Electronic-structure methods for twisted moiré layers

Abstract

When single layers of 2D materials are stacked on top of one another with a small twist in orientation, the resulting structure often involves incommensurate moiré patterns. In these patterns, the loss of angstrom-scale periodicity poses a significant theoretical challenge, and the new moiré length scale leads to emergent physical phenomena. The range of physics arising from twisted bilayers has led to significant advances that are shaping into a new field, twistronics. At the moiré scale, the large number of atoms in these systems can make their accurate simulation daunting, necessitating the development of efficient multiscale methods. In this Review, we summarize and compare such modelling methods — focusing in particular on density functional theory, tight-binding Hamiltonians and continuum models — and provide examples spanning a broad range of materials and geometries.

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Fig. 1: An introduction to twistronics.
Fig. 2: Examples of 2D materials.
Fig. 3: Local stacking order in twisted bilayers.
Fig. 4: Examples of atomic relaxations in moiré materials.
Fig. 5: Comparison of electronic band structures for 1.08° twisted bilayer graphene generated by different methods.
Fig. 6: Twist-induced electronic properties in 2D moiré heterostructures.

References

  1. 1.

    Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    CAS  Google Scholar 

  2. 2.

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

    CAS  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  Google Scholar 

  4. 4.

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

    CAS  Google Scholar 

  5. 5.

    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  Google Scholar 

  6. 6.

    San-Jose, P. & Prada, E. Helical networks in twisted bilayer graphene under interlayer bias. Phys. Rev. B 88, 121408 (2013).

    Google Scholar 

  7. 7.

    Ramires, A. & Lado, J. L. Electrically tunable gauge fields in tiny-angle twisted bilayer graphene. Phys. Rev. Lett. 121, 146801 (2018).

    CAS  Google Scholar 

  8. 8.

    Huang, S. et al. Topologically protected helical states in minimally twisted bilayer graphene. Phys. Rev. Lett. 121, 037702 (2018).

    CAS  Google Scholar 

  9. 9.

    Efimkin, D. K. & MacDonald, A. H. Helical network model for twisted bilayer graphene. Phys. Rev. B 98, 035404 (2018).

    CAS  Google Scholar 

  10. 10.

    Liao, M. et al. Twist angle-dependent conductivities across MoS2/graphene heterojunctions. Nat. Commun. 9, 4068 (2018).

    Google Scholar 

  11. 11.

    Vela, A., Moutinho, M. V. O., Culchac, F. J., Venezuela, P. & Capaz., R. B. Electronic structure and optical properties of twisted multilayer graphene. Phys. Rev. B 98, 155135 (2018).

    CAS  Google Scholar 

  12. 12.

    Ribeiro-Palau, R. et al. Twistable electronics with dynamically rotatable heterostructures. Science 361, 690–693 (2018).

    CAS  Google Scholar 

  13. 13.

    Finney, N. R. et al. Tunable crystal symmetry in graphene–boron nitride heterostructures with coexisting moiré superlattices. Nat. Nanotechnol. 14, 1029–1034 (2019).

    CAS  Google Scholar 

  14. 14.

    Gerber, E., Yao, Y., Arias, T. A. & Kim, E.-A. Ab initio mismatched interface theory of graphene on α–RuCl3: Doping and magnetism. Phys. Rev. Lett. 124, 106804 (2020).

    CAS  Google Scholar 

  15. 15.

    Li, Y. & Koshino, M. Twist-angle dependence of the proximity spin-orbit coupling in graphene on transition-metal dichalcogenides. Phys. Rev. B 99, 075438 (2019).

    CAS  Google Scholar 

  16. 16.

    David, A., Rakyta, P., Kormányos, A. & Burkard, G. Induced spin-orbit coupling in twisted graphene–transition metal dichalcogenide heterobilayers: twistronics meets spintronics. Phys. Rev. B 100, 085412 (2019).

    CAS  Google Scholar 

  17. 17.

    Zollner, K., Faria Junior, P. E. & Fabian, J. Proximity exchange effects in MoSe2 and WSe2 heterostructures with CrI3: twist angle, layer, and gate dependence. Phys. Rev. B 100, 085128 (2019).

    CAS  Google Scholar 

  18. 18.

    Cheon, G. et al. Data mining for new two- and one-dimensional weakly bonded solids and lattice-commensurate heterostructures. Nano Lett. 17, 1915–1923 (2017).

    CAS  Google Scholar 

  19. 19.

    Mounet, N. et al. Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246–252 (2018).

    CAS  Google Scholar 

  20. 20.

    Haastrup, S. et al. The computational 2D materials database: high-throughput modeling and discovery of atomically thin crystals. 2D Mater. 5, 042002 (2018).

