Presets
Playback
Lattice
Wells of width a(1 - f) separated by barriers of width f a.
State and display
Choose a band and a crystal momentum k.
Filling
Two electrons per cell fill one band.
Layers
Model checks
The science
Electrons in a periodic potential
In a crystal an electron moves in a potential V(x) that repeats every lattice constant a. Bloch's theorem says the stationary states have the form ψ(x) = eikx u(x) with u(x + a) = u(x): a plane wave modulated by the lattice. The wave number k is only defined up to 2π/a, so all states are labelled by k in the first Brillouin zone -π/a < k ≤ π/a.
The Kronig-Penney model
Kronig and Penney (1931) chose the simplest periodic potential that can be solved exactly: wells of width w with V = 0 separated by barriers of width b and height V₀ (a = w + b). Matching ψ and ψ' at every edge gives the dispersion relation
For E > V₀ the hyperbolic functions become cos and sin. Real wave numbers exist only where the right-hand side lies between -1 and +1: these are the allowed bands. Between them are gaps, energies at which no travelling wave fits the lattice. A gap opens where the Bragg condition k = nπ/a makes the forward and reflected waves combine into standing waves, one piling up charge in the wells (lower energy) and one in the barriers (higher energy).
Reading the picture
The group velocity is v = (1/ℏ) dE/dk, zero at band edges; the effective mass m* = ℏ² / (d²E/dk²) is large where the band is flat. With two electrons per cell (spin up and down) a full band means an insulator or semiconductor when a gap lies above it, and a partly filled band means a metal.
Honest simplifications
The potential is a one-dimensional square comb, not the real three-dimensional ion potential; electrons do not interact with each other; the wave animation is slowed down and its phase rate is only proportional to the energy (real periods are femtoseconds); band edges are found on a 0.01 eV grid and the finite crystal drawn is exactly periodic, so there are no surface states.
What to try
- Set the barrier height to 0.5 eV: the bands merge into the free-electron parabola; raise it and gaps open at the zone edges.
- Move to
k = 1(the zone edge) in band 1 and then band 2: two standing waves, one in the wells and one on the barriers, with different energies. - Use the tight-binding preset: bands narrow and the effective mass grows because electrons hardly hop between wells.
- Compare one and two electrons per cell to see a metal turn into an insulator.
- Watch the group velocity fall to zero at the band edges.