An exactly solvable tight-binding billiard in graphene
DOI:
https://doi.org/10.31349/RevMexFis.71.050401Keywords:
Graphene, Triangulene, Edge States, Exact solutions, GraphsAbstract
A triangular graphenic billiard is defined as a planar carbon polymer in the Hückeloid approximation of π−band electrons. It is shown that the equilateral triangle of arbitrary size and zig-zag edges allows for exact solutions of the associated spectral problem. This is done by a construction of wave superpositions similar to the Lamé solution of the Helmholtz equation in a triangular cavity, revisited by Pinsky. Exact wave functions, eigenvalues, degeneracies, and edge states are provided. The edge states are also obtained by a non-periodic construction of waves with vanishing energy. A comment on its connection with recent molecular models, such as triangulene, is given.
References
A.K. Geim and K. S. Novoselov, The rise of graphene, Nature Materials 6 (2007) 183, https://doi.org/10.1038/nmat1849
M. I. Katsnelson, The Physics of Graphene (Cambridge University Press, 2020), https://doi.org/10.1017/9781108617567
S. Rathinavel, K. Priyadharshini, and D. Panda, A review on carbon nanotube: An overview of synthesis, properties, functionalization, characterization, and the application, Materials Science and Engineering: B 268 (2021) 115095, https://doi.org/10.1016/j.mseb.2021.115095
R. Martel, H. R. Shea, and P. Avouris, Rings of single-walled carbon nanotubes, Nature 398 (1999) 299, https://doi.org/10.1038/18589
J. Cai et al., Atomically precise bottom-up fabrication of graphene nanoribbons, Nature 466 (2010) 470, https://doi.org/10.1038/nature09211
E. Sadurní et al., Hidden duality and accidental degeneracy in cycloacene and Möbius cycloacene, Journal of Mathematical Physics 62 (2021) 052102, https://doi.org/10.1063/5.0031586
D. Condado, E. Sadurní, and R. A. Méndez-Sánchez, Algebraically solvable model for electron-phonon interactions in cycloacene molecules, Phys. Rev. A 108 (2023) 052823, https://doi.org/10.1103/PhysRevA.108.052823
H. Kono et al., Methylene-Bridged [6]−, [8]−, and [10] Cycloparaphenylenes: Size-Dependent Properties and Paratropic Belt Currents, Journal of the American Chemical Society 145 (2023) 8939, https://doi.org/10.1021/jacs. 2c13208
W. Yang, F. Zhang, and D. J. Klein, Benzenoid links, Journal of Mathematical Chemistry 47 (2009) 457, https://doi.org/10.1007/s10910-009-9583-8
A. Misra, T. G. Schmalz, and D. J. Klein, Clar Theory for Radical Benzenoids, Journal of Chemical Information and Modeling 49 (2009) 2670, https://doi.org/10.1021/ci900321e
M. V. Berry and R. J. Mondragon, Neutrino billiards: timereversal symmetry-breaking without magnetic fields, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 412 (1987) 53, https://doi.org/10.1098/rspa.1987.0080
O. Klein, Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac, Zeitschrift für Physik 53 (1929) 157, https://doi.org/10.1007/bf01339716
G. Breit, An Interpretation of Dirac’s Theory of the Electron, Proceedings of the National Academy of Sciences 14 (1928) 553, https://doi.org/10.1073/pnas.14.7.553
M. R. Setare, P. Majari, C. Noh and S. Dehdashti, Photonic realization of the deformed Dirac equation via the segmented graphene nanoribbons under inhomogeneous strain, Journal of Modern Optics 66 (2019) 1663, https://doi.org/10.1080/09500340.2019.1656783
E. J. Squires, The bag model of hadrons, Reports on Progress in Physics 42 (1979) 1187, https://doi.org/10.1088/0034-4885/42/7/003
Q. Zhang et al., Nuclear matter as an MIT bag crystal, Journal of Physics G: Nuclear Physics 12 (1986) L19, https://doi.org/10.1088/0305-4616/12/1/004
