Fig. 1. Schematic diagrams of a SWNT and roll-up vector. (a) Graphene strip rolled into a seamless tube. (b) 2D graphene sheet illustrating lattice vectors a 1 and a2, and the roll-up vector Ch = na 1 + ma2. The limiting, achiral cases of (n, 0) zigzag and (n, n) armchair are indicated with dashed lines. The translation vector T is along the nanotube axis and defines the 1D unit cell. The shaded, boxed area represents the unrolled unit cell formed by T and Ch. The diagram is constructed for (n, m) = (4, 2)

where q is an integer. If one of these allowed subbands passes through one of the K points, the nanotube will be metallic, and otherwise semiconducting. Thus to first order, zigzag (n, 0) or chiral (n, m) SWNTs are metallic when (n — m)/3 is an integer and otherwise semiconducting.

Independent of helicity, the energy gaps of the semiconducting (n, 0) and (n, m) tubes should depend inversely on diameter. In addition, the finite curvature of the tubes also leads to mixing of the n/a bonding and n*/a* antibonding orbitals in carbon. This mixing should cause the graphene band crossing (kF) to shift away from the K point and should produce small gaps in (n, 0) and (n, m) metallic tubes with the magnitude of the gap depending inversely with the square of the diameter [7,14]. However, (n, n) armchair tubes are expected to be truly metallic since kF remains on the subband of the nanotube [15].

Away from the K point, signature features in the Density of States (DOS) of a material appear at the band edges, and are commonly referred to as van Hove Singularities (VHS). These singularities are characteristic of the dimension of a system. In three dimensions, VHS are kinks due to the increased degeneracy of the available phase space, while in two dimensions the VHS

appear as stepwise discontinuities with increasing energy. Unique to one-dimensional systems, the VHS are manifested as peaks. Hence, SWNTs and other 1D materials are expected to exhibit spikes in the DOS due to the 1D nature of their band structure.

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