Valley coupling in finite-length metallic single-wall carbon nanotubes

摘要

Degeneracy of discrete energy levels of finite-length, metallic single-wallcarbon nanotubes depends on type of nanotubes, boundary condition, length ofnanotubes and spin-orbit interaction. Metal-1 nanotubes, in which twonon-equivalent valleys in the Brillouin zone have different orbital angularmomenta with respect to the tube axis, exhibits nearly fourfold degeneracy andsmall lift of the degeneracy by the spin-orbit interaction reflecting thedecoupling of two valleys in the eigenfunctions. In metal-2 nanotubes, in whichthe two valleys have the same orbital angular momentum, vernier-scale-likespectra appear for boundaries of orthogonal-shaped edge or cap-terminationreflecting the strong valley coupling and the asymmetric velocities of theDirac states. Lift of the fourfold degeneracy by parity splitting overcomes thespin-orbit interaction in shorter nanotubes with a so-called minimal boundary.Slowly decaying evanescent modes appear in the energy gap induced by thecurvature of nanotube surface. Effective one-dimensional model reveals the roleof boundary on the valley coupling in the eigenfunctions.