Zhen Ma1, Shuai Li1, Meng-Meng Xiao1, Ya-Wen Zheng1, Ming Lu3,2, Haiwen Liu5, Jin-Hua Gao1(), X. C. Xie2,4
1. School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China 2. International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 3. Beijing Academy of Quantum Information Sciences, Beijing 100193, China 4. CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 5. Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China
We report that the twisted few layer graphite (tFL-graphite) is a new family of moiré heterostructures (MHSs), which has richer and highly tunable moiré flat band structures entirely distinct from all the known MHSs. A tFL-graphite is composed of two few-layer graphite (Bernal stacked multilayer graphene), which are stacked on each other with a small twisted angle. The moiré band structure of the tFL-graphite strongly depends on the layer number of its composed two van der Waals layers. Near the magic angle, a tFL-graphite always has two nearly flat bands coexisting with a few pairs of narrowed dispersive (parabolic or linear) bands at the Fermi level, thus, enhances the DOS at EF . This coexistence property may also enhance the possible superconductivity as been demonstrated in other multiband superconductivity systems. Therefore, we expect strong multiband correlation effects in tFL-graphite. Meanwhile, a proper perpendicular electric field can induce several isolated nearly flat bands with nonzero valley Chern number in some simple tFL-graphites, indicating that tFL-graphite is also a novel topological flat band system.
Cao Y., Fatemi V., Demir A., Fang S., L. Tomarken S., Y. Luo J., D. Sanchez-Yamagishi J., Watanabe K., Taniguchi T., Kaxiras E., C. Ashoori R., Jarillo-Herrero P.. Correlated insulator behavior at half-filling in magic-angle graphene superlattices. Nature, 2018, 556(7699): 80 https://doi.org/10.1038/nature26154
2
Cao Y., Fatemi V., Fang S., Watanabe K., Taniguchi T., Kaxiras E., Jarillo-Herrero P.. Unconventional superconductivity in magic-angle graphene superlattices. Nature, 2018, 556(7699): 43 https://doi.org/10.1038/nature26160
3
Yankowitz M., Chen S., Polshyn H., Zhang Y., Watanabe K., Taniguchi T., Graf D., F. Young A., R. Dean C.. Tuning superconductivity in twisted bilayer graphene. Science, 2019, 363(6431): 1059 https://doi.org/10.1126/science.aav1910
4
Lu X., Stepanov P., Yang W., Xie M., Aamir M., Das I., Urgell C., Watanabe K., Taniguchi T., Zhang G., Bachtold A., MacDonald A., Efetov D.. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature, 2019, 574(7780): 653 https://doi.org/10.1038/s41586-019-1695-0
5
Sharpe A., Fox E., Barnard A., Finney J., Watanabe K., Taniguchi T., Kastner M., Goldhaber-Gordon D.. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science, 2019, 365(6453): 605 https://doi.org/10.1126/science.aaw3780
6
Codecido E., Wang Q., Koester R., Che S., Tian H., Lv R., Tran S., Watanabe K., Taniguchi T., Zhang F., Bockrath M., Lau C.. Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle. Sci. Adv., 2019, 5(9): eaaw9770 https://doi.org/10.1126/sciadv.aaw9770
7
Jiang Y., Lai X., Watanabe K., Taniguchi T., Haule K., Mao J., Andrei E.. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature, 2019, 573(7772): 91 https://doi.org/10.1038/s41586-019-1460-4
8
