Intrinsic Carrier Concentration Calculator

Formula

How this is calculated

In an intrinsic (undoped) semiconductor at thermal equilibrium the number of electrons promoted from the valence to the conduction band equals the number of holes left behind. Statistical mechanics gives nᵢ = √(Nc · Nv) · exp(−Eɡ / 2kT), where Nc and Nv are the effective densities of states in the conduction and valence bands (in cm⁻³), Eɡ is the band-gap energy in eV, k is Boltzmann's constant (8.617×10⁻⁵ eV/K) and T is the absolute temperature in kelvin.

Nc and Nv themselves scale as T^(3/2). The reference values used here are the widely-cited 300 K figures: for silicon Nc = 2.80×10¹⁹ cm⁻³, Nv = 1.04×10¹⁹ cm⁻³, Eɡ = 1.12 eV, giving nᵢ ≈ 1.5×10¹⁰ cm⁻³ at 300 K. For germanium (Eɡ = 0.66 eV) nᵢ is roughly 2.4×10¹³ cm⁻³ — much higher because the narrower gap is easier to bridge thermally. For GaAs (Eɡ = 1.42 eV) nᵢ is around 2×10⁶ cm⁻³.

The exponential dependence on temperature means nᵢ rises extremely rapidly with T. The log-scale chart shows this: at high temperatures every semiconductor eventually becomes intrinsic, swamping intentional doping and degrading device performance. This is the physical reason silicon devices typically lose specifications above ~150°C. These values assume the simple parabolic-band approximation; corrections for temperature-dependent band-gap narrowing exist but are omitted here.