Kronig–Penney Delta Bands

Worked Example: Band Classification

Consider a delta-barrier lattice with dimensionless barrier strength P = 1 and a chosen free phase z = 0.5π. The allowed and forbidden bands are determined by evaluating the right side of the dispersion relation, F(z) = cos(z) + P * sin(z) / z.

1. Identify z = 0.5π = 1.570796...
2. Evaluate cos(0.5π) = 0
3. Evaluate sin(0.5π) = 1
4. Calculate F(z) = 0 + 1 * 1 / (0.5π) = 2 / π ≈ 0.63662
5. Since |F(z)| < 1, this energy level lies within an allowed band.
6. The Bloch real phase is acos(0.63662) ≈ 0.88069.

Experiment: Exploring the Bands

You can interact with the Kronig–Penney Delta Bands module to visualize how the barrier strength affects the allowed energies.

1. Set the Barrier strength to 1.
2. Change the Free phase over pi to 0.25.
3. Observe that the Dispersion right side becomes ≈ 1.6074.
4. Since this value is greater than 1, the Band classification correctly shows "Forbidden".
5. Notice the Bloch decay phase becomes non-zero, indicating exponential decay of the wavefunction.

Limitations and Assumptions

The delta-function barrier is a mathematical idealization. Real crystals have finite-width, continuous potentials (like Coulomb potentials of nuclei). While the Kronig–Penney model captures the fundamental origin of band gaps—Bragg reflection of electron waves—it does not predict the exact energy levels of real materials. Furthermore, this 1D model does not capture 3D effects like overlapping bands or complex Fermi surfaces.

References

The concepts discussed in this interactive module are based on standard solid-state physics curricula and educational materials used to introduce energy bands to university students.

• Maslov, D. "The Kronig-Penney Model". Physics 6426, University of Florida. Available at: https://www.phys.ufl.edu/~maslov/phz6426/dkp.pdf