---
title: "Intrinsic concentration and its temperature dependence"
subject: "Electrical Engineering Material"
chapter: 6
key: "semester_3/electrical_engineering_material:intrinsic-concentration-derivation"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/intrinsic-concentration-derivation
site: BCT Engineering
---

# Intrinsic concentration and its temperature dependence

Electrical Engineering Material · Chapter 6: Semiconductors

- Asked in: 2069 Chaitra, 6 marks
- Video: [Intrinsic Concentration in Semiconductor Material | Electronics Devices and Circuits - EDC](https://www.youtube.com/watch?v=SOXrmHw_flM) — 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/intrinsic-concentration-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6

## Question

Derive the relation for intrinsic concentration and explain how temperature effects intrinsic concentration with necessary diagram.

## Short answer

Intrinsic concentration ni = √(Nc Nv) exp(-Eg/2kT), with Nc, Nv ∝ T^(3/2). ni increases rapidly with temperature due to the exponential term; a semilog plot of ni vs 1/T yields slope -Eg/2k. Resistance falls sharply (negative temperature coefficient).

## Answer

**Intrinsic concentration** $n_i$ is the number of electrons per unit volume in the conduction band (equal to the number of holes in the valence band) of a pure semiconductor at temperature $T$.

#### Electron concentration in the conduction band
The number of electrons is the density of states $g_c(E)$ times the probability of occupancy $f(E)$, integrated over the band:
$$
n = \int_{E_c}^{\infty} g_c(E)\, f(E)\, dE, \qquad g_c(E) = \frac{8\sqrt{2}\,\pi\, m_e^{*3/2}}{h^3}\sqrt{E-E_c}
$$
For $E - E_F \gg kT$ (non-degenerate case), $f(E) \approx \exp[-(E-E_F)/kT]$ (Boltzmann approximation). Integrating gives
$$
n = N_c \exp\left[-\frac{E_c - E_F}{kT}\right], \qquad N_c = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{3/2}
$$
where $N_c$ is the effective density of states at the conduction band edge.

#### Hole concentration in the valence band
In the same way, using $1 - f(E)$ as the probability that a state is empty:
$$
p = N_v \exp\left[-\frac{E_F - E_v}{kT}\right], \qquad N_v = 2\left(\frac{2\pi m_h^* kT}{h^2}\right)^{3/2}
$$

#### Intrinsic concentration (mass action law)
Multiplying $n$ and $p$, the Fermi level cancels:
$$
\begin{aligned}
np &= N_c N_v \exp\left[-\frac{E_c - E_v}{kT}\right] = N_c N_v \exp\left(-\frac{E_g}{kT}\right) \\
\text{For intrinsic: } n = p = n_i \;&\Rightarrow\; n_i^2 = N_c N_v e^{-E_g/kT} \\
n_i &= \sqrt{N_c N_v}\, \exp\left(-\frac{E_g}{2kT}\right)
\end{aligned}
$$
Since $np = n_i^2$ does not depend on $E_F$, it holds for doped semiconductors too (mass action law). Setting $n = p$ also gives the intrinsic Fermi level
$E_{Fi} = \frac{E_c+E_v}{2} + \frac{3}{4}kT\ln\left(\frac{m_h^*}{m_e^*}\right)$, which is close to mid-gap.

#### Effect of temperature
As $N_c, N_v \propto T^{3/2}$,
$$
n_i = C\, T^{3/2} \exp\left(-\frac{E_g}{2kT}\right)
$$
- The exponential term dominates, so $n_i$ rises very rapidly with $T$. For Si ($E_g$ = 1.1 eV), $n_i$ is about $1.45\times10^{10}$ cm⁻³ at 300 K and rises by roughly 2 times for every 8–10 K rise near room temperature.
- A plot of $\ln n_i$ against $1/T$ is nearly a straight line with slope $-E_g/2k$; this is used to measure $E_g$.
- Larger band gap means smaller $n_i$ (Ge > Si > GaAs at the same $T$).
- Because conductivity $\sigma = n_i e(\mu_e + \mu_h)$, an intrinsic semiconductor's resistance falls sharply with temperature (negative temperature coefficient, used in thermistors).

```text
 ln(ni)
   |\
   | \   slope = -Eg/2k
   |  \
   |   \     Ge
   |    \     \
   |     \ Si  \
   +------------------> 1/T
```

High $n_i$ at high temperature is also why doped devices stop working properly above a certain temperature: intrinsic carriers swamp the dopant carriers.

## Check yourself

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

1. What is n in the non-degenerate case?
   A. Nc exp[+(Ec - EF)/kT]
   B. Nv exp[-(EF - Ev)/kT]
   C. Nc kT
   D. Nc exp[-(Ec - EF)/kT]
   Answer: D. Integrating g(E) f(E) with Boltzmann statistics gives this. (Key point: Electron concentration)

2. Which gives ni?
   A. sqrt(Nc Nv) exp(-Eg/2kT)
   B. Nc Nv exp(-Eg/kT)
   C. sqrt(Nc Nv) exp(-Eg/kT)
   D. (Nc + Nv) exp(-Eg/2kT)
   Answer: A. Setting n = p = ni gives ni^2 = Nc Nv exp(-Eg/kT). (Key point: Intrinsic formula)

3. How does ni vary with temperature?
   A. It is independent of T
   B. As T^(3/2) exp(-Eg/2kT), rising rapidly
   C. It decreases with T
   D. Linearly with T
   Answer: B. The exponential term dominates, so ni rises sharply. (Key point: Temperature dependence)

4. What does a plot of ln ni against 1/T give?
   A. A straight line of slope +Eg/k
   B. A parabola
   C. A straight line of slope -Eg/2k
   D. A horizontal line
   Answer: C. The slope gives the band gap. (Key point: ln ni vs 1/T)

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