Localized States at Zigzag Edges of Multilayer Graphene and Graphite   Steps
  We report the existence of zero energy surface states localized at zigzag edges of $N$-layer graphene. Working within the tight-binding approximation, and using the simplest nearest-neighbor model, we derive the analytic solution for the wavefunctions of these peculiar surface states. It is shown that zero energy edge states in multilayer graphene can be divided into three families: (i) states living only on a single plane, equivalent to surface states in monolayer graphene; (ii) states with finite amplitude over the two last, or the two first layers of the stack, equivalent to surface states in bilayer graphene; (iii) states with finite amplitude over three consecutive layers. Multilayer graphene edge states are shown to be robust to the inclusion of the next nearest-neighbor interlayer hopping. We generalize the edge state solution to the case of graphite steps with zigzag edges, and show that edge states measured through scanning tunneling microscopy and spectroscopy of graphite steps belong to family (i) or (ii) mentioned above, depending on the way the top layer is cut. 
Eduardo V. Castro, N. M. R. Peres, J. M. B. Lopes dos Santos