# Intrinsic Carrier Concentration Calculator

Created by Purnima Singh, PhD
Reviewed by Wojciech Sas, PhD and Steven Wooding
Last updated: Dec 21, 2022

Use Omni's intrinsic carrier concentration calculator to calculate the electron/hole concentration in intrinsic semiconductors at a given temperature. If you want to know what intrinsic semiconductors are or the formula for intrinsic carrier concentration, continue reading. You will also find the difference between intrinsic and extrinsic semiconductors and a discussion on the effect of temperature on the conductivity of the intrinsic semiconductor.

If you are interested in semiconductor devices, you should also check out our Shockley diode calculator and MOSFET calculator.

## What are intrinsic semiconductors?

An intrinsic semiconductor is just a pure semiconductor without any significant defects or external impurities. The electrical conductivity of intrinsic semiconductors depends strongly on temperature. At absolute zero ($T = 0\ \rm K$), semiconductors behave like insulators.

However, as we increase the temperature, the electrons in the conduction band gain thermal energy and jump from the valence band to the conduction band. The excitation of electrons to the conduction band creates holes in the valence band. When we apply an electric field to such a material, the electrons and holes move towards the opposite electrodes. This movement of electrons and holes is responsible for the electrical conductivity of semiconductors. The conductivity of semiconductors increases with increasing temperature.

Silicon and germanium are the most common examples of intrinsic semiconductors.
These are some of their common applications:

1. To manufacture electronic components like transistors, diodes, and solar cells; and

## Intrinsic carrier concentration formula

The formula to calculate the intrinsic carrier concentration is:

$N_i = \sqrt {N_c\ N_v}\ \exp\left ( \frac{-E_g}{2 \text{k} T} \right )$

where:

• $N_i$ — Intrinsic carrier concentration or the intrinsic carrier density;
• $N_c$ — Effective density of states in the conduction band;
• $N_v$ — Effective density of states in the valence band;
• $E_g$ — Band gap energy;
• $T$ — Absolute temperature; and
• $\rm k$ — Boltzmann constant.

We also have a tool that can estimate the charge carrier number density of metals. For more information, visit the number density calculator.

Table I shows the value of $N_c$, $N_v$, and $E_g$ at 300 K for some common intrinsic semiconductors.

Table I: Table of values for $N_c$, $N_v$, and band gap energy ($E_g$) at 300 K.

Semiconductor

$N_C\ (\rm cm^{-3})$

$N_v\ (\rm cm^{-3})$

$E_g\ \rm(eV)$

Silicon

$2.82 \times 10^{19}$

$1.83 \times 10^{19}$

1.12

Germanium

$1.02 \times 10^{19}$

$5.65 \times 10^{18}$

0.66

Gallium Arsenide (GaAs)

$4.35 \times 10^{17}$

$7.57 \times 10^{18}$

1.424

## How to use the intrinsic carrier concentration calculator?

Let us see how to calculate the intrinsic carrier concentration of silicon at $T = 400\ \rm K$ using our intrinsic carrier concentration calculator:

1. Choose the semiconductor material, i.e., silicon, from the drop-down menu.

2. The calculator will autofill the values of effective density of states in the conduction band ($N_c = 2.82 \times 10 ^{19}\ \rm cm^{-3}$) and valence band ($N_v = 1.83 \times 10 ^{19}\ \rm cm^{-3}$) as well as the band gap energy ($E_g = 1.12\ \rm eV$) at $300\ \rm K$. You can also input these values if you know them.

3. Enter the temperature, i.e., $T = 400\ \rm K$.

4. The calculator will evaluate the intrinsic carrier concentration of silicon at $400\ \rm K$ ($N_i = 4.56 \times 10 ^{12}\ \rm cm^{-3}$). It will also display the value of band-gap energy at $400\ \rm K$ ($E_g = 1.1\ \rm eV$).

5. For a given temperature $T$, our tool scales up/down the effective values of $N_c$, $N_v$, and $E_g$. It will use these values to determine the intrinsic carrier concentration at $T$.

6. You can also use our calculator in Advanced mode to get a more realistic value of the intrinsic carrier concentration in silicon at different temperatures. The Advanced mode uses :

$\quad \scriptsize N_i = 5.29 \times 10^{19} \left ( \frac{T}{300} \right )^{2.54} \exp \left( \frac{-6726}{T} \right)$

## Temperature dependence of the energy band gap in intrinsic semiconductors

As the electrical properties of an intrinsic semiconductor depend strongly on the temperature, we should scale up/down the values mentioned in table I according to the temperature.

The experimentally determined relation between the energy band gap and temperature is:

$\scriptsize E_g = E_g(0) - \frac{\alpha T^2}{T + \beta}$

where $E_g(0)$, $\alpha$, and $\beta$ are fitting parameters. Table II shows the value of these parameters for some common semiconductors.

Table II: Table of values for $E_g(0)$, $\alpha$, and $\beta$.

