Aharonov-Bohm oscillation and chirality effect in optical activity of single-wall carbon nanotubes

We study the Aharonov-Bohm effect in the optical phenomena of single-wall carbon nanotubes (SWCN) and also their chirality dependence. Especially, we consider the natural optical activity as a proper observable and derive its general expression based on a comprehensive symmetry analysis, which reveals the interplay between the enclosed magnetic flux and the tubule chirality for arbitrary chiral SWCN. A quantitative result for this optical property is given by a gauge invariant tight-binding approximation calculation to stimulate experimental measurements.

Figure 1

(Color online) For given ${\mathbf{R}}_0[n_1,n_2]$, there are five chiral vectors, ${\mathbf{R}}_1[n_1+n_2,-n_1]$, ${\mathbf{R}}_2[n_2,-n_1-n_2]$, ${\mathbf{R}}_3[-n_1,-n_2]$, ${\mathbf{R}}_4[-n_1-n_2,n_1]$, and ${\mathbf{R}}_5[-n_2,n_1+n_2]$, denoting exactly the same SWCN with their chiral angle deferred $\pi ∕3$ from each other successively. The tubule described by ${\mathbf{R}}_0^{\prime }[n_1+n_2,-n_2]$ is the mirror image of that described by ${\mathbf{R}}_0\bullet {\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are 2D graphite lattice basis vectors.

Figure 2

(Color online) The dependence of the ORP on magnetic flux and chiral angle is plotted in (a) for $\nu =1$ and in (b) for $\nu =0$. The normalized incident light frequency is $\overline{\omega }=0.5$ and the tubule diameter is around $37\mathrm{nm}$. In (a) the ORP for $\nu =1$ tubules is a periodic function of chiral angle with period $2\pi ∕3$ and magnetic flux with period ${\varphi }_0$. The ORP for $\nu =-1$ tubules can be obtained through the translation of chiral angle by $\pi ∕3$ in (a). In (b), the ORP for the $\nu =0$ tubules is a periodic function of chiral angle with period $\pi ∕3$ and magnetic flux with period ${\varphi }_0$.

Figure 3

Coefficients $a_1^{+}$, $a_2^{+}$, and $a_2^0$ are plotted as functions of magnetic flux $\varphi$ for different values of renormalized frequency $\overline{\omega }$ in (a), (b), and (c), respectively. The filled square, open circle and cross symbol correspond to $\overline{\omega }=0.4$, 0.5 and 0.6, respectively. Figure (d) is the plot of $a_1^{+}$, $a_2^{+}$, and $a_2^0$ vs $\overline{\omega }$ denoted by triangle, circle and square, respectively, for different values of magnetic flux. The open symbols are the data for $\varphi =0$ and the filled symbols for $\varphi =0.5{\varphi }_0$.