Edge states and the quantized Hall effect in graphene

Abstract

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.

Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models

“…The existence of localized edge states was studied by Fujita et al [26], who showed that edge states appear for semi-infinite graphene with zigzag termination, whereas armchair termination does not lead to edge states. Brey and Fertig [27] have used the DE to study the electronic states of GNRs, and by using appropriate boundary conditions, they arrived at the same conclusion. Localized edge states in GALs have previously been studied by Vanević et al [28].…”

Edge states in graphene-like systems

“…Graphene ribbons are interesting by their own sake, and there has been enormous progress in the fabrication of ribbons with smooth edges [53][54][55][56][57] , so that inter-edge coupling is also an interesting topic [30][31][32]35,37,38,40,42,51,57,58 .…”

“…Therefore, the localization length can be written as λ = q −1 , is maximal for the Dirac point, q = 0, and decreases (the state becomes more localized) as q increases. A more complete analysis along this line, including the calculation of the inter edge coupling in the case of graphene ribbons, can be found in the seminal work of Brey and Fertig 58 . The validity of the effective mass approximation to describe states abrupt perturbations, such as an edge, is questionable a priori.…”

“…This occurs because the edge wave function associated to a given edge lives in the opposite sublattice than the edge state coming from the opposite edge. An important property of the electronic states of zigzag ribbons is that they do not mix valleys 58 . Actually, the confined bulk modes are cuts of the two Dirac cones, whereas the E 0 flat bands join the otherwise disconnected valleys.…”

An analysis of the zero energy states in graphene

“…Bound states in graphene can be created using either magnetic fields [1][2][3] or scalar potentials [4][5][6]. In particular, zero energy states in graphene created by using magnetic fields [7][8][9][10][11] as well as scalar potentials [12][13][14] have been studied. Here we shall propose a new potential to confine the electrons (and the holes) and examine in detail the corresponding zero energy states.…”