Band Structure

Start with the free electron. Pick the first preset. The solid curve lies exactly on the dashed parabola, folded back into the zone at ±π/a the way a road map folds. Nothing is forbidden; the bands touch. The table says every gap is closed.

Switch the potential on. Go to “Nearly free” and press the button. The moment V₀ leaves zero, the curve tears at the fold — at k = ±π/a first, then at k = 0 for the next band up. The rose gaps open in proportion to V₀ while V₀ is small: the table quotes the first-order prediction beside the computed number, and for the nearly-free preset they agree to a few percent.

Set b/a to exactly 0.50. The second gap closes again while the first stays open. A rectangular barrier half a cell wide has no second Fourier component, and the n-th gap is set by the n-th Fourier component alone — this is the sharpest single check that the picture is perturbation theory and not decoration.

Then push V₀ to 30. The bands stop being bent parabolas and flatten into thin lines at the energies of a particle trapped in one well. Those are the atomic levels; a band is what one becomes when the atom has neighbours to tunnel to. The gap prediction now fails, and the caption says so.

Finally, count electrons. Back on “Nearly free”, slide the electron count from 1 to 4. Odd numbers make a metal every time: the Fermi level cuts through a band, and the teal half of the curve sits under a violet half with no gap between them. Two electrons fill band 1 exactly and the label reads semiconductor, because the gap above it is about 1.3 eV on this page’s scale. Now raise V₀ to 6 with the count still at 2. Same electrons, same filling — but the gap has grown past 3 eV and the label flips to insulator. The electrons did not change; the gap did.

A wave with wavelength exactly twice the lattice spacing — k = π/a — reflects off every atom in phase. Bragg reflection. It cannot travel, so it stands, and there are two ways to stand: with the crests on the atoms or with the crests between them. Those two standing waves feel different potential energies, and the difference between them is the gap. Everything in the left panel follows from that one sentence: the tear appears at the zone boundary because that is where the reflection is total, and its size is twice the Fourier component of V at 2π/a because that is how much the two standing waves differ.

The right panel is the same information read the other way. Each stripe is a range of energy in which an electron can move through the barriers from cell to cell forever; each gap is a range in which anything you put in decays within a few cells. The tight-binding preset makes the origin plain — the stripes sit at the bound-state energies of one well, thickened by a tunnelling rate — and the nearly-free preset makes the other limit plain, where they are the parabola with slits cut in it.

Conduction needs an empty state a tiny energy above a filled one. Halfway up a band there are always some. At the top of a filled band, with a gap above it, there are none within reach, and the solid is an insulator — or a semiconductor if the gap is small enough that heat or light can bridge it. Odd electron counts give metals; even ones give gaps. That the periodic table’s alkali metals conduct and its noble-gas solids do not is this slider.

The potential is the Kronig–Penney cell: a flat well of width a − b followed by a barrier of height V₀ and width b, repeated forever. In each piece the wave is a plain cosine or a plain exponential, and carrying ψ and ψ′ across one cell is a 2×2 matrix M(E). Bloch’s theorem says a crystal state at energy E exists exactly when M(E) has an eigenvalue e^ika of unit size — and since det M = 1, that is the single condition cos(ka) = ½ Tr M(E). Bands are where the right-hand side lies in [−1, 1]; gaps are where it does not; E(k) is that equation solved for E.

This is done inlib/quantum/bands.tson top of wave-1’s Schrödinger engine, and checked inbands.test.tsagainst the free parabola at V₀ = 0, against the Kronig–Penney condition as printed in Kittel, against 2|Vₙ| for weak potentials, against the tabulated Mathieu characteristic values for a cosine potential sampled on the engine’s grid, and against a hard-wall supercell run through the engine’s finite-difference solver — which finds the same N states per band, plus a surface state in the gap that the infinite crystal does not have.

One dimension, one electron, a rectangular barrier. Real crystals are three-dimensional and their gaps can be indirect — the top of the valence band and the bottom of the conduction band at different k — which is why silicon is a poor light emitter and gallium arsenide is a good one. Nothing here can show that; there is only one k.

The electrons do not interact. In a real metal they screen each other and the band picture survives as a description of quasiparticles; in some oxides it fails outright and a half-filled band insulates (a Mott insulator). The rule “odd count means metal” is the independent-electron rule, and it is what this page computes.

The eV scale is a stated convention, one length unit = 2 Å, chosen so the lattice-spacing slider covers real interatomic distances. The metal / semiconductor / insulator labels are exact for the first split — a band is or is not partly full — and conventional for the second, which is a line drawn at about 3 eV.