We're now up to the 18th lecture of the commencement, of course.
And over the last couple of weeks, we have been building up our knowledge of how to understand electrons in solids.
We know about the structure. We know about electrons and holes in our electron effect, the masses.
We know about doping. And at this point,
we actually have arrived at enough knowledge that we can actually think about putting together materials in order to make real devices,
and in particular, semiconductor devices, semiconductor devices,
which is in some way the crowning glory of the entire field of condensed matter physics.
It is because we can make semiconductor devices that we can build things like computers, we can build cell phones,
we can build iPads and all these other things that we take for granted and completely change our, you know, the way we live these days.
It is all because we understand semiconductor physics extremely well. The good news is that it is totally unexamined, but that means that we can't.
The bad news is that means we can't spend a whole lot of time on it. Even though it is an extremely interesting subject.
You could spend the rest of your entire life studying semiconductor devices and building new ones, and probably some people in this room really well.
The semiconductor industry is a huge industry that employs many, many physicists.
So people will be thinking about semiconductor devices in the future.
I promise you someone will. But for us, it is going to be limited to about 40 minutes.
But we'll see how far we can get in 40 minutes. So the simplest semiconductor device we're going to discuss is the quantum wealth,
which is nothing more than a a sandwich of one material inside another material.
So imagine having a three layer material over here.
It's over here. We'll put aluminium arsenide here is gallium arsenide and here's aluminium arsenide.
So it's a thin layer of gallium on our side in between lemon mars and this is some
position coordinate and we have actually studied this material before gallium arsenide
it remember it is a it has zinc structures FCC with the basis aluminium arsenide
is just like gallium arsenide except you replace the gallium with the aluminium.
So it's also FCC with basis. And the reason they have chosen these these materials, aluminium arsenide,
gallium arsenide is actually the there is a little bit of a gift of nature here.
It turns out that the lattice constant the size of the in itself for gallium arsenide in aluminium outside is extremely, extremely precisely the same.
It's just something that randomly occurs which makes these very desirable materials to work with.
The reason that makes things very desirable is means that you can line up the atoms directly
between the gallium arsenide and the aluminium arsenide and basically have just one giant crystal.
And as you walk through the crystal, the only thing that changes is in going from this material to this material.
All the aluminiums get secretly replaced with gallium, so it just looks like one big crystal of one material.
But over here you have aluminium, here you have gallium, and then you back here and it's aluminium is again,
the difference between the different materials is that the bandgap in aluminium arsenide is actually bigger than the bandgap in gallium arsenide.
So you might have a band structure that looks like this. It's a big band gap out here, small band gap here, big band gap here.
So if we plot the conduction, band energy conduction, it will be big here.
Then it drops down in this region here, which is known as the quantum well, and then it comes back up over here.
So this is the lunar arsenide, here's the gallium arsenide, here's the aluminium arsenide.
So if you're an electron living in that conduction band, what you see is you basically see a well, some sort of confining potential,
like a particle in a box, and you would form a wave function in that particle in a box in the usual way.
And of course you can have exciton state wave functions also in that particle in a box.
Also, the only difference between this and the kind of particle in the box problems
that you solved in your quantum mechanics courses is that it's the M star,
not m the effect a mass of the electron is important.
The effect of massive electron in the conduction man is the important thing that shows up in in Schrödinger's equation,
not the real mass of the electron. And the other thing that's different is that this V of x, this potential is actually due to the band structure.
Due to band structure. So that is what's causing the potential to go up and down to the bottom of the conduction band is moving is moving up and down.
Now, if you think about the Valence Band. It actually looks the opposite.
The Valence Band E Valence, the top of the Valence Band starts low, goes up high in the middle, and then comes back down like this.
And if you are a hole living in the Valence band and remember the holes try to go up instead of coming down like electrons try to come down.
So in fact, a hole can get stuck in this or upside down.
Well, and it would also form a particle in a box wave function inside are inside this
quantum wall so you can trap both an electron up here and a hole down here.
Now, this very simple structure is actually used quite frequently in in the semiconductor industry, very frequently for making lasers.
What happens is you imagine putting an electron up here and a hole down here, and then an electron can fall into that hole while emitting a photon.
And it comes out.
And if you learn some laser physics last time, you understand how if you have transitions like this, you can build you can build lasers out of them.
The thing that is neat about this structure is you can change the energy of the iron states by changing the width of the well.
