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>> [Slide 1] So hello.
This lecture is a part of Unit 1,
but it's really a different --
it's a supplemental part of Unit 1.
What we're assuming in this course is
that you have a basic understanding
of semiconductor physics.
If you have that basic understanding, you'll be able
to follow the course and, I think, learn some new
and interesting things about nano devices
and nano transistors in particular.
What I'd like to do in this special lecture is to quickly go
through some highlights of semiconductor physics.
Now, this is material that it takes me five or six weeks to go
through when I teach my introductory
semiconductor course.
So it's not going to be a lecture
that can teach semiconductor physics.
My intent is just to quickly review some quick --
some key concepts that we'll be using throughout the course.
If you've seen those before or if there are one or two
that you think, well, maybe I need to refresh my memory
and review that a little more, then you're in good shape.
If the material that I present
in this quick summary is completely brand new to you,
then you probably don't have the background that you're going
to need to be successful in this course.
[Slide 2] So let's take a quick look at some key concepts
in semiconductor physics.
These are the seven topics that I'd like to go over.
And I'm going to do them very quickly.
We're not teaching this material.
We're just quickly reviewing what some of the key points are.
[Slide 3] All right.
Let's begin at a very basic level; silicon atoms.
Silicon is the most common semiconductor used
for electronic devices.
You'll remember from your freshman chemistry course
that atoms have energy levels.
We label them like 1S, 2S, 2P, 3S, 3P,
etc. Silicon has Atomic Number 14,
which means it has 14 electrons that have to be accommodated
in those energy levels.
And you might remember from your freshman chemistry
that S orbitals can hold two electrons.
So each of these N equal 1
and N equals 2 orbitals hold two electrons.
In P orbitals we have a Px, a Py, and a Pz orbital.
Each one of those can hold two electrons,
spin up and spin down.
So the P orbitals can hold six electrons.
And we just go through and we fill up all the energy levels
until all 14 electrons are accounted for.
And if we do that, we'll find that we fill up the 3S level
and we put two electrons in the 3P level.
So we have four valence electrons.
But if you look at those uppermost energy levels,
the N equals 3 states, there are eight states there; two S states
and there are six P states.
And we've used four of those eight states.
The lower states here, lower energy states,
we call these the core levels.
There's not much that we can do to interact with them.
They're shielded from the outside world.
The chemistry in semiconductor physics all has to do
with the valence electrons, so that's what we focus on.
[Slide 4] Now, we're going to take a crystal chunk of silicon
where we have a large number of atoms
that are covalently bonded together.
So we're going to be talking
about a very large number of atoms.
The density of silicon is about 5 times 10
to the 22nd per cubic centimeter.
This is the crystal structure.
We call this a diamond crystal structure
because diamond also crystallizes in this structure.
And the key point is that each atom forms covalent bonds
to four nearest neighbors, with those four valence electrons
that it shares with its four nearest neighbors.
So that's the crystal structure.
Only the valence electrons are going to be --
in the valence states are going to be important to us.
We have eight states for every atom.
And if we have N atoms, 5 times 10
to the 22nd per cubic centimeter, then we're going
to have 8 times N atoms states in this solid that are going
to be of concern to us.
But when we put these atoms very closely together
and form these covalent bonds,
the electron wave functions overlap and things change.
The energy levels change.
[Slide 5] Energy levels become energy bands.
So if we just look at those valence electrons,
the four electrons in the eight states, if we put them
in a silicon crystal the energy levels are going to broaden
and smear out into a range of energy bands.
And what we'll find if we do the quantum mechanics properly is
that half of the states lower in energy
and form those covalent bonds.
We call that the valence band of energies.
So all of the energies within that range,
half of the states are there and they're all filled
with the four electrons from each of the silicon atoms.
Now, the other four states that were empty,
they go to higher ranges of energies.
And we create a band of energies we call the conduction band.
And at T equals 0, 0 temperature, they're all empty.
In between the valence band and the conduction band,
there is a region where there are no states
for electrons to reside.
That's called the "forbidden gap."
We can't have electrons there.
So this is the situation at 0 Kelvin.
All the electron -- valence electrons are
in the valence band.
