Welcome back. It's now the 16th lecture of the condensed matter course.
When last we left off, we were talking about banned structure of solids.
And in particular, we were thinking about the nearly free electron model as a way of understanding the band structure.
A summary of the most important things we learned is really given by this one picture here.
So you have the free electron parabola and due to the periodic potential,
you can scatter from bronze on boundary to bronze on boundary and you open up a gap at the bronze on boundary.
You can also view it in a reduced zone scheme, if you prefer.
Now, one thing that's really important in this picture is to realise that states that are in the extended zones game,
states that are inside the first boron zone are pushed down, whereas states that are outside the bronze zone are pushed up.
So maybe I'll write that generally states inside, inside these boundary by the boundary pushed down and outside are pushed up outside and pushed up.
And the closer you are to the brownstone boundary, the more your energy is changed from the free electron model.
So let's see if we can take this principle and understand what happens to various Fermi surfaces when we go to higher dimensions.
So let's consider two dimensions the square lattice.
So we have a square brown zone. Square bronze zone.
I'll draw a square bronze zone. Here it is. Square K equals zero is in the middle.
This point over here is pi over a pi over a the corner.
And let's consider first the case of one electron per unit cell.
So that means we should have a one half filled, drawn zone, half filled lowest band or half filled bronze zone.
So what happens? Well, in the absence of scattering potential, you just have a nice Fermi circle.
Are Fermi from a sphere on two dimensions? It's a disk, I guess.
And this is actually a fairly good representation of many monovalent materials.
Materials that have one electron per unit cell, for example, sodium.
So here's the two dimensional analogue. You just have a a disk filling exactly half of the area of the barren zone.
So this circle is supposed to fill exactly half of the area of of that square.
Sodium is a basic material. So it's Braun Zone.
Is this funny wired shape. It's a trunk rama.
He draw dodecahedron or some strange shape like that.
But inside it's filling exactly half of the volume is a spherical Fermi surface,
which tells us that sodium is nearly a free electron system because its Fermi surface is almost perfectly spherical.
Now what happens when we add a stronger periodic potential where we let the periodic
potential start moving around the energy of certain states in the bronze zone?
Well, what happens is the states near the bronze zone boundary, but inside the bronze on boundary get pushed down in energy.
And the states that are closest to the bronze zone boundary get pushed down the most.
So if you consider a state around here, this is pretty close to the bronze on boundary.
So it's energy is getting pushed down more than a state here whose energy is getting pushed down less.
As a result, the Fermi surface wants to deform in such a way that more states around here are going to fill.
So it's going to sort of deform like this. So that more states near the bronze on boundary are filling because they're getting pushed
down more and energy than the states which are farther from the Browns own boundary here.
So they'll get a Fermi surface that kind of looks a little bit more like this,
a little bit deformed due to the shifting around of the energies of the states in India in the bronze zone.
And in fact, that's what we get for lithium, which is again, a monovalent material.
So it's Fermi surface should really fill half of the first boron zone.
Again, the same shape of the bronze zone as a BC material. And you can see here's the 2D analogue drawn a little bit better than I do on the board.
And you can see that what's happened is where the Fermi surface, which is basically spherical, comes close to the brown zone.
It's pulled out towards the bronze zone boundary because those states which are close to the brown zone
boundary have been pushed down more in energy than the states which are further from the bronze zone boundary.
So that makes sense. All right. Now, in fact, if the period potentially becomes extremely strong,
what can happen is the states right at the bronze zone boundary can be pushed down so much that they go below the Fermi surface,
in which case the Fermi surface deforms until it actually touches the brown zone boundary.
And you get something that looks more like this. And the states inside that shape are filled in.
The seats outside that shape are empty. And that's exactly what happens with them.
Terrell Copper, also a monovalent material is actually an FCC material, so the shape of its bronze zone is different.
It is a truncated octahedron, all shape that we've seen before.
And you can see that the here's a picture in 2D which is drawn better than I've been able to draw on on the board.
You can see that the Fermi surface is basically spherical, roughly spherical,
but and fills exactly half of the Bronx zone because it's monovalent and one electron per unit cell should fill half of the first boron zone.
But where the sphere comes close to the barren zone boundary,
the states near the bronze zone boundary get pushed down an energy and those states preferentially
fill compared to the other states which are farther from the bronze zone boundary,
which are over here, which are pushed down less and those states have become empty.
So is that clear how the shape of the Fermi surface is getting deformed? All right.
So that's that's one interesting case. Let's consider another interesting case where we have two electrons per unit cell.
So here two electrons per a cell. Again, we have the the the bronze on his K equals zero on the middle.
Now, in the absence of a periodic potential, again, we should just have a Fermi circle.
