Diffusion current and concentration gradient: Question bank

Before the traps, let us pin down every symbol used on this page, in plain words. Do not skip this — the traps below only work if these are second nature.

Definition The symbols on this page (each earned before use)

$n$ — the electron concentration: how many electrons sit in one cubic centimetre, in $cm^{- 3}$ . Picture dots per box. $p$ is the same idea for holes.
$\frac{d n}{d x}$ — the concentration gradient, i.e. the slope of the dots-per-box curve as you walk along $x$ . Steep curve = big number; flat curve = zero.
$F$ — the particle flux: how many carriers cross a unit area per second, in $cm^{- 2} s^{- 1}$ . It counts particles, not charge.
$q$ — the elementary charge, the size of the charge on one carrier, $q = 1.6 \times 1 0^{- 19} C$ (always positive; the sign of an electron is put in by hand as $- q$ ).
$J$ — the current density: charge crossing a unit area per second, in $A/cm^{2}$ . This is $F$ turned into charge flow.
$D$ — the diffusion coefficient, $cm^{2} / s$ : how vigorously carriers spread. Built from the hopping picture as $D = \frac{1}{2} v_{t h} ℓ$ .
$v_{t h}$ — the thermal speed, $cm/s$ : the average random speed a carrier moves between collisions.
$ℓ$ — the mean free path, $cm$ : the average distance a carrier travels before it collides and forgets which way it was going.
$μ$ — the mobility, $cm^{2} / (V\cdots)$ : how fast carriers drift per unit electric field, i.e. drift speed $= μ E$ . It measures how easily a field pushes them.

Now the three anchors to hold in your head while answering:

Recall The three anchors (peek only if stuck)

Flux is always $F = - D \frac{d n}{d x}$ for both carriers — the sign of current comes only from the charge factor in $J = q F$ .
Diffusion depends on the slope $d n / d x$ , never on the raw density $n$ .
No electric field is required; random thermal motion + a concentration imbalance is enough.

The two figures below are the mental pictures every trap refers back to.

Figure — Diffusion current and concentration gradient

True or false — justify

A crowded region always carries a large diffusion current.

False — if the density is high but uniform (

d n / d x = 0

), equal numbers cross each way, so the net diffusion current is exactly zero.

Diffusion current can flow with no electric field present.

True — it is driven by random thermal motion down a concentration gradient, so an unbiased sample's neutral region can still carry diffusion current.

Electrons and holes diffusing the same physical direction produce currents in the same direction.

False — they carry opposite charge, so in

J = q F

the same

F

gives opposite-sign currents.

Doubling the carrier concentration everywhere doubles the diffusion current.

False — shifting the whole profile up by a constant leaves the slope

d n / d x

unchanged, so the diffusion current is unchanged.

The diffusion coefficient

D

has units of

cm^{2} / s

True — from

D = \frac{1}{2} v_{t h} ℓ

, speed

\times

length gives

(cm/s) (cm) = cm^{2} / s

, an area per time.

At thermal equilibrium the diffusion current inside a pn junction is zero.

False — it is large but cancelled by an equal and opposite drift current, so only the net current is zero, not each part.

Making the concentration profile steeper increases the diffusion current.

True — current

\propto ∣ d n / d x ∣

, so a steeper slope (more crowding difference over the same distance) drives a larger net flux.

The minus sign in

F = - D d n / d x

means diffusion current is always negative.

False — the sign of

F

depends on the sign of the slope; the minus just encodes "carriers flow down the gradient" (from high to low).

D

and mobility

μ

are independent material parameters that happen to be tabulated together.

False — the Einstein relation

D / μ = k T / q = V_{T}

ties them, because both

D

(spreading) and

μ

(field-driven drift) arise from the same random thermal collisions.

For a uniform-slope (linear) profile the diffusion current is the same at every position.

True — a straight line has one constant slope, so

d n / d x

and hence

J

do not vary with

x

Spot the error

J_{n} = q D_{n} d n / d x

has a plus sign, so electrons always give positive current."

The plus comes from

(- q) (- D_{n} d n / d x)

J = q F

; the sign of $J_{n}$ still follows the sign of the slope

d n / d x

, so it can be negative when the electron density falls in

+ x

J_{p} = - q D_{p} d p / d x

, so holes make negative current."

With a falling hole profile (

d p / d x < 0

) the two minus signs cancel and

J_{p} > 0

; holes diffusing in

+ x

carry positive charge in

+ x

"Diffusion is faster where there are more carriers, so current scales with

n

That describes drift (

J_{d r i f t} = q n μ E

, which uses mobility

μ

); diffusion scales with the gradient

d n / d x

, so a dense but flat region gives no diffusion current.

"Because carriers move randomly, there can be no directed net current."

The individual motion is random, but more carriers cross from the crowded side than the sparse side, and this number imbalance is the net flux.

"We can use

F = - D d n / d x

for holes by writing

F = - D_{p} d n / d x

For holes you must use the hole density:

F_{p} = - D_{p} d p / d x

; mixing the electron symbol

n

into the hole flux is a bookkeeping error.

"The Einstein relation gives

D / μ = V_{T}

, so at higher temperature

D

shrinks."

V_{T} = k T / q

rises with temperature, so for fixed

μ

the diffusion coefficient

D = μ V_{T}

actually grows with

T

"In

D = \frac{1}{2} v_{t h} ℓ

, the

\frac{1}{2}

is arbitrary."

It comes from the counting step: only half the carriers in each slab (separated by one mean free path

ℓ

) happen to be heading toward the plane, so exactly half contribute to one-way flux.

Why questions

Why does diffusion current depend on the slope rather than the value of

n

The Taylor expansion of the two crossing fluxes cancels the

n (x_{0})

term and leaves only the difference

- ℓ d n / d x

, so only the local slope survives.

Why is a minus sign needed in Fick's first law?

Carriers move from high to low concentration, i.e. against an increasing

n

, so the flux

F

must be opposite in sign to

d n / d x

Why do

J_{n}

and

J_{p}

carry opposite algebraic signs even though both fluxes are

- D d / d x

J = q F

you multiply flux by charge: electrons contribute

(- q)

and holes

(+ q)

, flipping one sign relative to the other.

Why can diffusion happen with no force pushing the carriers?

The "drive" is statistical, not mechanical — random walks simply spread a crowd out because there are more ways to move into empty space than into full space.

Why are

D

and

μ

linked at all?

Both are set by the mean free path

ℓ

and collision time of the same thermal scattering events, so their ratio reduces to

k T / q

via the Einstein relation.

Why does the crowded-classroom analogy correctly predict a current?

More kids wander out of the packed corner than back in, and if the kids carry charge that net wandering is precisely a diffusion current — larger when the crowding difference is steeper.

Why is only the region within one mean free path

ℓ

of the plane counted in the derivation?

Carriers farther than

ℓ

collide (and randomize direction) before reaching the plane, so they cannot contribute a clean one-way crossing.

Why does drift current, unlike diffusion current, need an electric field?

Drift is carriers being pushed by the field force

q E

at speed

μ E

; remove the field and there is no directed push, whereas diffusion needs only a concentration imbalance.

Edge cases

What is the diffusion current where the concentration profile is flat (

d n / d x = 0

Zero — equal numbers cross each way, so despite possibly high density there is no net diffusion flux.

What happens to the diffusion current at the peak of a bump-shaped profile?

It is zero at the peak because the slope momentarily passes through zero, then reverses sign on the two sides of the peak.

For an exponentially decaying excess profile, where is the diffusion current largest?

At the injection edge — the boundary

x = 0

where excess carriers are pumped in (e.g. the edge of a diode's neutral region) — since the profile is steepest there and

d p / d x

decays with

x

If the concentration increases in the

+ x

direction, which way do electrons physically diffuse?

In the

- x

direction — toward lower density — regardless of any sign convention on current.

What is the net current in a neutral region where diffusion and drift are equal and opposite?

Zero net current, even though each mechanism separately moves many carriers — this is the equilibrium condition behind the Einstein relation.

As temperature approaches zero, what happens to diffusion?

Thermal speed

v_{t h} \to 0

D \to 0

; with no random motion there is no diffusion, and

V_{T} = k T / q \to 0

consistently.

What does the Continuity equation add that Fick's law alone misses?

It tracks how carrier density changes in time when flux is non-uniform (plus generation/recombination), whereas Fick's law only gives the instantaneous flux from the present gradient.