The conduction band is completely empty.
[Slide 6] Now, if we go to room temperature, 300 Kelvin,
there is a small amount of thermal energy,
say, .026 electron volts.
You know, this forbidden gap is about 1 electron volt.
But there is a small probability
that there is enough thermal energy
to kick us some small fraction of the electrons
from the valence band and move them into the conduction band.
When we do that, we're left behind a hole
in the valance band; you know, a state that is now empty
in the valance band, and an electron in the conduction band.
We've now moved it up in energy into the conduction band.
We'll find that those holes are mobile and they behave
like positive charge carriers.
We'll use those for P channel transistors.
And the electrons are mobile.
They behave as charged carriers, and we'll use those
for N channel devices.
[Slide 7] So we're going to be talking a lot in this course
about energy band diagrams.
An energy band diagram is a plot of energy versus position.
Remember, only the top of the valence band is of interest
to us because deep down below that, all the states are filled.
Only the bottom of the conduction band is of interest
to that because way above the bottom all
of the states are empty.
But very near the bottom
of the conduction band we can have a few electrons.
Very near the top of the valence band we can have a few empty
states or holes.
So that's what we focus on.
If this is silicon,
the forbidden gap is 1.1 electron volts wide.
So room temperature we have some thermal energy due
to the jiggling of all of the atoms due
to the random thermal motion.
It's relatively small compared to the band gap.
But there is an exponentially small probability that some
of these bonds will be broken and an electron can be moved
from the valence band to the conduction band.
And that small probability times a large number of states
that there are there means that we're going
to have some small probability of creating electron hole pairs
in pure silicon at room temperature.
It turns out that the number in silicon is almost exactly 10
to the 10th electrons and holes per cubic centimeter
at room temperature.
Now, that may seem like a large number.
But remember, there are 5 times 10
to the 22nd atoms per cubic centimeter, and there are 10
to the 10th of these electron hole pairs.
So it's really a very small concentration.
[Slide 8] Now, there's another way that we like to look at this
and another picture that we like to draw.
It's a little complicated
to draw these three-dimensional diagrams
of how the crystal structure actually looks
and how the four nearest neighbors
of silicon arrange themselves in these tetrahedral bonds.
So frequently we'll draw a 2-D picture which is just meant
to represent the fact that each atom is surrounded
by four nearest neighbors
and that each one has four valence electrons.
It shares those four valence electrons
with its four nearest neighbors, completes its valence show
of states, and that forms covalent bonds.
Now, in this picture, if we look
at room temperature there is some small probability that one
of those bonds can be broken.
And if one of those bonds is broken,
we're left behind an empty state.
Now, if there's an empty state there, then an electron
from a nearby bond can hop into that
and the empty state can move.
And then an electron
from another nearby state can hop into that hole.
And what we find is that the hole then can move
about the crystal lattice much like a charge carrier.
It's an absence of a negative charge;
it behaves like a positive charge carrier.
Now, when we broke the bond we also released an electron
that used to be bound in the conduction band.
Now it's in the valence band.
That electron is free to wander around and carry current.
So that thermal energy creates a --
breaks a small number of bonds
and creates these electron hole pairs
which create mobile charge carriers
that are positive and negative.
[Slide 9] Now, a MOSFET, the basic material is a semiconductor.
It also makes uses of insulators and metals.
So we should just remind you briefly what an insulator,
a metal, and a semiconductor is.
Insulators don't conduct electricity very well.
They usually don't conduct heat very well either.
Metals conduct electricity very well.
And they usually conduct heat very well also.
Semiconductors aren't good metals
and they aren't good semiconductors,
but they have a very important feature.
You can control their properties.
We can make them reasonably metallic,
we can make them reasonably insulating, and we can do
that by adding a small number of impurities to the pure silicon.
That's what makes them so useful.
[Slide 10] That's why we use semiconductors to build electronic devices.
Now, on energy band diagrams, insulators, metals,
and semiconductors look something like this.
The key feature for an insulator is
that the band gap is very, very wide.
So if I look at silicon dioxide or glass, this is a material
that is commonly used in integrated circuits.
It has a very wide band gap.
