Spectrum of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi></mml:math>Electrons in Graphene as a Macromolecule

Abstract

We report the exact solution of the spectral problem for a graphene sheet framed by two armchair- and two zigzag-shaped boundaries. The solution is found for the pi electron Hamiltonian and gives, in particular, a closed analytic expression of edge-state energies in graphene. It is shown that the lower symmetry of graphene, in comparison with C6h of 2D graphite, has a profound effect on the graphene band structure. This and other results obtained have far-reaching implications for the understanding of graphene electronics. Some of them are briefly discussed.