Optical scattering from a planar array of finite-length metallic carbon nanotubes

A model is developed for optical scattering from planar arrays of finite-length single-wall metallic carbon nanotubes. The scattered field is predicted using a periodic Green’s function for the array, which includes all electromagnetic interactions, and a quantum conductance function ${\sigma }_{\mathit{cn}}\left(\omega \right)$ for the carbon nanotubes. It is found that for both individual carbon nanotubes and nanotube arrays, the optical far scattered field is proportional to ${\sigma }_{\mathit{cn}}\left(\omega \right)$, so that scattering characteristics are governed by effects associated with electronic transitions. This is in strong distinction to the case for far-infrared arrays, where mutual electromagnetic coupling effects were previously found to be very important for a wide range of broadside nanotube spacings. Furthermore, due to strong damping, single-wall nanotubes do not exhibit longitudinal current resonances (optical antenna effects) associated with the finite length of the tubes, which is also quite different from the far-infrared result.

Figure 1

Infinite planar array of finite-length carbon nanotubes. Each tube has length $2L$, radius $a$, and the array periods along the $x$ and $z$ axes are $D_x$ and $D_z$, respectively.

Figure 2

(Color online) Magnitude of the current at the center $(z=0)$ of the center nanotube in an infinite array of (10, 10) nanotubes, calculated from the Born approximation (3) and from the numerical solution of the integral equation for the array case.

Figure 3

(Color online) Comparison between the IE solution and the Born approximations ${\mathbf{E}}_{\mathit{BA}}^s$ and ${\mathbf{E}}_{\mathit{BA},0}^s$ for the far scattered field from an infinite planar array of (10, 10) carbon nanotubes. The field is incident from ${\theta }^i=90°$ and ${\varphi }^i=30°$, and the scattered field is determined at $\theta =90°$ and $\varphi =150°$ (i.e., specular reflection), at a distance of $r=1\mu \mathrm{m}$.

Figure 4

(Color online) Normalized scattered field intensity of an isolated (10, 10) carbon nanotube from simulation and measurement, and for an infinite array of nanotubes. The peak electric-field values for the isolated tube and for the infinite array of tubes are $4.153\times {10}^{-5}$ and $0.0626\mathrm{V}∕\mathrm{m}$, respectively. The Born approximation (9) for the infinite array is also shown.

Figure 5

(Color online) Normalized scattered field intensity of an isolated (11, 8) carbon nanotube from simulation (with $\tau =2.2\mathrm{fs}$) and measurement.

Figure 6

(Color online) Scattered electric field from an array of (10, 10) carbon nanotubes ($D_x=8\mathrm{nm}$ and $D_z=3\mathrm{nm}+2L$). The nanotubes have half lengths $L=50$, 150, and $500\mathrm{nm}$.

Figure 7

(Color online) The scattered electric field from several different array configurations of (10, 10), $L=150\text{−}\mathrm{nm}$ carbon nanotubes. In all cases the end-to-end spacing $({\delta }_z=D_z-2L)$ is $3\mathrm{nm}$, and the broadside spacing $D_x$ is varied.

Figure 8

(Color online) The scattered electric field from several arrays of (10, 10) carbon nanotubes. In all cases $L=150\mathrm{nm}$ and $D_x=50\mathrm{nm}$, and $D_z$ is varied.