---
title: "Generation of carriers and conductivity"
subject: "Electrical Engineering Material"
chapter: 6
key: "semester_3/electrical_engineering_material:generation-conduction-conductivity"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/generation-conduction-conductivity
site: BCT Engineering
---

# Generation of carriers and conductivity

Electrical Engineering Material · Chapter 6: Semiconductors

- Asked in: 2075 Chaitra, 6 marks
- Video: [ECE Purdue Semiconductor Fundamentals L1.5: Materials Properties - Free Carriers in Semiconductor](https://www.youtube.com/watch?v=TKCWJLVIwoE) — nanohubtechtalks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/generation-conduction-conductivity
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6

## Question

Describe the phenomenon of generation of electrons and holes, and conduction in semiconductor. Also derive equation for conductivity.

## Short answer

In a semiconductor, thermal energy breaks covalent bonds, creating free electrons and holes. Conduction occurs when both carriers drift under an electric field. Conductivity σ = e(nμ_e + pμ_h), summing electron and hole contributions.

## Answer

#### Generation of electrons and holes

In a pure semiconductor such as Si, every atom forms four covalent bonds with its neighbours. At 0 K all valence electrons are held in bonds; the valence band is full, the conduction band is empty, and the crystal is an insulator.

At a higher temperature the lattice vibrates. When a bond electron gains energy $\ge E_g$ (1.1 eV for Si) from thermal vibration (or from a photon, $h\nu \ge E_g$), it **breaks free** and jumps to the conduction band. This leaves an empty bond, a **hole**, which acts as a positive charge $+e$ in the valence band. So carriers are always produced in pairs: **electron–hole pair generation**.

```text
 Ec ----- e-  (free electron) ----
          ^
          |  thermal energy >= Eg
          |
 Ev ----- h+  (hole) -------------
```

At the same time, free electrons meet holes and **recombine**. In equilibrium the generation rate equals the recombination rate, giving a steady concentration $n = p = n_i$, where $n_i \propto T^{3/2}e^{-E_g/2kT}$ grows rapidly with temperature.

#### Conduction

When a field $\mathcal{E}$ is applied:
- Free electrons in the CB drift opposite to $\mathcal{E}$.
- A neighbouring bond electron jumps into a hole, so the hole moves in the direction of $\mathcal{E}$. Holes behave like positive particles.

Both movements give current in the same direction (along $\mathcal{E}$), so the currents add.

#### Derivation of conductivity

Let $n$, $p$ be the concentrations and $v_{de}$, $v_{dh}$ the drift velocities. The current density due to electrons is $J_e = env_{de}$, and due to holes $J_h = epv_{dh}$. With $v_{de} = \mu_e\mathcal{E}$ and $v_{dh} = \mu_h\mathcal{E}$:

$$
\begin{aligned}
J &= J_e + J_h = en\mu_e\mathcal{E} + ep\mu_h\mathcal{E}\\
&= e(n\mu_e + p\mu_h)\mathcal{E}
\end{aligned}
$$

Comparing with Ohm's law $J = \sigma\mathcal{E}$:

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

Special cases:
- Intrinsic: $\sigma_i = en_i(\mu_e+\mu_h)$
- n-type: $\sigma \approx eN_d\mu_e$
- p-type: $\sigma \approx eN_a\mu_h$

Example: intrinsic Si, $\sigma_i = 1.45\times10^{10}\times1.6\times10^{-19}\times1800 \approx 4.2\times10^{-6}$ S cm⁻¹.

## Check yourself

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

1. How are electron-hole pairs generated?
   A. An electron gaining energy at least Eg (thermal or photon) leaves a bond and a hole is left
   B. Electrons gain less than Eg
   C. Holes are donated by donors only
   D. At 0 K carriers are plentiful
   Answer: A. Carriers always appear in pairs. (Key point: Generation)

2. What is the equilibrium condition?
   A. Generation is zero
   B. Generation rate equals recombination rate, giving n = p = ni, which rises with T
   C. ni falls with T
   D. Recombination stops
   Answer: B. ni is proportional to T^(3/2) exp(-Eg/2kT). (Key point: Equilibrium)

3. How do electrons and holes move in a field?
   A. Both along E, cancelling
   B. Electrons along E, currents cancel
   C. Electrons opposite to E, holes along E, and their currents add in the same direction
   D. Holes do not move
   Answer: C. Opposite charge and opposite motion give same-direction current. (Key point: Conduction directions)

4. How is sigma derived?
   A. J = en mu_e E - ep mu_h E
   B. J = (n + p) E / e
   C. sigma = e/(n mu_e)
   D. J = en mu_e E + ep mu_h E, compared with J = sigma E
   Answer: D. sigma = e(n mu_e + p mu_h). (Key point: Conductivity derivation)

## Related in Electrical Engineering Material

- [Mechanisms generating only electrons or holes](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ageneration-electrons-holes-mechanisms) — 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
- [Why n-type has many conduction electrons](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Antype-electrons-conduction-band) — Electrical Engineering Material, chapter 6
- [Compensated semiconductor concentrations and conductivity](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acompensated-conductivity) — Electrical Engineering Material, chapter 6
- [Si with 10¹⁶ As: carriers, conductivity, Fermi shift](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aarsenic-1e16-carriers-fermi) — Electrical Engineering Material, chapter 6
- [p-type minority carrier suppression and conductivity](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aptype-minority-suppression) — Electrical Engineering Material, chapter 6

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