So if you make it a thinner layer in inside this of gallium arsenide, inside aluminium arsenide,
then the the energy ion states change just because the energy I can say so is depend on the width of the box.
And so you can tune the energy of the upcoming phonon by just changing the width of the well.
And that's why this kind of device is frequently used in in Optoelectronics.
Okay. All right. So this is a very simplest device that we ever need to think about.
We don't have to worry about any opens. Now we're going to think about adding doping, which will enable us to make some much more complicated devices.
So doping and although I said at the beginning of the lecture,
the semiconductor devices is not examined of all doping is examined, all we are we discuss this at the end of last lecture.
I'll remind you of some of the results that if we drop a semiconductor,
the density of electrons in the conduction band is some constants which I want write down times an exponential factor,
even minus beta e conduction minus mu. So basically the big part of this,
the exponential factor tells you that you have to activate an electron up into the conduction band from the chemical potential.
It's exponentially activated up into the conduction band.
And then there's a couple of three factors out front which aren't that interesting.
Similarly, and actually it should be clear that as you raise the chemical potential towards the conduction band,
the density of electrons in the conduction band goes up.
Similarly, if we want to know about the density of holes in the valence band P that some constants also even minus beta nu minus V.
So similarly we have the holes being activated down into the valence band
because remember it requires energy to push a hole down into the valence band.
The holes want to bubble up and again it's activated.
If you move the chemical potential down towards the valence band, then the density of holes goes up.
The law of mass action.
Mass Action tells us that the product of these two things and of t p of t equals is a couple of these these two sets of constants.
Then this is the minus beta e conduction minus valence, and this is independent of the chemical potential.
So what I want you to take away from this,
these equations which have to do with the density of electrons and the conduction intensive holes in the valence band is that if we dope,
if doped heavily with n,
in other words at a lot of donors, then chemical potential goes up, up to the conduction band energy,
it goes up towards the conduction band energy because we have to raise the number of electrons while keeping the product here fixed.
And the only way that happens is by raising the chemical potential.
Similarly, if we do not heavily, if doped heavily with P with acceptors adding holes to the valence band,
then you goes down to the balance band energy that much good.
That is everything that is examined of all of our device physics is these two two boards right here.
So now what we can do is we can think about what happens if we put some dope ends in our semiconductor.
And this brings us to our first interesting device, which is the Junction P and junction,
which was surprisingly discovered or essentially discovered in 1874 by Ferdinand Brown.
Ferdinand Brown actually had no idea what it was he discovered,
but he was a clever enough physicist to realise that what he discovered was really interesting.
In fact it became an extremely important thing. It was a little electronic widget that enabled the development of the radio.
Without brands discovering of this device, we wouldn't have been able to build the radio.
And of course, the radio is one of the inventions that really changed communications and changed the world in many ways.
So Junction changed the world back in the 1800s or beginning of 1900s.
But actually it was not really understood why the Junction behave the way it did for about 70 years after Brown's discovery.
So let's see if we can understand what it was that Brown didn't understand. So, first of all, what is the structure?
So you start with a big piece of semiconductor and you just up one side of it and you end up the other side and dope the other side.
Now, when Braun actually built this, he took two pieces of materials and just stuck them together.
In the modern era,
you take one big crystal and you add some droppings over here and you add some dummies over here much more control than what Brown did,
but generally the same idea. And so let us draw a picture of where the conduction band and the Valence band are.
So the conduction band energy might be up here. The Valence band energy might be down here in Valence down here.
And here's our division in the middle. Now, over on this side, we're doped.
So that means we have a bunch of electrons up in the conduction band here and the chemical potential
is way up here mu up here near the conduction band because we have doped it with the electrons.
So the chemical potential went up towards the conduction band energy and now over down here we have a bunch of
holes in the valence band and their chemical potential has been pulled down to near the valence band energy.
So far, so good. All right. Now, the way it's drawn, this looks like you have negative charges on the left and positive charges on the right.
That is not actually true because we had left out the charges on the nuclei.
So the nuclei have charges are also nuclei nukes.
We and they have compensating charges like this and competition charges over here.
Because you remember when you add a for example, when you add a phosphorus to silicon, you add a extra nuclear charge, you also add an extra electron.
The whole thing is neutral. The electron goes off into the conduction band.
But you left a positive nuclear charge behind as well.
So everything here is electrically neutral. No electric fields, electron negative charge is compensated by the nuclear positive charge,
and the whole positive charge is compensated by the R by the acceptor negative charge here.
