Analytic approach to the linear susceptibility of zigzag carbon nanotubes

A general analytical expression for the linear optical susceptibility of single walled zigzag carbon nanotubes for light polarized parallel to the nanotube axis has been obtained. Using a simple orthogonal tight binding model with nearest neighbor interactions, the electric dipole matrix element and subsequently the susceptibility as a function of optical frequency, have been investigated. For light polarized perpendicular to the nanotube axis, no analytic solution for the susceptibility has been found. However, a closed-form expression for the electric dipole matrix element has been obtained. Hence, numerical evaluation of the perpendicular susceptibility is greatly simplified.

Figure 1

Rolled-up (left) and unrolled (right) unit cell of (4, 0) CN. For an $(n,0)$ CN, the chain (1, 2, 3, 4) of carbon atoms is repeated $n$ times around the circumference.

Figure 2

$k$ dependence of the long-axis momentum matrix elements $P_z$ for a (6, 0) CN. All allowed interband transitions are included.

Figure 3

The real $\left({\chi }_{zz}^{\prime }\right)$ and imaginary $\left({\chi }_{zz}^{″}\right)$ parts of the susceptibility of semiconducting zigzag CNs for parallel polarization of the light.

Figure 4

Imaginary part of the interband long-axis susceptibility for two metallic CNs. The inset illustrates the common resonance at $\hslash \omega =4{\gamma }_0$.

Figure 5

$k$ dependence of $\mid P_y\mid$ for a (10, 0) CN for all allowed transitions.

Figure 6

The real $\left({\chi }_{yy}^{\prime }\right)$ and imaginary $\left({\chi }_{yy}^{″}\right)$ parts of the susceptibility for zigzag CNs for perpendicular polarization.

Figure 7

(a) The real $\left({\chi }^{\prime }\right)$ and imaginary $\left({\chi }^{″}\right)$ parts of the susceptibility for (160, 0) CN for both parallel and perpendicular polarizations and (b) their imaginary $\left({\chi }^{″}\right)$ parts in comparison with graphite. The three curves nearly coincide and therefore they are hardly distinguishable.