PLEASE NOTE: This version of the course has been formed from an earlier version, which was actively run by the instructor and his teaching assistants. Some of what is mentioned in the video lectures and the accompanying material regarding logistics, book availability and method of grading may no longer be relevant to the present version. Neither the instructor nor the original teaching assistants are running this version of the course. There will be no certificate offered for this course. Learn how MOS transistors work, and how to model them. The understanding provided in this course is essential not only for device modelers, but also for designers of high-performance circuits.
Conduction – Part 2
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En provenance du cours de Columbia University
MOS Transistors
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Yannis TsividisCharles Batchelor Professor of Electrical Engineering
Fu Foundation School of Engineering and Applied Science
In this video, we will talk about the second conduction mechanism, namely
diffusion. First of all, we will assume that only
electrons are present for now. And, in this figure, in the upper part of
this figure, we represent electrons by little dots.
And as you can see, towards the left, the concentration of electrons is high and it
gradually goes down as we go towards the right.
Electrons then tend to diffuse towards to the right from regions of high, high
concentration to regions of low concentration.
You would have gotten this effect even if these little dots were smoke particles.
In other words diffusion has nothing to do with the fact that electrons are charged.
It only has to do with the fact that the concentration varies with distance.
Diffusion is simply a statistical phenomenon.
If you take a narrow slab to the left of the cross section shown in the middle of
the bar electrons due to thermal agitation will be moving in all directions and some
of them will happen to cross towards the right.
If you now repeat this experiment and, and you consider a slab to the right of this
cross-section, then some of the electrons you need because of thermal agitation will
cross to the left. Because there are fewer electrons to the
right of this cross-section than to the left, more electrons on average crawls
towards the right than electrons from the other side crawls towards the left.
So, there is a net move, movement of electrons from left to right and this is
diffusion. Diffusion turns out to be proportional to
the gradient of the concentration. So let us look at the equation that
describes diffusion. It is shown here.
The more negative the concentration gradient is, the steeper this curve is and
the more the diffusion. Of course, the current is also
proportional to b times c, which is simply the cross-sectional area of the bar
through which current flows and it is proportional to the charge that each
carrier carries with it. And finally, we have a constant of
proportionality, d, which is the diffusion constant or diffusivity.
And the so-called Einstein relation tells us that the diffusion constant is the
mobility times the thermal voltage. I remind you that thermal voltage is kt
over q. Now, I would like to relate the current
not to the concentration of carriers, but directly to the concentration of the
charge per unit area with respect to position x.
What is the charge per unit area? If we have the uniform distribution, we
could take the total charge and divide by the [inaudible], by the area as we look at
it from the top. But the concentration here is non-uniform,
so we have to do this for a narrow slab of the semi-conductor, which has a total area
of b, the distance here times delta x. The small amount of length that we are
considering and the area is this. So the charge per unit area is the total
charge inside that region divided by the area.
What is the total charge in the region? It is the volume of this region, which is
b times c times delta x multiplied by the volume concentration of electrons, n of x
at that point times the charge of that each of these electrons carries.
So this is the total charge contained in this region and we divide by the n of the
area, as we look at it from the top, and these, and the delta x's cancel out.
So we finally get that Q prime, the charge per unit area that is now a function of
position, is minus Q times c times n of x. And, because of the minus sign in it, it
varies in the opposite direction from n of x.
It goes up, as you see here. So now, if you replace n of x from this
equation we just derived as a function of Q prime of x, you find this equation shown
here in the frame. And this tells us that the current is the
mobility times the thermal voltage times b, the width of the bar, times the rate of
change of Q prime, the charge per unit area with respect to position.
We will use this equation when we derive the general current equations for the MOS
transistor. Just to finish this, there is a special
case I would like to mention. If at a, in other words, at this point
here, the charge per unit area happens to go to zero, so we, we have this graph
here. Then it turns out you can derive using the
similar concepts as we've done before, that the transit time is a squared divided
by mobility times twice the thermal voltage as we see here.
And the reasons that you have a square over here are similar to those that I
mentioned in the case of drift. You can also have hole current.
The total current would be sum of the electron current plus the hole current.
The electron current can have a drift, drift component and a diffusion component,
and the hole current can have a drift component and a diffusion component as
well. So the total current in general can
consist of four components, but very often, it turns out that two or more of
these are negligible and it makes our life easier.
Finally, I will only mention that the total electron and hole currents, drift
plus diffusion can be related to quasi-Fermi potential gradients.
I will not discuss this, but you can find the results on the appendix B of the book.
So in this video, we thought about the fusion current, the current the current
due to concentration gradients in other words, and about total currents.
In the next video, we will discuss an important phenomenon, the conduct
potentials, which are potentials that happen when you join two of the similar
materials together. And this later will lead us to the
discussion of p-n junctions
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