Foundations — Crystal lattice structure of silicon

This page assumes nothing. Before we can read the parent note on the silicon lattice, we need to earn every symbol it throws at us. We build them in an order where each one leans only on the ones before it.

0. What is an atom, for our purposes?

1. Valence electrons — the "hands"

The picture: four short line segments leaving the atom-dot. Each one is a hand looking for another hand to hold.

Why the topic needs it: those 4 hands are the reason silicon bonds to 4 neighbours, and 4 neighbours is what forces the whole geometry. Deep dive: Valence electrons and the octet rule.

$\underset{\text{silicon’s own}}{\underbrace{4}}+\underset{\text{one from each neighbour}}{\underbrace{4}}=8$

2. What an angle is, and how we measure it

Before the parent note says "bond angle $\approx 109.5^{\circ }$", you must know what an angle is.

• A $90^{\circ }$ angle is a perfect square corner (look at the amber corner marker in the figure).
• An angle bigger than $90^{\circ }$ is called obtuse — it looks "opened up," leaning backwards.
• The silicon bond angle $109.5^{\circ }$ is obtuse: the bonds lean away from each other. That "leaning away" is the physical fact that atoms want maximum space.

3. Vectors — arrows with direction and length

The parent note writes things like $\mathbf{a}=(1,1,1)$. That bold letter is a vector.

The picture: an arrow from the origin (the corner $0,0,0$) to the point $(x,y,z)$.

Why the topic needs it: each of silicon's 4 bonds is a direction. To find the angle between two bonds, we treat them as two vectors — and there's exactly one tool for the angle between vectors.

4. Length of a vector — why the square root appears

The parent note writes $|\mathbf{a}|=\sqrt{3}$. Here's where that comes from.

$|\mathbf{a}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$

• Why the topic needs it: the angle formula divides by both vectors' lengths, so we must be able to compute them.

5. Cosine — a dial that turns an angle into a number

The very next tool, the dot product, secretly carries an angle inside it. But an angle is a turn, and the dot product is a plain number — so we first need the gadget that converts one into the other. That gadget is the cosine.

6. The dot product — the tool that measures angles

This is the heart of the derivation, so we build it slowly.

• When two arrows point the same way, $\cos \theta =1$ and the dot product is large positive.
• When they are perpendicular ($90^{\circ }$), $\cos \theta =0$ and the dot product is exactly $0$.
• When they point apart (obtuse), $\cos \theta$ is negative — and this is exactly what happens for silicon's bonds.

7. arccos — undoing the cosine to get the angle back

The dot product hands us $\cos \theta =-\frac{1}{3}$. But we wanted the angle $\theta$, not its cosine. We need the reverse button.

$\theta =\arccos \left(-\frac{1}{3}\right)\approx 109.47^{\circ }$

Because $-\frac{1}{3}$ is negative, $\theta$ lands between $90^{\circ }$ and $180^{\circ }$ — i.e. obtuse, i.e. bonds spread apart. Every case checks out.

8. The tetrahedron — the shape of "spread out"

The picture: put an atom at the centre, its 4 bonds reaching to the 4 corners of this tent. The angle between any two of those bonds is the $109.5^{\circ }$ we just derived.

9. The cube, its edge $a$, and its diagonals

10. Sharing atoms — fractions and the counting

$N=\underset{\text{corners}}{\underbrace{8\times \frac{1}{8}}}+\underset{\text{faces}}{\underbrace{6\times \frac{1}{2}}}+\underset{\text{interior}}{\underbrace{4\times 1}}=1+3+4=8$

• Why fractions at all? If every cube claimed its corner atoms fully, we'd count the same atom 8 times over. Fractions stop the double-counting.

11. Number density $n$ and scientific notation

$n=\frac{8}{(5.43\times 10^{-8}\text{cm}{)}^3}\approx 5.0\times 10^{22}{\text{atoms/cm}}^3$

• Why the topic needs it: later, when we add impurities (Doping of silicon), we compare against this $\sim 5\times 10^{22}$ benchmark to see how few dopant atoms we actually add.

12. Where these foundations lead (FCC, band gap, carriers)

You'll meet three more names built on top of the above:

• Face-centred cubic (FCC) lattice — a cube with atoms at corners and face-centres. Silicon = two of these interlocked.
• Energy band gap in semiconductors — because the lattice repeats perfectly, electron energies clump into allowed "bands" with a forbidden gap $E_g$ between them.
• Holes and electrons as charge carriers — when a bond breaks, the freed electron and the empty spot ("hole") both carry current.
• Miller indices and crystal planes and Single-crystal silicon wafer manufacturing — how we name directions and grow the perfect crystal.

Prerequisite map

Equipment checklist

Test yourself — you're ready for the parent note when you can answer every line.