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

Degeneracy of discrete energy levels of finite-length, metallic single-wall carbon nanotubes depends on the type of nanotubes, boundary condition, length of nanotubes, and spin-orbit interaction. Metal-1 nanotubes, in which two nonequivalent valleys in the Brillouin zone have different orbital angular momenta with respect to the tube axis, exhibit nearly fourfold degeneracy and small lift of the degeneracy by the spin-orbit interaction reflecting the decoupling of two valleys in the eigenfunctions. In metal-2 nanotubes, in which the two valleys have the same orbital angular momentum, vernier-scale-like spectra appear for boundaries of orthogonal-shaped edge or cap termination reflecting the strong valley coupling and the asymmetric velocities of the Dirac states. Lift of the fourfold degeneracy by parity splitting overcomes the spin-orbit interaction in shorter nanotubes with a so-called minimal boundary. Slowly decaying evanescent modes appear in the energy gap induced by the curvature of nanotube surface. Effective one-dimensional lattice model reveals the role of boundary on the valley coupling in the eigenfunctions.