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In the opposite case, when you have very small distances, or very small times,

then, between scattering events, between collisions, the velocity can be higher

than the drift velocity, and this is what we call velocity overshoot.

Also when the distances are very small, there may be situations where you go from

the source to the drain, without a significant number of collisions, or

without any collisions. This is demonstrated very well using

Monte Carlo analysis. Monte Carlo is an analysis that tracks

individual carriers. It describes the individual physics of

each carrier. And then derives statistical results

based on the behavior of a very large number of carriers.

It's a very detailed tool that is computationally very intensive.

It is very good for theoretical investigations but not appropriate for

compact models for cab. Nevertheless it is is instructive to see

a result from such an analysis. This is a Monte-Carlo simulation by

Frank, Laux and Fischetti. The reference is in the book.

This is the position along the channel. Roughly, the source extends oh, up to

this point, the drain is here, and the rest here is the channel.

And here is E sub C, the bottom of the conduction band, edge.

I remind you that the bottom of the conduction band is, corresponds to the

minimum potential energy of. To the potential energy, rather of

electrons, it is the minimum energy they have to have, in order to be in the

conduction mat, and therefore be free. The individual spots here are electrons.

There are many electrons in the source. Now, these electrons move because of

random thermal motion, and diffuse, some of them may find themselves to the right

of the peak here. Once they are here, then, they encounter,

towards the right, a decreasing E sub C, which is equivalent to saying an

increasing potential Therefore there is a field that attracts these negative

charges towards the drain. So they start going towards the drain.

Some of them encounter collisions, and lose their kinetic energy, they fall

down, like this. And, this for, this electron here, barely

has a potential energy enough to, to keep it free.

Then it starts accelerating again, it might encounter another collision, go

down again, and so on. But there are some other electrons that

can keep going, and they're lucky, in the sense that they haven't encountered any

collision. For example, this guy here, has

practically the same energy as it had over here.

It hasn't encountered any collisions, so it hasn't fallen down.

And there are a few of those electrons that go all the way to the drain, without

encounter any collisions. This is called ballistic operation, for

obvious reasons. It can be encountered in extremely small

lengths, for a given technology. Now the velocity of these electrons that

make it all the way to the drain due to ballistic operation without any collision

can be higher then thermal velocity and the saturation velocity.

And by thermal velocity I mean the average velocity due to thermal motion in

the lattice. If you increase, V D S, then E sub C

becomes steeper. And the field near the sources increases,

the hump here becomes narrower. And we have discussed related effects

when we discussed dibble. So now, it is more likely that some

electrons will find themselves To the right of the hump, and we'll go.

And that means then that, the current, due to these electrons, increases.

Such effects, can be taken into account to describe the current, using the

equation we have shown again and again. The current is W times minus the

inversion layer Charge density, times the drift velocity here.

Because of these effects we have to replace the drift velocity by an

effective velocity. And, we look at the effective velocity

and the charge at x 0, where x 0 is the position of the peak of the E C curve

over here. So this is called the virtual source, it

occurs at x equal x 0. And, with an equation like this, you can

predict the current. And you can extend this type of analysis,

even to include current from the drain towards the source.

For, after all, if there is no drain source voltage, there are two currents,

one from the virtual source and another rom the virtual drain, and the two cancel

each other out. And you get zero current.

There are concepts like that, that people use to model a device.

emphasizing the shape of v's of c and you can find those in the references in the

book. So we have seen very briefly the concepts

of velocity over suit and bolistic operation.

In this video. most CAD models do not incorporate these

effects. For now these are mostly of theoretical

interest, but they are intensely studied, and of course, it is good to be familiar

with these terms, in case you encounter them in the literature.

In the next video, we'll, we'll consider another type of effect, polysilicon

depletion. And in contrast to the effects we've seen

so far, that had to do with a small horizontal dimensions, this one is of a

different nature, as you will see. And it can occur even in long tunnel

devices.