Diffusion current and concentration gradient

WHAT is diffusion current?

Key contrast with drift:

	Drift current	Diffusion current
Cause	Electric field $E$	Concentration gradient $dn/dx$
Needs field?	Yes	No
Proportional to	$n$ (carrier density)	$dn/dx$ (slope)

WHY does a net current flow? (Feynman-level)

HOW to derive Fick's law from first principles

We derive the flux from a simple "hopping" picture.

Setup: Carriers hop left or right by a mean free path $\ell$ every collision time $\tau$ . Let $v_{th}=\ell/\tau$ be thermal speed. Consider a plane at $x_0$ .

Step 1 — Count carriers that can cross. Why this step? Only carriers within one mean free path $\ell$ of the plane can reach it before their next collision. Half of them (by symmetry) head toward the plane.

From the left slab (centered at $x_0-\ell/2$ ): flux rightward $=\tfrac12 n(x_0-\tfrac{\ell}{2})\,v_{th}$
From the right slab (centered at $x_0+\ell/2$ ): flux leftward $=\tfrac12 n(x_0+\tfrac{\ell}{2})\,v_{th}$

Step 2 — Net particle flux $F$ (rightward positive). $F = \tfrac12 v_{th}\Big[\,n(x_0-\tfrac{\ell}{2}) - n(x_0+\tfrac{\ell}{2})\,\Big]$ Why this step? Net flow = (going right) − (going left).

Step 3 — Taylor expand around $x_0$ . $n(x_0\pm\tfrac{\ell}{2}) \approx n(x_0) \pm \frac{\ell}{2}\frac{dn}{dx}$ So the bracket becomes $-\ell\,\dfrac{dn}{dx}$ , giving $F = -\tfrac12 v_{th}\,\ell\,\frac{dn}{dx} \equiv -D\,\frac{dn}{dx}$ Why this step? The difference of the two Taylor terms isolates the slope, proving flux depends on the gradient. We named $D=\tfrac12 v_{th}\ell$ the diffusion coefficient.

Step 4 — Turn flux into current. Multiply by charge. Why this step? Current density $J=(\text{charge})\times(\text{particle flux})$ .

Electrons (charge $-q$ ): $J_n = (-q)\,F_n = (-q)\Big(-D_n\dfrac{dn}{dx}\Big)$ $\boxed{J_n = +q D_n \frac{dn}{dx}}$
Holes (charge $+q$ ): $J_p = (+q)\,F_p = (+q)\Big(-D_p\dfrac{dp}{dx}\Big)$ $\boxed{J_p = -q D_p \frac{dp}{dx}}$

Figure — Diffusion current and concentration gradient

Einstein relation (connecting diffusion to drift)

Sketch of why: In equilibrium, diffusion current exactly cancels drift current ( $J=0$ ). Setting $qD_n\frac{dn}{dx}=-qn\mu_n E$ and using $n\propto e^{qV/kT}$ (Boltzmann) forces $D/\mu = kT/q$ .

Worked examples

Worked example Example 1 — Electron diffusion current

A silicon sample has electrons varying linearly from $n=10^{17}\text{ cm}^{-3}$ at $x=0$ to $n=10^{16}\text{ cm}^{-3}$ at $x=2\,\mu m$ . Given $D_n=25\text{ cm}^2/\text{s}$ . Find $J_n$ .

Step 1 — Gradient. Why: linear profile ⇒ constant slope. $\frac{dn}{dx}=\frac{10^{16}-10^{17}}{2\times10^{-4}\text{cm}}=\frac{-9\times10^{16}}{2\times10^{-4}}=-4.5\times10^{20}\text{ cm}^{-4}$ Step 2 — Apply formula. Why: $J_n=qD_n\,dn/dx$ . $J_n=(1.6\times10^{-19})(25)(-4.5\times10^{20})= -1800\ \text{A/cm}^2$ Interpretation: electrons diffuse in $+x$ (high→low), so conventional current is in $-x$ — hence the negative sign. Magnitude $1800\text{ A/cm}^2$ .

Worked example Example 3 — Exponential profile (real diodes)

Excess holes in an n-region decay as $\delta p(x)=\delta p(0)e^{-x/L_p}$ with $L_p=5\,\mu m$ , $\delta p(0)=10^{14}\text{ cm}^{-3}$ , $D_p=12\text{ cm}^2/\text{s}$ . Find $J_p$ at $x=0$ .

