The Tight-Binding Approach and the Resulting Electronic Structure

  • Marcin Mucha-Kruczynski
Part of the Springer Theses book series (Springer Theses)


In this chapter, we describe the crystal and reciprocal lattices of bilayer graphene. We also discuss briefly the symmetry of the crystal lattice. We then introduce the tight-binding model for \(\pi \) electrons in bilayer graphene. We start with a general formulation valid for all points in the Brillouin zone and the resulting electronic structure. Next, we concentrate on the linear approximation of that model around the corners of the Brillouin zone.


Brillouin Zone Neutrality Point Bilayer Graphene Interlayer Coupling Electronic Dispersion 
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  1. 1.
    P.R. Wallace, The band theory of graphite. Phys. Rev. 71, 622 (1947)ADSzbMATHCrossRefGoogle Scholar
  2. 2.
    J.W. McClure, Band structure of graphite and de Haas-van Alphen effect. Phys. Rev. 108, 612 (1957)ADSCrossRefGoogle Scholar
  3. 3.
    J.C. Slonczewski, P.R. Weiss, Band structure of graphite. Phys. Rev. 109, 272 (1958)ADSCrossRefGoogle Scholar
  4. 4.
    J.W. McClure, Theory of diamagnetism of graphite. Phys. Rev. 119, 606 (1960)ADSCrossRefGoogle Scholar
  5. 5.
    R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998)CrossRefGoogle Scholar
  6. 6.
    C. Bena, G. Montambaux, Remarks on the tight-binding model of graphene. New J. Phys. 11, 095003 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    J.D. Bernal, The structure of graphite. Proc. R. Soc. A 106, 749 (1924)ADSCrossRefGoogle Scholar
  8. 8.
    M.S. Dresselhaus, G. Dresselhaus, Intercalation compounds of graphite. Adv. Phys. 30, 139 (1981)ADSCrossRefGoogle Scholar
  9. 9.
    F. Varchon, R. Feng, J. Hass, X. Li, B. Ngoc Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, E.H. Conrad, L. Magaud, Electronic structure of epitaxial graphene layers on SiC: effect of the substrate. Phys. Rev. Lett. 99, 126805 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    S.B. Trickey, F. Müller-Plathe, G.H.F. Diercksen, Interplanar binding and lattice relaxation in a graphite dilayer. Phys. Rev. B 45, 4460 (1992)ADSCrossRefGoogle Scholar
  11. 11.
    H. Ajiki, T. Ando, Electronic states of carbon nanotubes. J. Phys. Soc. Jpn. 62, 1255 (1993)ADSCrossRefGoogle Scholar
  12. 12.
    T. Ando, Theory of electronic states and transport in carbon nanotubes. J. Phys. Soc. Jpn. 74, 777 (2005)ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    J.M. Luttinger, W. Kohn, Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97, 869 (1955)ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    D.P. DiVincenzo, E.J. Mele, Self-consistent effective-mass theory for intralayer screening in graphite intercalation compounds. Phys. Rev. B 29, 1685 (1984)ADSCrossRefGoogle Scholar
  15. 15.
    T. Ohta, A. Bostwick, T. Seyller, K. Horn, E. Rotenberg, Controlling the electronic structure of bilayer graphene. Science 313, 951 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    L.M. Zhang, Z.Q. Li, D.N. Basov, M.M. Fogler, Z. Hao, M.C. Martin, Determination of the electronic structure of bilayer graphene from infrared spectroscopy. Phys. Rev. B 78, 235408 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    A.B. Kuzmenko, E. van Heumen, D. van der Marel, P. Lerch, P. Blake, K.S. Novoselov, A.K. Geim, Infrared spectroscopy of electronic bands in bilayer graphene. Phys. Rev. B 79, 115441 (2009)ADSCrossRefGoogle Scholar
  18. 18.
