# Semiconductor intrinsic carrier concentration is given by ni=BT^(3/2)*exp(-E/2kT), how is this derived?

The glorious book Sedra/Smith Microelectronic Circuits states that for a semiconductor the intrinsic charge concentration is is given by:

$$n_i = BT^{3/2}e^{-E_g/2kT}$$

Where $n_i$ is the intrinsic carrier concentration in the semiconductor material, $B$ is a material dependent parameter in $cm^{-3}K^{-3/2}$, $T$ is temperature in Kelvin, $E_g$ is bandgap energy in electron volts ($eV$ ), $k$ is the Boltzmann's constant.

What is this equation called and how is it derived? I REALLY want to know that :)

Sze's book 'Physics of Semiconductor Devices' has a derivation in section 1.4 (of the second edition). One finds the effective density of states in the conduction band, the effective density of states near the top of the valence band, use those to get the carrier concentrations $n$ and $p$, solve for the Fermi level by equating the two, and then use $np = n^{2}_{i}$ to relate $n_{i}$ back to the density of states (conduction and valence) and the Fermi level. The $T^{3/2}$ you are probably wondering about comes from the effective density of states in the valence band.