- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Chern mosaic and ideal flat bands in equal-twist trilayer graphene

Daniele Guerci1, Yuncheng Mao2, and Christophe Mora2

  • 1Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
  • 2Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, 75013 Paris, France

PDF

Phys. Rev. Research 6, L022025 – Published 30 April, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L022025

Abstract

We study trilayer graphene arranged in a staircase stacking configuration with equal consecutive twist angle. On top of the moiré crystalline pattern, a supermoiré long-wavelength modulation emerges that we treat adiabatically. For each valley, we find that the two central bands are topological with Chern numbers C=±1 forming a Chern mosaic at the supermoiré scale. The Chern domains are centered around the high-symmetry stacking points ABA or BAB and they are separated by gapless lines connecting the AAA points where the spectrum is fully connected. In the chiral limit and at a magic angle of θ∼1.69∘, we prove that the central bands are exactly flat with ideal quantum curvature at ABA and BAB. Furthermore, we decompose them analytically as a superposition of an intrinsic color-entangled state with ±2 and a Landau level state with Chern number ∓1. To connect with experimental configurations, we also explore the nonchiral limit with finite corrugation and find that the topological Chern mosaic pattern is indeed robust and the central bands are still well separated from the remote bands.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (97)

  1. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  2. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  3. R. Bistritzer and A. H. MacDonald, Moiré butterflies in twisted bilayer graphene, Phys. Rev. B 84, 035440 (2011).
  4. 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).
  5. 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).
  6. E. Suárez Morell, J. D. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, Flat bands in slightly twisted bilayer graphene: Tight-binding calculations, Phys. Rev. B 82, 121407(R) (2010).
  7. E. J. Mele, Commensuration and interlayer coherence in twisted bilayer graphene, Phys. Rev. B 81, 161405(R) (2010).
  8. 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).
  9. 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).
  10. X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold, A. H. MacDonald, and D. K. Efetov, Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature (London) 574, 653 (2019).
  11. 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).
  12. 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).
  13. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  14. S. L. Tomarken, Y. Cao, A. Demir, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and R. C. Ashoori, Electronic compressibility of magic-angle graphene superlattices, Phys. Rev. Lett. 123, 046601 (2019).
  15. H. Polshyn, M. Yankowitz, S. Chen, Y. Zhang, K. Watanabe, T. Taniguchi, C. R. Dean, and A. F. Young, Large linear-in-temperature resistivity in twisted bilayer graphene, Nat. Phys. 15, 1011 (2019).
  16. D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Cascade of electronic transitions in magic-angle twisted bilayer graphene, Nature (London) 582, 198 (2020).
  17. U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern, E. Berg, P. Jarillo-Herrero, and S. Ilani, Cascade of phase transitions and Dirac revivals in magic-angle graphene, Nature (London) 582, 203 (2020).
  18. Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, Strange metal in magic-angle graphene with near planckian dissipation, Phys. Rev. Lett. 124, 076801 (2020).
  19. 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).
  20. A. Jaoui, I. Das, G. Di Battista, J. Díez-Mérida, X. Lu, K. Watanabe, T. Taniguchi, H. Ishizuka, L. Levitov, and D. K. Efetov, Quantum critical behaviour in magic-angle twisted bilayer graphene, Nat. Phys. 18, 633 (2022).
