Crystal lattice structure of silicon: Question bank

Every answer is one or two sentences of real reasoning. Never accept a bare yes/no from yourself.

True or false — justify

Each line: claim, then :::, then the verdict with the reason that makes it true or false.

Silicon is a metal because its outer shell fills up to 8 electrons.

False — those 8 electrons are shared in covalent bonds, locked between atoms, not free to roam; a metal has a sea of free electrons, silicon has almost none at room temperature. See Valence electrons and the octet rule.

The diamond cubic structure is just a simple cube with an atom at each corner.

False — that is simple cubic (6 neighbours at

9 0^{\circ}

). Silicon needs 4 neighbours at

109. 5^{\circ}

, so it uses two interpenetrating FCC lattices (the Diamond cubic structure), 8 atoms per cell.

The lattice constant

a = 5.43 \overset{˚}{A}

is the distance between two bonded silicon atoms.

False —

a

is the cube edge. The bond length is a quarter of the body diagonal,

\frac{3}{4} a \approx 2.35 \overset{˚}{A}

, which is shorter than

a

0 K

pure silicon conducts electricity well.

False — at

0 K

every valence electron is trapped in a bond, so there are zero free carriers and silicon behaves as a perfect insulator.

Because each atom shares an electron with 4 neighbours, each bond contains 4 electrons.

False — each bond is a shared pair (2 electrons, one from each atom); 4 bonds × 2 electrons = 8 electrons around the atom, giving the full octet.

The tetrahedral bond angle

109. 5^{\circ}

is a coincidence of silicon's chemistry.

False — it is pure geometry: 4 repelling bonds spread as far apart as possible on a sphere, giving

arccos (- \frac{1}{3}) \approx 109.4 7^{\circ}

for any group-IV element (carbon in diamond has the same angle).

There are 8 atoms fully inside one diamond-cubic unit cell.

False — only 4 atoms sit fully inside (the tetrahedral/interior atoms); the count reaches 8 only after adding the shared corner (

8 \times \frac{1}{8}

) and face (

6 \times \frac{1}{2}

) contributions.

A single crystal and a lump of many small silicon crystals behave identically in a chip.

False — grain boundaries between small crystals break the periodicity, degrading the band structure and carrier flow; chips demand single-crystal wafers.

Silicon's energy band gap exists because silicon atoms are heavy.

False — the band gap arises from the periodic potential of the ordered lattice; break the periodicity and the clean gap degrades regardless of atomic mass.

Spot the error

Each line states a flawed argument; the reveal names the exact broken step.

"Silicon has 4 valence electrons and wants 8, so it must gain 4 electrons like chlorine gains one."

The error is assuming ionic behaviour. Group IV sits in the middle — gaining or losing 4 electrons costs too much energy, so silicon shares instead, forming 4 covalent bonds.

"Four bonds could sit flat in a square (90° apart), which packs neater than a tetrahedron."

A flat square keeps bonds only

9 0^{\circ}

apart; the tetrahedron spreads them to

109. 5^{\circ}

, which is farther apart on a sphere, minimising electron-pair repulsion (VSEPR). Flat is not the lowest-energy arrangement.

"Each corner atom is one whole atom of the cell, so 8 corners = 8 atoms from corners alone."

A corner atom is shared by 8 neighbouring cubes, so only

\frac{1}{8}

belongs to this cell; 8 corners give

8 \times \frac{1}{8} = 1

atom, not 8.

"Silicon barely conducts, so it must have no valence electrons to share."

Backwards — silicon conducts poorly because it has exactly 4 valence electrons, all tied up in bonds. Having zero would mean no bonding at all.

"Heating silicon adds electrons, which is why it conducts more when hot."

No electrons are added; heat breaks bonds, freeing electrons already present and creating holes. The atom count is unchanged.

"Doping works by physically adding free electrons into empty space in the lattice."

Doping replaces silicon atoms with group-III or group-V atoms; the mismatch in valence electrons is what donates a carrier — nothing floats in empty space.

