Page 118 - Principles and Applications of NanoMEMS Physics

106 Chapter 3

n
qui tessential nanotechnology device. Their properties are related to those of
a 2D perodic graphite sheet, see Figs. 3-13.
y y y

x x x
a a 1 1

a a 2 2
C C

Figure 3-13. Sketch of a graphene lattice, a single sheet of carbon atoms arranged in the
honeycomb structure, showing vectors utilized in describing the lattice. In this case, the vector
C is defined by the pair n=4, m=4, i.e., (4, 4).

The graphene lattice is defined by a vector C of the form C = na + ma ,
1 2
where a and a are the unit cell base vectors of the graphene sheet, Fig. 3-
1 2
13, with a = a = . 0 246 nm . The pair of integers (n, m), where n ≥ m ,
1 2
is used to represent a possible CNT structure [46]. Three types of CNT
structures are typically identified according to how the conceptual graphene
rolling into a cylinder is effected, namely, the armchair, the zigzag, and the
chiral CNT structures, see Fig. 3-14 [143]. The chiral angle, θ , of the
wrapping vectors describing these CNTs are related to the indices n and m
by the equation [46],

θ = sin − 1 3 m (78)
2 n 2 + nm + m 2

with θ = 0 for the Zigzag CNT, θ = 30 for the Armchair CNT, and
D
0 θ < 30 for the Chiral CNT. The corresponding CNT diameter is given
<
D
by,

2
2
d CNT () 0Å = . 783 n + nm + m . (79)