Q:

Is the intrinsic carrier concentration for semiconductors an average value?

I am learning about intrinsic semiconductors, and have just come across the intrinsic carrier concentration equation - 
$n_{i}=BT^{3/2}e^{\cfrac{-E_{g}}{2kT}}$
Which is supposed to give the number of conduction band carriers per cubic centimeter. Is this value an average? Surely the number of carriers in the conduction band cannot be the same everywhere in each $cm^3$ within the semiconductor at any given time?

A:

The formula you mention for $n_i$ is empirically supported (i.e., it is an empirical fit to experimental data for a solid material) and is indeed an average in a cubic centimeter of material. $E_g$ is the band gap energy for the material of interest and B is an empirically determined constant. Thus, normally, with access to a table of such constants and band gap energies, you can determine the intrinsic carrier concentration for a given temperature of interest.
$n_i$ plays a significant role in the design of semiconductors devices because of the relationship, $n_i^2 = np$ (law of mass action) which holds in doped and undoped material. This is important for, among other reasons, doping profiles and for understanding how a material acts a different temepratures. However, given that most semiconductor components are much much smaller than a $cm^3$ in volume (state of the art transistors have a linewidth of $14-22 nm$ and the junction depth is a few hundred angstrom), you can see that the of scale for $n_i$ is rather meaningless to an individual component and an average would be necessary anyway.
One last point, the reason why the equilibrium relation, $n_i^2 = np$, holds true is because in a crystal structure, the generation of hole-electron pairs and recombination is constantly occurring in the crystal because of thermal agitation in the material. Take the equation for $n_i$ and let the temperature go to absolute zero, you find that the intrinsic carrier concentration also becomes 0. The meaning of the temperature increasing is an increase in kinetic energy/vibration of atoms or equivalently an average increase in the internal energy in the crystal. This is because temperature is a function of the internal energy (U), $T = (\frac{\partial{S}}{\partial{U}})^{-1} = (\frac{Nk_b}{U})^{-1}$. So, there are more opportunities for bonds in crystal to break and reform. 

