Expectations from Theory

SWNTs can be viewed as an extension in one-dimension (1D) of different fullerene molecular clusters or as a strip cut from an infinite graphene sheet and rolled up to form a tube (Fig. 1a). Major characteristics of their electronic properties can be built up from relatively simply Huckel-type models using p(n) atomic orbitals. The diameter and helicity of a SWNT are uniquely characterized by the roll-up vector Ch = na1 + ma2 = (n, m) that connects crystallographically equivalent sites on a two-dimensional (2D) graphene sheet, where ai and a2 are the graphene lattice vectors and n and m are integers. The limiting, achiral cases, (n, 0) zigzag and (n, n) armchair are indicated with dashed lines in Fig. 1b.

Electronic band structure calculations predict that the (n,m) indices determine whether a SWNT will be a metal or a semiconductor [7,8,9]. To understand this unique ability to exhibit distinct electronic properties within an all-carbon sp2 hybridized network, it is instructive to consider the 2D energy dispersion of graphite. Graphite is a semi-metal or zero-gap semiconductor whose valence and conduction bands touch and are degenerate at six K(kF) points; these six positions define the corners of the first Brilluion zone. As a finite piece of the 2D graphene sheet is rolled up to form a 1D tube, the periodic boundary conditions imposed by Ch can be used to enumerate the allowed 1D subbands-the quantized states resulting from radial confinement-as follows:

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