Bidimensional honeycomb materials: a graph model through Dirac operator. (English) Zbl 07538763
Summary: We model a bidimensional honeycomb material with one atom at each vertex by a quantum graph with the Dirac operator on each edge. We consider at most two kinds of atoms and they are distinguished by a family of Robin boundary conditions at vertices. It is shown that Dirac cones, i.e. points with no spectral gap and (locally) linear dispersion relations, are present if, and only if, the boundary condition parameter is constant (in particular, if such parameter is zero one has the usual Neumann condition). So, Dirac cones occur only in case of just one kind of atom (as graphene). Corresponding nanotubes are briefly addressed. Similar results are known for the Schrödinger operator.
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 47E05 | General theory of ordinary differential operators |
| 81U30 | Dispersion theory, dispersion relations arising in quantum theory |
References:
| [1] | Katsnelson, M. I., Graphene: carbon in two dimensions, Materials Today, 10, 20 (2007) |
| [2] | Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E., Electronic transport in two-dimensional graphene, Rev. Mod. Phys., 83, 407 (2011) |
| [3] | Castro Neto, A. H.; Guinea, F.; Peres, N. M.R.; Novoselov, K. S.; Geim, A. K., The electronic properties of graphene, Rev. Mod. Phys., 81, 109 (2009) |
| [4] | DiVincenzo, D. P.; Mele, E. J., Self-consistent effective mass theory for intralayer screening in graphite intercalation compounds, Phys. Rev. B, 29, Article 1685 pp. (1984) |
| [5] | Zhou, S. Y., First direct observation of Dirac fermions in graphite, Nature Phys., 2, 595 (2006) |
| [6] | Wallace, P. R., The band theory of graphite, Phys. Rev., 71, 622 (1947) · Zbl 0033.14304 |
| [7] | Coulson, C. A., Note on the applicability of the free-electron network model to metals, Proc. Phys. Soc. A, 67, 608 (1954) · Zbl 0055.44403 |
| [8] | Fefferman, C. L.; Weinstein, M. I., Honeycomb lattice potentials: Dirac cones, J. Amer. Math. Soc., 25, Article 1169 pp. (2012) · Zbl 1316.35214 |
| [9] | Kuchment, P.; Post, O., On the spectra of carbon nano-structures, Commun. Math. Phys., 275, 805 (2007) · Zbl 1145.81032 |
| [10] | Amovilli, C.; Leys, F.; March, N., Electronic energy spectrum of two-dimensional solids: a chain of C atoms from a quantum network model, J. Math. Chem., 36, 93 (2004) · Zbl 1052.81689 |
| [11] | Ruedenberg, K.; Scherr, C. W., Free-electron network model for conjugated systems. I Theory, J. Chem. Phys., 21, Article 1565 pp. (1953) |
| [12] | Shipman, S. P., Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators, J. Spectr. Theory, 10, 33 (2020) · Zbl 1505.47040 |
| [13] | Fisher, L.; Li, W.; Shipman, S. P., Reducible Fermi surface for multi-layer quantum graphs including stacked graphene, Commun. Math. Phys., 385, Article 1499 pp. (2021) · Zbl 1468.81052 |
| [14] | Eastham, M. S.P., The Spectral Theory of Periodic Differential Equations (1973), Scottish Acad. Press: Scottish Acad. Press Edinburgh · Zbl 0287.34016 |
| [15] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics IV: Analysis of Operators (1978), Academic Press: Academic Press San Diego · Zbl 0401.47001 |
| [16] | Berkolaiko, G.; Comech, A., Symmetry and Dirac points in graphene spectrum, J. Spectr. Theory, 8, Article 1099 pp. (2018) · Zbl 1411.35092 |
| [17] | Benguria, R.; Fournais, S.; Stockmeyer, E.; Van Den Bosch, H., Self-adjointness of two-dimensional Dirac operators on domains, Ann. Henri Poincaré, 18, Article 1371 pp. (2017) · Zbl 1364.81117 |
| [18] | Freitas, P.; Siegl, P., Spectra of graphene nanoribbons with armchair: zigzag boundary conditions, Rev. Math. Phys., 26, Article 1450018 pp. (2014) · Zbl 1309.47088 |
| [19] | Fefferman, C. L.; Weinstein, M. I., Wave packets in honeycomb structures: two-dimensional Dirac equations, Commun. Math. Phys., 326, 251 (2014) · Zbl 1292.35195 |
| [20] | Jakubský, V.; Krejčiřfk, D., Qualitative analysis of trapped Dirac fermions in graphene, Ann. Phys., 349, 268 (2014) · Zbl 1343.82073 |
| [21] | Thaller, B., The Dirac Equation (1992), Springer: Springer Berlin · Zbl 0881.47021 |
| [22] | Bolte, J.; Harrison, J., Spectral statistics for the Dirac operator on graphs, J. Phys. A: Math. Gen., 36, Article 2747 pp. (2003) · Zbl 1038.05057 |
| [23] | Hunt, B., Massive Dirac fermions: Hofstadter butterfly in a van der Waals heterostructure, Science, 340, Article 1427 pp. (2013) |
| [24] | Watanabe, K.; Taniguchi, T.; Kanda, H., Direct-bandgap properties: evidence for ultraviolet lasing of hexagonal boron nitride single crystal, Nature Matter, 3, 404 (2004) |
| [25] | Li, L. H.; Chen, Y., Atomically thin boron nitride: Unique properties: applications, Adv. Funct. Mater., 26, Article 2594 pp. (2016) |
| [26] | Brown, M. B.; Eastham, M. S.P.; Schmidt, K. M., Periodic Differential Operators (2013), Birkhäuser: Birkhäuser Basel · Zbl 1267.34001 |
| [27] | Kuchment, P., Floquet Theory for Partial Differential Equations (1993), Birkhäuser: Birkhäuser Basel · Zbl 0789.35002 |
| [28] | Levitan, B. M.; Sargsjan, I. S., Sturm-Liouville: Dirac Operators (1991), Kluwer Academic: Kluwer Academic Dordrecht |
| [29] | Shcherbakov, A. O., Regularized trace of the Dirac operator, Mathematical Notes, 98, 168 (2015) · Zbl 1357.47048 |
| [30] | Do, N. T.; Kuchment, P., Quantum graph spectra of a graphyne structure, Nanoscale systems MMTA, 2, 107 (2013) · Zbl 1273.81067 |
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