Intrinsic Carrier Concentration Calculator

Background.

A perfectly pure semiconductor still conducts — a little. Thermal agitation constantly shakes electrons across the band gap, leaving holes behind, and the equilibrium density of those thermally created pairs is the intrinsic carrier concentration: nᵢ = √(N_cN_v)·exp(−E_g/2k_BT). It is the reference against which all doping is measured and the quantity that sets how ‘semi’ a semiconductor is.

The exponential is the whole story of semiconductor diversity. At room temperature, k_BT ≈ 25.9 meV, so a band gap of an electronvolt costs the exponential a factor of e⁻²² or so. Silicon (E_g = 1.12 eV) lands near 10¹⁰ carriers per cm³ — one thermally freed electron per ten trillion atoms. Germanium's smaller 0.66 eV gap yields a thousandfold more; wide-gap GaN (3.4 eV) essentially none, which is why it withstands high temperatures and voltages that would turn silicon intrinsically conductive.

The same exponential makes nᵢ ferociously temperature-sensitive: silicon's roughly doubles every 8–11 K near room temperature. That sensitivity is put to work in thermistors and betrays devices as leakage — a junction's reverse current rides on nᵢ or nᵢ², which is why hot electronics leak and why power devices must be derated with temperature.

The prefactor √(N_cN_v) collects the effective densities of states of the two bands — how many states each band offers within thermal reach of its edge — themselves T^(3/2) functions of temperature and effective mass. This page evaluates the nondegenerate equilibrium expression from the four entered parameters; choosing those parameters for a real material at a real temperature, and everything doping adds, lies outside it, as the scope note beside the result states.

What is intrinsic carrier concentration calculator?

The intrinsic carrier concentration nᵢ is the density of electrons (equal to the density of holes) present in a pure, undoped semiconductor at thermal equilibrium, given in the nondegenerate limit by nᵢ = √(N_cN_v)·exp(−E_g/2k_BT): the geometric mean of the conduction- and valence-band effective densities of states, discounted by half the band gap measured in thermal-energy units. For silicon at 300 K it is about 10¹⁰ cm⁻³ — tiny beside the 5×10²² atoms/cm³, and beside typical doping of 10¹⁵–10¹⁹ cm⁻³, which is exactly why doping controls silicon so decisively.

How to use this calculator.

1. 01Enter N_c and N_v, the effective densities of states, in m⁻³ — silicon at 300 K uses 2.8×10²⁵ and 1.04×10²⁵ (handbook cm⁻³ values ×10⁶).
2. 02Enter the band gap in eV at your temperature: 1.12 for Si, 0.66 for Ge, 1.42 for GaAs, 3.4 for GaN — gaps themselves shrink slightly as temperature rises.
3. 03Enter the absolute temperature in kelvin; 300 K is the conventional ‘room temperature’ of datasheets.
4. 04Read nᵢ in m⁻³ and divide by 10⁶ for the cm⁻³ units most literature quotes.
5. 05For a T sweep, re-enter N_c and N_v scaled by (T/300)^1.5 as well as the new T — holding them fixed captures only the exponential part of the true temperature dependence.

The formula.

In a nondegenerate semiconductor the electron density is n = N_c·exp(−(E_c−E_F)/k_BT) and the hole density p = N_v·exp(−(E_F−E_v)/k_BT): each band's effective states N discounted by how far the Fermi level sits from that band edge. Multiply them and the unknown Fermi level cancels — np = N_cN_v·exp(−E_g/k_BT), the mass-action law, valid doped or undoped. Purity adds the constraint n = p (every freed electron leaves one hole), and taking the square root yields nᵢ = √(N_cN_v)·exp(−E_g/2k_BT) — the half in the exponent exists because the gap's cost is shared between creating an electron and a hole. The mass-action law is also nᵢ's practical power: in doped material np = nᵢ² still holds, so knowing nᵢ and one carrier density immediately gives the other — the minority-carrier arithmetic underneath diode saturation currents and transistor gain. The engine evaluates the square root and exponential in Decimal arithmetic, rounding once to twelve significant digits; the exponential's argument uses k_B in eV/K so the entered eV gap divides cleanly.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Sze and Ng, Physics of Semiconductor Devices, 3rd ed. (PRINT).
2. [2]NIST, 2022 CODATA Boltzmann constant.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.