Our Fermi desk, I guess, whose volume is exactly the same as the first bronze zone.
So area in this case equals the same as bronze on busy area.
If there's no periodic potential, then of course, the brownstone doesn't matter.
There's no periodicity at all. You don't have to pay attention to the Bronze Age. You just get a perfect, a perfect circle.
However, if the periodic potential is strong, then what happens is all the states inside the first bronze are pushed down and energy.
The states outside the first bronze zone are pushed up in energy.
And instead of having a Fermi circle, what you get is you fill the entire first bronze zone and you empty the entire second bronze zone.
You get a filled band and a gap and an empty band.
The gap opens up so much that all the states in the first bronze zone are lower than all the states.
In the second bronze zone, you get a filled band, a gap, an empty band, and you get an insulator.
So these are the two possibilities with no periodic potential. It's a perfect circle, which you can view, if you like,
as filling part of the first bronze zone that's inside the square and part of the
second bronze zone that's outside the square with a strong periodic potential.
Everything in the bronze zone is pushed down in energy, so everything fills in the first bronze zone filled band.
Then there's a gap. Everything, the second bronze zone is pushed up, and energy says gap between the filled band and the empty band.
And so we have an insulator. Okay, so that's the simple part of it.
The more complicated part of it is what happens when the potential where the periodic potential is weak or intermediate intermediately strong.
In that case, what happens is it's some of the states in the first bronze,
a push down in energy and some of the states and the well, they're all pushed down, but they only push down a little bit.
So only some additional ones fill. So the ones that are are lowest in energy that haven't been filled are states here.
So these states will start to fill in addition to the states that were already filled in,
some of these states will start to fill in addition to the states that were already filled, some of these states up here and so forth.
And we'll start filling in the corners that we haven't filled in already.
But not what if the potential isn't sufficiently strong.
We won't fill in the entire first bronze zone. We'll leave a couple of states in the corners which are empty here, are still empty, still empty.
And the reason these states in the corners are the ones that are still empty are
because these are actually the highest kinetic energy states in the first bronze,
only the furthest from K equals zero. So those should be the last ones to fill them because they have the highest kinetic energy.
They would have to be pushed down the most before they go below the Fermi surface.
Similarly, in the second bronze zone, all the states are pushed up in energy.
The ones that remain filled are the ones with the lowest kinetic energy.
These little slivers here, everyone else empties, but these little slivers remains filled.
So, okay, so now I have a picture of it which is better drawn than I can draw. So it will look something like this that the.
Most of the first bronze zone is filled. The corners are empty because those are the highest kinetic energy states in the fresh bronze zone.
Most of the second bronze zone is empty,
except these little slivers here which remain fill because those are lowest kinetic energy states that are closest to k equals zero.
In the second bronze zone. So the this is still a medal.
It still has a Fermi surface, a point where the empty states meet the filled states within the bronze zone.
So it's still a medal. Here you can have low energy excitations.
Now, the way this is typically drawn is to view each band one at a time or each bronze zone, one at a time.
So you just draw the first bronze zone. You have this picture here, it's mostly filled with just some corners which remain empty.
And the second bronze zone where we draw it in reduced zone scheme, we can sort of translate everything back to within a single bronze zone.
So this piece here gets translated to over here by two pi over a.
This piece here again is translated to here by two over eight.
This piece here is translated to down here. This piece here is translated to up there.
This is the usual reduced zone skin. You move everything over until it's all within a single square bronze zone.
And it would be only these little pockets that remain filled in the second band or the second bronze zone.
Now, this may look a little bit unnatural,
but this is in fact exactly what happens in many di valence materials with two electrons per unit cell, for example, calcium.
This may look a little complicated, but let me walk you through it. This is the actual Fermi surface of calcium in the second batch.
So here's the it's an FCC material. So it has this truncated octahedron shape.
Here's the second bronze zone written in reduced stone scheme.
So it looks exactly like the first bronze zone.
And you can see that what's going on is there is just these little yellow regions where electrons are still filled day.
They're just like these little pieces here peeking into the second zone,
just little, little caps, little regions which still haven't quite been emptied.
And these pieces are actually the lowest kinetic energy points in the second bronze zone, just like these pieces here,
where actually the lowest kinetic energy points in the second bronze zone there are closest to K equals zero.
They're in the centre of these faces. Now, this picture is a little bit harder to to pass, but to see what's going on here,
the places which are transparent are where the where the Fermi surfaces actually hit the brown zone boundary and then it's not drawn.
So this great big hole here is where the Fermi surface, where the electrons fill all the way until they hit the bronze zone boundary.
And the places where it's not filled all the way up to the bronze zone boundary is where it's drawn yellow.
