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Engineering symmetry breaking in 2D layered materials

Nature Reviews Physics volume 3, pages 193–206 (2021)Cite this article

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Abstract

Symmetry breaking in 2D layered materials plays a significant role in their macroscopic electrical, optical, magnetic and topological properties, including, but not limited to, spin-polarization effects, valley-contrasting physics, nonlinear Hall effects, nematic order, ferroelectricity, Bose–Einstein condensation and unconventional superconductivity. Engineering symmetry breaking of 2D layered materials not only offers extraordinary opportunities to tune their physical properties but also provides unprecedented possibilities to introduce completely new physics and technological innovations in electronics, photonics and optoelectronics. Indeed, over the past 15 years, a wide variety of physical, structural and chemical approaches have been developed to engineer the symmetry breaking of 2D layered materials. In this Technical Review, we focus on the recent progress on engineering the breaking of inversion, rotational, time-reversal and gauge symmetries in 2D layered materials, and present our perspectives on how these may lead to new physics and applications.

Key points

  • Most of the fascinating physical phenomena in 2D materials are dictated by their underlying symmetry breaking, namely, the breaking of inversion, rotational, time-reversal and gauge symmetries.

  • The symmetry breaking in 2D materials can be engineered by a wide variety of physical and chemical approaches. This opens the possibilities to manipulate the internal quantum degrees of freedom (such as spin, valley and layer pseudospin) for the emerging fields of spintronics, valleytronics and twistronics.

  • Engineering symmetry breaking in 2D materials can create unique opportunities to integrate different broken symmetries within one system, providing an unprecedented path to underpin new physics and transform the landscape of technological innovations.

  • Through engineering spontaneous symmetry breaking in magic-angle moiré superlattices, semimetallic graphene can be turned into a series of quantum states (such as a band insulator, Mott-like correlated insulator, quantum anomalous Hall insulator or superconductor), potentially offering new insights into strongly correlated physics, such as high-temperature unconventional superconductors and quantum spin liquids.

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Fig. 1: Symmetry breaking enabled various intriguing physical phenomena in 2D materials.
Fig. 2: Engineering inversion symmetry breaking.
Fig. 3: Engineering C3 rotational symmetry breaking.
Fig. 4: Engineering time-reversal symmetry breaking.
Fig. 5: Engineering spontaneous gauge symmetry breaking.

References

  1. Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

    ADS  Google Scholar 

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

    Article  Google Scholar 

  3. Sun, Z., Martinez, A. & Wang, F. Optical modulators with 2D layered materials. Nat. Photonics 10, 227–238 (2016).

    Article  ADS  Google Scholar 

  4. Mak, K. F., Xiao, D. & Shan, J. Light–valley interactions in 2D semiconductors. Nat. Photonics 12, 451–460 (2018).

    Article  ADS  Google Scholar 

  5. Gong, C. & Zhang, X. Two-dimensional magnetic crystals and emergent heterostructure devices. Science 363, eaav4450 (2019).

    Article  Google Scholar 

  6. Sun, Z. et al. Giant nonreciprocal second-harmonic generation from antiferromagnetic bilayer CrI3. Nature 572, 497–501 (2019). This paper demonstrated the nonreciprocal second-order nonlinear optical effect in 2D materials enabled by the inversion symmetry breaking of the magnetic point group.

    Article  ADS  Google Scholar 

  7. Xu, S.-Y. et al. Spontaneous gyrotropic electronic order in a transition-metal dichalcogenide. Nature 578, 545–549 (2020).

    Article  ADS  Google Scholar 

  8. Autere, A. et al. Nonlinear optics with 2D layered materials. Adv. Mater. 30, 1705963 (2018).

    Article  Google Scholar 

  9. Seyler, K. L. et al. Electrical control of second-harmonic generation in a WSe2 monolayer transistor. Nat. Nanotechnol. 10, 407–411 (2015).

    Article  ADS  Google Scholar 

  10. Huang, B. et al. Tuning inelastic light scattering via symmetry control in the two-dimensional magnet CrI3. Nat. Nanotechnol. 15, 212–216 (2020).

    Article  ADS  Google Scholar 

  11. Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. Nature 565, 337–342 (2019). This work represents one of the earliest experimental studies of nonlinear Hall effect enabled by the combination of inversion symmetry breaking and C3 rotational symmetry breaking.

    Article  ADS  Google Scholar 

  12. Deng, Y. et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895–900 (2020).

    Article  ADS  Google Scholar 

  13. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).

    Article  ADS  Google Scholar 

  14. Kang, K., Li, T., Sohn, E., Shan, J. & Mak, K. F. Nonlinear anomalous Hall effect in few-layer WTe2. Nat. Mater. 18, 324–328 (2019).

    Article  ADS  Google Scholar 

  15. Schaibley, J. R. et al. Valleytronics in 2D materials. Nat. Rev. Mater. 1, 16055 (2016).

    Article  ADS  Google Scholar 

  16. Langer, F. et al. Lightwave valleytronics in a monolayer of tungsten diselenide. Nature 557, 76–80 (2018).

    Article  ADS  Google Scholar 

  17. Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014).

    Article  ADS  Google Scholar 

  18. Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76–79 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Song, P. et al. Coexistence of large conventional and planar spin Hall effect with long spin diffusion length in a low-symmetry semimetal at room temperature. Nat. Mater. 19, 292–298 (2020).

    Article  ADS  Google Scholar 

  20. Shi, Y. et al. Imaging quantum spin Hall edges in monolayer WTe2. Sci. Adv. 5, eaat8799 (2019).

    Article  ADS  Google Scholar 

  21. Zhu, H. et al. Observation of chiral phonons. Science 359, 579–582 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  22. Huang, L. et al. Spectroscopic evidence for a type II Weyl semimetallic state in MoTe2. Nat. Mater. 15, 1155–1160 (2016).

