Comparative study of the
phonon dispersion of single-walled carbon nanotubes (SWNTs)
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Fig. 1. Phonon dispersion of three SWNTs calculated within a valence force field model[1]. |
The phonon dispersion of any SWNT can be calculated within a symmetry-adapted model of the lattice dynamics based either on the use of force-constants or on the use of the linear-response approximation in the tight-binding or the ab-initio approaches. As noted before, the phonon dispersion of a SWNT consists of 6N branches, a large number of which are doubly degenerate. The number of carbon pairs N in the unit cell of the nanotube can be as large as several thousands for the experimentally synthesized SWNTs. Therefore, for better illustration, we chose to show the phonon dispersion of three SWNTs with small N. The phonon dispersion, calculated within a valence force field model[1], is presented in Fig. 1. The numerous branches can be put in three categories with respect to the character of the atomic displacements: 1) highest-frequency modes with tangential bond-stretching character, 2) medium-frequency modes with out-of-plane displacement, and 3) low-lying optical branches and four acoustic branches originating from folded acoustic branches of graphene with in-phase movement of the adjacent atoms. The zone-center phonons of practical importance will be discussed on a separate page. It is important to note that the slope of the longitudinal and the twist acoustic branches can be associated with the in-plane Young's and shear moduli of the tube. The values of these two moduli are similar to in-plane ones of graphene with softening for narrow tubes[1]. The phonon dispersion, calculated within a non-orthogonal tight-binding model in the linear-response approximation[2], is shown in Fig. 2. The phonon frequencies generally overestimate the experimental and ab-initio values by about 12% but, apart from this, they have the same general appearance as the force-constant ones in Fig. 1. Metallic tubes show additionally Kohn anomaly at the zone center and in a point inside the Brillouin zone, arising from strong scattering of electrons at the Fermi surface by phonons. For the truly metallic tubes (armchair tubes), the anomaly can lead to Peierls instability of the tube at low temperature. References: 1. V. N. Popov, V. Van Doren, and M. Balkanski, Phys. Rev. B 61 (2000) 3078. 2. V. N.
Popov and Ph. Lambin, Phys. Rev. B 73
(2006) 085407. |
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Fig. 2. Phonon dispersion of the same three SWNTs as in Fig. 1, calculated within a non-orthogonal tight-binding model[2]. The phonon frequency are higher than those in Fig. 1 roughly by 12%. However, they agree fairly well with the experimental and ab-initio ones when downshifted by a factor of 0.9. |
Valentin Popov
February 22, 2006