Electron-hole pair generation: Foundations — Hardware deep dive

Before you can read the parent topic comfortably, you must know every squiggle it throws at you. This page defines all of them from nothing, in the order that each one leans on the one before it. If a symbol scared you on the parent page, it is unpacked here.

0 · The picture the whole topic lives inside

Every symbol below is a label on one diagram: two horizontal shelves with a gap between them. The lower shelf holds electrons stuck in bonds; the upper shelf is where a free electron can roam; the gap is the price of the jump.

Keep this picture in your head. Each new symbol is just a name for a piece of it.

1 · Electron and its charge $q$

The picture: a small dot on the lower shelf. Why the topic needs it: generation is the story of one electron moving, so you must know what carries the charge before anything else.

2 · The two shelves: valence band and conduction band

The word band means "a group of energy levels so close together they act like one continuous shelf an electron can sit on."

Why two bands? Because a material can only conduct if electrons have somewhere free to move. The gap between "stuck" and "free" is the entire drama of generation. (Deeper build: Energy Bands & Band Gap.)

3 · Energy, and measuring it in electron-volts (eV)

The picture: the vertical axis of the shelf diagram is labelled $E$ . Moving a dot upward costs energy; letting it fall releases energy. Why eV and not joules? A joule is enormous for one electron — like measuring a grain of rice in tonnes. eV keeps the numbers human-sized ( $\sim 1$ instead of $\sim 1 0^{- 19}$ ).

4 · The band gap $E_{g}$

The picture: the vertical arrow spanning the gap in figure s01 — its length is $E_{g}$ . Why the topic needs it: $E_{g}$ is the entrance fee for making a pair. No jump costs less than $E_{g}$ ; that single fact drives the cutoff wavelength and the temperature behaviour later.

5 · The hole and its charge $+ q$

Why invent the hole at all? A valence band has billions of electrons; tracking them all is hopeless. It is far easier to track the handful of empty seats. Watch the figure: as electrons shuffle left to fill the gap, the empty seat slides right — so the hole "moves" the opposite way to the electrons and looks positive. (Full story: Recombination & Carrier Lifetime.)

6 · Counting carriers: $n$ , $p$ , and concentration

Why "per volume" and not a plain count? A big crystal has more carriers than a small one, but the crowding is the same — and crowding is what determines current. Concentration measures crowding, so it is the fair thing to compare.

7 · The intrinsic carrier concentration $n_{i}$

The picture: in figure s01, count the dots on the top shelf ( $n$ ) and the empty seats on the bottom shelf ( $p$ ) — in a pure crystal they are always equal, and that equal count is $n_{i}$ . Why the topic needs it: $n_{i}$ is the single number that tells you "how conducting is this pure material at this temperature." (Compare pure vs doped in Intrinsic vs Extrinsic Semiconductors.)

8 · Temperature $T$ , the thermal energy $k T$ , and $k$

The picture: heat makes the shelves vibrate; $k T$ is a measure of how violently. Why the topic needs it: to break a bond you must supply $E_{g}$ , and thermal vibration is one way to supply it. Comparing $E_{g}$ against $k T$ tells you how rare a spontaneous jump is — that comparison is the heart of the next symbol.

9 · The exponential $e^{- E_{g} /2 k T}$

This is the scariest-looking symbol on the parent page, so we build it in three tiny pieces.

Why the $2$ in $2 k T$ ? In a pure crystal the "starting height" for the jump (the Fermi level, symbol below) sits halfway up the gap, so a carrier only climbs $E_{g} /2$ , not the full $E_{g}$ . That halved climb is why the denominator is $2 k T$ . (This is the single most common mistake on the topic — see the parent's mistake callout.)

10 · The Fermi level $E_{F}$

The picture: a dashed line halfway up the gap in figure s04. Why the topic needs it: it is the "sea level" from which every carrier measures its climb — and its mid-gap position is what puts the factor of $2$ in the exponent above. (Deeper: Fermi-Dirac Distribution & Fermi Level.)

11 · Densities of states $N_{c}$ , $N_{v}$ and the square root

12 · Photons: $h ν$ , $h c / λ$ , and the symbols in them

Why two forms of the same thing? $h ν$ says "energy per wiggle × wiggles-per-second." Substituting $ν = c / λ$ rewrites it in terms of wavelength, which is what we actually measure for coloured light. Why the topic needs it: to free an electron with light you need $E_{photon} \geq E_{g}$ ; that inequality gives the cutoff wavelength. (Application: Photodiodes & Solar Cells.)

13 · Generation & recombination: $G$ , $R$ , $r$

The picture: dots jumping up ( $G$ ) and falling back down ( $R$ ) in figure s01; when the up-rate equals the down-rate the populations hold steady. Why the topic needs it: $n p = n_{i}^{2}$ is the rule that survives even after doping — it is why adding electrons pushes holes down. (Currents that follow: Drift & Diffusion Currents; adding carriers: Doping — Donors & Acceptors.)

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

What does

q

stand for and what is an electron's charge?

q

is the size of one unit of charge; an electron carries

- q

Which band is "bound" and which is "free"?

Valence band = bound (lower shelf); conduction band = free (upper shelf).

What is energy measured in here, and why not joules?

Electron-volts (eV); joules are absurdly large for a single particle.

Define the band gap

E_{g}

in one line.

The energy height an electron must be given to jump from the valence edge to the conduction edge.

Is a hole a real particle? What charge does it act like?

No — it is a missing electron (an empty seat) that behaves like a

+ q

carrier.

What do

n

and

p

count?

n

= free electrons per volume;

p

= holes per volume.

In a pure crystal, how do

n

p

, and

n_{i}

relate?

n = p = n_{i}

What does

k T

represent physically?

The typical energy of thermal jiggling;

\approx 0.026

eV at 300 K.

Why is the exponent

- E_{g} /2 k T

and not

- E_{g} / k T

for

n_{i}

The Fermi level sits mid-gap, so a carrier climbs only half the gap.

Where does the Fermi level

E_{F}

sit in a pure semiconductor?

Near the middle of the band gap.

Why does

n_{i}

contain a square root?

n_{i}

is one side of the pair, so we take the root of the electron×hole product

N_{c} N_{v} e^{- E_{g} / k T}

Two ways to write a photon's energy?

E = h ν = h c / λ

What must a photon satisfy to make a pair?

E_{photon} \geq E_{g}

, i.e.

λ \leq h c / E_{g}

State the mass-action law and when it holds.

n p = n_{i}^{2}

; holds in equilibrium, even after doping.

Connections

1.3.06 Electron-hole pair generation (Hinglish)
Energy Bands & Band Gap
Intrinsic vs Extrinsic Semiconductors
Doping — Donors & Acceptors
Recombination & Carrier Lifetime
Fermi-Dirac Distribution & Fermi Level
Photodiodes & Solar Cells
Drift & Diffusion Currents