lett. 121, 146801 (2018).
  79. A. Ramires and J. L. Lado, Impurity-induced triple point fermions in twisted bilayer graphene, Phys. Rev. B 99, 245118 (2019).
  80. S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117, 1 (1995).
  81. Y. Li, Z. Zhan, X. Kuang, Y. Li, and S. Yuan, TBPLaS: A tight-binding package for large-scale simulation, Comput. Phys. Commun. 285, 108632 (2023).
  82. A. Hams and H. De Raedt, Fast algorithm for finding the eigenvalue distribution of very large matrices, Phys. Rev. E 62, 4365 (2000).
  83. E. Polizzi, Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B 79, 115112 (2009).
  84. P. A. Pantaleón, V. T. Phong, G. G. Naumis, and F. Guinea, Interaction-enhanced topological Hall effects in strained twisted bilayer graphene, Phys. Rev. B 106, L161101 (2022).
  85. M. Oliva-Leyva and G. G. Naumis, Understanding electron behavior in strained graphene as a reciprocal space distortion, Phys. Rev. B 88, 085430 (2013).
  86. M. Oliva-Leyva and G. G. Naumis, Generalizing the Fermi velocity of strained graphene from uniform to nonuniform strain, Phys. Lett. A 379, 2645 (2015).
  87. M. Koshino, N. F. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene, Phys. Rev. X 8, 031087 (2018).
  88. M. Koshino and N. N. Nam, Effective continuum model for relaxed twisted bilayer graphene and moiré electron-phonon interaction, Phys. Rev. B 101, 195425 (2020).
  89. S. Carr, S. Fang, Z. Zhu, and E. Kaxiras, Exact continuum model for low-energy electronic states of twisted bilayer graphene, Phys. Rev. Res. 1, 013001 (2019).
  90. 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).
  91. M. M. A. Ezzi, G. N. Pallewela, C. De Beule, E. Mele, and S. Adam, Analytical model for atomic relaxation in twisted moiré materials, Phys. Rev. Lett. 133, 266201 (2024).
  92. A. Ceferino and F. Guinea, Pseudomagnetic fields in fully relaxed twisted bilayer and trilayer graphene, 2D Mater. 11, 035015 (2024).
  93. V. H. Nguyen and P. Dollfus, Strain-induced modulation of Dirac cones and van Hove singularities in a twisted graphene bilayer, 2D Mater. 2, 035005 (2015).
  94. M. Long, Z. Zhan, P. A. Pantaleón, J. A. Silva-Guillén, F. Guinea, and S. Yuan, Electronic properties of twisted bilayer graphene suspended and encapsulated with hexagonal boron nitride, Phys. Rev. B 107, 115140 (2023).
  95. M. Long, P. A. Pantaleón, Z. Zhan, F. Guinea, J. A. Silva-Guillén, and S. Yuan, An atomistic approach for the structural and electronic properties of twisted bilayer graphene-boron nitride heterostructures, npj Comput. Mater. 8, 73 (2022).
  96. G. Li, R. Krishna Kumar, P. Stepanov, P. A. Pantaleón, Z. Zhan, H. Agarwal, A. Bercher, J. Barrier, K. Watanabe, T. Taniguchi, A. B. Kuzmenko, F. Guinea, I. Torre, and F. H. L. Koppens, Infrared spectroscopy for diagnosing superlattice minibands in twisted bilayer graphene near the magic angle, Nano Lett. 24, 15956 (2024).
  97. F. Escudero, Diagrammatic perturbation approach to moiré bands in twisted bilayer graphene, Phys. Rev. B 110, 045442 (2024).
  98. S.-M. Choi, S.-H. Jhi, and Y.-W. Son, Controlling energy gap of bilayer graphene by strain, Nano Lett. 10, 3486 (2010).
  99. The relevant effect of the scalar potential is brought to our attention by discussions with Haim Beidenkopf's group.
  100. F. Guinea and N. R. Walet, Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters, Phys. Rev. B 99, 205134 (2019).
  101. I. V. Lebedeva and A. M. Popov, Two phases with different domain wall networks and a reentrant phase transition in bilayer graphene under strain, Phys. Rev. Lett. 124, 116101 (2020).
  102. F. Mesple, N. R. Walet, G. Trambly 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).
  103. V. Hung Nguyen, D. Paszko, M. Lamparski, B. Van Troeye, V. Meunier, and J.-C. Charlier, Electronic localization in small-angle twisted bilayer graphene, 2D Mater. 8, 035046 (2021).
  104. A. Timmel and E. Mele, Dirac-Harper theory for one-dimensional moiré superlattices, Phys. Rev. Lett. 125, 166803 (2020).
  105. 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).
  106. S. Fang, S. Carr, Z. Zhu, D. Massatt, and E. Kaxiras, Angle-dependent ab initio low-energy Hamiltonians for a relaxed twisted bilayer graphene heterostructure, arXiv:1908.00058.
  107. Y. H. Kwan, G. Wagner, T. Soejima, M. P. Zaletel, S. H. Simon, S. A. Parameswaran, and N. Bultinck, Kekulé spiral order at all nonzero integer fillings in twisted bilayer graphene, Phys. Rev. X 11, 041063 (2021).
  108. O. Vafek and J. Kang, Continuum effective Hamiltonian for graphene bilayers for an arbitrary smooth lattice deformation from microscopic theories, Phys. Rev. B 107, 075123 (2023).
  109. J. Kang and O. Vafek, Analytical solution for the relaxed atomic configuration of twisted bilayer graphene including heterostrain, Phys. Rev. B 112, 125138 (2025).
  110. V. Hsieh, D. Halbertal, N. R. Finney, Z. Zhu, E. Gerber, M. Pizzochero, E. Kucukbenli, G. R. Schleder, M. Angeli, K. Watanabe, et al., Domain-dependent surface adhesion in twisted few-layer graphene: Platform for moiré-assisted chemistry, Nano Lett. 23, 3137 (2023).
  111. P. Cazeaux, D. Clark, R. Engelke, P. Kim, and M. Luskin, Relaxation and domain wall structure of bilayer Moiré systems, J. Elast. 154, 443 (2023).
  112. B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries, Phys. Rev. B 103, 205413 (2021).
  113. L. Rademaker and P. Mellado, Charge-transfer insulation in twisted bilayer graphene, Phys. Rev. B 98, 235158 (2018).
  114. T. Cea and F. Guinea, Band structure and insulating states driven by Coulomb interaction in twisted bilayer graphene, Phys. Rev. B 102, 045107 (2020).
  115. M. M. A. Ezzi, L. Peng, Z. Liu, J. H. Z. Chao, G. N. Pallewela, D. Foo, and S. Adam, A self-consistent Hartree theory for lattice-relaxed magic-angle twisted bilayer graphene, arXiv:2404.17638.
  116. P. A. Pantaleón, T. Low, and F. Guinea, Tunable large Berry dipole in strained twisted bilayer graphene, Phys. Rev. B 103, 205403 (2021).
  117. I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials, Phys. Rev. Lett. 115, 216806 (2015).
  118. T. Low, Y. Jiang, and F. Guinea, Topological currents in black phosphorus with broken inversion symmetry, Phys. Rev. B 92, 235447 (2015).
  119. S. Sinha, P. C. Adak, A. Chakraborty, K. Das, K. Debnath, L. D. V. Sangani, K. Watanabe, T. Taniguchi, U. V. Waghmare, A. Agarwal, and M. M. Deshmukh, Berry curvature dipole senses topological transition in a moiré superlattice, Nat. Phys. 18, 765 (2022).
  120. T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005).
  121. J. Pan, H. Wang, L. Zou, X. Wang, L. Zhang, X. Dong, H. Xie, Y. Ding, Y. Zhang, T. Taniguchi, et al., Topological valley transport in bilayer graphene induced by interlayer sliding, Phys. Rev. Lett. 135, 126603 (2025).
  122. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  123. R. Battilomo, N. Scopigno, and C. Ortix, Berry curvature dipole in strained graphene: A Fermi surface warping effect, Phys. Rev. Lett. 123, 196403 (2019).
  124. C. Moulsdale, A. Knothe, and V. Fal’ko, Engineering of the topological magnetic moment of electrons in bilayer graphene using strain and electrical bias, Phys. Rev. B 101, 085118 (2020).
  125. P. A. Pantaleón, T. Cea, R. Brown, N. R. Walet, and F. Guinea, Narrow bands, electrostatic interactions and band topology in graphene stacks, 2D Mater. 8, 044006 (2021).
  126. K. Cuypers, R. Smeyers, B. Jorissen, and L. Covaci, Evolution of the Berry curvature dipole in uniaxially strained bilayer graphene, arXiv:2601.05962.
  127. X. Gao, A. Jimeno-Pozo, P. A. Pantaleon, E. Codecido, D. L. Sharifi, Z. Zhang, Y. Liu, K. Watanabe, T. Taniguchi, M. W. Bockrath, F. Guinea, and C. N. Lau, Double-edged role of interactions in superconducting twisted bilayer graphene, arXiv:2412.01578.
  128. R. Dutta, A. Ghosh, S. Mandal, K. Watanabe, T. Taniguchi, H. Krishnamurthy, S. Banerjee, M. Jain, and A. Das, Electric field-tunable superconductivity with competing orders in twisted bilayer graphene near the magic angle, ACS Nano 19, 5353 (2025).
  129. Z.-D. Song and B. A. Bernevig, Magic-angle twisted bilayer graphene as a topological heavy fermion problem, Phys. Rev. Lett. 129, 047601 (2022).
  130. H. Shi and X. Dai, Heavy-fermion representation for twisted bilayer graphene systems, Phys. Rev. B 106, 245129 (2022).

Outline

Information

Phys. Rev. Research 8, 023225– Published 29 May, 2026
  • Received 26 November 2025
  • Accepted 6 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation