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  • Letter
  • Open Access

Topological piezoelectric response in moiré graphene systems

Ran Peng

Jianpeng Liu*

  • School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China

  • School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China and ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China

  • *liujp@shanghaitech.edu.cn
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Phys. Rev. Research 4, L032006 – Published 11 July, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L032006

Abstract

We theoretically study the piezoelectric effects in moirè graphene systems. Since the strain couples to the electrons in the system as an effective vector field, which has opposite signs for the K and K′ valleys of graphene, its effects on the two valleys with opposite Chern numbers do not cancel out, but adds up. As a result, some components of the piezoelectric tensor (PET) in these systems, which typically have nontrivial topology in their flat bands, are nearly quantized in terms of the valley Chern numbers. The quantization of the PET is topologically protected by valley U(1) symmetry of the system. Such a conclusion is verified by numerical calculations of the in-plane piezoelectric response of hBN-aligned twisted bilayer graphene (TBG), twisted bilayer-monolayer graphene (TBMG), and twisted double bilayer graphene (TDBG) systems using both continuum model and atomistic tight-binding model. We propose that such nearly quantized piezoelectric response may serve as a direct experimental probe for the valley Chern numbers of the flat bands in moiré graphene systems.

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References (69)

  1. 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).
  2. 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).
  3. E. Codecido, Q. Wang, R. Koester, S. Che, H. Tian, R. Lv, S. Tran, K. Watanabe, T. Taniguchi, F. Zhang et al., Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle, Sci. Adv. 5, eaaw9770 (2019).
  4. 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).
  5. P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. L. Koppens, J. Lischner, L. Levitov, and D. K. Efetov, Untying the insulating and superconducting orders in magic-angle graphene, Nature (London) 583, 375 (2020).
  6. 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).
  7. X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. Li, Tuning electron correlation in magic-angle twisted bilayer graphene using coulomb screening, Science 371, 1261 (2021).
  8. 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).
  9. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras et al., Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  10. A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean et al., Maximized electron interactions at the magic angle in twisted bilayer graphene, Nature (London) 572, 95 (2019).
  11. Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule, J. Mao, and E. Y. Andrei, Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene, Nature (London) 573, 91 (2019).
  12. Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene, Nature (London) 572, 101 (2019).
  13. Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora, R. Polski, Y. Zhang, H. Ren, J. Alicea, G. Refael et al., Electronic correlations in twisted bilayer graphene near the magic angle, Nat. Phys. 15, 1174 (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 (2019).
  15. 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).
  16. 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, Phys. Rev. Lett. 127, 197701 (2020).
  17. H. Polshyn, J. Zhu, M. Kumar, Y. Zhang, F. Yang, C. Tschirhart, M. Serlin, K. Watanabe, T. Taniguchi, A. MacDonald et al., Electrical switching of magnetic order in an orbital chern insulator, Nature (London) 588, 66 (2020).
  18. S. Chen, M. He, Y.-H. Zhang, V. Hsieh, Z. Fei, K. Watanabe, T. Taniguchi, D. H. Cobden, X. Xu, C. R. Dean, and M. Yankowitz, Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene, Nat. Phys. 17, 374 (2021).
