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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.

>> [Slide 1] So hello.

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