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
  • Editors' Suggestion
  • Letter
  • Open Access

Massive Dirac fermions in moiré superlattices: A route towards topological flat minibands and correlated topological insulators

Ying Su1,2, Heqiu Li3,4, Chuanwei Zhang2, Kai Sun3, and Shi-Zeng Lin1

  • 1Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
  • 2Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
  • 3Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
  • 4Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada

Phys. Rev. Research 4, L032024 – Published 12 August, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L032024

Abstract

We demonstrate a generic mechanism to realize topological flat minibands by confining massive Dirac fermions in a periodic moiré potential, which can be achieved in a heterobilayer of transition metal dichalcogenides. We show that the topological phase can be protected by the symmetry of moiré potential and survive to arbitrarily large Dirac band gap. We take the MoTe2/WSe2 heterobilayer as an example and find that the topological phase can be driven by a vertical electric field. By projecting the Coulomb interaction onto the topological fat minibands, we identify a correlated Chern insulator at half filling and a quantum valley-spin Hall insulator at full filling which explains the topological states observed in the MoTe2/WSe2 in the experiment. Our work clarifies the importance of Dirac structure for the topological minibands and unveils a general strategy to design topological moiré materials.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (101)

  1. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  2. K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically Thin MoS2: A New Direct-Gap Semiconductor, Phys. Rev. Lett. 105, 136805 (2010).
  3. D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled Spin and Valley Physics in Monolayers of MoS2 and Other Group-VI Dichalcogenides, Phys. Rev. Lett. 108, 196802 (2012).
  4. M. Katsnelson, K. Novoselov, and A. Geim, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys. 2, 620 (2006).
  5. D. Xiao, W. Yao, and Q. Niu, Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Transport, Phys. Rev. Lett. 99, 236809 (2007).
  6. K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, The valley Hall effect in MoS2 transistors, Science 344, 1489 (2014).
  7. W. Yao, D. Xiao, and Q. Niu, Valley-dependent optoelectronics from inversion symmetry breaking, Phys. Rev. B 77, 235406 (2008).
  8. T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu et al., Valley-selective circular dichroism of monolayer molybdenum disulphide, Nat. Commun. 3, 887 (2012).
  9. A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science 365, 605 (2019).
  10. Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, Nearly flat Chern bands in moiré superlattices, Phys. Rev. B 99, 075127 (2019).
  11. Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All Magic Angles in Twisted Bilayer Graphene are Topological, Phys. Rev. Lett. 123, 036401 (2019).
  12. Y.-H. Zhang, D. Mao, and T. Senthil, Twisted bilayer graphene aligned with hexagonal boron nitride: Anomalous Hall effect and a lattice model, Phys. Rev. Research 1, 033126 (2019).
  13. J. Y. Lee, E. Khalaf, S. Liu, X. Liu, Z. Hao, P. Kim, and A. Vishwanath, Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene, Nat. Commun. 10, 1 (2019).
  14. M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020).
  15. P. Stepanov, M. Xie, T. Taniguchi, K. Watanabe, X. Lu, A. H. MacDonald, B. A. Bernevig, and D. K. Efetov, Competing zero-field Chern insulators in superconducting twisted bilayer graphene (2020), arXiv:2012.15126 [cond-mat.mes-hall]
  16. M. Xie and A. H. MacDonald, Nature of the Correlated Insulator States in Twisted Bilayer Graphene, Phys. Rev. Lett. 124, 097601 (2020).
  17. N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene, Phys. Rev. Lett. 124, 166601 (2020).
  18. F. Wu and S. Das Sarma, Collective Excitations of Quantum Anomalous Hall Ferromagnets in Twisted Bilayer Graphene, Phys. Rev. Lett. 124, 046403 (2020).
  19. Y. Su and S.-Z. Lin, Current-Induced Reversal of Anomalous Hall Conductance in Twisted Bilayer Graphene, Phys. Rev. Lett. 125, 226401 (2020).
  20. K. P. Nuckolls, M. Oh, D. Wong, B. Lian, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Strongly correlated Chern insulators in magic-angle twisted bilayer graphene, Nature (London) 588, 610 (2020).
  21. 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).