    Article  ADS  Google Scholar 

  23. Belopolski, I. et al. Signatures of a time-reversal symmetric Weyl semimetal with only four Weyl points. Nat. Commun. 8, 942 (2017).

    Article  ADS  Google Scholar 

  24. Lv, B. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).

    Google Scholar 

  25. Wu, W. et al. Piezoelectricity of single-atomic-layer MoS2 for energy conversion and piezotronics. Nature 514, 470–474 (2014).

    Article  ADS  Google Scholar 

  26. Zhu, H. et al. Observation of piezoelectricity in free-standing monolayer MoS2. Nat. Nanotechnol. 10, 151–155 (2015).

    Article  ADS  Google Scholar 

  27. Ares, P. et al. Piezoelectricity in monolayer hexagonal boron nitride. Adv. Mater. 32, 1905504 (2020).

    Article  Google Scholar 

  28. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018). This paper engineered the spontaneous gauge symmetry breaking through creating flat bands in moiré superlattices and demonstrated the unconventional superconductivity.

    Article  ADS  Google Scholar 

  29. Xi, X. et al. Ising pairing in superconducting NbSe2 atomic layers. Nat. Phys. 12, 139–143 (2016).

    Article  Google Scholar 

  30. Wang, Z. et al. Evidence of high-temperature exciton condensation in two-dimensional atomic double layers. Nature 574, 76–80 (2019).

    Article  ADS  Google Scholar 

  31. Lu, J. et al. Evidence for two-dimensional Ising superconductivity in gated MoS2. Science 350, 1353–1357 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Manchon, A., Koo, H. C., Nitta, J., Frolov, S. & Duine, R. New perspectives for Rashba spin–orbit coupling. Nat. Mater. 14, 871–882 (2015).

    Article  ADS  Google Scholar 

  33. Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nat. Phys. 10, 343–350 (2014).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  35. Liu, Y. et al. Van der Waals heterostructures and devices. Nat. Rev. Mater. 1, 16042 (2016).

    Article  ADS  Google Scholar 

  36. Jariwala, D., Marks, T. J. & Hersam, M. C. Mixed-dimensional van der Waals heterostructures. Nat. Mater. 16, 170–181 (2017).

    Article  ADS  Google Scholar 

  37. Castellanos-Gomez, A. et al. Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping. 2D Mater. 1, 011002 (2014).

    Article  Google Scholar 

  38. Wu, S. et al. Electrical tuning of valley magnetic moment through symmetry control in bilayer MoS2. Nat. Phys. 9, 149–153 (2013).

    Article  Google Scholar 

  39. Zhang, Y. et al. Direct observation of a widely tunable bandgap in bilayer graphene. Nature 459, 820–823 (2009).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  41. Sui, M. et al. Gate-tunable topological valley transport in bilayer graphene. Nat. Phys. 11, 1027–1031 (2015).

    Article  Google Scholar 

  42. Castro, E. V. et al. Biased bilayer graphene: semiconductor with a gap tunable by the electric field effect. Phys. Rev. Lett. 99, 216802 (2007). This study represents one of the earliest reports to engineer inversion symmetry breaking through out-of-plane electric fields.

    Article  ADS  Google Scholar 

  43. McCann, E. & Fal’ko, V. I. Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006).

    Article  ADS  Google Scholar 

  44. Oostinga, J. B., Heersche, H. B., Liu, X., Morpurgo, A. F. & Vandersypen, L. M. Gate-induced insulating state in bilayer graphene devices. Nat. Mater. 7, 151–157 (2008).

    Article  ADS  Google Scholar 

  45. Weitz, R. T., Allen, M., Feldman, B., Martin, J. & Yacoby, A. Broken-symmetry states in doubly gated suspended bilayer graphene. Science 330, 812–816 (2010).

    Article  ADS  Google Scholar 

  46. Velasco, J. et al. Transport spectroscopy of symmetry-broken insulating states in bilayer graphene. Nat. Nanotechnol. 7, 156–160 (2012).

    Article  ADS  Google Scholar 

  47. Ohta, T., Bostwick, A., Seyller, T., Horn, K. & Rotenberg, E. Controlling the electronic structure of bilayer graphene. Science 313, 951–954 (2006).

    Article  ADS  Google Scholar 

  48. Shimazaki, Y. et al. Generation and detection of pure valley current by electrically induced Berry curvature in bilayer graphene. Nat. Phys. 11, 1032–1036 (2015).

    Article  Google Scholar 

  49. Mak, K. F., Lui, C. H., Shan, J. & Heinz, T. F. Observation of an electric-field-induced band gap in bilayer graphene by infrared spectroscopy. Phys. Rev. Lett. 102, 256405 (2009).

    Article  ADS  Google Scholar 

  50. Taychatanapat, T. & Jarillo-Herrero, P. Electronic transport in dual-gated bilayer graphene at large displacement fields. Phys. Rev. Lett. 105, 166601 (2010).

    Article  ADS  Google Scholar 

  51. Yan, J. et al. Dual-gated bilayer graphene hot-electron bolometer. Nat. Nanotechnol. 7, 472–478 (2012).

    Article  ADS  Google Scholar 

  52. Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

    Article  ADS  Google Scholar 

  53. Ju, L. et al. Tunable excitons in bilayer graphene. Science 358, 907–910 (2017).

    Article  ADS  Google Scholar 

  54. Ju, L. et al. Unconventional valley-dependent optical selection rules and landau level mixing in bilayer graphene. Nat. Commun. 11, 2941 (2020).

