Page 121 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 109

ν 2π
ν , (81)
k x =
N x 3 a
for ν = 1 ,..., N , where N is the number of unit cells spanning the
x x
circumference. Thus, it can be shown that an armchair CNT rolled such that
its circumference lies along k and the transport longitudinal axis is along
x
k , would have longitudinal 1D band structures at each of the discrete
y
values of k given by (81), see Figure 3-18. Similarly, a zigzag CNT has its
x
circumferential momentum vector quantized according to,

ν ν 2 π
k = , (82)
y
N a
y
for ν = 1 ,..., N . In this case, the resulting CNT may be either metallic or
y
semiconducting. Metallic, when its index n is divisible by three, in which
case a slice passes through a K-point and the tube behaves as a 1D metal

with Fermi velocity v = 8× 10 5 m s / [144], and otherwise,
F
semiconducting. In the context o ballistic CNTs, their conductance is given
f
by Landauer’s formula, G = (Ne 2 h )T , where N, the number of one-
dimensional channels is four, due to electron spin degeneracy and the two
bands at K- and K’-points, see Fig. 3-17(a). This works out to
G = ( e4 2 h )= /1 k 5 . 6 Ω , assuming T=1. The energy gap of semiconducting
CNTs is related to their diameter by [144], [145],

4= v 9 . 0 eV
E = F ≈ . (83)
GAP [ ]
3 d d nm
CNT CNT
In the general case of a chiral CNT, Dresselhaus et al. [146], [147] have
shown that a metallic CNT is obtained whenever,
n − m = 3q , (84)

where q is an integer. In summary, the current knowledge of electronic-
structural properties of SWNTs is as follows [46]: all armchair tubes are
expected to be metallic, one-third of zigzag and chiral tubes are expected to
be metallic, and the rest are expected to be semiconducting [46].