Ultraviolet $\pi$-plasmon contribution to the transverse optical response of doped single-walled carbon nanotubes

A model for the effective permittivity of a doped single-walled carbon nanotube (CNT) thin film in the optical range is proposed. The permittivity of CNT walls is calculated from the quantum theory of $\pi$-electron transitions. The contributions from $\sigma$ electrons and ultraviolet $\pi$ plasmon are taken into account phenomenologically using experimental data obtained for graphene and CNT film. These contributions lead to an enhancement of the depolarization effect thereby strongly suppressing the transverse response of the CNTs. They also cause a decrease in both the frequency and height of the absorption peak associated with the azimuthal intersubband plasmon in doped CNTs. This eliminates the existing discrepancy between experimental and previous theoretical data. The azimuthal plasmon response is studied in a bundle of doped CNTs.

Figure 1

(a) Frequency dependencies of the real and imaginary parts of ${\overline{\varepsilon }}_{\left|\right|,\perp }-2.4$ and ${\varepsilon }_{pl}$. (b) Frequency dependencies of the real and imaginary parts of the CNT film permittivity. Symbols: experimental data taken from Fig. 1(c) of Ref. [3]. Solid and dashed lines: theoretical results. The inset show the comparison of experimental data (symbols) for the permittivity of graphene on ${\text{SiO}}_2/\text{Si}$ substrate from Table A1 of Ref. [17] and calculated data (lines) for the average permittivity of CNT wall ${\overline{\varepsilon }}_{\perp }+{\varepsilon }_{pl}$ along circumferential direction.

Figure 2

(a) Frequency dependencies of Re ${\sigma }_{\mathrm{in},\mathrm{out}}$ and its components presented in (8) at $\kappa =1$. (b) Frequency dependencies of $0.25\mathrm{Re}{\overline{\sigma }}_{\perp }$ (in units $e^2/h$) with (${\varepsilon }_{pl}\ne 0$) and without (${\varepsilon }_{pl}=0$) $\pi$-plasmon contribution for CNTs in the air ($\kappa =1$) and in the dielectric host medium ($\kappa =2$). $e$ is an electron charge; $h$ is the Planck constant.

Figure 3

Schematic illustration of the charge distribution, current oscillations $j_{\theta ,z}$, and nonzero field components $E_{\theta ,\rho ,z}$ and $H_{\theta ,z}$ for the azimuthal (a) and axial (b) TM modes excited in CNT by the incident field $E_0$.

Figure 4

Calculated frequency dependencies of the optical density (OD) of CNT film with thickness of 80 nm at various doping levels $E_F\in \{0,0.7,0.8,0.9,1.0,1.1\}\mathrm{eV}$ at $\kappa =1$ (a), $\kappa =2$ (b) and (c). The arrow indicates a direction of Fermi energy increasing. The $\pi$-plasmon and $\sigma$-electron transitions are not taken into account in (c), ${\sigma }_{pl}=0, {\varepsilon }_{\sigma }=0$.

Figure 5

Frequency dependencies of the normalized absorption cross section of a doped individual CNT, pair of CNTs, and bundle of seven CNTs. All the CNTs are identical with tubule index (0,17) at doping level $E_F=1.1\mathrm{eV}$ and intertube distance 0.34 nm. The orientations of the incident field ${\mathbf{E}}_0$ are indicated with arrows.

Figure 6

Geometry of $m\mathrm{th}$ and $n\mathrm{th}$ CNTs.