    CAS  Google Scholar 

  21. 21.

    Kindermann, M., Uchoa, B. & Miller, D. L. Zero-energy modes and gate-tunable gap in graphene on hexagonal boron nitride. Phys. Rev. B 86, 115415 (2012).

    Google Scholar 

  22. 22.

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

    CAS  Google Scholar 

  23. 23.

    Wallbank, J. R., Patel, A. A., Mucha-Kruczyński, M., Geim, A. K. & Fal’ko., V. I. Generic miniband structure of graphene on a hexagonal substrate. Phys. Rev. B 87, 245408 (2013).

    Google Scholar 

  24. 24.

    Mucha-Kruczyński, M., Wallbank, J. R. & Fal’ko, V. I. Heterostructures of bilayer graphene and h-BN: Interplay between misalignment, interlayer asymmetry, and trigonal warping. Phys. Rev. B 88, 205418 (2013).

    Google Scholar 

  25. 25.

    Moon, P. & Koshino, M. Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice. Phys. Rev. B 90, 155406 (2014).

    Google Scholar 

  26. 26.

    Song, J. C. W., Samutpraphoot, P. & Levitov, L. S. Topological Bloch bands in graphene superlattices. Proc. Natl Acad. Sci. USA 112, 10879–10883 (2015).

    CAS  Google Scholar 

  27. 27.

    Jung, J., DaSilva, A. M., MacDonald, A. H. & Adam, S. Origin of band gaps in graphene on hexagonal boron nitride. Nat. Commun. 6, 6308 (2015).

    CAS  Google Scholar 

  28. 28.

    Jung, J., Laksono, E., DaSilva, A. M., MacDonald, A. H. & Mucha-Kruczyński, M. Moiré band model and band gaps of graphene on hexagonal boron nitride. Phys. Rev. B 96, 085442 (2017).

    Google Scholar 

  29. 29.

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

    CAS  Google Scholar 

  30. 30.

    Chen, X. et al. Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices. Phys. Rev. B 89, 075401 (2014).

    Google Scholar 

  31. 31.

    Zhou, S., Han, J., Dai, S., Sun, J. & Srolovitz, D. J. van der Waals bilayer energetics: Generalized stacking-fault energy of graphene, boron nitride, and graphene/boron nitride bilayers. Phys. Rev. B 92, 155438 (2015).

    Google Scholar 

  32. 32.

    Shirodkar, S. N. & Kaxiras, E. Li intercalation at graphene/hexagonal boron nitride interfaces. Phys. Rev. B 93, 245438 (2016).

    Google Scholar 

  33. 33.

    Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).

    CAS  Google Scholar 

  34. 34.

    Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

    CAS  Google Scholar 

  35. 35.

    Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).

    CAS  Google Scholar 

  36. 36.

    Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    CAS  Google Scholar 

  37. 37.

    Chen, G. et al. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys. 15, 237–241 (2019).

    CAS  Google Scholar 

  38. 38.

    Chen, G. et al. Signatures of tunable superconductivity in a trilayer graphene moiré superlattice. Nature 572, 215–219 (2019).

    CAS  Google Scholar 

  39. 39.

    Wang, L. et al. New generation of moiré superlattices in doubly aligned hBN/graphene/hBN heterostructures. Nano Lett. 19, 2371–2376 (2019).

    CAS  Google Scholar 

  40. 40.

    Xian, L., Kennes, D. M., Tancogne-Dejean, N., Altarelli, M. & Rubio, A. Multiflat bands and strong correlations in twisted bilayer boron nitride: doping-induced correlated insulator and superconductor. Nano Lett. 19, 4934–4940 (2019).

    CAS  Google Scholar 

  41. 41.

    Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: Electronic structure. Phys. Rev. Lett. 99, 256802 (2007).

    CAS  Google Scholar 

  42. 42.

    Campanera, J. M., Savini, G., Suarez-Martinez, I. & Heggie, M. I. Density functional calculations on the intricacies of moiré patterns on graphite. Phys. Rev. B 75, 235449 (2007).

    Google Scholar 

  43. 43.

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

    CAS  Google Scholar 

  44. 44.

    Suárez Morell, E., Pacheco, M., Chico, L. & Brey, L. Electronic properties of twisted trilayer graphene. Phys. Rev. B 87, 125414 (2013).

    Google Scholar 

  45. 45.

    Correa, J. D., Pacheco, M. & Morell, E. S. Optical absorption spectrum of rotated trilayer graphene. J. Mater. Sci. 49, 642–647 (2014).

    CAS  Google Scholar 

  46. 46.

