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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 :)

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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.

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  • $\begingroup$ Does this equation not have a name assigned to it? $\endgroup$
    – quantum231
    Commented Feb 12, 2015 at 22:32
  • $\begingroup$ wow dude, I am from an electronic engineering background. It just had a look at that book. It sure goes into full depth on the matter of semiconductors. I am blown away as now I don't know how to comprehend even more equations that Sedra/Smith book does not even mention at all. lol it is. $\endgroup$
    – quantum231
    Commented Feb 12, 2015 at 22:41
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    $\begingroup$ Not all equations get a name assigned - perhaps the physics/EE community should get in the habit of selling naming rights! As for Sze, I think it remains the gold standard for a university senior / graduate level course in device physics (which is indeed how it ended up on my shelf years ago). It is one that many people keep around as the go-to reference on device physics. $\endgroup$
    – Jon Custer
    Commented Feb 12, 2015 at 22:52

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