Intuition The one core ideaA semiconductor crystal is always trading electrons between two energy "floors": some jump up (generation), some fall down (recombination). Recombination and generation is the study of how fast the crystal restores balance whenever we push it away from that resting stalemate.
Before you can read a single formula in the parent note, you need to know what each letter means , what picture it stands for, and why the topic can't do without it . We build them in an order where each one leans only on the ones before it. Nothing is used before it is earned.
Everything in this topic lives inside one picture — a band diagram . Let us draw it first, because every symbol that follows is a label on this picture.
Definition Conduction band, valence band, band gap
The valence band is the "ground floor" — where electrons sit when they are bound to atoms.
The conduction band is the "upper floor" — where an electron is free to roam and carry current.
The band gap is the empty stairwell between them: a range of energies no electron is allowed to have.
E g E g height of that stairwell — the minimum energy an electron must gain to jump from the ground floor to the upper floor.
Picture: the vertical arrow spanning the gap in the figure above.
Why the topic needs it: every recombination event releases roughly E g Where that energy goes (light? heat? another electron?) is exactly what splits the three mechanisms apart.
The $ signs just mean "this is a mathematical symbol"; $E_g$ is read "E-sub-g". Nothing more mysterious than a name.
When an electron leaves the ground floor, it leaves behind an empty seat . That empty seat behaves like a positive particle that can also move around. We give it a name.
A hole is the absence of an electron in the valence band — an empty seat that drifts like a positive charge.
Picture: the pale-pink circle with a "+" in the figure; when a neighbouring electron slides into it, the empty seat appears to move the other way.
Now we need to count these particles, because rates depend on how many there are.
product n p n p pairings is n × p n p n + p
Left alone in the dark at a fixed temperature, the crystal settles into a steady resting state. The counts there get a subscript "0".
n 0 p 0 n 0 p 0 at thermal equilibrium (the resting stalemate, no light, no injected current).
Picture: the same slab as before, but frozen in its calm resting state — a fixed number of blue dots and pink seats.
n i perfectly pure crystal, every electron on the upper floor left exactly one seat behind, so n 0 = p 0 n i Picture: a pristine crystal where blue dots and pink seats come in matched pairs — count either one, that's n i Why: n i
Intuition Why the product isfixed (a quick rationale)
Think of it as a chemical balance: thermal energy keeps generating pairs at a steady rate G 0 ∝ n 0 p 0 n 0 p 0 name that constant n i 2 product can't budge, exactly like squeezing a balloon: push one side in, the other bulges out, volume fixed.
this product is the topic's compass
n i 2 n p n p − n i 2 which way the crystal is out of balance :
n p − n i 2 > 0 recombine to remove them.n p − n i 2 < 0 generate to make them.n p − n i 2 = 0 ( n p − n i 2 ) every rate formula in the parent note.
Real devices deliberately add impurities to tilt the balance — this is doping . There are two flavours, and they are perfect mirror images.
N D N D donor atoms per cm − 3 electron . This makes an n-type crystal, where electrons vastly outnumber holes: n 0 ≈ N D p 0 = n i 2 / N D Picture: sprinkle extra blue dots onto the upper floor without adding any new seats.
N A N A acceptor atoms per cm − 3 creates one extra free hole . This makes a p-type crystal, where holes vastly outnumber electrons: p 0 ≈ N A n 0 = n i 2 / N A Picture: punch extra pink empty seats into the ground floor without adding any new upper-floor dots.
Definition Majority vs minority carrier
Majority carrier = the plentiful one — electrons in n-type, holes in p-type.Minority carrier = the scarce one — holes in n-type, electrons in p-type.
Why: recombination bottlenecks on the scarce partner — there are so few of them that the rate is limited by how fast a minority carrier finds a plentiful majority carrier. That is why the parent note keeps talking about "minority-carrier lifetime."
When we shine light or inject current, we add carriers on top of the equilibrium counts. The Greek letter Δ
Δ n Δ p n = n 0 + Δ n , p = p 0 + Δ p Δ n above equilibrium, Δ p above equilibrium.
Picture: the extra dots and seats stacked on top of the calm baseline in the figure.
Why: since light creates electrons and holes in pairs , usually Δ n = Δ p
Definition Low injection vs high injection
Low injection: the excess is tiny compared to the majority count (Δ p ≪ n 0 linear and lifetime is a constant.High injection: the excess rivals or exceeds the majority count. The math turns nonlinear.
