Having developed a general analysis for the MOS Structure, it is now time to simplify the results and obtain simple equations that we can use in the specific case of strong inversion. Let us first, look at an MOS Structure. And the plot of the charge density, the electric field, and the potential versus y. I'm assuming that the gate charges, all pile up in a very, very shallow regions. So, rho is very large, but the width of this distribution here is very small. The same is true for the interface charge, here. The electrons also exist in a very shallow region. Actually called the charge sheet, shown here, and it's negative. And then, you have a constant density of charges, due to acceptor, atoms that are ionized, in the substrate like that. It is easy to show that the area under these things, this is the charge per unit area of Q'G prime, this is the effective interface charge per unit area of Q'0 zero prime, this is Q'I prime. The inversion limit charge per unit area and this is the depletion region charge per unit area, QB prime. So now, you will recall that we can integrate the charge density, taking into account the permittivity to produce the electric field, as we reviewed during our background review last week. So, we have. We start from a, a neutral point where there's no electric field, then we integrate this charge so the field goes up, and then, we enter the oxide. As you go from the gate to the oxide, you go from the material of a certain permittivity, here the material of a different permittivity. So, basic electrostatic show that there will be a jump in the field from here to there due to this fact. And again, you may have to review one of the appendices in the book if you are not current with such things or if you want, you can take what I'm saying for granted. Now, in the oxide, there is no charge, when you integrate, the field does not change. Then, you reach the interface charge, so it goes up. The integral of this charge here, shown here, goes up by that. Then, this is another jump. Why? Because you go from the oxide, which has a certain permittivity to the semiconductor that has a different permittivity. So, the ratio of the two permittivities will determine this jump over here. Then, you integrate the inverse or layer charge, go down like this, you integrate the depletion region charge like this and so on. Finally, you take the integral of this with the minus sign and get the, surf the potential. So, the potential, since this is positive, goes down here like this linearly because this is a cons. And then, over here, this one, this linear variation gives rise to a quadratic variation like this. So, this actually is the potential versus distance y, and we have seen it before. And we have said that the sum of oxide potential, surface potential, and contact potential difference, is equal to VGB, the total gate body applied voltage. Let's now continue with inversion. First of all, in inversion we're above depletion, so I assume that Psi S is larger than the [unknown] potential. And then, the general equation I had shown you for the total charge in the semi conductors simplifies to this. We saw how it simplifies for the case of deep depletion. Now, if you do the same thing and you neglect very small terms in it, it turns out you get this and this is an exercise that would, I would advice you to go through. So now, we have this structure, the general charge in the semiconductor is splitting to 2 charges, inverse layer charge and depletion region charge, QI and QB. I'm going to assume that all of the inversion layer charges are piled up in a very, very shallow region. This is the so-called charge sheet approximation, as I have already mentioned. And that means that this depletion region is purely a depletion region without any other charge in it than charges that I show here, immobile except or atoms. So, we can use the same approximation for Q'B that we have used for PN junctions in the P type of in the P type of the PN junction, we have derived this result, we can still use it for this case here. So now, what is the inversion layer charge, QI, which is in fact the charge that would be carrying current once this structure becomes part of an MOS transistor. Well, we know that Q'I plus Q'B is the total charge Q'C, therefore, Q'I is Q'C, minus Q'B. So, all you have to do is take this and subtract this from it and we end up with this. So, finally we have the inversion layer charge per unit area as a function of surface potential, Psi S, which appears in these places. And of course, it depends on the permittivity or a semiconductor and it depends on the substrate [unknown]. This equation is crucial to us. It is general, in the sense that it will predict that charge in inversion, meaning, strong inversion, moderate inversion and weak inversion. In special cases, such as strong inversion and weak inversion, this equation simplifies further as we will see pretty soon. So now, here is the same structure that we have been discussing. A repeat of the equations I showed you for Q'B and Q'I. Let's plot this one. Here is the result. This is the surface potential. This is the bulk charge, Q'B, like this. It goes as the square root of Psi S. That's why it has this shape. Q'i, in the beginning is very small because this term is very small, and this root of Psi S cancels this root of Psi S, so it starts with practically zero. And then, eventually the exponential takes off and Q'I really, takes off like this. The total charge Q'C is the sum of Q'B plus Q'I. So, you'll see it over here. So, practically, the total charge coincides with the depletion region charge for low surface potential values and becomes practically equal to Q'I for large, surface potential values. Now, I will not go through a detailed discussion of the properties of this plot. But I will only mention that before this really takes off, you have the weak inversion region, after it really takes off and has a large slope, you have the strong inversion region. And in between, you have the moderate inversion region. These regions can be defined precisely. And I will say something about this a little later. So, what do we have now? We have the potential balance equation that we have written before. We have the charge balance equation, which now we write as gate, plus interface, plus inversion layer, plus depletion region charge, add up to 0. We still have the linear relation between the gate charge above the oxide and the oxide potential. We have the bulk charge which is proportional to the square root of the surface potential, all of that we have shown. And finally, we have the equation for Q'I prime, the inversion layer charge per unit area, which we have shown is given by this. These equations are fundamental in inversion. They are five equations, in five unknowns. What are the unknowns? The oxide potential, the surface potential, the gate charge, the bulk charge, and the inversion layer charge. So, in principal, you have enough information to solve for any of these, but again, this one complicates things and that this equation cannot be solved explicitly for Psi S. Nevertheless, we can use this equations to derive some useful results. This is one of them. So, by eliminating certain unknowns here, playing with these equations, it's very easy to end up with this. This is now an equation relating the surface potential to the externally applied gate body voltage. You can solve for it, but not analytically. You have to do it numerically, and another equation you can derive by playing with these equations is this one. That gives you the inverse layer charge, in terms of VGB and Psi S. We will have use for both of these equations pretty soon, here they will make our life easier in discussing inversion and in fact particular regions of inversion. The same structure shown here, the same equation I just arrived shown here and here is a plot. This is the surface potential versus VGB. As you raise VGB you go from depletion to weak inversion. Then you go to moderate inversion and then you go to strong inversion, and again, this limits can defined precisely. Notice that the weak inversion is between phi F or surface potential and 2 phi F and here for a given VGB you have the resulting Psi S. Now, with respect to Psi S you, can plot the charges. And these charges here are the charges I have shown you in a plot previously, only this plot has been now rotated 90 degrees counterclockwise, so that this axis Psi S aligns with this psi s. So for a given value of VGB, let's say here, you can find the corresponding Psi S over here, and then, go here and find the charges. As I said before, the limits between depletion and weak inversion, weak inversion, moderate inversion and so on, can be defined precisely. These limits will be denoted by VLO, VMO And VHO. There are reason for these names, I don't want to waste time here, again, there is much more information about them in the book. You can find these limits precisely, you can define them and develop equations for them. The reason I will not do so in this lecture is, first of all, we do not have time and second, we're emphasizing models that are valid throughout. The inversion region, whether it is weak or moderate or strong. So, for such models you don't really need to know what the values of these things are. In the next video, I will show you how these general results that were valid in all regions of inversion simplify in the case of strong inversion.