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.