    Article  ADS  Google Scholar 

  55. Cao, T., Wu, M. & Louie, S. G. Unifying optical selection rules for excitons in two dimensions: Band topology and winding numbers. Phys. Rev. Lett. 120, 087402 (2018).

    Article  ADS  Google Scholar 

  56. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  58. Ju, L. et al. Topological valley transport at bilayer graphene domain walls. Nature 520, 650–655 (2015).

    Article  ADS  Google Scholar 

  59. Martin, I., Blanter, Y. M. & Morpurgo, A. Topological confinement in bilayer graphene. Phys. Rev. Lett. 100, 036804 (2008).

    Article  ADS  Google Scholar 

  60. Li, J. et al. Gate-controlled topological conducting channels in bilayer graphene. Nat. Nanotechnol. 11, 1060–1065 (2016).

    Article  ADS  Google Scholar 

  61. Xu, S.-Y. et al. Electrically switchable Berry curvature dipole in the monolayer topological insulator WTe2. Nat. Phys. 14, 900–906 (2018).

    Article  Google Scholar 

  62. You, J.-S., Fang, S., Xu, S.-Y., Kaxiras, E. & Low, T. Berry curvature dipole current in the transition metal dichalcogenides family. Phys. Rev. B 98, 121109 (2018).

    Article  ADS  Google Scholar 

  63. Lee, J., Mak, K. F. & Shan, J. Electrical control of the valley Hall effect in bilayer MoS2 transistors. Nat. Nanotechnol. 11, 421–425 (2016).

    Article  ADS  Google Scholar 

  64. Lui, C. H., Li, Z., Mak, K. F., Cappelluti, E. & Heinz, T. F. Observation of an electrically tunable band gap in trilayer graphene. Nat. Phys. 7, 944–947 (2011).

    Article  Google Scholar 

  65. Khodkov, T., Khrapach, I., Craciun, M. F. & Russo, S. Direct observation of a gate tunable band gap in electrical transport in ABC-trilayer graphene. Nano Lett. 15, 4429–4433 (2015).

    Article  ADS  Google Scholar 

  66. Zhang, Y., van den Brink, J., Felser, C. & Yan, B. Electrically tuneable nonlinear anomalous Hall effect in two-dimensional transition-metal dichalcogenides WTe2 and MoTe2. 2D Mater. 5, 044001 (2018).

    Article  Google Scholar 

  67. Yuan, H. et al. Zeeman-type spin splitting controlled by an electric field. Nat. Phys. 9, 563–569 (2013).

    Article  Google Scholar 

  68. Klein, J. et al. Electric-field switchable second-harmonic generation in bilayer MoS2 by inversion symmetry breaking. Nano Lett. 17, 392–398 (2016).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  70. Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  72. Gorbachev, R. et al. Detecting topological currents in graphene superlattices. Science 346, 448–451 (2014).

    Article  ADS  Google Scholar 

  73. Giovannetti, G., Khomyakov, P. A., Brocks, G., Kelly, P. J. & Van Den Brink, J. Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations. Phys. Rev. B 76, 073103 (2007).

    Article  ADS  Google Scholar 

  74. Stepanov, E. A. et al. Direct observation of incommensurate–commensurate transition in graphene-hBN heterostructures via optical second harmonic generation. ACS Appl. Mater. Interfaces 12, 27758–27764 (2020).

    Article  Google Scholar 

  75. Wang, E. et al. Gaps induced by inversion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride. Nat. Phys. 12, 1111–1115 (2016).

    Article  Google Scholar 

  76. Zhu, M. J. et al. Edge currents shunt the insulating bulk in gapped graphene. Nat. Commun. 8, 14552 (2017).

    Article  ADS  Google Scholar 

  77. Chen, Z.-G. et al. Observation of an intrinsic bandgap and Landau level renormalization in graphene/boron-nitride heterostructures. Nat. Commun. 5, 4461 (2014).

    Article  ADS  Google Scholar 

  78. 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).

    Article  ADS  Google Scholar 

  79. Yankowitz, M. et al. Dynamic band-structure tuning of graphene moiré superlattices with pressure. Nature 557, 404–408 (2018).

    Article  ADS  Google Scholar 

  80. Yankowitz, M., Watanabe, K., Taniguchi, T., San-Jose, P. & LeRoy, B. J. Pressure-induced commensurate stacking of graphene on boron nitride. Nat. Commun. 7, 13168 (2016).

    Article  ADS  Google Scholar 

  81. Zhou, S. Y. et al. Substrate-induced bandgap opening in epitaxial graphene. Nat. Mater. 6, 770–775 (2007). This study represents one of the earliest reports to engineer inversion symmetry breaking through staggered sublattice potential.

    Article  ADS  Google Scholar 

  82. Pletikosić, I. et al. Dirac cones and minigaps for graphene on Ir(111). Phys. Rev. Lett. 102, 056808 (2009).

    Article  ADS  Google Scholar 

  83. Varykhalov, A. et al. Intact Dirac cones at broken sublattice symmetry: photoemission study of graphene on Ni and Co. Phys. Rev. X 2, 041017 (2012).

    Google Scholar 

  84. Lin, C.-L. et al. Substrate-induced symmetry breaking in silicene. Phys. Rev. Lett. 110, 076801 (2013).

    Article  ADS  Google Scholar 

  85. Xu, X. et al. Interfacial engineering in graphene bandgap. Chem. Soc. Rev. 47, 3059–3099 (2018).

    Article  Google Scholar 

  86. Jiang, T. et al. Valley and band structure engineering of folded MoS2 bilayers. Nat. Nanotechnol. 9, 825–829 (2014).

    Article  ADS  Google Scholar 

  87. Hsu, W.-T. et al. Second harmonic generation from artificially stacked transition metal dichalcogenide twisted bilayers. ACS Nano 8, 2951–2958 (2014). This study represents one of the earliest reports to engineer inversion symmetry breaking through interlayer twist angle.