  19. S. Xu, M. M. Al Ezzi, N. Balakrishnan, A. Garcia-Ruiz, B. Tsim, C. Mullan, J. Barrier, N. Xin, B. A. Piot, T. Taniguchi et al., Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene, Nat. Phys. 17, 619 (2021).
  20. X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, Y. Ronen, H. Yoo, D. Haei Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Tunable spin-polarized correlated states in twisted double bilayer graphene, Nature (London) 583, 221 (2020).
  21. C. Shen, Y. Chu, Q. Wu, N. Li, S. Wang, Y. Zhao, J. Tang, J. Liu, J. Tian, K. Watanabe, T. Taniguchi, R. Yang, Z. Y. Meng, D. Shi, O. V. Yazyev, and G. Zhang, Correlated states in twisted double bilayer graphene, Nat. Phys. 16, 520 (2020).
  22. Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene, Nature (London) 583, 215 (2020).
  23. C. Rubio-Verdú, S. Turkel, L. Song, L. Klebl, R. Samajdar, M. S. Scheurer, J. W. F. Venderbos, K. Watanabe, T. Taniguchi, H. Ochoa et al., Universal moiré nematic phase in twisted graphitic systems, Nat. Phys. 18, 196 (2020).
  24. 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).
  25. G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, B. L. Chittari, K. Watanabe, T. Taniguchi, Z. Shi, J. Jung et al., Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice, Nat. Phys. 15, 237 (2019).
  26. H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99, 195455 (2019).
  27. J. Liu, J. Liu, and X. Dai, Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase, Phys. Rev. B 99, 155415 (2019).
  28. N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene, Phys. Rev. Lett. 124, 166601 (2020).
  29. 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).
  30. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle, Phys. Rev. X 9, 021013 (2019).
  31. 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).
  32. J. Liu, Z. Ma, J. Gao, and X. Dai, Quantum Valley Hall Effect, Orbital Magnetism, and Anomalous Hall Effect in Twisted Multilayer Graphene Systems, Phys. Rev. X 9, 031021 (2019).
  33. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  34. J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Continuum model of the twisted graphene bilayer, Phys. Rev. B 86, 155449 (2012).
  35. H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Origin of Mott Insulating Behavior and Superconductivity in Twisted Bilayer Graphene, Phys. Rev. X 8, 031089 (2018).
  36. A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry, Phys. Rev. B 100, 195135 (2019).
  37. M. Koshino, Band structure and topological properties of twisted double bilayer graphene, Phys. Rev. B 99, 235406 (2019).
  38. J. Wang and Z. Liu, Hierarchy of Ideal Flatbands in Chiral Twisted Multilayer Graphene Models, Phys. Rev. Lett. 128, 176403 (2022).
  39. P. J. Ledwith, A. Vishwanath, and E. Khalaf, Family of Ideal Chern Flatbands with Arbitrary Chern Number in Chiral Twisted Graphene Multilayers, Phys. Rev. Lett. 128, 176404 (2021).
  40. R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 (1993).
  41. R. Resta and D. Vanderbilt, Theory of polarization: A modern approach, in Physics of Ferroelectrics: A Modern Perspective, edited by K. M. Rabe, C. H. Ahn, and J.-M. Triscone (Springer-Verlag, Berlin, 2007).
  42. S. Coh and D. Vanderbilt, Electric Polarization in a Chern Insulator, Phys. Rev. Lett. 102, 107603 (2009).
  43. D. Vanderbilt, Berry-phase theory of proper piezoelectric response, J. Phys. Chem. Solids 61, 147 (2000).
  44. H. Suzuura and T. Ando, Phonons and electron-phonon scattering in carbon nanotubes, Phys. Rev. B 65, 235412 (2002).
  45. V. M. Pereira and A. H. Castro Neto, Strain Engineering of Graphene's Electronic Structure, Phys. Rev. Lett. 103, 046801 (2009).
  46. N. N. T. Nam and M. Koshino, Lattice relaxation and energy band modulation in twisted bilayer graphene, Phys. Rev. B 96, 075311 (2017).
  47. Z. Bi, N. F. Q. Yuan, and L. Fu, Designing flat bands by strain, Phys. Rev. B 100, 035448 (2019).
