So let us discuss the band structure of graphene, so these words, band structure may sound a bit cryptic
to you.
And for this reason, I will start this section with discussing a bit.
Why do we have band structures and what our band structures?
So first of all, as in almost every notebook, we are going to import the nonpaying module and also
the iPod lip module.
And since I will create and show you several figures here, I will also increase the figure size and
this notebook by writing down this command.
So as I said, I want to start with a simple example that explains to you why we have band structures
and what band structures actually are.
And this is the model of free electrons, which then leads to the quasi free electrons in a periodic
crystal.
So free electrons, you probably know?
I think so.
If you have just an electron in vacuum, for example, you can describe its energy by the kinetic energy,
which is the momentum p squared over two times the mass.
This is basically the same as the kinetic energy, which is am over two times the velocity squared.
So what we can then do is we can express the momentum p by h bar times the wave factor K.
So this is at this point, just some quantity that we have introduced.
So if we write that E is equal to a small square of a 2m times square, then this is actually also the
correct solution of the stationary shooting equation.
If we do not consider any potential.
So you probably know this energy here from classical physics.
So as I said, just particles, just electrons in the vacuum, for example.
But it turns out this is also the correct solution in quantum mechanics.
So here you see the shooting equation, and if we do not consider any potential, it is pretty simple.
It's just this one here.
So you have a pretty.
Then you have two second derivative of the wave function.
You have the energy and you have to wave function again.
And what you tried to do now in quantum mechanics is you try to solve this equation, which means you're
trying to determine at the same time the wave function 5X and also the energy, which will be a number.
And it turns out that this solution here is correct.
It solves this Schrodinger equation if the wave function is a plane wave.
So it's something like a constant times the exponential function with K x as the exponent, so we can
check if this is really true.
So just take this function here.
Put it in here.
So the second derivative of exponential function to the power i.e. K X is the function itself times
the the exponent here.
So we have to calculate the inner derivative, so basically the derivative of the exponent.
And since it's the second derivative, we have to do it twice.
So what we get is I k and both of them squared, so I squared is minus one and k squared is just case
quiet.
So we get for the second derivative minus k squared times the function itself.
So what we have now is minus h bar squared over two m times, K squared times and other minus one from
the ice squared.
So we have H Bar Square over two and Times Square five is equal to ify.
And this is why the only solution for E can then be exactly the prefecture's h by square of a two m
times square.
So you see that our classical solutions, our observations for electrons are also fulfilled and reproduced
in quantum mechanics.
So basically, what this tells us is that the SweeTango equation works just fine and that the quantum
mechanics can re establish the results from classical physics.
But now let's finally get to our example.
So how do we get banned structures or why do we need band structures at all?
Well, to be honest, it's not really that easy to see and not really that easy to explain.
But the reason why these man structures occur is that we have now not a vacuum.
We do not have free electrons, but we have electrons that live in the periodic potential.
So we are considering here a solid or a periodic crystal.
For example, take a piece of iron.
You have.
The iron atoms and the form of periodic letters and of course, the electrons, which are in the letters,
they will feel a certain interaction with the iron atoms.
So it turns out that in order to describe this, it's not that important to know exactly what the potential
looks like.
It's more important to just know the periodicity.
And of course, the periodicity of the letters is the same as for what was already période does.
A T of the potential V is the same as the periodic periodicity of the crystal itself.
So here in this example, I want to do it as simple as possible, so I went to one dimensions.
So what we can say is.
Or Crystal has a certain lattice constant.
So basically the distance between two atoms, which I call a.
And so the potential will have the same value at each of these positions x plus end times a wire and
is just some integer number.
That's pretty clear whenever the electron is directly at the position of an atom.
Then it will have the same potential no matter at which atom it is, because all atoms are the same.
And of course, if we include not as potential in treating equation, it will look a bit different.
And also, this will, of course, change the way functions PHI.
It will change the energies.
And this is what we are going to discuss next.
So corresponding to my explanation, I will create now a figure.
And of course, I could have created this figure in advance.
But I think it's a nice exercise to which we can do together to practice on how to create figures.
So what we're trying to do here is we are trying to generate a crystal.
So basically just some dots that represent atoms.
And then we are going to plot the potential V above or in the same plot.
You could say.
