The Delta Function Model of a Crystal *

The Kronig-Penny model of a solid crystal contains an infinite array of repulsive delta functions.

$$\bgroup\color{black} V(x)=aV_0\sum\limits_{n=-\infty}^\infty \delta(x-na) \egroup$$

Our states will have positive energy.

This potential has a new symmetry, that a translation by the lattice spacing $a$ leaves the problem unchanged. The probability distributions must therefore have this symmetry

$$\bgroup\color{black} \vert\psi(x+a)\vert^2=\vert\psi(x)\vert^2 ,\egroup$$

which means that the wave function differs by a phase at most.

$$\bgroup\color{black} \psi(x+a)=e^{i\phi}\psi(x) \egroup$$

The general solution in the region $(n-1)a<x<na$ is

$$\bgroup\color{black} \psi_n(x)=A_n\sin(k[x-na])+B_n\cos(k[x-na]) \egroup$$

$$\bgroup\color{black} k=\sqrt{2mE\over\hbar^2} \egroup$$

By matching the boundary conditions and requiring that the probability be periodic, we derive a constraint on $k$ similar to the quantized energies for bound states.

$$\bgroup\color{black} \cos(\phi)=\cos(ka)+{maV_0\over\hbar^2k}\sin(ka) \egroup$$

Since $\cos(\phi)$ can only take on values between -1 and 1, there are allowed bands of $k$ and gaps between those bands.

The graph below shows $\cos(ka)+{maV_0\over\hbar^2k}\sin(ka)$ as a function of $k$. If this is not between -1 and 1, there is no solution, that value of $k$ and the corresponding energy are not allowed.

This energy band phenomenon is found in solids. Solids with partially filled bands are conductors. Solids with filled bands are insulators. Semiconductors have a small number of charge carriers (or holes) in a band.