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 Nlayer graphene. Working within the tight-binding approximation, and using the simplest nearestneighbor 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 nearestneighbor 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. Introduction. – In the past few years carbon physics presented new challenges to the scientific community, increasing the list of rather unusual phenomena occurring in this life support element. On one hand, the discovery of metal free carbon-based magnetism open a new research field in fundamental physics, with possible applications in spin electronics [1–3]. On the other, the isolation of a single graphite layer – graphene – revealed an ultrarelativistic system full of unconventional electronic properties, and regarded with great expectation from the point of view of applications [4–6]. The origin of the observed magnetism in carbonbased materials is still under debate, but the presence of open edges seem to be an ubiquitous feature [3]. In proton bombarded graphite, which shows room temperature ferromagnetism, proton irradiation induces hydrogen-terminated edges [7, 8]. In activated carbon fibers and graphitized nanodiamond particles – known as nanographite – Curie-Weiss behavior and an enhanced paramagnetic susceptibility has been reported [2]. In these nanographites edges play a predominant role due to the built-in nano-dimension. Edges are assumed to induce πlocalized electrons due to surface (edge) states, which has been seen as a key ingredient to understand carbon’s magnetic behavior [1, 3]. Indeed, the existence of edge states localized at zigzag edges of single layer graphene, induced either by extended defects or vacancies, is now well documented and their magnetic behavior has been extensively reported [3, 9–12]. Despite the positive correlation between edge state magnetism in graphene single layer and magnetic phenomena in graphite and nanographite, strictly speaking, neither of them are a single layer of graphene. Although the interlayer coupling is known to be very small, its effect is not negligible. To give an example, massless Dirac fermions in single layer graphene turn out to be massive in bilayer graphene [6]. This brings about the question whether edge states are robust to stacking, or in other words, whether multilayer graphene can support edge states localized on zigzag edges. Moreover, with the advent of graphene physics, also graphene multilayers (bilayer, trilayer, ...) were isolated. These graphene multilayers show interesting properties on their own [4, 6], dissimilar from their single layer constituent, and can be even more suitable for