Semiconductor Bandstructure

Abstract

In order to understand the electronic and related optical properties of semiconductors one needs to analyze the details of their bandstructure. We will give a short review of some basic approaches to theoretically treat the bandstructure starting, e.g., from free electrons or from atomic orbitals, including the basic concepts of $\varvec{k}\cdot \varvec{p}$ perturbation theory. With this understanding of the general properties of bandstructures we then take a look at some prominent substances. We will derive the detailed properties of the highest valence bands and lowest conduction band in GaAs, Si and other cubic semiconductors with tetrahedral coordination of the atoms/ions. This is followed by a discussion of wurtzite-type semiconductors like ZnO and GaN. We will close this chapter with some properties of further semiconductors, which have important applications in devices like narrow-gap semiconductors, oxides and new absorber materials for thin-film solar cell.

Problems

15.1

Find out about the electron configuration of valence electrons in the atoms Si, Ge, Ga, In, Al, Zn, Cd, Cu, As, Se, S, O, N, Cl ... and determine the orbitals forming the $\Gamma$-point valence and conduction bands in the related semiconductors (Si, Ge, GaAs, GaN, ZnSe, Cu$_2$O, ...).

15.2

Make a sketch of surfaces of constant energy in a two- (or even three-) dimensional $\varvec{k}$-space for spherical, simple cubic, and hexagonal (plane $\varvec{k}\bot \varvec{c}$) symmetries. Is spherical symmetry compatible with cubic and hexagonal symmetry?

15.3

Show that the wave vectors for which the scattered waves interfere constructively in one- and two-dimensional square lattices in the concept of nearly free electrons are just the borders of the Brillouin zones.

15.4

Verify that (15.2a) fulfils the Bloch criterion, i.e., that the wave function is transformed onto itself under a translation $\varvec{R}_{i}$ apart from a phase factor.