Electronic states of graphene nanoribbons studied with the Dirac equation

We study the electronic states of narrow graphene ribbons (nanoribbons) with zigzag and armchair edges. The finite width of these systems breaks the spectrum into an infinite set of bands, which we demonstrate can be quantitatively understood using the Dirac equation with appropriate boundary conditions. For the zigzag nanoribbon we demonstrate that the boundary condition allows a particlelike and a holelike band with evanescent wave functions confined to the surfaces, which continuously turn into the well-known zero energy surface states as the width gets large. For armchair edges, we show that the boundary condition leads to admixing of valley states, and the band structure is metallic when the width of the sample in lattice constant units has the form 3M+1, with M an integer, and insulating otherwise. A comparison of the wave functions and energies from tight-binding calculations and solutions of the Dirac equations yields quantitative agreement for all but the narrowest ribbons.