D. Vasak et al., Deformed solutions of the MIT quark bag model, Journal of Physics G: Nuclear Physics 9 (1983) 511, https://doi.org/10.1088/0305-4616/9/5/004
G. Hall, On the eigenvalues of molecular graphs, Molecular Physics 33 (1977) 551, https://doi.org/10.1080/ 00268977700100471
D. J. Klein, T. H. Seligman, and E. Sadurní, Eigen-Persistence in Graphs, Match Communications in Mathematical and in Computer Chemistry 92 (2024) 339, https://doi.org/10.46793/match.92-2.339k
E. Hückel, Quantentheoretische Beiträge zum Benzolproblem, Z. Physik 70 (1931) 204, https://doi.org/10.1007/bf01339530
E. Hückel, Quanstentheoretische Beiträge zum Benzolproblem, Z. Physik 72 (1931) 310, https://doi.org/10.1007/bf01341953
R. Band, O. Parzanchevski, and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, Journal of Physics A: Mathematical and Theoretical 42 (2009) 175202, https://doi.org/10.1088/1751-8113/42/17/175202
T. Sunada, Riemannian Coverings and Isospectral Manifolds, Annals of Mathematics 121 (1985) 169, https://doi.org/10.2307/1971195
G. Lamé, Memoire sur la propagation de la chaleur dans les polyedres, Journal de l’Ecole Polytechnique 22 (1833) 194
G. Lamé, Lecons sur la Theorie Analytique de la Chaleur (MalletBachelier, 1861)
E. Sadurní, T. H. Seligman, and F. Mortessagne, Playing relativistic billiards beyond graphene, New Journal of Physics 12 (2010) 053014, https://doi.org/10.1088/1367-2630/12/5/053014
W. Lenz, Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern, European Physical Journal A 21 (1920) 613
E. Sadurní, E. Rivera-Mociños, and A. Rosado, Discrete symmetry in graphene: the Dirac equation and beyond, Rev. Mex. Fís. 61 (2015) 170
M. A. Pinsky, The Eigenvalues of an Equilateral Triangle, SIAM Journal on Mathematical Analysis 11 (1980) 819, https://doi.org/10.1137/0511073
M. Bellec et al., Manipulation of edge states in microwave artificial graphene, New Journal of Physics 16 (2014) 113023, https://doi.org/10.1088/1367-2630/16/11/113023
U. Kuhl et al., Dirac point and edge states in a microwave realization of tight-binding graphene-like structures, Phys. Rev. B 82 (2010) 094308, https://doi.org/10.1103/PhysRevB.82.094308
K. Nakada et al., Edge state in graphene ribbons: Nanometer size effect and edge shape dependence, Phys. Rev. B 54 (1996) 17954, https://doi.org/10.1103/PhysRevB.54.17954
I. Kleftogiannis, I. Amanatidis, and V. A. Gopar, Conductance through disordered graphene nanoribbons: Standard and anomalous electron localization, Phys. Rev. B 88 (2013) 205414, https://doi.org/10.1103/PhysRevB.88.205414
P. Omling, H. Linke, and L. C. Lindelof, Transport Properties of a Triangular Electron Billiard, Japanese Journal of Applied Physics 36 (1997) 3996, https://doi.org/10.1143/JJAP.36.3996
G. Casati and T. c. v. Prosen, Mixing Property of Triangular Billiards, Phys. Rev. Lett. 83 (1999) 4729, https://doi.org/10.1103/PhysRevLett.83.4729
Z. Ge et al., Direct visualization of relativistic quantum scars in graphene quantum dots, Nature 635 (2024) 841, https://doi.org/10.1038/s41586-024-08190-6
E. J. Heller, Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits, Phys. Rev. Lett. 53 (1984) 1515, https://doi.org/10.1103/PhysRevLett.53.1515
A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi 44 (2001) 131, https://doi.org/10.1070/1063-7869/44/10S/S29
E. Y. Andrei et al., The marvels of moiré materials, Nature Reviews Materials 6 (2021) 201, https://doi.org/10.1038/s41578-021-00284-1
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 D. Condado, E. Sadurní
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
