Effects of the boundary geometry on the edge current in the two dimensional topological insulator

We study the effects of the boundary shape on the edge transport of the two dimensional topological insulator described by Kane-Mele model. The edge state is robust against all time-reversal invariant defects. However, when we consider an arbitrary sample, the edge is not straight and consists of various types of boundaries. Actually, the transport property of the edge-state in the Kane-Mele model depends on the boundary type of the edge such as zigzag and armchair edges. Therefore, the edge-transport can be affected by a corner connecting two different types of edges. Here, we investigate the energy spectrum of the various shapes of finite-size honeycomb lattice with corners along the edge. We also calculate the transport properties on the edges by applying an artificial gauge field which drives a persistent current along the edges. Although the corner of the edge seems a geometrical defects and is expected to have a little effect on the transport, our results show that the geometrical defects strongly affect the edge current depending on the corner types.