So you see these little corners here similar to these corners here along these edges,
these edges are the points which are furthest from k equals zero in the in the first brian zone.
So zero is in the centre. The places closest are all filled up just like these are filled up all the way to the bronze zone boundary.
The points farthest are the points along these edges and that's where it's painted in yellow,
where the where the Fermi surface has not yet hit the bronze zone boundary.
Let me draw it a little bit bigger there. There it is.
So, in fact, with calcium, even though it's di valent, it has two electrons per unit cell.
It actually has to two bands which are partially filled.
The first band is partially filled or mostly filled in. The second band, although mostly empty, still has a couple of electrons in it.
So it's actually metallic. Okay, so this is sort of the general structure of a lot of bands in in many, many materials.
So for the next for this lecture and for a couple of the next lectures,
we're going to talk an awful lot about band theory and the ramifications of band theory and
the physical things that you can deduce about actual materials by thinking about band theory.
And remember that what we're doing in band theory is where we're categorised.
We're giving energies to every crystal wave vector in the bronze zone,
and we know that these crystal wave vectors actually correspond to modified plane waves or what are known as Bloch wave functions in Bloch's theorem.
And this is a very good way to describe the physics in many of these materials, thinking in terms of these modified plane waves.
But actually, we should be warned that there are some failures of band theory, and we'll discuss some of them now.
And just briefly and then sort of in the last couple of lectures of the term, we'll come back and talk about some of them.
Again, band theory, just to be aware that these exist.
One major failure of band theory is that sometimes, although maybe rarely, monovalent monovalent material's violent unit cells.
Can be insulators. And according to band theory, that should never happen.
Because if you have a monovalent or a trivalent unit cell,
if you ever have an odd number of electrons per unit cell, maybe it should say odd valent in general.
Odd valent any odd valent unit cell can be can be insulators in general.
In band theory, this should never happen. If you have an odd valence unit cell, you should have a partially filled band and so it should be metal.
So we have to ask ourselves, what did we leave out? Well, what we left out is interaction between electrons.
Interaction? Why? Why? Why?
Because we left out. We left out. Out.
Electron. Electron interaction. Electron. Electron interactions.
If these are strong, they can make the material insulator, even though by counting electrons you would think that it's a metal.
And let's try to understand roughly why that is.
So imagine you have a bunch of atoms lined up in a row like this and assume it's monovalent monovalent.
So it's one electron per unit cell. But suppose the interaction is very strong.
Suppose v coulomb de coulomb a very strong, strong.
If the Coulomb interaction is extremely strong.
Then you might have a situation where there should be no two electrons on one atom, on one, on one atom, one atom.
So never should you find two electrons on one atom, because the Coulomb interaction would make that energy extremely high and very unfavourable.
So if that's true, then what you have to have is you have to have a situation where there's one electron on every site.
Okay? One electron, one electron, one electron on every single site.
Then you ask, can these electrons move around? Can they hop from one side to the other?
Well, imagine this guy trying to hop with some hopping matrix on t to his neighbour.
Can you hop to his neighbour?
He can't because there's someone already sitting there and the Coulomb interaction is preventing you from having two electrons on one site,
high energy from having two electrons on one side because of the common direction.
So you can't hop. So what you get is effectively sometimes called a traffic jam of electrons.
Jam of electrons are more frequently known as a mott insulator after Nevill Mott, a British Nobel laureate who described it for the first time,
and roughly in order to get a mott insulator, you have to have the Coulomb much stronger than the hopping and sort of understanding why this is.
Remember. The reason electrons hop in form plane ways is because they can lower their
kinetic energy by spreading out their wave function from one side to another.
This is the usual Heisenberg uncertainty.
If you can spread out your wave function, you can use delta x, delta p, you can make delta p lower, and then you can make your kinetic energy lower.
But. If the so it wants to form it to the electrons want to spread out because of the hopping.
How much does it lower its energy? It lowers its energy about.
T if it hops from one site to another, if it's allowed to spread out its way, wave, wave function.
But if the Coulomb interaction is much, much stronger than the hopping,
then the electron would gladly sacrifice the amount of kinetic energy it's gaining by forming a plane
wave in order to prevent itself from ever sitting on a site where there's already another electron.
In which case it will set up one of these. It's a very classical picture of just one electron sitting on each site.
And, you know, quantum mechanics has to be considered at all. Says one electron in each site.
And it's an insulator because no one can move because no one wants to sit on the neighbouring site because there's an electron there.
Okay, so this effect is completely left out of band theory and so it's kind of a failure of
bang theory at the very end of the course we'll come back and analyse this again.
A second failure of band theory. Failure to failure number two is magnetism magnetism.
Such as? Or maybe I'll say feral magnetism, which we haven't defined yet.