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  89. Du, L., Dai, Y. & Sun, Z. Twisting for tunable nonlinear optics. Matter 3, 987–988 (2020).

    Article  Google Scholar 

  90. Yang, F. et al. Tunable second harmonic generation in twisted bilayer graphene. Matter 3, 1361–1376 (2020).

    Article  Google Scholar 

  91. Kim, C.-J. et al. Stacking order dependent second harmonic generation and topological defects in h-BN bilayers. Nano Lett. 13, 5660–5665 (2013).

    Article  ADS  Google Scholar 

  92. Yasuda, K., Wang, X., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Stacking-engineered ferroelectricity in bilayer boron nitride. Preprint at arXiv https://arxiv.org/abs/2010.06600 (2020).

  93. Suzuki, R. et al. Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry. Nat. Nanotechnol. 9, 611–617 (2014).

    Article  ADS  Google Scholar 

  94. Scuri, G. et al. Electrically tunable valley dynamics in twisted WSe2/WSe2 bilayers. Phys. Rev. Lett. 124, 217403 (2020).

    Article  ADS  Google Scholar 

  95. Du, L. et al. Robust circular polarization of indirect Q-K transitions in bilayer 3R–WS2. Phys. Rev. B 100, 161404 (2019).

    Article  ADS  Google Scholar 

  96. Yao, K. et al. Nonlinear twistoptics at symmetry-broken interfaces. Preprint at arXiv https://arxiv.org/abs/2006.13802 (2020).

  97. Kim, K. et al. van der Waals heterostructures with high accuracy rotational alignment. Nano Lett. 16, 1989–1995 (2016). This paper demonstrated the ‘tear-and-stack’ technique to control the interlayer twist angle to within 0.1° accuracy.

    Article  ADS  Google Scholar 

  98. Kim, K. et al. Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene. Proc. Natl Acad. Sci. USA 114, 3364–3369 (2017).

    Article  ADS  Google Scholar 

  99. Du, L. et al. Modulating PL and electronic structures of MoS2/graphene heterostructures via interlayer twisting angle. Appl. Phys. Lett. 111, 263106 (2017).

    Article  ADS  Google Scholar 

  100. Liao, M. et al. Precise control of the interlayer twist angle in large scale MoS2 homostructures. Nat. Commun. 11, 2153 (2020).

    Article  ADS  Google Scholar 

  101. Manzeli, S., Ovchinnikov, D., Pasquier, D., Yazyev, O. V. & Kis, A. 2D transition metal dichalcogenides. Nat. Rev. Mater. 2, 17033 (2017).

    Article  ADS  Google Scholar 

  102. Yang, H., Kim, S. W., Chhowalla, M. & Lee, Y. H. Structural and quantum-state phase transitions in van der Waals layered materials. Nat. Phys. 13, 931–937 (2017).

    Article  Google Scholar 

  103. Du, L. et al. Giant valley coherence at room temperature in 3R WS2 with broken inversion symmetry. Research 2019, 6494565 (2019).

    Google Scholar 

  104. Li, Y., Duerloo, K.-A. N., Wauson, K. & Reed, E. J. Structural semiconductor-to-semimetal phase transition in two-dimensional materials induced by electrostatic gating. Nat. Commun. 7, 10671 (2016).

    Article  ADS  Google Scholar 

  105. Wang, Y. et al. Structural phase transition in monolayer MoTe2 driven by electrostatic doping. Nature 550, 487–491 (2017). This study represents one of the earliest reports to control the crystal phase through electrostatic doping.

    Article  ADS  Google Scholar 

  106. Kappera, R. et al. Phase-engineered low-resistance contacts for ultrathin MoS2 transistors. Nat. Mater. 13, 1128–1134 (2014).

    Article  ADS  Google Scholar 

  107. Lin, Y.-C., Dumcenco, D. O., Huang, Y.-S. & Suenaga, K. Atomic mechanism of the semiconducting-to-metallic phase transition in single-layered MoS2. Nat. Nanotechnol. 9, 391–396 (2014).

    Article  ADS  Google Scholar 

  108. Zhu, J. et al. Argon plasma induced phase transition in monolayer MoS2. J. Am. Chem. Soc. 139, 10216–10219 (2017).

    Article  Google Scholar 

  109. Zhang, K. et al. Raman signatures of inversion symmetry breaking and structural phase transition in type-II Weyl semimetal MoTe2. Nat. Commun. 7, 13552 (2016).

    Article  ADS  Google Scholar 

  110. Keum, D. H. et al. Bandgap opening in few-layered monoclinic MoTe2. Nat. Phys. 11, 482–486 (2015).

    Article  Google Scholar 

  111. Chen, S.-Y., Goldstein, T., Venkataraman, D., Ramasubramaniam, A. & Yan, J. Activation of new Raman modes by inversion symmetry breaking in type II Weyl semimetal candidate T′-MoTe2. Nano Lett. 16, 5852–5860 (2016).

    Article  ADS  Google Scholar 

  112. 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).