So far so good. Okay, now here we have electrons up here in the conduction band and we have holes here in the valence band.
And in fact, this electron, if it falls down to the Valence Band annihilating its hole, he can let off energy.
He can mirror photon, for example, and lower the energy.
So what happens is that the electron here are now a little bit of a razor transform, annihilates the whole electron,
annihilates the hole, and you end up with a recently annihilate one more here, annihilate one more.
And we end up with this region in the middle from here to here, which is known as the depletion region.
Depletion region where there are no electrons and no holes, there's no mobile charge carriers anymore.
You only have the static nuclear charges left. Nothing moving around.
So you might think. This annihilation will keep going because the electrons are up at high energy and the holes are down at low energy.
So we can just keep annihilating electrons with holes, keep emitting photons, and lower and lower and lower the energy.
But that is not true. In fact, it does not keep going.
And the reason it does not keep going is because we are leaving a net charge behind.
This region now is charged. This is net positive charge because of the doping and this is net negative charge, net minus charge.
So as we start annihilating, we leave actually a charged region here and a charge region here.
And that means we have an electric field against an electric field, points from the minus to the positive and minus divisor positive.
Negative. Which way? I see the direction of a positive charge is the negative.
Points this way. The electric field goes from positive.
No test charge goes towards the negative. Yes. Okay. Got it. Right. So so there's an electric field here that we're building up.
And as we annihilate more more charges here, we end up with more and more electric field.
And that means if we want to take one more electron here and annihilate one more hole here,
we first must we must climb up a potential hill and electrostatic potential hill to get past this electric field to reach the hole here.
At some point, this electrostatic potential hill becomes too big and we can't get over it and we can't have any more annihilation.
So the depletion with depletion.
Depletion with. Mission with.
Is that by selling the gap, energy gap, the amount of energy you would you would gain by annihilating one more energy,
with one more electron, with more whole, it has to equal two times Delta PHI.
The amount of energy you have to put into it to get the electron over the electrostatic
hill that you're fighting against to move the electron past the discharge region.
Is that clear? Roughly someone?
Yes. Okay, good. At least someone said yes. So this this potential here is just the integral of EDL from well, across the system.
All right. So so that generally we we do not believe that this depletion with is going to be infinitely big.
It's going to stop at some region. But this is not a good way to draw what goes on because this looks a little it looks a little strange.
It looks it looks incorrect. I mean, it looks like the electron here is at higher energy and the whole is at lower energy.
So it kind of looks like the electrons and still want to annihilate the hole.
But we know that there's an electric field that's making it difficult.
So we what we want to do is we want to redraw this picture in a way that will make it obvious that the electron does not want to come in,
annihilate one more hole. So what we do is we draw things in terms of the so-called electrochemical potential.
Electrochemical potential. Potential, which is the combination of the regular chemical potential and the electrostatic potential.
Okay. So we're going to plot that instead,
which is going to combine together the so the band structure diagram as well as the Coulomb effects of the space charge that we left behind.
So over on this side, draw the same picture we had before.
So we have the conduction band of the energy up here, the electrochemical potential input here.
So this is a new minus EFI over here.
We have the valence band energy over here in Valence.
Now over here on the other side, we are going to line up the electrochemical potential, exactly the same.
But here the valence band energy will be here in Valence and the conduction band energy will be up here.
The conduction and these get connected together like this.
Okay. That's. Those are supposed to be parallel lines. Not very parallel.
Uh, improve it. The improvement rises up a little bit more, so they look very vaguely parallel.
So these are supposed to be. Okay.
Good. So all I have done is I've taken that picture up there and I have twisted it.
I have of bent it so as to account for the additional electrostatic potential.
So now the on the left, in the right, the electrochemical potential lines up with the electric chemical potential over here.
And what that tells us is that there is no net driving force for moving an electron from this side to this side,
that there is a change in the chemical potential,
but there is are equal change in the electrostatic potential that compensated the left hand side and the right hand side are completely lined up.
As far as the electron is concerned, it does not want to move to the left. The electron didn't want to go left, don't want to go to the right either.
It just wants to say stay put. So this is the equilibrium situation.
Is that sort of clear how this has been drawn? Okay. All right.
As a terrible picture, isn't it? Okay. But this this device here, this is basically the physics of the junction.
And even as it stands, just like that, without doing anything further, it's an interesting device because it's basically a solar cell.