  48. F. Rost, R. Gupta, M. Fleischmann, D. Weckbecker, N. Ray, J. Olivares, M. Vogl, S. Sharma, O. Pankratov, and S. Shallcross, Nonperturbative theory of effective Hamiltonians for deformations in two-dimensional materials: Moiré systems and dislocations, Phys. Rev. B 100, 035101 (2019).
  49. J. Yu and C. Liu, Piezoelectricity and topological quantum phase transitions in two-dimensional spin-orbit coupled crystals with time-reversal symmetry, Nat. Commun. 11, 2290 (2020).
  50. J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Graphene Bilayer with a Twist: Electronic Structure, Phys. Rev. Lett. 99, 256802 (2007).
  51. D. Weckbecker, S. Shallcross, M. Fleischmann, N. Ray, S. Sharma, and O. Pankratov, Low-energy theory for the graphene twist bilayer, Phys. Rev. B 93, 035452 (2016).
  52. P. Moon and M. Koshino, Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice, Phys. Rev. B 90, 155406 (2014).
  53. J. Jung, A. Raoux, Z. Qiao, and A. H. MacDonald, Ab initio theory of moiré superlattice bands in layered two-dimensional materials, Phys. Rev. B 89, 205414 (2014).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L032006 for: (a) the detailed presentation of the continuum models for the twisted bilayer, and twisted multilayer graphene systems; (b) the derivations for the strain induced pseudo vector potentials in graphene; (c) the results of piezoelectric response from atomistic tight-binding calculations; and (d) the piezoelectric response of hBN-aligned TBG in the presence of local strain, heterostrain, and intervalley coupling.
  55. N. N. Klimov, S. Jung, S. Zhu, T. Li, C. A. Wright, S. D. Solares, D. B. Newell, N. B. Zhitenev, and J. A. Stroscio, Electromechanical properties of graphene drumheads, Science 336, 1557 (2012).
  56. N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. H. C. Neto, and M. F. Crommie, Strain-induced pseudo-magnetic fields greater than 300 tesla in graphene nanobubbles, Science 329, 544 (2010).
  57. L. Meng, W.-Y. He, H. Zheng, M. Liu, H. Yan, W. Yan, Z.-D. Chu, K. Bai, R.-F. Dou, Y. Zhang, Z. Liu, J.-C. Nie, and L. He, Strain-induced one-dimensional landau level quantization in corrugated graphene, Phys. Rev. B 87, 205405 (2013).
  58. H. Yan, Y. Sun, L. He, J.-C. Nie, and M. H. W. Chan, Observation of Landau-level-like quantization at 77 k along a strained-induced graphene ridge, Phys. Rev. B 85, 035422 (2012).
  59. P. Moon and M. Koshino, Optical absorption in twisted bilayer graphene, Phys. Rev. B 87, 205404 (2013).
  60. X. He, N. Tang, X. Sun, L. Gan, F. Ke, T. Wang, F. Xu, X. Wang, X. Yang, W. Ge et al., Tuning the graphene work function by uniaxial strain, Appl. Phys. Lett. 106, 043106 (2015).
  61. R. Gupta, F. Rost, M. Fleischmann, S. Sharma, and S. Shallcross, Straintronics beyond homogeneous deformation, Phys. Rev. B 99, 125407 (2019).
  62. N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath, and M. P. Zaletel, Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling, Phys. Rev. X 10, 031034 (2020).
  63. Y. Zhang, K. Jiang, Z. Wang, and F. Zhang, Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study, Phys. Rev. B 102, 035136 (2020).
  64. B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yazdani, and B. A. Bernevig, Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram, Phys. Rev. B 103, 205414 (2021).
  65. B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries, Phys. Rev. B 103, 205413 (2021).
  66. Y. H. Kwan, G. Wagner, T. Soejima, M. P. Zaletel, S. H. Simon, S. A. Parameswaran, and N. Bultinck, Kekulé Spiral Order at All Nonzero Integer Fillings in Twisted Bilayer Graphene, Phys. Rev. X 11, 041063 (2021).
  67. G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Global Phase Diagram of the Normal State of Twisted Bilayer Graphene, Phys. Rev. Lett. 128, 156401 (2022).
  68. M. He, Y. Li, J. Cai, Y. Liu, K. Watanabe, T. Taniguchi, X. Xu, and M. Yankowitz, Symmetry breaking in twisted double bilayer graphene, Nat. Phys. 17, 26 (2021).
  69. J. Xue (private communication).

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Phys. Rev. Research 4, L032006– Published 11 July, 2022
  • Received 26 September 2021
  • Accepted 25 May 2022
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