Band theory of single-walled carbon nanotubes

In this paper, a curvilinear coordinate system is used in space and in k-space to study the energy band of single-walled carbon nanotubes wrapped at a helical angle. Using this method, a general function of the bandgap associated with the radius of the tube and the helical angle is derived based on the tight-binding theory. The three-dimensional hexagonal Brillouin zone of the tube is on the surface of cylinder in the k-space. For two tubes with different diameters, there is a distance between the cylindrical Brillouin zones in the radial direction. The Brillouin zone varies with the radius of the tube and the number of cells on the circumference. For the metallic zigzag tubes, the bandgaps decrease discretely to zero at the corners of the Brillouin zones, and those corners are singular points of zero gaps. With the transformation of coordinates, the metallic zigzag type is proven to be equivalent to an armchair configuration. Electrical characteristics of the chiral effects are briefly highlighted.