Ab initio study of the dielectric response of crystalline ropes of metallic single-walled carbon nanotubes: Tube-diameter and helicity effects

The dielectric-response functions of crystalline ropes of metallic single-walled carbon nanotubes were determined from time-dependent density-functional theory in the random-phase approximation. Interband transitions and plasmonic excitations were studied as a function of momentum transfer. The impact of the tube diameter was shown for the $\left(n,n\right)$ armchair-type series ($n$ ranging from 3 to 8) covering a diameter range from 4 to $11\text{Å}$. Helicity effects were examined for the thinnest tubes, the armchair (3,3) versus the zigzag (5,0) configurations. Our results give detailed insight into the various kinds of excitations that can be observed in ropes of nanotubes. Recent experimental findings and trends by electron energy loss and Raman spectroscopies are reproduced and explained.

Figure 1

Hexagonal unit cell for the (3,3) carbon tubes. ${\text{S}}_t$ denotes the interwall separation.

Figure 2

(Color online) Dielectric (imaginary part) and energy-loss function for the (3,3), (4,4), and (5,5) tubes calculated for a small $\left(0.23{\text{Å}}^{-1}\right)$ momentum-transfer ${\mathbf{q}}_{\parallel }$ parallel to the tubes axis. The $\text{Im}\left[{\epsilon }_M\right]$ is also plotted for the (6,6) tubes. The results were obtained without LFE. The insets contain the corresponding results for graphite (from Ref. 26). Broadening is set to 0.5 eV.

Figure 3

(Color online) $q$-dependent energy-loss function for the (3,3), (5,5), and (7,7) tubes calculated for $\mathbf{q}$-orientation parallel to the tubes axis. $q$ is given in units of ${\text{Å}}^{-1}$. The results were obtained without LFE. Broadening is set to 0.1 eV.

Figure 4

Dispersion of the $\pi$ plasmon for $\mathbf{q}$-orientation parallel to the tubes axis. The momentum-transfer $\mathbf{q}$ has units of ${\text{Å}}^{-1}$. The dotted lines are guides for the eyes.

Figure 5

(Color online) Dielectric function of the tubes in the 0–10 eV range for small $q$ (equal to $0.06{\text{Å}}^{-1}$) of orientation parallel to the tubes axis. Results obtained without LFE. Full curves denote corresponding results with a broadening of 0.1 eV.

Figure 6

$q$-dependent energy-loss function of the (3,3) and (5,0) tubes. $\mathbf{q}$ is parallel to the tube axis. Solid (dashed) curves denote results obtained with (without) LFE. Broadening is set to 0.5 eV.

Figure 7

(Color online) Dielectric function of the (3,3) tubes in the 0–10 eV range for three $\mathbf{q}$ vectors of orientation parallel to the tubes axis. Results obtained without LFE. Full curves denote corresponding results with a broadening of 0.1 eV.

Figure 8

(Color online) Dielectric function of the (5,0) tubes in the 0–10 eV range for three $\mathbf{q}$ vectors of orientation parallel to the tubes axis. Results obtained without LFE. Full curves denote corresponding results with a broadening of 0.1 eV.