Silicon Photonics: Fundamentals and Devices

2.2 Band Structure

2.2.1 E–k Diagram: General Considerations

Semiconductors comprising only Group IV elements like Si or Ge have a diamond crystal structure. The cubic cell consists of two interpenetrating face-centered cubic (fcc) lattices displaced with respect to each other along the (111) direction by one quarter of the body diagonal of the elemental cube. A primitive unit cell has two Si atoms, and therefore there is a center-of-inversion midway between the two nearest neighbors.

On the other hand, semiconductors made of Group III and V elements, like InSb or GaAs, have a zinc-blende crystal structure, which is derived from the diamond structure by making the two interpenetrating fcc lattices different. In GaAs, for example, one fcc lattice contains the Ga atom, while the other fcc lattice contains the As atom. Since the two nearest neighbors are different, there is no inversion symmetry. This difference from Si as well as the different bonding mechanisms is responsible for the changes in band structure of the two materials.

To understand the transport and optical processes in a semiconductor, it is sufficient to know the behavior of outermost valence electrons in a crystal. A perfect crystal consists of many valence electrons and ions. The many-electron problem is reduced to a single-electron problem by using various levels of approximation, and finally the one particle Schrödinger equation is solved. The wave function of an electron is written as

(2.1)

where Ψ is the wave ...