Introduction: A Solvable Crystal Model
The Kronig-Penney model replaces the smooth periodic potential of a real crystal with periodic delta functions. Despite its simplicity, it captures the essential physics of energy bands and band gaps.
The Model
\[V(x) = \sum_{n=-\infty}^{\infty} \alpha\delta(x - na)\]Delta function barriers (or wells) at regular intervals \(a\).
The Key Equation
Applying Bloch's theorem and matching conditions at the deltas:
\[\cos(Ka) = \cos(ka) + \frac{m\alpha}{k\hbar^2}\sin(ka)\]where \(K\) is the crystal momentum and \(k = \sqrt{2mE}/\hbar\).
Energy Bands
The right side must lie between -1 and +1 for solutions to exist. When it's outside this range, there are no solutions—these are band gaps.
- Allowed bands: RHS in \([-1, 1]\)
- Forbidden gaps: RHS outside \([-1, 1]\)
The Quantum Connection
The band structure explains all of solid-state electronics. Metals have partially filled bands; insulators have full bands with large gaps; semiconductors have small gaps that can be controlled by doping or electric fields. The Kronig-Penney model shows these features with elementary mathematics.