Band Theory of Solids: Conductors, Insulators, Semiconductors

Why Copper Conducts and Glass Does Not

The resistivity of copper is about 10⁻⁸ Ω m; that of glass is over 10¹⁰ Ω m, a difference of more than 10¹⁸ times. Silicon sits in between, and, unlike a metal, it conducts better when it is heated or when light shines on it. Simple ideas about "free electrons" cannot explain these huge differences. The explanation comes from quantum physics: the band theory of solids. This tutorial explains how energy levels in atoms spread into bands in a solid, the valence and conduction bands, the band gap, why materials are conductors, insulators or semiconductors, how temperature and light affect them, and how doping creates the n-type and p-type semiconductors used in every electronic device.

From Energy Levels to Energy Bands

An isolated atom has sharp, separate electron energy levels, as shown by its line spectrum. When two identical atoms are brought close together, their outer electrons interact and each energy level splits into two slightly different levels. With N atoms, each level splits into N levels.

A small crystal contains around 10²³ atoms, so each atomic level splits into so many closely spaced levels that they form an effectively continuous energy band. Between the bands there may be ranges of energy that no electron can have, called band gaps (or forbidden gaps).

Two bands matter most for electrical properties:

the valence band, the highest band that is normally filled with electrons (the electrons that form bonds);
the conduction band, the next band above, in which electrons are free to move through the crystal.

Why a Full Band Cannot Conduct

To carry a current, electrons must gain a little energy from an applied electric field and move to slightly higher energy states. In a completely full band, every state is already occupied (the Pauli exclusion principle allows only two electrons per state), so electrons have nowhere to move to and cannot respond to the field. A full band carries no current. Conduction needs either a partly filled band, or electrons that have been lifted into an otherwise empty band.

Band Theory of Solids: Conductors, Insulators and Semiconductors

Conductors (metals). In a metal, the highest occupied band is only partly filled, or the valence and conduction bands overlap. There are huge numbers of empty states just above the occupied ones, so electrons respond easily to an electric field. Metals conduct very well at any temperature.

Insulators. The valence band is full and the conduction band is empty, separated by a large band gap, typically more than about 4 eV. Diamond's gap is about 5.5 eV. At room temperature, the typical thermal energy of an electron is only about 0.025 eV, so almost no electrons can jump the gap. Insulators conduct almost nothing.

Semiconductors. The arrangement is the same as in an insulator, but the band gap is small: about 1.1 eV for silicon and 0.67 eV for germanium. At room temperature, a small but significant number of electrons gain enough thermal energy to jump into the conduction band, so the material conducts a little.

Electrons and Holes

When an electron jumps from the valence band to the conduction band, it leaves behind an empty state in the valence band called a hole. A neighbouring valence electron can move into the hole, leaving a new hole behind it, so the hole effectively moves through the crystal. A hole behaves like a positive charge carrier.

In a pure (intrinsic) semiconductor, electrons and holes are always created in pairs, so their numbers are equal. Both contribute to the current: electrons moving in the conduction band, and holes moving (in the opposite direction) in the valence band.

The Effect of Temperature

Metals: the number of free electrons is fixed. Heating makes the lattice ions vibrate more, so electrons collide more often and the resistance increases.
Semiconductors: heating lifts many more electrons across the gap, creating more electron–hole pairs. This increase in charge carriers far outweighs the extra collisions, so the resistance decreases sharply as temperature rises.

This is how a thermistor (NTC type) works: its resistance falls as it warms up, making it an ideal temperature sensor in thermometers, fire alarms and battery chargers.

The Effect of Light

A photon of light can lift an electron across the band gap if its energy hf is at least the gap energy Eg. This is the basis of:

light-dependent resistors (LDRs), whose resistance falls in bright light;
photodiodes and solar cells, which turn light into current.

Example 1. Find the longest wavelength that can create electron–hole pairs in silicon (Eg = 1.1 eV).

λ = hc/Eg = (6.63 × 10⁻³⁴ × 3.0 × 10⁸)/(1.1 × 1.6 × 10⁻¹⁹) ≈ 1.13 × 10⁻⁶ m = 1,130 nm, in the near infrared. All visible light (400 to 700 nm) has enough energy, which is why silicon makes good solar cells and camera sensors.

The reverse process explains LEDs: when an electron falls from the conduction band back into a hole, it can emit a photon with energy about equal to the band gap. Materials with different gaps give different colours: gallium arsenide (about 1.4 eV) emits infrared, while gallium nitride-based LEDs with wider gaps emit blue light.

Example 2. Why is glass transparent? Its band gap is large (several eV), so photons of visible light (1.8 to 3.1 eV) don't have enough energy to lift electrons across it. They are not absorbed and pass through. Metals, with no gap, absorb and re-emit visible light at the surface, which makes them shiny and opaque.

Doping: n-Type and p-Type Semiconductors

Pure silicon conducts too poorly for most uses, so tiny amounts of impurity are added in a process called doping, typically about one impurity atom per million silicon atoms.

n-type: silicon (4 outer electrons) is doped with a group 5 element such as phosphorus or arsenic (5 outer electrons). Four electrons form bonds; the fifth is very loosely held. Its energy level lies just below the conduction band, so almost every such "donor" electron enters the conduction band at room temperature. Electrons are the majority carriers.
p-type: silicon is doped with a group 3 element such as boron or gallium (3 outer electrons). One bond in each impurity is missing an electron. Its energy level lies just above the valence band, and valence electrons easily jump into it, leaving holes in the valence band. Holes are the majority carriers.

Doped (extrinsic) semiconductors still contain small numbers of the other kind of carrier (minority carriers), created thermally. Note that n-type and p-type materials are still electrically neutral overall: the extra electrons are balanced by the positive charge of the donor nuclei.

Joining p-type and n-type material makes a p–n junction, the basis of diodes, LEDs, solar cells and transistors, covered in the tutorials on semiconductors and diodes and on transistors.

Summary

In a solid, atomic energy levels spread into bands separated by band gaps.
Conductors have partly filled or overlapping bands; insulators have a full valence band and a large gap; semiconductors have a small gap (about 1 eV).
A full band cannot conduct; conduction needs electrons in the conduction band or holes in the valence band.
Heating lowers a semiconductor's resistance but raises a metal's.
Photons with hf ≥ Eg create electron–hole pairs; doping with group 5 or group 3 elements makes n-type and p-type semiconductors.