Okay. Welcome back. This is now the eighth lecture of the condensed matter course.
One of the more wonderful things about quantum mechanics is that particles are waves and waves are particles.
And that means the things that we learn when we study vibrational waves in solids can often be applied to other types of waves in solids,
such as electron waves or even electromagnetic waves.
So today we're going to be studying electron waves in solids, and this lecture is a little bit out of place because later on in the course,
we're going to spend several days studying electron band structure, electron waves and solids in some amount of depth.
But the reason I am inserting this lecture here is to make the point that we're really studying the same thing.
Whether we're studying vibrational waves or we're studying electron electron waves, it's really very, very similar.
We're also going to see that much of the calculation we do today is similar to what we did when we studied the covalent bond,
which was a wave just between two atoms. So the picture we're going to look at today is a one dimensional, one dimensional, tight, binding chain.
And it's going to be very analogous to the one dimensional vibrational change that we looked at in the last couple of lectures,
also similar to the covalent bond. So we're going to imagine having a bunch of nuclei in a chain like this,
and it will give them a lattice constant, a distance between identical nuclei.
And we're going to add one electron to this chain of nuclei,
and we're going to see what happens as we allow the electron to hop back and forth between the different nuclei.
So of course, we have to start with the Hamiltonian as the usual piece squared over to term.
Plus, it will have an interaction with all of the different nuclei of our minus our sub j where our sub j is position, position of nucleus.
J And as I did when I when we studied the covalent bond, we're going to abbreviate these terms for convenience.
This term will be called K for kinetic energy and these terms will be called V sub J for interaction with the nucleus.
Now, similar to what we did when we started the covalent bond,
it's useful to think first about an electron only interacting with a single nuclei, not with any of the other nuclei.
So we'll write h oops or i k plus v some m as our Hamiltonian.
And that means that the electron has this kinetic energy and it's interacting with the nucleus only.
And we'll give it the atomic when it interacts with the nucleus.
M only. So the eigen state of the Hamiltonian, which has it interacting with the nucleus,
we'll call that the cat M and that will put the electron on the end of the nucleus, as if all of the other nuclei were not there at all.
So a label this cat one if the electron is sitting here as if it's not interacting with any of the other nuclei,
we'll call this one to this one, three and so forth.
Okay. Happy so far. Okay.
All right. Now, as we did when we started the covalent bond, we're going to make a bad assumption,
that assumption, which is that these cats and an m are or the normal.
This is not too bad if the nuclei are far apart from each other because an electron sitting over here,
an electron sitting way over there, are pretty much orthogonal to each other.
But when the nuclei get close together, then they're not orthogonal anymore.
And the reason we make this approximation, even those of an approximation, is for simplicity.
A lot of the calculation just gets a lot easier if we make assumption and we don't learn a whole lot more from doing more properly.
It is not that much harder to do it properly. There's an exercise in the book that walks you through it and you can go through it if you want,
but you'll get most of the interesting physics out of just this simplified approximation.
Okay. So once we have made this that assumption, we can write down our trial wave function,
trial wave function very similar to what we did with the covalent bond, which will have the form psi equals sum over n phi n n.
A linear combination of atomic orbitals,
which is a word we used equivalent to type binding linear combination of atomic orbitals, of atomic orbitals or LCA.
Oh, we're making, we have a bunch of atomic orbitals electrons sitting on nuclear.
So we're going to make a linear combination of them with coefficients vice some n.
The reason people love this type of approximation is because you can make it more and more accurate
by just adding more things to the right hand side with variational parameters phi in front of them.
So for example, you could have an electron sitting on site n in excited state alpha.
So this could be 1s2 as to P, so forth and so on.
And we can just make our bases state bigger and bigger. Bases get bigger and bigger and bigger and give all of these coefficients.
And as we make the bases set bigger and bigger and bigger and bigger, we get a more and more accurate approximation of the true wave function.
