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Bands, Gaps & the Great Divide: Metals, Insulators, Semiconductors

Restore the periodic potential the free-electron model ignored and electrons organize into bands split by gaps — the single idea that explains the 10^30-fold range in electrical conductivity and every transistor ever built.

The free-electron model's blind spot

The free-electron model explains metals beautifully but is blind to a glaring fact: some solids don't conduct at all. Electrical conductivities span more than thirty orders of magnitude from a good metal to a good insulator — a range no theory of a featureless electron gas could ever produce. The missing ingredient is exactly the thing the model threw away in Guide 3: the periodic potential of the ion cores.

Bloch's theorem

The master key is Bloch's theorem. In a periodic potential the electron eigenstates are not plain plane waves but plane waves modulated by a function u_{n\mathbf{k}} that shares the lattice's periodicity. Each state carries a crystal momentum \hbar\mathbf{k} and a band index n; the allowed energies E_n(\mathbf{k}) form continuous bands.

\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_{n\mathbf{k}}(\mathbf{r}), \qquad u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})

A Bloch state: a plane wave times a lattice-periodic modulation.

Bands and gaps

Two complementary pictures give the same band structure. In the nearly-free-electron view, the weak periodic potential Bragg-reflects electrons at the Brillouin-zone boundary, opening a forbidden band gap where no propagating states exist. In the tight-binding view, we start from isolated atomic levels; bringing atoms together lets electrons hop with amplitude t, broadening each sharp level into a band of width set by t. For a 1D chain the tight-binding band is a simple cosine.

E(k) = \varepsilon_0 - 2t\cos(ka)

The 1D tight-binding band: an atomic level ε₀ broadened into a band of width 4t.

Valence and conduction bands separated by a gap; the size of the gap sets whether the solid is a metal, a semiconductor or an insulator.

The great divide

Now count electrons. A completely filled band carries no net current — for every electron moving one way there is one moving the other, and they cancel. So everything hinges on how the electrons fill the bands. A partly filled band means empty states are available right at the Fermi level: the solid is a metallic conductor. A completely filled valence band with an empty conduction band above it means an electron must jump the gap to move: with a large gap the solid is an insulator; with a small gap (around 1\,\mathrm{eV}) it is a semiconductor, an insulator at T=0 that conducts modestly once thermal energy lifts a few electrons across.

Slide the gap from zero (metal) through small (semiconductor) to large (insulator); toggle n- and p-type doping to see how the Fermi level moves into the gap.

Semiconductors and the p-n junction

Semiconductors are useful precisely because their conductivity is tunable. Intrinsically, the density of thermally excited carriers rises exponentially with temperature, following a Boltzmann factor set by half the gap (an electron and a hole are created together). But the real power comes from doping: adding donor atoms injects mobile electrons (n-type), acceptors inject mobile holes (p-type). Join a p-type and an n-type region and you get the p-n junction — carriers diffuse across, leaving a depletion region with a built-in field that lets current flow one way only. That diode is the atom of all electronics.

n_i \propto T^{3/2}\,e^{-E_g/2k_B T}

Intrinsic carrier density: exponentially sensitive to the band gap relative to k_BT.

  1. Silicon has a gap E_g ≈ 1.1 eV; diamond (also a group-IV crystal) has E_g ≈ 5.5 eV. At T = 300 K, k_BT ≈ 0.0259 eV. Compare their intrinsic carrier densities through the exponent E_g/2k_BT.
  2. Silicon: E_g/2k_BT = 1.1 / (2 × 0.0259) ≈ 21.2, so the Boltzmann factor is e^(−21.2) ≈ 6×10^(−10).
  3. Diamond: E_g/2k_BT = 5.5 / (2 × 0.0259) ≈ 106, so the factor is e^(−106) ≈ 10^(−46). Utterly negligible.
  4. The ratio is e^(106−21.2) = e^(84.8) ≈ 10^(37). Silicon has about 10^37 times more intrinsic carriers than diamond at room temperature — which is why one is the backbone of electronics and the other is a superb insulator, from a gap difference of a factor of five.
  5. Lesson: because the gap sits inside an exponential, a modest change in E_g produces an astronomical change in conductivity. This exponential sensitivity is the whole reason band gaps matter so much.