Materials such as iron,
where the number of other spin electrons is greater than the number of down spin electrons are is not equal to the number of downstream electrons.
You can have a different number of up spin and down spin electrons.
A nonzero magnetisation. Even in the absence of external magnetic field, even when even when B equals zero.
The external equals zero, the external. And this is something that should never happen in band theory.
In band theory. You should fill up both the up electrons and the down electrons.
The same Fermi Energy. If they were filled up differently,
you could save energy by flipping over some up electrons and turning them into a down electrons and levelling out the two Fermi surfaces.
It would save you kinetic energy to do this. The reason this happens is something will come back to at the end of the course.
This is also due to interactions. Interactions plus quantum mechanics plus quantum.
And it's totally outside of what on of traditional bound theory where we're treating only playing
way wave states and energies of up electrons and down electrons should be treated on even footing.
Okay, so these are things we should just keep in the back of our head places where band theory is going to fail.
Okay. But for most materials, we're going to think about band theory as a very good way to think about it.
So we're going to go ahead, blaze on with band theory and see how much we can understand about materials by thinking in this band theory language.
So the first thing we're going to think about in band theory is optical properties,
optical properties of solids, of solids and band theory via band theory theory.
So first of all, what I mean by by optical properties, what I mean by optical properties.
Optical properties means what wavelengths of light?
What wavelength of light? Of light.
Is absorbed, absorbed, transmitted or reflected?
Transmitted or reflected.
In other words, if you put your material in white light, which is where we usually put it,
what is what what wavelengths of light actually reflect off of it and go into your eye and give that material the colour that you actually see.
So let's remind ourselves a couple of things about light. Light, visible, light.
Visible light, you might recall, ranges from wavelengths of 740 nanometres,
which we which is reddish all the way up to about 400 nanometres, which is violet.
And this is all you can see, or most people can see, at any rate.
And if you think about energies of photons, this is about 1.7 electron volt photon and this is about 3.1 electron volts for photon.
And then between red and violet is the whole rainbow spectrum. Red, orange, yellow, green, blue, indigo by the usual thing.
Now let's think about how it is that light is going to interact with this with some solid in band theory,
how this type of light will interact with our material.
So we draw some sort of band diagram. So here's E, here's K.
I'm not going to draw the whole bronze on.
I'm just going to sort of sketch some diagrams and this will go off in to all directions in case space and maybe a diagram like this.
So this will be a filled fill valence band.
So imagine you have a filled valence band here. Filled valence band.
We have a gap gap here.
E gap. And then we have an empty conduction band.
Maybe I should define what I mean by e gap generally. So e gap equals main conduction band energy.
Conduction band energy.
Minus Maxim. Minus minus Max.
Max Valence Band and energy.
So difference between this energy here and this energy here.
So if this is a big gap, it's an insulator. If it's small gap, it's a semiconductor.
The chemical potential is somewhere in the middle of the gap.
So when now let's ask, how is it that visible light is going to interact with our insulator here?
Well, some photon comes in with some energy bar omega h, bar omega.
And if that photon energy is bigger than the gap, then it can excite an electron out of the field valence band up to the empty conduction band.
It absorbs the electron and that energy is taken up by the transition between the low energy state and the high energy state.
Sorry, they say it absorbs the electron, absorbs the photon,
and the electron gets excited from the bike by jumping up from the filled valence band up to the empty conduction band.
So what this means is that an insulator in soup insulator insulator with gap e gap greater than 3.1 EV is transparent.
Why is that transparent? Well, if the gap is greater than three point 1ed, then any photon you can see,
any visible photon does not have the energy to excite an electron to the valent from the valence band to the conduction band.
Okay, that's the key point. If the photon doesn't have enough energy to excite the electron between the two bands,
then basically the photon can't interaction with the electrons at all. The other photon just has no choice,
but it could just go completely through the solid without interacting at all because it can't make any of these transitions.
So we know plenty of materials like this where the where the gap is greater than 3.1 EV and therefore they look transparent.
Diamond quartz window glass although it's not crystalline, alumina, gallium, nitride,
all these materials which are transparent, they have energy gaps greater than 3.1 ev.
Now if you have a smaller band gap, smaller gaps, smaller E gap, then light is absorbed.
Only photons. Only photons with h bar omega greater than e gap can be absorbed, can be absorbed.
So some colours are absorbed and some colours are not absorbed.
And this gives many of the many materials, the colours that we actually see.
So let me show you a couple examples. So here is down here is our our usual rainbow with the energies of the photons written on them.
1.75 up to 2.6. We out here is about three point 1dd if you look at this material.
Cinnabar It's mercury sulphide. It's gap is about to Evie And this material looks red because all the all frequencies
except for red red has a low enough frequency that it cannot make any of,
of these excitation so,
so that the material is transparent to red gets frequency is lower than e gap so it must go through whereas other frequencies are just absorbed.
So the material looks red because the light coming through that material is red.
All the other all the other light frequencies have been absorbed.
If you look at this material we algar, it has a slightly larger gap.
2.4 EV Well, all the way out here and it looks a little bit more orange because the red orange,
the yellow and even some of the green gets through the material and all the higher frequencies,
the blues in the violets do not get through the material. They get absorbed.
So this actually ends up looking kind of orange.
Now you might say, well, how come it doesn't look a little bit more green than orange, but it's a little bit more complicated.
You have to sort of average over over all these colours that get through.
So the middle of the colours that get through is somewhere around orange. And plus there is there can be some other effects going on.
But roughly the reason this looks orange is because the blues and the violets have been absorbed and the colours
that actually go through without making any excitations and without getting absorbed are the reds and the oranges.
Similarly, sulphur has an even larger gap, still 2.6, and now the the colour that you see is pushed even further out.
It looks more yellow because the reds, the oranges, yellows, some of the greens and even a little bit of the blues get through that material.
And so it sort of on average looks kind of yellowish. The violets have not gotten through that.
They've been absorbed. So this gives many materials, their colours.
The story is a little bit more complicated than I've told you. Because the structure,
the actual details of the band structure can matter a lot as to how strongly something is absorbed or not absorbed in the most important issue.
And let me draw again here is going to draw a random band structure.
So here's a filled valence band, part of a filled valence band.
Now you can have a conduction band here that looks kind of like this.
I think that. And so you can have a situation where you have a where you can make a transition from here to here.
For every different case, this is known as an indirect transition or an indirect gap gap.
Maybe I'll write down the definition over here.
So an indirect gap or indirect transition? Indirect gap equals gap between different case.
And then there you can have a direct transition like this.
This is known as the Direct Gap. Direct gap.
I've also drawn a direct gap over here, which is a gap between the same case.
Direct. Gap equals gap between same case and same case.
And it turns out to matter a whole lot.
Whereas for its interaction with light, whether the gap is direct or indirect, and let's think about why why this is.
Well, let's recall that light frequency of light is C times K and C is huge.
We've run into this problem before. So if we're trying to conserve both energy and momentum by absorbing a photon.
The photons for any reasonable frequency. That's not like x ray frequency for any light frequency.
If you put in a light frequency here, a k is tiny.
So when when a photon is absorbed, the wilczek, if you're conserving momentum, the electron basically has to go straight up.
It can't if the photons do not carry large amounts of momentum.
They only carry tiny amounts of momentum because the speed of light is so large.
Is that clear? Obvious? Yes. Okay. Good.
Thank you. Thank you for answering. Should even give you a chocolate for answering.
Okay. So. So what we have is that a direct, direct gap?
Transitions have strong light absorption.
Strong light absorption. Whereas.
Indirect gap transitions. Indirect transitions.
Are very weak absorption. Now you might wonder.
Weak absorption. Why it is we get any absorption at all with an indirect gap transition.
Okay. There's a couple of of reasons you might think if you have to conserve energy and momentum,
if you're if you can't conserve energy momentum, there should be no absorption at all.
But in fact, there's more complicated ways where you can conserve energy of momentum.
For example, you can get one photon in.
Photon in. Those two one electronic sided plus one phone on excited.
And if you do this, then the momentum change of the electron can cancel the momentum of the phone on.
You have the electron going off one way and the phone on going off the other way. And you can still conserve both energy and momentum.
But this is a much more complicated process and it occurs much more rarely than just the straight, direct gap absorption.
Another way that you can get weak absorption is imperfect, imperfect momentum conservation.
K Conservation. Now why is it that you should conserve k only imperfectly?
Remember that the reason that we the k this crystal momentum is conserved in the first place,
that all conservation laws come from symmetries and the symmetry that that forces us to have.
K Conservation is translational symmetry by a lattice vector.
So that assumes that you have a perfect crystal. So in a perfect crystal K is absolutely conserved.
But if there's disorder, this is free from disorder or impurities disorder or impurities.
From disorder and impurities in K is not perfectly conserved anymore because the system isn't translational invariant anymore.
So the K will be not perfectly conservative. You'll get some amount of of absorption via the indirect gap.
So the actual physical absorption as a function of energy.
So H for omega here here is absorption. This is going to be a very cartoon, right.
Cartoon on it. Are that when you get to the frequency, which is the indirect frequency,
if there is an indirect transmission frequency, it could be the case that the lowest energy excitation is direct.
In which case, you know, this just occurs when you when you get up to that up to that frequency and at any higher frequency,
you'll get this plus plus other things as well.