Okay, this is sort of a variational approach so we can solve for the ground state by finding the best r coefficients.
Then once we have the ground state, we can solve for the first excited state by finding the best coefficients,
the lowest energy coefficients, subject to being orthogonal to the first thing that we just found and so forth and so on.
So now what we have to do is we have to write down an equation for these coefficients.
And the equation, again, something that you will solve for in your your homework will be of the form h and am phi m equals e phi n and you know,
it looks like a shorter equation. It quacks like a Schrodinger equation.
It probably is a shortening equation. It's not the real Schrodinger equation because the real equation has a full Hamiltonian sitting here, here.
Instead we just have a matrix and h m and you can think of this as being the projection of the true Schroder
equation onto the basis said made up of these cats and that we that we are working with does that make sense.
It looks like a Schrödinger equation. It's going to act like a shorthand equation and you'll derive it also.
Okay. So given our an equation over here, we have to now calculate these matrix elements that go into our ordering equation.
So let's do that. It is useful to take our Hamiltonian and divide it up into pieces.
So first we will remove the m the interaction with the nucleus from the interaction with all the other nuclei.
So this will be j not equal to m the sum j.
The reason we do this is because then we can write h on the cat m equals k plus we m on the cat.
M plus the interaction with all the other nuclei.
The sub j and cat. M. Okay. Now we defined the cat m to be the eigen state of k plus v m.
So this thing here is just atomic on cat m.
Good. Happy with that. So then we can take the inner product,
close up the inner product with the head and over here and we get the atomic delta and m plus and some over j not equal to m v some j m.
And this is the interesting term here. So this term here just tells us that no matter what side the electron is sitting on,
which nucleus, the electron sitting sitting on it has energy, atomic.
And this is all the interaction with all of the other nuclei. Now, there's a couple of things that can happen with this term.
One possibility is that any equals M, in which case this is what this is telling you,
is that there is some change in its energy sitting on Nucleus M due to its interaction with all of the other atoms, not M okay.
So this will be will give this thing an energy. We'll call it V, not equals R and some big j, not equal to M.
And I guess these can be ends like that. And this is so this is interacting with all the nuclei, not including M and its expectation of its energy.
So it's just shifting its energy on that particular site. That's not particularly interesting.
The more interesting thing is what happens is and not equal to M.
So in this case you have a this term is maybe I should call this term something first.
This is what we call direct before and I guess I call it V Cross when we talked about the covalent bond for the energy equal to M term,
this is what we call hopping before. And the reason we call it hopping was because it will give it the value minus T, minus T.
The reason we call it hopping is if you think in the time dependent Schrödinger equation type of way,
you can take an electron sitting inside M and have it end up onsite rn.
So this off diagonal term in the Hamiltonian allows an electron to move from one side to another.
Hence we call it hopping. Now an approximation, which is actually a fairly good approximation.
Is that n minus M greater than one hopping is zero, hopping equals zero or is approximately zero.
And the reason for that is because it's very hard for an electron to hop very far in one step.
If you think about it for a second,
what we're really calculating when we take this matrix columns is some sort of when we write out the matrix element explicitly,
it's something like this. Right.
This is what matrix elements look like. A bra, a cat, and an interaction.
Now, if I n if an m are far apart, these wave functions decay very quickly as you go away from the nucleus.
So there there'll be no point in space where both this is large and this is large.
If the two nuclei are very far apart.
So that's why we can we can assume that this matrix element is going to be zero in less and an m are essentially neighbours.
Okay, good. So what we have now is we have this a in the end we have that we can write and sum over j not equal to m v sub j.
M equals the direct term v not if and equals m we'll call it minus t if n equals m plus or minus one.
So we can have one side only and zero. Otherwise.
Okay. Good people happy. Fairly happy with that.
Ken? Yes. Someone had someone say. Someone say yes.
Yes. Thank you. I would send you another chocolate, but I gave you one yesterday, so.