A solar or photoelectric cell. Photoelectric cell.
So. And the idea of this is that you put in photons.
Put in photons, get our current.
Get current. And this idea of putting in photons and getting our current is a $20 billion industry right now with people trying to build for
ourselves that are going to replace our fossil fuels and potentially save our world from global warming and everything else.
So I encourage all of you to go into the semiconductor industry and try to figure out how we're going to make solar cells more
efficient so that we can get all of our power from these things instead of from burning coal and oil and things like that.
At any rate. Let us try to figure out why it is that this device has this property that if you put in light, you'll get out current.
So if you put in light into this into this picture, you can excite an electron from the valence band up to the conduction band.
So you leave a hole behind and you excite an electron up to the conduction band from the valence band.
Now, in this depletion region, there's an electric field, there's a strong electric field in this depletion region.
You can see it in this picture by the curvature of these bands.
And what happens is the electron wants to it is pushed by the electric field in this direction because it wants to go down to lower energy.
And the hole is pushed in this direction, wants to go up to higher energy because holes always try to go up in the in the band structure.
And either way, this is always current. Going to the right is either positive, current, positive charge going to the right.
This is plus going to the right or negative charge going to the left.
And both of those are current in the same direction. So just by putting in a photon here.
Photon. Which creates this exhortation.
You managed to create current.
So this is the, you know, the the physics is going to save us from global warming one of these days.
So please, you know, go out into the world and figure out how to build this better.
You know, they're still not quite as cheap as burning fossil fuels that people aren't willing to use them yet.
But maybe someday we will. Okay.
This is not what first Brown discovered in 1874, but he discovered with something very with the same device.
But he discovered another effect that these things have, which is maybe equally interesting, which is the effect of rectification.
Which means net current flows one way low, one way, one way, but not the other.
And if you're a good with your circuits,
you might be able to figure out why it is they're having something with this property which rectifies enables you to build a radio,
which is what he realised very quickly that if you have something that rectifies allowing content to flow one way but but not the other.
You can build a radio with it. So. Why?
Why is this? Why does it have that physique? So let's take this device and let us imagine.
Well, let me draw the same picture again,
except what I am going to do now is I'm going to put a voltage across this picture to try to get current to flow.
So let's do that. So exactly the same picture you see here is the electrochemical potential meu minus e ify on the left.
And then down here is the valence band, Energy Valence.
And then over here, instead of lining up the electrochemical potentials exactly on the left and the right,
I'm going to shift the chemical potential down to the electrochemical potential down here.
So this is if it was lined up. Exactly. It would be exactly the same lined up here.
And now I'm going to shift it downwards by easy here, by some energy.
So this is the electrochemical potential on the right. So now because I'm putting a voltage across the whole thing.
So then we have to put in the valence band energy against new minus EFI here, it's here and valence band energy is here.
Valence and then the conduction band energy will be here and then in the depletion region have electric field which connects these together like that.
Okay. So is it clear what I have drawn here? I've taken exactly the same picture.
I've just applied a voltage across the to to shift this chemical potential down to here.
And. All right. So now now that I have applied this this voltage, how much current flows?
Well, there are several ways you can get current. One way to get current with this process.
One is that an electron in the Valence band could get excited up to the conduction band and then eventually find its way down the hill.
So let's call that process one. And process two is sort of the reverse of that,
which is the hole up in the conduction band could get excited down into the Valence Band because remember,
it requires positive energy to push a hole down into valence band and then find its way up the hill.
So process one and two, one and two. This either an electron going to the left or a hole going to the right.
And either way, its rate is either the minus beta e gap,
because the limiting step is to get the electrons up into the conduction band or get the hole down into the valence band.
And that costs the gap energy. So it's activated. Okay.
So that is one thing that can happen that can cause current. But another thing that can happen that can cause current is you have some number of
electrons up here in the conduction band already and they can just climb up the hill.
Let's call it three or we have some holes here in the Valence Band and they can just climb down the hill, let's call it four.
So three and four, three and four, we have electrons going to the right or a holes going to the left,
and they happen at a rate of either the minus E gap minus EV.
Why is that? Well, the height of this hill from here to here.
Is Gab minus eve. If we had not applied this way, it would be exactly the same gap energy that we would need to climb up the hill.
But now we have reduced that hill. We reduce the size of the hill by ev by applying the voltage across the system.
So the total current total j j equals well is going to be a combination of these two things.
I call it A1 to either the minus beta E gap minus a34.
Those are just coefficients either minus beta B gap minus any which.
Okay. Then we need to know what these coefficients are.
But actually there's sort of a trick for that that the current when there zero voltages should be zero.
If you do not apply any voltage in the system, you should not get any current.
So that means a one to an 84 should be equal.
So j is proportional to each the minus beta e gap factor that out times are either the EV or Katie t that is beta minus one.
Okay. So this is what's known as the diode equation. Diode equation.
The injunction or the Rectifier is sometimes known as a diode and we can plot its behaviour here.
So we have. Current voltage on this axis and current on this axis and is exponential in.
In voltage. So this is this is zero current here and this is negative current here.
So maybe I should have drawn it like the zero is here and negative is here.
So with positive voltage it goes up exponentially, whereas negative voltage it goes to a constant which is extremely small constant.
So this is frequently drawn figuratively as being straight up like this and straight flat like this,
which the APR APR say it allows current to flow in one direction and allows no current to flow in the other direction.
Frequently we draw in a circuit diagram.
We draw a diode like this diode, which means current flows only in one direction.
It's pretty neat device to put into your circuits when you're designing circuits.
So this was what Vernon Brown discovered experimentally, more or less,
and it was not understood until the 1940s when people really developed a good understanding of semiconductor physics.
And by that time when they finally understood semiconductor physics, they realised they could do a lot more with it than just building diodes.
And this enabled them to start working on the next big invention that changed the world,
which is the transistor invented in 1947 by the Bell Labs team of John Bardeen, Walter Bratton and William Shockley.
Bardeen is definitely a name that everyone should know because he won two Nobel Prizes,
one for the transistor, one for the theory of Superconductivity.
A very bright but pretty much unknown person.
And this device, the transistor is the elementary that the smallest piece, the elementary piece, the building block of every computer ever made.
Now it is in your average cell phone or laptop or iPad.
There are billions of these transistors sitting on, literally billions of them in there.
It is the elementary switch or elementary amplifier that enables everything to work.
So transistors come in many different varieties. The one we're going to discuss is actually known as the MOSFET.
MOSFET, which stands for a metal oxide oxide semiconductor field effect transistor.
And the star. And I explain what that means as we as we get to it.
Although I should comment that the original transistor that they built in 1947 was not of this design.
They were aware that they could build this design, but for various reasons they decided that was not the one they were going to try to build first.
This is the type of transistor that is most frequently in circuits now.
So that's why explaining that one is also easier to understand, except nicer to talk about.
The way it works is fairly simple device. You take a big piece of a semiconductor and you drop it and it has a surface like this.
You can do it backwards, you can open if you want it to, but let's imagine it's doped.
And then we take two small regions and we end up them, make it a little bigger,
this region here and up this region here, and then we'll end up this region here, this and this is all doped.
Okay. So what we have now is we have a surface with an end to end and a big bulk with p doped.
We'll put contacts here on the wires coming in. This will call the source source wire and this is called the drain wire drain.
And then we try to run current from the source to the drain.
Now if you look at this structure here, you'll realise that current is not going to flow from the source to the drain.
You'll have a depletion region at the interface between an MP.
There's an MP junction there and PNG junction here.
And one of the PM junctions wants the current to only flow one way and the other one wants it to flow only the other way.
You can kind of think of it figuratively as being, you know, two back to back diodes pointing in opposite directions.
So one of them only allows the current to flow to the left, the other one only allows the current to flow the right.
So between the two of them, you can't get any current through it at all.
Okay, so this looks like a boring device, but now we add the last part of the device.
We put a big piece of metal on top like this, which is known as the gate.
That metal metal gate.
And then we attach it to the source via a little bit of battery here.
So voltage plus battery like that, attach the metal gate and then we put an insulator in between here, which is usually an oxide layer.
And this is what gives the name of the device metal oxide semiconductor.
So the gate is metal, the oxide is in the middle and the semiconductor is on the bottom.
The purpose of the oxide is just to make a small insulating layer so the metal doesn't touch the semiconductor.
You can make it of anything you want. It's just typically made of oxides, which are very nice insulators and they're easy to build.
You can make out of any insulator you wanted to put there.
You could make it a diamond, you can make it, you know, it could be just vacuum if you wanted.
But the point is you have to have the metal not touching the semiconductor.
So it's usually called metal oxide semiconductor. In fact, the gate does not have to be metal either.
