This video, we will discuss intrinsic semiconductors. Now, we now know how to calculate the electron concentration in the conduction band and the whole concentration in the valence band. The only missing information is the Fermi level position, Fermi level EF relative to the band edges Ec, conduction band edge, and then Ev, the valence band edge. Now in other to evaluate that, we need to know more about the semiconductor that we're dealing with. And we're going to classify semiconductors into two different ways. One is intrinsic semiconductor, which we will discuss today. And the other is the extrinsic semiconductors, which we will cover in the next couple of videos. Now, what is an intrinsic semiconductor? Intrinsic semiconductor is a semiconductor that does not contain any impurities. It's a pure semiconductor. Now, in an intrinsic semiconductor at a finite temperature, some of the electrons near the top of the valence band are excited by thermal energy to the conduction band. And as I mentioned earlier in an earlier video, this produces two partially filled energy bands, valence band and conduction band, both of which contribute to the electrical conduction. Now, this thermal excitation process, if you think about this, one electron being promoted up to the conduction band, leaves one empty state behind. So this process produces one electron in the conduction band, one hole in the valence band. If this is all that's happening, then the total number of electrons at conduction band must be equal to the total number of holes in the valence band. So in an intrinsic semiconductor, electron concentration always equal to the whole concentration. Now, you may ask, what is this process of thermal excitation really, physically? Well, you can think of this in a bond diagram here. So this is a simplified diagram of silicone crystal. We projected it on to the plane. So this is not a real crystal structure. So this is just a schematic. Each silicone atom is bonded to four neighboring silicone atom here. And they are in a covalent bond, so each bond contains two electrons. These electrons in this covalent bondings, are in this energy level diagram, are the electrons in the valence band. Now, when the semiconductor is at a finite temperature, then the thermal energy will agitate everything, agitate the atoms, agitate the electrons. And some of the electrons may receive, may have, high enough thermal energy to break free from this bond. And then, this electron that has broken free can freely move around the crystal. And that electron represents the electron in the conduction band, in this energy band diagram. And the one broken band here, that is left behind by this electron that has broken free, is the hole in the valence band. So this is a physical picture. And this is more abstract picture that we will more often use. It's very convenient in the energy level scheme. Now, as I said, in an intrinsic semiconductor, the only mechanism that can produce electrons in the conduction band holding the valence band is a thermal excitation. And this external excitation process is inherently producing one electron and one hole at the same time. And that means that in any situation, in an intrinsic semiconductor, the electron concentration in the whole conduction band, whole concentration in the valence band must be equal to each other. We're going to call that ni, intrinsic carrier concentration. This intrinsic carrier concentration obviously, is a function of temperature. At a higher temperature, you could imagine that more electrons will be promoted. So the concentration in the conduction band, electron concentration in the conduction band, the whole concentration in the valence band, are higher at a lower temperature, then this process becomes more difficult. And the carrier concentration will be lower. The intrinsic carrier concentration can be easily calculated by using the law of mass action. Law of mass action says that np product, electron concentration and whole concentration product, is given by this equation here. Product of the densities of states, effective densities of states, times the exponential factor containing the band gap energy. And by plugging in the expressions that we derive for the Nc and Nv, we can derive this. The key feature here is that it depends exponentially to the band gap energy. There is a negative sign, so larger band gap energy means exponentially small carrier concentration. Now this makes sense because, if you have a larger band gap, then you need that much more energy in order to overcome the band gap, and promote an electron to the conduction band from valence band. It also shows that it depends on the effective masses. So effective mass of the electron and the effective mass of hole, where did they come from? They came from these effective densities of states. So if there's a greater density of state, that means generally there are more carriers. And therefore, there is a greater chance to have a larger number of carriers in any situation. So if you plot the intrinsic carrier concentration as a function of temperature, in this case, 1/T, in a semi log plot, this is the log scale here, then you get this straight line. Meaning that in individual, all these lines represent an exponential function, as shown here. And the slope of this individual line corresponds to the band gap energy. So the germanium, which has the smallest band gap, has the smallest slope, and silicone has a larger slope, slightly larger band gap energy. Gallium arsenate has the largest band gap and has the largest slope. Now, we can calculate the Fermi level of these intrinsic semiconductor. How? Now we know the carrier concentration, and we can use the equations for non-degenerate semiconductor that we derived in the previous video. And the carrier concentration here is ni, because we have an intrinsic carrier concentration. And then plug in ni here, and then solve this equation for EF. If you do that, then you get this equation here. So what is this? Well, it starts from Ev, and then you go up half the band gap. So this first two terms represents half middle of the band gap. And then you have this third term, which is proportional to kT, and natural log of the ratio of the effective masses. Ratio of the effective masses here. Now typically, the whole effective mass is higher than the electron effective mass. However, they don't differ by orders of magnitude. So this logarithm of the ratio of the effective mass, generally, is a small number. This guy here, generally the small number. Small number multiplied by kT, which is small, 0.029 electron volt for room temperature. And typical band gap of a semiconductor for silicone for example is 1.1 electron volts. So this last term is generally a very small number compared to the band gap. So a lot of times, we ignore it. And we can say that this energy just corresponds to the middle of the band gap. Now, this quantity here, Fermi level of an intrinsic semiconductor, we write it as Ei, and call it intrinsic Fermi level. And the intrinsic Fermi level is located at the middle of the band gap, slightly above if you want to be accurate. But that quantity is small. Now in an intrinsic semiconductor, the carrier concentration is intrinsic carrier concentration, and the Fermi level is intrinsic Fermi level. Now, it is true that the intrinsic carrier concentration represents the actual carrier concentration in an intrinsic semiconductor. Intrinsic Fermi level is the actual Fermi level in an intrinsic semiconductor. However, we don't really use intrinsic semiconductor. Intrinsic semiconductors don't really have much application. As you will see later, a lot of the semiconductor devices use extrinsic semiconductor. Nevertheless, intrinsic carrier concentration and intrinsic Fermi level become very, very important parameters, with which we characterize the properties of extrinsic semiconductors. So it is very important to know that the intrinsic carrier concentration and intrinsic Fermi levels are material parameters. If you look at the expressions that we derive for ni and Ei, they depend on the effective densities of states, and therefore, effective masses of the conduction band and the valence band and temperature. So these are some numbers that will be determined, if you pick what semiconductor you're dealing with. So these are material parameters. And we will use this intrinsic carrier concentration and intrinsic Fermi levels as a material parameter to describe the properties of semiconductors that are not necessarily intrinsic. So for example, you can rewrite the n and p, electron concentration and whole concentration, in terms of ni and Ei. And by using the definitions of ni and Ei, that we derived in the previous slides, and if you plug that in to the expression for n and p that we derived in the previous video for non-degenerate semiconductor, you can derive this expression. Those are equivalent expressions, and it is a very convenient expression. We will be using this a lot later on in this course, as we describe various different properties of extrinsic semiconductors and various devices made of extrinsic semiconductors. Now lastly, we can rewrite the law of mass action in a very compact form, as n, p product is equal to ni squared intrinsic carrier concentration squared.