    Article  ADS  Google Scholar 

  113. Sie, E. J. et al. An ultrafast symmetry switch in a Weyl semimetal. Nature 565, 61–66 (2019).

    Article  ADS  Google Scholar 

  114. Cho, S. et al. Phase patterning for ohmic homojunction contact in MoTe2. Science 349, 625–628 (2015).

    Article  ADS  Google Scholar 

  115. Wang, J., Zhu, B.-F. & Liu, R.-B. Second-order nonlinear optical effects of spin currents. Phys. Rev. Lett. 104, 256601 (2010).

    Article  ADS  Google Scholar 

  116. Werake, L. K. & Zhao, H. Observation of second-harmonic generation induced by pure spin currents. Nat. Phys. 6, 875–878 (2010).

    Article  Google Scholar 

  117. Balog, R. et al. Bandgap opening in graphene induced by patterned hydrogen adsorption. Nat. Mater. 9, 315–319 (2010).

    Article  ADS  Google Scholar 

  118. Gibertini, M., Koperski, M., Morpurgo, A. & Novoselov, K. Magnetic 2D materials and heterostructures. Nat. Nanotechnol. 14, 408–419 (2019).

    Article  ADS  Google Scholar 

  119. Burch, K. S., Mandrus, D. & Park, J.-G. Magnetism in two-dimensional van der Waals materials. Nature 563, 47–52 (2018).

    Article  ADS  Google Scholar 

  120. Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).

    Article  ADS  Google Scholar 

  121. Jiang, S., Shan, J. & Mak, K. F. Electric-field switching of two-dimensional van der Waals magnets. Nat. Mater. 17, 406–410 (2018).

    Article  ADS  Google Scholar 

  122. Huang, B. et al. Electrical control of 2D magnetism in bilayer CrI3. Nat. Nanotechnol. 13, 544–548 (2018).

    Article  ADS  Google Scholar 

  123. Jiang, S., Li, L., Wang, Z., Mak, K. F. & Shan, J. Controlling magnetism in 2D CrI3 by electrostatic doping. Nat. Nanotechnol. 13, 549–553 (2018).

    Article  ADS  Google Scholar 

  124. Zhang, Y. et al. Switchable magnetic bulk photovoltaic effect in the two-dimensional magnet CrI3. Nat. Commun. 10, 3783 (2019).

    Article  ADS  Google Scholar 

  125. Klein, D. R. et al. Probing magnetism in 2D van der Waals crystalline insulators via electron tunneling. Science 360, 1218–1222 (2018).

    Article  ADS  Google Scholar 

  126. Song, T. et al. Giant tunneling magnetoresistance in spin-filter van der Waals heterostructures. Science 360, 1214–1218 (2018).

    Article  ADS  Google Scholar 

  127. Wang, Z. et al. Very large tunneling magnetoresistance in layered magnetic semiconductor CrI3. Nat. Commun. 9, 2516 (2018).

    Article  ADS  Google Scholar 

  128. Xia, F., Wang, H., Hwang, J. C. M., Neto, A. H. C. & Yang, L. Black phosphorus and its isoelectronic materials. Nat. Rev. Phys. 1, 306–317 (2019).

    Article  Google Scholar 

  129. Li, L. et al. Emerging in-plane anisotropic two-dimensional materials. InfoMat 1, 54–73 (2019).

    Article  Google Scholar 

  130. Autere, A. et al. Rapid and large-area characterization of exfoliated black phosphorus using third-harmonic generation microscopy. J. Phys. Chem. Lett. 8, 1343–1350 (2017).

    Article  Google Scholar 

  131. Sturman, B. I. & Fridkin, V. M. Photovoltaic and Photo-refractive Effects in Noncentrosymmetric Materials Vol. 8 (CRC Press, 1992).

  132. Zhang, Y. et al. Enhanced intrinsic photovoltaic effect in tungsten disulfide nanotubes. Nature 570, 349–353 (2019). This study represents one of the earliest reports to engineer C3 rotational symmetry breaking through reducing the effective dimensionality.

    Article  ADS  Google Scholar 

  133. Osterhoudt, G. B. et al. Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal. Nat. Mater. 18, 471–475 (2019).

    Article  ADS  Google Scholar 

  134. Yuan, H. et al. Polarization-sensitive broadband photodetector using a black phosphorus vertical p–n junction. Nat. Nanotechnol. 10, 707–713 (2015).

    Article  ADS  Google Scholar 

  135. Sodemann, I. & Fu, L. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015). This study predicted the nonlinear Hall effect and Berry curvature dipole in materials where both inversion and C3 rotational symmetries are broken.

    Article  ADS  Google Scholar 

  136. Lee, J., Wang, Z., Xie, H., Mak, K. F. & Shan, J. Valley magnetoelectricity in single-layer MoS2. Nat. Mater. 16, 887–891 (2017).

    Article  ADS  Google Scholar 

  137. Son, J., Kim, K.-H., Ahn, Y. H., Lee, H.-W. & Lee, J. Strain engineering of the Berry curvature dipole and valley magnetization in monolayer MoS2. Phys. Rev. Lett. 123, 036806 (2019).

    Article  ADS  Google Scholar 

  138. Zelisko, M. et al. Anomalous piezoelectricity in two-dimensional graphene nitride nanosheets. Nat. Commun. 5, 4284 (2014).

    Article  ADS  Google Scholar 

  139. Ho, S.-C. et al. Zero-magnetic-field Hall effects in artificially corrugated bilayer graphene. Preprint at arXiv https://arxiv.org/abs/1910.07509 (2019).

  140. Alexeev, E. M. et al. Emergence of highly linearly polarized interlayer exciton emission in MoSe2/WSe2 heterobilayers with transfer-induced layer corrugation. ACS Nano 14, 11110–11119 (2020).