Okay. Good. So we can take a Hamiltonian and rewrite it as h as a big matrix.
H&M equals. What is it?
It's atomic. Plus the interaction with all the other nuclei, not including the site itself.
If you're sitting on one site interacting with all the other ones, not including yourself.
Plus there's going to be another term which delta and and plus one plus delta and comma and minus one.
So there's this additional term here which allows you to hop one step to the left or one step to the right.
With an amplitude. A t. Okay.
Good. So this is a great big matrix.
If we have let's say we have NW and nuclei to begin with, then H is an end by and matrix and an end by an Hamiltonian matrix.
And we need to find its eigenvalues. So how do we do that?
That looks like a complicated problem if and is a is a pretty large number.
Well, again, we can solve this very similar to what we did for the vibrational chains we use.
And on that, which is an English word. And that's.
And the answers we use is via is a plane way vanguards I k and a like this.
Now a couple of comments about this. First of all, you may be expecting an the Iomega T from what we did with the vibrational chain.
The reason there's no Iomega t is because we're solving the time independent shortening equation, not the time dependent shortening equation.
If we were solving the time dependent shortening equation, there'd be an either the Iomega t as well.
Okay. That's why it's not there. In this case.
It's just simpler in quantum mechanics to work with time independent agencies than it is to work with time dependent wave functions.
So that's the first thing.
Second thing is that you probably if you're careful, you put now a square root of capital N downstairs so that this is a normalised wave function.
The normalisation is not going to matter much for us, but strictly speaking, it should probably be there.
And the third thing to note is that this this wave is the same.
If you shift K to K plus to pi over A is something we discovered last time.
The thing that is important is not the momentum but the crystal momentum.
If you're shifting k by two pi over a, get back exactly the same the same wave.
Okay, so let's take our and plug it into that Hamiltonian one step here we get epsilon,
not even minus i k and a minus t into the minus i k and plus one a plus e to the minus i k and minus one a equals E in
the minus i k and a So it's just plugging the on that into the shortening equation using that form of the Hamiltonian.
Okay. So it's this is the oops. I didn't tell you what this I didn't tell you what it is.
Sorry about that. This thing here I called in not.
So this is the energy on site. This allows you to hot to let hop to the right.
And this is the eigen energy on the other side. People happy.
Okay, good. Thank you. Good.
So then you just cancel out a bunch of factors, a bunch of exponential factors, and you get E is e, not minus two t cosine k.
So let's actually plot that. You go?
So here's pi over a year, minus pi over a year.
This is energy on this axis. Energy.
This is not. And then we'll have a nice cosine form that looks kind of like this looks kind of like this.
And if you wanted to, you can continue it out periodically further.
But what we're really interested in is the ways within the Bruins zone, one zone from here to here,
because once we go outside of the Brown Zone, we're just reproducing the same waves over and over again.
And if we want different waves, we have to be all the ways within the bronze zone in different from each other.
But once we go outside of the bronze zone, we start repeating things that we've already considered.
Okay. Just a bit of nomenclature, which is which is fairly useful.
May I put it over here? An energy band. Energy.
Banned means one case, one iron state at each.
Each kick in the bronze arm.
So here we've drawn an energy band.
Now, you'll remember this looks a little bit like what we got when we.
When we solved the monotonic harmonic chain.
The vibrations of a single chain of where every atom is exactly the same and every spring is exactly the same.
I last lecture when we when we solved the diatomic chain or the alternating chain, we found that there were two branches of excitations.
There were two normal modes at each k vector K in this direction.
If we had something similar in this picture where there are two eigen states, a low energy one and a high energy one,
we would say that there are two energy bands aligned, one in a high energy band.
We do not use the words acoustic and optical when we're talking about electrons for reasons that will become clear in a moment.
But it is useful to compare this this dispersion to what we got for vibrations, vibrations,
the atomic chain one atomic we had omega squared equals R I guess it was two kappa
over M minus two kappa over m cosine k and I believe it was equivalent to that.
