Edge states for the n = 0 Laudau level in graphene

Abstract

In the anomalous quantum Hall effect (QHE), a hallmark of graphene, nature of the edge states in magnetic fields poses an important question, since the edge and bulk should be intimately related in QHE. Here we have theoretically studied the edge states, focusing on the E = 0 edge mode, which is unusual in that the mode is embedded right within the n = 0 bulk Landau level, while usual QHE edge modes reside across adjacent Landau levels. Here we show that the n = 0 Landau level, including the edge mode, has a wave function amplitude accumulated along zigzag edges whose width scales with the magnetic length, lB. This contrasts with the usual QHE where the charge is depleted from the edge. The implications are: (i) The E = 0 edge states in strong magnetic fields have a topological origin in the honeycomb lattice, so that they are outside the continuum ("massless Dirac") model. (ii) The edge-mode contribution decays only algebraically into the bulk, but this is "topologically" compensated by the bulk contribution, resulting in the accumulation over lB. (iii) The real space behavior obtained here should be observable in STM experiments.