So the first thing that I'm going to do is I'm going to generate an X-ray.
So these will be our values on the x axis.
And this corresponds to the values of our coordinate x that enters the potential, for example.
And as we do, it's most often for these plots.
We will be generated by writing and pitot and then space.
And now we have to decide which range we want to use.
And of course, I tried this already before, and I think if we go, if we want to plot three atoms.
So basically going from minus three to plus three, it looks good if we have a bit of a margin here.
So I go from minus three point five to plus three point five.
And we take one hundred and one points in between.
So that's the function will then look smooth.
So now we have our X-ray and what we can do next is we can just plot the the atom positions and we can
do this by writing, Please don't get her.
And then we just write.
Let me do it like this, and we just write.
Our MP darts in space, so we create a new land space and it goes from minus three to three.
We have seven points, so we have basically here a list starting from minus three and then minus two
minus one zero one two three.
And then for the Y coordinates, I say the microphone should just be zero.
So most easy would be to just right and zeros and then the number of points that we want to have.
And you see, we have here our chain of atoms and we have plotted seven of these atoms.
So now we can use our X-ray to plot on top of the potential.
So I will write peeled dot plots.
And now we wrote X Ray, and we plot to potential.
Of course, we don't know what the potential looks like exactly, but here I want to just use a function
which gets to periodicity, right?
So I write x ray and then we multiply by two point.
So I think this should work.
Yes.
So you see now, OK, it's a bit choppy, actually.
Some maybe increase the number here.
Yeah, it looks much better.
So you see we have a cosine function and the potential at the position of these atoms is always at the
minimum.
So it would be good for the electron to be positioned right at the atom position and then in between,
the potential is positive.
So this makes sense.
And also it's of course, reasonable that we have this periodicity.
So now this plot looks a bit ugly.
So let me try to improve a bit the visuals here.
So we did this already several times, so I will just write and try to talk a bit while I do this.
So first of all, we must create a figure and then add a subplot because what I want to do is I want
to change the aspect ratio.
And apparently this is only possible if we create such a figure with a subplot.
So once again, this one means we have a grid of one time small plots and this is the first plot.
So, yeah, it's a bit useless because we only are going to use one plot.
So I think this is a bit tedious about Python two always having to write this just to set the aspect
ratio.
But it is what it is.
And now we can finally write, said aspect ratio.
And you see how it looks.
I think it looks now much better.
And not the only thing that I'm going to do is I'm going to write X label, which is the position which
is X, and the unit here is eight.
So we are plotting multiples of eight, which is two letters constant, and the white label is just
a potential three.
And I think this is a really nice figure.
I mean, it's still pretty simplistic, but it allows us to understand what I explained before.
So we have no our potential v of X, which fulfills this condition.
It has the same value no matter which position we look at if we shift the position by multiples of the
letters constant.
For example, we are here, we are shifting by two letters constants than we are here, and it's the
same value.
All right.
So now we must explore how do the wave functions and especially how do the ion energies change?
So the way functions, I can tell you, they've become periodic block functions, so I will not go into
the details here.
But it turns out that these functions are, of course, also periodic in space, which I think is quite
clear.
If the potential is periodic, then the wave function is also periodic, but they are also periodic
in K.
So here we have the solution of plane waves e to the power ICAC's.
And so what happens here is we get another factor here, which is some function that's periodic in X
and in K. So the only thing that we care about is the energies, and I will show you how this changes.
So to get an idea on what the ion energies look like is we could just say, OK, we don't really care
so much about the actual potential here.
We just care about the periodicity and we take our free electrons which have this energy and position
this parabola at every position of the atoms.
And we know that we have this periodicity in X and K.
So this means we can also do the same thing in K.
So yeah, sounds a bit difficult, but here I'm going to show you what's going to happen.
So first of all, we are going to create and carry so some number of cape points, which is again a
linear space.
So remember, once again, the key value basically determines the momentum of the electrons because
p is equal to each bar.
Time is key.
So here we could start, for example, from PI and go to point sorry, like this.
And then we generate an energy ari, which is one half times.
Or actually it is H Bach Square over two m Times Square.
But for simplicity, here I will use H Bar and both of them will be equal.
All to one.
So this means we just have one half times square.