    Shen, C. et al. Correlated states in twisted double bilayer graphene. Nat. Phys. 16, 520–525 (2020).

    CAS  Google Scholar 

  47. 47.

    Liu, X. et al. Spin-polarized correlated insulator and superconductor in twisted double bilayer graphene. Preprint at arXiv http://arxiv.org/abs/1903.08130 (2019).

  48. 48.

    Cao, Y. et al. Electric field tunable correlated states and magnetic phase transitions in twisted bilayer-bilayer graphene. Preprint at arXiv http://arxiv.org/abs/1903.08596 (2019).

  49. 49.

    Lee, J. Y. et al. Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene. Nat. Commun. 10, 5333 (2019).

    Google Scholar 

  50. 50.

    Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted double bilayer graphene. Phys. Rev. B 99, 235417 (2019).

    CAS  Google Scholar 

  51. 51.

    Koshino, M. Band structure and topological properties of twisted double bilayer graphene. Phys. Rev. B 99, 235406 (2019).

    CAS  Google Scholar 

  52. 52.

    Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearly flat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127 (2019).

    CAS  Google Scholar 

  53. 53.

    Tarnopolsky, G., Kruchkov, A. J. & Vishwanath, A. Origin of magic angles in twisted bilayer graphene. Phys. Rev. Lett. 122, 106405 (2019).

    CAS  Google Scholar 

  54. 54.

    Liu, J., Ma, Z., Gao, J. & Dai, X. Quantum valley Hall effect, orbital magnetism, and anomalous Hall effect in twisted multilayer graphene systems. Phys. Rev. X 9, 031021 (2019).

    CAS  Google Scholar 

  55. 55.

    Ahn, S. J. et al. Dirac electrons in a dodecagonal graphene quasicrystal. Science 361, 782–786 (2018).

    CAS  Google Scholar 

  56. 56.

    Moon, P., Koshino, M. & Son, Y.-W. Quasicrystalline electronic states in 30° rotated twisted bilayer graphene. Phys. Rev. B 99, 165430 (2019).

    CAS  Google Scholar 

  57. 57.

    Amorim, B. & Castro, E. V. Electronic spectral properties of incommensurate twisted trilayer graphene. Preprint at arXiv http://arxiv.org/abs/1807.11909 (2018).

  58. 58.

    Mora, C., Regnault, N. & Bernevig, B. A. Flatbands and perfect metal in trilayer moiré graphene. Phys. Rev. Lett. 123, 026402 (2019).

    CAS  Google Scholar 

  59. 59.

    Zuo, W.-J. et al. Scanning tunneling microscopy and spectroscopy of twisted trilayer graphene. Phys. Rev. B 97, 035440 (2018).

    CAS  Google Scholar 

  60. 60.

    Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019).

    CAS  Google Scholar 

  61. 61.

    Carr, S. et al. Ultraheavy and ultrarelativistic Dirac quasiparticles in sandwiched graphenes. Nano Lett. 20, 3030–3038 (2020).

    CAS  Google Scholar 

  62. 62.

    Cea, T., Walet, N. R. & Guinea, F. Twists and the electronic structure of graphitic materials. Nano Lett. 19, 8683–8689 (2019).

    CAS  Google Scholar 

  63. 63.

    Kang, J., Li, J., Li, S.-S., Xia, J.-B. & Wang, L.-W. Electronic structural moiré pattern effects on MoS2/MoSe2 2D heterostructures. Nano Lett. 13, 5485–5490 (2013).

    CAS  Google Scholar 

  64. 64.

    Fang, S. et al. Ab initio tight-binding Hamiltonian for transition metal dichalcogenides. Phys. Rev. B 92, 205108 (2015).

    Google Scholar 

  65. 65.

    Gani, Y. S., Steinberg, H. & Rossi, E. Superconductivity in twisted graphene NbSe2 heterostructures. Phys. Rev. B 99, 235404 (2019).

    CAS  Google Scholar 

  66. 66.

    Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol. 7, 699–712 (2012).

    CAS  Google Scholar 

  67. 67.

    Butler, S. Z. et al. Progress, challenges, and opportunities in two-dimensional materials beyond graphene. ACS Nano 7, 2898–2926 (2013).

    CAS  Google Scholar 

  68. 68.

    Ataca, C., Sahin, H. & Ciraci, S. Stable, single-layer MX2 transition-metal oxides and dichalcogenides in a honeycomb-like structure. J. Phys. Chem. C 116, 8983–8999 (2012).

    CAS  Google Scholar 

  69. 69.

    Duerloo, K.-A. N., Li, Y. & Reed, E. J. Structural phase transitions in two-dimensional Mo- and W-dichalcogenide monolayers. Nat. Commun. 5, 4214 (2014).

