EC-5.3 Crystals, Lattices, and How Bands Form

What this is and why it exists

This is the topic the whole area has been quietly assuming.

Bring two identical atoms close and each shared energy level splits in two. Bring a huge number close, as in a crystal, and each level becomes a dense band of levels. Between those bands are ranges of energy no electron can occupy.

That picture explains conduction, insulation, doping, junctions, why a diode conducts one way, and why silicon does not make light. It is the single most load-bearing diagram in electronics, and until now nothing here taught it.

The examination syllabus names it as the very first line of its Electronic Devices section. So does every other one.

The vocabulary

• Lattice — the repeating arrangement of atoms that makes a crystal.
• Unit cell — the smallest block that, repeated, builds the whole lattice.
• Energy band — a range of allowed energies so densely packed it behaves as continuous.
• Valence band — the band holding the bonding electrons.
• Conduction band — the empty band above it, where an electron is free to move.
• Band gap — the forbidden range between the two.
• Band diagram — energy plotted up the page against position across it.
• Hole — a vacancy in the valence band, followed as though it were a positive particle.
• Direct gap — a gap an electron can cross without changing momentum.

The mental model

A crystal repeats one arrangement of atoms in every direction. Describe a small cell and the repeat rule and you have described the whole solid. Silicon's arrangement has every atom bonded to four neighbours, which is where its four valence electrons go.

Now the central argument, and it is worth following slowly.

Two isolated atoms have identical energy levels. Bring them close enough to interact and those two levels cannot both stay identical, so each one splits into two slightly different ones. Three atoms give three. A crystal has an enormous number of atoms, so each original level becomes an enormous number of levels spread over a narrow range.

Those levels are so close together that nothing is gained by counting them. Treat the range as continuous and call it a band.

Between the bands, the forbidden ranges survive. They were forbidden for a single atom and they are still forbidden now. That surviving gap is the band gap, and its width is the number that decides what kind of material you are holding.

Two bands matter. The valence band holds the bonding electrons and is full. The conduction band above it is empty, and an electron up there is not tied to any particular bond and can move.

A band diagram puts energy up the page and position across it. That one convention carries a lot. A sloped band means an electric field, a step means a junction, and a bend means charge has moved. Learning to read this picture is most of what device physics asks of you. Draw it by hand rather than only looking at it.

Then there is the hole. When an electron leaves the valence band it leaves a vacancy, and a neighbouring electron slides into it, leaving a vacancy where it came from. Rather than follow a near-full band of electrons, follow the vacancy and treat it as a positive particle moving the other way. It is a bookkeeping convenience rather than a real particle, and it is the right one — every device calculation is shorter for it.

One last piece, which decides which materials can make light. Crossing the gap requires the right energy, and it also requires momentum to work out. In some materials an electron can drop across the gap and give up its energy as a photon. In others it cannot do so without a lattice vibration taking up momentum too, and the energy leaves as heat instead. Silicon is the second kind. That is why silicon does not make light, and why a light-emitting device is made of something else entirely.

What you should now be able to explain or do

• Explain how discrete atomic levels become bands when atoms are brought together.
• Name the valence band, the conduction band and the gap, and say what each is.
• Read a band diagram: say what a slope, a step and a bend each mean.
• Explain what a hole is and why following one is easier than following electrons.
• Say why some materials emit light and silicon does not.

Check yourself