Comparative study of the optical properties of single-walled carbon nanotubes within orthogonal and nonorthogonal tight-binding models

The dielectric response function of single-walled carbon nanotubes is calculated within tight-binding models with different levels of complexity. First, the effects of the orbital basis set of the model, the orbitals overlap, and the structural optimization on the electronic band structure and the dielectric function of three small-radius nanotubes are investigated in detail. Second, the optical transition energies for a large number of nanotubes are derived from the peak positions of the imaginary part of the dielectric function for parallel and perpendicular light polarization. These results can be useful for the assignment of absorption spectra of nanotube samples and for the determination of the conditions for resonant Raman scattering from nanotubes. The obtained results are compared to recent spectrofluorimetric data on isolated single-walled carbon nanotubes.

Figure 1

Calculated electronic band structure of nanotubes (5,0), (3,3), and (4,2) in the energy range between $-4$ and $4\mathrm{eV}$ with respect to the Fermi energy within the $o$-TB model (full lines) and the $\pi$-TB model (dotted lines)

Figure 2

Same as for Fig. 1 but for the $n$-TB1 model (full lines) and the $o$-TB model (dotted lines).

Figure 3

Same as for Fig. 1 but for the $n$-TB2 model (full lines) and the $n$-TB1 model (dotted lines).

Figure 4

Calculated dielectric function ${\epsilon }_2$ of nanotubes (5,0), (3,3), and (4,2) for parallel and perpendicular polarization in the energy range from $0\text{to}6\mathrm{eV}$ within the $\pi$-TB, $o$-TB, $n$-TB1, and $n$-TB2 models. The ab initio data (Ref. 8) are denoted by asterisks.

Figure 5

Calculated dielectric function ${\epsilon }_2$ of 26 nanotubes with $N_a<800$ (Ref. 22) for parallel and perpendicular polarization in the energy range from $0\text{to}3.5\mathrm{eV}$ within the $n$-TB2 model. The curves are arranged upwards in order of increasing tube radius from about $3.1\text{to}6.1\mathrm{\AA }$.

Figure 6

Calculated optical transition energies $E_{11}$ and $E_{22}$ for parallel polarization versus tube radius $R$ for all armchair and zigzag nanotubes in the range $2\mathrm{\AA }<R<15\mathrm{\AA }$ within the $\pi$-TB, $o$-TB, $n$-TB1, and $n$-TB2 models.

Figure 7

Calculated optical transition energies $E_{ii}$ for parallel polarization for all nanotubes in the range $2\mathrm{\AA }<R<15\mathrm{\AA }$ and $N_a<800$ within the $o$-TB model (left panel, full symbols) and the $n$-TB2 model (right panel, full symbols) in comparison with those of the $\pi$-TB model (empty symbols). Data for metallic and semiconducting tubes are designated by triangles and circles, respectively.

Figure 8

Calculated optical transition energies $E_{ii}$ for perpendicular polarization for all nanotubes as in Fig. 7.

Figure 9

Comparison between spectrofluorimetric data on SWNTs for the transition energies $E_{11}$ and $E_{22}$ (Ref. 22) and corresponding $\pi$-TB and $n$-TB2 results shifted up uniformly by $0.3\mathrm{eV}$. It is clearly seen that the latter points coincide well with the experimental data. The upward shift can be attributed to self-energy and excitonic effects that are not included in the band structure picture.