Fig. 19.29. Plot of the energy bands E(k) for the ID tubule for values of the energy, in dimensionless units E(k)/y0, for the case of the metallic carbon nanotube specified by integers (n, m) = (9,6) for which the largest common divisor is d = 3, and the Fermi level is at EF = 0. The general behavior of the four energy bands intersecting at k = 0 is typical of the case where ti — m / 3rd, and r is an integer [19.101], states is averaged out, and this 2D value has been found to yield good agreement with first principles calculations [19.102],

Another important result pertains to the semiconducting tubules, showing that their energy gap depends upon the reciprocal tubule diameter d„

a, independent of the chiral angle of the semiconducting tubule, where ac_c is the nearest-neighbor C-C distance on a graphene sheet. A plot of Eg vs. 100/d, is shown in Fig. 19.30 for the graphite overlap integral taken as y0 = 3.13 eV. The results in Fig. 19.30 are important for testing the ID model for the electronic structure of carbon nanotubes, because this result allows measurements to be made on many individual semiconducting tubules, which are characterized only with regard to tubule diameter without regard for their chiral angles (see §19.6). Using a value of y0 = 2.5 eV as given by the local density functional calculation [19.102], Eq. (19.33) suggests that the band gap exceeds thermal energy at room temperature for tubule diameters d, < 140 Â. Furthermore, since about one third of the cylinders of a multiwall nanotube are conducting, certain electronic properties of nanotubes are dominated by contributions from these conducting constituents.

Metallic ID energy bands are generally unstable under a Peierls distortion. However, the Peierls energy gap obtained for the metallic tubules is found to be greatly suppressed by increasing the tubule diameter, so that the Peierls gap quickly approaches the zero-energy gap of 2D graphite [19.46,

Fig. 19.30. The energy gap Eg calculated by a tight binding model for a general chiral carbon nanotubule as a function of 100 kjdn where d, is the tube diameter in A. It is noted that the relationship becomes linear for large values ofd, [19.101].

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