Why: the neat formula τ = 1/ ( B n 0 )
Now the counts become movement : how many pairs vanish or appear each second in each cubic centimetre.
R G U
R recombination rate — pairs destroyed per second per cm 3 cm − 3 s − 1 G generation rate — pairs created per second per cm 3 U = R − G net recombination rate.
Picture: two opposing streams of traffic (down = recombine, up = generate); U difference — the leftover flow after cancellation.
Intuition Why we must use thenet U R and generating — but the two flows are equal, so nothing observably changes. Only the imbalance R − G U U = 0 R = 0
If we stop injecting and let the crystal relax, the excess doesn't vanish instantly — it fades. The speed of that fade is captured by one number.
τ τ minority carrier survives before recombining.
Picture: the time it takes the excess-decay curve to fall to a fraction 1/ e ≈ 0.37
Intuition Why we track theminority carrier (and write Δ p Δ n τ p Δ p p-type material electrons are the scarce ones, so there we watch Δ n τ n always follow the minority carrier , because it is the one whose disappearance you can actually measure. Choosing Δ p n ↔ p
Intuition WHY an exponential — and WHY toole fixed fraction of whatever excess remains: d t d Δ p = − τ Δ p only tool that undoes this equation. We reach for e − t / τ Continuity Equation .)
B B cm 3 / s n p R r a d = B n p Why these units work: [ cm 3 / s ] × [ cm − 3 ] × [ cm − 3 ] = [ cm − 3 s − 1 ] τ = 1/ ( B n 0 )
Definition Trap constants:N t σ v t h
N t trap (defect) density — number of mid-gap traps per cm − 3 cm − 3 σ capture cross-section — the effective "catch area" one trap presents to a passing carrier (units cm 2 v t h thermal velocity — the average speed a carrier zips around at (units cm / s Why: together they build the trap lifetime τ = 1/ ( σ v t h N t ) [ cm 2 ] [ cm/s ] [ cm − 3 ] = [ s − 1 ]
Definition Auger coefficients:C n C p three carriers to meet, so its rate has three concentration factors and two possible line-ups:
C n cm 6 / s electron–electron–hole event: two electrons plus a hole, giving the term C n n 2 p C p cm 6 / s hole–hole–electron event: two holes plus an electron, giving the term C p n p 2 Units check: [ cm 6 / s ] [ cm − 3 ] 3 = [ cm − 3 s − 1 ] τ A ug er = 1/ ( C n n 0 2 )
Band diagram: two floors + gap Eg
Equilibrium counts n0 p0 and ni
Law of Mass Action np = ni squared
Doping ND and NA: majority and minority
Distance from balance: np minus ni squared
Excess carriers delta n delta p
Lifetime tau and exponential decay
Recombination and Generation mechanisms
Every arrow means "you need the box on the left to understand the box on the right." Notice how two independent chains — the "balance" chain (mass action → n p − n i 2 U
Cover the right-hand side and test yourself — if any answer is a blank, re-read its section above.
What does E g The band-gap energy — the height of the forbidden stairwell between valence and conduction bands.
What is a hole, physically? The empty seat left when an electron leaves the valence band; it drifts like a positive charge.
What do n p Free electrons and holes per unit volume, in cm − 3
Why does the product n p n + p Because n × p
What is n i The carrier count in a perfectly pure crystal where n 0 = p 0
State the Law of Mass Action and why the product is fixed. n 0 p 0 = n i 2
What is the difference between N D N A N D N A
Difference between majority and minority carriers? Majority = the plentiful carrier from doping; minority = the scarce one that bottlenecks recombination.
What do Δ n Δ p The excess carriers above equilibrium, n = n 0 + Δ n p = p 0 + Δ p
Why must we use U = R − G R At rest R = G = 0 U U = 0
What is τ The carrier lifetime; the minority carrier is scarce, so its disappearance is what we can measure (Δ p , τ p Δ n , τ n
What rate law does B R r a d = B n p B cm 3 / s
Which Auger term is which? C n n 2 p C p n p 2 C cm 6 / s
Why is the decay exponential? Because the crystal removes a fixed fraction of the remaining excess each instant: d Δ p / d t = − Δ p / τ
