Nearly Free Electron Model

If the electrons in a metal experience no periodic potential, the properties of the metal can be calculated with the free electron model, and the band structure is given by the empty lattice approximation. If a small periodic potential is added, this is called the Nearly free electron model. In this case, small band gaps appear at the Brillouin zone boundaries, and some of the bands that have the same energies in the empty lattice approximation split apart from each other. There are still some bands with degenerate energies in the Nearly free electron model due to symmetry.

One way to add a periodic potential is to use the Plane wave method. If the amplitude of the periodic potential is set to zero, the plane wave method reproduces the results of the empty lattice approximation. As the amplitude of the periodic potential increases, gaps open up at the Brillouin zone boundaries and the lowest band splits off. The upper bands reconfigure so that they take nearly the form of the empty lattice approximation again. Some animations of how the band structure evolves as the amplitude of the periodic potential increases are linked below.

In the plane wave method, it is assumed that the potential can be written as a Fourier series and that the wavefunction has Bloch form. Substituting this form for the wavefunction and the potential into the Schrödinger equation results in a set of equations called the central equations,

$$\dfrac{\hbar^2(\vec{k}+\vec{G}')^2}{2m}C_{\vec{G}'} + \sum_{\vec{G}} U_{\vec{G}} C_{\vec{G}'-\vec{G}} = E C_{\vec{G}'}.$$

There is an equation like this for every reciprocal lattice vector. In principle, there are an infinite number of reciprocal lattice vectors but it is usually a reasonable approximation to use a few hundred. The minimum number of reciprocal lattice points that could be used is 2. This is a crude approximation that only includes one Fourier coefficient but this approximation has the advantage that there is an analytic solution at the Brillouin zone boundary. In the case that only two reciprocal lattice vectors are included, the central equations written in matrix form are,

\begin{equation} \left[\begin{matrix} \frac{\hbar^2\vec{k}^2}{2m}-E & U \\ U & \frac{\hbar^2(\vec{k} -\vec{G})^2}{2m}-E \end{matrix}\right] \left[\begin{matrix} C_0 \\ C_{-\vec{G}} \end{matrix}\right] =0. \end{equation}

At the Brillouin zone boundary, $k = \frac{G}{2}$ and the equations become,

\begin{equation} \left[\begin{matrix} \frac{\hbar^2(G/2)^2}{2m}-E & U \\ U & \frac{\hbar^2(G/2)^2}{2m}-E \end{matrix}\right] \left[\begin{matrix} C_0 \\ C_{-\vec{G}} \end{matrix}\right] =0, \end{equation}

and the energies are,

$$E = \frac{\hbar^2\left(\frac{G}{2}\right)^2}{2m} \pm U.$$

The result is that the size of the band gap at the Brillouin zone boundary is approximately $2U$. In other words, the size of the band gap at the Brillouin zone boundary is approximately the amplitude of the periodic potential.