Page 120 - Principles and Applications of NanoMEMS Physics

108 Chapter 3

points (the corners of the first Brilloin zone) there is zero gap between
conduction and valence bands in graphene.

y
r r
a 2 a 1

r r
b 2 b 1
x
Reciprocal lattice points
r
1 Brillouin zone K point
st

G
st
Figure 3-15 (a) Reciprocal lattice of graphene with the 1 Brilloin zone (shaded). b and
1
G
b are the primitive lattice vectors. The K point lies at the edge of the BZ. 2D grapheme
2
sheets “rolled” around the y axis, will give rise to armchair CNTs. (b) LCAO bandstructure of
grapheme. The Fermi level lies at E=0. Courtesy of Prof. Christian Schönenberger, University
of Basel, Switzerland].
The effect of rolling the graphene sheet to form the CNT manifests itself
in the band structure as follows. On the one hand, the momentum of
electrons along the circumference of the cylinder becomes quantized. On the
other, propagation is now only possible along the cylinder axis, i.e., in one
dimension, thus the concomitant CNT band structure corresponds to slices of
the 2D graphene structure. When the slice passes through a K-point, the
CNT is metallic since, at these points, the gap is zero; when it doesn’t, it is
semiconducting. In particular, CNT structure type and its electronic
properties are related as follows [46]. For armchair CNTs, the
circumferential momentum vector is quantized according to,