Edge states in a honeycomb lattice: Effects of anisotropic hopping and mixed edges

We study the edge states in graphene in the presence of a magnetic field perpendicular to the plane of the lattice. Most of the work done so far discusses the edge states in either zigzag or armchair edge graphene considering an isotropic electron hopping. In practice, graphene can have a mixture of armchair and zigzag edges and the electron hopping can be anisotropic, which is the subject of this article. We predict that the mixed edges smear the enhanced local density of states (LDOS) at E=0 of the zigzag edge and, on the other hand, the anisotropic hopping gives rise to the enhanced LDOS at E=0 in the armchair edge. The behavior of the LDOS can be studied using scanning tunneling microscopy (STM) experiments. We suggest that care must be taken while interpreting the STM data, because the clear distinction between the zigzag edge (enhanced LDOS at E=0) and armchair edge (suppressed LDOS at E=0) can be lost if the hopping is not isotropic and if the edges are mixed.