    CAS  Google Scholar 

  70. 70.

    Rasmussen, F. A. & Thygesen, K. S. Computational 2D materials database: electronic structure of transition-metal dichalcogenides and oxides. J. Phys. Chem. C 119, 13169–13183 (2015).

    CAS  Google Scholar 

  71. 71.

    Navarro-Moratalla, E. et al. Enhanced superconductivity in atomically thin TaS2. Nat. Commun. 7, 11043 (2016).

    CAS  Google Scholar 

  72. 72.

    Yang, Y. et al. Enhanced superconductivity upon weakening of charge density wave transport in 2H–TaS2 in the two-dimensional limit. Phys. Rev. B 98, 035203 (2018).

    CAS  Google Scholar 

  73. 73.

    Liu, K. et al. Evolution of interlayer coupling in twisted molybdenum disulfide bilayers. Nat. Commun. 5, 4966 (2014).

    CAS  Google Scholar 

  74. 74.

    Zhang, C. et al. Interlayer couplings, moiré patterns, and 2D electronic superlattices in MoS2/WSe2 hetero-bilayers. Sci. Adv. 3, e1601459 (2017).

    Google Scholar 

  75. 75.

    Yu, H., Wang, Y., Tong, Q., Xu, X. & Yao, W. Anomalous light cones and valley optical selection rules of interlayer excitons in twisted heterobilayers. Phys. Rev. Lett. 115, 187002 (2015).

    Google Scholar 

  76. 76.

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

    CAS  Google Scholar 

  77. 77.

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

    CAS  Google Scholar 

  78. 78.

    Wu, F., Lovorn, T. & MacDonald, A. H. Topological exciton bands in moiré heterojunctions. Phys. Rev. Lett. 118, 147401 (2017).

    Google Scholar 

  79. 79.

    Wu, F., Lovorn, T., Tutuc, E. & MacDonald, A. H. Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018).

    CAS  Google Scholar 

  80. 80.

    Naik, M. H. & Jain, M. Ultraflatbands and shear solitons in moiré patterns of twisted bilayer transition metal dichalcogenides. Phys. Rev. Lett. 121, 266401 (2018).

    CAS  Google Scholar 

  81. 81.

    Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).

    CAS  Google Scholar 

  82. 82.

    Sivadas, N., Okamoto, S., Xu, X., Fennie, C. J. & Xiao, D. Stacking-dependent magnetism in bilayer CrI3. Nano Lett. 18, 7658–7664 (2018).

    CAS  Google Scholar 

  83. 83.

    Jiang, P. et al. Stacking tunable interlayer magnetism in bilayer CrI3. Phys. Rev. B 99, 144401 (2019).

    CAS  Google Scholar 

  84. 84.

    Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).

    CAS  Google Scholar 

  85. 85.

    Lazić, P. CellMatch: Combining two unit cells into a common supercell with minimal strain. Comput. Phys. Commun. 197, 324–334 (2015).

    Google Scholar 

  86. 86.

    Koda, D. S., Bechstedt, F., Marques, M. & Teles, L. K. Coincidence lattices of 2D crystals: Heterostructure predictions and applications. J. Phys. Chem. C 120, 10895–10908 (2016).

    CAS  Google Scholar 

  87. 87.

    Frisenda, R. et al. Recent progress in the assembly of nanodevices and van der Waals heterostructures by deterministic placement of 2D materials. Chem. Soc. Rev. 47, 53–68 (2018).

    CAS  Google Scholar 

  88. 88.

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

    CAS  Google Scholar 

  89. 89.

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

    CAS  Google Scholar 

  90. 90.

    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).

    Google Scholar 

  91. 91.

    Carr, S. et al. Twistronics: manipulating the electronic properties of two-dimensional layered structures through their twist angle. Phys. Rev. B 95, 075420 (2017).

    Google Scholar 

  92. 92.

    Massatt, D., Luskin, M. & Ortner, C. Electronic density of states for incommensurate layers. Multiscale Model. Simul. 15, 476–499 (2017).

    Google Scholar 

  93. 93.

    Bernal, J. D. The structure of graphite. Proc. R. Soc. Lond. A Math. Phys. Sci 106, 749–773 (1924).

    CAS  Google Scholar 

  94. 94.

    Kumar, H., Er, D., Dong, L., Li, J. & Shenoy, V. B. Elastic deformations in 2D van der Waals heterostructures and their impact on optoelectronic properties: predictions from a multiscale computational approach. Sci. Rep. 5, 10872 (2015).

    CAS  Google Scholar 

  95. 95.

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

    CAS  Google Scholar 

  96. 96.

