Electron-hole pair generation: Question bank — Hardware deep dive

Read each prompt, commit to an answer out loud with a reason, then reveal. A bare "true/false" scores zero; the reasoning is the point.

Definition Symbol glossary — read this first

Every symbol used on this page, in plain words:

$E_{g}$ — band gap, the energy an electron must gain to jump from bound (valence) to free (conduction). Measured in electron-volts (eV).
$k$ — Boltzmann's constant, the exchange rate between temperature and energy: $k \approx 8.617 \times 1 0^{- 5} eV/K$ . It tells you "how much energy a degree of warmth is worth."
$T$ — absolute temperature in kelvin (K). $k T$ is the typical thermal energy jostling each electron.
$n$ , $p$ — number of free electrons ( $n$ ) and holes ( $p$ ) per unit volume.
$n_{i}$ — intrinsic carrier concentration: the value of $n = p$ in a pure material.
$N_{c}$ , $N_{v}$ — effective density of states at the conduction-band edge ( $N_{c}$ ) and valence-band edge ( $N_{v}$ ): roughly "how many seats are available" for carriers at each band edge. Both grow as $T^{3/2}$ (more thermal room → more available seats).
$h$ — Planck's constant; $ν$ (Greek "nu") — light's frequency. A photon's energy is $h ν$ .
$c$ — speed of light; $λ$ — light's wavelength. Since $c = λ ν$ , photon energy $= h c / λ$ .
$G$ — generation rate (pairs born per second per volume); $R$ — recombination rate (pairs destroyed per second). $G_{t h}$ is the thermal part of $G$ .
$r$ — recombination coefficient, how eagerly an electron and hole annihilate; $R = r n p$ . It sets the carrier lifetime — see Recombination & Carrier Lifetime.

Why questions

Why are electrons and holes always created together, never singly?

Freeing a bound electron necessarily leaves an empty state behind. The "leaving" and the "vacancy" are two descriptions of one event — you cannot have one without the other.

Why does a hole behave as a positive charge carrier?

When valence electrons shift to fill the vacancy, the empty spot appears to move the opposite way to the electrons; a missing negative charge drifting one way is equivalent to a positive charge drifting that way.

Why is the factor of 2 in

E_{g} /2 k T

physically meaningful, not just algebra?

The intrinsic Fermi level sits mid-gap, so a carrier only needs to climb

E_{g} /2

from the Fermi level to the band edge — the exponent measures that half-climb.

Why does raw silicon need doping to be a useful device material?

Intrinsic

n_{i}

is exponentially temperature-sensitive, so device behaviour would drift wildly with temperature; doping fixes the majority carrier concentration so it barely depends on thermal generation.

Why does silicon look opaque to visible light but transparent to far-infrared?

Visible photons exceed

E_{g}

and get absorbed by creating EHPs; far-IR photons have energy below

E_{g}

, cannot free electrons, and pass through unabsorbed.

Why does the mass-action law force holes to drop when you add donor electrons?

The product

n p

must stay at the fixed value

n_{i}^{2}

; if

n

rises,

p

must fall proportionally to keep the product constant — extra electrons speed up recombination of the existing holes.

Edge cases

What happens to

n_{i}

T \to 0

The exponent

- E_{g} /2 k T \to - \infty

, so the exponential crushes

n_{i}

0

(the

T^{3/2}

prefactor also vanishes): no thermal carriers, and the material becomes a perfect insulator.

What is the generation rate for a photon with

h ν

just below

E_{g}

Zero band-to-band generation — the photon lacks the energy to lift an electron across the gap and is not absorbed for EHP creation (it may only be absorbed by other, weaker mechanisms).

For a hypothetical material with

E_{g} = 0

(a semimetal-like limit), what does

n_{i}

do?

The exponential becomes

e^{0} = 1

, so

n_{i} \to N_{c} N_{v}

. But

N_{c}, N_{v} \propto T^{3/2}

, so

n_{i}

is not a fixed number — it still grows with temperature through the density-of-states prefactor, giving abundant carriers that behave like a conductor.

If you shine light with

h ν = 2 E_{g}

, do you get two EHPs per photon?

Not in the simple picture — one photon typically makes one EHP, with the excess energy lost as heat (thermalisation). Extra pairs per photon require special high-energy or multi-exciton effects, not the standard threshold model.

In a heavily doped n-type sample, is thermal generation of pairs switched off?

No. Thermal generation

G_{t h}

still makes pairs, but the huge donor electron population makes recombination

R = r n p

fast (short carrier lifetime), so the equilibrium hole count

p = n_{i}^{2} / n

becomes tiny while pairs are still constantly born and destroyed — see Recombination & Carrier Lifetime.

Recall

Recall One-line self-test

Full formula for $n_{i}$ ? ::: $n_{i} = N_{c} N_{v} e^{- E_{g} /2 k T}$ , with $N_{c}, N_{v} \propto T^{3/2}$ .
Why the exponent $- E_{g} /2 k T$ ? ::: Half the gap, because the intrinsic Fermi level sits mid-gap.
Does $n p = n_{i}^{2}$ survive doping? ::: Yes, in equilibrium — holes adjust to keep the product fixed.
Cutoff condition for optical generation? ::: $h ν \geq E_{g}$ , i.e. $λ \leq h c / E_{g}$ .
Is a hole a real particle? ::: No — it is a collective vacancy that acts like a $+ q$ carrier.
What sets recombination rate and lifetime? ::: $R = r n p$ ; the coefficient $r$ fixes the carrier lifetime.

Connections

1.3.06 Electron-hole pair generation (Hinglish) — parent topic
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

Question bank — Electron-hole pair generation

True or false — justify

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