  21. 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).
  22. 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).
  23. 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).
  24. M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020).
  25. J. H. Pixley and E. Y. Andrei, Ferromagnetism in magic-angle graphene, Science 365, 543 (2019).
  26. G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi, Z. Shi, T. Senthil, D. Goldhaber-Gordon, Y. Zhang, and F. Wang, Tunable correlated Chern insulator and ferromagnetism in a Moirésuperlattice, Nature (London) 579, 56 (2020).
  27. S.-Y. Li, Y. Zhang, Y.-N. Ren, J. Liu, X. Dai, and L. He, Experimental evidence for orbital magnetic moments generated by moiré-scale current loops in twisted bilayer graphene, Phys. Rev. B 102, 121406(R) (2020).
  28. S. Grover, M. Bocarsly, A. Uri, P. Stepanov, G. D. Battista, I. Roy, J. Xiao, A. Y. Meltzer, Y. Myasoedov, K. Pareek, K. Watanabe, T. Taniguchi, B. Yan, A. Stern, E. Berg, D. K. Efetov, and E. Zeldov, Chern mosaic and berry-curvature magnetism in magic-angle graphene, Nat. Phys. 18, 885 (2022).
  29. N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for anomalous Hall ferromagnetism in twisted bilayer graphene, Phys. Rev. Lett. 124, 166601 (2020).
  30. C. Repellin, Z. Dong, Y.-H. Zhang, and T. Senthil, Ferromagnetism in narrow bands of moiré superlattices, Phys. Rev. Lett. 124, 187601 (2020).
  31. W.-Y. He, D. Goldhaber-Gordon, and K. T. Law, Giant orbital magnetoelectric effect and current-induced magnetization switching in twisted bilayer graphene, Nat. Commun. 11, 1650 (2020).
  32. P. Potasz, M. Xie, and A. H. MacDonald, Exact diagonalization for magic-angle twisted bilayer graphene, Phys. Rev. Lett. 127, 147203 (2021).
  33. 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, K. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo-Herrero, and A. Yacoby, Fractional Chern insulators in magic-angle twisted bilayer graphene, Nature (London) 600, 439 (2021).
  34. C. Repellin and T. Senthil, Chern bands of twisted bilayer graphene: Fractional Chern insulators and spin phase transition, Phys. Rev. Res. 2, 023238 (2020).
  35. D. Parker, P. Ledwith, E. Khalaf, T. Soejima, J. Hauschild, Y. Xie, A. Pierce, M. P. Zaletel, A. Yacoby, and A. Vishwanath, Field-tuned and zero-field fractional Chern insulators in magic angle graphene, arXiv:2112.13837.
  36. G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett. 122, 106405 (2019).
  37. 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).
  38. F. K. Popov and A. Milekhin, Hidden wave function of twisted bilayer graphene: The flat band as a Landau level, Phys. Rev. B 103, 155150 (2021).
  39. P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional Chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Res. 2, 023237 (2020).
  40. O. Vafek and J. Kang, Renormalization group study of hidden symmetry in twisted bilayer graphene with coulomb interactions, Phys. Rev. Lett. 125, 257602 (2020).
  41. B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. I. matrix elements, approximations, perturbation theory, and a k·p two-band model, Phys. Rev. B 103, 205411 (2021).
  42. 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).
  43. E. Khalaf, A. J. Kruchkov, G. Tarnopolsky, and A. Vishwanath, Magic angle hierarchy in twisted graphene multilayers, Phys. Rev. B 100, 085109 (2019).
  44. A. Abouelkomsan, Z. Liu, and E. J. Bergholtz, Particle-hole duality, emergent fermi liquids, and fractional Chern insulators in moiré flatbands, Phys. Rev. Lett. 124, 106803 (2020).
  45. Y. Ren, Q. Gao, A. H. MacDonald, and Q. Niu, WKB estimate of bilayer graphene's magic twist angles, Phys. Rev. Lett. 126, 016404 (2021).
  46. 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).
  47. P. J. Ledwith, E. Khalaf, and A. Vishwanath, Strong coupling theory of magic-angle graphene: A pedagogical introduction, Ann. Phys. 435, 168646 (2021).
