Reasoning behind the density of free electrons/holes being constant with respect to volume of a semiconductor material

Question

Consider a silicon semiconductor material which is having a cuboid shape of dimensions \$2\times 2\times 1\ \mathrm{cc}\$, suppose the atomic density of silicon is \$10^{22}/\mathrm{cc}\$, so in all there are \$18\times 10^{22}\$ atoms. Band theory says each discrete energy level for a single silicon atom would be split into \$18\times 10^{22}\$ levels so that the Pauli principle is followed.

My question is that if this is the case then when we find the hole or free electron concentration using Fermi distribution and density of states, why that result is a constant quantity i.e. not depending on which volume region you are considering it (considering temperature is constant). Since one can observe that there may be a situation that in the first location of volume 1 cc that the corresponding free electron energy levels is in the range of E1 to E2 and in the second location of volume 1 cc its in the E3 to E4 range, so why we expect the electron concentration to be remaining same per unit volume?? Whose expression is a constant being equal to \$N_c(\frac{e^{E_c - E_f}}{KT})\$ ??

Or is the reason it turns out to be constant because it indicates average concentration of free electrons and not exact? As the integration was done for whole energy levels of conduction band for free electron concentration calculation.

Answer 1

Like many other concepts in Physics, "being constant" is also a relative one. When a fluctuation of a parameter is much faster than the considering frequency range, the mean of fluctuation over time can be considered as being constant.

It's true that free electron or hole density is not constant over every block of the semiconductor. And it is changing fast. This is the reason of Johnson-Nyquist noise.

The magnitude of the heat noise decreases as the frequency increases. At high frequencies, the noise is dominated by other types of noise such as shot noise, and is no longer considered as thermal noise.

To summarise, we can consider it's "being constant". But there are some exceptional cases.

You can consider it being constant:

• operation temperature is not so high
• signal bandwidth is narrow (because heat noise spectrum is spreaded widely)

You cannot ignore its effect when:

• working with low-noise, high-precision system, with wide bandwidth. (e.g. measurement analog frontend amplifier)