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We study the electronic states of narrow graphene ribbons (``nanoribbons'')with zigzag and armchair edges. The finite width of these systems breaks thespectrum into an infinite set of bands, which we demonstrate can bequantitatively understood using the Dirac equation with appropriate boundaryconditions. For the zigzag nanoribbon we demonstrate that the boundarycondition allows a particle- and a hole-like band with evanescent wavefunctionsconfined to the surfaces, which continuously turn into the well-known zeroenergy surface states as the width gets large. For armchair edges, we show thatthe boundary condition leads to admixing of valley states, and the bandstructure is metallic when the width of the sample in lattice constant units isdivisible by 3, and insulating otherwise. A comparison of the wavefunctions andenergies from tight-binding calculations and solutions of the Dirac equationsyields quantitative agreement for all but the narrowest ribbons.Comment: 5 pages, 6 figure

Electronic states of graphene nanoribbons studied with the Dirac equation