It just has to be conducting. So it can be whatever you want as long as it's conducting.
At any rate, it's called metal oxide semiconductor because it's almost always made of metal oxide and semiconductor.
All right. So what's the purpose of this metal? This metal gate, you charge it up.
So it has plus charges on it. And this plus charges act like a capacitor and they're going to attract negative charges on the semiconductor here.
Now, this region is semiconductor, so we're basically just charging up a capacitor plate between the metal and the semiconductor.
This region here on the top of the semiconductor, it used to be p do not even have holes there.
But now we have attracted a lot of electrons into that region.
So instead of being p doped, it is now majorly doped because we have we just attracted electrons there.
So now it's n so instead of having something that looks like that picture I had originally, it now looks more like this.
This whole region here all the way from here to here is negative, is n, is n,
doped, has has electrons in it, and the depletion region sort of makes a big.
Sort of environment like this. Now, here you can get current directly from this end region to this end region via this electron channel.
So you've added electrons to this layer, so the electrons can travel directly from the source to the drain without having to go through a junction.
That's the point. That good? Happy with that?
All right, good. So the characteristics of such a device look more or less like this,
that if you plot the current versus the voltage, this is the voltage between the source and the drain.
This is the current that flows from the source to the drain. If you put no gate voltage on V8 equals zero, you get almost no current flowing.
But if you put a little bit of gate voltage on the gate equals say two volts equals two volts.
You all of a sudden get a lot of source drain current. So this enables you to take a small voltage on the gate and control a very large current.
So this is basically why this device is so useful.
It acts as an amplifier. It can amplify a very small signal on the gate,
and then you can do interesting things with it because now it's become a big current that you can work with.
The elementary circuit, that element that builds all the computers in the modern age.
All right. Well, I hope many of you will go on and study more semiconductor device physics more than these last 35 minutes.
But this is all we have time for in this course.
Unfortunately, we have to move on to the the final segment of our course and the final segment of a course.
Is there leaving all this stuff behind us? And we move on to study magnetism, magnetism.
So take a deep breath and we move on to magnetism.
And and the first question one probably wants to ask is, why are we studying magnetism and why are you studying magnetism now?
The whole course, pretty much from day one, all the way up to lecture 18 halfway through,
which is right now we've really been studying waves in solids one way or the other,
whether they're vibrational waves or whether they're electron waves or the x ray waves.
We really have just been studying waves in solids in one way or the other, and magnetism is a little bit different.
We're not really studying waves and solids when we're studying magnetism. So why is it that we're studying magnetism?
There's a lot of reasons why we want to study magnetism. The first reason why we want to study magnetism is it certainly is an effect that has
been known about since antiquity and the records of people knowing about magnetism.
1000 B.C. in Greece, thousand B.C. in China.
They knew how to build compasses and well, probably predates that by by thousands of years.
Probably they were cavemen who knew about magnetism. You stuck notes on the refrigerator.
The theory anyway. But.
But despite the fact that this effect was was known about thousands and thousands years ago,
there is absolutely no understanding of it whatsoever until people understood quantum mechanics.
And that is one thing that makes this very interesting. In fact, there's a theorem by Baur and Van Nguyen, known as the Bois Van Buren Theorem.
Niels Bohr I think everyone knows, but Enrico Van Leone was one of his students.
We Win Theorem. And their content and the more value and theorem is basically no magnetism without quantum sim without quantum.
Nothing in classical physics can ever give you magnetism in any sense.
So that is kind of an interesting statement. So it tells us that, you know, that magnetism is a really good place to go to understand,
you know, the macroscopic ramifications of microscopic magnetism.
So that's one of the reasons why we want to study it.
Another reason we want to study magnetism is because it is a rather spectacular failure of everything we've been doing so far.
That magnetism, the band theory we've been studying so far will always predict that you have the same number of ups,
spin electrons as down spin electrons, whereas in magnetism it's just not true.
You have magnetic substances where we have a different number of spin up electrons and spin
down electrons is completely outside of band theory and it's completely not predicted.
So magnetism is a way to understand with the limitations of what we've already been doing.
The final reason, which is maybe not so obvious, is that magnetism is a really great place to study.
Phase transitions between phases of matter so everyone's familiar with in phase transitions between ice and water, water and steam.
And you've probably studied these things in your thermodynamics classes,
but when one really gets down to understanding the details of these phase transitions,
most of what we know about phase transitions comes from studying magnets, where things are much better controlled and much better understood.
And much of what we know about the melting of ice into water,
we actually know because we studied magnets and then we translated our knowledge into understanding ice and water.
That is not an obvious statement. We will make some statements about phase transitions in this course.
If you go on to study other courses, you'll learn a lot more about it.
Okay, so let's start by defining a couple of types of magnetism, types of magnetism that we're going to be working on of magnetism,
or at least some of the types of magnetism we're going to need to know about.
Usually what we do is we apply a magnetic field to some object and we measure groups,
we measure a magnetisation m and there's a proportion if there's a proportionality constant,
we call the proportionality chi over new night where chi is known as the susceptibility or magnetic susceptibility susceptibility.
I'm going to spell that wrong as you see ability susceptibility.
And Moonshot is the usual magnetic permeability, the thing that's permeability, the thing that's on your data sheet and your exams.
So given that as a definition, there is a couple of kinds of magnetism that you can have.
One is power magnetism, which means or a paramagnetic is any material with greater than zero.
In other words, you apply a magnetic field and you get a magnetic moment in the same direction as a field that you applied,
which seems to make some amount of sense. And we have know a couple of examples of that.
Example example one is a free spin.
Probably studied this in statistical mechanics last year.
If you have a free stance or is sitting in outer space, if you apply a magnetic field to it, its energy is lower.
If the magnetic moment points in the same direction as magnetic field.
So the spin will try to flip over and point in the same direction as the magnetic field.
Hence the free spin is a power magnet.
Example to which we studied earlier in this term is a Fermi Gas or electron gas which is studying the Sommerfeld model.
And we found Pauli power magnetism, power magnetism.
And there was some calculation which went along with that and it is basically exactly the same physics.
You have a bunch of electrons, you apply magnetic field. The moments of the electrons want to align with the magnetic field to lower that energy,
but they can't all flip over because of the exclusion principle, so only a few of them flip over.
But nonetheless the magnetisation that develops is in the same direction as the magnetic field that's been applied.
So this is a power magnet, albeit a weaker one, than the free spin, because only some of the electrons will flip over in the case of a Fermi Gas.
We'll see more about those in a while. So another type of magnetism which is common is dear magnetism.
Which basically means Kai is less than zero.
In other words, when you apply a magnetic field, the magnetic moment wants to point in the opposite direction as the magnetic field.
Now, we may not know any examples of that immediately,
but there is something that's qualitatively similar that we do now qualitatively like lenses law,
which you may remember from first year in lenses law,
to remind you if you have a loop of wire, if you apply a magnetic field to the loop of wire,
currents will flow in that loop of wire to oppose the magnetic field.
So you're developing a magnetic moment in the opposite direction of the change in magnetic field.
Our loop of wire, however, is not a dire magnet.
The reason for this is because that will develop a magnetic moment.
But eventually the the current in the wire wall will decay and go down to zero and you'll have no magnetic moment left in the wire.
So, in fact, it's not magnetic at all. It has no magnetic moment unless the wire is a superconductor which has no resistance or current never decay,
and the current will keep flowing around forever and ever and ever. And and the magnetic moment would stay there forever and ever.
And that's a real die magnet. We do not study superconductors this year.
It's just nice to know that. That you could do this with a superconducting, get a real die magnet anyway,
that the physics of magnetism is actually quite similar qualitatively to the physics of lenses law.
But when you add quantum mechanics to it, you kind of, you know,
in some interpretation you can think of these currents as being being stuck there forever and always opposing the magnetic field.
Even when you when you do it with lenses like you have to keep changing the magnetic field to get the current to keep flowing in a die a magnet.
As long as the magnetic field stays on the current, the current should keep flowing.
So we'll see how that happens in a bit. The third case that we need to know about is the case of a ferromagnetic,
which is when magnetisation is not equal to zero, even when B equals zero, something like iron.
Has this property, the thing that sticks to your refrigerator. This is the thing that we usually think of as a magnet when we think about magnets.
So a good place to start. Our study of magnetism is one atom at a time.
So we'll start with atomic magnetism, atomic magnetism, and we'll build up to magnetism of real materials.
And I should put it in this caveat that we always will we will ignore ignore the magnetism of nuclei, nuclear nuclei all together.
And the reason why we ignore the nuclei is that if you think about the magnetic moment of some object, it's given by the born magnets on where to.
Of the object for an electron. This is roughly the the up to the factor.
This is the magnetic moment of an electron.
