Einstein relation between mobility and diffusion

2.1.10 · Hardware › Band Theory & Carrier Physics

KYA relate ho raha hai?

KAISE derive karte hain? (first principles se)

YE derivation kyun kaam karti hai: Thermal equilibrium mein, jab koi external wire nahi hai, net current har jagah zero honi chahiye. Drift aur diffusion currents bilkul cancel hone chahiye. Woh cancellation force karne se, aur known equilibrium carrier distribution se, $D / μ$ ratio nikalta hai.

Step 1 — Dono currents likho (electrons). $J_{n} = drift q n μ_{n} E + diffusion q D_{n} \frac{d n}{d x}$

Ye step kyun? Total electron current = drift + diffusion. Dhyan raho diffusion term $+ q D_{n} d n / d x$ hai kyunki electron charge $- q$ times flux $- D_{n} d n / d x$ deta hai $+ q D_{n} d n / d x$ .

Step 2 — Equilibrium condition. Equilibrium mein $J_{n} = 0$ : $q n μ_{n} E = - q D_{n} \frac{d n}{d x}$

Ye step kyun? Koi battery nahi, koi net current nahi — dono microscopic drives balance karte hain.

Step 3 — Field ko potential se relate karo. $E = - \frac{d ψ}{d x}$ , toh: $n μ_{n} \frac{d ψ}{d x} = D_{n} \frac{d n}{d x}$

Ye step kyun? Field, electrostatic potential ka negative slope hota hai; isse hum equilibrium $n (ψ)$ plug in kar sakte hain.

Step 4 — Equilibrium carrier distribution use karo. Equilibrium mein electron density potential energy $- q ψ$ mein Boltzmann statistics follow karti hai: $n (x) = n_{0} exp (\frac{q ψ ( x )}{k _{B} T}) \Rightarrow \frac{d n}{d x} = n \cdot \frac{q}{k _{B} T} \frac{d ψ}{d x}$

Ye step kyun? Ye physical input hai. Temperature $T$ par, carriers higher-potential-energy states mein exponentially kam rehte hain. $n$ ka gradient poori tarah potential ki wajah se hai.

Step 5 — Substitute karo aur cancel karo. $n μ_{n} \frac{d ψ}{d x} = D_{n} \cdot n \frac{q}{k _{B} T} \frac{d ψ}{d x}$ Dono sides se $n \frac{d ψ}{d x}$ cancel karo: $\frac{D _{n}}{μ _{n}} = \frac{k _{B} T}{q}$

Ye step kyun? Har position-dependent cheez cancel ho jaati hai — ratio ek pure constant hai jo sirf temperature se set hoti hai. Holes ke liye same argument deta hai $D_{p} / μ_{p} = k_{B} T / q$ .

Figure — Einstein relation between mobility and diffusion

YE sach kyun hai, intuitively (kinetic view)

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek playground mein bacche hain. Agar ek teacher hawa chalaye (ek "field"), bacche usi direction mein drift karte hain — woh hai drift/mobility. Bina hawa ke bhi, bacche random daud-te daud-te dheere-dheere bheed wale kone se khaali jagah mein spread ho jaate hain — woh hai diffusion. Wahi daudne wale pair dono cause karte hain! Toh jo baccha jaldi spread hota hai (bada $D$ ) woh wahi baccha hai jise hawa asaani se push karti hai (bada $μ$ ). Bacche kitne hot/energetic hain ( $k_{B} T$ ) yeh set karta hai ki random spreading, wind push ke comparison mein kitni strong hai. Yehi poora raaz hai.

Active recall

Einstein relation formula

D / μ = k_{B} T / q

(thermal voltage

V_{T}

)

Isse derive karne ke liye kaun si condition use hoti hai

Thermal equilibrium mein zero net current — drift aur diffusion currents cancel ho jaate hain

Thermal voltage

k_{B} T / q

ki value 300 K par

≈ 25.9 mV (~0.026 V)

Derivation mein physical input (Step 4)

Boltzmann distribution:

n = n_{0} exp (q ψ / k_{B} T)

D / μ

scattering details se independent kyun hai

Same

τ

aur

m^{*}

drift aur diffusion dono ko govern karte hain, isliye ratio mein cancel ho jaate hain

D_{n}

μ_{n}

ke terms mein express karo

D_{n} = (k_{B} T / q) μ_{n}

Common unit error jo avoid karni hai

k T / q

ki jagah

k T

use karna;

D / μ

volts mein hona chahiye

Classical Einstein relation kab FAIL karta hai

Degenerate (heavily doped) semiconductors mein — Fermi–Dirac chahiye; tab

D / μ > k_{B} T / q

Si electrons ka

D_{n}

(

μ_{n} = 1350

) 300 K par

0.0259 \times 1350 \approx 35 cm^{2} / s

D / μ

temperature ke saath kaise scale karta hai

Linearly,

D / μ \propto T

Connections

Drift current and mobility
Diffusion current and Fick's law
Boltzmann distribution in semiconductors
Thermal voltage V_T
Continuity equation and carrier transport
Degenerate semiconductors & generalized Einstein relation
Fermi-Dirac vs Maxwell-Boltzmann statistics