2.5.2 Phonon Dispersion Relations of SWCNTs
The phonon dispersion relations for a SWCNT 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
threedimensional vibrations of atoms are achieved.
The corresponding onedimensional phonon energy dispersion relation for the CNT
is given by

(2.12) 
where
denotes the polarization,
is the azimuthal quantum number, and
is the wavevector
of phonons. However, the zonefolding method does not always give the correct
dispersion relation for a CNT, especially in the
low frequency region. For example, the outofplane 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 outofplane radial acoustic vibration, which corresponds to a
breathing mode (RBM) with a nonzero frequency [37]. To avoid these
difficulties, one can directly diagonalize the dynamical matrix (see Fig. 2.11a).
Fundamental phonon polarizations in CNTs are radial (R), transverse (T), and
longitudinal (L). As shown in Fig. 2.11b, 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
outofplane 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

(2.13) 
where
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 [55].
Figure 2.11:
The phonon dispersion relations of a) a
armchair CNT [12] and b) a zigzag CNT with
, see (2.5.2) [55].

M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors