One-dimensional semirelativity for electrons in carbon nanotubes

It is shown that the band structure of single-wall semiconducting carbon nanotubes (CNT) is analogous to relativistic description of electrons in vacuum, with the maximum velocity $u={10}^8\mathrm{cm}∕\mathrm{s}$ replacing the light velocity. One-dimensional semirelativistic kinematics and dynamics of electrons in CNT is formulated. Two-band $\mathbf{k}\bullet \mathbf{p}$ Hamiltonian is employed to demonstrate that electrons in CNT experience a zitterbewegung (trembling motion) in absence of external fields. This zitterbewegung should be observable much more easily in CNT than its analogue for free relativistic electrons in vacuum. It is argued that in the lowest subband of metallic CNT, where the rest effective mass of electrons vanishes, the zitterbewegung should not occur.