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

    Google Scholar 

  97. 97.

    Carr, S. et al. Relaxation and domain formation in incommensurate two-dimensional heterostructures. Phys. Rev. B 98, 224102 (2018).

    CAS  Google Scholar 

  98. 98.

    Bistritzer, R. & MacDonald, A. H. Transport between twisted graphene layers. Phys. Rev. B 81, 245412 (2010).

    Google Scholar 

  99. 99.

    Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory of moiré superlattice bands in layered two-dimensional materials. Phys. Rev. B 89, 205414 (2014).

    Google Scholar 

  100. 100.

    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  Google Scholar 

  101. 101.

    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  Google Scholar 

  102. 102.

    Fang, S., Carr, S., Zhu, Z., Massatt, D. & Kaxiras, E. Angle-dependent ab initio low-energy Hamiltonians for a relaxed twisted bilayer graphene heterostructure. Preprint at arXiv https://arxiv.org/abs/1908.00058 (2019).

  103. 103.

    Morsch, O. & Oberthaler, M. Dynamics of Bose-Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179–215 (2006).

    CAS  Google Scholar 

  104. 104.

    Forsythe, C. et al. Band structure engineering of 2D materials using patterned dielectric superlattices. Nat. Nanotechnol. 13, 566–571 (2018).

    CAS  Google Scholar 

  105. 105.

    Shi, L., Ma, J. & Song, J. C. W. Gate-tunable flat bands in van der Waals patterned dielectric superlattices. 2D Mater. 7, 015028 (2019).

    Google Scholar 

  106. 106.

    Carr, S., Fang, S., Jarillo-Herrero, P. & Kaxiras, E. Pressure dependence of the magic twist angle in graphene superlattices. Phys. Rev. B 98, 085144 (2018).

    CAS  Google Scholar 

  107. 107.

    Chittari, B. L., Leconte, N., Javvaji, S. & Jung, J. Pressure induced compression of flatbands in twisted bilayer graphene. Electron. Struct. 1, 015001 (2018).

    Google Scholar 

  108. 108.

    Li, L. J. et al. Controlling many-body states by the electric-field effect in a two-dimensional material. Nature 529, 185–189 (2016).

    CAS  Google Scholar 

  109. 109.

    Amorim, B. et al. Novel effects of strains in graphene and other two dimensional materials. Phys. Rep. 617, 1–54 (2016).

    CAS  Google Scholar 

  110. 110.

    Naumis, G. G., Barraza-Lopez, S., Oliva-Leyva, M. & Terrones, H. Electronic and optical properties of strained graphene and other strained 2D materials: a review. Rep. Prog. Phys. 80, 096501 (2017).

    Google Scholar 

  111. 111.

    Fang, S., Carr, S., Cazalilla, M. A. & Kaxiras, E. Electronic structure theory of strained two-dimensional materials with hexagonal symmetry. Phys. Rev. B 98, 075106 (2018).

    CAS  Google Scholar 

  112. 112.

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

    CAS  Google Scholar 

  113. 113.

    Shao, X., Wang, K., Pang, R. & Shi, X. Lithium intercalation in graphene/MoS2 composites: First-principles insights. J. Phys. Chem. C 119, 25860–25867 (2015).

    CAS  Google Scholar 

  114. 114.

    Wan, J. et al. Tuning two-dimensional nanomaterials by intercalation: materials, properties and applications. Chem. Soc. Rev. 45, 6742–6765 (2016).

    CAS  Google Scholar 

  115. 115.

    Bediako, D. K. et al. Heterointerface effects in the electrointercalation of van der Waals heterostructures. Nature 558, 425–429 (2018).

    CAS  Google Scholar 

  116. 116.

    Larson, D. T., Fampiou, I., Kim, G. & Kaxiras, E. Lithium intercalation in graphene–MoS2 heterostructures. J. Phys. Chem. C 122, 24535–24541 (2018).

    CAS  Google Scholar 

  117. 117.

    Lin, Z. et al. Defect engineering of two-dimensional transition metal dichalcogenides. 2D Mater. 3, 022002 (2016).

    Google Scholar 

  118. 118.

    Ramires, A. & Lado, J. L. Impurity-induced triple point fermions in twisted bilayer graphene. Phys. Rev. B 99, 245118 (2019).

    CAS  Google Scholar 

  119. 119.

    Jones, R. O. Density functional theory: Its origins, rise to prominence, and future. Rev. Mod. Phys. 87, 897–923 (2015).

    Google Scholar 

  120. 120.

    Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).

    Google Scholar 

  121. 121.

    Mele, E. J. Commensuration and interlayer coherence in twisted bilayer graphene. Phys. Rev. B 81, 161405 (2010).