You'll see that the two dispersions look awfully similar, but there is a notable difference.
And the notable difference is that with the shorter equation,
we got e on the left hand side and with the vibrational chain we had omega squared on the left hand side.
The origin of this difference is that the shorter equation, if you think about the time dependent shortening equation,
it has one time derivative, whereas Newton's equations F equals May, the acceleration is two time derivatives.
So that's why we're getting two frequencies over the left hand side, but one only one energy on the right hand side.
When we're thinking about Schrodinger, that actually makes a rather important distinction between the two.
When we had the vibrational chain at low energy, we got linear modes coming in down to zero frequency at zero wave vector in this picture here.
In the in the electrons.
In the case of electrons, if you expand near zero wave vector, it's not minus two T and then low substituted for small wave vector k a squared
over two like this we see that we get to see here what we get is constant,
which I'm not interested in plus k.
A. Squared times t. It is quadratic at low energy, as you would expect from a cosine.
And that is why we do not use the word acoustic for these low energy modes for vibrations.
Acoustic modes, by definition, sound modes have linear dispersion that the frequency should be proportional to wave vector.
That's the definition of sound. So here is quadratic.
So we don't call it acoustic and we don't call it optical either if there is higher energy brand branches.
But this this picture of the energy being quadratic in wave vector should be fairly familiar to you from other contexts.
For example, if you have a free electron free electrons, an electron in outer space flying around or something,
it's energy is some constant plus h bar squared k squared over two m.
K squared over to our right. It is also quadratic in in k.
You might choose C to be zero if you wanted to, or you could choose it to be M.C. squared if you're thinking about it's mass energy as well.
But the important thing is that it's quadratic and in a way vector the same as this electron over here.
So it might be useful for us to sort of think in terms of free electrons for a second.
And so what we do is we define an effective mass.
Effective mass and the star such that h squared over two and a star equals A squared times t.
With this definition. Then for our energy band, for an energy band, for electrons in our energy band, energy band,
we still have energy equals a constant, same constant plus h bar squared k squared over two m star.
It looks just like free electrons is still quadratic in k, at least a small k,
but now it's an effective mass m star rather than the real physical mass of the electron.
M okay, now this. You should really think about this for a second.
The effective mass that we're getting here in this tight binding chain that we're solving has nothing to do with the actual mass of the electron.
It actually has to do with the hopping between nuclei for greater hopping.
The mass gets smaller for smaller hopping, the mass gets larger.
I think. And so it has.
It's totally unrelated to the to the actual physical mass of the electron.
You will find sample systems, actual physical systems, metals or whatnot,
where the mass is 100 times less than the physical mass of the electron or 100 or a thousand times more than the physical mass of the electron.
Although it's not uncommon to have effective masses in real materials which are on the order of the actual mass of the physical electron.
The other thing to keep in mind here is that the K we're talking about in the energy band is not momentum, but is crystal momentum.
Crystal momentum or.
Or Bar-Kays Christian momentum if you put the H brian which means only that we are only defining our momentum modulo the crystal wave vector,
the reciprocal lattice wave vector to pi over a If I shifted everything by two pi over a, I get back the same physical wave.
So it is slightly different from the free electron waves that were that we are familiar with.
The last thing that I want to emphasise here are the Eigen states.
I can states our waves, our plane waves are just waves.
Now why is that interesting?
It means that I can take an electron and if I put it in some state here, it's sort of a left, a right moving wave at right moving wave.
And in that right moving wave extends clear across the system is in the IK an our form of the wave which I guess scrolled off the top of the board.
But it's a plane way that goes clear across the system and that might be surprising for a second.
Y Well, remember, when we when we studied Sommerfeld theory,
there was this issue that the scattering length of electrons in SA seemed unreasonably long.
You have all these nuclei with positive charges and they're all over the place.
