Intrinsic Carrier Concentration Calculator

Intrinsic carrier concentration (nᵢ) is the number of free electrons (and holes) per unit volume in a pure, undoped semiconductor at thermal equilibrium. It depends on temperature and the semiconductor's band gap energy. For silicon at 300 K, nᵢ ≈ 8.89×10⁹ cm⁻³.

The standard formula is nᵢ = √(Nc × Nv) × exp(−Eg / 2kT), where Nc and Nv are the effective densities of states in the conduction and valence bands, Eg is the band gap energy, k is Boltzmann's constant (8.617×10⁻⁵ eV/K), and T is the absolute temperature in Kelvin.

At 300 K, silicon has a band gap energy of approximately 1.12 eV and germanium has a band gap of approximately 0.66 eV. GaAs has a larger band gap of about 1.42 eV. These values decrease slightly with increasing temperature.

At very low temperatures (near 0 K), there is insufficient thermal energy to excite electrons across the band gap from the valence band to the conduction band. With essentially no free carriers, the material cannot conduct electricity and behaves like an insulator. As temperature rises, thermal energy generates electron-hole pairs and conductivity increases.

An intrinsic semiconductor is a pure material with no intentional dopants; its carrier concentration is determined solely by temperature and band gap. An extrinsic semiconductor has been intentionally doped with impurity atoms (donors or acceptors) to dramatically increase its electron or hole concentration beyond the intrinsic level, enabling precise control of electrical properties in devices.

Carrier concentration increases strongly with temperature because the exponential term exp(−Eg/2kT) grows as T increases. Higher temperatures provide more thermal energy to promote electrons across the band gap, generating more electron-hole pairs. The Nc and Nv prefactors also increase with temperature, scaling roughly as T^(3/2).

At 300 K, silicon has Nc ≈ 2.82×10¹⁹ cm⁻³ and Nv ≈ 1.83×10¹⁹ cm⁻³. Germanium has Nc ≈ 1.04×10¹⁹ cm⁻³ and Nv ≈ 6.00×10¹⁸ cm⁻³. GaAs has Nc ≈ 4.35×10¹⁷ cm⁻³ and Nv ≈ 9.14×10¹⁸ cm⁻³. These values scale with temperature as T^(3/2).

Both semiconductors and insulators have a band gap separating the valence and conduction bands, but the key difference is the size of that gap. Semiconductors typically have band gaps below ~3 eV (e.g., Si: 1.12 eV, Ge: 0.66 eV), while insulators have much larger gaps (e.g., diamond: ~5.5 eV, SiO₂: ~9 eV) that thermal energy cannot practically overcome.