It's very hard to create these electron hole pairs
and to create carriers.
Now, metals -- so it turns out in metals that when you fill
up all of the energy levels and account for all
of the electrons, the top-most energy level is
in the middle of a band.
That means we're not -- in an insulator or a semiconductor,
you fill it up -- you fill up all the states below a band,
and then the next state above that is completely empty.
In a metal, you simply fill up a band half way.
That means that if I apply a field it's very easy now
for these electrons to move because there's a state
for them to move up in energy.
So this is an energy band diagram for a metal.
Now, in a semiconductor, it looks just like an insulator,
but the semiconductor is smaller.
Significantly smaller so that we can create some electron
hole pairs.
We can create charge carriers by other means as well.
[Slide 11] Okay. Now, I've been talking about silicon.
Silicon is a Column IV semiconductor.
The IV means there are four valence electrons.
You can see there's germanium below silicon.
That's also Column IV.
Germanium is also a semiconductor.
Carbon is above it.
If carbon crystallizes in the diamond structure,
then it could also be considered a semiconductor although the
band gap is quite wide and you might think
of it as an insulator.
Now, it's interesting to look at Column III, boron, or aluminum,
or gallium; and Column V, phosphorus or arsenic,
and ask what would happen if we could insert a Column III
or a Column V element in the silicon crystal lattice?
That turns out to do something very, very useful for us.
And that's called doping.
[Slide 12] So here's our cartoon picture of the silicon lattice structure
with its four nearest neighbors.
Let's say that we replace one of the silicon atoms
with a phosphorus or an arsenic atom.
Phosphorus or arsenic is Column 5,
so it has five valence electrons.
When we put it in the silicon lattice,
four of those five valence electrons form covalent bonds
with the four silicon nearest neighbors.
In gallium, gallium is a Column 3 element.
When we put gallium in, we only have three valence electrons
to form covalent bonds.
We're missing a covalent bond with one
of the four silicon nearest neighbors.
[Slide 13] So let's see what would happen there if we put phosphorus in.
We put phosphorus in; four
of the five valence electrons form covalent bonds
with the four nearest neighbors.
The fifth electron is just weakly bound.
There is a net charge of plus 1 on the phosphorus atom.
The remaining electron sees that plus charge.
This looks like a little hydrogen atom except, you know,
in a hydrogen atom, we learn
in freshman chemistry what the binding energy is
of the electron in a hydrogen atom.
It's minus 13.6 electron volts in the lowest state.
We can think of this phosphorus atom
in silicon like a hydrogen atom.
The only difference is when we compute the binding energy
for this extra electron, we have to take account of the fact
that it's inside the silicon material
which has a dielectric constant that is ten times --
a little more than ten times higher than vacuum.
And that means since it's squared in the denominator
that the binding energy is a little more
than a hundred times smaller.
So it's very easy to break that bond
and for this additional electron to be free --
become a free electron.
It's weakly-bound; thermal energy can break it.
Now we have an electron that can wander
around in the conduction band and dope the semiconductor.
[Slide 14] So if we -- if we insert a concentration of dopants,
every time that dopant is ionized,
meaning this bond is broken, we get an electron
in the conduction band.
So it's a very easy way for us to control the number
of electrons in the conduction band simply by the number
[Slide 15] of phosphorus atoms that we put into the silicon lattice.
Now, we can do the same complementary thing with boron.
If we put boron in the silicon lattice, we're missing --
we only have three valence electrons,
so we're missing a bond.
So we have a missing place there.
[Slide 16] Now, one of its neighbors can hop over and fill that up.
And now I have a hole in one of its neighbors.
And now one of its neighboring electron covalent bonds can hop
in and fill that, and the hole moves.
So if I put a Column III element in,
something very similar happens.
It's just that I get a concentration of holes
in the valence band, positive charge carriers equal
to the concentration of the number of dopants that I put in.
Now, I'm assuming room temperature
so that there's enough thermal energy to break this weak bond.
[Slide 17] Okay. So we've been talking about what happens
when we put Column III or Column V elements in silicon.
We dope the semiconductor, and that's what makes semiconductors
so interesting and useful.
Gallium is a Column III element.
It would also be a P-type dopant in silicon.
But arsenic is a Column V. It would be an N-type dopant
in silicon.
But I could also make compounds of gallium arsenide
which are ionic -- slightly ionic,
but on average they have four valence electrons.
They behave as semiconductors as well.
They have different properties, different band gaps.
They're useful in certain situations.
We will talk from time to time on these type
of semiconductors as well.
They're called III-V semiconductors;
compound semiconductors made
from Column III and Column V. Okay.
[Slide 18] Now, let's talk about how we fill states and talk
about the number of electrons in the conduction band.
So if I look at the conduction band I could ask myself,
what is the probability
that those empty states there get filled with an electron?
Or if I'm in the valence band, I start out with them all filled.
And then at room temperature I'm interested
in what is the probability that they're empty?
Because that gives me holes that are positive charge carriers.
[Slide 19] Now, if I go back to my silicon atom, I could look at --
all I have to do is fill up the energy levels
with the 14 electrons, and I could sort of draw a line there.
And I could say below that line there's a high probability
that the states are occupied.
Above that line, there's a lower probability.
If I go way above the line, there's no probability
that the states are occupied.
[Slide 20] Now, I could do the same thing in a solid with energy bands.
I could draw a line; and I could say that above
that line there is a decreasing probability
that states are occupied.
Lower -- below that line,
there's an increasing probability
that the states are occupied.
The further below I go, the higher the probability is
that they're occupied.
We call this line the Fermi level.
And the probability that a state is occupied is related
to the relation of the energy to the Fermi energy; and it's given
by this famous Fermi function.
So if you look at this function, you can see what it does.
If the energy is way above the Fermi level,
this is E to a big number, the probability goes to 0.
So if I'm way up here in energy, there's almost no probability
that the state is occupied.
If I go to energies that are way below the Fermi energy,
this is e to the minus a very large number; it's almost 0.
The probability that the state way down here
at low energy is occupied is almost 1.
It's almost certain to be occupied.
So it does qualitatively what we expect,
and quantitatively it's the correct answer.
So conduction band states that are
above the Fermi energy have a small probability
of being occupied.
Valence band states which are much lower below the Fermi
energy have an even smaller probability that they're empty.
[Slide 21] Okay. So using the Fermi energy we can think
about how we compute the carrier density in a semiconductor.
So let's look at a particular band of energies
in the conduction band and ask ourselves how many electrons are
there in that range of energy?
Well, first of all, I have
to know how many states there are in that range.
And to find the states in that energy, we multiply the density
of states times the width of the energy, DE.
So we know there are four n states where n is the number
of atoms per cubic centimeter.
Half the states went into the conduction band.
But they're distributed over the range of energies as described
by this density of states.
It's the number of states per unit energy per unit volume.
So that's a known quantity for all the common semiconductors.
We know how to compute it.
So I have the number of states in that energy range,
and then I simply multiply by the probability
that those states are occupied.
And that depends -- is given by the Fermi function,
and it depends on where the energy is
with respect to the Fermi energy.
Now, if I'm in the valence band, what I'm interested
in is what is the probability that the state is empty?
What's the probability
that there's a hole in the valence band?
So again, there's a density of states for the valence band.
They might be different; there's the same total number of states
in the valence band, but they might be distributed
in energy differently.
And then there's a probability now --
I'm interested in the probability
that the state is empty.
So that's 1 minus the probability that it's filled.
Okay. Now, if I want the total number of electrons
in the conduction band and the total number of electrons
in our holes in the valence band,
I simply perform these integrals.
And you can work these integrals out, and we can get the answer.
By the way, the 0 on these quantities indicates
that I'm talking about equilibrium now.
We haven't applied any voltages or shined --
[Slide 22] we're not shining light on the semiconductor
or anything like that.
Okay. You work out that integral;
this is the answer that you get.
So if you've taken a basic semiconductor course,
you probably work these integrals out.
And the key point is that the electron density is related
to the location of the Fermi energy with respect to the bond
on the conduction band.
The higher the Fermi energy, the more electrons
in the conduction band.
The number of holes in the valence band is related
to the location of the Fermi level with respect to the top
of the valence band, EV.
