Optical properties of $\mathrm{SiC}$ nanotubes: An ab initio study

The band structure and optical dielectric function $ϵ$ of single-walled zigzag $\left[\right(3,0),(4,0),(5,0),(6,0),(8,0),(9,0),(12,0),(16,0),(20,0),(24,0\left)\right]$, armchair $\left[\right(3,3),(4,4),(5,5),(8,8),(12,12),(15,15\left)\right]$, and chiral $\left[\right(4,2),(6,2),(8,4),(10,4\left)\right]$ $\mathrm{SiC}\text{−}\mathrm{NTs}$ as well as the single honeycomb $\mathrm{SiC}$ sheet have been calculated within density-functional theory with the local-density approximation. The underlying atomic structure of the $\mathrm{SiC}\text{−}\mathrm{NTs}$ is determined theoretically. It is found that all the $\mathrm{SiC}$ nanotubes are semiconductors, except the ultrasmall (3,0) and (4,0) zigzag tubes which are metallic. Furthermore, the band gap of the zigzag $\mathrm{SiC}\text{−}\mathrm{NTs}$ which is direct, may be reduced from that of the $\mathrm{SiC}$ sheet to zero by reducing the diameter $\left(D\right)$, though the band gap for all the $\mathrm{SiC}$ nanotubes with a diameter larger than $\sim 20\mathrm{\AA }$ is almost independent of diameter. For the electric field parallel to the tube axis $(E\Vert \widehat{z})$, the ${ϵ}^{″}$ for all the $\mathrm{SiC}\text{−}\mathrm{NTs}$ with a moderate diameter (say, $D>8\mathrm{\AA }$) in the low-energy region $(0–6\mathrm{eV})$ consists of a single distinct peak at $\sim 3\mathrm{eV}$. However, for the small diameter $\mathrm{SiC}$ nanotubes such as the (4,2), (4,4) $\mathrm{SiC}\text{−}\mathrm{NTs}$, the ${ϵ}^{″}$ spectrum does deviate markedly from this general behavior. In the high-energy region (from $6\mathrm{eV}$ upwards), the ${ϵ}^{″}$ for all the $\mathrm{SiC}\text{−}\mathrm{NTs}$ exhibit a broad peak centered at $\sim 7\mathrm{eV}$. For the electric field perpendicular to the tube axis $(E\perp \widehat{z})$, the ${ϵ}^{″}$ spectrum of all the $\mathrm{SiC}\text{−}\mathrm{NTs}$ except the (4,4), (3,0), and (4,0) nanotubes, in the low-energy region also consists of a pronounced peak at around $3\mathrm{eV}$ while in the high-energy region is roughly made up of a broad hump starting from $6\mathrm{eV}$. The magnitude of the peaks is in general about one-half of the magnitude of the corresponding ones for $E\Vert \widehat{z}$. Interestingly, the calculated static dielectric constant $ϵ\left(0\right)$ for all the $\mathrm{SiC}$ nanotubes is nearly independent of diameter and chirality with $ϵ\left(0\right)$ for $E\Vert \widehat{z}$ being only about 30% larger than for $E\perp \widehat{z}$. The calculated electron energy loss spectra of all the $\mathrm{SiC}$ nanotubes for both electric field polarizations are rather similar to that of $E\perp \widehat{c}$ of the $\mathrm{SiC}$ sheet, being dominated by a broad $(\pi +\sigma )$-electron plasmon peak at near $21\mathrm{eV}$ and a small $\pi$-electron plasmon peak at $\sim 4\mathrm{eV}$.

Figure 1

(Color online) Band structure, density of states (DOS), dielectric function, and energy loss spectrum of the $\mathrm{SiC}$ sheet. In (a), the zero energy is at the top of the valence band. In (e), the spectra below $15\mathrm{eV}$ has been multiplied by 5.

Figure 2

(Color online) Calculated band gaps of the $\mathrm{SiC}$ nanotubes vs diameter. For comparison, the band gap of the $\mathrm{SiC}$ sheet is shown as the dashed-dotted line.

Figure 3

Energy bands and density of states of the zigzag (3,0), (5,0), (8,0) $\mathrm{SiC}$ nanotubes. $\Gamma Z$ is the 1D Brillouin zone of length $\pi$/$T$ where $T$ is the lattice constant (Table ). In (a) and (b), the dashed line indicates the Fermi level $\left(0\mathrm{eV}\right)$. In (c) the top of the valance band is at $0\mathrm{eV}$.

Figure 4

Energy bands and density of states of the armchair (8,8), (12,12) and chiral (6,2) $\mathrm{SiC}$ nanotubes. The top of the valence band is at $0\mathrm{eV}$. $\Gamma Z$ is the 1D Brillouin zone of length $\pi ∕T$ where $T$ is the lattice constant (Table ).

Figure 5

(Color online) Calculated dielectric functions of the zigzag $\mathrm{SiC}$ nanotubes. “Parallel” and “perpendicular” represent the electric field polarized parallel and perpendicular to the tube axis, respectively.

Figure 6

(Color online) Calculated dielectric functions of the armchair $\mathrm{SiC}$ nanotubes. “Parallel” and “perpendicular” represent the electric field polarized parallel and perpendicular to the tube axis, respectively.

Figure 7

(Color online) Calculated dielectric functions of the chiral $\mathrm{SiC}$ nanotubes. “Parallel” and “perpendicular” represent the electric field polarized parallel and perpendicular to the tube axis, respectively.

Figure 8

(Color online) Static dielectric constant $ϵ\left(0\right)$ of the $\mathrm{SiC}$ nanotubes as a function of diameter. The dashed line is a constant least-squares fit.

Figure 9

(a) ${\alpha }_{xx}\left(0\right)$ and (b) ${\alpha }_{zz}\left(0\right)$ vs diameter $D$ for the $\mathrm{SiC}$ nanotubes. The dashed line is a linear least-squares fit.

Figure 10

(Color online) Calculated energy loss function of the selected zigzag, armchair, and chiral $\mathrm{SiC}$ nanotubes. “Parallel” and “perpendicular” denote the electric field polarized parallel and perpendicular to the tube axis, respectively.