Interdisciplinary Applied Mathematics

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13.2.1 Carbon Nanotubes

Carbon nanotubes were discovered by (Iijima, 1991). Since their discovery, nanotubes have aroused great excitement because of their unique physical properties, which span a wide range, from structural to electronic. For example, nanotubes have a light weight and a record-high elastic modulus, and they are one of the strongest fibers that can be made. There are two main types of carbon nanotubes (Saito et al., 1998; Dekker, 1999; Baughman et al., 2002):

A single-walled carbon nanotube (SWNT), made of a single atomic layer thick of graphite (called graphene) rolled into a seamless cylinder.

FIGURE 13.8. Schematic of a two-dimensional graphene sheet illustrating lattice vectors ai and a2, and the chiral vector ch = nai + ma2.

The diameter of SWNT, dcnt, is given by L/п, where L is the circumferential length of the carbon nanotube given by

L = |Ch| = л/сь • Ch = ил/n? + m2 + nm,

where a = 2.49 A is the lattice constant of the honeycomb lattice. The chiral angle в is defined as the angle between the vectors ch and ai, with values of в in the range 0 < в < 30°, because of the hexagonal symmetry of the honeycomb lattice. The chiral angle в denotes the tilt angle of the hexagons with respect to the direction of the nanotube axis, and the angle в specifies the chiral symmetry. The indices (n, m) can also be used to compute the chiral angle, i. e. ,

2n + m

cos 0 = —    .

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