    Article  Google Scholar 

  141. Yoo, H. et al. Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene. Nat. Mater. 18, 448–453 (2019).

    Article  ADS  Google Scholar 

  142. Weston, A. et al. Atomic reconstruction in twisted bilayers of transition metal dichalcogenides. Nat. Nanotechnol. 15, 592–597 (2020).

    Article  ADS  Google Scholar 

  143. Bai, Y. et al. Excitons in strain-induced one-dimensional moiré potentials at transition metal dichalcogenide heterojunctions. Nat. Mater. 19, 1068–1073 (2020).

    Article  ADS  Google Scholar 

  144. Frisenda, R. et al. Symmetry breakdown in franckeite: spontaneous strain, rippling, and interlayer moiré. Nano Lett. 20, 1141–1147 (2020).

    Article  ADS  Google Scholar 

  145. Guinea, F., Katsnelson, M. & Geim, A. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nat. Phys. 6, 30–33 (2010).

    Article  Google Scholar 

  146. Levy, N. et al. Strain-induced pseudo-magnetic fields greater than 300 tesla in graphene nanobubbles. Science 329, 544–547 (2010).

    Article  ADS  Google Scholar 

  147. Yang, R., Shi, Z., Zhang, L., Shi, D. & Zhang, G. Observation of Raman g-peak split for graphene nanoribbons with hydrogen-terminated zigzag edges. Nano Lett. 11, 4083–4088 (2011).

    Article  ADS  Google Scholar 

  148. Wu, J. B. et al. Monolayer molybdenum disulfide nanoribbons with high optical anisotropy. Adv. Opt. Mater. 4, 756–762 (2016).

    Article  Google Scholar 

  149. Cui, X. et al. Rolling up transition metal dichalcogenide nanoscrolls via one drop of ethanol. Nat. Commun. 9, 1301 (2018).

    Article  ADS  Google Scholar 

  150. Cook, A. M., Fregoso, B. M., De Juan, F., Coh, S. & Moore, J. E. Design principles for shift current photovoltaics. Nat. Commun. 8, 14176 (2017).

    Article  ADS  Google Scholar 

  151. Thiel, L. et al. Probing magnetism in 2D materials at the nanoscale with single-spin microscopy. Science 364, 973–976 (2019).

    Article  ADS  Google Scholar 

  152. Song, T. et al. Switching 2D magnetic states via pressure tuning of layer stacking. Nat. Mater. 18, 1298–1302 (2019).

    Article  ADS  Google Scholar 

  153. Li, T. et al. Pressure-controlled interlayer magnetism in atomically thin CrI3. Nat. Mater. 18, 1303–1308 (2019).

    Article  ADS  Google Scholar 

  154. Yan, J. et al. Stacking-dependent interlayer coupling in trilayer MoS2 with broken inversion symmetry. Nano Lett. 15, 8155–8161 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  156. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).

    Article  ADS  Google Scholar 

  157. Novoselov, K. S. et al. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nat. Phys. 2, 177–180 (2006).

    Article  Google Scholar 

  158. Taychatanapat, T., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene. Nat. Phys. 7, 621–625 (2011).

    Article  Google Scholar 

  159. Li, L. et al. Quantum Hall effect in black phosphorus two-dimensional electron system. Nat. Nanotechnol. 11, 593–597 (2016).

    Article  ADS  Google Scholar 

  160. Bandurin, D. A. et al. High electron mobility, quantum Hall effect and anomalous optical response in atomically thin InSe. Nat. Nanotechnol. 12, 223–227 (2016).

    Article  ADS  Google Scholar 

  161. Aivazian, G. et al. Magnetic control of valley pseudospin in monolayer WSe2. Nat. Phys. 11, 148–152 (2015).

    Article  Google Scholar 

  162. Srivastava, A. et al. Valley Zeeman effect in elementary optical excitations of monolayer WSe2. Nat. Phys. 11, 141–147 (2015).

    Article  Google Scholar 

  163. Zhang, X.-X. et al. Zeeman-induced valley-sensitive photocurrent in monolayer MoS2. Phys. Rev. Lett. 122, 127401 (2019).

    Article  ADS  Google Scholar 

  164. Li, Y. et al. Valley splitting and polarization by the Zeeman effect in monolayer MoSe2. Phys. Rev. Lett. 113, 266804 (2014).

    Article  ADS  Google Scholar 

  165. Wang, Z., Shan, J. & Mak, K. F. Valley- and spin-polarized Landau levels in monolayer WSe2. Nat. Nanotechnol. 12, 144–149 (2017).

    Article  ADS  Google Scholar 

  166. Roch, J. G. et al. First-order magnetic phase transition of mobile electrons in monolayer MoS2. Phys. Rev. Lett. 124, 187602 (2020).

    Article  ADS  Google Scholar 

  167. Back, P. et al. Giant paramagnetism-induced valley polarization of electrons in charge-tunable monolayer MoSe2. Phys. Rev. Lett. 118, 237404 (2017).

    Article  ADS  Google Scholar 

  168. Zhang, J. et al. Enhancing and controlling valley magnetic response in MoS2/WS2 heterostructures by all-optical route. Nat. Commun. 10, 4226 (2019).

    Article  ADS  Google Scholar 

  169. Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).

    Article  ADS  Google Scholar 

  170. Sie, E. J. et al. Large, valley-exclusive Bloch-Siegert shift in monolayer WS2. Science 355, 1066–1069 (2017).

    Article  ADS  Google Scholar 

  171. Sie, E. J. et al. Valley-selective optical Stark effect in monolayer WS2. Nat. Mater. 14, 290–294 (2015).

    Article  ADS  Google Scholar 

  172. Kim, J. et al. Ultrafast generation of pseudo-magnetic field for valley excitons in WSe2 monolayers. Science 346, 1205–1208 (2014). This study represents one of the earliest reports to engineer time-reversal symmetry breaking through coherent light–matter interaction.

