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.