  22. S. Wu, Z. Zhang, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Chern insulators, van hove singularities and topological flat bands in magic-angle twisted bilayer graphene, Nat. Mater. 20, 488 (2021).
  23. I. Das, X. Lu, J. Herzog-Arbeitman, Z.-D. Song, K. Watanabe, T. Taniguchi, B. A. Bernevig, and D. K. Efetov, Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene, Nat. Phys. 17, 710 (2021).
  24. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Flavour Hund's coupling, Chern gaps and charge diffusivity in moiré graphene, Nature (London) 592, 43 (2021).
  25. Y. Wang, J. Herzog-Arbeitman, G. W. Burg, J. Zhu, K. Watanabe, T. Taniguchi, A. H. MacDonald, B. A. Bernevig, and E. Tutuc, Topological edge transport in twisted double-bilayer graphene (2021), arXiv:2101.03621 [cond-mat.mes-hall].
  26. A. T. Pierce, Y. Xie, J. M. Park, E. Khalaf, S. H. Lee, Y. Cao, D. E. Parker, P. R. Forrester, S. Chen, K. Watanabe et al., Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene, Nat. Phys. 17, 1210 (2021).
  27. M. He, J. Cai, Y.-H. Zhang, Y. Liu, Y. Li, T. Taniguchi, K. Watanabe, D. H. Cobden, M. Yankowitz, and X. Xu, Chirality-dependent topological states in twisted double bilayer graphene (2021), arXiv:2109.08255 [cond-mat.mes-hall].
  28. B. L. Chittari, G. Chen, Y. Zhang, F. Wang, and J. Jung, Gate-Tunable Topological Flat Bands in Trilayer Graphene Boron-Nitride Moiré Superlattices, Phys. Rev. Lett. 122, 016401 (2019).
  29. Y.-H. Zhang and T. Senthil, Bridging Hubbard model physics and quantum Hall physics in trilayer graphene/h−BN moiré superlattice, Phys. Rev. B 99, 205150 (2019).
  30. G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi et al., Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice, Nature (London) 579, 56 (2020).
  31. F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers, Phys. Rev. Lett. 122, 086402 (2019).
  32. 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).
  33. 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).
  34. G. Chen, A. L. Sharpe, P. Gallagher, I. T. Rosen, E. J. Fox, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi et al., Signatures of tunable superconductivity in a trilayer graphene moiré superlattice, Nature (London) 572, 215 (2019).
  35. 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).
  36. P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K. Efetov, Untying the insulating and superconducting orders in magic-angle graphene, Nature (London) 583, 375 (2020).
  37. H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi, H. Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu et al., Superconductivity in metallic twisted bilayer graphene stabilized by WSe2, Nature (London) 583, 379 (2020).
  38. 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).
  39. 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).
  40. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene, Nature (London) 590, 249 (2021).
  41. Z. Hao, A. Zimmerman, P. Ledwith, E. Khalaf, D. H. Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene, Science 371, 1133 (2021).
  42. Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant superconductivity in moiré graphene, Nature (London) 595, 526 (2021).
  43. H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Spectroscopic signatures of strong correlations and unconventional superconductivity in twisted trilayer graphene (2021), arXiv:2109.12127 [cond-mat.mes-hall].
  44. C. Xu and L. Balents, Topological Superconductivity in Twisted Multilayer Graphene, Phys. Rev. Lett. 121, 087001 (2018).
  45. H. Guo, X. Zhu, S. Feng, and R. T. Scalettar, Pairing symmetry of interacting fermions on a twisted bilayer graphene superlattice, Phys. Rev. B 97, 235453 (2018).
  46. F. Wu, A. H. MacDonald, and I. Martin, Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene, Phys. Rev. Lett. 121, 257001 (2018).
  47. M. Fidrysiak, M. Zegrodnik, and J. Spałek, Unconventional topological superconductivity and phase diagram for an effective two-orbital model as applied to twisted bilayer graphene, Phys. Rev. B 98, 085436 (2018).
  48. Y. Su and S.-Z. Lin, Pairing symmetry and spontaneous vortex-antivortex lattice in superconducting twisted-bilayer graphene: Bogoliubov-de Gennes approach, Phys. Rev. B 98, 195101 (2018).
  49. C.-C. Liu, L.-D. Zhang, W.-Q. Chen, and F. Yang, Chiral Spin Density Wave and d+id Superconductivity in the Magic-Angle-Twisted Bilayer Graphene, Phys. Rev. Lett. 121, 217001 (2018).