    Article  ADS  Google Scholar 

  173. Ye, Z., Sun, D. & Heinz, T. F. Optical manipulation of valley pseudospin. Nat. Phys. 13, 26–29 (2017).

    Article  Google Scholar 

  174. Scharf, B., Xu, G., Matos-Abiague, A. & Žutić, I. Magnetic proximity effects in transition-metal dichalcogenides: converting excitons. Phys. Rev. Lett. 119, 127403 (2017).

    Article  ADS  Google Scholar 

  175. Zhong, D. et al. Layer-resolved magnetic proximity effect in van der Waals heterostructures. Nat. Nanotechnol. 15, 187–191 (2020).

    Article  ADS  Google Scholar 

  176. Wang, Z., Tang, C., Sachs, R., Barlas, Y. & Shi, J. Proximity-induced ferromagnetism in graphene revealed by the anomalous Hall effect. Phys. Rev. Lett. 114, 016603 (2015). This study represents one of the earliest reports to engineer time-reversal symmetry breaking through magnetic proximity effects.

    Article  ADS  Google Scholar 

  177. Wei, P. et al. Strong interfacial exchange field in the graphene/EuS heterostructure. Nat. Mater. 15, 711–716 (2016).

    Article  ADS  Google Scholar 

  178. Zhao, C. et al. Enhanced valley splitting in monolayer WSe2 due to magnetic exchange field. Nat. Nanotechnol. 12, 757–762 (2017).

    Article  Google Scholar 

  179. Zhong, D. et al. Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics. Sci. Adv. 3, e1603113 (2017).

    Article  ADS  Google Scholar 

  180. Ciorciaro, L., Kroner, M., Watanabe, K., Taniguchi, T. & Imamoglu, A. Observation of magnetic proximity effect using resonant optical spectroscopy of an electrically tunable MoSe2/CrBr3 heterostructure. Phys. Rev. Lett. 124, 197401 (2020).

    Article  ADS  Google Scholar 

  181. Yu, R. et al. Quantized anomalous Hall effect in magnetic topological insulators. Science 329, 61–64 (2010).

    Article  ADS  Google Scholar 

  182. Zhang, J. et al. Topology-driven magnetic quantum phase transition in topological insulators. Science 339, 1582–1586 (2013).

    Article  ADS  Google Scholar 

  183. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    Article  ADS  Google Scholar 

  184. Zhou, J. et al. Synthesis of co-doped MoS2 monolayers with enhanced valley splitting. Adv. Mater. 32, 1906536 (2020).

    Article  Google Scholar 

  185. Li, B. et al. A two-dimensional Fe-doped SnS2 magnetic semiconductor. Nat. Commun. 8, 1958 (2017).

    Article  ADS  Google Scholar 

  186. Han, W., Kawakami, R. K., Gmitra, M. & Fabian, J. Graphene spintronics. Nat. Nanotechnol. 9, 794–807 (2014).

    Article  ADS  Google Scholar 

  187. González-Herrero, H. et al. Atomic-scale control of graphene magnetism by using hydrogen atoms. Science 352, 437–441 (2016).

    Article  ADS  Google Scholar 

  188. Tilley, D. R. Superfluidity and Superconductivity (Routledge, 2019).

  189. Ye, J. et al. Superconducting dome in a gate-tuned band insulator. Science 338, 1193–1196 (2012).

    Article  ADS  Google Scholar 

  190. Costanzo, D., Jo, S., Berger, H. & Morpurgo, A. F. Gate-induced superconductivity in atomically thin MoS2 crystals. Nat. Nanotechnol. 11, 339–344 (2016).

    Article  ADS  Google Scholar 

  191. Saito, Y. et al. Superconductivity protected by spin–valley locking in ion-gated MoS2. Nat. Phys. 12, 144–149 (2015).

    Article  Google Scholar 

  192. Lu, J. et al. Full superconducting dome of strong Ising protection in gated monolayer WS2. Proc. Natl Acad. Sci. USA 115, 3551–3556 (2018).

    Article  ADS  Google Scholar 

  193. Sohn, E. et al. An unusual continuous paramagnetic-limited superconducting phase transition in 2D NbSe2. Nat. Mater. 17, 504–508 (2018).

    Article  ADS  Google Scholar 

  194. Tsen, A. et al. Nature of the quantum metal in a two-dimensional crystalline superconductor. Nat. Phys. 12, 208–212 (2016).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  197. Shimazaki, Y. et al. Strongly correlated electrons and hybrid excitons in a moiré heterostructure. Nature 580, 472–477 (2020).

    Article  ADS  Google Scholar 

  198. Cao, Y. et al. Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature 583, 215–220 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  200. Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    Article  ADS  Google Scholar 

  201. Liu, X. et al. Tunable spin-polarized correlated states in twisted double bilayer graphene. Nature 583, 221–225 (2020).

    Article  ADS  Google Scholar 

  202. Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  204. Chen, G. et al. Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature 579, 56–61 (2020).

    Article  ADS  Google Scholar 

  205. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  207. Xie, Y. et al. Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene. Nature 572, 101–105 (2019).

    Article  ADS  Google Scholar 

  208. Kerelsky, A. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

    Article  ADS  Google Scholar 

  209. Zhang, Y.-H., Po, H. C. & Senthil, T. Landau level degeneracy in twisted bilayer graphene: Role of symmetry breaking. Phys. Rev. B 100, 125104 (2019).

