The Delta Function Model of a Crystal *

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

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

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

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

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

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

The general solution in the region \bgroup\color{black}$(n-1)a<x<na$\egroup is

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


\begin{displaymath}\bgroup\color{black} k=\sqrt{2mE\over\hbar^2} \egroup\end{displaymath}

By matching the boundary conditions and requiring that the probability be periodic, we derive a constraint on \bgroup\color{black}$k$\egroup similar to the quantized energies for bound states.

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

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

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

\epsfig{file=figs/kronig.eps,width=5in}

\epsfig{file=figs/bands.eps,height=4in}

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.

Jim Branson 2013-04-22