Topological Properties of Carbon Nanotube


Mikio Eto, Keio University

In spite of a lot of work on the Carbon Nanotubes (CNTs), it has not been well known that they can be a Topological Insulator (TI). In TIs, the electronic state in the bulk is characterized by a topological number. If the topology is different from that in the vacuum, the band gap is closed at the surface since the continuous change of electronic states is impossible from the vacuum to the bulk. This corresponds to a metallic surface state (bulk-edge correspondence), which shows unique properties such as quantum spin Hall effect and Majorana fermion.
CNT is a quasi-one-dimensional system with sub-lattice structure consisting of A and B sites. The character is determined by the chiral vector which indicates the circumference on the graphene. The CNT is semiconducting or metallic. Even in metallic CNTs, the finite curvature results in a small band gap. Such a system is classified into BDI or AIII TI in the absence and presence of magnetic field, respectively. The topological number, or “winding number” in this case, is an integer. We have obtained the topological number for all the CNTs and shown that almost the CNTs are TIs. The topological CNTs have the edge states which are localized around the edges of the tube from the bulk-edge correspondence. The edge states are observable by the experiment of the Coulomb oscillation using the CNT as a quantum dot.