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.