Figure 1

Schematic diagram of the geometry involved in helical wrapping. (a) A single strand is assumed to wind around a cylinder of radius $R$ at an angle $\theta$. (b) In the two-dimensional depiction, the unwrapped tube is represented by a stripe of width $2\pi R$ and the coiling angle $\theta$ defines a unit cell of length $2\pi R∕\tan \theta$.Reuse & Permissions

Figure 2

Relative energy change per perturbation for a one-dimensional free electron gas perturbed by a series of equally spaced delta-function scatterers for (i) two scatterers (dashed line); (ii) seven scatterers (continuous line); (iii) infinite number of scatterers (dotted line). The amplitude of the scattering potential is $\lambda =-0.3$ in all cases. The distance between successive minima (maxima) is $\pi ∕k_F$, where $k_F$ is the Fermi wave number of the gas.Reuse & Permissions

Figure 3

Relative energy change per unit length of perturbation for (a) a two-dimensional free electron gas perturbed by an infinite array of equally spaced delta-function scatterers $(E_F=2.0)$ and (b) a free electron gas cylinder in the presence of a coiling perturbation for two separate Fermi levels ($E_F=0.5$, solid line and $E_F=1.0$, dashed line). A series of energy minima is obtained, indicating the existence of energetically favorable coiling angles. Inset in (b) is a graph of ${\Delta }_{\infty }$ as a function of the coiling angle $\theta$. The amplitude of the scattering potential is $\lambda =-0.3$ in all cases. The distance between successive minima (maxima) is again seen to be $\pi ∕k_F$, where $k_F$ is the Fermi wave number of the gas.Reuse & Permissions

Figure 4

(Color online) Angular dependence of the relative energy change ${\Delta }_N$ for helical wrapping on a (9,0) zigzag nanotube. An overall nonmonotonic trend depicted by the dashed line is superimposed to short-period fluctuations that correspond to high-symmetry angles of the perturbation.Reuse & Permissions