By considering the electrons in a solid as a free electron gas, that is, the electrons are free to wander around the crystal without being influenced by the potential of the atomic nuclei, we can obtain a relationship for the number of available states in a solid. A free electron has a velocity v and a momentum p=mv. Its energy consists entirely of kinetic energy (V=0) therefore,

De Broglie hypothesised that if waves could exhibit a particle-like properties, then might particles also exhibit wave-like properties. This idea is expressed as particle wave duality and allows us to give the electron a wave number k.

In this way the electron can be represented by a vector in velocity, momentum or k-space. If we choose to represent the electron state as a vector it points in a direction given by the components magnitude of the basis vectors in k-space. It should be apparent that vectors of the same magnitude have the same energy forming spherical shells. This can be understood better, if we consider the equation for the energy of the electron in terms of k.

Classically, all values of energy would be allowed and there would be no restriction on the number of electrons with the same value of k. However, at atomic scales, quantum mechanical effects dominate and two further famous principles come into play. These are the Heisenburg uncertainty principle and the Pauli exclusion principle. Together these two rules mean that the wavefunction for the electron must satisfy the Schrödinger equation, subject to boundary conditions. The solution of the Schrödinger equation leads to wavefunctions of the form:

This is only valid if and only if

where n_x, n_y, n_z are integers.

With this restriction in k-space, only certain values of k-space lead to acceptable electron wave-functions. k-space would be filled if each position was filled with a cubic unit cell of volume.

The diagram shows 1/8 of the total spherical shells. The problem of finding the number of allowed states between k and k+dk amounts to finding the number of these allowed states between spheres of radius k and k+dk. The volume between the two shells is given by:

Therefore the number of states is given simple by dividing this volume by the volume of a single energy state.

The density of states is usually given as the number of states per unit volume. In factor the number of states is increased by a factor of two because we take into account the Pauli exclusion principle, which allows the an electron spin states of ± 1/2 to share the same k.

Finally we can use the above relationship and the equation linking energy and wave-vector to eliminate k and obtain a relationship for the density of states per unit volume, in terms of the energy of the electron. The density of states per unit energy, per unit volume is finally:

Using our knowledge of the bandstructure of the semiconductor we can define a density of state functions for the conduction and valence band. For the conduction band is given by

Similarly for the valence band

Where we have taken the conventional zero of energy at the top of the valence band.

The density of states for a 3-dimensional semiconductor has the following square root dependence. What happen to the density of states in lower dimensional materials? When the electrons are confined in one direction, as in a quantum, within the well the energy of the electrons is restricted in one of the three dimensions, making the material behave like a 2-dimensional crystal. In this situation, the allowed values of k are points in a plane and equal energies describe circles within the plane.

A similar argument, finding the number of allowed points within an annulus of thickness dk leads to

In this case the density of states remains constant with energy. However, in actual semiconductors there is more than one conduction or valence band. This leads to a staircase like density of states.

In one-dimension, the allowed k states become points on a straight line. The density of states is thus the number of allowed states from k to k+dk in both positive and negative directions. Therefore