Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. It is often expressed as a function g(E) of the internal energy E, or a function g(k) of the wavevector k. It is usually used with electronic energy levels in a solid. In 3 dimensions, for example, the density of states in reciprocal space (k-space) is$g(k)\backslash ,dk\; =\backslash frac\{V\}\{2\backslash pi^2\}\backslash ,k^2\backslash ,dk$where V is the volume of the solid.

A more precise definition is as follows: g(E) dE is the number of allowed energy levels per unit volume of the material, within the energy range E to E + dE (and equivalently for k).

Calculation of the density of states

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To calculate the density of energy states of a particle we first calculate the density of states in reciprocal space (momentum- or k-space). The separation between states is fixed by the boundary conditions. For free electrons and photons within a box of size L, and for electrons inside a crystal lattice with lattice constant L, periodic Born-von Karman boundary conditions can be applied. Using the free particle wavefunction of a plane wave this implies

where n is any integer and dk the separation of different k states.

The total number of k-states available to a particle is the volume of k-space accessible to it divided by the volume of a single k-state. The volume accessible is simply the integral from $k\; =\; 0$ to $k\; =\; k$, and the volume of a k-state is $\backslash partial\; k^n$.

In 1D this is $G(k)\; =\; \backslash frac\{\; \backslash int\backslash limits\_0^k\backslash ,\{d\{\backslash mathbf\{k\}\}\}\; \}\{\backslash partial\; k\}$

and in 3D $G(k)\; =\; \backslash frac\{\; \backslash int\backslash limits\_0^k\backslash ,\{d^3\{\backslash mathbf\{k\}\}\}\; \}\{\backslash left(\; \{\backslash partial\; k\}\; \backslash right)^3\}$

These expressions can then be differentiated with respect to k to give the density of states at a given k value: $g(k)\backslash ,dk\; =\; \backslash frac\{dG(k)\}\{dk\}\backslash ,dk$

To find the density of energy states, the relation between energy and momentum for a particular particle is used, to express k and dk in g(k)dk in terms of E and dE. For example for a free electron: $E\; =\; \backslash frac\{p^2\}\{2m\}\; =\; \backslash frac\{(\backslash hbar\; k)^2\}\{2m\}$, $dE\; =\; \backslash frac\{\backslash hbar^2\; k\}\{m\}\backslash ,dk$

This gives a density of states at energy E per unit volume, $g(E)\; =\; \backslash frac\{d\}\{dk\}\backslash left(\{\; \backslash frac\{\; \backslash int\backslash limits\_0^k\backslash ,\{d^3\{\backslash mathbf\{k\}\}\}\; \}\{\backslash left(\; \{\backslash partial\; k\}\; \backslash right)^3\; \}\; \}\backslash right)\backslash frac\{dk\}\{dE\}.$

Density of states of the free electron gas in one dimension

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The use of the term gas here means that it is not allowed for the electrons to interact with one another. We assume that we have $N$ electrons constrained to exist in a circular geometry, with radius $L$. This has the effect of imposing periodic boundary conditions; this is physically acceptable for a system that is very long.

The eigenfunctions are then indexed by the label $n$, which takes on all integer values: $\backslash psi\_n(\backslash theta)=\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}e^\{i\; n\; \backslash theta\}$

This eigenfunction has energy eigenvalue $E\_n\; =\; \backslash frac\{\backslash hbar^2\; n^2\}\{2mr^2\}$

It should be remembered that is a gas of electrons, not just a single one, and one starts to fill the states.

External link

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• Britney Spears' Guide to Semiconductor Physics

See also

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• van Hove singularity

Condensed matter physics | Statistical mechanics

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