    Article  ADS  Google Scholar 

  210. Chatterjee, S., Bultinck, N. & Zaletel, M. P. Symmetry breaking and skyrmionic transport in twisted bilayer graphene. Phys. Rev. B 101, 165141 (2020).

    Article  ADS  Google Scholar 

  211. Liu, X., Watanabe, K., Taniguchi, T., Halperin, B. I. & Kim, P. Quantum Hall drag of exciton condensate in graphene. Nat. Phys. 13, 746–750 (2017).

    Article  Google Scholar 

  212. Cao, Y. et al. Nematicity and competing orders in superconducting magic-angle graphene. Preprint at arXiv https://arxiv.org/abs/2004.04148 (2020).

  213. Jin, C. et al. Stripe phases in WSe2/WS2 moiré superlattices. Preprint at arXiv https://arxiv.org/abs/2007.12068 (2020).

  214. Liu, C.-C., Zhang, L.-D., Chen, W.-Q. & Yang, F. Chiral spin density wave and d + id superconductivity in the magic-angle-twisted bilayer graphene. Phys. Rev. Lett. 121, 217001 (2018).

    Article  ADS  Google Scholar 

  215. Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).

    Article  ADS  Google Scholar 

  216. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).

    Article  ADS  Google Scholar 

  217. Zhou, Y. et al. Signatures of bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Preprint at arXiv https://arxiv.org/abs/2010.03037 (2020).

  218. Smoleński, T. et al. Observation of Wigner crystal of electrons in a monolayer semiconductor. Preprint at arXiv https://arxiv.org/abs/2010.03078 (2020).

  219. Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  221. Mak, K. F. & Shan, J. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. Nat. Photonics 10, 216–226 (2016).

    Article  ADS  Google Scholar 

  222. Delaney, P., Choi, H. J., Ihm, J., Louie, S. G. & Cohen, M. L. Broken symmetry and pseudogaps in ropes of carbon nanotubes. Nature 391, 466–468 (1998).

    Article  ADS  Google Scholar 

  223. Chen, Y.-J., Cain, J. D., Stanev, T. K., Dravid, V. P. & Stern, N. P. Valley-polarized exciton–polaritons in a monolayer semiconductor. Nat. Photonics 11, 431–435 (2017).

    Article  ADS  Google Scholar 

  224. Sun, Z. et al. Optical control of room-temperature valley polaritons. Nat. Photonics 11, 491–496 (2017).

    Article  Google Scholar 

  225. Lundt, N. et al. Optical valley Hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor. Nat. Nanotechnol. 14, 770–775 (2019).

    Article  ADS  Google Scholar 

  226. Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2017).

    Article  Google Scholar 

  227. Li, F., Huang, X., Lu, J., Ma, J. & Liu, Z. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30–34 (2018).

    Article  Google Scholar 

  228. Dong, J.-W., Chen, X.-D., Zhu, H., Wang, Y. & Zhang, X. Valley photonic crystals for control of spin and topology. Nat. Mater. 16, 298–302 (2017).

    Article  ADS  Google Scholar 

  229. Noh, J., Huang, S., Chen, K. P. & Rechtsman, M. C. Observation of photonic topological valley Hall edge states. Phys. Rev. Lett. 120, 063902 (2018).

    Article  ADS  Google Scholar 

  230. Lin, Y.-J., Jiménez-García, K. & Spielman, I. B. Spin–orbit-coupled Bose–Einstein condensates. Nature 471, 83–86 (2011).

    Article  ADS  Google Scholar 

  231. Ji, S.-C. et al. Experimental determination of the finite-temperature phase diagram of a spin–orbit coupled Bose gas. Nat. Phys. 10, 314–320 (2014).

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the financial support by the Academy of Finland (grant nos. 314810, 333982, 336144, 336818 and 333099), the Academy of Finland Flagship Programme (PREIN), the EU H2020-MSCA-RISE-872049 (IPN-Bio), the National Key R&D Program of China (grant nos. 2020YFA0308800), the NSF of China (grants nos. 11734003, 12061131002), the Strategic Priority Research Program of Chinese Academy of Sciences (grant no. XDB30000000), EPSRC (grant no. EP/T014601/1), the European Union’s Horizon 2020 research and innovation programme (grant no. 820423, S2QUIP) and the European Research Council (ERC) (grant no. 834742). C.N.L. acknowledges the support of DOE BES DE-SC0020187, NSF DMR 1807928 and 1922076.

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Authors and Affiliations

  1. Department of Electronics and Nanoengineering, Aalto University, Aalto, Finland

    Luojun Du & Zhipei Sun

  2. Cambridge Graphene Centre, University of Cambridge, Cambridge, UK

    Tawfique Hasan

  3. Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Cientificas (CSIC), Madrid, Spain

    Andres Castellanos-Gomez

  4. Key Lab of Advanced Optoelectronic Quantum Architecture and Measumrement (MOE), School of Physics, Beijing Institute of Technology, Beijing, China

    Gui-Bin Liu & Yugui Yao

  5. Department of Physics, The Ohio State University, Columbus, OH, USA

    Chun Ning Lau

  6. QTF Centre of Excellence, Department of Applied Physics, Aalto University, Aalto, Finland

    Zhipei Sun

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L.D. and Z.S. conceived the idea and led the writing. T.H. contributed substantially to the discussion of the content. All authors reviewed and edited the manuscript before submission.

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Du, L., Hasan, T., Castellanos-Gomez, A. et al. Engineering symmetry breaking in 2D layered materials. Nat Rev Phys 3, 193–206 (2021). https://doi.org/10.1038/s42254-020-00276-0

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