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

2. Near zeros concentrations the longitudinal conductivity is of the order of Independent of temperature and magnetic field 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)

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. K. Nomura and A. H. MacDonald, cond-mat/0606589. L. A. Falkovsky and A. A. Varlamov, cond-mat/0606800. Short range scattering Coulumb scattering

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

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

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

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

8. Result per valley per spin

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

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. The conductivity proportional with number of layers (J) Single layer graphene (J=1): Our result using the 2 nd 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 Conclusions