Tong Q., Yu H., Zhu Q., Wang Y., Xu X., Yao W.. Topological mosaics in moiré superlattices of van der Waals heterobilayers. Nat. Phys., 2017, 13(4): 356 https://doi.org/10.1038/nphys3968
9
M. B. Lopes dos Santos J., M. R. Peres N., H. Castro Neto A.. Graphene bilayer with a twist: Electronic structure. Phys. Rev. Lett., 2007, 99(25): 256802 https://doi.org/10.1103/PhysRevLett.99.256802
10
Bistritzer R., H. MacDonald A.. Moiré bands in twisted double-layer graphene. Proc. Natl. Acad. Sci. USA, 2011, 108(30): 12233 https://doi.org/10.1073/pnas.1108174108
11
M. B. Lopes dos Santos J., M. R. Peres N., H. Castro Neto A.. Continuum model of the twisted graphene bilayer. Phys. Rev. B, 2012, 86(15): 155449 https://doi.org/10.1103/PhysRevB.86.155449
Koshino M., F. Q. Yuan N., Koretsune T., Ochi M., Kuroki K., Fu L.. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X, 2018, 8(3): 031087 https://doi.org/10.1103/PhysRevX.8.031087
14
Kang J., Vafek O.. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X, 2018, 8(3): 031088 https://doi.org/10.1103/PhysRevX.8.031088
15
A. Gonzalez-Arraga L., L. Lado J., Guinea F., San-Jose P.. Electrically controllable magnetism in twisted bilayer graphene. Phys. Rev. Lett., 2017, 119(10): 107201 https://doi.org/10.1103/PhysRevLett.119.107201
C. Po H., Zou L., Vishwanath A., Senthil T.. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X, 2018, 8(3): 031089 https://doi.org/10.1103/PhysRevX.8.031089
18
Isobe H., F. Q. Yuan N., Fu L.. Unconventional superconductivity and density waves in twisted bilayer graphene. Phys. Rev. X, 2018, 8(4): 041041 https://doi.org/10.1103/PhysRevX.8.041041
19
Padhi B., Setty C., Phillips P.. Doped twisted bilayer graphene near magic angles: Proximity to Wigner crystallization, not Mott insulation. Nano Lett., 2018, 18(10): 6175 https://doi.org/10.1021/acs.nanolett.8b02033
20
Guinea F., Walet N.. Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers. Proc. Natl. Acad. Sci. USA, 2018, 115(52): 13174 https://doi.org/10.1073/pnas.1810947115
21
C. Liu C., D. Zhang L., Q. Chen W., Yang F.. Chiral spin density wave and d+id superconductivity in the magic-angle-twisted bilayer graphene. Phys. Rev. Lett., 2018, 121(21): 217001 https://doi.org/10.1103/PhysRevLett.121.217001
22
Guo H., Zhu X., Feng S., T. Scalettar R.. Pairing symmetry of interacting fermions on a twisted bilayer graphene superlattice. Phys. Rev. B, 2018, 97(23): 235453 https://doi.org/10.1103/PhysRevB.97.235453
23
P. Lin Y., M. Nandkishore R.. Kohn−Luttinger superconductivity on two orbital honeycomb lattice. Phys. Rev. B, 2018, 98(21): 214521 https://doi.org/10.1103/PhysRevB.98.214521
24
Wu F., H. MacDonald A., Martin I.. Theory of phonon-mediated superconductivity in twisted bilayer graphene. Phys. Rev. Lett., 2018, 121(25): 257001 https://doi.org/10.1103/PhysRevLett.121.257001
J. Peltonen T., Ojajärvi R., T. Heikkilä T.. Mean-field theory for superconductivity in twisted bilayer graphene. Phys. Rev. B, 2018, 98(22): 220504 https://doi.org/10.1103/PhysRevB.98.220504
27
M. Kennes D., Lischner J., Karrasch C.. Strong correlations and d+id superconductivity in twisted bilayer graphene. Phys. Rev. B, 2018, 98(24): 241407 https://doi.org/10.1103/PhysRevB.98.241407
28
Z. You Y., Vishwanath A.. Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene. npj Quantum Mater., 2019, 4(1): 16 https://doi.org/10.1038/s41535-019-0153-4
29
Liu X., Hao Z., Khalaf E., Y. Lee J., Ronen Y., Yoo H., Haei Najafabadi D., Watanabe K., Taniguchi T., Vishwanath A., Kim P.. Tunable spin-polarized correlated states in twisted double bilayer graphene. Nature, 2020, 583(7815): 221 https://doi.org/10.1038/s41586-020-2458-7
30
Shen C., Chu Y., Wu Q., Li N., Wang S., Zhao Y., Tang J., Liu J., Tian J., Watanabe K., Taniguchi T., Yang R., Y. Meng Z., Shi D., V. Yazyev O., Zhang G.. Correlated states in twisted double bilayer graphene. Nat. Phys., 2020, 16(5): 520 https://doi.org/10.1038/s41567-020-0825-9
31
Cao Y., Rodan-Legrain D., Rubies-Bigorda O., M. Park J., Watanabe K., Taniguchi T., Jarillo-Herrero P.. Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature, 2020, 583(7815): 215 https://doi.org/10.1038/s41586-020-2260-6
32
W. Burg G., Zhu J., Taniguchi T., Watanabe K., H. MacDonald A., Tutuc E.. Correlated insulating states in twisted double bilayer graphene. Phys. Rev. Lett., 2019, 123(19): 197702 https://doi.org/10.1103/PhysRevLett.123.197702
33
H. Zhang Y., Mao D., Cao Y., Jarillo-Herrero P., Senthil T.. Nearly flat Chern bands in moiré superlattices. Phys. Rev. B, 2019, 99(7): 075127 https://doi.org/10.1103/PhysRevB.99.075127
Y. Lee J., Khalaf E., Liu S., Liu X., Hao Z., Kim P., Vishwanath A.. Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer grapheme. Nat. Commun., 2019, 10: 5333 https://doi.org/10.1038/s41467-019-12981-1
37
Haddadi F., Wu Q., J. Kruchkov A., V. Yazyev O.. Moiré flat bands in twisted double bilayer graphene. Nano Lett., 2020, 20(4): 2410 https://doi.org/10.1021/acs.nanolett.9b05117
38
Wu F., Das Sarma S.. Ferromagnetism and superconductivity in twisted double bilayer graphene. Phys. Rev. B, 2020, 101(15): 155149 https://doi.org/10.1103/PhysRevB.101.155149
39
J. Culchac F.B. Capaz R.Chico L. Suarez Morell E., Flat bands and gaps in twisted double bilayer grapheme, arXiv: 1911.01347 (2019)
40
Ma Z.Li S. W. Zheng Y.M. Xiao M.Jiang H.H. Gao J.C. Xie X., Topological flat bands in twisted trilayer grapheme, arXiv: 1905.00622 (2019)
41
J. Zuo W., B. Qiao J., L. Ma D., J. Yin L., Sun G., Y. Zhang J., Y. Guan L., He L.. Scanning tunneling microscopy and spectroscopy of twisted trilayer graphene. Phys. Rev. B, 2018, 97(3): 035440 https://doi.org/10.1103/PhysRevB.97.035440
42
Suárez Morell E., Pacheco M., Chico L., Brey L.. Electronic properties of twisted trilayer graphene. Phys. Rev. B, 2013, 87(12): 125414 https://doi.org/10.1103/PhysRevB.87.125414
43
Li X.Wu F. H. MacDonald A., Electronic structure of single-twist trilayer graphene, arXiv: 1907.12338 (2019)
44
Carr S., Li C., Zhu Z., Kaxiras E., Sachdev S., Kruchkov A.. Ultraheavy and ultrarelativistic Dirac quasiparticles in sandwiched graphenes. Nano Lett., 2020, 20(5): 3030 https://doi.org/10.1021/acs.nanolett.9b04979
45
Xu S., M. Al Ezzi M., Balakrishnan N., Garcia-Ruiz A., Tsim B., Mullan C., Barrier J., Xin N., A. Piot B., Taniguchi T., Watanabe K., Carvalho A., Mishchenko A., K. Geim A., I. Fal’ko V., Adam S., H. C. Neto A., S. Novoselov K., Shi Y.. Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene. Nat. Phys., 2021, 17(5): 619 https://doi.org/10.1038/s41567-021-01172-9
46
