Why a Pinch of Impurity Makes Silicon Conduct

Every phone, laptop and LED depends on silicon — a material that conducts far worse than copper but far better than glass, and whose conduction can be controlled precisely by adding tiny amounts of impurity.

This part covers energy bands, intrinsic semiconductors with electrons and holes, and doping to make n-type and p-type semiconductors.

**In a solid, atomic energy levels spread into bands, and the gap between the filled valence band and the conduction band above it decides conduction — no gap in metals, a small gap of about $1$ eV in semiconductors, and a gap of more than $3$ eV in insulators.

The three cases:

- Metals — the conduction band is partly filled or overlaps the valence band, so electrons move freely
- Semiconductors** — a small band gap $E_g$, about $1.1$ eV for silicon and $0.7$ eV for germanium; some electrons cross it at room temperature
- Insulators — a large gap, about $5.4$ eV for diamond; almost no electrons cross

Worked example. The longest wavelength of light that can lift an electron across silicon's gap is

$$\lambda =\frac{hc}{E_g}=\frac{1240}{1.1}\approx 1130\text{ nm}$$

in the infrared, so visible light easily creates charge carriers in silicon. For diamond, $\lambda =\frac{1240}{5.4}\approx 230$ nm, in the ultraviolet — which is why pure diamond is transparent.

An everyday example. Copper wires carry current into your home, the plastic coating keeps it from your hand, and silicon chips in your phone control it — one material from each group.

The substance. Heating a semiconductor increases its conductivity — the opposite of a metal, whose resistance rises when heated.

**A pure semiconductor such as silicon is intrinsic: when heat breaks a covalent bond, a free electron and a vacancy called a hole are created together, so $n_e=n_h=n_i$, and both carry current in an electric field.

How carriers form:

- Each silicon atom shares its four valence electrons in covalent bonds with four neighbours
- Thermal energy frees an electron, leaving a hole that behaves like a positive charge
- A neighbouring bound electron can jump into the hole, so the hole moves the opposite way

Current.** The total current is the sum of the electron and hole currents, $I=I_e+I_h$.

Worked example. Pure silicon at room temperature has $n_i\approx 1.5\times 10^{16}$ m${}^{-3}$ and about $5\times 10^{28}$ atoms m${}^{-3}$:

$$\frac{n_i}{N}=\frac{1.5\times 10^{16}}{5\times 10^{28}}=3\times 10^{-13}$$

so only about three atoms in every ten million million have a broken bond.

An everyday example. Cars in a traffic queue each move forward into the gap ahead, so the empty space appears to travel backwards — just like a hole.

The substance. A hole is not a particle — it is a missing electron, and its motion is really bound electrons moving the other way.

Adding a tiny amount of a pentavalent impurity such as arsenic supplies extra free electrons and makes an n-type semiconductor, while a trivalent impurity such as boron supplies extra holes and makes a p-type semiconductor; the more numerous carrier is the majority carrier.

n-type — pentavalent dopants such as phosphorus, arsenic and antimony:

- Four electrons bond with silicon; the fifth is loosely held and easily freed
- The dopant is a donor; electrons are majority carriers and holes minority carriers

p-type — trivalent dopants such as boron, aluminium and indium:

- Only three bonds form, leaving a hole
- The dopant is an acceptor; holes are majority carriers and electrons minority carriers

Mass action law at equilibrium: $n_e{n}_h=n_i^2$

Worked example. Silicon with $n_i=1.5\times 10^{16}$ m${}^{-3}$ is doped with $5\times 10^{22}$ arsenic atoms m${}^{-3}$, about one per million silicon atoms:

$$n_e\approx 5\times 10^{22}{\text{ m}}^{-3},n_h=\frac{(1.5\times 10^{16}{)}^2}{5\times 10^{22}}=4.5\times 10^9{\text{ m}}^{-3}$$

An everyday example. Rooftop solar cells are made of a thin layer of n-type silicon on p-type silicon.

The substance. A doped semiconductor stays electrically neutral — each extra free electron in n-type material is balanced by a fixed positive donor ion.