Radius and chirality dependence of the radial breathing mode and the $G$-band phonon modes of single-walled carbon nanotubes

Radius and chirality dependence of the radial breathing mode and the $G$ -band phonon modes of single-walled carbon nanotubes

Valentin N. Popov and Philippe Lambin

Phys. Rev. B 73, 085407 – Published 10 February 2006

Abstract

The radial breathing and $G$ -band vibrational modes of all 300 single-walled carbon nanotubes in the radius range from $2 to 12 Å$ were calculated within a symmetry-adapted nonorthogonal tight-binding model. The dynamical matrix was calculated within this model using the linear-response approximation. The obtained phonon frequencies show well-expressed radius and chirality dependence and family behavior. The curvature-induced effects on the frequencies are found to be important for small- and moderate-radius tubes. The strong electron-phonon interactions in metallic tubes bring about Kohn anomalies of certain phonon branches. Among the Raman-active phonons, these interactions have strongest effect on the longitudinal tangential $A_{1}$ phonons of metallic tubes, whose frequency becomes lower than that of the transverse tangential $A_{1}$ phonons. The calculated frequencies are compared to available theoretical and experimental data.

Received 25 August 2005

DOI:https://doi.org/10.1103/PhysRevB.73.085407

Authors & Affiliations

Valentin N. Popov^* and Philippe Lambin

Laboratoire de Physique du Solide, Facultés Universitaires Notre-Dame de la Paix, Rue de Bruxelles 61, B-5000 Namur, Belgium

^*Permanent address: Faculty of Physics, University of Sofia, BG-1164 Sofia, Bulgaria.

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Vol. 73, Iss. 8 — 15 February 2006

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Images

Figure 1
The calculated phonon dispersion of graphene within the NTB model (crosses) in comparison with available experimental data (solid symbols). The phonon curves, downshifted by a factor of 0.9, are shown by empty circles. The discontinuous first derivative of the LO branch at the $Γ$ point and the TO branch at the $K$ point are signatures of Kohn anomaly at these points (Ref. 26).Reuse & Permissions
Figure 2
The phonon dispersion of three narrow nanotubes (11,0), (12,0), and (7,7) calculated within the NTB model. Certain branches of tubes (12,0) and (7,7) show Kohn anomaly at the zone center and at $k$ points inside the Brillouin zone (bold lines) which can be attributed to strong electron-phonon interactions.Reuse & Permissions
Figure 3
The eigenvectors of the soft phonons at points $q = 0$ and $q = 2 k_{F}$ of nanotube (7,7) calculated within the NTB model. These phonons are either $A_{1}$ ones, or belong to $B_{1}$ phonon branches. The upper figures correspond to higher phonon frequency and vice versa, as indicated by the side arrow.Reuse & Permissions
Figure 4
The RBM frequency of all nanotubes in the radius range from $2 to 12 Å$ , obtained by fitting Eq. (18) to the NTB results, reduced by a factor of 0.9. The data were fitted by the power law $1141 ∕ R$ (inset). The RBM frequencies in the small-radius region are shown in comparison with available ab initio results (crosses) (Ref. 27).Reuse & Permissions
Figure 5
The $G$ -band mode frequencies for all nanotubes in the radius range from $2 to 12 Å$ obtained by fitting Eqs. (19, 20) to the NTB results, reduced by a factor of 0.9. The corresponding frequencies, obtained by zone-folding of the NTB phonon dispersion of graphene, are provided for comparison. Major differences between both sets of data are observed for $R < 6 Å$ . All curves tend to the optical phonon frequency of graphene of $1582 {cm}^{- 1}$ (dashed line) in the large-radius limit. The $A_{1}^{LO}$ mode frequency of metallic tubes exhibits large softening due to strong electron-phonon interactions.Reuse & Permissions
Figure 6
The small-radius region of the graph in Fig. 5 showing the strong chirality dependence of the $G$ -band mode frequencies (solid circles). The empty circles on the left panel only mark the positions of the $A_{1}^{LO} (M)$ frequencies as obtained from Eq. (19). The bottom strip of points on the same panel gives the $A_{1}^{LO} (M)$ frequencies derived by fitting Eq. (20) to the NTB results. The circles on all panels show characteristic family patterns for $L_{1} + L_{2} = const$ and $2 L_{1} + L_{2} = const$ illustrated with solid and dotted lines, respectively.Reuse & Permissions

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