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Minimal Conductivity in Bilayer Graphene

József Cserti Eötvös University Department of Physics of Complex Systems. Minimal Conductivity in Bilayer Graphene. J. Cs.: cond-mat/0608219. International School, MCRTN’06, Keszthely, Hungary , Aug. 27- Sept. 1, 2006. Minimal Conductivity in Bilayer Graphene.

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Minimal Conductivity in Bilayer Graphene

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  1. József Cserti Eötvös University Department of Physics of Complex Systems Minimal Conductivity in Bilayer Graphene J. Cs.: cond-mat/0608219 International School, MCRTN’06, Keszthely, Hungary, Aug. 27- Sept. 1, 2006.

  2. Minimal Conductivity in Bilayer Graphene K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal'ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A. K. Geim, Nature Physics 2, 177 (2006) Near zeros concentrations the longitudinal conductivity is of the order of Independent of temperature and magnetic field

  3. Theoretical results for single layer graphene Single layer graphene: • A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, PRB 50, 7526 (1994) • E. Fradkin, PRB 63, 3263 (1986) • P. A. Lee, PRL 71, 1887 (1993) • E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, PRB 66, 045108 (2002) • V. P. Gusynin and S. G. Sharapov, PRL 95, 146801 (2005) • N. M. R. Peres, F. Guinea, and A. H. Castro Neto, PRB 73, 125411 (2006) • M. I. Katsnelson, Eur. J. Phys B 51, 157 (2006) • J. Tworzyd lo, B. Trauzettel, M. Titov, A. Rycerz, C.W.J. Beenakker, PRL 96, 246802 (2006) K. Ziegler, cond-mat/0604537. L. A. Falkovsky and A. A. Varlamov, cond-mat/0606800. K. Nomura and A. H. MacDonald, cond-mat/0606589. Short range scattering Coulumb scattering

  4. Theoretical results forbilayer graphene M. Koshino and T. Ando, cond-mat/0606166 strong-disorder regime weak-disorder regime M. I. Katsnelson, cond-mat/0606611

  5. Hamiltonian for bilayer graphene E. McCann and V. I. Fal'ko, Phys. Rev. Lett. 96, 086805 (2006) J=1 single layer J=2 bilayer graphene Equivalent form: Pseudo spin, Pauli matrices

  6. Plane wave solution: Eigenvalues: Dirac cone Green’s function: 2 by 2 matrix

  7. Kubo formula • Bernevig, PRB 71, 073201 (2005) (derived for spintronic systems) conductivity tensor: correlation function: where Fermi function:

  8. Result per valley per spin

  9. Second method A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, PRB 50, 7526 (1994) Equivalent form: where

  10. Result per valley per spin

  11. Including the two valleys and the electron spin (factor of 4) Kubo formula Second method The two definitions yield two different results for the longitudinal conductivity of perfect graphenes But numerically they are close to each other

  12. Conclusions • The conductivity proportional with number of layers (J) • Single layer graphene (J=1): • Our result using the 2nd method agrees with • many earlier predictions • Our result for bilayer is close to the experimental one • Our result agrees with M. Koshino and T. Ando (cond-mat/0606166) • result derived for the case of strong disorder • The two methods give two different results for the longitudinal • conductivity !?! • The minimal conductivity in graphene systems still remains • a theoretical challenge in the future

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