Electronic Band Structure of Carbon Nanotubes in Equilibrium and None-Equilibrium Regimes

Abstract

The exploration of CNTs was a great contribution to the world of science and technology. After its exploration in 1991 by Iijima [1], extensive practical and theoretical researches about its nature gradually began to develop [2-6]. Today, we know about CNTs much more about its chemical, mechanical, optical and electrical properties than before. The methods of fabrication have also progressed. Due to their electrical and optical properties, CNTs are the subject of studies about their usage in new electronic and optoelectronic devices. In this chapter we will focus on their electronic band structure, because it is the most important characteristic of a solid that should be studied to be used in determination of its electronic, optical and optoelectronic properties. In order to investigate the electronic band structure of a solid, it is first necessary to have a good understanding of its crystal lattice and atomic structure. Therefore, as the first step of this chapter we will begin with the investigation of the geometry of SWCNTs. Then we will continue with the calculation of allowed wave vectors for the electronic transport. Having finished this step, we will introduce the electronic band structure of SWCNTs. As is known, single walled carbon nanotube or SWCNT consists of grephene sheet that is rolled into a cylinder over a vector called “chiral vector” (Fig. 1(a)) so that the beginning and the end of this vector join to form the circumstantial circle of the cylinder Fig. 1(b). As is shown in Fig. 1(a) the chiral vector may be written in terms of unit vectors a1 and a2 , therefore C may be written as: