Kronig-Penney Band Structure

What you are seeing: what happens to electron energies when you put them inside a 1D crystal instead of free space. A free electron has $E = \hbar^2 k^2 / 2 m$ , a smooth parabola. Once you turn on a periodic potential (here: a comb of delta-function spikes, period $a$ , dimensionless strength $P$ ), the parabola gets cut into allowed bands separated by forbidden gaps. The dispersion is $\cos(k a) = \cos(q a) + (P / q a)\sin(q a)$ with $q = \sqrt{2 m E}/\hbar$ .

Left panel: the right-hand side $f(q a)$ plotted vs. dimensionless energy $\varepsilon = (q a)^2$ . The horizontal band marks $|f| \le 1$ : inside it, real $k a$ exists; outside, only forbidden (gap) states. Right panel: the resulting band structure $\varepsilon(k a)$ in the reduced zone, with allowed bands in blue.

Figure 1. Kronig-Penney delta-comb dispersion (left) and band structure (right).

P (strength)4.0

electron energy9.0

eps max60

WHAT TO TRY

Sweep the electron energy: when it lands in an allowed band the wave at the top sails through the barrier comb as a blue Bloch state; drop it into a gap and the same wave turns red and decays away, evanescent because the lattice Bragg-reflects it.
Turn up the barrier strength P: the free-electron parabola fractures into allowed bands split by forbidden gaps, the moment a periodic potential can turn a metal into an insulator.
At P = 0 the gaps close and free electrons return; crank P high and the bands narrow toward flat atomic levels, the tight-binding limit.
The gaps open exactly at the Brillouin zone boundaries, where the electron wave Bragg-reflects off the lattice, the origin of every semiconductor band gap.