  48. J. Wang, Y. Zheng, A. J. Millis, and J. Cano, Chiral approximation to twisted bilayer graphene: Exact intravalley inversion symmetry, nodal structure, and implications for higher magic angles, Phys. Rev. Res. 3, 023155 (2021).
  49. Y. Sheffer and A. Stern, Chiral magic-angle twisted bilayer graphene in a magnetic field: Landau level correspondence, exact wave functions, and fractional Chern insulators, Phys. Rev. B 104, L121405 (2021).
  50. Z.-D. Song and B. A. Bernevig, Magic-angle twisted bilayer graphene as a topological heavy fermion problem, Phys. Rev. Lett. 129, 047601 (2022).
  51. Y. Sheffer, R. Queiroz, and A. Stern, Symmetries as the guiding principle for flattening bands of Dirac fermions, Phys. Rev. X 13, 021012 (2023).
  52. M. Barkeshli and X.-L. Qi, Topological nematic states and non-abelian lattice dislocations, Phys. Rev. X 2, 031013 (2012).
  53. Y.-L. Wu, N. Regnault, and B. A. Bernevig, Bloch model wave functions and pseudopotentials for all fractional Chern insulators, Phys. Rev. Lett. 110, 106802 (2013).
  54. Y.-L. Wu, N. Regnault, and B. A. Bernevig, Haldane statistics for fractional Chern insulators with an arbitrary Chern number, Phys. Rev. B 89, 155113 (2014).
  55. J. Wang, S. Klevtsov, and Z. Liu, Origin of model fractional Chern insulators in all topological ideal flatbands: Explicit color-entangled wavefunction and exact density algebra, Phys. Rev. Res. 5, 023167 (2023).
  56. X. Zhang, K.-T. Tsai, Z. Zhu, W. Ren, Y. Luo, S. Carr, M. Luskin, E. Kaxiras, and K. Wang, Correlated insulating states and transport signature of superconductivity in twisted trilayer graphene superlattices, Phys. Rev. Lett. 127, 166802 (2021).
  57. A. Uri, S. C. de la Barrera, M. T. Randeria, D. Rodan-Legrain, T. Devakul, P. J. D. Crowley, N. Paul, K. Watanabe, T. Taniguchi, R. Lifshitz, L. Fu, R. C. Ashoori, and P. Jarillo-Herrero, Superconductivity and strong interactions in a tunable moiré quasiperiodic crystal, arXiv:2302.00686.
  58. 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).
  59. 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).
  60. Z. Hao, A. M. 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).
  61. H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Evidence for unconventional superconductivity in twisted trilayer graphene, Nature (London) 606, 494 (2022).
  62. X. Liu, N. J. Zhang, K. Watanabe, T. Taniguchi, and J. I. A. Li, Isospin order in superconducting magic-angle twisted trilayer graphene, Nat. Phys. 18, 522 (2022).
  63. S. Carr, C. Li, Z. Zhu, E. Kaxiras, S. Sachdev, and A. Kruchkov, Ultraheavy and ultrarelativistic Dirac quasiparticles in sandwiched graphenes, Nano Lett. 20, 3030 (2020).
  64. D. Călugăru, F. Xie, Z.-D. Song, B. Lian, N. Regnault, and B. A. Bernevig, Twisted symmetric trilayer graphene: Single-particle and many-body hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field, Phys. Rev. B 103, 195411 (2021).
  65. F. Xie, N. Regnault, D. Călugăru, B. A. Bernevig, and B. Lian, Twisted symmetric trilayer graphene. II. Projected hartree-fock study, Phys. Rev. B 104, 115167 (2021).
  66. D. Guerci, P. Simon, and C. Mora, Higher-order Van Hove singularity in magic-angle twisted trilayer graphene, Phys. Rev. Res. 4, L012013 (2022).
  67. M. Christos, S. Sachdev, and M. S. Scheurer, Correlated insulators, semimetals, and superconductivity in twisted trilayer graphene, Phys. Rev. X 12, 021018 (2022).
  68. 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).
  69. M. He, Y.-H. Zhang, Y. Li, Z. Fei, K. Watanabe, T. Taniguchi, X. Xu, and M. Yankowitz, Competing correlated states and abundant orbital magnetism in twisted monolayer-bilayer graphene, Nat. Commun. 12, 4727 (2021).
  70. H. Polshyn, Y. Zhang, M. A. Kumar, T. Soejima, P. Ledwith, K. Watanabe, T. Taniguchi, A. Vishwanath, M. P. Zaletel, and A. F. Young, Topological charge density waves at half-integer filling of a moiré superlattice, Nat. Phys. 18, 42 (2022).
