[Slide 1] Welcome back. So we're going to
continue our discussion in this lecture of this
traditional MOSFET model that we developed in the
previous lecture, and I'm going to show how
that simple two-piece model can be extended into
a real model that does a good job
of matching the IV characteristics of present-day transistors.
This virtual source model is going to provide
a conceptual framework that we're going to use
as we go throughout the remainder of the
course. [Slide 2] So the model itself was
developed at MIT and this is an example
of the kind of fits that it provides
to current day technology and 32 nm technology,
and it really has proven to be remarkably
successful at describing a wide variety of not
only a silicon but also 35 and even
graphine transistors. So although it is semi empirical,
it seems to capture something about the essential
physics of MOSFET that make it widely applicable.
[Slide 3] So let's recall what we were
doing in the last lecture. We developed this
very simple two-piece approximation for the IV characteristic.
We had a simple expression for the linear
current, with simple expression for the saturated current.
Those two currents intersected a critical voltage we'll
call VDsat, and we had when a simple
expression for VDsat. And now our question is,
can we extend this model and make it
a smooth and continuous model that describes the
IV characteristics better? [Slide 4] So the two-piece
model, we can easily apply it, we just
apply drain to source voltage and we ask
are we less than VDsat, are we bigger
than VDsat? And we use the appropriate expression,
but we'd like a smooth connection between those
two regimes. And the question is how we
do that, how do we connect those two
pieces smoothly? Now we could do this mathematically,
but then it gets it very complicated and
involves additional simplifying assumptions. Or we could do
it empirically, and that's the essence of the
virtual source model, it makes this connection empirically.
So we have some physical expressions for the
linear and saturated current and we're now going
to empirically connect these two. [Slide 5] So
let's see how we do that. Let's go
back to the small drain voltage. This was
our expression for the linear current, the current
for small drain voltage. If I just rewrite
that a little bit. It's current per unit
width, that's why we always express these currents
as microamps per micrometer or whatever. And I
can pull the Cox VG minus VT out
front because that's the charge. And then I
can pull the mew and drain voltage divided
by L, that's the electric field. So when
I write it that way, what we've written
it as is current per unit width is
charge times velocity, very simple expression. And the
velocity under low drain to source voltage is
just mobility times the constant electric field in
the channel. [Slide 6] Let's look at large
drain voltage. Our saturated current expression looks like
this. We can rewrite that as the current
per unit width. Bring the Cox VG minus
VT out front because that's the charge, and
then we have Vsat. So again I can
write the current per unit width as charge
times average velocity, and the average velocity in
this case is just the Highfield saturated velocity
in this traditional model. [Slide 7] So we
have an IV expression, then we can always
write the current as charge times velocity. Above
threshold, we'll use this simple expression for charge.
Below threshold, we'll ignore the charge and just
assume that there is no charge. Later on
we'll have to go back next week when
we talk about MOS electrostatics and try to
do a better job at that so that
we can describe the sub threshold characteristics. But
for now I'll just assume that there is
no charge below threshold. And then we have
expressions for the velocity under small drain voltage
and under high drain voltage, and we'd like
to smoothly make that transition in average velocity
from one to the other. [Slide 8] And
here's the way that we do it in
a virtual source model. We introduce an empirical
function, it's physics motivated but the physical underpinnings
are just a little bit weak, but the
expression looks like this, we'll discuss in a
slide or two to why that expression makes
some sense. But it's an empirical expression, it's
drain voltage divided by Vsat, and we have
expression for that, over one plus drain voltage
divided by VDsat to some power beta, and
then the whole quantity to the power one
over beta. So that's an empirical expression, and
let's take a look and see what that
does. The drain voltage is very small, then
in the denominator I can ignore it with
regard to the one, and this Fsat function
just ends up being drain voltage divided by
VDsat for small drain voltage. That means that
the average velocity is just VD divided by
VDsat, drain voltage saturation, divided by the Highfield
saturation velocity, but we have an expression for
the drain saturation voltage. Plug the expression in,
and we end up having the average velocity
is mobility times electric field, and that's exactly
what it's supposed be. SO this empirical expression
for small drain voltage gives us the correct
answer. What about large drain voltage? So for
large drain voltage, you can see that we
can ignore the one in the denominator, and
then the beta to the power one over
beta will just give us one, and this
saturation function will approach one. Now the average
velocity is just one times Vsat, it's Vsat,
and that's exactly what it's supposed to be
under high drain bias. So this empirical saturation
function has the right value at the two
limits, and this empirical parameter beta, which is
typically 1.5 to 2 or so, controls the
transition between the two regimes. [Slide 9] Now
let's see if we can understand a little
bit about what this saturation function is all
about. Physically I would think that whichever velocity
is the smaller is the one that matters
and is going to limit the current. So
I could say that one over the actual
average velocity is one over the linear regime
velocity plus one over the satuated velocity. Whichever
one of those two is smaller is going
to be the one that's important in this
sum and that's going to be the one
that dominates. So this would be a plausible
way to get a smooth connection between the
two regimes. And if you simply solve this
for the average velocity, you'll get an expression
that looks like that, that's very much like
that Fsat function that we saw previously, the
only difference now is that we're missing the
beta and one over beta, and that empirical
parameter beta just gives an adjustment that controls
the sharpness of the transition between the two
regimes and allows us to fit a wide
variety of transistors. Okay so this extra parameter,
it typically doesn't vary a lot, typically between
1.6 and 1.8 for n-channel MOSFETs, and roughly
1.4 for p-channel MOSFETs, but it's a parameter
that we typically adjust for the particular device
to get the best fit to the IV
characteristics. [Slide 10] So although this is an
empirical parameter, it does produce a smooth curb
that does the right thing, and in practice
it works remarkably well. [Slide 11] So I'm
going to call this the level zero virtual
source model. Step-by-step we're going to make this
more sophisticated. The level zero model writes the
current as charge times velocity. We use a
very simple expression for the charge, ignoring the
charge below the threshold voltage. We use an
empirical expression for the average velocity, it an
empirical parameter one over beta. There are only
five parameters in this model and these five
parameters have very clear physical significance. There's a
gate oxide capacitance per square centimeter, there's a
threshold voltage, there is a Highfield saturation velocity,
there is the effected mobility, and there's a
length of the channel. So with those five
parameters it turns out that in practice, you
can do a very good job of fitting
the IV characteristics of modern transistors if we're
allowed to adjust Vsat and V effective and
think of them as semi-empirical. The five steps
are a current, charge, average velocity, saturation function,
and VDsat. [Slide 12] Okay so let's think
about how we would improve this model and
make it better, and we can't do it
all in this lecture but we'll do it
step-by-step. So one of our assumptions is that
we have no charge below threshold. The leakage
current when the device is off is very
important these days. We're going to need to
develop an expression for the charge below threshold,
that means we're going to need to understand
MOS electrostatics in a little bit of detail,
that's what the next couple of lectures are
all about. We're also going to find that
above threshold, the slope of this line is
not the oxide capacitance, it's the gate capacitance
and it's somewhat less than the oxide capacitance.
We'll have to understand what that's all about
too. [Slide 13] In the output regime, we
don't have a zero slope we have a
finite slope, we have to make sure that
our model does that too and that'll be
relatively easy to do when the physics is
easy to understand. [Slide 14] One of the
things that we're going to have to be
very careful about is the fact that any
real device has series resistance. We put metal
contacts on the source and drain there is
inevitably some series resistance that's added there. There's
some resistance in the source and the drain
themselves before the current gets out the metal
contact. What we've been talking about really are
the IV characteristics of the intrinsic device, assuming
that there are none of those extra resistances.
So let me just label now the voltages
that I've been talking about the voltages on
the intrinsic terminals of the of the device,
so I'll label them with primes now. And
we have an expression for the drain current
in terms of the intrinsic voltages that get
into the heart of the device. [Slide 15]
The real transistor has these extra resistances, and
I'm not going to draw one for the
gate because there's really no significant current flowing
through the gate terminal so there's no significant
voltage drop there. Under AC conditions, that resistance
is important, but not under DC conditions. So
the actual voltages that we apply to the
drain source and gate terminal are different than
the voltages that actually get into the heart
of the device, and we have to account
for that. [Slide 16] I won't go through
the details, but it's just a little bit
of circuit analysis and it's relatively straightforward. There
is no voltage drop here so the gate
voltage is the intrinsic gate voltage. The intrinsic
drain voltage is just the drain voltage minus
the voltage drop due to the drain current
flowing through that drain series resistance. And the
intrinsic source voltage is the source voltage plus
the voltage drop across the series resistance in
the source. So there is a simple relation,
the drain current is in terms of the
intrinsic voltages, and we can relate the intrinsic
voltages to the extrinsic voltages with some simple
circuit analysis. Well actually this is two equations
and two unknowns, so given the drain, the
applied voltages on the actual drain terminal, source
terminal, and gate terminal, we can solve these
two equations for the two unknowns, which are
the intrinsic voltages across the source and the
drain, and that requires some non-linear Newton iteration,
but it's a relatively straightforward process so we
can do the correction numerically and it's easy
to add the effects of these series resistances.
