[Slide 1 L2.6] Well we've come to the end of Unit 2
so let's just quickly summarize the main topics
that we've been discussing in Unit 2
and then we'll be prepared to move on to the next unit.
[Slide 2] So the main topics that we talked about were
after a quick review of energy bands
we went to the energy band in view of a MOSFET.
This is a really important thing for you to understand
because if you understand how MOSFETs function
in terms of energy band diagrams,
you really know most of the story.
We then very quickly reviewed traditional MOSFET I-V theory,
going in a little more detail,
the classic square law theory.
And we put it all together in something we call
the Virtual Source Model,
a model that we're going to continue to develop and refine
as we proceed throughout this course.
[Slide 3] So in terms of energy band diagrams,
an energy band diagram,
the first task when analyzing any device problem
or semiconductor problem typically
is to draw an energy band diagram.
Bands bend when the electrostatic potential
is changing with position.
The second task is to read the energy band diagram.
To read the energy band diagram,
if we want the electrostatic potential
we just flip one of the bands upside down.
If we want the electric field
we just take the slope of one of the bands.
And we recognize that the electron and hole densities
depend exponentially on the separation
of the Fermi level from the intrinsic level.
Or the Fermi level from the conduction band
or the Fermi level from the valence band.
So we can easily read off qualitatively
the hole and electron density that way.
Now we could also get the space charge density.
The Poisson equation says
that the slope of the electric field
is related to the space charge density
so the electric field
is related to the slope of the energy band.
We could take a second derivative.
Or it turns out it's usually easier
to deduce what the space charge density is
by knowing what the hole
and electron concentrations are versus position
and presumably we know how we've doped the semiconductor
so we can put that together
and deduce the space charge density.
[Slide 4] So how transistors operate
in terms of energy band diagrams is a very simple story.
In the linear region we recognize
that when we draw the energy band diagram
there is a potential energy barrier for electrons
between the source and the drain.
That barrier is in the channel.
We apply a small voltage to the drain
and pull the conduction band in the drain down a little bit
and then the gate voltage,
a positive gate voltage lowers the energy.
As we push the barrier down,
more and more current flows
and we get the linear region current.
Up here in the saturated region,
that's the region where the drain voltage is high.
We pull the conduction band to the drain way down,
but the height of the barrier
is controlled by the gate voltage
in a well-designed transistor
and only leakage current flows unless we apply
a positive gate voltage, push the barrier down
and then electrons flow out of the source,
over the small potential barrier downhill
all the way to the drain and out the contact.
So this is a very simple but very physical picture
of how a transistor operates.
[Slide 5] So we talk briefly then about how we can
understand some things we had discussed in Unit 1
in terms of these energy band diagrams.
So for example when we see the transfer characteristics
plotted on a log plot and we remember how to extract DIBL
and DIBL is a reflection of the fact
that the threshold voltage is decreasing
as the drain voltage increases
so that leads to a translation
of the transfer characteristic in the sub-threshold region.
We pull that number off and quote it
in millivolts per volt of translation.
But the physics behind that,
now we can understand in terms of energy band diagrams.
[Slide 6] Now if this is our energy band diagram,
zero gate voltage and low drain voltage,
there's a large energy barrier,
electrons can't flow from the source
across the channel and out the drain.
If we apply a large drain voltage and pull the barrier down,
the electrons still have a hard time flowing
because of the large energy barrier.
But if the drain voltage
pulls the barrier down a little bit,
that helps turn the device on,
helps make it easier for electrons to hop over the barrier.
Effectively that lowers the threshold voltage.
That's the physics of DIBL and a transistor designer
would want to minimize this effect,
would want the drain voltage
to pull that barrier down as little as possible.
We would like it in ideal transistor
for the height of that energy barrier
to be fully controlled by the gate voltage
and not by the drain voltage.
[Slide 7] Alright so DIBL physics is easy to understand
in terms of energy band diagrams.
Well then we turn to deriving a few equations.
So the equations,
the physical picture is more easily conveyed
by the energy band diagram,
but we want some simple physical equations
that tell us the magnitude of the current.
We did the linear region first.
We wrote the current is charge times velocity.
We wrote the charge is Cox VG minus VT.
We wrote the velocity is mobility times electric field.
We said the electric field is just the voltage
across the channel divided by the channel length.
We put it all together and we got a very simple
but very good description of the linear region current.
[Slide 8] Okay, we also looked at the high voltage region.
In the high drain voltage region we consider two cases,
a long channel device which behave as square law devices
and a short channel device.
In the long channel device
something called velocity pinch-off occurs.
The mobile charge goes approximately to zero
when we hit the pinch-off point.
The current doesn't stop flowing, but it saturates.
We write the current the same way,
proportional to charge times velocity.
