A systematic ab initio study of the optical as well as structural and electronic properties of the carbon nanotubes within density-functional theory in the local-density approximation has been performed. Highly accurate full-potential projected augmented wave method was used. Specifically, the optical dielectric function $\varepsilon$ and second-order optical susceptibility ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ as well as the band structure of a number of the armchair [(3,3),(5,5),(10,10),(15,15),(20,20)], zigzag [(5,0),(10,0),(15,0),(20,0)] and chiral [(4,2),(6,2),(6,4),(8,4), (10,5)] carbon nanotubes have been calculated. The underlying atomic structure of the carbon nanotubes was determined theoretically. It is found that for the electric field parallel to the nanotube axis $\left(E\Vert \mathrm{ẑ}\right),$ the absorptive part ${\varepsilon }^{″}$ of the optical dielectric function for the small nanotubes (the diameter being smaller than, say, 25 Å) in the low-energy range (0–8 eV) consists of a few distinct peaks. Furthermore, the energy position, the shape, and the number of the peaks depend rather strongly on the diameter and chirality. This suggests that one could use these distinct optical features to characterize the chirality and diameter of the grown nanotubes. In contrast, for the electric field perpendicular to the nanotube axis $\left(E\perp \mathrm{ẑ}\right),$ the ${\varepsilon }^{″}$ spectrum of all the nanotubes studied except the three 4 Å nanotubes in the low-energy region is made up of a broad hump. The bandwidth of the hump increases with the nanotube diameter and the magnitude of the hump is in general about half of that of the ${\varepsilon }^{″}$ for $E\Vert \mathrm{ẑ}.$ Surprisingly, given their one-dimensional character, the optical anisotropy of the nanotubes is smaller than that of graphite. For the nanotubes with a moderate diameter (say, 30 Å) such as the (20,20) nanotube, the optical anisotropy is not large and the ${\varepsilon }^{″}$ spectrum for both electric-field polarizations becomes rather similar to that of graphite with electric-field parallel to the graphene layers $\left(E\perp c\right).\mathrm{}$ The calculated static polarizability $\alpha \mathrm{}\left(0\right)\mathrm{}$ for the semiconducting nanotubes is rather anisotropic with $\alpha \mathrm{}\left(0\right)\mathrm{}$ for $E\Vert \mathrm{ẑ}$ being several times larger than that for $E\perp \mathrm{ẑ}.$ For both electric-field polarizations, $\alpha \mathrm{}\left(0\right)\mathrm{}$ is nearly proportional to the square of the tube diameter. The calculated electron energy loss spectra of all the nanotubes studied here for both electric field polarizations are similar to that of $E\perp c$ of graphite, being dominated by a broad $(\pi +\sigma )$-electron plasmon peak at near 27 eV and a small $\pi$-electron plasmon peak at 5–7 eV. Only the chiral nanotubes would exhibit second-order nonlinear optical behavior. Furthermore, only two tensor elements ${\chi }_{\mathrm{xyz}}^{\mathrm{}\left(2\right)\mathrm{}}$ and ${\chi }_{\mathrm{yzx}}^{\mathrm{}\left(2\right)\mathrm{}}$ are possibly nonzero with ${\chi }_{\mathrm{xyz}}^{\mathrm{}\left(2\right)\mathrm{}}=-{\chi }_{\mathrm{yzx}}^{\mathrm{}\left(2\right)\mathrm{}}.$ For all the chiral nanotubes studied here, both the real and imaginary parts of ${\chi }_{\mathrm{xyz}}^{\mathrm{}\left(2\right)\mathrm{}}(-2\omega ,\omega ,\omega )$ show an oscillatory behavior. The absolute value of ${\chi }_{\mathrm{xyz}}^{\mathrm{}\left(2\right)\mathrm{}}(-2\omega ,\omega ,\omega )$ of all the chiral nanotubes in the photon energy range of 0.1–3.0 eV is huge, being up to ten times larger than that of GaAs, suggesting that chiral nanotubes have potential applications in nonlinear optics, e.g., second-harmonic generation and sum-frequency generation. Nevertheless, the static value of both ${\chi }_{\mathrm{xyz}}^{\mathrm{}\left(2\right)\mathrm{}}$ and ${\chi }_{\mathrm{yzx}}^{\mathrm{}\left(2\right)\mathrm{}}$ is zero.

• Received 16 October 2003

DOI:https://doi.org/10.1103/PhysRevB.69.205416