Mixing of the fully symmetric vibrational modes in carbon nanotubes

We study the mixing of the fully symmetric modes in single-walled carbon nanotubes with ab initio calculations. With a variational model, we confirm the results from finite-difference calculations. We further analyze the effect of the mixing on the calculation of phonon frequencies and electron-phonon coupling matrix elements ${\mathcal{M}}_{e\text{−}\mathit{ph}}$. We find that neglecting the mixing leads to errors of up to 60% for ${\mathcal{M}}_{e\text{−}\mathit{ph}}$ for the radial breathing mode and up to $50{\mathrm{cm}}^{-1}$ difference for the high-energy mode frequency.

Figure 1

(Color online) Angle $\vartheta$ as function of the inverse nanotube diameter for several nanotubes (open symbols denote metallic, while filled symbols denote semiconducting NTs). The full line shows a linear fit for all zigzag tubes. The metallic $(n,0)$ tubes are above the line, while the semiconducting $(n,0)$ are below. Thus, the RBM is more radial for metallic zigzag tubes than for semiconducting zigzag tubes. Inset: A section of a (5,0) nanotube with five C atoms illustrating the definition of $\phi$ and $\vartheta$.

Figure 2

(Color online) Contour plot of the total energy for a full spherical fully symmetric displacement of a (4,1) nanotube. The minima (bright yellow) are slightly displaced from the points (90°, 0°) and (90°, 180°) corresponding to outward and inward radial displacements, respectively.

Figure 3

(Color online) Angle $\phi$ as a function of the inverse nanotube diameter. We use the same symbols as in Fig. 1.

Figure 4

(Color online) Normalized nonradial component of the RBM eigenvector $z_{\mathrm{RBM}}$ for zigzag NTs compared with that of Dobardžić et al. (Ref. 5) and our simple graphene model. The data of the metallic tubes fit well the model of Dobardžić et al., whereas the semiconducting tubes show larger components. We use the same symbols as in Fig. 1.

Figure 5

(Color online) Frequency ${\omega }_{\mathrm{HEM}}$ from finite-difference calculation including mixing (circles) and for the nonmixed frozen-phonon calculation (squares). Open (closed) symbols denote metallic (semiconducting) NTs. Frequencies of the nonmixed modes are underestimated, except for most metallic zigzag tubes.

Figure 6

Figure illustrating the mixing between the radial and the tangential components of the RBM in zigzag and armchair tubes when the radial component points outward.