Electronic Structure and Density of States of SWNTs

We now introduce the basic definitions of the carbon nanotube structure and of the calculated electronic and phonon energy bands with their special Density of States (DOS). The structure of a SWNT is specified by the chiral vector Ch

where a1 and a2 are unit vectors of the hexagonal lattice shown in Fig. 1. The vector Ch connects two crystallographically equivalent sites O and A on a two-dimensional (2D) graphene sheet, where a carbon atom is located at each vertex of the honeycomb structure . When we join the line AB' to the parallel line OB in Fig. 1, we get a seamlessly joined SWNT classified by the integers (n, m), since the parallel lines AB' and OB cross the honeycomb lattice at equivalent points. There are only two kinds of SWNTs which have mirror symmetry: zigzag nanotubes (n, 0), and armchair nanotubes (n, n). The other nanotubes are called chiral nanotubes, and they have axial chiral symmetry. The general chiral nanotube has chemical bonds that are not Fig. 1. The unrolled honeycomb lattice of a nanotube. When we connect sites O and A, and sites B and B', a nanotube can be constructed. OA and OB define the chiral vector Ch and the transla-tional vector T of the nanotube, respectively. The rectangle OAB'B defines the unit cell for the nanotube. The figure is constructed for an (n,m) = (4, 2) nanotube 

parallel to the nanotube axis, denoted by the chiral angle 9 in Fig. 1. Here the direction of the nanotube axis corresponds to OB in Fig. 1. The zigzag, armchair and chiral nanotubes correspond, respectively, to 9 = 0°, 9 = 30°, and 0 <|9|< 30°. In a zigzag or an armchair nanotube, respectively, one of three chemical bonds from a carbon atom is parallel or perpendicular to the nanotube axis.

The diameter of a (n, m) nanotube dt is given by dt = Ch/n = VZac-ci'm2 + mn + n2)1/2/V (2)

where aC-c is the nearest-neighbor C-C distance (1.42 A in graphite), and Ch is the length of the chiral vector Ch. The chiral angle 9 is given by d = tan"1 [V3m/{m + 2 n)}. (3)

The 1D electronic DOS is given by the energy dispersion of carbon nano-tubes which is obtained by zone folding of the 2D energy dispersion relations of graphite. Hereafter we only consider the valence n and the conduction n* energy bands of graphite and nanotubes. The 2D energy dispersion relations of graphite are calculated  by solving the eigenvalue problem for a (2 x 2) Hamiltonian H and a (2 x 2) overlap integral matrix S, associated with the two inequivalent carbon atoms in 2D graphite,

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