the Fermi level for the (9,0) tubules and a vanishing density of states for the (10,0) tubules, consistent with the metallic nature of the (9,0) tubules and the semiconducting nature of the (10,0) zigzag tubules. Referring to Fig. 19.27, a singularity in the density of states appears as each energy band contributes to the density of states as a function of energy. An energy dependence of (E — E()) 'l/1 is associated with each ID singularity in the density of states shown in Fig. 19.27. Also shown in the figure (dashed curves) is the density of states for the 2D valence and conduction bands before zone folding. It is interesting to note that the k = 0 energy gap for the zigzag (n, 0) tubules decreases with increasing n, as shown in Fig. 19.27(c), where it is noted that the tubules for which n = 3q' (q' being an integer) are metallic and have no energy gap.
19.5.2. Single-Wall Nonsymmorphic Chiral Tubules
For the case of a chiral tubule, the generalization of the boundary conditions given in Eqs. (19.24) and (19.27) becomes [19.6]:
in which Nya0 corresponds to the translation r, and */3NxaQ corresponds to the rotation i/» in the space group symmetry operation R = (i//|t). The chiral vector Ch in Eq. (19.29) is defined by Eq. (19.1), and q is an integer for specifying inequivalent energy bands. Solutions for the ID energy band structure for chiral tubules yield [19.6]
where —it < ka0 < it, the ID wave vector k = ky, while kx in Eq. (19.23) is determined by Eq. (19.29).
The calculated results for the ID electronic structure show that for small diameter graphene tubules, about one third of the tubules are metallic and two thirds are semiconducting, depending on the tubule diameter d, and chiral angle 0. Metallic conduction in a general fullerene tubule is achieved when
where n and m are pairs of integers (n, m) specifying the tubule diameter and chiral angle through Eqs. (19.2) and (19.3) and q is an integer. Tubules satisfying Eq. (19.31) are indicated in Fig. 19.2(b) as dots surrounded by large circles and these are the metallic tubules. The smaller dark circles in this figure correspond to semiconducting tubules. Figure 19.2(b) shows that all armchair tubules are metallic, but only one third of the possible zigzag and chiral tubules are metallic, as discussed above [19.6]).
A more detailed analysis of the E(k) relations for the chiral nanotubes shows that they can be classified into three general categories [19.87,101], First, if Eq. (19.7.1) is not satisfied, or n - m ^ 3q, then the tubule is semiconducting with an energy band gap. This is the situation for approximately two thirds of the tubules [see Fig. 19.2(b)] and will be discussed further below. For those tubules where n - m = 3q and where metallic conduction is thus expected, two further cases follow. If, in addition, n-m — 3rd is satisfied where d is the largest common divisor of n and m and r is an integer, then band degeneracies between the valence and conduction bands at the Fermi level develop at k = ±(2tt/3T) where T is the length of the lattice vector of the ID unit cell. The tight binding energy band structure for this case is illustrated in Fig. 19.28 for the (7,4) tubule for which d = 1 and r — 1 [19.87]. Armchair tubules denoted by (n, n) also fall into the category exemplified in Fig. 19.28. The third category satisfies both n — m = 3q, which is the condition for metallic tubules, and n — m ^ 3rd. In this third case, the band degeneracy at the Fermi level occurs at k — 0 and involves a fourfold band degeneracy. This case is illustrated in Fig. 19.29 for a (8,5) tubule [19.87]. From Eq. (19.31), metallic zigzag tubules (0,3q) are in this category.
For all metallic tubules, independent of their diameter and chirality, it follows that the density of states per unit length along the tubule axis is a constant given by where a0 is the lattice constant of the graphene layer and ya is the nearest-neighbor C-C tight binding overlap energy [19.87]. While the value of y0 for 3D graphite is 3.13 eV, a value of 2.5 eV is obtained for this overlap energy in the 2D case when the asymmetry in the bonding and antibonding
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