  50. D. M. Kennes, J. Lischner, and C. Karrasch, Strong correlations and d+id superconductivity in twisted bilayer graphene, Phys. Rev. B 98, 241407(R) (2018).
  51. H. Isobe, N. F. Q. Yuan, and L. Fu, Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene, Phys. Rev. X 8, 041041 (2018).
  52. B. Roy and V. Juričić, Unconventional superconductivity in nearly flat bands in twisted bilayer graphene, Phys. Rev. B 99, 121407(R) (2019).
  53. T. Huang, L. Zhang, and T. Ma, Antiferromagnetically ordered Mott insulator and d+ id superconductivity in twisted bilayer graphene: A quantum monte carlo study, Sci. Bull. 64, 310 (2019).
  54. S. Ray, J. Jung, and T. Das, Wannier pairs in superconducting twisted bilayer graphene and related systems, Phys. Rev. B 99, 134515 (2019).
  55. B. Lian, Z. Wang, and B. A. Bernevig, Twisted Bilayer Graphene: A Phonon-Driven Superconductor, Phys. Rev. Lett. 122, 257002 (2019).
  56. J. González and T. Stauber, Kohn-Luttinger Superconductivity in Twisted Bilayer Graphene, Phys. Rev. Lett. 122, 026801 (2019).
  57. Y.-Z. You and A. Vishwanath, Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene, npj Quantum Mater. 4, 16 (2019).
  58. T. Cea and F. Guinea, Coulomb interaction, phonons, and superconductivity in twisted bilayer graphene, Proc. Natl. Acad. Sci. 118, e2107874118 (2021).
  59. R. M. Fernandes and L. Fu, Charge-4e Superconductivity from Multicomponent Nematic Pairing: Application to Twisted Bilayer Graphene, Phys. Rev. Lett. 127, 047001 (2021).
  60. E. Khalaf, S. Chatterjee, N. Bultinck, M. P. Zaletel, and A. Vishwanath, Charged skyrmions and topological origin of superconductivity in magic-angle graphene, Sci. Adv. 7, eabf5299 (2021).
  61. W. Qin and A. H. MacDonald, In-Plane Critical Magnetic Fields in Magic-Angle Twisted Trilayer Graphene, Phys. Rev. Lett. 127, 097001 (2021).
  62. E. Lake and T. Senthil, Re-entrant superconductivity through a quantum Lifshitz transition in twisted trilayer graphene (2021), arXiv:2104.13920 [cond-mat.mes-hall].
  63. P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional Chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Research 2, 023237 (2020).
  64. C. Repellin and T. Senthil, Chern bands of twisted bilayer graphene: Fractional Chern insulators and spin phase transition, Phys. Rev. Research 2, 023238 (2020).
  65. Z. Liu, A. Abouelkomsan, and E. J. Bergholtz, Gate-Tunable Fractional Chern Insulators in Twisted Double Bilayer Graphene, Phys. Rev. Lett. 126, 026801 (2021).
  66. H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices, Phys. Rev. Research 3, L032070 (2021).
  67. Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester et al., Fractional Chern insulators in magic-angle twisted bilayer graphene (2021), arXiv:2107.10854 [cond-mat.mes-hall].
  68. F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands, Phys. Rev. Lett. 121, 026402 (2018).
  69. Y. Zhang, N. F. Q. Yuan, and L. Fu, Moiré quantum chemistry: Charge transfer in transition metal dichalcogenide superlattices, Phys. Rev. B 102, 201115(R) (2020).
  70. Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan, and K. F. Mak, Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices, Nature (London) 579, 353 (2020).
  71. E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta et al., Mott and generalized wigner crystal states in WSe2/WS2 moiré superlattices, Nature (London) 579, 359 (2020).
  72. 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).
  73. Z. Chu, E. C. Regan, X. Ma, D. Wang, Z. Xu, M. I. B. Utama, K. Yumigeta, M. Blei, K. Watanabe, T. Taniguchi, S. Tongay, F. Wang, and K. Lai, Nanoscale Conductivity Imaging of Correlated Electronic States in WSe2/WS2 Moiré Superlattices, Phys. Rev. Lett. 125, 186803 (2020).
  74. X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao et al., Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice, Nat. Phys. 17, 715 (2021).
  75. H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe et al., Imaging two-dimensional generalized wigner crystals, Nature (London) 597, 650 (2021).
  76. T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. Devakul, K. Watanabe, T. Taniguchi, L. Fu et al., Quantum anomalous Hall effect from intertwined moiré bands, Nature (London) 600, 641 (2021).
  77. G.-B. Liu, D. Xiao, Y. Yao, X. Xu, and W. Yao, Electronic structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides, Chem. Soc. Rev. 44, 2643 (2015).
  78. C. Fang, M. J. Gilbert, and B. A. Bernevig, Bulk topological invariants in noninteracting point group symmetric insulators, Phys. Rev. B 86, 115112 (2012).
