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From this website (link: http://hyperphysics.phy-astr.gsu.edu/HBASE/solids/fermi3.html#c1 ), we get the following expression:

[tex]Population Density of Conduction Electron = \int ^{\infty}_{Egap}N(E)d(E) = \frac{2^{5/2}(m \pi k T)^{3/2} exp (-E gap/2kT)}{h^3}[/tex]

My questions are based on the derivation given by the website as follow:-

1. EF = Egap / 2

Is this a defined property of semiconductors?

2. Is there an error in the following expression?

[tex]N(E)d(E) = \frac{8\sqrt{2} \pi m^{3/2}}{h^3} \sqrt{E-Egap}\ e^{-(E-Egap/2)}[/tex]

I mean originally it has kT in the power of exponential term. Is it mistakenly dropped?

3. Is "integration by parts" applied on the following expression?

[tex]N(E)d(E) = \frac{8\sqrt{2} \pi m^{3/2}}{h^3} \sqrt{E-Egap}\ e^{-(E-Egap/2)}[/tex]

Somehow, I can't solve the integration.

Thanks.