But if the indirect like that picture there, if the indirect gap frequency is lower, you'll get weak absorption at the indirect frequency weak.
But when you get up to the direct gap frequency, direct gap frequency below the indirect frequency, you get no absorption.
Then when you get to the direct gap frequency, you get strong absorption when you get above this direct gap frequency.
So this is sort of a cartoon of the absorption is a function of frequency when there's an indirect gap and a direct gap.
So let's actually look at real data. The real data isn't quite so nice, but you can actually you can see a little bit what's going on.
This is silicon.
Silicon has an indirect gap and a direct gap with an indirect frequency which is lower than the direct gap frequency like I've drawn up there.
And the energy is increasing as you go to the left.
Wavelength is getting smaller as you go to the left and you see that the absorption actually keeps increasing as you change the frequency.
But when you get to the indirect gap, there's a little bit of a glitch. Now, this looks like a small glitch, but it's a logarithmic plot.
So if you plotted linearly, it would actually be a fairly big step then.
But it keeps increasing, increasing, increasing, increasing. But then when you get to the direct gap frequency, there's a really big glitch.
Again, this is more or less a factor of ten from here to here.
And, you know, so you get a big step when you get to the direct frequency.
There's lots of other processes that give you weak absorption everywhere else,
more complicated processes involving phonons and evolving other stuff too.
So it's not quite as simple as I've drawn here, very figuratively, but at least you get the general idea of of the those two pieces of physics.
Okay, so the physics and in metals you've studied metals a little bit before, probably in your in M courses they appear shiny and reflective.
And the reason they appear shiny and reflective is because they have high conductivity, high conductivity.
And I think you probably studied this in your electromagnetism courses by metals reflect.
Did you do that? Yes, yes. Hopefully. Okay. So really what you're asking about is the conductivity as a function of Omega.
And from that you can get optical properties. Optical properties.
But again, you know, we have. Let me draw.
You have some bronze on here. Here's here's a medal. It's a battle that has slowed down.
So we might have a medal like this. Now, the optical properties at some high frequency will depend a lot on where the second band is.
If you can make transitions up to a second band, that will change Sigma over Omega.
Even though at low frequency this thing may be a very good metal and the electrons will move very freely.
If you're thinking about the conductivity at high frequency,
you do have to worry about whether the transition lines up with an inter band transition or it doesn't line up with an intra band transition.
Whether you can make these transitions or not will affect the optical properties.
And it's actually the detailed band structure that makes copper the copper gold,
the gold and silver look silver that has to do with which transitions can be made in between bands and in those different materials.
Won't go into that in too much more detail.
Now, in any of these materials, it's always important to realise that details matter matter for optical properties.
Actually, details matter for many things, so I'll just write that down. Details matter.
Sort of a general principle to live by one. One detail that matters a lot is the surface.
When you look at the material, you see the surface of the material.
And very frequently the surface of the material is not like the rest of the material for one reason or another.
A really good example of that is a material like sodium.
Sodium is a really good metal. But if you look at its surface, here's a chunk of sodium and the surface of sodium looks kind of grey.
It's not reflective. Why is that? Well, because you're not actually looking at sodium.
You're looking at sodium oxide. You put the sodium out on the table within about an hour, the oxygen in the air has attacked.
The surface of the sodium is formed. Sodium oxide. Sodium oxide is an insulator.
So you're actually seeing in insula a thin layer of insulator on the surface.
In this picture, the sodium has been cut in half to open up a metallic surface.
And so for about an hour, that surface will look metallic.
And then after an hour, it will look grey again because you're forming an oxide layer on top of it.
So the surfaces matter an awful lot. Frequently the surfaces are not like the rest of the material.
Another thing that matters an awful lot is impurities.
And we'll discuss impurities a lot in next lecture as well. Maybe at the end of this lecture if we have time.
But for the optical properties, let's imagine we have, okay, here's a empty conduction band.
Empty conduction band.
And then here we have some filled valence band filled valence band.
And that's what the our insulator or semiconductor would look like in the absence of impurities.
But if I add some foreign impurities into the material,
it can actually add eigen states in the middle of the gap at energies that are somewhere in the middle of the gap.
So these are impurity ion states ise due to the presence of the impurities and the absence of those impurities.
This material might be transparent if it has a big gap, but when the impurities are there,
then you can make some number of transitions of some frequency up to the impurity and back so that you can get certain colours.
So certain colours sort of a, you know, modifying the transparency of your of your material.
So even with a weak, very small number of impurities, this can make a huge difference.
So here's a classic example is a material diamond which everyone's familiar with.
It's diamonds are a girl's best friend or whatever the phrases. Isn't that a James Bond movie or something?
Anyway, Diamonds are forever. What was the James one area in their mind? So?
So with diamonds pure, a diamond is a is completely clear.
Whereas if you add just a couple of impurities, this is one boron impurity per 10 million carbon atoms, and it turns the diamond somewhat blue.
This is a rather famous blue diamond. It's the hope diamond estimated value of $250 million.
It's sitting in the Smithsonian Museum in Washington. It's also supposed to have a curse associated with it.
So don't go try to steal it. It's very bad for you or something.
Anyway, if you if you add a nitrogen atom from 1 million carbons, it turns the diamond a rather yellow colour.
This is the Tiffany Diamond, also highly valuable, owned by the Tiffany Company.
I believe it's in their store in New York. Estimated value.
Well, it's probably a lot more than 12 million now. It's a rather large, beautiful diamond and it's coming entirely.
The colour of the diamond isn't coming entirely from impurities. Now, these these diamonds were actually mined.
They were dug out of the ground, out of out of mines.
But in fact, in the modern era, we're very good at making materials synthetically in laboratories.
You can make a perfectly good diamond with no impurities whatsoever in your laboratory.
There's known as synthetic diamonds. They're a lot cheaper than mining them out of the ground.
And if you want it to be blue, you just add boron.
If you want it to be yellow, you add nitrogen. If you want to see some other colours, you can you can make it pink, you can make it,
you can make it whatever colour you want by adding the appropriate impurities that put it in the right purity.
Iron States. The advantage of the synthetic diamonds are well, they're cheaper, they're actually more.
There have fewer. You can design the impurities, you can put whatever impurity you want in it.
You can keep the impurities out so it can be as pure as you want. And in fact, if you buy synthetic diamonds, you do not support any wars in Africa.
So if you're out, if you're planning on buying diamonds, you know, please, you know,
somehow that the diamond industry wants you to believe that somehow the synthetic
diamonds are inferior to the to the natural diamonds dug out of a mine.
It's totally not true. It's the same carbon. In fact, the carbon that you grow in your lab is probably a lot purer than the stuff you pull out of.
Pull out of the mine. So. So buy synthetic diamonds. I don't own stock and said it's inside diamonds.
But full disclosure, I don't.
Okay, so let's go on and start talking about what happens once you once you make excitations of electrons in materials.
So so again here is energy will draw the top of a here's the top of a valence band filled valence band top of valence band valence band filled.
And the bottom of a conduction band I've drawn. This is a direct gap material, but it could be indirect.
It's not so, so crucial and it's empty conduction.
Anyway. Okay, so there's some gap energy in between the two.
And if I put in a photon in order to excite an electron out of the conduction band, out of the dance band of the conduction band,
I then get an electron up here in the conduction band, but I get the absence of electron down here in the Valence Band.
And that is an important thing to have a name for, which is known as a hole.
Also the name of a good rock band. Absence of of.
Of electron. Electron in otherwise.
Wise filled valence band.
So when you put in this energy to excite the electrons up to the conduction band,
it could be photon or it could be actually thermal energy too can excite.
If the gap is small enough so then your temperature is high enough,
you can thermally excite an electron from the valence band up to the conduction band.
Then what you've done is you actually pair created an electron in a hole.
It's entirely analogous to a positron. Analogous to analogous.
Analogous to positron. We're in outer space, you know, in the vacuum of outer space,
if you put in a high enough frequency photon, you could pair create an electron positron pair,
you'd have to put in the 2mc squared energy, the mass energy of the electron in the positron, you could create them.
And at some time later the electron in the positron could come back together and pair annihilate.
And the minute photon are two photons as the case may be.
And the same thing here. You have to put in the gap energy and you pair create an electron and a hole
and these things run around in the in the conduction band or the valence band.
And at some time later they can come back together and emit their energy in photons or in
some other way they can get rid of their energy and they come back and they re annihilate.
So it's entirely analogous to the idea of an electron positron pair.
In fact, when Dirac wrote down his theory of positrons, he was well aware of what was going on in semiconductors.
And he was using this as as more or less a model of the picture of positrons.
So this is one way you can have you can arrange to get holes in the valence band or electrons in the conduction band by adding
energy either either light energy or or thermal energy to excite someone up from the valence band into the conduction band.
There's another way you could get holes in your valence band and electrons in your conduction band, which is to get the bands to overlap.
See if I've drawn this right. So here's my otherwise filled valence band.
If the bands overlap by a little bit, here's F, then these states will be filled in the valence band.
But the states on the top of the Valence Band will unfilled and fill the lowest states in the conduction band.
And this is exactly what we saw in the case of calcium earlier today,
that the the lowest band sort of is not quite filled and the second band is just starting to fill.
And so this becomes a little bit metallic because you have some holes in the valence band and some electrons in the conduction band.
Now actually some useful nomenclature, which is the density of holes in a system, density of well, maybe density of electrons.
Electrons in conduction band in conduction band.
It's usually called MN and it's called MN actually for not for a number,
but for negative charge and for negative because the electrons have negative charge.
Whereas the density of holes, density of holes in valence band equals p p for positive charge.
That's the usual nomenclature. And whether we create our electrons in the conduction band, in our holes,
in our valence band by adding energy like this or by overlapping the bands like that, it appears that any goals.
P So maybe write that down does an equal.
P Generally, because it's every time I created an electron, I created a hole.
And in that picture up there, every time I transferred one electron from one band to the other,
I created both electron in the connection band and a hole in the Valence band.
So it looks like the answer is yes, but it's going to be sort of a qualified yes.
Yes, if in pure material.
No, no, with impurities.
So we have to think about how it is that impurities change the story here.
So let's consider the world's most important semiconductor silicon.
Silicon is what all electronics, more or less, are made out of.
So I'm going to draw it as a square lattice, even though it's not a square large enough to see last with the basis.
But this will, just for figurative purposes, will draw it as a square lattice here.
These are all silicon atoms. Then we're going to imagine taking one silicon atom and replacing it with another atom.
Which atom are we going to use? We'll use something right next to Silicon on the periodic table.
Will you take phosphorus now? So we'll take this atom here and make it into a phosphorus atom fast.
I'm going to write out fast so it doesn't get confused with P.
It's unfortunate that the symbol for phosphorus is P and P is also the symbol for the density of holes.
Those are unrelated. It's just sort of random happens to be inconvenient, but we have to deal with it.
So phosphorus equals it's basically it's a silicon atom plus one proton.
It's on the left. It's on the right of silicon in the periodic table plus one electron plus one electron.
It has some neutrons to which we're not so concerned about.
So I can take this phosphorus and view it as a silicon plus an additional positive charge plus one additional electron.
Okay. So it has an additional positive charge here. And one electron sitting up there, one e sitting somewhere else.
Now, what happens to that one electron? This one, too. Okay, well, maybe I should talk about the proton first.
So this proton turns out, and we'll justify this in the next lecture.
You can more or less ignore the presence of that proton and just think of it as a silicon nucleus.
Yes, it has a charge, but that charge is not so important. What's important is this electron here.
Where does it go? Well, the valence band is already filled.
We started with silicon. It had a filled valence band. Pure Silicon is a Phil Downes band and a gap and then an empty conduction band.
So this one electron must go into conduction band.
So here we have added an electron into the conduction band without adding a hole into the valence band.
Let me let me show you a better cartoon of this. So in this picture, the electrons are drawn as little circles around the silicon atoms.
And the fact that there are two electrons next to each other is supposed to depict a covalent bond that holds the two silicon atoms together.
The fact that all everywhere, except for where the phosphorus is,
all the electrons are tied up in these covalent bonds is another way of saying that we have a filled band.
All the electrons are accounted for and there's a gap to putting the electrons anywhere else.
Now, when you put in the phosphorus. The phosphorus can make the same for covalent bonds.
But then there's one electron left over.
And since the entire valence band is filled, the only place that electron can go is up into the conduction band.
Okay, so that's what happens. You can do the same and play the same game with a different doping.
Let's back up to doping. On the other side of silicon is aluminium.
We could use aluminium, aluminium, which is silicon minus one proton.
And again, we don't have to worry about the proton too much. We can ignore this. We'll justify that later.
Ignore. And then it's minus one electron. So with aluminium, we're missing one electron that we need to fill the valence band.
So this adds a whole adds whole as a whole.
And actually let me drop okay here that using boron not aluminium boron is right above aluminium in the periodic table.
So it's chemically exactly the same. So it's the same story.
Boron is trivalent. It can make three covalent bonds, but there's one electron missing in order to fill the entire valence band.
So we're missing one electron. Let me just give a couple terms of nominal.
I'm sorry. I'm going over before we end. So phosphorus is known as a known as a N doping because it adds an N and a negative
charge or a donor impurity or a donor impurity because it donates an electron,
whereas aluminium or boron is known as as PE doping because it adds a hole or an
acceptor impurity because it accepts or acceptor impurity accepts an external electron.
And maybe I should define the word dope end before ending.
So dope and doping is a general word, which means the a foreign chemical.
Foreign chemical introduced introduced into an otherwise pure to otherwise pure compound, otherwise pure pure material or pure substance.
I guess. To change its properties. To change its properties.
Rather general word. Of particular importance these days, considering the Olympics are going on.
Okay. So I guess we'll end there and we'll start again tomorrow.
Sorry for going over.