  79. F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ”Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  80. In triangular lattice, there is one site per unit cell and the PT and C3 symmetries can not stabilize any Dirac cone because there is only one single band.
  81. L. Meckbach, J. Hader, U. Huttner, J. Neuhaus, J. T. Steiner, T. Stroucken, J. V. Moloney, and S. W. Koch, Ultrafast band-gap renormalization and build-up of optical gain in monolayer MoTe2, Phys. Rev. B 101, 075401 (2020).
  82. N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Cepellotti, G. Pizzi, and N. Marzari, Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds, Nat. Nanotechnol. 13, 246 (2018).
  83. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L032024 for (1) Wannier orbitals of moiré minibands, (2) comparison between the two continuum models in the trivial phase, (3) local stacking configurations in AA- and AB-stacked TMD heterobilayers, (4) details of Hartree-Fock calculations, (5) topological phase protected by the C4 symmetry of moiré potential, and (6) potential realizations of massive Dirac fermions confined in a moiré potential, which includes Refs. [31, 95, 96, 97, 98, 99, 100, 101].
  84. Y. Zhang, T. Devakul, and L. Fu, Spin-textured Chern bands in Ab-stacked transition metal dichalcogenide bilayers, Proc. Natl. Acad. Sci. 118, e2112673118 (2021).
  85. C. Zhang, C.-P. Chuu, X. Ren, M.-Y. Li, L.-J. Li, C. Jin, M.-Y. Chou, and C.-K. Shih, Interlayer couplings, Moiré patterns, and 2D electronic superlattices in MoS2/WSe2 hetero-bilayers, Sci. Adv. 3, e1601459 (2017).
  86. S. Shabani, D. Halbertal, W. Wu, M. Chen, S. Liu, J. Hone, W. Yao, D. N. Basov, X. Zhu, and A. N. Pasupathy, Deep moiré potentials in twisted transition metal dichalcogenide bilayers, Nat. Phys. 17, 720 (2021).
  87. W. T. Geng, V. Wang, Y. C. Liu, T. Ohno, and J. Nara, Moiré potential, lattice corrugation, and band gap spatial variation in a twist-free MoTe2/MoS2 heterobilayer, J. Phys. Chem. Lett. 11, 2637 (2020).
  88. F. Wu, T. Lovorn, and A. H. MacDonald, Topological Exciton Bands in Moiré Heterojunctions, Phys. Rev. Lett. 118, 147401 (2017).
  89. T. Li, S. Jiang, L. Li, Y. Zhang, K. Kang, J. Zhu, K. Watanabe, T. Taniguchi, D. Chowdhury, L. Fu, J. Shan, and K. F. Mak, Continuous Mott transition in semiconductor moiré superlattices, Nature (London) 597, 350 (2021).
  90. C. Robert, R. Picard, D. Lagarde, G. Wang, J. P. Echeverry, F. Cadiz, P. Renucci, A. Högele, T. Amand, X. Marie, I. C. Gerber, and B. Urbaszek, Excitonic properties of semiconducting monolayer and bilayer MoTe2, Phys. Rev. B 94, 155425 (2016).
  91. There are two degenerate ground states with the top valence band from either the +K or −K valley is empty at ν=1.
  92. Q. Tong, H. Yu, Q. Zhu, Y. Wang, X. Xu, and W. Yao, Topological mosaics in moiré superlattices of van der Waals heterobilayers, Nat. Phys. 13, 356 (2017).
  93. Y.-M. Xie, C.-P. Zhang, J.-X. Hu, K. F. Mak, and K. T. Law, Valley-Polarized Quantum Anomalous Hall State in Moiré MoTe2/WSe2 Heterobilayers, Phys. Rev. Lett. 128, 026402 (2022).
  94. M. A. Cazalilla, H. Ochoa, and F. Guinea, Quantum Spin Hall Effect in Two-Dimensional Crystals of Transition-Metal Dichalcogenides, Phys. Rev. Lett. 113, 077201 (2014).
  95. Y. Xu, C. Horn, J. Zhu, Y. Tang, L. Ma, L. Li, S. Liu, K. Watanabe, T. Taniguchi, J. C. Hone et al., Creation of moiré bands in a monolayer semiconductor by spatially periodic dielectric screening, Nat. Mater. 20, 645 (2021).
  96. M. Koshino, N. F. Q. 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).
  97. J. Kang and O. Vafek, Symmetry, Maximally Localized Wannier States, and a Low-Energy Model for Twisted Bilayer Graphene Narrow Bands, Phys. Rev. X 8, 031088 (2018).
  98. K. Sun, W. V. Liu, A. Hemmerich, and S. D. Sarma, Topological semimetal in a fermionic optical lattice, Nat. Phys. 8, 67 (2012).
  99. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  100. M. Mogi, M. Kawamura, R. Yoshimi, A. Tsukazaki, Y. Kozuka, N. Shirakawa, K. Takahashi, M. Kawasaki, and Y. Tokura, A magnetic heterostructure of topological insulators as a candidate for an axion insulator, Nat. Mater. 16, 516 (2017).
  101. D. Xiao, J. Jiang, J.-H. Shin, W. Wang, F. Wang, Y.-F. Zhao, C. Liu, W. Wu, M. H. W. Chan, N. Samarth, and C.-Z. Chang, Realization of the Axion Insulator State in Quantum Anomalous Hall Sandwich Heterostructures, Phys. Rev. Lett. 120, 056801 (2018).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation