Degeneracy of discrete energy levels of finite-length, metallic single-wall carbon nanotubes depends on type of nanotubes, boundary condition, length of nanotubes and spin-orbit interaction. Metal-1 nanotubes, in which two non-equivalent valleys in t
he Brillouin zone have different orbital angular momenta with respect to the tube axis, exhibits 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 model reveals the role of boundary on the valley coupling in the eigenfunctions.
A phonon frequency shift of the radial breathing mode for metallic single wall carbon nanotubes is predicted as a function of Fermi energy. Armchair nanotubes do not show any frequency shift while zigzag nanotubes exhibit phonon softening, but this s
oftening is not associated with the broadening. This chirality dependence originates from a curvature-induced energy gap and a special electron-phonon coupling mechanism for radial breathing modes. Because of the particle-hole symmetry, only the off-site deformation potential contributes to the frequency shift. On the other hand, the on-site potential contributes to the Raman intensity, and the radial breathing mode intensity is stronger than that of the $G$ band. The relationship between the chirality dependence of the frequency shift of the radial breathing mode and the $Gamma$ point optical phonon frequency shift is discussed.
The quantum corrections to the frequencies of the $Gamma$ point longitudinal optical (LO) and transverse optical (TO) phonon modes in carbon nanotubes are investigated theoretically. The frequency shift and broadening of the TO phonon mode strongly d
epend on the curvature effect due to a special electron-phonon coupling in carbon nanotubes, which is shown by the Fermi energy dependence of the frequency shift for different nanotube chiralities. It is also shown that the TO mode near the $Gamma$ point decouples from electrons due to local gauge symmetry and that a phonon mixing between LO and TO modes is absent due to time-reversal symmetry. Some comparison between theory and experiment is presented.
