---
title: "Einstein relation; mobility, conductivity, resistivity"
subject: "Electrical Engineering Material"
chapter: 6
key: "semester_3/electrical_engineering_material:einstein-mobility-conductivity"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/einstein-mobility-conductivity
site: BCT Engineering
---

# Einstein relation; mobility, conductivity, resistivity

Electrical Engineering Material · Chapter 6: Semiconductors

- Asked in: 2075 Chaitra, 5+3 marks
- Video: [Einstein Relation in Semiconductor (Basics, Statement, Meaning & Proof) Explained](https://www.youtube.com/watch?v=t2sRYkVDZaQ) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/einstein-mobility-conductivity
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6

## Question

Derive Einstein's relation between mobility and diffusion co-efficient. Also define the terms electron mobility, conductivity and resistivity.

## Short answer

Equating drift and diffusion currents in equilibrium with flat Fermi level gives D/μ = kT/e. Mobility μ = v_d/E = eτ/m*, conductivity σ = e(nμ_n + pμ_p), resistivity ρ = 1/σ.

## Answer

#### Einstein relation

The **Einstein relation** links the diffusion coefficient $D$ and the drift mobility $\mu$ of a carrier: $\dfrac{D}{\mu} = \dfrac{kT}{e}$.

**Derivation.** Take a non-uniformly doped n-type bar in **thermal equilibrium** (no external voltage). The donor and electron concentration $n(x)$ falls with $x$.

```text
 n(x) high ---> low
 ---------------------------
 |  diffusion: e- move ->   |
 |  field E builds up        |
 |  drift:   e- move <-     |
 ---------------------------
```

1. Electrons diffuse from high to low concentration. Electron diffusion current density:

$$
J_{diff} = eD_e\frac{dn}{dx}
$$

2. The donors left behind are positive, so a built-in field $\mathcal{E}$ appears; it causes a drift current

$$
J_{drift} = en\mu_e\mathcal{E}
$$

3. In equilibrium there is no net current:

$$
en\mu_e\mathcal{E} + eD_e\frac{dn}{dx} = 0
$$

4. In equilibrium the Fermi level is flat, so $n(x) = N_c\exp[-(E_c(x)-E_F)/kT]$ and

$$
\frac{dn}{dx} = -\frac{n}{kT}\frac{dE_c}{dx}
$$

5. The field bends the band: $dE_c/dx = e\mathcal{E}$. Hence

$$
\frac{dn}{dx} = -\frac{ne\mathcal{E}}{kT}
$$

6. Substitute in step 3:

$$
\begin{aligned}
en\mu_e\mathcal{E} - eD_e\frac{ne\mathcal{E}}{kT} &= 0\\
\frac{D_e}{\mu_e} &= \frac{kT}{e}
\end{aligned}
$$

Similarly for holes $D_h/\mu_h = kT/e$. At 300 K, $kT/e = 0.0259$ V; e.g. for Si electrons $D_e = 1350\times0.0259 \approx 35\ \text{cm}^2\text{s}^{-1}$.

#### Electron mobility

**Drift mobility** is the drift velocity per unit applied field:

$$
\mu_e = \frac{v_d}{\mathcal{E}} = \frac{e\tau}{m_e^*}
$$

Unit: m² V⁻¹ s⁻¹ (or cm² V⁻¹ s⁻¹). It shows how easily electrons move; it falls with lattice scattering (higher T) and impurity scattering (heavier doping).

#### Conductivity

**Conductivity** $\sigma$ is the ratio of current density to electric field, $J = \sigma\mathcal{E}$:

$$
\sigma = e(n\mu_e + p\mu_h)
$$

Unit: S m⁻¹ (Ω⁻¹ m⁻¹).

#### Resistivity

**Resistivity** is the reciprocal of conductivity, i.e. the resistance of a unit cube of material:

$$
\rho = \frac{1}{\sigma} = \frac{RA}{L}
$$

Unit: Ω m.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is Einstein's relation?
   A. D/mu = kT/e
   B. D mu = kT/e
   C. D/mu = e/kT
   D. D/mu = kT
   Answer: A. At 300 K, kT/e = 0.0259 V. (Key point: Relation)

2. What condition is used in the derivation?
   A. At equilibrium net current is zero: drift balances diffusion
   B. A large applied voltage
   C. A uniform carrier concentration
   D. No built-in field exists
   Answer: A. en mu E + e D dn/dx = 0, with dn/dx = -n e E/kT. (Key point: Derivation basis)

3. What is the conductivity expression?
   A. sigma = e(n + p)
   B. sigma = e(n mu_e + p mu_h)
   C. sigma = (n mu_e + p mu_h)/e
   D. sigma = e(n mu_e - p mu_h)
   Answer: B. From J = sigma E with both carriers drifting. (Key point: Conductivity)

4. What is resistivity?
   A. rho = sigma
   B. rho = RL/A
   C. rho = 1/sigma = RA/L, resistance of a unit cube
   D. rho = R/(AL)
   Answer: C. Units ohm m. (Key point: Resistivity)

## Related in Electrical Engineering Material

- [Einstein relation between diffusion and mobility](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aeinstein-diffusion-mobility) — Electrical Engineering Material, chapter 6, asked 4×
- [Diffusion process and Einstein relation](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adiffusion-einstein-relation) — Electrical Engineering Material, chapter 6, asked 9×
- [Intrinsic conductivity and resistivity of GaAs](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Agaas-intrinsic-conductivity) — Electrical Engineering Material, chapter 6
- [Diffusion of carriers and conductivity](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adiffusion-conductivity) — Electrical Engineering Material, chapter 6
- [Drift velocity, mobility and conductivity in metals](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adrift-mobility-conductivity-metals) — Electrical Engineering Material, chapter 2

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