3

Elementary Theory of Electronic Band Structure in Semiconductors

In this chapter, we derive the Hamiltonian using the multiband effective theory. To understand the optical properties of semiconductors, we need to know the electronic band structure. In particular, the band structure calculations of the nanostructures, such as the quantum well, quantum wire, and quantum dot, are of great interest, because these structures have existing and potential applications for many important optoelectronic devices. First, we review several basic theories for band structure calculation, in particular, the effective mass theory. Next, we introduce the band structure calculation for nanostructures. Specially, we focus on the band structure of the quantum well and the crystal orientation effect on electronic properties.

3.1 BLOCH THEOREM AND EFFECTIVE MASS THEORY

3.1.1 Bloch Theorem

A fundamental theorem concerning electrons in a crystal was proved by Bloch in 1928. It states that the wave functions of the electrons in a crystal have the Bloch form

where is a function that has the periodicity of the lattice, that is, where is any primitive lattice translation. The Bloch function, in general, ...