Analytical approach to phonons and electron-phonon interactions in single-walled armchair carbon nanotubes

Phonon spectra are calculated in single-walled armchair carbon nanotubes within a mass and spring model. Classical Hamiltonian for the lattice vibrations that include nearest-neighbor, next-nearest-neighbor, and bond bending interactions has been quantized in the usual way, and then the phonon Hamiltonian is diagonalized by canonical transformations. Resolvent formalism is used to obtain phonon frequencies in analytical forms, where a procedure of the Fano problem is employed to choose correct phonon modes. The force constants are chosen from previous works, and the Raman mode at $1600{\mathrm{cm}}^{-1}$ is set to obtain the other modes. The electron-phonon interaction is investigated by phonon modulation of the hopping interaction, where the same canonical transformations are used in accordance with the phonon part. The electron-phonon coupling strengths in intraband and interband scattering for all modes within nearest-neighbor and radial bond bending interactions are found in terms of the $q$ wave vectors and other parameters. Further, a different approach for the diagonalization of the electronic part arising from the tight-binding Hamiltonian is also presented in accordance with the electron-phonon interaction part.

Figure 1

A three-dimensional plot of a (10,10) armchair SWCNT showing the nearest- and next-nearest neighbors of carbon atom $A$ with the related angles used in the text.

Figure 2

Graphical picture of how the atomic layers of the graphene are labeled. Graphical illustrations of nearest- and next-nearest neighbors of $A$ $\left(B1\right)$ atoms are also given in the same picture together with the relevant phases.

Figure 3

For $\alpha =0$, $q$ dependence of the phonon spectra according to Eq. (20) (left panel). In the right panel, again $q$ dependence of the phonon spectra, but after the second unitary transformation, according to Eq. (33), together with their comparison with those found by Mahan and Jeon [the solid and dashed lines correspond to the results of Eq. (33), and the dotted lines to those of Ref. 16]. To make the comparison easy, the results of Ref. 16 are also given in the inset.

Figure 4

Plot of ${\tilde{\omega }}_i\left(\mathit{q}\right)$ according to Eq. (33) for (a) $\alpha =0,\pm 1,\pm 2,\pm 3,\pm 4,\pm 5$ and (b) $\alpha =\pm 6,\pm 7,\pm 8,\pm 9,10$ and their collection into a single panel.