Ab initio phonon dispersions of single-wall carbon nanotubes

Phonon-dispersion curves for a series of single-wall carbon nanotubes (SWCNT’s) have been obtained by ab initio supercell approach. Force constants are calculated using norm-conserving pseudopotential plane-wave method in the framework of the density-functional theory and local-density approximation. Cumulant force-constant method is used instead of interatomic force-constant method to construct the dynamical matrices. We get much better low-frequency dispersions than existing results with no soft phonon mode presented. Good accuracy is shown in the full frequency range. Residue forces in the relaxed SWCNT structures are filtered out by a linear fitting force-constant scheme. Raman and IR active modes are reanalyzed by $D_{2nd}$ group symmetry. The frequencies are found to agree excellently with the experimental observations. Our results confirm the reported frequency drop of the $A_{1g}$ mode. Some issues in the low-temperature specific-heat experiment and the neutron-scattering experiment are well explained by our phonon dispersions. Phonon dispersions of graphite are also calculated using the same method, being in good agreement with experiment. Theoretical analysis is also presented to explain the advantages of cumulant force-constant method to get better low-frequency dispersions.

Figure 1

The supercell approach for calculating interatomic force constants. In the central primitive cell, a source atom $\left(0j\right)$ is displaced away from its equilibrium position. Forces induced on the destination atom $\left(mk\right)$ in the $m$th primitive cell are then evaluated by ab initio Hellman-Feynman force method. The ratio of $\delta f∕\delta u$ is the interatomic force constant in the harmonic approximation. Interatomic interactions are usually spherically truncated in the dynamical matrices.

Figure 2

Linearity of the forces on the destination atom over the displacement of the source atom. Forces are drawn for the (10,10) SWCNT. Nonzero residue forces will cause the lines missing the origin but will not affect the slopes of the lines, where the cumulant force constant are extracted.

Figure 3

Phonon dispersions of graphite.

Figure 4

Phonon-dispersion curves for (10,10) nanotube.

Figure 5

Phonon-dispersion curves for (5,5) and (6,6) nanotubes.

Figure 6

Frequencies of Raman active modes and IR active modes for armchair tubes $(n,n)(n=5–12)$.

Figure 7

Phonon-dispersion curves for (10,0) nanotube by the CFC based dynamical matrix method of Eq. (4).

Figure 8

Phonon-dispersion curves for (11,0) and (7,0) tubes.

Figure 9

Frequencies of Raman active modes and IR active modes for zigzag tubes $(n,0)(n=8–12)$. Note the mode softening effect of the $A_{1g}$ mode shown for (9,0) and (12,0) nanotubes.