Post on 19-Jan-2016
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
Near zeros concentrations the longitudinal conductivity is of the order of
Independent of temperature and magnetic field
Minimal Conductivity in Bilayer GrapheneK. 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)
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 scatteringCoulumb scattering
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
E. McCann and V. I. Fal'ko, Phys. Rev. Lett. 96, 086805 (2006)
Hamiltonian for bilayer graphene
J=1 single layerJ=2 bilayer graphene
Equivalent form:
Pseudo spin, Pauli matrices
Plane wave solution:
Eigenvalues:
Green’s function:
Dirac cone
2 by 2 matrix
Kubo formula
conductivity tensor:
correlation function:
where
Fermi function:
A. Bernevig, PRB 71, 073201 (2005) (derived for spintronic systems)
Result
per valley per spin
where
Equivalent form:
A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, PRB 50, 7526 (1994)
Second method
Result
per valley per spin
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
• 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
Conclusions