Advertisement

On the graphene Hamiltonian operator

  • C. Conca
  • R. Orive
  • J. San Martín
  • V. SolanoEmail author
Article
  • 18 Downloads

Abstract

We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805–826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.

Keywords

Periodic solutions General spectral theory Spectral theory and eigenvalue problems Graphene Honeycomb structure 

Mathematics Subject Classification

34L05 82D80 34B60 34B45 47A10 34D20 

Notes

Acknowledgements

C. Conca and J. San Martín were partially supported from PFBasal-01 (CeBiB), PFBasal-03 (CMM) projects. C.Conca also received partial support from Ecos-Conicyt Grant C13E05 and by Fondecyt Grant 1140773. J. San Martín also received partial support from Fondecyt Grant 1180781. V. Solano was partially supported by Scholarship Program of CONICYT-Chile, Folio Number 21110749, by PFBasal-03 (CMM) project and by the Grants SEV-2011-0087 from Ministerio de Ciencia e Innovación (MICINN) of Spain. We would like to thank M. Solano for valuable comments on the manuscript.

References

  1. Aguirre F, Conca C (1988) Eigenfrequencies of a tube bundle immersed in a fluid. Appl Math Optim 18(1):1–38MathSciNetCrossRefGoogle Scholar
  2. Alexander S (1983) Superconductivity of networks. A percolation approach to the effects of disorder. Phys Rev B 27(3):1541MathSciNetCrossRefGoogle Scholar
  3. Allaire G, Orive R (2005) On the band gap structure of Hill’s equation. J Math Anal Appl 306(2):462–480MathSciNetCrossRefGoogle Scholar
  4. Amovilli C, Leys FE, March NH (2004) Electronic energy spectrum of two-dimensional solids and a chain of c atoms from a quantum network model. J Math Chem 36(2):93–112MathSciNetCrossRefGoogle Scholar
  5. Avron JE, Raveh A, Zur B (1988) Adiabatic quantum transport in multiply connected systems. Rev Mod Phys 60(4):873MathSciNetCrossRefGoogle Scholar
  6. Bloch F (1929) Über die quantenmechanik der elektronen in kristallgittern. Z Phys 52(7–8):555–600CrossRefGoogle Scholar
  7. Conca C, Planchard J, Vanninathan M (1995) Fluids and periodic structures. Wiley, ChichesterzbMATHGoogle Scholar
  8. Conca C, Vanninathan M (1997) Homogenization of periodic structures via bloch decomposition. SIAM J Appl Math 57(6):1639–1659MathSciNetCrossRefGoogle Scholar
  9. Eastham MS (1973) The spectral theory of periodic differential equations. Scottish Academic Press, EdinburghzbMATHGoogle Scholar
  10. De Gennes PG (1981) Champ critique d’une boucle supraconductrice ramefieé. C R Acad Sci Paris 292B:279–282Google Scholar
  11. Harris PJF (2002) Carbon nano-tubes and related structures: new materials for the twenty-first century. AAPTGoogle Scholar
  12. Katsnelson MI (2007) Graphene: carbon in two dimensions. Mater Today 10(1):20–27CrossRefGoogle Scholar
  13. Korotyaev E, Lobanov I (2006) Zigzag periodic nanotube in magnetic field. arXiv:math/0604007 (arXiv preprint)
  14. Korotyaev E, Lobanov I (2007)Schrödinger operators on zigzag nanotubes. In: Annales henri poincare, vol 8, no 6. Birkhuser-Verlag, pp 1151–1176Google Scholar
  15. Kuchment P (2002) Graph models for waves in thin structures. Waves Rand Media 12(4):R1–R24MathSciNetCrossRefGoogle Scholar
  16. Kuchment P (2004) Quantum graphs and their applications in special issue of waves in random media 14(1):S107–S128Google Scholar
  17. Kuchment P, Post O (2007) On the spectra of carbon nano-structures. Commun Math Phys 275(3):805–826MathSciNetCrossRefGoogle Scholar
  18. Leys FE, Amovilli C, March NH (2004) Topology, connectivity and electronic structure of C and B cages and the corresponding nanotubes. J Chem Inf Comput Sci 44(1):122–135CrossRefGoogle Scholar
  19. Mills RGJ, Montroll EW (1970) Quantum theory on a network. II. A solvable model which may have several bound states per node point. J Math Phys 11(8):2525–2538CrossRefGoogle Scholar
  20. Montroll EW (1970) Quantum theory on a network. I. A solvable model whose wavefunctions are elementary functions. J Math Phys 11(2):635–648CrossRefGoogle Scholar
  21. Pauling L (1936) The diamagnetic anisotropy of aromatic molecules. J Chem Phys 4(10):673–677CrossRefGoogle Scholar
  22. Ruedenberg K, Scherr CW (1953) Free-electron network model for conjugated systems. I. Theory. J Chem Phys 21(9):1565–1581CrossRefGoogle Scholar
  23. Saito R, Dresselhaus G, Dresselhaus MS (1998) Physical properties of carbon nanotubes. World Scientific, SingaporeCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Mathematical Engineering Department, Center for Mathematical Modeling-UMI 2807 CNRS-UChile and Center for Biotechnology and BioengineeringUniversidad de ChileSantiagoChile
  2. 2.Mathematics Department, Science FacultyUniversidad Autónoma de MadridMadridSpain
  3. 3.Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UC3M-UCM)MadridSpain
  4. 4.Mathematical Engineering Department and Center for Mathematical Modeling UMI 2807 CNRS-UChileUniversidad de ChileSantiagoChile
  5. 5.Engineering FacultyUniversidad del DesarrolloSantiagoChile

Personalised recommendations