Diffusion current and concentration gradient

2.1.9 · Hardware › Band Theory & Carrier Physics

Diffusion current KYA hai?

Drift se key contrast:

	Drift current	Diffusion current
Cause	Electric field $E$	Concentration gradient $d n / d x$
Field chahiye?	Haan	Nahi
Proportional to	$n$ (carrier density)	$d n / d x$ (slope)

Net current flow KYUN hota hai? (Feynman-level)

Fick's law ko first principles se KAISE derive karte hain

Hum flux ko ek simple "hopping" picture se derive karte hain.

Setup: Carriers left ya right mean free path $ℓ$ se hop karte hain har collision time $τ$ mein. Mano $v_{t h} = ℓ / τ$ thermal speed hai. Ek plane $x_{0}$ par socho.

Step 1 — Count karo kitne carriers cross kar sakte hain. Yeh step kyun? Sirf woh carriers jo plane se ek mean free path $ℓ$ ke andar hain, apne next collision se pehle wahan pahunch sakte hain. Unme se aadhe (symmetry se) plane ki taraf ja rahe hain.

Left slab se (centered at $x_{0} - ℓ /2$ ): rightward flux $= \frac{1}{2} n (x_{0} - \frac{ℓ}{2}) v_{t h}$
Right slab se (centered at $x_{0} + ℓ /2$ ): leftward flux $= \frac{1}{2} n (x_{0} + \frac{ℓ}{2}) v_{t h}$

Step 2 — Net particle flux $F$ (rightward positive). $F = \frac{1}{2} v_{t h} [n (x_{0} - \frac{ℓ}{2}) - n (x_{0} + \frac{ℓ}{2})]$ Yeh step kyun? Net flow = (right jaana) − (left jaana).

Step 3 — $x_{0}$ ke around Taylor expand karo. $n (x_{0} \pm \frac{ℓ}{2}) \approx n (x_{0}) \pm \frac{ℓ}{2} \frac{d n}{d x}$ Toh bracket ban jaata hai $- ℓ \frac{d n}{d x}$ , jisse milta hai $F = - \frac{1}{2} v_{t h} ℓ \frac{d n}{d x} \equiv - D \frac{d n}{d x}$ Yeh step kyun? Dono Taylor terms ka difference slope ko isolate karta hai, proving karta hai ki flux gradient par depend karta hai. Humne $D = \frac{1}{2} v_{t h} ℓ$ ko diffusion coefficient naam diya.

Step 4 — Flux ko current mein convert karo. Charge se multiply karo. Yeh step kyun? Current density $J = (charge) \times (particle flux)$ .

Electrons (charge $- q$ ): $J_{n} = (- q) F_{n} = (- q) (- D_{n} \frac{d n}{d x})$ $J_{n} = + q D_{n} \frac{d n}{d x}$
Holes (charge $+ q$ ): $J_{p} = (+ q) F_{p} = (+ q) (- D_{p} \frac{d p}{d x})$ $J_{p} = - q D_{p} \frac{d p}{d x}$

Figure — Diffusion current and concentration gradient

Einstein relation (diffusion ko drift se connect karna)

Kyun ka sketch: Equilibrium mein, diffusion current drift current ko exactly cancel karta hai ( $J = 0$ ). $q D_{n} \frac{d n}{d x} = - q n μ_{n} E$ set karke aur $n \propto e^{q V / k T}$ (Boltzmann) use karne par $D / μ = k T / q$ force hota hai.

Worked examples

Common mistakes

Active recall

Recall Quick self-test (hide karke answer karo)

Diffusion current kis quantity par depend karta hai? ⇒ gradient $d n / d x$ par, $n$ par nahi.
Fick's law mein minus sign kyun hai? ⇒ flow gradient ke neeche hota hai.
$J_{n}$ aur $J_{p}$ ke opposite signs kyun hain? ⇒ opposite carrier charge.
$D$ aur $μ$ ko kya link karta hai? ⇒ Einstein relation $D / μ = k T / q$ .

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek classroom jahan saare bachche ek corner mein thuse hue hain aur baaki room khali hai. Kisi ko order nahi diya move karne ka — lekin jab bachche randomly wander karte hain, toh bheed wale corner se OUT jaane wale zyada hote hain andar wapas aane waalon se, sirf isliye kyunki corner packed hai. Dheere dheere room evenly fill ho jaata hai. Agar bachche charged badges pehne hों, toh yeh wandering ek electric current hai — diffusion current. Do jagahon ke beech crowding ka jitna zyada farq ho (jitni steep "crowdedness slope"), utna tez flow hoga.

Flashcards

Diffusion current kis quantity ke proportional hota hai?

Concentration gradient

d n / d x

ke (concentration

n

ke nahi).

Particle flux ke liye Fick's first law batao.

F = - D \frac{d n}{d x}

(flow gradient ke neeche hota hai).

Electron diffusion current density likho.

J_{n} = + q D_{n} \frac{d n}{d x}

Hole diffusion current density likho.

J_{p} = - q D_{p} \frac{d p}{d x}

J_{n}

aur

J_{p}

ke opposite signs kyun hain jabki physical flow direction same hai?

Opposite charge: electron flux times

- q

sign flip karta hai; hole flux times

+ q

rakhta hai.

Diffusion ki physical origin kya hai?

Random thermal motion combined with concentration imbalance; koi field required nahi.

Einstein relation batao.

D / μ = k T / q = V_{T} \approx 25.9

mV at 300 K.

Diffusion coefficient

D

ki units kya hain?

cm^{2} / s

Kya zero electric field mein diffusion current exist kar sakta hai?

Haan — ise sirf gradient chahiye, field nahi.

Agar

n

uniform hai, toh diffusion current kya hai?

Zero, kyunki

d n / d x = 0

Fick's law mein minus sign physically kahan se aata hai?

Net flow high se low concentration ki taraf jaata hai, yaani increasing-

n

direction ke opposite.

Hopping model mein

D

kya hai?

D = \frac{1}{2} v_{t h} ℓ

(aadhi thermal speed times mean free path).

Connections

Drift current and mobility — field-driven counterpart; dono milke total $J$ banate hain.
Einstein relation — $D$ ko $μ$ se $k T / q$ ke zariye tie karta hai.
Continuity equation — diffusion + drift + recombination ko combine karta hai.
PN junction diode — minority carriers ka diffusion diode current set karta hai.
Fick's laws of diffusion — semiconductors se aage general framework.
Thermal voltage $V_T$ — Einstein relation aur diode equation mein appear hota hai.