    A.B. Kuzmenko, I. Crassee, D. van der Marel, P. Blake, K.S. Novoselov, Determination of the gate-tunable band gap and tight-binding parameters in bilayer graphene using infrared spectroscopy. Phys. Rev. B 80, 165406 (2009)ADSCrossRefGoogle Scholar
  19. 19.
    Z.Q. Li, E.A. Henriksen, Z. Jiang, Z. Hao, M.C. Martin, P. Kim, H.L. Stormer, D.N. Basov, Band structure asymmetry of bilayer graphene revealed by infrared spectroscopy. Phys. Rev. Lett. 102, 037403 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    L.M. Malard, J. Nilsson, D.C. Elias, J.C. Brant, F. Plentz, E.S. Alves, A.H. Castro Neto, M.A. Pimenta, Probing the electronic structure of bilayer graphene by Raman scattering. Phys. Rev. B 76, 201401(R) (2007)Google Scholar
  21. 21.
    J. Yan, E.A. Henriksen, P. Kim, A. Pinczuk, Observation of anomalous phonon softening in bilayer graphene. Phys. Rev. Lett. 101, 136804 (2008)ADSCrossRefGoogle Scholar
  22. 22.
    A. Das, B. Chakraborty, S. Piscanec, S. Pisana, A.K. Sood, A.C. Ferrari, Phonon renormalization in doped bilayer graphene. Phys. Rev. B 79, 155417 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    D.L. Mafra, L.M. Malard, S.K. Doorn, H. Htoon, J. Nilsson, A.H. Castro Neto, M.A. Pimenta, Observation of the Kohn anomaly near the K point of bilayer graphene. Phys. Rev. B 80, 241414(R) (2009)Google Scholar
  24. 24.
    J.B. Oostinga, H.B. Heersche, X.L. Liu, A.F. Morpurgo, L.M.K. Vandersypen, Gate-induced insulating state in bilayer graphene devices. Nat. Mater. 7, 151 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    Y. Zhang, T.-T. Tang, C. Girit, Z. Hao, M.C. Martin, A. Zettl, M.F. Crommie, Y.R. Shen, F. Wang, Direct observation of a widely tunable bandgap in bilayer graphene. Nature 459, 820 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    K.F. Mak, C.H. Lui, J. Shan, T.F. Heinz, Observation of an electric-field-induced band gap in bilayer graphene by infrared spectroscopy. Phys. Rev. Lett. 102, 256405 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    M. Mucha-Kruczyński, E. McCann, V.I. Fal’ko, Electron-hole asymmetry and energy gaps in bilayer graphene. Semicond. Sci. Technol. 25, 033001 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    E. McCann, V.I. Fal’ko, Landau level degeneracy and quantum hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006)ADSCrossRefGoogle Scholar
  29. 29.
    D.S.L. Abergel, V.I. Fal’ko, Optical and magneto-optical far-infrared properties of bilayer graphene. Phys. Rev. B 75, 155430 (2007)ADSCrossRefGoogle Scholar
  30. 30.
    M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2, 620 (2006)CrossRefGoogle Scholar
  31. 31.
    K. Kechedzhi, E. McCann, V. Fal’ko, B. Altshuler, Influence of trigonal warping on interference effects in bilayer graphene. Phys. Rev. Lett. 98, 176806 (2007)ADSCrossRefGoogle Scholar
  32. 32.
    K.S. Novoselov, E. McCann, S.V. Morozov, V.I. Fal’ko, M.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A.K. Geim, Unconventional quantum Hall effect and Berry’s phase of 2pi in bilayer graphene. Nat. Phys. 2, 177 (2006)CrossRefGoogle Scholar
  33. 33.
    M.I. Katsnelson, Minimal conductivity in bilayer graphene. Eur. Phys. J. B 52, 151 (2006)ADSCrossRefGoogle Scholar
  34. 34.
    E. McCann, D.S.L. Abergel, V.I. Fal’ko, Electrons in bilayer graphene. Solid State Commun. 143, 110 (2007)ADSCrossRefGoogle Scholar
  35. 35.
    I. Lifshitz, Anomalies of electron characteristics of a metal in the high pressure region. Sov. Phys. J. Exp. Theor. Phys. 11, 1130 (1960)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of PhysicsLancaster UniversityLancasterUK

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