Conductivity of bilayer graphene: Sublattice coherent contributions within Boltzmann and Kubo approaches
Final Report Abstract
The electrical conductivity is obviously the most important quantity characterizing semiconductors in both fundamental research and applied studies. The conductivity of graphene - a few atom thick layer of carbon atoms isolated recently in Manchester - does not vanish at the neutrality point where the sample is completely undoped. The latter is certainly one of the most mysterious phenomena observed in graphene. Indeed since the density of electrons vanishes as the Fermi energy approaches zero (i. e. the sample becomes completely undoped), the natural expectation is that the conductivity should also vanish in this limit. However, graphene has never exhibited the conductivity smaller than a few of conductance quantum. The conventional explanation of this "conductivity without electrons" employs so called charge inhomogeneities - the conductive electron and hole puddles allowing the current to flow even though the overall charge carrier concentration remains zero. This explanation seems to be incomplete because the suspended graphene samples still exhibit the conductivity minimum although the charged impurities are removed upon annealing and the puddle formation should therefore be suppressed. The main outcomes of the present project includes an alternative explanation of the non-vanishing conductivity phenomenon. The explanation is based on the very fact that the electrons and holes are not independent in graphene because of its crystallographic structure. In the framework of this project it has been shown that this entanglement is responsible for the non-vanishing conductivity minimum in bilayer graphene. If the valence band is full, and the conduction band is empty, then the momentum space drift due to the electric field does not repopulate states in a full band, as emphasized in text-book transport theory, and the conductivity should be zero. However, the electric field does drive the valence band from equilibrium because it alters the mometum-dependent coupling term between the conduction and valence bands. Qualitatively the valence band is still completely occupied, but it no longer has the equilibrium valence band wavefunction. The carriers are no longer well defined in the classical sense, they turn out to be the quantum particles being in the superposition of the conduction and valence band states. Such carriers can conduct even though the valence band is full, and the conduction band is empty. This explains the non-vanishing conductivity minimum in suspended undoped graphene. The surprising result obtained theoretically within this project is the non-vanishing conductivity at the neutrality point even in the multilayer graphene with the number of layers larger than one. The conductivity minimum turns out to be universal for monolayer, bilayer, and even for trilayer graphene. The measurements are awaited to prove this prediction for the trilayer samples. As a hint for a possible report about this study in the public media I would like to refer to the progress article by K. Novoselov and Andre Geim [Novoselov&Geim Nature Materials vol.6, p.183 (2007)], in particular to the section "Conductivity without charge carriers", where they emphasize the importance of the conductivity minimum problem. I believe, that the results obtained within the project can be seen as a valuable contribution to the solution of the problem discussed there at the general level. Now one can understand how the conductivity "without" charge carriers is possible.
Publications
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Finite Conductivity Minimum in Bilayer Graphene without Charge Inhomogeneities. Phys. Rev. B vol. 82, 155308 (2010)
Maxim Trushin, Janik Kailasvuori, John Schliemann, A.H. MacDonald
