Compound semiconductors (GaN, GaAs, SiC) overview: Foundations

This page assumes nothing. Before you meet $E_{g}$ , $μ$ , $λ$ , or the phrase "valence-averaging rule," we build every piece from a picture. Read top to bottom; each idea is the floor the next one stands on.

1. Atom, valence electron, and the group number $g$

Think of the valence electrons as an atom's "hands." An atom is happiest (lowest energy) when its outer shell is full — for the atoms we care about, that means 8 shared electrons around it, i.e. it wants to be holding 4 bonds, each bond being a shared pair.

Figure — Compound semiconductors (GaN, GaAs, SiC) overview

Gallium (Ga) is group III → $g = 3$ → 3 hands. Arsenic (As) is group V → $g = 5$ → 5 hands. Alone, neither makes a clean 4-bond lattice. Together, average $= (3 + 5) /2 = 4$ . Balance restored.

2. Crystal lattice and the lattice constant $a$

Because the pattern repeats, we only need one number to describe its size: the length of the repeating box's edge.

Why does $a$ matter for our topic? Because when you grow one crystal on top of another (which is how compound semiconductors are made — see Epitaxy and crystal growth), the two crystals must line their atoms up. If their $a$ values differ, the atoms don't match, and the mismatch tears the film. That single idea powers the mismatch formula:

Formula Lattice mismatch

$f$ (with sign convention) $f = \frac{a _{layer} - a _{substrate}}{a _{substrate}}$ In words: "by what fraction is the top crystal's spacing bigger or smaller than the base crystal's spacing?" We divide by $a_{substrate}$ because we measure strain relative to the surface the film must conform to. Sign convention — cover every case:

$f = 0$ : perfect match, no strain.
$f > 0$ (layer's natural spacing is bigger than the substrate): the film is squeezed inward to fit → compressive strain in the film.
$f < 0$ (layer's natural spacing is smaller): the film is stretched outward to fit → tensile strain in the film. Either way, if $∣ f ∣$ is large, the strain energy eventually rips the film into dislocations. GaN on sapphire has $∣ f ∣ \approx 0.16$ ( $16%$ ) — a big mismatch, so it needs buffer tricks.

Recall Why divide by

$a_{substrate}$ and not by the layer? Because we ask "how strained is the film forced to be, relative to the surface it must sit on." The substrate is the fixed reference the film must conform to, so it goes on the bottom. ::: The substrate sets the spacing; the film stretches to match it, so we measure strain relative to the substrate.

3. Energy, the electron-volt (eV), and the hill picture

Before "bandgap" can mean anything, we need a way to measure energy at the scale of a single electron.

4. Energy bands and the bandgap $E_{g}$

Why the topic lives or dies on $E_{g}$ :

Light: an electron falling back down across the gap releases roughly $E_{g}$ of energy as a photon → the gap sets the colour (but only if the gap is "direct" — see §5).
Heat/voltage survival: a wide gap resists being broken by heat or strong fields → tough power devices.

5. Direct vs indirect gap — why $E_{g}$ alone does not make light

Here is the subtlety the headline formula hides: having a bandgap is necessary for emitting light, but not sufficient. Whether an electron falling across the gap actually turns into a photon depends on a second property — the momentum alignment of the bands.

6. Photon, frequency $ν$ , wavelength $λ$ , and why $λ = h c / E_{g}$

Why the other two symbols appear:

$h$ = Planck's constant, the fixed conversion between a photon's frequency $ν$ and its energy (energy $= h ν$ ). We need it because it is the link between "how energetic" and "how fast it wiggles."
$c$ = speed of light, the fixed link between $ν$ and $λ$ .

7. Electric field, breakdown field $E_{cr i t}$ , and mobility $μ$

Prerequisite map

Read it bottom-up: atoms give valence counts and lattices; lattices give the mismatch that makes growth hard; bands give the gap, which (together with direct-vs-indirect) feeds colour, and also feeds toughness; mobility feeds speed. All these streams pour into the topic.

Equipment checklist

Cover the right side; can you answer each before revealing?

A group-IV element brings how many valence electrons, and why does that make silicon a clean crystal?

4 — every atom can form exactly 4 bonds with no leftovers.

What does the bar in

\overset{g}{ˉ}

mean?

"The average of" — here the weighted average number of valence electrons per atom.

For a 50/50 III–V compound, what is

\overset{g}{ˉ}

and does it form a silicon-like lattice?

\overset{g}{ˉ} = 0.5 (3) + 0.5 (5) = 4

, so yes.

What does the lattice constant

a

physically measure?

The edge length of the smallest repeating cube — the atom spacing.

Write the lattice mismatch

f

and say what

f > 0

versus

f < 0

means.

f = (a_{layer} - a_{substrate}) / a_{substrate}

;

f > 0

→ film compressed,

f < 0

→ film stretched (tensile),

f = 0

→ perfect match.

What is one electron-volt?

The energy one electron gains crossing a 1-volt difference.

Define the bandgap

E_{g}

in one sentence.

The forbidden energy range (in eV) an electron must be given to jump from the valence band to the conduction band.

Does a wider gap make a material more or less conductive at room temperature?

Less conductive — harder to free carriers.

Why can silicon have a bandgap yet not emit light?

It is indirect-gap — falling electrons must involve a phonon, so they release energy as heat, not light.

The letter

E

means two things — how do you tell them apart?

By subscript and units:

E_{g}

= energy (eV);

E

E_{cr i t}

= electric field (MV/cm or V/cm).

Give the pocket formula linking

E_{g}

to emitted wavelength.

λ [nm] = 1240/ E_{g} [eV]

State the relation defining mobility

μ

v_{drift} = μ E

— drift speed per unit field.

Which single number sets switching speed, and which sets voltage tolerance?

Mobility

μ

sets speed; breakdown field

E_{cr i t}

sets voltage.