Chen S., He M., H. Zhang Y., Hsieh V., Fei Z., Watanabe K., Taniguchi T., H. Cobden D., Xu X., R. Dean C., Yankowitz M.. Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene. Nat. Phys., 2021, 17(3): 374 https://doi.org/10.1038/s41567-020-01062-6
47
Polshyn H., Zhu J., A. Kumar M., Zhang Y., Yang F., L. Tschirhart C., Serlin M., Watanabe K., Taniguchi T., H. MacDonald A., F. Young A.. Electrical switching of magnetic order in an orbital Chern insulator. Nature, 2020, 588(7836): 66 https://doi.org/10.1038/s41586-020-2963-8
48
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, 2019, 9(3): 031021 https://doi.org/10.1103/PhysRevX.9.031021
49
L. Chittari B., Chen G., Zhang Y., Wang F., Jung J.. Gate-tunable topological flat bands in trilayer graphene boron-nitride moiré superlattices. Phys. Rev. Lett., 2019, 122(1): 016401 https://doi.org/10.1103/PhysRevLett.122.016401
50
Chen G., Jiang L., Wu S., Lyu B., Li H., L. Chittari B., Watanabe K., Taniguchi T., Shi Z., Jung J., Zhang Y., Wang F.. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys., 2019, 15(3): 237 https://doi.org/10.1038/s41567-018-0387-2
51
Chen G., Sharpe A., Gallagher P., Rosen I., Fox E., Jiang L., Lyu B., Li H., Watanabe K., Taniguchi T., Jung J., Shi Z., Goldhaber-Gordon D., Zhang Y., Wang F.. Signatures of tunable superconductivity in a trilayer graphene moiré superlattice. Nature, 2019, 572(7768): 215 https://doi.org/10.1038/s41586-019-1393-y
52
Wu F., Lovorn T., Tutuc E., H. MacDonald A.. Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett., 2018, 121(2): 026402 https://doi.org/10.1103/PhysRevLett.121.026402
53
H. Naik M., Jain M.. Ultraflatbands and shear solitons in moiré patterns of twisted bilayer transition metal dichalcogenides. Phys. Rev. Lett., 2018, 121(26): 266401 https://doi.org/10.1103/PhysRevLett.121.266401
54
Pan Y., Fölsch S., Nie Y., Waters D., C. Lin Y., Jariwala B., Zhang K., Cho K., Robinson J., Feenstra R.. Quantum-confined electronic states arising from the moiré pattern of MoS2–WSe2 heterobilayers. Nano Lett., 2018, 18(3): 1849 https://doi.org/10.1021/acs.nanolett.7b05125
55
Wu F., Lovorn T., Tutuc E., Martin I., H. Mac-Donald A.. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett., 2019, 122(8): 086402 https://doi.org/10.1103/PhysRevLett.122.086402
56
Conte F., Ninno D., Cantele G.. Electronic properties and interlayer coupling of twisted MoS2/NbSe2 heterobilayers. Phys. Rev. B, 2019, 99(15): 155429 https://doi.org/10.1103/PhysRevB.99.155429
57
Javvaji S., H. Sun J., Jung J.. Topological flat bands without magic angles in massive twisted bilayer graphenes. Phys. Rev. B, 2020, 101(12): 125411 https://doi.org/10.1103/PhysRevB.101.125411
58
M. Park J., Cao Y., Watanabe K., Taniguchi T., Jarillo-Herrero P.. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature, 2021, 590(7845): 249 https://doi.org/10.1038/s41586-021-03192-0
59
Hao Z., M. Zimmerman A., Ledwith P., Khalaf E., H. Najafabadi D., Watanabe K., Taniguchi T., Vishwanath A., Kim P.. Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science, 2021, 371(6534): 1133 https://doi.org/10.1126/science.abg0399
60
Liang M., M. Xiao M., Ma Z., H. Gao J.. Moiré band structures of the double twisted few-layer graphene. Phys. Rev. B, 2022, 105(19): 195422 https://doi.org/10.1103/PhysRevB.105.195422
61
See the Supplemental Material for more details.