"The bond angle comes from

arccos (+ \frac{1}{3}) \approx 70. 5^{\circ}

The dot product of vectors to alternate cube corners is $- 1$ , giving

cos θ = - \frac{1}{3}

and an obtuse angle

109. 5^{\circ}

; the positive sign gives the wrong (acute) angle between the wrong pair of directions.

Why questions

Why does silicon form exactly 4 bonds and not 3 or 5?

It has 4 valence electrons and needs 4 more to complete an octet of 8; sharing one electron with each of 4 neighbours does exactly that — no more, no fewer. See Valence electrons and the octet rule.

Why is the tetrahedron the geometry, not any other shape?

4 electron pairs repel; the tetrahedron is the unique arrangement that maximises the minimum angle between 4 points on a sphere, so repulsion is minimised.

Why do we divide corner atoms by 8 and face atoms by 2 when counting?

Because those atoms are shared: a corner is split among 8 abutting cubes and a face among 2, so each cell owns only its fair fraction (

\frac{1}{8}

\frac{1}{2}

Why does the periodicity of the lattice matter, not just the bonding?

The repeating potential is what forces electron energies into allowed bands separated by a gap; a single bond gives energy levels, but only a periodic array gives the band gap that makes silicon a semiconductor.

Why is the atomic density (

\sim 5 \times 1 0^{22} cm^{- 3}

) worth memorising?

It is the reference against which doping concentrations are compared — knowing dopants are millions of times rarer than host atoms explains why they perturb, not replace, silicon's structure.

Why does breaking one bond create two charge carriers, not one?

A freed electron leaves behind an empty bonding spot — a hole — that also carries current; so one broken bond yields an electron and a hole together. See Holes and electrons as charge carriers.

Why can we describe diamond cubic using Miller indices at all?

Because the structure is perfectly periodic, its atomic planes repeat at fixed orientations and spacings, which Miller indices exist to label.

Edge cases

What happens to silicon's conductivity in the limit

T \to 0 K

All bonds stay intact, so the free-carrier count goes to zero and silicon becomes a perfect insulator — the ideal limiting behaviour.

What happens as temperature rises far above room temperature?

More bonds break, freeing many electron-hole pairs, and silicon conducts increasingly well — the opposite of a metal, whose conductivity drops with heat.

If you removed the "two-interpenetrating-FCC" offset and used a single FCC lattice, what breaks?

Each atom would have 12 nearest neighbours (FCC coordination), impossible to satisfy with only 4 valence electrons — the octet cannot be completed, so the covalent picture collapses.

What is the coordination number of silicon, and does it change at the crystal's surface?

Inside, it is 4; at a surface, some atoms have fewer than 4 neighbours, leaving unsatisfied ("dangling") bonds — a boundary case that matters for device interfaces.

If an atom is exactly on the body diagonal but not at a corner, is it counted whole?

The 4 tetrahedral atoms sit inside the cube (along body diagonals but not at corners/faces), so each is counted as a full 1 — they are shared with no other cell.

Does carbon (also group IV) form the same lattice, and does that prove the geometry is chemistry-independent?

Yes — diamond is the same diamond-cubic structure with the same

109. 5^{\circ}

angle, confirming the geometry follows from group-IV bonding, not from silicon specifically.

Recall Quick self-test

Diamond cubic = ? interpenetrating FCC lattices ::: Two, offset by $\frac{1}{4}$ of the body diagonal. Corner atom belongs to how many cells? ::: 8, so it counts as $\frac{1}{8}$ per cell. Sign of $cos θ$ for the bond angle? ::: Negative ( $- \frac{1}{3}$ ), because the angle is obtuse. Free carriers at $0 K$ ? ::: Zero — pure silicon is an insulator there.

Connections

Face-centred cubic (FCC) lattice
Diamond cubic structure
Valence electrons and the octet rule
Energy band gap in semiconductors
Holes and electrons as charge carriers
Doping of silicon
Single-crystal silicon wafer manufacturing
Miller indices and crystal planes