The density of states \( D(W) \) is the derivative of the number of states \( N \) with respect to the energy \(W\):

Definition of the density of states

Formula anchor$$ \begin{align} D(W) ~=~ \frac{\text{d}N}{\text{d}W} \end{align} $$

Sometimes we consider the density of states \( g(W) \) per volume \( V \) of the crystal:

Definition: density of states per volume

Formula anchor$$ \begin{align} g(W) ~=~ \frac{1}{V} \, \frac{\text{d}N}{\text{d}W} \end{align} $$

In the following, we will consider the density of states of a free electron gas that is spatially confined:

Density of state of a two-dimensional electron gas. This occurs in 2d materials, such as graphene or in the quantum Hall effect.

Density of state of a three-dimensional electron gas. This occurs, for example, in metals. The density of states can be used to determine the charge carrier density in the metal.

Density of state of a one-dimensional electron gas. 1d density of states is useful for materials where electrons can propagate in only one dimension. 1d materials include, for example, nanotubes and nanowires.

Density of states (2d)

The smallest reciprocal area \( \mathcal{A}_1 \) (in k-space) occupied by one single state is:

Formula: Reciprocal area of a k-state

Formula anchor$$ \begin{align} \mathcal{A}_1 ~=~ \frac{(2\pi)^2}{A} \end{align} $$

Here \( A \) is the area of a real two-dimensional crystal lattice.

To find out the 2d density of states, according to the definition 1, we must first find out the number of available states \( N^{\small 2\text d} \) in the 2d electron gas. In most cases, however, the exact number \( N^{\small 2\text d} \) of states is not known. We can determine it, however, if in addition to Eq. 3 we provide the reciprocal area \( \mathcal{A}_N \) occupied by these \( N^{\small 2\text d} \) states.

Since the states are very close to each other (see the dimension of the 1st Brillouin zone), you can approximate the reciprocal area \( \mathcal{A}_N \) occupied by states \( N^{\small 2\text d} \) by a circular area. Each circle inside of this circular area is a line of constant energy. Why? Because the distance from the coordinate origin to all \(k\) values on the circle line is given by the radius \( k \). The area of a circle with radius \(k\) is given by the following formula:

Area of a circle in k-space

Formula anchor$$ \begin{align} \mathcal{A}_N ~=~ \pi \, k^2 \end{align} $$

To find out the number \( N^{\small 2\text d} \) of states, you have to find out how often the area \( \mathcal{A}_1 \) fits into \( \mathcal{A}_N \). So divide eq. 4 by eq. 3:

Because the electrons have a spin (spin-up and spin-down), not only one state fits into the area \( \mathcal{A}_1 \), but two! Equation 5 gives only half of the states (e.g. only spin-up states). So multiply equation 5 by 2 to consider also the other half of the spin states:

Remember: Our goal is to find out the density of states using the derivative 1. We have now determined the number of states \( N^{\small 2\text d}(k) \), but as a function of the wavenumber \(k\). But we need a number of states \( N^{\small 2\text d}(W) \) expressed with the energy \(W\). Only then we can differentiate \( N^{\small 2\text d}(W) \) with respect to the energy.

To express the number of states \( N^{\small 2\text d}(W) \) using the energy, we need to know the relation between the energy \( W \) and the wavenumber \( k \). This relation is given by the dispersion relation \( W(k) \). Depending on the system, it can be linear in \( k \), quadratic in \( k \), or have a complicated form that can be determined by experiment. We look here at the quadratic dispersion relation of a free electron gas.

For a free electron gas the following quadratic dispersion relation holds:

Dispersion relation: free electron gas

Formula anchor$$ \begin{align} W(k) ~=~ \frac{\hbar^2}{2m} \, k^2 \end{align} $$

Here \( m \) is the mass of the electron (in practice the effective mass \( m^* \)) and \( \hbar \) is the reduced Planck's constant. Wavenumber \( k \) is the magnitude of the wave vector \( \boldsymbol{k} \). Rearrange the dispersion relation 7 for \( k \):

Dispersion relation of a free electron gas rearranged for wavenumber

Formula anchor$$ \begin{align} k ~=~ \sqrt{ \frac{2m \, W}{\hbar^2} } \end{align} $$

Put Eq. 8 into Eq. 6. Thus you make the number of states \( N^{\small 2\text d} \) independent of the wavenumber \( k \), but dependent on the energy \(W\):

Number of states in k-space using quadratic dispersion relation.

As you can see in the formula 10, the 2d density of states is independent of the energy \( W \). Thus, the 2d density of states is constant for all energies that an electron can have.

Density of states (3d)

The smallest reciprocal volume \( \mathcal{V}_1 \) occupied by a single state is:

Volume in k-space of a single state

Formula anchor$$ \begin{align} \mathcal{V}_1 ~=~ \frac{(2\pi)^3}{V} \end{align} $$

The reciprocal volume of a sphere occupied by \( N^{\small 3\text d} \) states is:

Sometimes you are interested in the density of states \( g(W) \) per volume. If so, then divide 16 by the volume \( V \) to get the number of states per energy AND per volume:

Density of states (3d) per volume for a quadratic dispersion relation

A single state occupies the following reciprocal length \( \mathcal{L}_1 \):

Reciprocal length of a single state (1d)

Formula anchor$$ \begin{align} \mathcal{L}_1 ~=~ \frac{2\pi}{L} \end{align} $$

Here \( L \) is the length of a one-dimensional system (e.g. of a quantum wire) in which there is a free electron gas.

In one-dimensional systems, \( N^{\small 1\text d} \) states take the following reciprocal length:

Reciprocal length taken by N states

Formula anchor$$ \begin{align} \mathcal{L}_N ~=~ 2k \end{align} $$

Why exactly \( 2k \)? Because \( k \) is the radius. The total reciprocal length is therefore \( 2k\)! Still unclear? Draw a circle with the radius \( k \). What is then its diameter?

Divide Eq. 19 by Eq. 18 to determine the number \( N^{\small 1\text d} \) of states that fit into the length section \( \mathcal{L}_N \):

Number of 1d states within a length section

Formula anchor$$ \begin{align} N^{\small 1\text d} &~=~ 2k \cdot \frac{L}{2\pi} \\\\
&~=~ \frac{L}{\pi} \, k \end{align} $$

Substitute the wavenumber 8 of the quadratic dispersion relation into Eq. 20:

Number of 1d states using quadratic dispersion relation

Now you have learned what the density of states is and how to find it for a free electron gas in different dimensions. In further lessons, we will look at how the density of states can be used to determine the charge carrier density and heat capacity of a material.

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