Let us consider an electron whose motion is governed by the potential generated by the ions located at the points of the crystal lattice . The Schroedinger equation is

(5.6) |

with the Hamiltonian given by

Then, this theorem states that the bounded eigenstates have the following form:

(5.7) |

and

(5.8) |

with beloging to . Furthermore, it is possible to prove the existence of an infinite sequence of eigenpairs of solutions

with belonging to the non negative integers set . The function describes the -th energy band of the crystal.

The energy band of crystals can be obtained at the cost of intensive numerical calculations by the quantum theory of solids. However, in order to describe electron and hole transport, for most applications, a simplified description is adopted which is based on simple analytical models. These are the effective mass approximation and the Kane dispersion relation, which are used in

(5.9) |

where is the non-parabolicity parameter.

It is possible to choose other energy band relations, but they are actually not implemented in