4.3 TightBinding HAMILTONian
The general form of the tightbinding HAMILTONian for electrons in a CNT can be written as

(4.13) 
The sum is taken over all rings , along the transport direction, which
is assumed to be the direction of the cylindrical coordinate system, and over all
atomic locations , in a ring. We use a nearestneighbor tightbinding
bond model [243,10]. Each atom in an
coordinated CNT has three nearest neighbors, located
away. The bandstructure consists of orbitals only, with the hopping
parameter
and zero onsite potential.
Furthermore, it is assumed that the electrostatic
potential rigidly shifts the onsite potentials. Such a tightbinding
model is adequate to model transport properties in undeformed CNTs.
In this work we consider zigzag CNTs. However, this method can be readily
extended to armchair or chiral CNTs. Within the nearestneighbor
approximation, only the following parameters are
non zero [10]

(4.14) 
Figure 4.6 shows that a zigzag CNT is composed of rings (layers)
of  and type carbon atoms, where and represent the
two carbon atoms in a unit cell of graphene. Each Atype ring is
adjacent to a Btype ring. Within nearestneighbor tightbinding
approximation the total HAMILTONian matrix is block tridiagonal [243]

(4.15) 
where the diagonal blocks,
, describe the coupling within an
Atype or Btype carbon ring and offdiagonal blocks,
and
, describe the coupling between adjacent rings.
It should be noted that the odd numbered HAMILTONian
refer to Atype rings
and the even numbered one to Btype rings. Each Atype ring couples to the
next Btype ring according to
and to the previous Btype ring
according to
. Each Btype ring couples to the next Atype ring
according to
and to the previous Atype ring according to
.
In a zigzag CNT, there are carbon atoms in each ring, thus, all the
submatrices in (4.15) have a size of .
In the nearestneighbor tight binding approximation, carbon atoms within a ring
are not coupled to each other so that
is a diagonal matrix. The
value of a diagonal entry is the potential at that carbon atom site. In the
case of a coaxially gated CNT, the potential is constant along the CNT
circumference. As a result, the submatrices
are given by the
potential at the respective carbon ring times the identity matrix

(4.16) 
Figure 4.6:
Layer layout of a
zigzag CNT. Circles are rings of Atype carbon atoms and squares
rings of Btype carbon atoms. The coupling coefficient
between nearest neighbor carbon atoms is . The coupling matrices
between rings are denoted by
and
, where
is a diagonal matrix and
is nondiagonal.

There are two types of coupling matrices between nearest carbon rings,
and
. As shown in Fig. 4.6, the first type,
, only couples an A(B) carbon atom to its B(A) counterpart in the
neighboring ring. The coupling matrix is just the tightbinding coupling
parameter times an identity matrix,

(4.17) 
The second type of coupling matrix,
, couples an A(B) atom to two
B(A) neighbors in the adjacent ring. The coupling matrix is

(4.18) 
The period of the zigzag CNT in the longitudinal direction contains four rings,
, and has a length of
. Therefore, the average distance
between the rings is

(4.19) 
M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors