In this video we will introduce the concept of quasi-Fermi level. Now, recall what the Fermi level was, it's something that we describe the equilibrium carrier concentration with. So when a semiconductor is at a thermal equilibrium, then we can use Fermi level to express the carrier concentrations. So for example, if you recall, for a non-degenerate semiconductor, we derive this very convenient exponential equation containing E sub f Fermi level. Now note that we have used a single Fermi level to describe both n and p, and this will be the case for equilibrium. Now, consider a semiconductor in a non-equilibrium situation, say, by light illumination. Then the light is going create this excess carrier concentration, delta n and delta p. Now assume low level injection condition again, so this excess carrier concentration is small compared to the majority carrier concentration, equilibrium majority carrier concentration. Then, the majority carrier concentration doesn't change, it still can be described by this equilibrium equation. However, your minority carrier concentration is going to be many orders of magnitude greater or smaller than the equal equilibrium value. So in that case, obviously, we can't use the same equation for the minority carrier concentrations. Now, it would be nice if we can use an expression as compact and as simple as this, so hence the concept of quasi-Fermi level. So if the semiconductor is not too far from equilibrium, we call that quasi-equilibrium. In that case, we can write down the majority and minority carrier concentration in the same form as the equilibrium case, except that we use two separate formula levels F sub n and F sub p. Two separate Fermi levels for electron and hole, for majority carriers and minority carriers, and these guys are called the quasi-Fermi level. For majority carrier concentration, if the semiconductor, if the excitation condition, satisfies the low level injection, then this thing will be very similar. Majority carrier concentration will be the same as the equilibrium value. In that case, your F sub n, the electron quasi-Fermi level, will be equal to the equilibrium Fermi level. But the hole quasi-Fermi level, or the quasi-Fermi level for holes, will be very different from the equilibrium Fermi level. Now let's consider an example here, so this is an example that we considered in the previous video as an example of a continuity equation. We were considering a p-type semiconductor with a doping density of 10 to the 17th, with a light illumination. And in this case, the excess carrier concentration generated by this light illumination is 10 to the 14th per cubic centimeters. And it's much smaller than the equilibrium majority carrier concentration, but it is much, much greater than the equilibrium minority carrier concentration. Now we can describe this as carrier concentration itself, but it is very convenient that we can describe the situation using Fermi levels as well, quasi-Fermi level, that is, because this is a non-equilibrium situation. So, here is the equilibrium case, before light illumination, there is a conduction band, there is a valance band. In the middle of band gap, you have an intrinsic Fermi level. Now, doping density is 10 to the 17th, and so majority carrier concentration is 10 to the 17th. And minority of concentration given by the law of mass action, so about 10 to the 3rd power. Now, Fermi level is down below here, because this is a p-type semiconductor, with a hole concentration much greater than the electron. Under light illumination, the carrier concentration changes like this, whole concentration remains the same, it changes only by 0.1%. Whereas the minority carrier concentration increase by 11 orders of magnitude. So you can describe these two cases, majority carrier concentration and the minority carrier concentration, in this non equilibrium situation using quasi-Fermi level, one for hole, quasi-Fermi level for hole, F sub p. Because the hole concentration didn't really change, your quasi-Fermi level for hole will basically be the same as the equilibrium Fermi level. For electrons, this guy 10 to the 14th is larger than the intrinsic carrier concentration. So if you express this number In terms of Fermi level, that Fermi level is going to be up above the intrinsic Fermi level, just like in the n-type semiconductor. So your quasi-Fermi level for hole will be located in the upper half of the band gap. So you can describe this non-equilibrium situation created by light illumination as a minority carrier increasing dramatically. Or equivalently, you can describe this situation as the original Fermi level split up into two quasi-Fermi levels, one for holes and one for electrons. Now, this quasi-Fermi level gives you a very convenient mathematical approach to describe current as well. So the hole current in general is combination of the drift current and the diffusion current, this is something that we talked about before. And use the non-degenerate semiconductor equation for hole concentration and that is, That was p is equal to ni exponential e sub i- fp kt, this was the equation that was in the previous slide. Now take a gradient of p, and then you will get this equation here. Now plug this expression back into your general current density equation above, then you get this. Now, you can work this out and show that this hole current here, Hole current here is given by simply a mobility times the carrier concentration times the gradient of your quasi-Fermi level. And similarly, the electron current also is given by the gradient of the quasi-Fermi level for electrons. So now you can imagine that whenever you have a current, then the current is necessarily accompanying some gradient of quasi-Fermi level.
