The phonon dispersion relations for a SW-CNT can be determined by folding that
of a graphene layer (see Section 2.4.2). Since there are carbon atoms
in the unit cell of a CNT, phonon dispersion branches for the
three-dimensional vibrations of atoms are achieved.
The corresponding one-dimensional phonon energy dispersion relation for the CNT
is given by
2.5.2 Phonon Dispersion Relations of SW-CNTs
denotes the polarization,
is the azimuthal quantum number, and
is the wave-vector
of phonons. However, the zone-folding method does not always give the correct
dispersion relation for a CNT, especially in the
low frequency region. For example, the out-of-plane tangential acoustic (ZA)
modes of a graphene sheet do not give zero energy at the when
rolled into a CNT. Here, at , all the carbon atoms of the CNT move
radially in and out-of-plane radial acoustic vibration, which corresponds to a
breathing mode (RBM) with a non-zero frequency . To avoid these
difficulties, one can directly diagonalize the dynamical matrix (see Fig. 2.11-a).
Fundamental phonon polarizations in CNTs are radial (R), transverse (T), and
longitudinal (L). As shown in Fig. 2.11-b, zone center phonons, also
referred to as -point phonons, can belong to the transverse acoustic
(TA), the longitudinal acoustic (LA), the radial breathing mode (RBM), the
out-of-plane optical branch (RO), the transverse optical (TO), or the
longitudinal optical (LO) phonon branch.
The LO phonon branch near the -point has an energy of
, whereas the energy of the RBM phonon branch is inversely
proportional to the CNT diameter
is the diameter of the CNT in nanometer [41,42].
Zone boundary phonons, also referred to as
-point phonons, are found to
be a a mixture of fundamental polarizations .
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors
The phonon dispersion relations of a) a
armchair CNT  and b) a zigzag CNT with
, see (2.5.2) .