Intrinsic Carrier Concentration Calculator

Omni's intrinsic carrier concentration calculator allows you to work out the carrier concentration in intrinsic semiconductors.

Intrinsic carrier concentration8.8801e+9 per cm³8.8801e+15 per m³ for Silicon at 300 K — silicon's accepted room temperature value is about 1.0×10¹⁰ per cm³

Thermal voltage kT/q25.852 mV25.85 mV at 300 K — the scale factor in every diode equation

Band gap1.12 eV43.3 times the thermal energy, which is why the exponential makes n_i so small

Longest wavelength that can excite a carrier1.107 µmhc/E_g — light of longer wavelength passes straight through, which is why silicon is opaque to visible light but transparent in the infrared

Conduction band density of states2.8165e+19 per cm³

Valence band density of states1.8294e+19 per cm³

At 10 K warmer1.8761e+10 per cm³the exponential dominates completely — a ten degree rise roughly doubles the intrinsic concentration in silicon, which is why leakage current climbs so steeply with temperature

Debye length408.8455 nm√(εkT/nq²) — beyond this a charge is screened by the rearrangement of everything around it and is effectively invisible

At four times the density204.4227 nmhalved — screening goes as the inverse square root of density, so denser plasmas hide charges more effectively

Carriers within a Debye sphere28.63the plasma parameter. Collective behaviour needs this to be large — with only a handful of particles in the sphere the medium is a set of individual collisions, not a plasma

The formula

n_i = √(N_cN_v) exp(−E_g/2kT); λ_D = √(εkT/nq²)

The exponential runs everything

Intrinsic carrier concentration carries a factor exp(−E_g/2kT), and at room temperature silicon's band gap is about 43 times the thermal energy. That exponential is why pure silicon has around 10¹⁰ carriers per cubic centimetre against copper's 10²³, and why doping — which adds carriers directly rather than thermally — changes conductivity by so many orders of magnitude.

It is also why semiconductors are temperature sensitive in a way metals are not. Roughly every ten degrees doubles the intrinsic concentration in silicon, so leakage current climbs steeply with temperature and a device that is stable at room temperature can run away when hot. Wide-gap materials like silicon carbide and gallium nitride exist largely to push that limit further out.

The band gap sets the colour

A photon can only excite a carrier if it carries at least the gap energy, so light with a wavelength longer than hc/E_g passes straight through. Silicon's 1.12 eV gap puts that threshold at about 1100 nm, which is why silicon is opaque to visible light and transparent in the near infrared — and why silicon photodetectors stop responding there.