I have 2 questions regarding the band gap of semiconductors. They are both regarding the following sentence from the lecture notes:
Firstly, we assume that the Fermi level is far from both bands E_F−E_v≫kT and E_c−E_F≫kT.
Question 1: How can we assume that the Fermi level is far from both bands? Isn’t the assumption of a semi conductor that the band gap is relatively small? Or else it would become an insulator right?
Question 2: If we assume that the Fermi level is far from both bands, how do when then know that E_F−E_v≫kT and E_c−E_F≫kT?
When we say things like “large” or “small”, we need to always keep in mind what we are comparing to. The assumption that we make here is that the bandgap of a semiconductor is way larger than the available thermal energy. At room temperature, the available thermal energy is around k_B T = 25.9 meV. On the other hand, semiconductors have a bandgap E_G of around eV. So we see that the band gap is something like 2 order of magnitude larger than thermal energy, and thus E_G \gg k_B T. Because the valance band of a semiconductor is full and the conduction band is empty (at T=0 that is), we expect the Fermi level to be somewhere in the middle of the bandgap. Therefore, a quantity like E_C - E_F is gonna have a similar order of magnitude as E_G, and we can thus use the condition that E_c−E_F≫kT.
Just to conclude, the semiconductors band gap is usually way larger than the thermal energy available. Of course, as you pointed out, we can say the same about insulators. The distinction between an insulator and a semiconductor is not a very solid one. Usually, we say that an insulator is something that has more than a couple of eV of the bandgap. So all the conditions we derived above also apply to insulators. However, we can distinguish the two in a more practical sense: in semiconductors, we can still measure currents, however, it is almost impossible to do so for insulators because the intrinsic density is so much smaller. For example, consider a semiconductor with band gap 1eV and an insulator with band gap 2 eV. If we take the ratio of their intrinsic carrier concentrations:
The semiconductor has 10^{16} times the carriers than insulator does! So a single eV makes a huge difference to the point that you cannot really measure the current in most insulators.