Energy Band Structures of Semiconductors

Abstract

The physical properties of semiconductors can be understood with the help of the energy band structures. This chapter is devoted to energy band calculations and interpretation of the band structures. Bloch theorem is the starting point for the energy band calculations. Bloch functions in periodic potentials is derived here and a periodic function is shown to be expressed in terms of Fourier expansion by means of reciprocal wave vectors. Brillouin zones are then introduced to understand energy band structures of semiconductors. The basic results obtained here are used throughout the text. Nearly free electron approximation is shown as the simplest example to understand the energy band gap (forbidden gap) of semiconductors and the overall features of the energy band structure. The energy band calculation is carried out first by obtaining free-electron bands (empty lattice bands) which are based on the assumption of vanishing magnitude of crystal potentials and of keeping the crystal periodicity. Next we show that the energy band structures are calculated with a good approximation by the local pseudopotential method with several Fourier components of crystal potential. The nonlocal pseudopotential method, where the nonlocal properties of core electrons are taken into account, is discussed with the spin–orbit interaction. Also \(\varvec{k}\cdot \varvec{p}\) perturbation method for energy band calculation is described in detail. The method is extended to obtain the full band structures of the elementary and compound semiconductors. Another method “tight binding approximation” will be discussed in connection with the energy band calculation of superlattices in Chap. 8.