Categories: Physics, Statistics.

# Density of states

The density of states $$g(E)$$ of a physical system is defined such that $$g(E) \dd{E}$$ is the number of states which could be occupied with an energy in the interval $$[E, E + \dd{E}]$$. In fact, $$E$$ need not be an energy; it should just be something that effectively identifies the state.

In its simplest form, the density of states is as follows, where $$\Gamma(E)$$ is the number of states with energy less than or equal to the argument $$E$$:

\begin{aligned} g(E) = \dv{\Gamma}{E} \end{aligned}

If the states can be treated as waves, which is often the case, then we can calculate the density of states $$g(k)$$ in $$k$$-space, i.e. as a function of the wavenumber $$k = |\vb{k}|$$. Once we have $$g(k)$$, we use the dispersion relation $$E(k)$$ to find $$g(E)$$, by demanding that:

\begin{aligned} g(k) \dd{k} = g(E) \dd{E} \quad \implies \quad g(E) = g(k) \dv{k}{E} \end{aligned}

Inverting the dispersion relation $$E(k)$$ to get $$k(E)$$ might be difficult, in which case the left-hand equation can be satisfied numerically.

Define $$\Omega_n(k)$$ as the number of states with a $$k$$-value less than or equal to the argument, or in other words, the volume of a hypersphere with radius $$k$$. Then the $$n$$-dimensional density of states $$g_n(k)$$ has the following general form:

\begin{aligned} \boxed{ g_n(k) = \frac{D}{2^n k_{\mathrm{min}}^n} \: \dv{\Omega_n}{k} } \end{aligned}

Where $$D$$ is each state’s degeneracy (e.g. due to spin), and $$k_{\mathrm{min}}$$ is the smallest allowed $$k$$-value, according to the characteristic length $$L$$ of the system. We divide by $$2^n$$ to limit ourselves to the sector where all axes are positive, because we are only considering the magnitude of $$k$$.

In one dimension $$n = 1$$, the number of states within a distance $$k$$ from the origin is the distance from $$k$$ to $$-k$$ (we let it run negative, since its meaning does not matter here), given by:

\begin{aligned} \Omega_1(k) = 2 k \end{aligned}

To get $$k_{\mathrm{min}}$$, we choose to look at a rod of length $$L$$, across which the function is a standing wave, meaning that the allowed values of $$k$$ must be as follows, where $$m \in \mathbb{N}$$:

\begin{aligned} \lambda = \frac{2 L}{m} \quad \implies \quad k = \frac{2 \pi}{\lambda} = \frac{m \pi}{L} \end{aligned}

Take the smallest option $$m = 1$$, such that $$k_{\mathrm{min}} = \pi / L$$, the 1D density of states $$g_1(k)$$ is:

\begin{aligned} \boxed{ g_1(k) = \frac{D L}{2 \pi} \: 2 = \frac{D L}{\pi} } \end{aligned}

In 2D, the number of states within a range $$k$$ of the origin is the area of a circle with radius $$k$$:

\begin{aligned} \Omega_2(k) = \pi k^2 \end{aligned}

Analogously to the 1D case, we take the system to be a square of side $$L$$, so $$k_{\mathrm{min}} = \pi / L$$ again. The density of states then becomes:

\begin{aligned} \boxed{ g_2(k) = \frac{D L^2}{4 \pi^2} \:2 \pi k = \frac{D L^2 k}{2 \pi} } \end{aligned}

In 3D, the number of states is the volume of a sphere with radius $$k$$:

\begin{aligned} \Omega_3(k) = \frac{4 \pi}{3} k^3 \end{aligned}

For a cube with side $$L$$, we once again find $$k_{\mathrm{min}} = \pi / L$$. We thus get:

\begin{aligned} \boxed{ g_3(k) = \frac{D L^3}{8 \pi^3} \:4 \pi k^2 = \frac{D L^3 k^2}{2 \pi^2} } \end{aligned}

All these expressions contain the characteristic length/area/volume $$L^n$$, and therefore give the number of states in that region only. Keep in mind that $$L$$ is free to choose; it need not be the physical size of the system. In fact, we typically want the density of states per unit length/area/volume, so we can just set $$L = 1$$ in our preferred unit of distance.

If the system is infinitely large, or if it has periodic boundaries, then $$k$$ becomes a continuous variable and $$k_\mathrm{min} \to 0$$. But again, $$L$$ is arbitrary, so a finite value can be chosen.

## References

1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.
2. B. Van Zeghbroeck, Principles of semiconductor devices, 2011, University of Colorado.