Lattice vibrations in single-wall carbon nanotubes

We present a study of lattice vibrations in single-wall carbon nanotubes. The study is based on a force-constant model, where the force constants are obtained from a general invariant potential that includes up to fourth-nearest-neighbor interactions. Applying the model, we obtain phonon dispersions and density of states for various types of nanotubes. General low-frequency modes such as ring modes and longitudinal modes are also identified and investigated. They are almost independent of the chiral angle but significantly dependent on the radius. The radius dependence is used for scaling, leading to a universal density of states at low frequencies.

Figure 1

Configurations containing up to $m\text{th}$-nearest-neighbor interactions.

Figure 2

Phonon dispersions of single-wall carbon nanotubes. The nanotubes in (a)–(c) are (17,1), (11,9), and (12,8), respectively.

Figure 3

Ring modes of a (17,1) nanotube. The ring modes obtained using the force-constant approach described in this paper are visually indistinguishable (not shown) from those obtained by Suzuura and Ando using a continuum model (Ref. 5).

Figure 4

Density of states per atom of three nanotubes with the same radius but different chiral angles. (a) shows the entire phonon density of states. A low-frequency magnification of the same dispersion is shown in (b). The van Hove singularities in (b) correspond to the radial breathing mode (RBM), ring modes $\lambda$, longitudinal modes ${\lambda }^{\prime }$, and higher-frequency modes ${\lambda }^{″}$.

Figure 5

Density of states per atom of three nanotubes with the same chiral angle. (a) shows the entire phonon density of states and (b) shows a magnification of the low-frequency region.

Figure 6

Scaled density of states per atom. By scaling the density of states and frequency with radius or radius squared, van Hove singularities from arbitrary nanotubes representing a certain mode could be mapped onto each other to a very good approximation. (a) illustrates how peaks from modes with frequencies with inverse radius dependence coincide for three different nanotubes. (b) Low-frequency ring modes can be scaled to produce a low-frequency universal density of states.