Low energy electronic phenomena in nanotubes: Piezoelectric nanotubes and helically wrapped nanotubes

We use a low energy, continuum theory to study two nanotube systems, piezoelectric nanotubes and helically wrapped carbon nanotubes. We find that the piezoelectric response of heteropolar nanotubes is controlled by the aspect ratio, the chiral angle, and two dimensionless parameters related to the relative contributions of the elastic and electrostatic energies. We study these systems under uniform and non-uniform applied stresses, and uniform and non-uniform applied electric fields. We find that the behavior of these quasi-one dimensional materials differs from the response of conventional, three dimensional piezoelectric materials. Piezoelectric nanotubes develop significant interior bound charge densities in addition to surface charges, and have a stretch-twist coupling of electrostatic origin. The theory is applied to estimate the piezoelectric response of boron-nitride nanotubes. We also study the single particle electronic spectrum of a carbon nanotube in a helical potential. The spectrum depends on the strength of the applied potential and on a dimensionless parameter, P, which is the ratio of the circumference of the nanotube to the pitch of the helix. We find that the minimum band gap of a semiconducting nanotube is reduced by an arbitrarily weak helical potential, and for a given field strength there is an optimal P which produces the biggest change in the band gap. For metallic nanotubes the Fermi velocity is reduced by this potential and for strong fields two small gaps appear at the Fermi surface in addition to the gapless Dirac point. A simple model is developed to estimate the magnitude of the field strength and its effect on DNA-CNT complexes in an aqueous solution. We find that under typical experimental conditions the predicted effects of a helical potential are likely to be small and we discuss several methods for increasing the size of these effects