# 2.2 Theoretical Background

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 SW-CNT is conveniently explained in terms of its one-dimensional 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 two-dimensional 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 one-dimensional translation vector . The unit cell of the one-dimensional 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 C-C 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 SW-CNTs [12,22].

Table 2.2: Structural properties for CNTs [12].
Symbol Description Formula
a length of unit vectors
Å,      Å
, unit vectors
, reciprocal lattice vectors
chiral vector
,
length of
diameter
chiral angle
gcd

gcd

translational vector

length of
number of hexagons in the unit-cell

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors