In this video, we will calculate the equilibrium carrier concentrations. So, the number of charge carriers (the electrons in the conduction band, and the holes in the valence band) can be found by multiplying the total number of allowed states by the probability function. So, n here is the density of electrons or concentration of electrons in the conduction band, as a function of energy and temperature, in general. And that is given by the density of states of the conduction band, which is in general a function of energy, and the probability of finding an electron at energy E at a given temperature T. Likewise, the concentration of holes in the valence band as a function of energy and temperature, is given by the density of states of valence band times the probability of not finding an electron at our energy E. And that is given by one minus the probability of finding an electron at energy E. So, this equation will specify electron concentration and hole concentration as a function of energy and temperature. g(E) is the density of states in each band, which we calculated in the previous video, and all we need to know now is the probability of finding an electron at energy E at a given temperature T. Now, once we know this, then the total number of carriers in the band is simply given by integrating this quantity here, n(E,T), p(E,T) across the entire band. For the conduction band, that means bottom of the conduction band E sub c to infinity, and then for the holes it starts from negative infinity to the top of the valence band, which is E sub v. Now, at equilibrium, the occupation probability for an electron for any electronic state is given by a function called the Fermi-Dirac distribution function or Fermi-Dirac probability function, as shown here. So, it's one over exponential, E minus Ef divided by kT plus 1. Ef here is a quantity called Fermi level, k sub E is a Boltzmann constant, and T is the temperature in Kelvin. The Fermi-Dirac function looks like this. So, it looks more like a step function, and it transitions over at Fermi level. When energy is below Fermi level, probability is very high, in general, and at energies above the Fermi level, probability is low, and at zero temperature, this is a rigorous step function, but at a finite temperature there is a gradual transition. At energy E equals Ef, your probability of finding an electron is always one half. So, you can do your own exercise here. So, suppose that you have a certain energy level distribution shown here, and if you calculate the possible configuration of electrons according to the Pauli exclusion principle, saying that you cannot have more than two electron at an energy level, and you can find all possible configuration of electrons here and calculate their energy, plot it, you will find that it follows the Fermi-Dirac distribution function very nicely. There are other probability functions. Bose-Einstein probability function, shown here. The only difference between the Fermi-Dirac function is that this exponential factor is the same but the sign in front of the additional plus one it now changed to minus one. Now, if you get rid of this one altogether, then you get the Maxwell-Boltzmann distribution function. So these Fermi-Dirac and Bose-Einstein distribution functions are quantum mechanical probability function. Fermi-Dirac function applies to particles called fermions. Electrons and holes are examples of fermions that we use Fermi-Dirac function. Bose-Einstein function applies to particles called bosons, and examples of boson is a photon, light quantum, and some other atomic species also follow this, are known to follow this Bose-Einstein distribution function. If the exponential factor here is very large, then you can obviously ignore this additional plus or minus one. And in that case, both of these two functions then get reduced to Maxwell-Boltzmann distribution function. This is the distribution function that describes classical particles. So, if you plot Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac all together, you can see that they all converge into the single function for every high energy. That's the classical limit. Now, in semiconductors, we are dealing with fermions, electrons and holes, so we'll use Fermi-Dirac distribution function. So, the total electron concentration in the conduction band, again, given by the integration over the entire conduction band of the product, g sub c times F. Density of states of the conduction band times the Fermi-Dirac distribution function. Hole concentration of the valence band, again same thing, integration across this entire valence band of the product of density of state of the valence band times one minus F, probability of not finding an electron at energy E. So, we already know what the density of states are, we calculated in the previous video. We now plug in the Fermi-Dirac probability function for F here, then you get these integration. Now, the integration is pictorially represented here. So, this here is the density of states that we calculated of the conduction band, which increases that square root of E. Same thing here in the valence band. Now, at very high energy, the density of state will collapse, and the reason for that is because the parabolic energy band approximation does not apply, it breaks down at energies far away from the band edge. But that is okay, because as you will see in the forming director solution function, the probability of finding an electron at this very high energy