The lower the energy is, the more holes in the valence band.
The particular constants out front are called
"effective densities of states,"
and they're known material constants
for common semiconductors.
Now, what's this function, F sub one half?
So this is just the function you get
from this numerical integration.
It turns out to be a function that pops up frequently
in semiconductor physics.
They're called Fermi-Dirac integrals of order one half.
They're a little bit complicated to deal with.
If you need to deal with them, I refer you to these notes.
You can even find an iPhone app in order
to evaluate them if you need to.
But it makes things a little bit complicated mathematically
because it's not a familiar function to most of us.
[Slide 23] So frequently in semiconductor physics we make use
of an approximation.
And we're going to do that for the most part in this course.
It turns out that if the Fermi level doesn't get too close
to the conduction band and if it doesn't get too close
to the valence band, and usually we say as long
as it stays a few kT below the conduction band or a few kT
above the valence band,
then these Fermi-Dirac integrals reduce
to a well-known function we're all familiar with.
They reduce to exponentials.
So these are the expressions that we're going
to primarily rely on in this course because they're easier
and they allow us to get the basic concepts across.
If we need to do quantitative calculations from time to time,
we might make use of Fermi-Dirac.
So the electron concentration is exponentially related
to the position of the Fermi level with respect
to the conduction band edge.
And same thing for the hole concentration.
The lower the Fermi level, the higher the hole concentration.
The higher the Fermi level,
the higher the electron concentration.
If you multiply these two quantities together,
you'll find that you get a material-dependent constant.
Effective density of states for the conduction band,
for the valence band, and E to the minus band gap over kT.
That's what gives us this quantity NI squared,
this intrinsic density 10 to the 10th squared,
for electrons in silicon.
So we're going to make use
of the so-called non-degenerate expressions
because it's much easier, and it allows us
to get the basic ideas across.
[Slide 24] So here's how that would all play
out for a typical semiconductor.
If I have a semiconductor that's N-type,
my Fermi level will be near the conduction band.
If it's really heavily doped N-type,
the Fermi level might be inside the conduction band.
If I doped it with some number of phosphorus atoms,
ND per cubic centimeter,
say 10 to the 18th per cubic centimeter, then I'll have 10
to the 18th electrons per cubic centimeter
because that fifth electron from each one of those will --
the bond will have been broken at room temperature,
and it will be free to wander around in the conduction band.
Now, if I'm interested in where the Fermi energy is,
I'll use this expression that relates the location
of the Fermi energy to the electron density.
So if I know the electron density
because I know how I built the semiconductor,
I can then determine the Fermi energy.
If I want to know the whole density, well,
I can determine it too because I now know the Fermi energy.
Or there's an even easier way to determine the hole density.
I can remember that n times p is equal
to ni squared in equilibrium.
So p is just ni squared over the electron density.
Ni is 10 to the 10th in silicon.
10 to the 10th squared is 10 to the 20th.
In this example, I said the doping density was 10
to the 18th; so I have 10 to the 18th electrons.
So I would have a hundred holes per cubic centimeter.
Very small number.
[Slide 25] Now, in a P-type semiconductor, my Fermi energy would be
down near the valence band.
So if I dope it with NA, boron atoms per cubic centimeter,
I'll get the same number of holes at room temperature.
If I want to know where the Fermi energy is,
I just use this relation for the --
between the Fermi energy and the hole density.
Now that I know the Fermi energy, I can figure
out how many electrons will be there.
It will be a very, very small number
because the Fermi energy is way below the bottom
of the conduction band.
Or I could find that number even easier by remembering
that np is equal to ni squared
for a nondegenerate semiconductor in an equilibrium.
And then I could simply solve for the electron density;
that's ni squared divided by the hole density.
[Slide 26] In an intrinsic semiconductor the Fermi level will be near the
middle of the band gap.
It won't be exactly near the middle
because the density states is a little different typically
in the conduction and valence band.
But it will be very close to the middle.
When the Fermi level is close to the middle in the band gap,
there is a small probability that states
in the conduction band will be occupied.
That gives me the 10 to the 10th intrinsic carriers per
cubic centimeter.
