The free-electron model is a triumph — it explains heat capacity, the shape of metals' electron gas, even the sharp Fermi surface. But taken at its word it predicts a catastrophe: if the electrons in a solid really glide through a smooth, featureless box, then every solid should be a metal. Push a voltage across it and the electrons should flow. Yet diamond, table salt and window glass are superb insulators — you could hang a power line on a diamond and it would carry nothing — while copper next to them conducts so freely we wire whole cities with it. Same electrons, same quantum mechanics. What is the difference?

The one thing the free-electron model throws away is the very thing that makes a crystal a crystal: the atoms sit on a periodic lattice, so an electron does not feel a flat potential but a landscape of regularly spaced dips, one at every ion. This page is about the astonishing consequence of that periodicity. It carves the once-continuous ladder of allowed electron energies into bands of permitted energies separated by gaps where no state can live. Whether the highest band an electron reaches is full or only partly full then decides, cleanly and completely, whether the material is a metal or an insulator. That single idea — energy bands from a periodic potential — is the foundation of all of solid-state electronics.

The potential repeats — so the physics must too

Line the ions up on a lattice and the potential energy an electron feels inherits the lattice's perfect periodicity. If \mathbf{R} is any lattice vector (a whole number of steps from one ion to another), then shifting by \mathbf{R} lands you on an identical piece of crystal:

This is a strong statement. It says the Hamiltonian H = -\dfrac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) is unchanged when you translate the whole world by a lattice vector. In quantum mechanics, whenever an operation leaves the Hamiltonian invariant, the energy eigenstates can be chosen to be simple (eigenstates) of that operation too. Translation by \mathbf{R} is exactly such a symmetry, and asking what it does to the wavefunction is what gives us Bloch's theorem.

A one-dimensional cartoon is enough to fix the idea: imagine a chain of atoms a distance a apart, so V(x) = V(x+a) = V(x+2a) = \dots. The potential is a row of identical wells. An electron travelling along this chain sees the same scenery over and over — and, as we will see, that repetition forces its wavefunction into a very particular shape.

Bloch's theorem: a plane wave wearing a lattice-periodic coat

Here is the central result of the whole subject. In a periodic potential, the energy eigenstates are not plain plane waves (those belong to the free electron), and they are not localised at single atoms either. They are a compromise between the two:

Read the form \psi_{\mathbf{k}} = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_{\mathbf{k}} slowly, because it is the whole idea in one line. The slowly rolling plane wave e^{i\mathbf{k}\cdot\mathbf{r}} carries the electron across the crystal — this is the part that looks free. But riding on top of it is u_{\mathbf{k}}(\mathbf{r}), a bumpy function that repeats atom by atom, piling up extra amplitude near the ions where the potential is deep. The electron is extended like a wave and textured like the lattice. It is neither trapped at one atom nor blind to the atoms; it is a wave that has learned the rhythm of the crystal.

Crystal momentum and the first Brillouin zone

There is a subtlety hidden in the label \mathbf{k}. Because only the phase e^{i\mathbf{k}\cdot\mathbf{R}} is physical, two wavevectors that differ by a reciprocal lattice vector \mathbf{G} (defined so that e^{i\mathbf{G}\cdot\mathbf{R}} = 1 for all \mathbf{R}) give exactly the same phase and hence label the same physical state:

So there is no point letting \mathbf{k} run over all of space; every distinct state is already caught inside one primitive cell of the reciprocal lattice. We conventionally take the cell centred on the origin — the first Brillouin zone. In our 1-D chain of spacing a the reciprocal spacing is G = 2\pi/a, and the first Brillouin zone is the interval

Every allowed electron state can be labelled by a k in this one interval, plus a band index n telling you which band it sits in. The energy of the states, E_n(k), is a set of curves — the band structure — each periodic in k with period G. The two complementary pictures below (nearly-free electrons and tight-binding) are just two ways of drawing those curves.

Picture one — nearly free electrons: a weak lattice opens gaps

Start from the free electron, whose energy is the smooth parabola E(k) = \dfrac{\hbar^2 k^2}{2m}, and switch on a weak periodic potential as a small perturbation. Almost everywhere it does very little. But at the edge of the Brillouin zone, k = \pm\pi/a, something special happens: the two plane waves e^{ikx} and e^{i(k-G)x} — that is, the states at +\pi/a and -\pi/a — have exactly the same free-electron energy. They are degenerate, and even a tiny potential mixes them strongly.

Degenerate perturbation theory says the two mixed states repel: one combination is pushed down in energy and the other pushed up, so a forbidden gap tears open right at the zone boundary. Its width is set by the relevant Fourier component V_G of the potential:

The once-continuous parabola is cut into bands, separated by gaps at every zone boundary. Drag the slider to turn the lattice on: at |V_G| = 0 the bands touch (a free-electron parabola folded into the zone), and as you strengthen the potential the gap of width 2|V_G| yawns open at k = \pm\pi/a.

