An Alternative Approach to the Problem of CNT Electron Energy Band Structure

Abstract

We intended to discuss in this chapter, TBM (tight binding method), APW (augmentedplane-wave), OPW (orthogonalized -plane-wave) methods and corresponding theoretical concepts. In particulars, we pay a great attention to the theory of CNT (Carbon Nano Tube), but discuss in less details some conventional band structure models, unless nearly electron approximation (NFA), TBM, APW and OPW models have been used for determining the electron energy band structure of solids. In fact, this chapter is partly based on the many – electron description of nano transistor – CNTFET (carbon nano tube field effect transistor), which was done with a number of MSC and PhD students for a number of years at university of Mazandaran in Iran (See our published papers [1-7] for more details). We hope this chapter can complete the present book and be of interest for researchers whom work in the nano technology and for beginners. Some part of the material may be used in lection course for students. There are actually two different approaches for studying the band spectrum of CNT. In the first view, some researchers believe that carbon atoms are as isolated atoms and consider the CNT potential of neighbor's atoms as a perturbation and neglect the intra atomic potential. The second approach is about the density functional theory (DFT), in that the exact exchange energy (EXX) instead of the exchange energy given by the local – density approximation (LDA). The EXX energy, which corresponds to the Fock term in the HartreeFock scheme, is treated as a function of electron densities via the eigenfunctions of the Kohn-Sham KS equations [8]. This approach cannot satisfy the electron behavior in CNT due to its self-interaction-free in its construction. Indeed, this chapter discusses about electronic band energy. It is an energy interval in which electronic states exist in the CNT. This energy structure has been usually obtained by solving the Schrodinger equation for electrons in the CNT. As usual, the electronic wave functions depend on both the wave vector and the spatial coordinates. The eigenvalues and eigenvectors have been determined by Fourier – transforming the differential equation into an algebraic equation. The solution of this equation can be used for some special cases with some reasonable approximation, such as NFE and TB methods. However, these approaches cannot be used for samples with critical dimensions of less than 100 nm due to overlap integrals in nano scale samples. The reason is that carbon atoms are not in fact stationary, but continually undergo vibrations (like thermal vibrations of ions in a crystal) about their positions, in where, the