Figure 4.8 shows carbon rings of A or B-type coupled to a semi-infinite CNT acting as
a contact. Each circle (rectangle) represents a carbon ring consisting of
A or B-type carbon atoms. The carbon ring couples to the nearest ring, with a
coupling matrix of
or
, and
is the surface GREEN's function for the
th ring in the left extension, ordered from the channel-contact interface.
The recursive relation (G.22) can be applied to the CNT in Fig. 4.8 and gives
(4.31)
where
(see Appendix G.1), and
and
are given
by (4.17) and (4.18), respectively. Since the potential is invariant inside the
contact,
. Furthermore,
due to the
periodicity of the CNT lattice. Using these relations, (4.31) represent
two coupled matrix equations with two unknowns,
and
,
which can be solved by iteration. However, in mode-space representation matrices
and
are replaced by the numbers and , respectively.
As a result, the surface GREEN's function for each mode can be calculated
analytically by solving a quadratic equation
(4.32)
The self-energy of the left contact for the th mode is therefore
given by
(4.33)
A similar relation holds for the right contact self-energy.
Figure 4.8:
Computing the
surface GREEN's function for the left contact. The surface GREEN's function
for the th ring inside the contact is .