    Google Scholar 

  122. 122.

    Shallcross, S., Sharma, S., Kandelaki, E. & Pankratov, O. A. Electronic structure of turbostratic graphene. Phys. Rev. B 81, 165105 (2010).

    Google Scholar 

  123. 123.

    Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Continuum model of the twisted graphene bilayer. Phys. Rev. B 86, 155449 (2012).

    Google Scholar 

  124. 124.

    Pan, D., Wang, T.-C., Xiao, W., Hu, D. & Yao, Y. Simulations of twisted bilayer orthorhombic black phosphorus. Phys. Rev. B 96, 041411 (2017).

    Google Scholar 

  125. 125.

    Uchida, K., Furuya, S., Iwata, J.-I. & Oshiyama, A. Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes. Phys. Rev. B 90, 155451 (2014).

    Google Scholar 

  126. 126.

    Lucignano, P., Alfè, D., Cataudella, V., Ninno, D. & Cantele, G. Crucial role of atomic corrugation on the flat bands and energy gaps of twisted bilayer graphene at the magic angle θ ~ 1.08°. Phys. Rev. B 99, 195419 (2019).

    CAS  Google Scholar 

  127. 127.

    Kang, P. et al. Moiré impurities in twisted bilayer black phosphorus: effects on the carrier mobility. Phys. Rev. B 96, 195406 (2017).

    Google Scholar 

  128. 128.

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

    Google Scholar 

  129. 129.

    Berland, K. et al. van der Waals forces in density functional theory: a review of the vdW-DF method. Rep. Prog. Phys. 78, 066501 (2015).

    Google Scholar 

  130. 130.

    Dion, M., Rydberg, H., Schröder, E., Langreth, D. C. & Lundqvist, B. I. Van der Waals density functional for general geometries. Phys. Rev. Lett. 92, 246401 (2004).

    CAS  Google Scholar 

  131. 131.

    Koshino, M. et al. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

    CAS  Google Scholar 

  132. 132.

    Goodwin, Z. A. H., Corsetti, F., Mostofi, A. A. & Lischner, J. Attractive electron-electron interactions from internal screening in magic-angle twisted bilayer graphene. Phys. Rev. B 100, 235424 (2019).

    CAS  Google Scholar 

  133. 133.

    Rademaker, L., Abanin, D. A. & Mellado, P. Charge smoothening and band flattening due to Hartree corrections in twisted bilayer graphene. Phys. Rev. B 100, 205114 (2019).

    CAS  Google Scholar 

  134. 134.

    Cea, T., Walet, N. R. & Guinea, F. 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).

    CAS  Google Scholar 

  135. 135.

    Roy, B. & Juričić, V. Unconventional superconductivity in nearly flat bands in twisted bilayer graphene. Phys. Rev. B 99, 121407 (2019).

    CAS  Google Scholar 

  136. 136.

    Das Sarma, S. & Wu, F. Electron–phonon and electron–electron interaction effects in twisted bilayer graphene. Ann. Phys. 417, 168193 (2020).

    Google Scholar 

  137. 137.

    Wu, F., MacDonald, A. H. & Martin, I. Theory of phonon-mediated superconductivity in twisted bilayer graphene. Phys. Rev. Lett. 121, 257001 (2018).

    CAS  Google Scholar 

  138. 138.

    Bultinck, N. et al. Ground state and hidden symmetry of magic angle graphene at even integer filling. Preprint at arXiv https://arxiv.org/abs/1911.02045 (2019).

  139. 139.

    Kang, J. & Vafek, O. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).

    CAS  Google Scholar 

  140. 140.

    Carr, S., Fang, S., Po, H. C., Vishwanath, A. & Kaxiras, E. Derivation of Wannier orbitals and minimal-basis tight-binding Hamiltonians for twisted bilayer graphene: First-principles approach. Phys. Rev. Res. 1, 033072 (2019).

    Google Scholar 

  141. 141.

    Suárez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic, Z. Flat bands in slightly twisted bilayer graphene: tight-binding calculations. Phys. Rev. B 82, 121407 (2010).

    Google Scholar 

  142. 142.

    Trambly de Laissardière, G., Mayou, D. & Magaud, L. Localization of Dirac electrons in rotated graphene bilayers. Nano Lett. 10, 804–808 (2010).

    Google Scholar 

  143. 143.

    Wang, Z. F., Liu, F. & Chou, M. Y. Fractal Landau-level spectra in twisted bilayer graphene. Nano Lett. 12, 3833–3838 (2012).

    CAS  Google Scholar 

  144. 144.

    Sboychakov, A. O., Rakhmanov, A. L., Rozhkov, A. V. & Nori, F. Electronic spectrum of twisted bilayer graphene. Phys. Rev. B 92, 075402 (2015).

    Google Scholar 

  145. 145.

    Lin, X. & Tománek, D. Minimum model for the electronic structure of twisted bilayer graphene and related structures. Phys. Rev. B 98, 081410 (2018).

    CAS  Google Scholar 

  146. 146.

    Moon, P. & Koshino, M. Energy spectrum and quantum Hall effect in twisted bilayer graphene. Phys. Rev. B 85, 195458 (2012).

    Google Scholar 

  147. 147.

    Gargiulo, F. & Yazyev, O. V. Structural and electronic transformation in low-angle twisted bilayer graphene. 2D Mater. 5, 015019 (2017).

    Google Scholar 

  148. 148.

    Lin, X., Liu, D. & Tománek, D. Shear instability in twisted bilayer graphene. Phys. Rev. B 98, 195432 (2018).

    CAS  Google Scholar 

  149. 149.

    McClure, J. W. Band structure of graphite and de Haas-van Alphen effect. Phys. Rev. 108, 612–618 (1957).

    CAS  Google Scholar 

  150. 150.

    Slonczewski, J. C. & Weiss, P. R. Band structure of graphite. Phys. Rev. 109, 272–279 (1958).

    CAS  Google Scholar 

  151. 151.

    Jung, J. & MacDonald, A. H. Accurate tight-binding models for the π bands of bilayer graphene. Phys. Rev. B 89, 035405 (2014).

    Google Scholar 

  152. 152.

    Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    CAS  Google Scholar 

  153. 153.

    Trambly de Laissardière, G., Mayou, D. & Magaud, L. Numerical studies of confined states in rotated bilayers of graphene. Phys. Rev. B 86, 125413 (2012).

    Google Scholar 

  154. 154.

    Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).

    CAS  Google Scholar 

  155. 155.

    Pizzi, G. et al. Wannier90 as a community code: new features and applications. J. Phys. Condens. Matter 32, 165902 (2020).

    CAS  Google Scholar 

  156. 156.

    Bakhta, A., Cancès, E., Cazeaux, P., Fang, S. & Kaxiras, E. Compression of Wannier functions into Gaussian-type orbitals. Comput. Phys. Commun. 230, 27–37 (2018).

    CAS  Google Scholar 

  157. 157.

    Tritsaris, G. A. et al. Perturbation theory for weakly coupled two-dimensional layers. J. Mater. Res. 31, 959–966 (2016).

    CAS  Google Scholar 

  158. 158.

    Weiße, A., Wellein, G., Alvermann, A. & Fehske, H. The kernel polynomial method. Rev. Mod. Phys. 78, 275–306 (2006).

    Google Scholar 

  159. 159.

    Le, H. A. & Do, V. N. Electronic structure and optical properties of twisted bilayer graphene calculated via time evolution of states in real space. Phys. Rev. B 97, 125136 (2018).

    CAS  Google Scholar 

  160. 160.

    Gonzalez-Arraga, L. A., Lado, J. L., Guinea, F. & San-Jose, P. Electrically controllable magnetism in twisted bilayer graphene. Phys. Rev. Lett. 119, 107201 (2017).

    Google Scholar 

  161. 161.

    Stuart, S. J., Tutein, A. B. & Harrison, J. A. A reactive potential for hydrocarbons with intermolecular interactions. J. Chem. Phys. 112, 6472–6486 (2000).

    CAS  Google Scholar 

  162. 162.

    Kolmogorov, A. N. & Crespi, V. H. Registry-dependent interlayer potential for graphitic systems. Phys. Rev. B 71, 235415 (2005).

    Google Scholar 

  163. 163.

    Los, J. H., Ghiringhelli, L. M., Meijer, E. J. & Fasolino, A. Improved long-range reactive bond-order potential for carbon. I. Construction. Phys. Rev. B 72, 214102 (2005).

    Google Scholar 

  164. 164.

    O’Connor, T. C., Andzelm, J. & Robbins, M. O. AIREBO-M: A reactive model for hydrocarbons at extreme pressures. J. Chem. Phys. 142, 024903 (2015).

    Google Scholar 

  165. 165.

    Wen, M., Carr, S., Fang, S., Kaxiras, E. & Tadmor, E. B. Dihedral-angle-corrected registry-dependent interlayer potential for multilayer graphene structures. Phys. Rev. B 98, 235404 (2018).

    CAS  Google Scholar 

  166. 166.

    Lin, M.-L. et al. Moiré phonons in twisted bilayer MoS2. ACS Nano 12, 8770–8780 (2018).

