Since there are two carbon atoms, 
and 
, in the unit cell of graphene, one must
consider 6 coordinates. The secular equation to be solved is thus a
dynamical matrix of rank 
, such that 6 phonon branches are
achieved.
The phonon dispersion relation of the graphene comprises three acoustic (A)
branches and three optical (O) branches. The modes are associated with
out-of-plane (Z), in-plane longitudinal (L), and in-plane transverse (T) atomic motions (Fig. 2.10-a).
Figure 2.10-b shows the phonon dispersion branches of graphene. The
three phonon dispersion branches, which originate from the 
-point of
the BRILLOUIN zone correspond to acoustic modes: an out-of plane mode (ZA), an
in-plane transverse mode (TA), and in-plane longitudinal (LA), listed in order
of increasing energy. The remaining three branches correspond to
optical modes: one out-of plane mode (ZO), and two in-plane modes (TO) and
(LO).
While the TA and LA modes display the normal linear dispersion around the
-point, the ZA mode shows a 
energy dispersion which is explained
in [12] as a consequence of the
point-group
symmetry of graphene. Another consequence of the symmetry are the linear
crossings of the ZA/ZO and the LA/LO modes
at the
-point.
Figure 2.10:
a) Atomic motions of carbon atoms in graphence
can be along the out-of-plane (Z), in-plane transverse (T), and in-plane longitudinal (L) direction.
b) The phonon dispersion branches for a graphene sheet, plotted along
high symmetry directions [12].
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M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors
![\includegraphics[width=\textwidth]{figures/FC-Eq.eps}](img304.png)