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In the previous video, we introduced a three-terminal to the two-terminal MOS

structure and discussed how applying a potential to that terminal with respect to

the body, modifies the electrical concentration in the channel, and showed

how various equations we have already derived for the two-terminal MOS structure

can be modified in a simple way to include this effect.

In this final video devoted to the, the three-terminal MOS structure, we will

consider specifically the body effect for this structure and also how strong

inversion and weak inversion equations get modified to be valid in that case.

Let's us begin with strong inversion. I am going to show you here the equations

that we already know for the two-terminal structure and how they get modified for

the three-terminal structure. The, I believe I have already made what

you're about to see plausible by what I showed you in the previous video, but, of

course, if you would like more details and more careful derivations, you're welcome

to look at the book. You will remember that in strong

inversion, for the two-terminal structure, we had found that the surface potential is

approximately pinned to a constant value which we had, we had called phi 0.

Well, for the three-terminal structure, instead of phi 0, we have phi 0 plus VCB.

And the reason is already, as described in the previous video.

This was the bulk charge or body charge for the two-terminal structure.

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And if we now have a third terminal with the voltage VCB applied, instead, we will

have this. This was the equation for the inversion

layer charge, where VT0 was the threshold voltage for the two-terminal structure.

In this case, we will have the same equation and instead of VTO, we'll, we'll

have something that is VTB. What was this threshold VT0 for the

two-terminal structure? It was just this, as you see it here.

So now, because of instead of phi 0, we now have phi 0 plus VCB here.

You see that the new [unknown] threshold will be as in the two-terminal case, but

it will have phi 0 plus VCB everywhere. Let us now look at a common practice.

We have seen the structure like this, where all voltages are referred to the

body terminal. This is our reference point.

So, we have gate body voltage and CB voltage.

Very often, instead of doing that, we refer voltages to the C terminal here.

So, we're talking about the gate to V voltage called VGC and we can talk about

the body to C voltage. But typically, we, we do the opposite.

We talk about C to B voltage, VCB. Now, it is clear that these two are

equivalent because VGC, VGC is the potential between gate and C terminal.

But, VGC is VGB minus VCB, right? So, if you take the difference between

this voltage and this voltage here, you will have VGC, okay?

So, the two are equivalent. Sometimes, we write equations for the top

way of doing things, and sometimes, we write equations for the bottom way of

doing things. So, for example, we know that for the top

structural, we had an inversion layer charge in strong inversion that was given

by this equation. Here, it would be given by this equation.

And what is VGC? It is simply VGB minus VCB, as we already

said here. And instead of VTB, we have a new

threshold which is, VTB minus VCB. Vtb is the threshold referred to the body

and VT Is the threshold referred to the C terminal.

Later, this will become the threshold referred to the source of the MOS

transistor which is a very widely used quantity.

Now, this is the threshold, if you take the expression role for VTB that I, I have

already shown you before and you subtract VCB from it, according to this, then you

get this equation for VT. And there's another common way of writing

it, it is this one, for VT0 stands for the 0 VCB threshold, in other words, if you

take this equation, you plug in VCB equal to 0, then what you get is this, VT0.

Vto coincides with the threshold of the two-terminal structure, and this shows you

how the new threshold, VT, goes above VT0 if you increase the voltage, VCB.

And this turns out to be one of the manifestations of the body effect that we

are about to study. The value of VT0, is given by this.

It is the value of VT, when VCB is 0. So, let's look at the body effect.

And sometimes, you hear that the body effect is that the fact that the threshold

changes when the body voltage changes. But in strong inversion, where you have

something called the threshold, it applies throughout, so this is a more general way

of defining the body effect and it goes like this.

Let us take this structure, with C being the reference terminal.

Let us keep VGC constant and let us increase VCB.

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Now, when you increase VCB, what will the charge of the electrons do here, what will

the inversion layer charge per unit area do?

Take a case in strong inversion, for example.

You have a lot of electrons in the inversion layer.

In a sense, the inversion layer is an extension of the n plus region.

And sometimes, the combination of the inversion layer and the body is called the

field induced n-p junction because the gate field created a lot of electrons here

and made the surface look like an n-type region.

So now, if you increase VCB, you are increasing the reverse bias between the n

and the p, both for the p-n junction and for the combination of inversion layer and

body. And what happens when you increase the

reverse bias, the depletion region width widens.

However, because V, VGC is kept constant, the voltage between the gate and the

inversion layers stays fixed. So, that means that across this capacitor,

you have a fixed potential. So, the amount of charges that we have on

the gate remain approximately fixed when you increase VCB, they don't change.

Now, these positive charges on the gate must be mirrored into negative charges, in

the substrate, in the body. Before, you have the bunch of electrons

universal layer and some accept our ions in depletion region.

Now, because you have increase the reverse bias of the p-n junction, you have more

negatively charged acceptor ions so you need fewer electrons in the inversion

layer, and therefore, the level of inversion will decrease.

So, we have seen, then, that when we keep VGC constant and when we increase VCB, QI,

the inversion layer transfer unit area goes down in magnitude.

And this is the definition of the body effect.

The body effect means that when the voltage between C and B goes up, the

magnitude of the inversion layer charge goes down, provided we keep VGC fixed.

And, of course, if we want to restore QI to the previous level, we must increase

VGC. So, here is then a set of curves versus

VCB. The VL, VM, and VH are the onset of weak,

moderate, and strong inversion. So, VL, VM, and VH are the values of VGC

at which you get to the beginning of weak inversion, the beginning of moderate

inversion, and the beginning of strong inversion.

They all go up with VCB as you can see, consistent with the body effect, and even

the body threshold goes up in the same way.

So, for example, if we take the beginning of moderate inversion VM, which is this

curve over here, it turns out to be given by this.

You may feel that already you already can predict this from what we have already

said. But a more careful derivation of this is

given in the book. The gamma quantity here is this, we have

already seen it before, and it is called the body effect coefficient because it is

instrumental in describing the body effect.

So, let's now look at the threshold. This is the threshold equation, we have

seen it before. The effect of VCB is shown here.

And you can see that the VCB goes up, the threshold goes up.

Vcb sometimes is called the back gate bias.

We will see later why this name has stuck. We also see that the influence of VCB on

the threshold is stronger if gamma is larger.

So, if you plug the increase of VT, compared to its value on VCB 0, this is,

we're plugging here VT minus VT0 versus VCB, you see that it goes up, and the

larger the value of gamma the worse things get.

Of course, such large values here are unreal, unrealistic.

There have been in the past devices with large gammas but of course, we try to keep

the body effect low for reasons that we will see when we get to the MOS

transistor. I would like to finish this slide by

saying that this is one of the manifestations of the body effect.

The body effect, as I already mentioned, is not only about threshold, it's

something more general. Okay.

Anew definition that we will need in our discussion of the MOS transistor is the

pinchoff voltage. So, let's take this structure, and now

back to referring everything to the body terminal.

The strong inversion approximation that we have shown is this one, QI is a

first-degree polynomial with VGB. And in this equation, we have the

threshold referred to the body which depends on VCB.

And I remind you, this was the expression for that threshold.

So now, you can see that if you keep VC, VGB the same, we keep VGB the same, but we

increase VCB, what will happen? The threshold will go up.

So, in this difference, it will have the effect of reducing the magnitude of QI and

eventually, this equation predicts that when VCB is so large that the VTB becomes

equal to VGB, QI will become 0. So, let's look at the plot.

This is what the plot looks like. This equation here is this plot, the

broken line. And, indeed, it shows that, eventually, QI

goes to 0. But when QI has a very small value, and it

is close to 0, we're not in strong inversion.

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We should not even be applying the strong inversion equation in that case.

But historically, that's how people did it, and so I have to show you that so you

can follow the literature. So then, again, QI goes to 0 only because

this strong inversion equation is being used outside its region of validity.

The region of validity of strong inversion is up to here where you have a significant

charts. Below it, it predicts wrong values for QI.

The accurate full inversion equation gives you this behavior, the solid line and here

you can see clearly that QI continues to be nonzero, and it only asymptotically

goes to 0 like this. Now, what is the pinchoff voltage?

The pinchoff voltage is the intercept of the strong inversion equation for QI.

So, the definition of the pinchoff voltage is this.

The pinchoff voltage is that value of VCB, that value of VCB, that makes the

threshold equal to the externally applied gated body, gate body voltage.

In other words, it makes this quantity equal to this quantity, and QI is

predicted to be 0. That is the pinchoff voltage.

So, it is easy to calculate the value of the pinchoff voltage.

All you have to do is take this definition and apply it.

You take VTB from here, you set it equal to VGB as the definition demands, and the

value of VCB that results as the solution of this equation is the pinchoff voltage,

and this algebra gives you this value for the pinchoff voltage.

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So, as a final topic for this video, we will talk about weak inversion.

I will not re-derive everything because the steps are similar to what you have

already seen for the two-terminal structure.

If you expand QI as we did for the two-terminal structure, you get the same

thing as before. But instead of 2phiF, we have 2phiF plus

VCB. Psi s is the surface potential inversion,

and in weak inversion, where the concentration of electrons is very, very

small, and the [unknown] of charge is much smaller than the depletion region charge,

we can still use the surface potential equation for depletion regions, which is

shown here. So, if you plug in psi s equal to psi sa

in the above equation, you can write it in this form.

And there's something very nice about this equation.

The influence of VGB is contained here, through psi sa, where psi sa is this.

And the influence of VCB, because I've separated the exponents in the top

equation, is described over here by a separate factor.

That will turn out to give us a very nice equation for the MOS transition on weak

inversion and which, in fact, will be similar to what you may have seen for

bipolar transistors. Continuing with the weak inversion, let us

assume we have a fixed given VCB. If you plug the surface potential versus

gate body voltage, you get the same type of shape as you got before only, instead

of the top of the weakened version being at 2phiF, it is now at 2phiF plus VCB.

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Now, to simplify the equation, you follow the same steps as we followed for the

two-terminal structure. We find the slope at the top of weak

inversion, and we call it 1 over n. So, what is n then?

It is the inverse of the slope. N is defined like this, okay?

And then, you make an approximation, instead of you, because this is

practically a straight line, instead of talking of changes along the surface

potential axis, we talk about changes along the VGB axis.

And following the same approach as for the two-terminal case, we get this equation.

So, VGB is the gate body voltage. Vmb is the onset of moderate inversion,

corresponds to this point here. It is the top of the weak inversion

region, this is VMB. And you have the differ, the difference

between VGB and VMB. If you want, you can refer everything to

the C terminal. And you can write in, instead of VGB minus

VMB, you can write VGC minus VM, where VGC is VGB minus VCB, and VM is VMB minus VC,

VS, VCB, like this. In this video, we have seen how the

presence of a third terminal in the three-terminal MOS structure, allows us to

control the, what happens in the channel. We discussed the body effect, we discussed

strong inversion and weak inversion. At this point we are done with both

two-terminal and three-terminal structures, and we are ready to begin our

discussion of the four-terminal MOS transistor.

Starting with the next video, this is all we will be doing for the rest of the

course.