And there's an equally small probability that states
in the valence band will be empty.
That gives me the 10 to the 10th holes in the valence band.
[Slide 27] Okay. So we've reviewed four key topics.
I want to quickly go through three more.
And then we will have completed this quick review
of semiconductor physics.
[Slide 28] So we want to talk now --
I've been talking about equilibrium semiconductors.
And we want to talk about how current flows.
So let's think about an electron in a vacuum
with a mass of m0 vacuum mass.
And you'll remember from freshman physics
and Newtonian mechanics if you exert a force on the electron --
force is mass times acceleration --
we could figure out the velocity of the electron.
We could figure out how far it goes as a function of time.
We could look at the relation between energy and momentum.
The kinetic energy is 1 Fmv squared.
The momentum is mass times velocity.
So I could write the energy as momentum squared,
divided by two times the mass of the electron.
And I would get a plot of energy versus momentum
that would look like this parabola.
That's what an electron looks like in vacuum.
[Slide 29] Now, we're going to be dealing with electrons
in the silicon crystal.
And this is actually a very, very complicated problem.
But people have learned --
and this was all sorted out during the early stages
of quantum mechanics and condensed matter of physics
in the early and mid parts of the 20th century.
People have learned that there are very simple ways to think
about how electrons move in crystals like semiconductors.
So what we find is that the lowest that the energy --
if the electron is in the conduction band,
the lowest its energy can be is at the bottom
of the conduction band.
But then it has an energy versus momentum relation
that looks very much like this electron in vacuum.
P squared divided by 2 times mass.
But it's an effective mass.
It's not the mass in the vacuum.
It's a mass that accounts for all of the interaction
with the silicon atoms in the crystal. Now, if I look at holes,
holes everything is flipped upside down.
When we plot these energy diagrams,
we're plotting electron energy.
So holes, actually the hole energy moves down.
The further down I go,
the higher the energy of the hole is.
But the same thing happens for holes.
It's just that everything is flipped upside down.
The highest the energy that the hole can have
in the valence band is the Ev, the top of the valence band.
And then it can have a lower energy.
And its energy momentum relation again will be parabolic.
But it will be given by an effective mass for holes.
It may be different from the effective mass for electrons.
And will be different from the effective mass
of an electron in vacuum.
So that's the energy versus momentum relation
for electrons in crystals.
And you might remember that this is not a --
really a momentum is something we call crystal momentum.
So we're really talking
about electron waves propagating through this crystal.
They have some wave vector k 2 pi over lambda.
So we're really talking about those waves.
Turns out that Plancks constant-- actually hbar, Plancks constant,
divided by 2 pi times k as the units of momentum.
And it really plays the role of momentum in --
for electrons in crystals.
We call that the crystal momentum.
So it means that we can use these very simple semi-classical
concepts to think about the motion of electrons
and holes in semiconductors.
We just have to -- we can use Newton's law.
We just have to replace the actual electron mass
by its effective mass.
[Slide 30] So that allows us to do things like compute the current
of electrons in semiconductors.
So if I have a hole, if I have an electron
in a semiconductor it will have some effective mass.
If I put a force on it, I can accelerate it;
and I can ask, what's its velocity?
So let's say I have an electric field pointing in the --
from left to right in the positive x direction.
I remember in freshman physics
that an electric field exerts a force.
It is minus q times the electric field on the electron,
minus sign because the electron is negatively charged.
So there's a force in the minus x direction.
So the electron will accelerate in the minus x direction,
but there'll be friction.
It'll accelerate and then it will scatter,
and it will reach a terminal velocity and then move
at a constant velocity.
And that constant velocity is minus mobility times
electric field.
If I double the electric field, they'll move twice as fast.
The mobility is some measure of the ease
with which the electron moves through the crystal lattice.
It's some measure of how much friction there is
that causes the velocity to come to some terminal value.
The mobility can be related to detailed physics.
You'll sometimes -- in a basic semiconductor course,
you derive this expression.
Mobility is q tau over the effective mass.
Tau is the average time an electron can move before it
scatters off of something and bounces off
in a random direction.
Now, I can calculate now the current
because current is basically charge times velocity.
