Energy Bands and the Classification of Solids

The free electron model has no answer to the most basic question in the subject: why is copper a conductor and diamond an insulator, when both are crystalline solids full of electrons? Bands answer it.

Where bands come from

An isolated atom has sharp, discrete energy levels. Bring two identical atoms close and their wavefunctions overlap; the exclusion principle forbids two electrons in the same state, so each level must split into two slightly different levels.

Bring $N$ atoms together — and in a solid $N \approx 10^{23}$ — and each atomic level splits into $N$ levels. They span a finite energy range, so $10^{23}$ levels are packed into a few electron-volts. The spacing between adjacent levels is around $10^{-23}\ \text{eV}$: utterly unresolvable, and for every practical purpose continuous.

That near-continuum is an energy band. The ranges between bands, where no states exist at all, are forbidden gaps.

Two bands matter:

• the valence band — the highest band that is filled or partly filled by electrons at absolute zero;
• the conduction band — the next band up.

The separation between the top of one and the bottom of the other is the band gap $E_g$.

Why a full band carries no current

This is the part worth getting right, because the classification follows from it immediately.

To carry current, electrons must gain a little energy from the field and move into states of slightly higher energy in the direction of drift. If a band is completely full, there is no such state — every one is occupied, and the exclusion principle forbids doubling up. The electrons are stuck. A full band carries no current no matter how large the field.

If a band is partly full, empty states sit immediately above the occupied ones. An arbitrarily small field can nudge electrons into them, and current flows.

So conduction requires not electrons but electrons with empty states adjacent in energy.

The classification

Conductors. The valence band is only partly filled, or the valence and conduction bands overlap so there is no gap at all. The Fermi level lies inside a band. Empty states sit immediately above the filled ones, so conduction happens at any temperature, including absolute zero. Sodium is the partly-filled case; magnesium, with an even number of valence electrons that would otherwise fill its band, conducts because its bands overlap.

Insulators. The valence band is completely full, the conduction band completely empty, and the gap is large — 5.5 eV for diamond. The Fermi level lies in the middle of the gap. To conduct, an electron must be lifted across the whole gap, and Topic 2 showed how improbable that is: at room temperature $k_B T$ is 0.026 eV, so an electron needs about 210 times its thermal energy in one go. The probability is vanishingly small, and the material does not conduct.

Semiconductors. Structurally identical to insulators — full valence band, empty conduction band, Fermi level in the gap — but the gap is small: 1.12 eV for silicon, 0.67 eV for germanium. At absolute zero a semiconductor is a perfect insulator. At room temperature a small but non-negligible number of electrons make it across, leaving vacancies behind, and both contribute to conduction.

The distinction between an insulator and a semiconductor is one of degree, not of kind. There is no sharp boundary in the definition — only the practical observation that a 1 eV gap gives useful conductivity at room temperature and a 5 eV gap does not.

The temperature signature

This gives the single most useful experimental test.

• In a metal, raising the temperature does not create new carriers — the band is already partly full. It only increases lattice vibrations, which scatter electrons more, shortening $\tau$. Resistance increases with temperature.
• In a semiconductor, raising the temperature promotes exponentially more electrons across the gap. The gain in carrier number overwhelms the loss in $\tau$. Resistance decreases with temperature, and steeply.

A material whose resistance falls as it warms is a semiconductor. This is what a thermistor exploits.