We study edges states of graphene ribbons in the quantized Hall regime, and show that they can be described within a continuum model (the Dirac equation) when appropriate boundary conditions are adopted. The two simplest terminations, zigzag and armchair edges, are studied in detail. For zigzag edges, we find that the lowest Landau level states terminate in two types of edge states, dispersionless and current-carrying surface states. The latter involve components on different sublattices that may be separated by distances far greater than the magnetic length. For armchair edges, the boundary conditions are met by admixing states from different valleys, and we show that this leads to a single set of edges states for the lowest Landau level and two sets for all higher Landau levels. In both cases, the resulting Hall conductance step for the lowest Landau level is half that between higher Landau levels, as observed in experiment.

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 particle-and a hole-like band with evanescent wavefunctions 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 is divisible by 3, and insulating otherwise. A comparison of the wavefunctions and energies from tight-binding calculations and solutions of the Dirac equations yields quantitative agreement for all but the narrowest ribbons.

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