Intrinsic Carrier Concentration Calculator (nᵢ vs temperature, silicon or custom)

Understanding the intrinsic carrier concentration calculator — concept and principle

This calculator finds the intrinsic carrier concentration nᵢ of a semiconductor, the number of free electrons (equal to the number of holes) per cubic centimetre in pure material, at any temperature. It uses Ioffe Institute parameters for silicon or your own band gap and effective densities of states. It is a basic tool in semiconductor device courses.

How it is calculated

nᵢ = √(N_c N_v) × exp(−E_g ÷ (2kT))

Silicon (Ioffe): E_g = 1.17 − 4.73 × 10⁻⁴ T² ÷ (T + 636) eV

N_c = 6.2 × 10¹⁵ T^(3/2) cm⁻³; N_v = 3.5 × 10¹⁵ T^(3/2) cm⁻³

• k = 8.617333262 × 10⁻⁵ eV/K; T in kelvin.
• The custom option scales your 300 K values of N_c and N_v by (T/300)^(3/2) and uses the band gap you enter.

The principle

In a pure semiconductor, thermal energy occasionally lifts an electron across the band gap, leaving a hole behind, so electrons and holes are created in pairs. The chance of crossing falls exponentially with the gap divided by the thermal energy, which is why nᵢ is extremely sensitive to temperature and to E_g. N_c and N_v count how many states are effectively available near each band edge; they grow as T^(3/2).

Worked example

Default inputs: T = 300 K, silicon.

1. E_g = 1.17 − 4.73 × 10⁻⁴ × 300² ÷ 936 = 1.1245 eV.
2. N_c = 6.2 × 10¹⁵ × 300^1.5 = 3.222 × 10¹⁹ cm⁻³; N_v = 1.819 × 10¹⁹ cm⁻³.
3. kT = 25.852 meV.
4. nᵢ = √(3.222 × 10¹⁹ × 1.819 × 10¹⁹) × exp(−1.1245 ÷ 0.051704) = 8.677 × 10⁹ cm⁻³ = 8.677 × 10¹⁵ m⁻³.

How to read the result

Silicon's nᵢ at 300 K is of the order of 10¹⁰ cm⁻³; Ioffe quotes 1 × 10¹⁰, and these formulas give about 0.9 × 10¹⁰. That is tiny compared with typical doping levels of 10¹⁵ cm⁻³ or more, which is why doped silicon's conductivity is controlled by its dopants. nᵢ roughly doubles for every 8 K or so of heating near room temperature, and when it approaches the doping level, a device stops working as designed.

Common mistakes

• Using the band gap at 0 K instead of at the operating temperature.
• Mixing cm⁻³ and m⁻³.
• Treating nᵢ as the carrier density of a doped sample.

Frequently asked questions

Why does nᵢ rise so fast with temperature?

It depends exponentially on −E_g/(2kT), so a small rise in T greatly increases the number of carriers.

How do electrons and holes relate in doped material?

Their product stays equal to nᵢ² in equilibrium (the mass-action law).

Why does germanium have a higher nᵢ than silicon?

Its band gap is smaller.

Sources: Ioffe Institute, NSM Archive, "Physical Properties of Semiconductors — Si: Band structure and carrier concentration"