Phonons in narrow carbon nanotubes

Accurate calculations of the phonon dispersion relations, phonon density of states, and phonon eigenvectors of the narrow single-wall carbon nanotubes in optimized geometry are carried out. The method applied is based on the force constants for graphene which reflect the long-range character of the dynamical matrix. Further, the relaxation and symmetry imposed modifications of the force constants are performed and the calculations are carried out by means of the fully symmetry implemented POLSym code. Shortcomings of the widely used frozen phonon model are overcome. The results obtained are compared to the Raman scattering measurements on the zeolite-grown nanotubes.

Figure 1

RBM longitudinal, $z$, and circumferential, $\varphi$, components. Left: $z$ and $\varphi$ dependence on the chiral angle, $\theta$. Overlap of the longitudinal and circumferential values in a case of the (5,2) tube is indicated. Right: $z∕\cos 3\theta$ and $\varphi ∕\sin 3\theta$ as a function of tube diameter. Solid curves are fits (1) and (2).

Figure 2

RBM frequencies of the narrow SWCNTs as a function of the inverse diameter. Solid symbols are chiral tubes, open ones are achiral tubes; linear fit is given by Eq. (3). The inset shows the RBM frequency overestimation $\Delta \omega$ which frozen phonon method gives.

Figure 3

Raman active HEM frequencies plotted as a function of the inverse radius. Triangles denote modes with $A_1$ and $A_{1\mathit{g}}$ symmetry, asterisks the ones with $E_1$ and $E_{1\mathit{g}}$ symmetry while the modes with $E_2$ and $E_{2\mathit{g}}$ symmetry are depicted as squares. Solid symbols are the circumferential modes, open ones are the longitudinal modes.

Figure 4

Phonon dispersions of the (5,0) and (3,3) nanotubes given by km quantum numbers. The bands with $m=0$ and $m=n$ (bold lines) are double-degenerate, with the vertical mirror parity (even and odd is indicated by $A$ and $B$). The remaining bands are fourfold degenerate. Circles denote the radial breathing and high energy totally symmetric modes.

Figure 5

Phonon dispersions of the (4,2) and (5,1) nanotubes given by $\tilde{k}\tilde{m}$ quantum numbers. For (4,2) tube, as $n=2$, six bands are assigned by $m=0$ (bold), and the other six by $m=1$ (dashed), while for (5,1) there are only six $m=0$ branches. All the bands are double degenerate. Circles denote the radial breathing mode and the high energy totally symmetric modes.

Figure 6

PDOS of the (4,2) and (5,1) SWCNTs in low- and high-energy regions.

Figure 7

PDOS of the (5,0) and (3,3) SWCNTs in low- and high-energy regions.