Every few angstroms you run into a nucleus with a positive charge,
and yet the electron has a scattering light that can be 100 angstroms, 1000 angstroms or a million angstroms long.
Now here we have a model where we have lots of nuclei lined up periodically, one after the other, after the other, after the other.
And the electron hops from one to the next to the next next and makes a wave that goes clear across the system.
No scattering at all. You would observe this as a perfectly good wave packet going entirely across the system without back scattering one bit.
So that's our first surprising result, and we're going to come back to that result very frequently later on in the term.
Is everyone fairly happy with that? It is, yes. Well, there's lots of other things that we haven't accounted for.
For example, I can we haven't talked about it. It could scatter into other electrons.
So that's actually in fact, that's something we're not going to discuss all term,
because the reason you can ignore scattering against other electrons is actually extremely subtle and you probably won't even learn it next year.
It wasn't understood properly until probably the 1980s or even later.
And it was it wasn't understood at all. Not even one little bit until the 1950s.
So that's a really tough question. But there's other things you can scatter.
There's impurities. So crystals aren't perfect. There's other junk in it and there's vibrations.
So you can have an electron moving along and they can hit a vibrational a phonon and scatter off of a phone on as well.
So to find a temperature, there's electron phonon scattering is frequently the limiting process.
Okay, good, good question. I give you another chocolate, but you already have one.
Okay. So we're going to do an exercise we did before counting counting states.
We did this for when we counted normal nodes in our vibrational chain.
So. Well, we have n nuclei. Nuclei we have.
So that means the length of our system is and times a and if the length of the system is in times a,
the case we are allowed must be of the form two pi over l times p, where p p is an integer.
An integer. You guys are familiar with that, that notation integer.
Yeah. Okay, good. P is integer. So the number of different keys, different case is equal to the range of k's that are different,
which is two pi over a the branch zone from here to here divided by the spacing between the adjacent case,
which is two pi over L which is l over a which is n not surprising.
The number of different k's were allowed in the brian zone is equal to the number of unit cells in the entire system.
In this case, it's easy to understand that in a different way.
We started with a bunch of cats.
M They were end of them and enemies and orbitals.
These were sitting on our nuclei in nucleus one, nucleus two, nucleus three and so forth.
And then we diagnosed our Hamiltonian and we get iron states.
K And there are unsurprisingly there are end of them and eigen states.
We had an M and Big N by a big A matrix and we analysed it.
We got eigenvalues and Eigen states. Okay, so you put in end state, you get out and states.
One other piece of nomenclature which is fairly useful here, which is this energy scale here,
which I guess for our last caller it off the top, it's actually 40.
C where was it? Yeah.
So it's a two times the cosine. The cosine two times the cosine is the, the range and cosine goes from plus 1 to -1.
So the total range from Emacs is Emacs here as in in here.
I'm in here. So. Emacs.
Emacs minus im in is known as the bandwidth.
With which in this case is 40.
Okay. Now, what do we what what is the bandwidth depend on?
It depends on how much hopping you have. The more hopping you have, the bigger the bandwidth.
So let's actually draw a little picture here.
What does the bandwidth actually depend on? What is what is t depend on?
Well, the thing we had at our disposal,
the thing we could change was the lattice constant a the distance between the nuclei as the distance between the nuclei goes up,
the hopping naturally goes down. It's harder to hop over a longer distance.
So t goes up, goes up, going this way.
Good people happy with that. Okay. And then energy is going to be this way.
And then we can draw a picture that kind of looks like this. We start with a bunch of atoms, all having eigen states with energy e atomic.
When the atoms are very far apart, they all have the same, the same energy.
The electron can sit here, it can sit here, it can sit on any one of the atoms and its energy is always e atomic.
And there are and eigen states, and I guess it can.
Sit on any one of the N atoms. Now, as we bring the atoms closer together, as we bring nuclei closer together.
T is going to increase and we're going to get a band that goes from Imean to Emacs here.
And there will be eigen states within this range between Imen and iMac.
So you pick any energy between Imean in Emacs and there will be some eigen states there.
So any energy between here and here, there will always be some k which has an state at that, at that energy.
Now, this will actually look a little bit familiar. From.
From when we studied the covalent bond. When we studied the covalent bond, we had two atoms with energy.
When the electrons sat on one, nuclear energy was inside. And when we sat on the other nucleus, this energy was inside.
Then we brought them together and we got an anti bonding orbital anti bond,
and we got a bonding orbital bond down here, one lower and one higher than the original energies.
And that's exactly what we have over here. We have a whole bunch of orbitals, all with the same energy.
We bring them together,
we allow the electron to hop back and forth and some of the energies go down and some of the energies go up and it spreads out into into a bath.
Could that make sense? Yes, hopefully.
So the spreading into a band is coming from the the hopping of the electron back and forth.
So. Right. So let's imagine now that we have not just a single electron that we want to consider,
but we have many electrons that we want to consider since we had any possible case dates.
Case dates. But each case date, each case can have two spins.
It can have spin up or spin down.
That means there's two. And total electrons can fit can fit in band.
So we fill up all the states. You can fill them up with either spin up or spin down.
So we can have a total of two n electrons fitting in that band.
So let's first consider a we are going to put this maybe here monovalent monovalent atom.
Which has one electron per nucleus.
So that will give us a total of nine electrons total and electrons total and that half fills the band, half filled band.
Okay. So if we have an electron's total or one electron per atom, we will fill up the band.
Halfway. Up to here.
And this is filled now with both spin up and spin down electrons.
Right now, this picture of a half filled band, half filled down.
It has some interesting properties. The first thing.
One, it has a Fermi surface has a Fermi surface surface.
With that Fermi surface, Fermi surface is this point here where the filled states meet the empty states.
And this point here where the fill states meet the empty state.
So there are just two.
Since it's one dimensional, it's actually just two Fermi points, two points where where you have the highest energy filled state.
And that is an empty state nearby because since it has a Fermi surface,
it is possible to make low energy excitations by taking some electron from
just below the Fermi surface and exciting it to just above the Fermi surface.
Okay. And that means that the heat capacity is going to be proportional to t like we calculated
in the Sommerfeld theory because you can make as many low energy excitations.
Well, you can make lots of low energy excitations with no problem. Maybe less obvious to is that this is a metal, it conducts electricity.
So that is maybe a little bit harder to imagine.
But let's think about it for a second. Here, what we have is we have over here we have right moving electrons.
Maybe I'll label them the sense this one these guys over here have positive k h bar-kays to the right.
So these are right movers. Right movers over here on this side of the diagram.
And over here, we have left the movers. Left movers over on this side.
So positive versus negative momentum.
And if we if we apply an electric field with very little energy cost, we can take some of the electrons from over here and move them to over here.
We can overpopulate the right hand side by a little bit and underpopulated the left hand side by a little bit.
It costs you only very little energy to do so.
Just shift the Fermi surface just a little bit and then you have a net current so metal can shift Fermi surface.
For me surface. And get current.
In fact, the way the way it actually happens is that you should really think about it.
You apply an electric field and each electron state accelerates a little bit in one direction.
So each electron changes its momentum a little bit until these guys get overpopulated, these guys get underpopulated, and then you have a net current.
So this is indeed a metal. And indeed it's very, very frequently the case that monovalent atoms are metals.
Often, I mean, a simple picture should always be true, but often monovalent monovalent materials.
Our metals. Now further.
Let us come back over here to the case, to this picture, and imagine what happens when we have fill the band.
So we're going to take this band. We're going to half fill it. Now, we filled these dates here and we've left these dates up here empty.
So this is a Hatfield band. Now, this is very similar to the covalent bond,
that there is a lower net energy when the electrons can localise between the two nuclei because they're filling only the bonding orbital.
