In 1998, Wilder et al. [23] conducted research into the fundamental gap of carbon nanotubes.
The study by Wilder et al. showed that nanotubes of type n-m=3l, where l is
zero or any positive integer, were metallic and therefore conducting. The fundamental gap (HOMO-LUMO)
would therefore be 0.0 eV. All other nanotubes, they showed, behaved as a semi-conductor. The
fundamental gap, they showed, was a function of diameter, where the gap was in the order of about
0.5 eV. Their data showed that the energy gap reflected the graph at right (adapted from
[23]. This graph can be modelled by the function:
Egap=2 y0 acc / d
Where y0 is the C-C tight bonding overlap energy (2.7
0.1 eV), acc is the
nearest neighbor C-C distance (0.142 nm), and d is the diameter. This shows that the fundamental gap
ranged from around 0.4 eV - 0.7 eV, which they said was in good agreement with the values obtained
from one-dimentional dispersement relations. They concluded that the fundamental gap of semi-conducting
nanotubes was determined by small variations of the diameter and bonding angle (determined by the twist,
see Equilibrium Structure for more information.)
In a study published at the same time by Odom, Huang, and Lieber [24],
they also agreed that the semiconducting properties of carbon nanotubes were determined by the formula
stated above. In addition, they suggested that a small gap would exist at the Fermi level in metallic
nanotubes. This would be because of the 
/
bonding orbitals and
*/* anti-bonding orbitals mixing due to the curvature in the
graphene sheet of a SWNT. They noted, however, that they had not observed any evidence to support this
as of the time of publishing (1998). More interesting to note, in the study by Wilder et al.,
they reported that the conducting nanotubes shows Egap to range from 1.7 - 2.0 eV, which could be the
evidence Odom et al. was predicting.
At the Fermi Energy (the highest occupied energy level), the density of states is finite for a
metallic tube (though very small), and zero for a semi-conducting tube. As energy is increased,
sharp peaks in the density of states, called Van Hove singularities, appear and specific energy levels.
The optical spectrum is given by:
x 
= 
E
where E is the energy difference between occupied and unoccupied
states, especially near the peaks. The spectrum is dominated by the transitions between
the Van Hove singularites. [13,19]