We focus on the charge at the beginning of the channel,
the so-called virtual source
because we know the charge there.
It's Cox VG minus VT.
The velocity is still mobility times electric field.
The electric field we would estimate
as the voltage across the channel.
Well the voltage at the pinch-off point
is not the drain voltage.
It's the pinch-off voltage, VG minus VT.
It's zero at the other end so the voltage
across the channel is the pinch-off voltage.
And minus zero at the source end.
Pinch-off voltage is VG minus VT.
So the average velocity in the channel is minus,
it's in the negative direction,
VG minus VT divided by the length of the channel.
But it turns out that the electric field
is really non-linear.
We need the electric field at the source
where we know the charge
and that turns out to be one half of the average value.
So we derived our classic square law MOSFET characteristic.
[Slide 9] Then we took the next step and we actually
went through a complete I-V derivation
so that we could not only do linear region,
saturated region.
We could do the triode region in between
and we saw how you can get a full description
of the I-V characteristic that is continuous
from linear to saturation.
[Slide 10] But then we switch to the case
that is more descriptive of modern transistors,
the so-called velocity saturation case.
And what we argued there is that the electric field
is very high in these very short channel transistors.
Even if the voltage on the drain is small,
the channel length is so incredibly short
that the electric field is very high.
Well we write the current as charge times velocity.
The charge is still minus Cox VG minus VT.
But now the velocity we argue,
the electric field is so high
the electric field will saturate
at its scattering limited velocity.
Okay now it's even easier to derive the I-V characteristic.
You put all those together and we get
the velocity saturation characteristic
which actually does a good job
of describing the I-V characteristics of modern transistors.
[Slide 11] Okay only one problem.
It's based on physics that we know is just wrong
and we know that from details numerical simulations
and a number of experiments as well
that show us that the average velocity
of electrons in the small channels can greatly exceed
the high field saturation, velocity in bulk silicon.
And it does that because there just isn't time
for electrons to scatter very much and slow down
as they zip across the channel and out the drain.
So this is an effect that we need to come back to
and understand more deeply.
[Slide 12] But we have developed a two piece model
and then the next step was to see if we could continuously
connect the linear region to the saturation region
so that we have a smooth model
that goes between all regions.
[Slide 13] So we developed a way to do that
and the way to do that we called a level zero VS model
and the equations are summarized here.
This is a model that you're going to see again and again
and we'll just,
we'll make it a little more sophisticated
as we learn more about MOSFET physics
and by the end of the course we'll have a model
that does a very good job of describing
the physics and the measured characteristics
of modern transistors with very short channel lenghts.
One of the attractive features about this model
is there are only a few parameters in the model.
Most of them have a clear physical significance.
Our problem right now is that two of them
don't seem to be physically valid,
but we will find that they can be reinterpreted
in a physically sensible way
and that's something we'll be discussing later.
And there's one empirical parameter
that we introduced as well
in this velocity saturation function.
[Slide 14] Alright so where we'll end up then
as we refine this model is something,
a form of the virtual source model
that I'll refer to as the MIT virtual source model
because it was developed by a group at MIT.
And it provides us with a very good way to describe,
a way to understand the physics of small transistors
and a way to analyze measured IV characteristics
and extract physically meaningful parameters
and understanding from those.
[Slide 15] What we are going to need to do though
is to take these two parameters,
mobility and saturated velocity,
whose physical significance
is unclear at the least at the nanoscale
and we're going to argue that they can be replaced
by something called an apparent mobility
and an injection velocity
and these are not fudged parameters,
not fitting parameters, not empirical.
They have sound physical basis
and we'll be discussing that physical basis
when we get to Unit 4.
So that pretty much wraps up the topics
that we discussed in Unit 2.
Now before we dive into this interesting transport physics,
we're going to spend some time in unit three
on MOS electrostatics.
[Slide 16] MOS electrostatics is a really important part
of transistor design and operation.
MOS electrostatics describes how the voltages
that we apply to the terminals
affect the height of these energy barriers
and raising and lowering those energy barriers
is how we manipulate the current.
Properly designed MOSFETs will have a
something we call electrostatic integrity.
They will be designed in a way
that the charge at the virtual source,
at the top of that energy barrier,
obey this very simple expression.
It's mostly controlled by the voltage on the gate
with a very weak dependence on the drain voltage
through the DIBL parameter.
Making that happen requires a good understanding
of both 1-dimensional
and 2-dimensional MOS electrostatics.
The other thing that we want to do
is to understand the charge below threshold.
We've assumed in Unit 2 that it was zero,
but it is actually non-zero.
It varies continuously below to above threshold,
below threshold that it counts for that important
leakage current that we see today.
So this whole important topic of MOS electrostatics
is something that we will spend some time trying to develop
a good sound understanding on in Unit 3.