And as long as you don't have to fill the anti bonding orbital. So here when you start letting the electrons hop back and forth,
they can lower their energy compared to this energy that they started with, this atomic energy they started with.
So there's a attractive force between the nuclei trying to get them closer together, trying to make the hopping much greater.
So that is what's forming the metallic bond between forms.
Metallic bond. Metallic bond between the nuclei is very, very similar to the covalent bond by letting the the electrons d localise,
by letting them reduce their kinetic energy, by spreading out their wave function, it forms a bonding force that holds the nuclei together.
Now, in this very simple picture, we have the same problem that we had when we started the covalent bond.
That by this picture, you would expect that the nuclei would be happier and happier and happier as they
got closer and closer together and everything would go down to zero distance.
And that's not true, because some of our approximations start to break down.
One of our approximations that this starts to break down is that we assumed orthogonal orbitals that simplifying assumption,
that that assumption that we started with another. So that's going to break down.
Another thing that breaks down is we forgot about the nuclear nuclear interaction,
which is more or less similar to the covalent bond case, is more or less cancelled by the direct interaction.
But when the nuclei start to get very close to each other, that cancellation is no good anymore and the nuclei start to repel.
So they're not going to get infinitely close anymore. But roughly, this is what causes the bonding in metals.
It's allowing the the electrons to spread out over many, over many atoms and lower their kinetic energy.
Okay. Happy. Okay, good.
So. We can now consider a more general case of dive in one's atoms.
Die valence atoms? I guess not more general.
This is just a different case. Diving into atoms like maybe helium.
Helium has two electrons. In this case we have two and electrons and two and eigen states we can fill.
So we get an entirely filled down. Filled then.
Well, okay, what happens then. Let's go back to this picture.
So now with the DI valent material, we completely fill this band and we completely fill like this.
Everything is filled. In this case, the fill band has is very boring.
The fill band is inert, filled and maybe inert.
Why is it a nerd? Well. When you think about can absorb, can, is it possible for the filled band to absorb any energy?
And the answer is no. It can't absorb any energy because all the states are already filled.
You can't transfer an electron back and forth from one state to another in order to give it energy or take it away.
All the states are filled is a unique state. There is nothing that can be moved around, cannot carry current.
It can't carry current because you can't change the number of left movers versus right movers.
They're all filled. So it has no heat capacity.
No heat capacity carries no current carries, no current.
And it also has no metallic bonding. And this is, again, very similar to what we had in the case of the other heat,
when we consider the possible bonding between this helium atom, between two helium atoms.
If you have to fill the bonding orbital and the anti bonding orbital, you don't gain any energy.
So here, if we have a fill ban, you had to you fill the lower energy states, but you had to fill the higher energy states too.
So it does not gain you anything to have the the hopping go up anymore.
Okay. So how are we doing?
Okay. Good. So as in the case of the vibrational change that we studied earlier this week,
you can also have a situation where not every atom or not every orbital you're considering is the same.
So let's now consider a case where there are two orbitals, two different orbitals per unit cell,
and this can kill can can occur in two different ways or several different ways.
One possible way is that you have two different atoms in the unit cell and they have different types of orbitals in them,
sort of a sodium in a chlorine atom or something like that in a new cell.
Another possible case is you have one atom but two different orbitals on that atom,
like an orbital and a p orbital or a1s orbital and a two orbital that you want to consider.
So without actually solving this problem in in detail, I can show you what the, what the spectrum looks like, what the dispersion looks like.
It's very similar to the case of the vibrational chain. So here's K, here's pi over a year, minus pi over A, here's E.
And you get a low energy band. Analogous to the acoustic mode that we got in the vibrational chain.
And you get a high energy band analogous to the optical mode that we had in the vibrational technology that is supposed to look symmetric.
Apologise about that. So now we have two energy bands, a lower energy band and a higher energy band.
We can also, if we want.
This is the reduced zone scheme, reduced zone scheme, and we can also draw it in the extended zone scheme which which I'll do over here actually.
