4.6.1 Scattering with Optical Phonons
In this section the selfenergies due to the interaction of electrons with
optical phonons (OP) are evaluated. As discussed in Section 2.6, the
phonon energy and the reduced electronphonon matrix elements for OP
phonons are approximately constant and independent of the phonon wavevector.
Under this assumption all terms except the exponential term
in (4.36) and (4.37) can be taken out of
the integral (4.38) and one obtains [55]

(4.39) 
where is an integer number. One has to multiply the above result by a
factor of , for the number of rings in the lattice period [55].
Equation (4.39) justifies the approximation which only considers
diagonal elements of the electronphonon selfenergy. As discussed
in Section 4.3, by employing the nearest neighbor tightbinding method
(block) tridiagonal matrices are achieved. Keeping only diagonal elements of
the electronphonon selfenergy, the matrices remain (block)
tridiagonal. Therefore, an efficient recursive method (Appendix H) can
be used to calculate the inverse matrices. This implies
considerable reduction of computational cost and memory requirement.
Using the result of (4.39) and the relations (2.15)
and (2.19) the selfenergy due to scattering with optical phonons can be written as

(4.40) 

(4.41) 
where
is given by

(4.42) 
where
(see (4.3)).
In the second line in (4.42) the
mass density of a zigzag CNT has been replaced
, where
is the mass of a
carbon atom.
The retarded selfenergy can be calculated as (3.76)

(4.43) 
where

(4.44) 
Since the lesser and greater selfenergies are assumed to be diagonal the
retarded selfenergy is also diagonal.
M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors