Density of states

The density of states and ( engl. density of states, abbreviated DOS) is a physical quantity that indicates how many states per energy interval and per frequency interval exist.

The following explanations refer to the solid state physics.

In 3-dimensional space, the density is constant and equal to, the length of one grating period.

The density of states can refer to different objects, for example, phonons, electrons, magnons or quasiparticles in superconductors.


: - Based on the volume - density of states for a countable number of energy levels defined by the General is


This is obtained by expanding with ( the smallest allowed change of k for a particle in a box of dimension d and length L) and transition to a Riemann integral ( Limes ), as to the volume density of states for continuous Energiniveaus:


  • The spatial dimension of the system considered
  • The magnitude of the wave vector
  • The delta distribution.

The density of states can be equivalently regarded as a derivation of the microcanonical partition function after the power: See Microcanonical partition.

The number of states with energy ( degeneracy ) is given by.


Clearly you count the allowed energy states for a given energy:

Considering a system with N discrete energy states, so the density will be described as the integral of the density of states is currently providing the number of states with energy:


In the above formula, the property of the delta distribution is at least important for the view, which should be noted here that this property is valid only for a finite number of simple zeros and of, see Delta Distribution.

N- dimensional electron gas

In an n-dimensional electron gas charge carriers can in dimensions 1, ..., n move freely. The appropriate proportion of energy is continuous and can be specified using the parabolic approximation:

It is

  • M * is the effective mass of the charge carrier in the solid state, more precisely, m * is the effective density of states mass
  • The Planck ( divided by ) the quantum of action.

In contrast, the energy component of the other dimensions is discretized in the values ​​.

The ( related to the volume V) density of states can be described in general:

This corresponds to

  • The pre-factor 2 the two possible spin states
  • The volume of the solid
  • N ( E) of the total number of states with energy less than or equal E (see: Microcanonical state total):
  • Describes in the n-dimensional k-space, the total volume of all states, which are accessible with the remaining energy
  • Is the volume of such a condition.
  • Here describes the Heaviside function.

In the Semiconductor

In semiconductor materials is due to the atomic nuclei periodically occurring, a similar approach for the conduction and valence bands made ​​(see band model ). The freely movable carriers in the two bands, ie electrons and holes, is an effective mass, and assigned to the assumed density of states for the two bands as described above as a square. The distance between the extremes of these two densities is called band gap. This is ( momentum space ) spoken at a displacement of the extrema in k-space of an indirect, with the same pulse difference from a direct semiconductor. The electrons and holes are trying to occupy a minimum of the energy in these possible states and strive towards the band edge, so to extremes. So it occurred, as far as possible, actually occupied states increasingly on there.

The energy of the conduction band lower edge is the top edge of the valence band, the difference is equal to the band gap energy. The density of states in the conduction band is ( is the density of states mass of the electron in the conduction band, ie, its average effective mass ):

The density of states in the valence band is ( is the density of states mass of the hole in the valence band ):

In doped semiconductors occur to these possible states or states in the band gap. These are close to n-type doping on the conduction band and close to p-type doping on the valence band. By supplying energy, the activation energy can be overcome and make it multiplies occupied states in conduction and valence band. In addition, changes by doping the position of the Fermi level: it is raised in n-type doping, or lower with p-type doping to the valence band to go on. Wherein an n -type doping are thus already at room temperature due to the thermal energy far more occupied states in the conduction band than that of an undoped material. The additional free charge carriers can thus increase the current transport.

The thermal population of the states is determined by the Fermi distribution. The probability density that a state is occupied with the energy that writes itself

The probability density that a state is occupied not occupied with the energy expressed or equivalent with a hole enrolls

This allows the carrier densities, ie electron density in the conduction band and holes in the valence band density, specify:

As well as

Actually, the limits of integration should not be extended to infinity, but only until the end of the respective band. However, there is the Fermi distribution already approximated zero - the chemical potential is namely in the area of the band gap - so that the error is negligible. To calculate the integrals see Fermi-Dirac integral.