Vibrational spectra of double-wall carbon nanotubes

The vibrational spectra of double-wall carbon nanotubes ($\mathrm{DWNTs}$) are studied theoretically within a continuum model in this paper. The phonon dispersion relations are derived numerically and analytically, and two radial breathing modes ($\mathrm{RBMs}$) are calculated analytically, which agree with experimental data [Phys. Rev. B 66, 075416 (2002)]. By comparison of the $\mathrm{RBM}$ frequencies with the ones of isolated single-wall carbon nanotubes, we find that there is a systematic upward shift for $\mathrm{DWNTs}$ $\mathrm{RBM}$ frequencies due to the interlayer van der Waals interactions. In case of counterphase modes, the upshift magnitude of $\mathrm{RBM}$ frequencies increases with an increase of the outer-layer radius $R$. However, for inphase $\mathrm{RBMs}$, the upshift magnitude of $\mathrm{RBM}$ frequencies may increase or decrease with an increase of $R$. We then discuss four acoustic phonons and find that there is no transverse acoustic phonon in $\mathrm{DWNTs}$. Finally, the general phonon dispersion relations are calculated.

Figure 1

(Color online) The upshift magnitude of frequency vs the outer layer radius $R$ is plotted. The solid symbols represent counterphase vibrational $\mathrm{RBM}$ frequencies, while the hollow ones denote inphase ones. Circles and triangles are calculated by use of $c_0$ and $c_0^{\prime }$, respectively.

Figure 2

(Color online) Frequency vs the wave vector $k$ is plotted to compare exact numerical results with approximate analytical ones. (a) and (b) show the exact numerical solutions and approximate analytical ones when $m=0,1,2,3$, respectively.