Semiconductor

$E_g(0)\ \rm (eV)$

$\alpha\ \rm (eV/K)$

$\beta\ \rm (K)$

Silicon

1.166

$4.73 \times 10^{-4}$

636

Germanium

0.7437

$4.77 \times 10^{-4}$

235

Gallium Arsenide (GaAs)

1.519

$5.41 \times 10^{-4}$

204

Just like band gap energy, the effective density of states also depends on the temperature. We can express the relation between the two as:

$\scriptsize N_c(T) = N_c(300 \text{K}) \left ( \frac{T}{300} \right )^{3/2} \\ \text {....and:} \\ N_v(T) = N_v(300 \text{K}) \left ( \frac{T}{300} \right )^{3/2}$

## Calculating carrier concentration in an intrinsic semiconductor — an example

Let us work out the carrier concentration in an intrinsic semiconductor, say germanium, at $T = 400\ \rm K$.

1. For germanium at $300\ \rm K$ (see Table I):
$\scriptsize \quad \quad \begin{split} N_c &= 1.02 \times 10^{19}\ \rm cm^{-3} \\ N_v &= 5.65 \times 10^{18}\ \rm cm^{-3} \\ E_g & = 0.66\ \rm eV \end{split}$
1. At $400\ \rm K$, we can evaluate the energy band gap for germanium as (see Table II):
$\scriptsize \quad \quad \begin{split} E_g &= E_g(0) - \frac{\alpha T^2}{T + \beta} \\ E_g (400\ \text{K}) &= 0.7437 - \frac{4.77 \times 10^{-4} \times 400^2}{400 + 235} \\ E_g &= 0.62\ \rm eV \end{split}$
1. The effective densities of states at $400\ \rm K$ are:
$\scriptsize \quad \quad \begin{split} N_c(T) &= N_c(300 \text{K}) \left ( \frac{T}{300} \right )^{3/2} \\ N_c(400 \ \text{K}) &= 1.02 \times 10^{19} \left ( \frac{400}{300} \right )^{3/2} \\ N_c(400 \ \text{K}) &= 1.57 \times 10^{19} \ \rm cm^{-3} \\ \text {....and:} &\\ N_v(T) &= N_v(300 \text{K}) \left ( \frac{T}{300} \right )^{3/2} \\ N_v(400 \ \text{K}) &= 5.65 \times 10^{18} \left ( \frac{400}{300} \right )^{3/2} \\ N_v(400 \ \text{K}) &= 8.70 \times 10^{18} \ \rm cm^{-3} \\ \end{split}$
1. Using the above values, and $k = 8.617 \times 10^{-5} \text{eV/K}$ we can calculate the carrier concentration in germanium at $400\ \rm K$ as:
$\scriptsize \quad \quad \begin{split} N_i &= \sqrt {N_c\ N_v}\ \exp\left ( \frac{-E_g}{2 \text{k} T} \right ) \\ N_i &= \sqrt {1.57 \times 10^{19} \times 8.70 \times 10^{18}}\ \\ &\exp\left ( \frac{-0.62 \ \text{eV}}{2 \times 8.617 \times 10^{-5}\ \text{eV/K} \times 400 \ \text {K}} \right ) \\ N_i &= 1.38 \times 10^{15}\ \rm cm^{-3} \\ \end{split}$

As you can see, estimating the carrier concentration in an intrinsic semiconductor at any given temperature is quite cumbersome. Hence, we recommend using our calculator so that you can do so in seconds. 🙂

## FAQ

### What is the difference between intrinsic and extrinsic semiconductors?

The main differences between intrinsic and extrinsic semiconductors are:

1. Intrinsic semiconductors are pure, i.e., they do not contain impurities. Extrinsic semiconductors are manufactured by adding small amounts of trivalent or pentavalent impurities to pure semiconductors.
2. The number of free electrons and holes is equal in intrinsic semiconductors. In extrinsic semiconductors, it is not the same and depends on the type of added impurity.
3. The electrical conductivity of intrinsic semiconductors is low, whereas extrinsic semiconductors have high electrical conductivity.
4. In intrinsic semiconductors, the electrical conductivity depends only on the temperature. On the other hand, extrinsic semiconductors' electrical conductivity depends on temperature and doping level.

### Why intrinsic semiconductors behave like insulators at low temperatures?

At absolute zero, the electrons do not have enough thermal energy to jump from the valence band to the conduction band. As a result, the valence band is full, and the conduction band is empty, and there are no electrons available to conduct electric current.

### What is the primary difference in the electronic structure of semiconductors as compared to insulators?

The magnitude of the energy gap between the valence band and the conduction band is different in semiconductors and insulators. The energy band gap in semiconductors is very small (~1 eV), whereas in insulators, the energy gap is large (~10 eV).

### What are the properties of intrinsic semiconductors?

Some properties of an intrinsic semiconductor are:

1. Its electrical conductivity is low and depends strongly on temperature.
2. At absolute zero, it behaves like an insulator.
3. The number of electrons in the conduction band is equal to the number of holes in the valence band.

### What is the bandgap energy at 300 K for silicon and germanium?

The band gap energy at 300 K for silicon is 1.12 eV, and for germanium, it is 0.66 eV. The energy band gap in semiconductors changes with temperature.

Purnima Singh, PhD
Intrinsic carrier density
Material (optional)
Silicon
DoS conduction
×10¹⁹
/cm³
DoS valence
×10¹⁹
/cm³
Band-gap energy (at 300 K)
eV
Temperature (T)
K
Band-gap energy (at T)
eV
Intrinsic carrier concentration
×10⁶