Effect of Curvature-Induced Superlattice Structures on Energy Band Structures of Helically Coiled Carbon Nanotubes

We studied electronic properties of a two-dimensional electron gas confined on the surface of helical curved tube, specifically, curvature induced by the geometry of helically coiled carbon nanotubes. The curvature generates two types of effective couplings that are (i) Penney-Kronig potential and (ii) a periodic spin-orbit potential. The energy-momentum dispersion relations are theoretically calculated using new variables to explain the electron spiraling around the circular helix line in counterclockwise and clockwise directions. Then we have to set up the formalism for dealing with a periodic potential as Bloch’s theorem. For calculation, we rewrite the Dirac equation in curved space-time in the form of a block matrix equation. Our numerical results demonstrate that there are pseudo-Landau quantization levels and the motion of an electron in circumference direction coupling with the arc length motion coupling. In addition, the one-dimensional dispersion relation lines present characteristics of time reversal symmetry and inversion symmetry. Finally, we also investigated the differential energy of band structures for the case where the periodic spin-orbit potential increased with respect to the increase of curvature. We found that the non-linearly increase of the differential energy was due to the periodic spin-orbit potential, corresponding to Zeeman effect.