2.6.1 ElectronPhonon Matrix Elements
An important case is the intrasubband scattering of electrons, ,
therefore, (Fig. 2.11b) and can be any of six different
phonon polarizations. One can omit the index and write the phonon
frequency as
and the reduced electronphonon matrix
element for a given band as
, where the
weak dependence on is neglected.
For intravalley processes, most of the phonons have
and are
referred to as point phonons. Near the point a linear
dispersion relation for acoustic phonons is assumed,

(2.16) 
where
is the acoustic phonon velocity. For
optical phonons the energy is assumed to be independent of the phonon
wavevector

(2.17) 
Near the
point the reduced electronphonon matrix elements can be approximated by

(2.18) 
for acoustic phonons and by

(2.19) 
for optical phonons [56].
Phonons inducing intervalley processes have a wavevector of
, where
is a wavenumber corresponding to the
point of the Brillouin
zone of graphite. For such phonons one can neglect the qdependence,
and
[56].
To calculate the electronphonon matrix elements one can employ the orthogonal
tightbinding [57], the nonorthogonal tightbinding [56], and
density functional theory [58] for the bandstructure and a
force constant model for the lattice dynamics [59,12].
Electronphonon matrix elements depend on the chirality and the diameter of the
CNT [57,56,58]. Figure 2.12 shows the
reduced matrix elements for intrasubband intravalley transitions in
semiconducting zigzag and chiral CNTs as a function of the CNT
radius [56].
Figure 2.12:
The calculated intravalley intrasubband
reduced electronphonon matrix elements of acoustic (in eV) and
optical phonons (in eV/Å) for zigzag (open circles) and chiral
(closed) CNTs with the radius range from 3.5 Å to 12 Å. The results for
CNTs with residuals 1 and 2 of the division by 3 are shown in the left
and right figures, respectively. Open and closed circles denote the
results for zigzag (Z) and chiral (C) CNTs, respectively. All
results are for the lowest conduction band [56].

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