Yeah. Extended zone scheme. Zone scheme.
We could draw the same thing, which would look kind of like this.
K. E. Here's pi over a.
Here is minus pi over a year to pi over a year minus two pi over a and the
idea of extended zone scheme is to spread out the bands into two boron zones.
So this is the first zone, first busy here and the second is out here and out here.
Second to busy and here. Let's see.
Okay. So all I did was I took. This piece and move over here by two pi over in this piece and moved over here
by two pi over a such at each possible k there's only one one excitation.
And in either case, this is supposed to line up right at the at the brand's own boundaries.
Gap comes in here. This gap comes in here at the bronze zone boundary.
Notice that there's a gap right at the zone boundary.
And it's not coincidental that if you sort of put this together and squint your eyes,
this looks like one overall parabola that a free electron might have,
but you've opened up gaps at the zone boundary, and we'll explain why that is later on in the term.
Now, one thing that you might consider here is suppose suppose the unit sell.
Unit sell has three electrons, has three electrons in it.
Well, then what happens? Then you fill up the entire lower band with both spin up and spin down electrons,
and you have half of the upper band filled to here and half of the upper band here is filled.
Half of the upper bands may fill the lower band and he filled half of the upper band.
Well, you'll notice that we have a entirely filled lower band.
The lower band is inert. Lower band is inert.
And we only need to worry if we're thinking about transport properties or heat properties that, you know, specific heat capacity, something like that.
We only have to worry about the upper band because the lower band is completely inert,
at least unless you give it an enormous energy to excite things out of the lower bound and up to the upper band.
Very high energy because these gaps are typically very large.
You can just ignore the lower band altogether and only worry about the electrons in the upper band.
But this is actually this is actually the answer to one of those questions, one of these puzzles that we wrote down earlier in the term.
And the puzzle from earlier in the term was why is it that you sometimes don't have to worry about core electrons?
If electrons are in core orbitals, why is it you can just throw them away when you're counting electrons for sodium,
you only count one electron per unit per per atom and not 11 electrons per atom when you're trying to calculate the metallic density.
And the reason is because basically those other electrons are completely filling bands and the bands
become inert and just aren't part of the interesting physics like heat capacity and and conduction.
Couple more comments here. So what we've had is that have these important principles.
Is it half filled band? Band is a metal.
Usually a metal. Usually. Medal.
And this is for one electron for unitel or actually any odd number of electrons present itself.
So maybe I'll say odd, odd number of electrons per unit cell,
whereas an even number of electrons buta cell, even number of electrons per unit cell per unit cell.
You might think that this is might or might be be an insulator because you would completely fill bands and then you have just an inert situation.
However, it turns out that there are many, many cases where you have an even number of electrons per unit cell, and yet it's a metal.
And the reason that that can occur is if you have a band structure, it looks as follows.
They have a more complicated band structure that looks like this.
So here's a K, here's pi over a, here's minus pi over a.
And here's the lower band like this. And then imagine that we have an upper band.
It does this. Okay. So it's it's dispersion and more complicated models.
You can have a dispersion that looks like this. In particular, the upper band slips below the top of the lower and the lower band.
So if I have two electrons per unit cell, in fact you could fill the entire lower band.
But it's lower energy. To partially fill this band and partially fill.
This ban. And that would that would make up your two electrons per unit.
So instead of entirely filling the lower band and leaving these states empty.
In that case, you have to partially fill bands instead of one completely filled bands.
And since you have to partially fill bands, you then have low energy excitations here and low energy excitations here.
And so this thing actually becomes a metal. So there are lots of cases of that.
Things like calcium has calcium is a metal, even though it has two electrons per unit itself.
And it's exactly this physics that's occurring. The two bands are actually crossing an energy.
So instead of completely filling one band, you have to partially fill bands.
Okay, I think maybe I better stop there and I see you, I guess tomorrow.