The charge is the charge on an electron times the density
of electrons that are there, and then times the velocity.
The faster they go, the more current.
The more there are of them, the more current.
If I simply plug in my relation for velocity,
we get an expression for the current due
to the electric field.
And we call this the drift current,
the current due to an electric field.
[Slide 31] So the picture is that electrons are in random thermal motion.
They are being bounced around and knocked
around as this lattice is jingling
with its kinetic energy due to its thermal energy.
But there is some small probability
when I apply electric field that there'll be some bias
to move the electrons from the left to right.
And the velocity, mu times the electric field,
is that small average that's superimposed upon this random
thermal motion.
Now, we know that the kinetic energy is three-halves kT,
and we know that it's --
that the kinetic energy is one-half m times the average
value of v squared.
So we can compute the rms average thermal velocity
and get an expression for it.
Turns out to be about 10 to the 7th centimeters per second
for electrons in silicon.
When we apply an electric field,
we typically give it a small average velocity superimposed
on this large random velocity.
[Slide 32] Now, something else can also happen.
Even if there is no electric field,
even if the electrons weren't charged,
whenever I have mobile particles in a concentration gradient,
I'll have a flux of particles diffusing
down a concentration gradient.
So this is Fick's Law, which says that particles flow
from high concentration to low concentration.
The flux of particles,
the number per square centimeter per second,
is minus the diffusion coefficient times the gradient
of the particle concentration.
So that diffusion coefficient has units
of centimeters squared per second, and this is known
as Fick's Law of diffusion.
Well, since I have charged particles
that are diffusing, I'll get a current.
And again, all I have to do is to take q times the flux
and I'll get a diffusion current.
So these are really two independent processes.
There is a diffusion current and there is a drift current.
[Slide 33] In order to get the total current,
I should add the two currents together.
And this gives me a basic current relation that those
of you who have had a course
in semiconductor physics before will have seen.
It's called Drift Diffusion equation.
Drift in an electric field;
diffusion down a concentration gradient.
Now, actually there's something very interesting
that Albert Einstein developed.
Even though these seem to be two independent processes,
the current must be 0 in equilibrium.
And that allows you to develop a relation
between the diffusion coefficient and the mobility.
And it turns out that D over mu is equal
to Boltzmann's constant, times temperature divided
by the charge on an electron.
This relation is called the Einstein relation,
and it's very important in semiconductor physics.
[Slide 34] Now, another topic that I want to mention is the fact
that energy bands can bend.
So if I have an energy band diagram,
I might show some energy band diagram that looks like this.
There's always a constant separation
between the conduction and the valence band,
because I'm talking about one semiconductor
that has a band gap that doesn't change.
There is an intrinsic level, which is just sort of the middle
of the band gap that I've shown as a dashed line there.
And I'm showing it varying with position.
But a very important fact that you want to remember is
that in equilibrium the Fermi level must be constant.
The bands can bend, but the Fermi level must be constant.
And you can see that because -- let's pick a particular energy.
If I ask, what's the probability that a state
at location A is occupied,
my Fermi function tells me the probability that it's occupied.
What's the probability that a state at location B
at the same energy, same total energy here, is also occupied?
Well, it's given by the same Fermi function.
And in order to make it the same probability,
the Fermi function has to be the same everywhere.
If it isn't, there would be some probability in equilibrium
of electrons moving from A to B or vice versa,
and that means I would have current flowing
and I wouldn't be in equilibrium.
So the Fermi level must be constant in equilibrium.
[Slide 35] Okay. Now, why do the bands bend?
Well, this is another thing that we learn in freshman physics,
that if we have an electron in vacuum
and we establish an electrostatic potential,
there will be an attraction that will lower its energy.
The energy will be lowered by an amount q times the voltage,
q times the electrostatic potential.
So positive electrostatic potentials lower the
electron energy.
[Slide 36] So let's see how that would play out in a typical semiconductor.
Let's say I have a semiconductor that starts at X equals 0
and goes off to infinity.
And on the surface of the semiconductor,
I'm going to put an insulating layer.
And then I'm going to put a metal plate.
This is like the gate at my MOSFET.
