Recent experiments have shown that a gate voltage applied perpendicular to the axis of a carbon nanotube can give rise to a spin–orbit interaction (SOI). This is of the same nature as the Rashba–Bychkov SOI at a semiconductor heterojunction and is due to the asymmetry in the confining potential of an electron on the surface of the nanotube. Using a continuum model, we obtain analytical expressions for the spin-split energy bands for electrons on the surface of nanotubes in the presence of SOI. Each energy level could then be used to accommodate electrons which could then be excited to yield intra-SO and inter-SO subband plasmons. Using numerical calculations, we present results for the plasmon dispersion relation. The anticrossing of these plasmon excitations and their group velocities are discussed.

We calculate the plasma excitations in a bundle as well as a two-dimensional (2D) periodic array of aligned parallel multishell nanotubes on a substrate. The carbon nanotubes are oriented perpendicular to the substrate. The model we use for the system is an electron gas confined to the surface of an infinitely long cylinder embedded in a background dielectric medium. Electron tunneling between individual tubules is neglected. We include the Coulomb interaction between electrons on the same tubule and on different tubules for the same nanotube and neighboring nanotubes. We present a self-consistent field theory for the dispersion equation for intrasubband and intersubband plasmon excitations. For both the bundle and 2D array of aligned parallel nanotubes, the dispersion relation of the collective modes is determined by a three-dimensional wave vector with components in the direction of the nanotube axes and in the transverse directions. The dispersion equation is solved numerically for a singlewall nanotube 2D array as well as a bundle, and the plasmon excitation energies are obtained as a function of wave vector. The intertube Coulomb interaction couples plasmons with different angular momenta m in individual nanotubes, lifting the ±m degeneracy of the single-nanotube modes. This effect is analyzed numerically as a function of the separation between the tubules. We show that the translational symmetry of the lattice is maintained in the plasmon spectrum for the periodic array, and the plasmon energies have a periodic dependence on the transverse wave vector q ⊥ . For the bundle, the Coulomb interaction between nanotubes gives rise to optical plasmon excitations.