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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.

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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.

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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.

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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.