The basic description of a semiconductor is its bandstructure, i.e. the variation of energy E with wave-vector k. The most important bands are:

Valence band: Last band at T=0

Conduction band: First unfilled band at T=0

In all technically important semiconductors, the valence band maximum is at k=0, known as the gamma point. In semiconductors where the conduction-band minimum occurs at k=0, the semiconductor is said to be a direct band semiconductor. At non-zero k=0, the semiconductor is an indirect-band semiconductor. In addition to these two main conduction bands other bands may also be present. In III-V semiconductors, Ge and Si there are 3 valence bands with maxima at k=0. These are the light-hole, heavy-hole and spin-orbit split off band.

The bands in a semiconductor material are approximated crudely by parabolic functions of k.

Conduction-band: (1)

Valence-band: (2)

The expression of the effective mass is found from the dynamics of a wave-packet, which represents a localised particle. The wave packet is a modulation envelope, with the carrier wave running through it. The packet is made up of a small spread of frequencies w around a central value w0; these are superimposed on one another. The wave packet moves at the group velocity vg

(3)

If an electric field Ef is applied, so that the wave packet moves a distance dx in time dt. The change in energy of the wave packet is

(4)

This change corresponds to a change dk of the central k value k0; it is given by

(5)

We convert this to a time derivative:

(6)

But , so (7)

The equation for the acceleration, can be calculated from

(8)

Substituting for from our first major result,

(9)

Comparing these forms we see ,

(10)

The dynamics of holes is more complicated. It is necessary to consider one unfilled state in the otherwise filled valence band. The result is that the hole mass acts like a particle with positive charge +e and mass m_h given by

(11)

Usually, only values of k near the bottom of the conduction band minimum and the valence band maximum are of interest; In this case , the is useful to draw the bandstructure energy as a function of position at k=0. In this representation of the energy bands, the donors and acceptors form levels in the energy gap region. At T=0K, any free carriers from donors and acceptors are bound to their atoms. So there is no conduction. For non-zero temperatures, the sites can be thermally ionised, releasing carriers in the bands so conduction can occur. These shallow level impurities are known as hydrogenic impurities. For donors an electron orbits a lattice site while for acceptors a hole orbits around a lattice site with residual negative charge. The energy required to ionise these carrier is much less than the binding energy of the hydrogen atom since the effective mass is smaller and the radius of the carrier orbit larger than that of the hydrogen atom.

(12)

A rough estimate for the temperature of ionisation is at room temperature. Initially when the temperature is low , excitation from donors and acceptors can be the only source of carriers: in this range the conductivity is extrinsic. In this regime, the doping of the semiconductor determine whether the semiconductor is n-type or p-type. At higher enough temperatures, direct thermal excitation from the valence band to the conduction band swaps the extrinsic density. There is then an equal number of electrons and holes; the conductivity is intrinsic with distinction between n and p.