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Eduardo Castro, Vitor M Pereira, J M B Lopes dos Santos.
Nuno M R Peres (U. Minho); Antonio H Castro Neto (Boston U).

In 2004 the Manchester group of Andre Geim produced, with an extremely unsophisticated method, the first samples of graphene, a flat single atom layer of sp2 bonded carbon and the building block for other allotropes of carbon (fullerenes, nanotubes and graphite). Graphene's remarkable mechanical and electrical properties hold the promise of new carbon based electronics, and other applications such as sensors, light sources or even hydrogen storage. The work of CFP in graphene started in June 2005, during a visit of CFP researchers to Boston University at a time when Paco Guinea (ICMM Madrid) and Nuno Peres (U. Minho) were also visiting Antonio Castro Neto at Boston University. An informal network to work on this theme was formed at the time and involves about a dozen physicists.

Vacancies in graphene

Graphene is a semi-metal, with a linearly vanishing density of states (DOS) at zero energy (the Fermi level for the undoped material). Our first work was a study of electronic properties of vacancy defects in graphene. It proved possible to construct a vacancy quasi-localized state of zero energy (the Fermi level) as linear combination of previously known zero energy edge states to the left and right of a line passing through the vacancy The many vacancy case was studied using the recursion method. Near the Fermi energy, the DOS fills up to a constant plateau (also predicted in a CPA) but in addition has a sharp peak closer to zero energy, reminiscent of the delta function of the quasi-localized state for a single impurity. A study of the inverse participation ratio showed that the states contributing to this feature have a real space decay similar to the single impurity state (~1/r, where r is the distance to the vacancy). This work in currently being extended to study long range (Coulomb) impurity potentials and more general impurities (which change site energies and hopping parameters).

The graphene bilayer: a tunable gap semiconductor.

Recently, we also developed a tight-binding study of the graphene bilayer, subject to a perpendicular electric field. This turns out to be a very interesting system for applications, since the bilayer opens a gap, tunable via the external electric field. We discussed the screening of the external field in terms of a Hartree theory (previously used by McCann in the context of a continuum description) and managed to obtain a consistent description of the measured cyclotron mass and energy gap as a function of the external field. Our results also clarify the differences in Quantum Hall effect plateaus in the biased and unbiased bilayers. Some preliminary studies of the disorder induced gap states were also conducted.

The bilayer with a twist

In a related work we showed that a relative rotation of the layers (a rather common stacking defect in graphite, often observed in STM studies of its surface) has profound effects on the electronic structure near the Fermi level:

* the low energy dispersion remains linear, as in a single layer; * the Fermi velocity can be significantly reduced with respect to the single layer value; * a perpendicular electric field does not open a gap. These predictions are in accord with several observations made in epitaxially grown few layer graphene, which have been puzzling to the community, in view of the predictions for *AB* stacked multilayers.

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