Drift current and electric field — Hinglish notes

2.1.8 · Hardware › Band Theory & Carrier Physics

WHY karte hain hum drift current ki baat?

WHAT hai problem? Ek semiconductor mein equilibrium par (koi field nahi), electrons aur holes constant thermal motion mein hote hain. Yeh high speed (~ $1 0^{5}$ m/s at room temperature) par random directions mein udte hain. Bahut saare carriers ko average karo, to net displacement zero hota hai — koi current nahi.

WHY field isse change karta hai? Ek electric field $E$ har charge par ek force $F = q E$ lagataa hai. Yeh force huge random thermal velocity ke upar ek tiny systematic velocity add karta hai. Woh small biased velocity hi drift velocity $v_{d}$ hai, aur wahi net current produce karta hai.

HOW drift velocity field se relate karta hai? (Derivation from scratch)

Step 1 — Newton between collisions. Do scattering events ke beech, charge $q$ aur effective mass $m^{*}$ wala carrier accelerate karta hai: $a = \frac{q E}{m ^{*}}$ Yeh step kyun? Collisions ke beech koi obstacle nahi hota, isliye sirf electric force hi kaam karta hai; $F = m^{*} a$ se acceleration milta hai.

Step 2 — Carriers har collision par randomize ho jaate hain. Collision ke baad velocity direction randomize ho jaati hai, isliye jo drift velocity gain hui thi woh kho jaati hai. Average mein ek carrier mean free time $τ$ tak travel karta hai collision se pehle.

Step 3 — Gained velocity ko average karo. Collision se just pehle gain ki hui velocity $a τ$ hoti hai; $0$ se $τ$ tak average karo to mean drift velocity hoti hai: $v_{d} = a τ = \frac{q E τ}{m ^{*}}$ Yeh step kyun? Har carrier collision ke baad random (net-zero) velocity se restart karta hai, isliye systematic part wahi hai jo ek free time $τ$ mein accumulate hua.

Step 4 — Mobility define karo. Hum saare material constants ko ek symbol mein bundle karte hain: $μ = \frac{q τ}{m ^{*}} \Rightarrow v_{d} = μ E$

HOW milta hai hume current density?

Step 1 — Ek area cross karne wale charge count karo. Density $n$ (per volume) wale electrons socho jo sab $v_{d}$ par drift kar rahe hain. Time $d t$ mein, ek cross-section $A$ ke $v_{d} d t$ distance ke andar ke saare electrons use cross kar lete hain. Cross kiya hua charge: $d Q = q n (A v_{d} d t)$

Step 2 — Current = charge per time, density = per area. $J = \frac{1}{A} \frac{d Q}{d t} = q n v_{d}$ Yeh step kyun? Current density $J = I / A$ natural quantity hai kyunki yeh sample size par depend nahi karta.

Step 3 — Drift velocity substitute karo. Electrons ke liye ( $q = - e$ , lekin woh $E$ ke opposite drift karte hain, isliye conventional current $E$ ke saath hota hai): $J_{n} = e n μ_{n} E$ aur holes ke liye: $J_{p} = e p μ_{p} E$

Kyun dono terms add hoti hain (subtract nahi)? Electrons ek taraf negative charge carry karte hain; holes doosri taraf positive charge carry karti hain. Dono same direction mein current dete hain ( $E$ ke along), isliye unke contributions add hote hain.

Figure — Drift current and electric field

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Drift velocity kya hai?

Woh net average velocity jo carriers field ki direction mein acquire karte hain, random thermal motion ke upar:

v_{d} = μ E

Mobility define karo aur uska formula do.

Drift velocity per unit field;

μ = q τ / m^{*}

, units cm²/(V·s).

Constant field ke under drift velocity limit ke bina kyun nahi badhti?

Mean free time

τ

par collisions velocity ko randomize kar dete hain, ek steady-state average

v_{d} = μ E

dete hain.

Total drift current density likho.

J = e (n μ_{n} + p μ_{p}) E

Electron aur hole drift currents add kyun hote hain cancel nahi?

Opposite charge AUR opposite drift direction ⇒ do sign flips ⇒ dono currents

E

ke along point karte hain.

Carriers ke terms mein conductivity kya hai?

σ = e (n μ_{n} + p μ_{p})

, aur

J = σ E

(microscopic Ohm's law).

Mobility effective mass aur scattering time par kaise depend karta hai?

μ = q τ / m^{*}

: zyada

τ

(kam collisions) aur kam

m^{*}

(lighter carrier) → zyada mobility.

Drift velocity thermal velocity se badi hai ya chhoti?

Bahut chhoti — thermal ~

1 0^{7}

cm/s, typical drift ~

1 0^{6}

cm/s ya usse kam.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek playground mein bahut saare bacche har random direction mein daud rahe hain — group mein koi actually kahin nahi jaata. Ab ek teacher chillata hai "ice-cream truck us taraf hai!" Har baccha abhi bhi doosron se takrata hai aur direction change karta hai, lekin average mein saari crowd slowly truck ki taraf drift karti hai. Woh slow crowd-drift hi drift velocity hai, aur charged bacchon ki moving crowd hi electric current hai. Zyada push karo (bada field) → woh faster drift karte hain → zyada current.

Connections

Effective Mass — $m^{*}$ jo $μ = q τ / m^{*}$ ke andar hai.
Scattering Mechanisms and Mean Free Time — $τ$ set karta hai aur $μ$ ki temperature dependence.
Diffusion Current and Carrier Gradients — doosra current mechanism; combined via Drift-Diffusion Equation.
Einstein Relation — $μ$ (drift) aur $D$ (diffusion) ko link karta hai.
Conductivity and Resistivity of Semiconductors — $σ = e (n μ_{n} + p μ_{p})$ .
Velocity Saturation and High-Field Effects — jahan $v_{d} = μ E$ break down karta hai.