It is similar for a nuclear on a proton or a neutron, but the mass that enters in here is bigger by a factor of a thousand or 2000 or something.
So that the magnetic moment that you get from from a nucleus is much, much, much, much, much smaller than the magnetic moment you get from electrons.
And that is why we're usually not concerned with the magnetism of nuclei.
It is there.
It sometimes turns out to be important, especially if you're thinking about things in extremely low temperatures or if you're looking at very,
very subtle and fine effects if you have to worry about the magnetism on nuclei.
But for us, we are only going to be worried about the magnetism of the electrons because it's just much bigger.
So we're going to ignore the nuclei and then we have to start the way.
We always start by writing down a Hamiltonian for the electrons in the atom.
So I write the Hamiltonian Hamiltonian, and it is conventional to use sort of a script for the Hamiltonian,
so you don't get it confused with the magnetic induction H you know, being H and new and that kind of stuff.
So we use a script H for the Hamiltonian and we'll write it in this hopefully ways that you've seen before.
So there's a a kinetic term, P plus A squared over two M where A is the vector potential double cross A is B,
whatever magnetic field I decide to apply to this system, then there's going to be.
So this is just the kinetic energy, then there's going to be an attraction to the nucleus.
V vrr And the last term we have seen before g you b b that sigma and this is the Zeeman term, zeeman arm two ns maybe one.
N Anyone now want one and I give them on term is my notes.
My notes say my notes. Don't say it. Okay? I don't know. All right.
So this is the term that tells you that the magnetic moment of the electron is lower energy.
If the electron points in the same direction at the magnetic moment, points in the same direction as magnetic field, this gets very confusing.
The signs get very confusing here. Remember that the magnetic moment of an electron spin points in the opposite
direction of its magnetic moment because the charge in the electron is negative.
So it looks as if the energy would be lower here when the so here,
here the energy is lower if sigma is in the opposite direction of B, so it's pointing in the opposite direction.
So it's negative. But if sigma is in the opposite direction is B, the magnetic moment is actually the opposite direction.
Sigma So the magnetic moment is pointing in the same direction is b minus signs.
Ah, you know the data. Me okay, maybe is the more magnetised for the electron energy is the g factor which will usually take to be just two.
All right, so what do we do with this?
Well, we're really would like to treat the magnetic field a little bit as a perturbation if we could serve.
You know, we'll start by rewriting this Hamiltonian as in the following way,
h equals P squared over two M plus V of R and that is just h zero or HB equals zero.
Separate out that term first. Then we have a couple terms left over that we didn't write yet.
So maybe that's writers here plus plus all this stuff we're going to write next.
So one is the cross term with the vector potentially over 2mp, a plus eight P, and then there's another term,
the vector potential squared over 2ma squared, and then we have the same on term left over g u b b that sigma.
And then we have to do a little bit of work here. And one piece of work we need to do is we need to simplify this term.
And the way we're going to simplify that is by choosing a gauge. Choose gauge for the vector potential.
And the gauge I'm going to choose is that the vector potential is one half the cross R And if you choose it like that,
this, this does give you del cross a equals B, if you choose it like this, then actually commute through with each other.
And it's maybe not so obvious to see if that is true. But it is true because that X component of A contains the y component of R,
so the x component of a can commute with the with the x component component of p and those are the ones that are multiplied by each other.
So it doesn't matter which order you put them in. So choosing this gauge, we can then we write this as E over to M to a dot p,
so those two right through each other and then we'll write in our expression for a while.
We cancel the two with the one half and we get B cross R dotted with P.
We use the triple product identity to rewrite that as E over to M the dot R across p,
r across p and then just absorbing a factor of h bar into the R across p will
recognise this thing as in the angular momentum and put the power up here.
So we get new B and D dot l where L is the orbital angular momentum equals orbital angular momentum.
So putting this all together, we end up with the final Hamiltonian that we're going to work with.
The Hamiltonian is the Hamiltonian at the equals zero plus couple terms.
You b be dotted with the orbital angular momentum plus g times the spin angular momentum plus this one more term they were left out.
E squared over to m e squared. And I guess maybe we should stop there and pick up next time with this Hamiltonian
and will analyse the effect of the magnetic field on this Hamiltonian. Okay.
See you tomorrow. Don't forget the meeting, meaning right after the lecture.
And then also the hands like this afternoon. 5 p.m. searching for extrasolar planets or something.
Okay. By.