  71. 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 (2022).
  72. Z. Zhu, S. Carr, D. Massatt, M. Luskin, and E. Kaxiras, Twisted trilayer graphene: A precisely tunable platform for correlated electrons, Phys. Rev. Lett. 125, 116404 (2020).
  73. Y. Mao, D. Guerci, and C. Mora, Supermoiré low-energy effective theory of twisted trilayer graphene, Phys. Rev. B 107, 125423 (2023).
  74. T. Cea, P. A. Pantaleón, and F. Guinea, Band structure of twisted bilayer graphene on hexagonal boron nitride, Phys. Rev. B 102, 155136 (2020).
  75. J. Shi, J. Zhu, and A. H. MacDonald, Moiré commensurability and the quantum anomalous Hall effect in twisted bilayer graphene on hexagonal boron nitride, Phys. Rev. B 103, 075122 (2021).
  76. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L022025 for provides an in-depth examination of the adiabatic Hamiltonian, covering its symmetries and notation. Additionally, it discusses fundamental properties of the Jacobi theta function and computes the derivative of the Wronskian.
  77. J. M. Luttinger and W. Kohn, Motion of electrons and holes in perturbed periodic fields, Phys. Rev. 97, 869 (1955).
  78. G. Bastard, Superlattice band structure in the envelope-function approximation, Phys. Rev. B 24, 5693 (1981).
  79. S. R. White and L. J. Sham, Electronic properties of flat-band semiconductor heterostructures, Phys. Rev. Lett. 47, 879 (1981).
  80. G. Bastard, Theoretical investigations of superlattice band structure in the envelope-function approximation, Phys. Rev. B 25, 7584 (1982).
  81. D. L. Smith and C. Mailhiot, Theory of semiconductor superlattice electronic structure, Rev. Mod. Phys. 62, 173 (1990).
  82. T. Devakul, P. J. Ledwith, L.-Q. Xia, A. Uri, S. C. de la Barrera, P. Jarillo-Herrero, and L. Fu, Magic-angle helical trilayer graphene, Sci. Adv. 9, eadi6063 (2023).
  83. N. Nakatsuji, T. Kawakami, and M. Koshino, Multiscale lattice relaxation in general twisted trilayer graphenes, Phys. Rev. X 13, 041007 (2023).
  84. F. K. Popov and G. Tarnopolsky, Magic angle butterfly in twisted trilayer graphene, Phys. Rev. Res. 5, 043079 (2023).
  85. F. K. Popov and G. Tarnopolsky, Magic angles in equal-twist trilayer graphene, arXiv:2303.15505.
  86. C. Mora, N. Regnault, and B. A. Bernevig, Flatbands and perfect metal in trilayer moiré graphene, Phys. Rev. Lett. 123, 026402 (2019).
  87. P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vortexability: A unifying criterion for ideal fractional Chern insulators, Phys. Rev. B 108, 205144 (2023).
  88. B. Mera and T. Ozawa, Uniqueness of Landau levels and their analogs with higher Chern numbers, arXiv:2304.00866.
  89. J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact Landau level description of geometry and interaction in a flatband, Phys. Rev. Lett. 127, 246403 (2021).
  90. A. Parhizkar and V. Galitski, A generic topological criterion for flat bands in two dimensions, arXiv:2301.00824.
  91. B. Estienne, N. Regnault, and V. Crépel, Ideal Chern bands are Landau levels in curved space, Phys. Rev. Res. 5, L032048 (2023).
  92. J. Dong, P. J. Ledwith, E. Khalaf, J. Y. Lee, and A. Vishwanath, Many-body ground states from decomposition of ideal higher Chern bands: Applications to chirally twisted graphene multilayers, Phys. Rev. Res. 5, 023166 (2023).
  93. J. Wang and Z. Liu, Hierarchy of ideal flatbands in chiral twisted multilayer graphene models, Phys. Rev. Lett. 128, 176403 (2022).
  94. 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).
  95. 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).
  96. J. Cano, S. Fang, J. H. Pixley, and J. H. Wilson, Moiré superlattice on the surface of a topological insulator, Phys. Rev. B 103, 155157 (2021).
  97. R. Resta, Geometry and topology in many-body physics (2020), arXiv:2006.15567.

Outline

Information

Phys. Rev. Research 6, L022025– Published 30 April, 2024
  • Received 12 May 2023
  • Accepted 25 January 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation