The structure of CNTs has been explored early by high resolution
transmission electron microscopy techniques yielding direct confirmation that
the CNTs are seamless cylinders derived from the honeycomb lattice representing
a single atomic layer of crystalline graphite, called a graphene sheet.
The structure of a SWCNT is conveniently explained in
terms of its onedimensional unit cell, defined by the vectors and
as shown in Fig. 2.2.
The circumference of any CNT is expressed in terms of the chiral vector
which connects two crystallographically equivalent
sites on a twodimensional graphene sheet [16]. The construction
in Fig. 2.2 depends uniquely on the pair of integers which
specify the chiral vector. The chiral angle is defined as the angle
between the chiral vector and the zigzag direction
(). Three distinct types of CNT structures
can be generated by rolling up the graphene sheet into a cylinder as
shown in Fig. 2.3. The zigzag and armchair
CNTs correspond to chiral angles of and
°, respectively, and chiral CNTs correspond to
°. The intersection of the vector
(which is normal to ) with the first lattice
point determines the fundamental onedimensional translation vector . The unit cell of the onedimensional lattice is the rectangle defined by
the vectors and .
The cylinder connecting the two hemispherical caps of the CNT
(see Fig. 2.3) is formed by superimposing the two ends of the vector
and the cylinder joint is made along the two lines
and
in Fig. 2.2. The lines
and
are both perpendicular to the
vector at each end of [16]. In the
notation for
, the vectors or
denote zigzag CNTs, whereas the vectors correspond to chiral
CNTs [21]. The CNT diameter
is given by

(2.1) 
where
is the length of and
is the CC
bond length (1.42 Å). The chiral angle is given by
. For the armchair CNT
° and for the zigzag CNT °.
From Fig. 2.2 it follows
that if one limits to the range
°, then
by symmetry, for a zigzag CNT. Both armchair and zigzag CNTs have a
mirror plane and thus are considered achiral. Differences in the CNT
diameter
and chiral angle give rise to different
properties of the various CNTs. The number of hexagons per unit cell of
a CNT, specified by integers , is given by

(2.2) 
where
if is not a multiple of , and
if is a multiple of , and is defined as the greatest common divisor
(gcd) of . Each hexagon in the honeycomb lattice contains two carbon atoms.
The unit cell area of the CNT is times larger
than that for a graphene layer and consequently the unit cell area for the CNT
in reciprocal space is correspondingly times smaller. Table 2.2
provides a summary of relations useful for describing the structure of
SWCNTs [12,22].
Figure 2.2:
The chiral vector
is defined on the honeycomb lattice of carbon atoms by unit
vectors and and the chiral angle with
respect to the zigzag axis (). The diagram is constructed for
.

Figure 2.3:
Schematic models of SWCNTs with the CNT axis
normal to the chiral vector. The latter is along (a) the
° direction for an armchair CNT, (b) the
direction for a zigzag CNT, and (c) a general
direction with
° for a chiral CNT.

Table 2.2:
Structural properties for CNTs [12].

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