Optical Properties of Carbon Nanotubes - Electronic Structure of Carbon Nanotube

Electronic Structure of Carbon Nanotube

A single-wall carbon nanotube can be imagined as graphene sheet rolled at a certain "chiral" angle with respect to a plane perpendicular to the tube's long axis. Consequently, SWCNT can be defined by its diameter and chiral angle. The chiral angle can range from 0 to 30 degrees.

However, more conveniently, a pair of indices (n, m) is used instead. The indices refer to equally long unit vectors at 60° angles to each other across a single 6-member carbon ring. Taking the origin as carbon number 1, the a₁ unit vector may be considered the line drawn from carbon 1 to carbon 3, and the a₂ unit vector is then the line drawn from carbon 1 to carbon 5. (See the upper right corner of the diagram at right.) To visualize a CNT with indices (n, m), draw n a₁ unit vectors across the graphene sheet, then draw m a₂ unit vectors at a 60° angle to the a₁ vectors, then add the vectors together. The line representing the sum of the vectors will define the circumference of the CNT along the plane perpendicular to its long axis, connecting one end to the other. In the diagram at right, C_h is a (4, 2) vector: the sum of 4 unit vectors from the origin directly to the right, then 2 unit vectors at a 60° angle down and to the right.

Tubes having n = m (chiral angle = 30°) are called "armchair" and those with m = 0 (chiral angle = 0°) "zigzag". Those indices uniquely determine whether CNT is a metal, semimetal or semiconductor, as well as its band gap: when |m – n| = 3k (k is integer), the tube is metallic; but if |m – n| = 3k ± 1, the tube is semiconducting. The nanotube diameter d is related to m and n as

In this equation, a = 0.246 nm is the magnitude of either unit vector a₁ or a₂.

The situation in multi-wall CNTs is complicated as their properties are determined by contribution of all individual shells; those shells have different structures, and, because of the synthesis, are usually more defective than SWCNTs. Therefore, optical properties of MWCNTs will not be considered here.