# Intrinsic Carrier Concentration Calculator

Created by Luis Hoyos
Based on research by
Misiakos K, Tsamakis D. Accurate measurements of the silicon intrinsic carrier density from 78 to 340 K Journal of Applied Physics (1993)
Last updated: Sep 07, 2022

Welcome to the intrinsic carrier concentration calculator, a tool created to calculate the carrier concentration of intrinsic semiconductors.

If you still don't know what we're talking about, in the following sections, we briefly explain what intrinsic carrier concentration is and the formula for its calculation.

🙋 In the advanced mode of this calculator, you can study the intrinsic carrier concentration vs. temperature relationship of silicon using an empirical formula that provides more accurate results.

## What is intrinsic carrier concentration?

Charge carrier density, also known as carrier concentration, is the number of charge carriers per volume. A charge carrier is any element that carries an electrical charge while moving. The most common charge carrier is the electron in metals. Even so, apart from electrons, we can find electron holes as charge carriers when dealing with semiconductors. The importance of carrier concentration relies on its relationship to electrical and thermal conductivity.

An intrinsic semiconductor is a semiconductor without any significant dopant species. In other words, it's a pure semiconductor without any significant defects or external impurities. Compared to other semiconductors, the electrical conductivity of semiconductors varies strongly with temperature.

Intrinsic carrier concentration, then, refers to the charge carrier density of an intrinsic semiconductor.

## Intrinsic carrier concentration formula

The formula to calculate the carrier concentration in an intrinsic semiconductor is:

Nᵢ = √(Nc Nv) × e-E₉/(2kT)

, where:

• Nᵢ — Semiconductor intrinsic carrier concentration, calculated as the number of carriers per cubic centimeter (cm⁻³);
• Nc — Effective density of states in the conduction band, in cm⁻³;
• Nv — Effective density of states in the valence band, in cm⁻³;
• Eg — Band gap energy, in electronvolts (eV); see what is an electronvolt;
• T — Absolute temperature, in kelvin (K); and
• kBoltzmann constant, whose value is 8.617333262 × 10⁻⁵ eV/K.

The density of states and the band gap energy of a system are temperature-dependent. At 300 K, the values for the three materials of this calculator are:

Effective density of states and band gap energy at 300 K.

Semiconductor

Nc (cm⁻³)

Nv (cm⁻³)

Eg (eV)

Silicon

2.82 × 10¹⁹

1.83 × 10¹⁹

1.12

Germanium

1.02 × 10¹⁹

5.65 × 10¹⁸

0.66

Gallium Arsenide (GaAs)

4.35 × 10¹⁷

7.57 × 10¹⁸

1.424

For example, to calculate the intrinsic carrier concentration of silicon at 300 K with the previous hole/electron concentration formula:

$\scriptsize N_i = \sqrt{(2.82 \times 10^{19} \text{ cm}^{-3}) (1.83 \times 10^{19} \text{ cm}^{-3})} \\\ \ \ \ \ \ \ \ \ \ \times e^{-\frac{1.12 \text{ eV}}{2(8.617333262 \times 10^{-5} \text{ eV/K})(300 \text{ K})}} \\\ N_i = 8.89 \times 10^9 \text{ cm}^{-3}$

## Temperature dependence of the energy band gap and density of states

Intrinsic semiconductor properties are highly dependent on temperature. The density of states varies with temperature (in K) in the following way:

Nc(T) = Nc,@300 K (T/300 K)3/2
Nv(T) = Nv,@300 K (T/300 K)3/2

For the energy band gap, we use the following experimental relationship:

Eg = Eg(0) - (α × T²)/(T + β)

where Eg, α, and β are fitting parameters of the experimental model that depend on the material. For the three materials of this calculator, these values and their units are:

Semiconductor

Eg(0) (eV)

α (eV/K)

β (K)

Silicon

1.166

4.73 × 10⁻⁴

636

Germanium

0.7437

4.77 × 10⁻⁴

235

Gallium Arsenide (GaAs)

1.519

5.41 × 10⁻⁴

204

## The empirical formula for intrinsic carrier concentration of silicon (advanced mode)

The advanced mode of this calculator allows studying the intrinsic carrier concentration vs. temperature dependence using a more realistic equation, empirically obtained by :

$\scriptsize N_i = 5.29 \times 10^{19} \times (T/300)^{2.54} \times e^{-6726/T} \text{ cm}^{-3}$

We can also calculate the intrinsic carrier concentration of silicon at 300K with this formula and obtain a more realistic result:

$\scriptsize N_i = 5.29 \times 10^{19} \times (300/300)^{2.54} \times e^\frac{-6726}{300} \text{ cm}^{-3} \\\ N_i = 5.29 \times 10^{19} \times 1 \times 183.28 \times 10^{28} \text{ cm}^{-3} \\\ N_i = 5.29 \times 10^{19} \times 183.28 \times 10^{28} \text{ cm}^{-3} \\\ N_i = 9.7 \times 10^9 \text{ cm}^{-3}$
Luis Hoyos
Intrinsic carrier density
Material (optional)
Silicon
DoS conduction (N꜀)
x10¹⁹
/cm³
DoS valence (Nᵥ)
x10¹⁹
/cm³
Band-gap energy (at 300 K)
eV
Temperature (T)
K
Band-gap energy (at T)
eV
Intrinsic carrier concentration (Nᵢ)
x10⁶
/cm³
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