Temperature Dependence of Conductivity of a Semiconductor In one of the physics books that I am following it states that:

If temperature is increases, the average energy exchanged in a collision increases. More valence electrons cross the gap and the number of electron hole pairs increases. It can be shows that the number of such pairs is proportional to the factor

$$T^{3/2}e^{\Delta E/2kT}$$
where $\Delta E$ is the band gap.
I have no idea where this came from, if anyone can show any resources or materials in which this is derived I would greatly appreciate it.
 A: Kittel (at least my 5th edition) goes through this derivation.  Refer to the diagram below and remember that in semiconductor physics the chemical potential $\mu$ is also the Fermi level $E_F$ below.

The derivation essentially involves calculating the concentration of electrons and holes at temperature T in the conduction and valence bands respectively, with appropriate approximations.
The concentration of electrons in the conduction band will be:  $$n=\int^{\infty}_{E_g}D(\epsilon)f(\epsilon)d\epsilon$$ where $D$ is the density of orbitals at $\epsilon$, $f$ is the Fermi-Dirac function and we are integrating from the top of the energy gap $E_g$ to infinity.
Kittel uses the free electron formula for $D$ [$=({\frac{2m}{\hbar^2}})^\frac{3}{2}\epsilon^\frac{1}{2}$], and approximates the F-D function as $$e^{\frac{\mu-\epsilon}{k_BT}}$$ since he assumes $\epsilon - \mu \gg k_BT$.
Plugging these into the integral and integrating, one gets: $$n=2(\frac{m_ek_BT}{2\pi\hbar^2})^{\frac{3}{2}}e^{\frac{\mu-E_g}{k_BT}}$$ A similar calculation for the number of holes, p (now integrating from $-\infty$ to 0), yields: $$p=2(\frac{m_hk_BT}{2\pi\hbar^2})^\frac{3}{2}e^{\frac{-\mu}{k_BT}}$$  (To get this you must also remember that the F-D for holes is 1 minus the F-D for electrons, which yields $e^{\frac{\epsilon-\mu}{k_BT}}$).  When you mulitply the expressions for n and p together, you see that you get an expression proportional to $T^3$ and $e^{\frac{-E_g}{k_BT}}$. Since in an intrinsic semiconductor, n=p, if you take the square root of this expression, you get the formula in your question (btw, you're missing a negative sign in your exponent).
A: In this resource there is at least a little bit more explanation. Especially the power of $T$ factor is explained as being the consequence of scattering as an influence on charge mobility. Maybe also the reference cited therein guides you further.
