4.4 ModeSpace Transformation
A mode space approach
significantly reduces the size of the HAMILTONian
matrix [9]. Due to quantum confinement along the CNT
circumference, circumferential modes appear and transport can be described in terms
of these modes. If modes contribute to transports, and if , then the size of
the problem is reduced from to
, where is number of carbon rings along the CNT. If
the potential profile does not vary sharply along the CNT, subbands are
decoupled [9] and one can solve onedimensional problems of
size .
Mathematically, one performs a basis transformation on the HAMILTONian of the zigzag
CNT to decouple the problem into onedimensional mode space lattices [243]

(4.20) 
with

(4.21) 
where
is the transformation matrix from the real space basis to the
mode space basis. The purpose is to decouple the modes after the
basis transformation, i.e., to make the HAMILTONian matrix blocks between
different modes equal to zero. This requires that after the transformation,
the matrices
,
, and
, become diagonal. Since
and
are identity matrices multiplied by a constant, they
remain unchanged and diagonal after any basis transformation,
and
.
To diagonalize
, elements of the transformation matrix
have to be the eigenvectors of
. These eigenvectors are plane
waves with wavevectors satisfying the periodic boundary condition around the
CNT. The eigenvalues are

(4.22) 
where
[243]. The phase factor in (4.22) has no
effect on the results such as charge and current density, thus it can be
omitted and
can be used instead.
Figure 4.7:
Zigzag
CNT and the corresponding onedimensional chain with two sites per unit
cell with hopping parameters and
.

After the basis transformation all submatrices,
,
, and
are diagonal. By reordering the basis according to
the modes, the HAMILTONian matrix takes the form

(4.23) 
where
is the HAMILTONian matrix for the th mode [243]

(4.24) 
The onedimensional tightbinding HAMILTONian describes a chain of atoms
with two sites per unit cell and onsite potential and hopping parameters
and (Fig. 4.7). The spatial grid used for device simulation
corresponds to the circumferential rings of carbon atoms. Therefore, the rank
of the matrices for each subband are equal to the total number of these rings .
Selfenergies can be also transformed into mode space
, see Section 4.5
and Section 4.6. The GREEN's functions can therefore be defined for
each subband (mode) and one can solve the system of transport equations for each subband
independently

(4.25) 

(4.26) 
M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors