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The Two-Terminal MOS Structure – General Analysis
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From the course by Columbia University
MOS Transistors
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From the lesson
The Two-Terminal and Three-Terminal MOS Structures
Meet the Instructors
Yannis TsividisCharles Batchelor Professor of Electrical Engineering
Fu Foundation School of Engineering and Applied Science
In this video, I will sketch for you, a way to do a general analysis of the
two-terminal MOS structure. We will develop equations that will be
valid for any surface condition. Be it accumulation, depletion, or
inversion. Here is the structure we have been
discussing. The general MOS structure with the general
voltage Vgb. The surface potential, between surface and
bulk, is Cs. The oxide potential is Cox.
And at any point y, the potential between that point and the bulk is C of y.
We need to write equations for this structure, that can predict what will
happen. For any value of V-gv, whether it is in
depletion, or in, at flat band, or in inversion or in accumulation.
We have to start like this. And once we have the general expressions
then we will simplify for particular regions, notably the regions of the
inversion that we are mostly interested in.
So we start with this equation, which that we have already used.
This relates, the, ratio of electron concentration at the point y, to, the
bulk, so n of y to n, n zero. And we have the exponential that we
involves the potential from the point we're interested in to the[INAUDIBLE] of y
divided by the thermal voltage. We already know that there is a
corresponding equation for holes, which is the same form, except it has a minor sign
here. And we also know that the total density at
any point y, in general Can consist of holes and electrons and ionized acceptor
atoms and ionized donor atoms with these sins.
We covered this in our background review. Finally, we have related charge density to
potential by using Poisson's equation, which is shown here.
Second derivative of, the potential with respect to position.
Rho is the total density over here. Epsilon sub s is the permittivity.
Now, if you manipulate these equations, you can derive the following result.
I'm going to bypass a lot of mathematical steps.
If you are interested, you can find them in Appendix C of the book.
So starting from this equations, we get this result.
It looks kind of very long but in fact it is written in a form that is probably
clearer than any form you may have seen in the semiconductor books.
The reason is that if you trace the development of this equation, you can
clearly identify where each term comes from.
This one comes from the hole contribution, if you have any holes in the surface, for
example, in the accumulation. This comes from, ionized acceptor atoms,
and we even allow for the presence of donor atoms, at the same time.
I forgot to say that this is possible in general because sometimes you may have a,
an n-type region, and to convert it into a p-type region You diffuse enough acceptor
atoms, so that the overall population of acceptors is larger than that of donors.
And the difference is the effective acceptor concentration.
So to allow for such a general situation, I have allowed both of our n sub a, and n
sub b. N sub d here.
And finally, this term is due to electrons.
This is the contribution of electrons to the whole chart.
So now we an equation that gives you the total charge Qc per unit area, that's what
the prime sign means here, as a function of the service potential.
This term here sign of Cs is simply one if Cs is positive.
It is, zero if Cs is zero, and it is equal to, minus one, if Cs is negative.
If you work with this equation. And, you make certain assumptions and
substitutions, again I have to skip the steps, but they are in the book.
You find this equation which is closer to the equation you see in semiconductor
device books. So here is a repeat of this equation.
It relates the total charge in the semiconductor to Cs,work Cs is the surface
potential. We also know that the gate charge is the
oxide capacitance times. The oxide potential.
This is the same as what you will have in the linear capacitor.
We also know that the sum of the gate charge, effective interface charge, and
space charge, is zero. Finally we have the, potential balance
equation, which we have shown before, and it is this one.
Now, if you look at this set of equations, what do you see?
You see four equations in foreign nodes. Vgb is applied externally, so this is
supposed to be a known voltage. The only unknowns are 2 charges.
[inaudible] charge, and semi conductor charge.
And two potentials. The ox side potential, and the surface
potential. Now in principle having four equations and
four unknowns you should be able to solve them.
Unfortunately because of this equation, it's not possible to really have explicit
solutions, so you must resort to numerical solutions for this.
Nevertheless, we can do something useful here.
We're going to eliminate some unknowns between these equations and end up with
this equation. What is this equation?
This equation relates the surface potential of Cs, which appears at various
places here To the externally applied voltage.
So in principle now we have finally a result that allows us to calculate the
surface potential as a faction of the externally applied voltage.
If we know the surface potential then we can find the[UNKNOWN] charged and we can
end up with the MS transistor equations. And that is why we are dealing with this.
This is probably the longest equation we'll ever see in this book, and pretty
soon I'm, I'm going to simplify it further.
This quantity here, the square root divided by C ox prime will be called
gamma. This will turn out to be the body effect
coefficient that you may have actually already used in simple models for
[unknown] transistors. We'll come back to that.
So we can solve this for cs, for a given Vgb, to find the surface potential.
And then, once we have the surface potential, we can plug it into this
equation at the top and find the space charge.
So we can plug these things. In fact, to plug this one, it turns out,
you don't even have to solve this numerically, because what you can do is,
you can give values to psi S, calculate the right-hand side, and find Vgb.
So now you can, create, psi S comma Vgb points and plot them.
So let's see what these plots look like. This is, what I just mentioned.
The plot of CS versus VGB, and I subtracted, the Flatman voltage here, to
make things cleaner. This is the total semiconductor charge, as
a function of the same quantity, Vgb minus the flat band voltage.
So let's interpret these plots, having in mind this general structure.
So let's start with the charge. It says that if Vgb minus Vfb is negative,
in other words Vgb is less than Vfb, we are in accumulation, and the charge is
positive. It should be because over here we would
get holes in accumulation as we have already discussed.
Then if Vgb starts becoming larger than Vfb, so when this is positive, we have
depletion. The charge becomes negative and increases,
because we are depleting acceptor atoms. And finally, when this is large enough,
you start getting electrons at the surface.
And now the charge becomes significantly more negative because of the presence of
electrons. Let's look at this plot.
When Vgb minus Vfb is negative, we are in accumulation, and the surface potential is
negative, as we have already mentioned. When we are in depletion, the surface
potential is positive, and it depletes the semiconductor.
You have ionized acceptor atoms And as you increase vgb, you deplete more and more.
And the surface potential goes up. And finally, the surface potential
flattens up, out like this. Why does this happen?
It happens because, at such values of vgb, the surface potential is so large over
here. That it's, the surface starts becoming
very attractive for electrons. You may recall that the concentration of
electrons at the surface is exponentially related to the surface potential.
So once that exponential takes off. Then, you can have plenty of extra charges
here without having to increase cs very much.
So as Vgb becomes large the charge, in the semiconductor which will start being
mostly due to electrons can become large without having to increase the surface
potential very much. Now, here this region is called inversion.
We can define it more precisely later on. I would like to mention this preliminary
fact here that the beginning part of the inversion is the weak inversion.
Start again. The beginning of this region is weak
inversion. Then we get into moderate inversion, and
when this flattens out we are in strong inversion.
We will come back and define these things more precisely later on.
Now we can simply these results in particular regions.
Let's for example assume that we are deep in depletion, where CS is positive, and
deep in depletion means we have a CS larger than several phi t, but not as high
as phi F. So now, certain, terms, become completely
negligible. And we also know the sign over here.
The sign of c s is positive, because we are in, depletion.
Therefore, the function sign[INAUDIBLE] s is equal to plus one.
This term is completely negligible because it is, ex-, an exponential, with a
negative quantity and the, absolute value of the quantity's large.
So this we can neglect. The same thing happens here.
This is an extremely small quantity multiplying this whole thing.
So you can forget about all this. And even phi T can be neglected compared
to Cs so sometimes we like to leave phi-T there but it really does not make much
difference for our purposes so I will. Be neglecting it.
You can do the same with the equation we had for VGB versus CS.
And we get the corresponding equation like this.
So now we have simple results for deep and depletion, let me summarize the results.
This equations after you eliminate these extremely small terms become like this.
This is the charge. And, this is, the gate body voltage.
Notice now we have very simple expressions.
The charge in the semiconductor in depletion, which is due to ionized
acceptor atoms, is proportional to, the square root of the surface potential.
And VGB is related to[INAUDIBLE] in this way.
Remember that in the general equation we had that related VGB to[INAUDIBLE], we
could not solve explicitly for[INAUDIBLE]. But this one of course we can solve
explicitly for[INAUDIBLE], and the result is this.
So this now gives you the surface potential for a given vgb, explicitly.
This expression, which is the surface potential in depletion, will be called csa
and we will have use for it pretty soon. Here is a plot of Cs versus Vgb minus the
flat[UNKNOWN] voltage. As you, we've seen in the previous slide
it flattens out like this in strong inversion.
And Csa is this expression, and it goes like that.
You can see now that in depletion, the two are extremely accurate.
They practically coincide, but not only do they coincide in depletion, they even
coincide in the, the beginning of the inversion region, which is the weak
inversion region. So in fact, what we will be doing later on
is we will approximate the surface potential in weak inversion by Csa.
So we have seen a way to analyze the semiconductor in a very general way.
Resulting in equations that are valued in accumulation, depletion, and inversion.
In the next video, we will focus on the most important region for us, inversion.
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