[Slide 17] Now just to talk about what
the resistances do. If I had a transistor
with no series resistance, I'd have an intrinsic
device and I'd have the IV characteristic and
the equations we have for the current refer
to this intrinsic device. So we have some
channel resistance, the slope in the small VDS
regime, and we have some on current. What
happens if there are series resistances present? [Slide
18] Well we'll get less current, and it's
relatively easy to understand what happens. The slope
in the low VDS regime is not just
the resistance of the channel, we also have
to add the series resistance of the source
and the drain, just gives us a higher
resistance in that regime. If I look at
the on current, it's lower than the intrinsic
on current, and the reason it's lower is
because the on current goes as W, oxide
capacitance Vsat times VGS minus VT, but the
actual voltage between the gate and the source
is the gate voltage minus the source voltage,
but the source voltage increased by this voltage
drop across the series resistance in the source,
so the gate to source voltage is smaller,
meaning I get a smaller drain current. So
that's what the extrinsic resistances do, just lower
the overall currents. In fact you can show
with just a little bit of algebra that
the fractional lowering of the measured on current
is just a ratio of the series resistance
to something I'll call the on resistance, and
the on resistance is just the drain voltage
minus the threshold divided by the on current.
So there are simple expressions that we can
estimate the effect of these series resistances are,
and if we want to calculate it properly
we just solve these two equations and two
unknowns and determine the intrinsic voltages and plug
them into our expressions. [Slide 19] So with
these kind of extensions, sub threshold conduction, which
we can't do until we talk about MOS
electrostatics in more detail, finite output resistance which
we talk about 2-D electrostatics we'll be able
to treat. And series resistance, which we've just
discussed, then this virtual source model is a
very good way to describe modern transistors. Now
we have to view this Highfield saturation velocity
as some kind of empirical parameter. We also
have to view this effective mobility as some
kind of empirical parameter because it really only
makes physical sense when the channel is many
mean free paths long. And we also need
to use the measured gate capacitance, which in
general is less than the oxide capacitance, we'll
talk about why in the next lecturer or
two. [Slide 20] Quite surprisingly this model with
just a little bit of adjustment of these
parameters that was developed on physics that was
very firm and solid in the 1960s 70s
and early 80s, still does a remarkably nice
job of describing modern day transistors. There's a
very simple reason that the transistors of barrier
control device and that was built into the
first model. It's just the magnitude of the
currents need to be adjusted a little bit
because of the breakdown, some of the physical
concepts. Now what we're going to find is
that these parameters acquire a very clear physical
meaning, and that's what the course is all
about. We will find that the saturation velocity
is replaced by something we'll call the injection
velocity, and we'll talk about what that means
later in the course. And that the actual
mobility of carriers in the inversion layer is
replaced by something we call the apparent mobility,
but the apparent mobility has a very clear
physical interpretation. [Slide 21] Now I might just
mention why this is called the virtual source.
So it's based on this idea of barrier
control, it's based on evaluating the current at
the top of the barrier. And the top
of the barrier is a place where the
charge is given by standard MOS electrostatics by
a very simple expression. The charge is Cox
VG minus VT not always but it's very
close to that if we design the transistor
well. So that point where the charge is
always maintained at that that very simple value,
is called the virtual source and that's where
we evaluate the current and do all of
our bookkeeping. [Slide 22] So the goal for
the rest of the course now is to
talk about the physics, to go beyond the
1960s 70s and early 80s physics and talk
about physics that is appropriate for these very
small devices, and to do this in a
way that we can take this virtual source
model and help evolve it into something that
is very typical and sound and applicable to
modern transistors. So the next task that we
need to address in this course is to
have a clearer understanding of MOS electrostatics, because
remember the most important thing about a transistor
is manipulating these energy barriers. Once we understand
how to manipulate energy barriers and how to
treat MOS electrostatics and then we'll look at
ballistic transport and quasi ballistic transport, but first
we'll discuss MOS electrostatics. So thank you and
we'll continue with a discussion on MOS electrostatics
in the next lecture.