    CAS  Google Scholar 

  167. 167.

    Koshino, M. & Son, Y.-W. Moiré phonons in twisted bilayer graphene. Phys. Rev. B 100, 075416 (2019).

    CAS  Google Scholar 

  168. 168.

    Gong, X. & Mele, E. J. Stacking textures and singularities in bilayer graphene. Phys. Rev. B 89, 121415 (2014).

    Google Scholar 

  169. 169.

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

    Google Scholar 

  170. 170.

    Mele, E. J. Band symmetries and singularities in twisted multilayer graphene. Phys. Rev. B 84, 235439 (2011).

    Google Scholar 

  171. 171.

    Kindermann, M. & Mele, E. J. Landau quantization in twisted bilayer graphene: the Dirac comb. Phys. Rev. B 84, 161406 (2011).

    Google Scholar 

  172. 172.

    Bistritzer, R. & MacDonald, A. H. Moiré butterflies in twisted bilayer graphene. Phys. Rev. B 84, 035440 (2011).

    Google Scholar 

  173. 173.

    Kariyado, T. & Vishwanath, A. Flat band in twisted bilayer Bravais lattices. Phys. Rev. Res. 1, 033076 (2019).

    Google Scholar 

  174. 174.

    de Gail, R., Goerbig, M. O., Guinea, F., Montambaux, G. & Castro Neto, A. H. Topologically protected zero modes in twisted bilayer graphene. Phys. Rev. B 84, 045436 (2011).

    Google Scholar 

  175. 175.

    Shallcross, S., Sharma, S. & Pankratov, O. Emergent momentum scale, localization, and van Hove singularities in the graphene twist bilayer. Phys. Rev. B 87, 245403 (2013).

    Google Scholar 

  176. 176.

    Walet, N. R. & Guinea, F. Lattice deformation, low energy models and flat bands in twisted graphene bilayers. Preprint at arXiv http://arxiv.org/abs/1903.00340 (2019).

  177. 177.

    Guinea, F. & Walet, N. R. Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers. Proc. Natl Acad. Sci. USA 115, 13174–13179 (2018).

    CAS  Google Scholar 

  178. 178.

    Brihuega, I. et al. Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis. Phys. Rev. Lett. 109, 196802 (2012).

    CAS  Google Scholar 

  179. 179.

    Stepanov, P. et al. The interplay of insulating and superconducting orders in magic-angle graphene bilayers. Preprint at arXiv http://arxiv.org/abs/1911.09198 (2019)

  180. 180.

    Saito, Y. et al. Independent superconductors and correlated insulators in twisted bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-020-0928-3 (2020).

    Article  Google Scholar 

  181. 181.

    Arora, H. S. et al. Superconductivity without insulating states in twisted bilayer graphene stabilized by monolayer WSe2. Preprint at arXiv http://arxiv.org/abs/2002.03003 (2020).

  182. 182.

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

    CAS  Google Scholar 

  183. 183.

    Jiang, Y. et al. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature 573, 91–95 (2019).

    CAS  Google Scholar 

  184. 184.

    Wang, L. et al. Magic continuum in twisted bilayer WSe2. Preprint at arXiv http://arxiv.org/abs/1910.12147 (2019).

  185. 185.

    Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 27, 1787–1799 (2006).

    CAS  Google Scholar 

  186. 186.

    Sun, J. et al. Semilocal and hybrid meta-generalized gradient approximations based on the understanding of the kinetic-energy-density dependence. J. Chem. Phys. 138, 044113 (2013).

    Google Scholar 

  187. 187.

    Klimeš, J., Bowler, D. R. & Michaelides, A. Chemical accuracy for the van der Waals density functional. J. Phys. Condens. Matter 22, 022201 (2009).

    Google Scholar 

  188. 188.

    Ranganathan, S. On the geometry of coincidence-site lattices. Acta Crystallogr. 21, 197–199 (1966).

    CAS  Google Scholar 

  189. 189.

    Koshino, M. Interlayer interaction in general incommensurate atomic layers. New J. Phys. 17, 015014 (2015).

    CAS  Google Scholar 

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Acknowledgements

The authors thank Z. Zhu, D. Larson and E. Kucukbenli for helpful discussions and reference recommendations. This work was supported in part by ARO MURI award no. W911NF-14-0247 and by the STC Center for Integrated Quantum Materials, NSF grant no. DMR-1231319. The tight-binding calculation shown in Fig. 5 was run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.

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Carr, S., Fang, S. & Kaxiras, E. Electronic-structure methods for twisted moiré layers. Nat Rev Mater (2020). https://doi.org/10.1038/s41578-020-0214-0

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