becomes very, very small. It decreases exponentially. So the density of state function not being accurate at very high energy is not relevant. It is not important. As long as you have an accurate description near the bandage, we are good. So, density of states times the probability function will give you this type of function in the conduction band. This is the function for the electron concentration as a function of energy. You can see that most of the electrons are concentrated at the bottom of the conduction band. Likewise, the density of state of valence band, which is near the top of the valence band goes as E squared and multiply by that by one minus F. Probability of finding a hole or probability of finding an empty state, not finding the electron, that product did you a function looking like this, this is the distribution of holes as a functional energy in the valence band. Now, if you integrate this whole thing across the energy that represents the energy colored red here, that represents the area colored red here. That area is the total number of electrons in the conduction band, or electron density in the entire conduction band and the area shaded blue here, it gives you the whole density in the entire valence band. Now, so if you perform the integral then you get that electron density in the conduction band and hole density in the valence band. But this integral can be done analytically, you have to use some numerical technique but there are some simple cases, important simple cases that you can do this integration analytically. So, that is the case when your Ec minus Ef conduction band bottom minus the formula level is much greater than kBT thermal energy, or Ef minus Ev is much greater in kBT. That is, in this case, it means that your formula level is far away from the bandage. If that's the case, then we can approximate the formula direct distribution function, or form of the probability function with a simple exponential factor which is the maximum Haltzman probability function. Then you can do the integration. And you can rewrite the integrate, integral like this. And you can re-express this integral and as this, you take the exponential factor out and you define N of c. So, what you do is you insert minus Ec equals Ec plus Ec inside and take Ec minus this part out because it does not depend on E, so you can take that outside the integral and you are left with this part inside the integral, which is this. And likewise, you can do the same trick with the hole density and you are left with these exponential factor containing formula level and the top of the valence band. And the integral of the remaining which is the product of density of state times the exponential factor containing energy. This quantity is called the effective density of states, it's not exactly the density of state. Density of states is these. It is a quantity closely related to the density of state but is multiplied with an exponential factor and integrate it throughout the band. So we call that an effective density of state for conduction band and the valence band. Now, you can do the integration here and you will get a quantity looking like this. And so this is the effective density of state for the conduction band this here, is the effective density of state for the valence band. Now, I want to point out that this is a quantity that depends on temperature. It is also a quantity that depends on the effective mass. So larger effective mass means higher density of state, effective density of state that is. And also, higher temperature, means higher effective density of state. Now, we can write down a Carrier Concentration in a simple compact form for a non-degenerate semiconductor. That is a semiconductor in which the formula is far away from the bandages. We can write down the equation for electron concentration and the conduction band like this. And all concentration in the valence band like this. Now, we know everything, we know the effective mass, if we know the band structure then we know the effective mass. In reality, you could just look up some references, we collect measures that calculated it, it's well known. All other parameters are either universal constants and the temperature, you know the temperature that you are operating at. So all we need to know is this: Ec minus Ef or Ef minus Ev. That is, where is the Formula level located relative to the bottom of the conduction band or the top of the valence band. That is something, that's the only thing that's missing and that prevents us from finally calculating the density of carrier concentrations in the conduction and valence band. Now, before we go on and attempt to calculate the Ef, we can derive a very useful expression that does not containing Ef. That is by noticing that this expression for electron contains negative Ef. This expression here, positive thought minus the minus here plus Ef this expression contains minus Ef. So if you multiply these two then Ef will cancel out and you are left with this. So the product of Np, the product of electron concentration in the conduction band in the whole concentration of the valence band that is equal to the product of the effective density of states times an exponential factor containing the band gap energy. This is called the Law of Mass Action. It is valid for any type of semiconductor, intrinsic or extrinsic, which we will introduce in the next video. So these is a very, very widely applicable expression that you will find very very useful.
