So the first plot shows you what happens if the main affect is global variations

in the average value of the physical parameters.

Whereas the second plot shows you where, what happens when local random variations

dominate, and you have large mismatch between adjacent Identically laid out

devices. This can be a serious issue especially in

analog design, where you really try to match the characteristics of two devices

next to each other. Now how do we model variability?

The ideal way is to focus on independent physical parameters, for example, oxide

thickness and substrate doping. And for some of these perimeters you use

relative variations, for example, the oxide thickness is equal to some nominal

value plus some deviation because of global affects, plus some deviation

because of local affects. For this local affects, we have already

seen that, in order to suppress them, you need to have a large gate area.

And infact it turns out that the variance of this is proportional to one over the

gate area, or variance is the square of the standard deviation.

You can do similar things for substrate doping and mobility, you can model them

again using relative variations just like we did here.

Now let us take some parameters, for example, the flatband voltage.

Now, it doesn't make sense to talk about relative variations, because let's say if

flatband voltage is equal to zero nominally, then any variation from it

would correspond to an infinite percent variation.

So, it really does not make sense to talk about relative variation, so we talk

instead about absolute variations. So VFB has a nominal value plus a change

due to global parameter, variation the change due to local variations, which are

not normalized to the mean. But the variance of the local variation

still turns out to be inversely proportional to the gate area.

Now let's take delta W as another example.

Delta W, I'll remind you, is the correction you need to apply to the mask

width of a transistor in order to arrive at it's real channel width.

And that delta W turns out, again, to have a nominal value, plus some change

due to global variations, and some change due to local variations.

So, let's say the device looks like that. Now the variations, the local variations

here cannot be expected to depend on the gate area, simply because of the nature

of delta W, were talking about how W is different.

If the device has some bumpiness along this edge, and this edge, the longer the

channel is, the more this this bumpiness will average out, and then you expect the

variation to, to local effects to be small.

So, it turns out then, that this variance, the variance of this quantity

is inversely portional to L, for the reason I just mentioned, not to WL.

Similarly, the delta L, which is the corresponding variant, correction we need

to apply to the master length, to arrive at the real length of the device has some

local variations that turn out to be to have a variance inversely proportional to

W. [COUGH] Now, for independent statistical

variables, we can add variances. And, for example, for the flat band

voltage, assuming that the global and local variations are independent, we can

take the sum of the two variances to arrive at the total variance of the flat

band voltage. Now, for two devices we can define a

correlation coefficient as it is done in statistics.

For example, we have the correlation between the flatband voltage of 2

devices, 1 and 2. If we have very large devices.

Then the local variations would be small, much smaller than the global variations.

And then the correlation coefficient between the two is approximately 1.

This is close to what we saw in the delta VT plots that I showed you a couple

slides ago for large devices. On the other hand, for very small

devices, the local variations become large.

And then you have almost 0 correlation coefficient.

Which is close to the case for the small devices, in the same delta VT plot, that

I showed you. Similarly for other parameters.

Now there's some important composite parameters which are not fundamental

parameters like the oxide thickness and substrate doping they're not independent

parameters. One is the threshold voltage.

The threshold voltage will depend on oxide thickness, substrate doping, flat

band voltage, and so on. So, the threshold voltage is modeled the

same way as the flat band voltage. It has a nominal value, plus a variation

due to global, effects, and a variation due to local effects.

The variation due to local effects, turns out to have a variance inversely

portional to the gate area for basically the reasons I mentioned before.

And this constant AVT is measured and it is an important parameter, at least in

analog design. Another important parameter, is the

so-called beta, which is W over mu CX prime.

This is the coefficient of proportionality in front of all of our

drain current equations. this one is modeled in the relative

sense, so you have the nominal value plus the nominal value times the relative

variation due to the global effects plus the nominal value times the relative

variation due to local variations. Now the local effects, again, have a

variance that is inversely proportional to the gate area.

And this A sub beta is an important perimeter, that circuit designers like to

know. Both of these lead to the conclusion that

if you want 2 devices matched well, both in terms of threshold and beta, you need

to make their dimensions large. This is why when you look at the layout

of an analog chip you'll find often, devices that are significantly larger

than the devices you find in digital circuits.

I would like to briefly mention about, something about the correlation between

different parameters. Let us take the threshold voltage.

The threshold voltage is given by this formula.

We have derived this formula. This is the body effect coefficient, and

it is inversely proportional to the oxide capacitance per unit area.

The beta parameter I showed you a moment ago is proportional to the oxide

parameter, to the oxide capacitance per unit area.

Now you can see that the two quantities VT 0 and beta are correlated because they

both depend on oxide capacitance per unit area and therefore on oxide thickness.