62
Guinea F., H. Castro Neto A., M. R. Peres N.. Electronic states and Landau levels in graphene stacks. Phys. Rev. B, 2006, 73(24): 245426 https://doi.org/10.1103/PhysRevB.73.245426
63
Koshino M., Ando T.. Orbital diamagnetism in multilayer graphenes: Systematic study with the effective mass approximation. Phys. Rev. B, 2007, 76(8): 085425 https://doi.org/10.1103/PhysRevB.76.085425
64
Min H., H. MacDonald A.. Chiral decomposition in the electronic structure of graphene multilayers. Phys. Rev. B, 2008, 77(15): 155416 https://doi.org/10.1103/PhysRevB.77.155416
Bussmann-Holder A., Keller H., Simon A., Bianconi A.. Multi-band superconductivity and the steep band/flat band scenario. Condens. Matter, 2019, 4(4): 91 https://doi.org/10.3390/condmat4040091
67
B. Wu J., Zhang X., Ijäs M., P. Han W., F. Qiao X., L. Li X., S. Jiang D., C. Ferrari A., H. Tan P.. Resonant Raman spectroscopy of twisted multilayer graphene. Nat. Commun., 2014, 5(1): 5309 https://doi.org/10.1038/ncomms6309
68
Zhang F., Sahu B., Min H., H. MacDonald A.. Band structure of ABC-stacked graphene trilayers. Phys. Rev. B, 2010, 82(3): 035409 https://doi.org/10.1103/PhysRevB.82.035409
69
Khalaf E., J. Kruchkov A., Tarnopolsky G., Vishwanath A.. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B, 2019, 100(8): 085109 https://doi.org/10.1103/PhysRevB.100.085109
70
Peng H., B. M. Schröter N., Yin J., Wang H., F. Chung T., Yang H., Ekahana S., Liu Z., Jiang J., Yang L., Zhang T., Chen C., Ni H., Barinov A., P. Chen Y., Liu Z., Peng H., Chen Y.. Substrate doping effect and unusually large angle van Hove singularity evolution in twisted bi- and multilayer graphene. Adv. Mater., 2017, 29(27): 1606741 https://doi.org/10.1002/adma.201606741
71
I. B. Utama M., J. Koch R., Lee K., Leconte N., Li H., Zhao S., Jiang L., Zhu J., Watanabe K., Taniguchi T.. et al.. Visualization of the flat electronic band in twisted bilayer graphene near the magic angle twist. Nat. Phys., 2021, 17: 184 https://doi.org/10.1038/s41567-020-0974-x
72
J. P. Thompson J., Pei D., Peng H., Wang H., Channa N., L. Peng H., Barinov A., B. M. Schröter N., Chen Y., Mucha-Kruczy’nski M.. Determination of interatomic coupling between two-dimensional crystals using angle-resolved photoemission spectroscopy. Nat. Commun., 2020, 11(1): 3582 https://doi.org/10.1038/s41467-020-17412-0
73
Lisi S., Lu X., Benschop T., A. de Jong T., Stepanov P., R. Duran J., Margot F., Cucchi I., Cappelli E., Hunter A., Tamai A., Kandyba V., Giampietri A., Barinov A., Jobst J., Stalman V., Leeuwenhoek M., Watanabe K., Taniguchi T., Rademaker L., J. van der Molen S., P. Allan M., K. Efetov D., Baumberger F.. Observation of flat bands in twisted bilayer grapheme. Nat. Phys., 2021, 17: 189 https://doi.org/10.1038/s41567-020-01041-x
74
Vela A., V. O. Moutinho M., J. Culchac F., Venezuela P., B. Capaz R.. Electronic structure and optical properties of twisted multilayer graphene. Phys. Rev. B, 2018, 98(15): 155135 https://doi.org/10.1103/PhysRevB.98.155135
Grushina A., K. Ki D., Koshino M., Nicolet A., Faugeras C., McCann E., Potemski M., Morpurgo A.. Insulating state in tetralayers reveals an even–odd interaction effect in multilayer graphene. Nat. Commun., 2015, 6(1): 6419 https://doi.org/10.1038/ncomms7419
77
Nam Y., K. Ki D., Soler-Delgado D., Morpurgo A.. A family of finite-temperature electronic phase transitions in graphene multilayers. Science, 2018, 362(6412): 324 https://doi.org/10.1126/science.aar6855
A. H. Goodwin Z., Klebl L., Vitale V., Liang X., Gogtay V., van Gorp X., M. Kennes D., A. Mostofi A., Lischner J.. Flat bands, electron interactions, and magnetic order in magic-angle mono-trilayer graphene. Phys. Rev. Mater., 2021, 5(8): 084008 https://doi.org/10.1103/PhysRevMaterials.5.084008
80
Zhang S.Xie B.Wu Q.Liu J.V. Yazyev O., Chiral decomposition of twisted graphene multilayers with arbitrary stacking, arXiv: 2012.11964 (2020)
81
Lin X., Zhu H., Ni J.. Emergence of intrinsically isolated flat bands and their topology in fully relaxed twisted multilayer graphene. Phys. Rev. B, 2021, 104(12): 125421 https://doi.org/10.1103/PhysRevB.104.125421