And I'm going to put a voltage on that plate.
So the voltage is positive.
I'm going to assume that I ground the end
of the semiconductor, so the voltage is 0 there.
And if I ask what's the voltage inside the semiconductor, well,
it's got to have some positive value here near the positive
voltage on the gate.
And eventually it will drop down and go to 0
when I get far away from the gate.
So the voltage versus position will look like that.
If I -- you know, the potential
at the surface we will call the surface potential.
[Slide 37] And that will be important for us later.
If I draw the energy band diagram,
deep in the bulk I've just got my valence band,
my conduction band, my P-dope.
My Fermi level is down near the valence band.
And this is what it looks like in the bulk.
But since the voltage is getting more
and more positive near the surface,
positive voltage lowers the energies
and the bands bend down.
So if I look at this, I would say I have some --
a semiconductor with some band bending.
But notice that the Fermi level is constant.
That tells me that I'm in equilibrium.
The bands bend because the electrostatic potential is
changing with position.
A gradient of an electrostatic potential is an electric field.
So the slope of the conduction band gives me the
electric field.
If I see a positive slope,
it means I have a positive electric field
in the semiconductor.
So we're going to be looking at energy band diagrams frequently
to try to understand what happens inside of MOSFETS.
[Slide 38] Okay. Now, the final topic that I want to mention
in this brief review
of semiconductor physics is quasi-Fermi levels.
Remember, in equilibrium we have a Fermi level that's constant
and cannot vary with position.
So let's say I have a semiconductor
and I ground one end,
and I apply a voltage to the other end.
Current will flow, and I will no longer be in equilibrium;
and I should no longer talk about a Fermi level.
But there will be something like a Fermi level there
that will tell me the probability that the electrons
in the conduction band at those states are occupied.
So what will happen?
So here's what happens.
[Slide 39] If you apply a positive voltage we first of all have
to replace the Fermi level
by an analogous quantity we call the quasi-Fermi level.
It's something like a Fermi level that can have a slope
out of equilibrium,
and a positive voltage pulls the quasi-Fermi level
at the right contact down.
We now have a slope to the quasi-Fermi level.
If I draw my energy band diagrams,
I'm going to have a slope to them as well.
And the reason is that I'm assuming
that there is a constant density of electrons
in this slab of semiconductor.
And if there's a constant density of electrons,
since this quasi-Fermi level determines the probability
that states in the conduction band are occupied,
it's got to be a constant distance below the
conduction band.
So the conduction band gets pulled down as well.
So when I apply a positive voltage on the right,
I lower the energy of the conduction band,
the valence band, and the quasi-Fermi level.
[Slide 40] So out of equilibrium, the message is we have
to replace Fermi levels by quasi-Fermi levels.
So out of equilibrium, we take these expressions
for the electron and hole densities,
and we replace the Fermi level
by a quasi-Fermi level for electrons.
If we're interested in holes, we replace the Fermi level
in the expression for the equilibrium hole density
with the quasi-Fermi level
to get the hole density out of equilibrium.
[Slide 41] So we have Fermi levels and quasi-Fermi levels.
The Fermi level is constant in equilibrium.
The quasi-Fermi level can have a slope.
And actually, one can show -- and it's very easy to show --
that one can write the current
as carrier density times mobility times gradient
of quasi-Fermi level.
That is mathematically identical
to this drift diffusion equation.
Actually, I've done it in the wrong way.
This is a much more general
and much more fundamental physical expression
for current flow than this is.
Now, this one requires several simplifying assumptions,
and we have to think about things.
This can be established
from some very general physical principles
and is really a better fundamental starting point
for current flow in semiconductors.
But we will refer to both throughout this course.
[Slide 42] So that is a very quick look at some basic concepts.
As I said, this takes me about five or six weeks to cover
in a typical semiconductor course.
But I wanted to go through these concepts with you all just
to be sure you're familiar with some basic concepts
in semiconductor physics that I'm going
to be using routinely throughout the remainder of this course.
If you're familiar with these or you're familiar with most
of these and have to go back and review one or two,
then you're ready to take the course and be successful.
Thanks a lot; and look forward to seeing you in the course.