This is the nearly-free-electron picture: bands are barely-bent free-electron parabolas, and gaps are small features that appear only where the lattice's periodicity resonates with the electron's wavelength. It is the natural language for simple metals, where the ions perturb an electron gas that is otherwise almost free.

Picture two — tight binding: atoms sharing electrons

The opposite limit is just as illuminating. Instead of starting from free electrons and adding a weak lattice, start from isolated atoms — each with a sharp atomic level of energy E_0 — and bring them close enough that an electron can hop from one atom to its neighbour. The single atomic level broadens into a band. For a 1-D chain with one orbital per atom, spacing a and nearest-neighbour hopping amplitude t, the band structure is a clean cosine:

Everything you want to know is in that formula. At the zone centre k = 0, \cos 0 = 1, giving the band bottom E(0) = E_0 - 2t. At the zone boundary k = \pi/a, \cos\pi = -1, giving the band top E(\pi/a) = E_0 + 2t. The total spread — the bandwidth — is therefore

A bigger hopping t (atoms closer together, orbitals overlapping more) makes a wider band; squeeze the atoms apart and t\to 0, the band collapses back to the isolated atomic level. Slide t and watch the cosine band stretch — its full height is always exactly 4t.

Tight binding is the natural language for narrow bands — the d-orbitals of transition metals, the electrons of insulators and molecular crystals — where electrons are nearly stuck to their atoms and only occasionally tunnel across. Nearly-free electrons and tight binding approach the same truth from opposite ends: both give bands separated by gaps.

Filling the bands: metal, insulator, or semiconductor

Now the payoff. Each band holds a fixed number of states — two electrons (spin up and down) for every primitive cell in the crystal. Fill the bands with the crystal's electrons from the bottom up, exactly as you fill atomic shells, and the character of the material is decided by where the filling stops:

• Metal. The topmost occupied band is only partly filled. Right above the highest filled state sit empty states at almost the same energy, so the smallest electric field nudges electrons into them and current flows freely. Copper, with its half-filled 4s band, is the classic case.
• Insulator. The electrons exactly fill a set of bands to the brim, and a large gap (several eV) separates the full valence band from the empty conduction band above. There are no nearby empty states, so a field pushes the electrons nowhere. Diamond, gap about 5.5\ \text{eV}, is an insulator for this reason.
• Semiconductor. Same full-band situation, but the gap is small — around 1\ \text{eV}. At room temperature a trickle of electrons is thermally kicked across, so the material conducts a little, and much more as it warms. Silicon (gap 1.1\ \text{eV}) is the workhorse of every chip.

Notice the deep simplicity: whether a band ends up full or partly full comes down to how many electrons each cell contributes. An odd number of electrons per cell leaves a band half-filled (a metal); an even number can exactly fill bands (an insulator or semiconductor). The whole grand distinction between a conductor and an insulator is, at bottom, a parity question about counting electrons — and it is invisible to the free-electron model.

Worked examples

Example 1 — bandwidth from hopping. A 1-D tight-binding chain has hopping amplitude t = 0.4\ \text{eV}. How wide is its band? The bandwidth is simply W = 4t, so

If we measured the band to be 2.0\ \text{eV} wide instead, we would read off t = W/4 = 0.5\ \text{eV}: the bandwidth is a direct measurement of how strongly neighbouring atoms share their electron.

Example 2 — the band edges. Take E(k) = E_0 - 2t\cos(ka) with E_0 = 3.0\ \text{eV} and t = 0.5\ \text{eV}. Evaluate the two ends of the band. At the zone centre, \cos(0) = 1:

At the zone boundary, \cos(\pi) = -1:

The band runs from 2.0 to 4.0\ \text{eV} — a spread of 2.0\ \text{eV} = 4t, just as promised.

Example 3 — the gap from the potential. A nearly-free-electron model of a crystal has a Fourier component |V_G| = 0.8\ \text{eV} at the reciprocal vector that folds the parabola. The gap it opens at the zone boundary is

That is a semiconductor-sized gap — a stronger lattice potential means a wider forbidden band and a more insulating material.

Example 4 — classify by filling. A crystal has a valence band completely full and the next band empty, with a gap of 0.7\ \text{eV} between them. Metal, insulator, or semiconductor? A full band cannot conduct, so it is not a metal. The gap is small — under an electron-volt — so thermal excitation can promote carriers across it: this is a semiconductor (in fact this is roughly germanium). Raise the gap to 5\ \text{eV} with the same full-band filling and it would be an insulator.