Step 1 — Gradient. Why: differentiate the exponential. $\frac{dp}{dx}=-\frac{\delta p(0)}{L_p}e^{-x/L_p}\Big|_{0}=-\frac{10^{14}}{5\times10^{-4}}=-2\times10^{17}\text{ cm}^{-4}$ Step 2 — Apply. $J_p=-qD_p\frac{dp}{dx}=-(1.6\times10^{-19})(12)(-2\times10^{17})=+0.384\text{ A/cm}^2$ Why positive: holes diffuse in $+x$ , and holes carry $+$ charge, so current is $+x$ . ✓

Common mistakes

Active recall

Recall Quick self-test (hide and answer)

What quantity does diffusion current depend on? ⇒ the gradient $dn/dx$ , not $n$ .
Why is there a minus sign in Fick's law? ⇒ flow is down the gradient.
Why do $J_n$ and $J_p$ have opposite signs? ⇒ opposite carrier charge.
What links $D$ and $\mu$ ? ⇒ Einstein relation $D/\mu=kT/q$ .

Recall Feynman: explain to a 12-year-old

Imagine a classroom where all the kids are squished into one corner and the rest of the room is empty. Nobody is ordered to move — but as kids randomly wander, more of them wander OUT of the crowded corner than wander back IN, simply because the corner is packed. Slowly the room fills up evenly. If the kids are wearing charged badges, this wandering is an electric current — diffusion current. The bigger the difference in crowding between two spots (the steeper the "crowdedness slope"), the faster the flow.

Flashcards

Diffusion current is proportional to which quantity?

The concentration gradient

dn/dx

(not the concentration

n

itself).

State Fick's first law for particle flux.

F=-D\,\dfrac{dn}{dx}

(flow down the gradient).

Write the electron diffusion current density.

J_n = +qD_n\,\dfrac{dn}{dx}

Write the hole diffusion current density.

J_p = -qD_p\,\dfrac{dp}{dx}

Why do

J_n

and

J_p

have opposite signs despite same physical flow direction?

Opposite charge: electron flux times

-q

flips sign; hole flux times

+q

keeps it.

What is the physical origin of diffusion?

Random thermal motion combined with a concentration imbalance; no field required.

State the Einstein relation.

D/\mu = kT/q = V_T \approx 25.9

mV at 300 K.

Units of the diffusion coefficient

D

\text{cm}^2/\text{s}

Can diffusion current exist with zero electric field?

Yes — it needs only a gradient, not a field.

n

is uniform, what is the diffusion current?

Zero, because

dn/dx=0

Where does the minus sign in Fick's law come from physically?

Net flow goes from high to low concentration, i.e., opposite to the increasing-

n

direction.

What is

D

in the hopping model?

D=\tfrac12 v_{th}\ell

(half thermal speed times mean free path).

Connections

Drift current and mobility — the field-driven counterpart; both sum to total $J$ .
Einstein relation — ties $D$ to $\mu$ via $kT/q$ .
Continuity equation — combines diffusion + drift + recombination.
PN junction diode — diffusion of minority carriers sets the diode current.
Fick's laws of diffusion — general framework beyond semiconductors.
Thermal voltage $V_T$ — appears in Einstein relation and diode equation.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, diffusion current ka concept bilkul simple hai. Socho ek jagah pe electrons bahut zyada crowded hain aur paas wali jagah pe kam. Random thermal motion ke wajah se har electron idhar-udhar bhaag raha hai — koi force nahi laga raha. Lekin kyunki ek side pe zyada electrons hain, us side se zyada electrons doosri side jaate hain bajaye wapas aane ke. Iss net flow ko hi hum diffusion current kehte hain, aur ye chalti hai concentration gradient ( $dn/dx$ ) ki wajah se — electric field ki zaroorat hi nahi.

Sabse important baat: diffusion current $n$ (kitne carriers hain) pe depend nahi karti, balki slope $dn/dx$ pe karti hai. Agar carriers uniform hain (slope zero), toh chahe density kitni bhi ho, diffusion current zero hogi. Fick's law kehta hai flux $F=-D\,dn/dx$ — minus sign isliye kyunki flow hamesha high se low taraf jaata hai. Phir current banane ke liye charge se multiply karo: electrons ke liye $J_n=+qD_n\,dn/dx$ aur holes ke liye $J_p=-qD_p\,dp/dx$ . Signs alag isliye hain kyunki electron ka charge negative hai, holes ka positive.

Ek aur cheez yaad rakho — Einstein relation $D/\mu = kT/q = V_T \approx 26$ mV (300 K pe). Kyunki $D$ aur mobility $\mu$ dono same random collisions se aate hain, isliye ye ek doosre se juda hue hain. Ye relation diode current samajhne mein bahut kaam aati hai.

Ye topic itna important kyun hai? Kyunki PN junction diode ka current mostly diffusion se hi banta hai — minority carriers junction ke paas gradient banate hain aur diffuse karte hain. Toh agar tumhe diode, BJT, ya solar cell samajhna hai, diffusion current ki foundation solid honi chahiye.

Diffusion current and concentration gradient

WHAT is diffusion current?

WHY does a net current flow? (Feynman-level)

HOW to derive Fick's law from first principles

Einstein relation (connecting diffusion to drift)

Worked examples

Common mistakes

Active recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Band Theory & Carrier Physics

Connections