So Kerry Square, and now we can plot this is, of course, really easy.
We just write guilty to the plot and we write Kerry come up and are great.
So this is our parabola.
So now we can, of course, gift the axis some label.
So I copy this and I write this here the way factor, which is proportional to the momentum of the electrons
in units of one over, actually, but it's not so important.
And then we have on the y axis the energy.
And now what happens is I said it was already that the energy will be also periodic in key space.
So here on this axis, we have the key factor on the value of Kate and the energy will be periodic and.
And it will be periodic at the unit of two PI over eight.
So this means when we shift this or when we go to PT. one minus two PI, we will see the same parabola.
So this is what I'm going to do now.
First of all, let me increase here the range for which we are plotting, and now I'm going to add more
of these parabolas.
But first of all, let me change that star to black and then we can just copy this paste and just right
here, plus two times pipe.
And you see, now we have a second parabola shifted by two pi over eight.
And this happens, of course, for all multiples of two PI.
So of course, I cannot do all of them here, but let me just do a few.
So minus two pi and we do the same thing for four PI.
Do the same thing for six pie, and I'd say, let's also do it for eight pie and we have quite a few
parabolas.
So you see, of course, here I would have I would need to have more parabolas here, but you can see
pretty nicely what's happening here.
So we have all of these parabolas which are shifted by K, which basically is due to the periodicity
of the letters.
So every atom position we get, we have an electron.
And this means and in case bays, we have these parabolas, which are shifted by multiples of two pi
over eight.
So no, I told you that this party is not really reasonable because we missed the parabola instead of
further to the left and to the right.
So I would say, let's zoom in here in this range and we can do this by using the commands plus thoughts.
Excellent.
And we will limit this to minus.
I can just copy here minus one point five PI and we're going to 1.5 Pi and the and the energy I will
also limit and I will go from zero to sixty five.
So you see, now the band structure looks much better or yeah, I don't know, it looks better, but
at least we have now this.
Yeah, we do not have this artifact of the left and the right.
Maybe for the moment, let me increased as a bit.
And here you can nicely see that we really have some periodicity.
So the band structure is still periodic with respect to two pi over.
So here, here and here it's periodic.
So basically, this means it is sufficient to just look at the band structure in this range here from
minus PI over eight to plus PI over eight.
And to visualize this, let me just add a few lines here so we can do this by writing minus and P Pi
to minus and P Pi.
This is the change of the X coordinate, which doesn't change at all, and the line goes from minus
f sorry from zero to sixty five.
And I will plot this graph.
So you see, we have now this line.
Let's add another one
four plus point.
And let's change the plot range to 1.5.
So here we are.
This is now the cell, or we call this in physics, the burial zone.
But it's not so important what we call it.
But this is like some characteristic range in K. that is able to describe all of the electrons.
So we have now this relation of free electrons that are in a periodic lattice and we have this relation
of the energy with respect to K, which is once again proportional to the momentum.
So so far, they are still free electrons.
They are still a quadratic their energy.
It still depends quadratic on the momentum, quadratic on the velocity.
So, yeah, nothing really new here.
But the thing is the potential does not only provide the periodicity, but it also provides some perturbation.
So you see for the tweeting equation we have included here does value V and of course, it will do something
to the energies.
And this is a bit hard to understand without me having to talk for an hour, which I don't want.
So I'm going to tell you what's going to happen.
So the potential break some symmetry in the system.
And whenever this happens in a physical system, the general is degeneracy will split up.
So this means all of these points were two bands across the bands will actually split up.
I'm going to show you what this means.
So I have prepared this already in advance.
So let me just copy here a few lines of code that you can also write down if you want.
And let me run this, so I'm adding here for more functions.
And you see, these are these RET functions here.
So this is what is going to happen when you consider now the actual potential.
So of course, this may not be the absolute correct solution, but it describes qualitatively what is
going to happen.
So these crossings here, they will split up due to due to the potential.
And so the initially quadratic band dispersions become more something like cosine functions.
And also, what you can see is that now the bands are the parabolas do not go from to not go from zero
to infinity.
But we have really some well-defined bands here.
So these red bands, they just oscillate in some range and between the bands, we even have band gaps
where there are no states anymore.
So this means due to this periodicity and due to this potential, the relation of energy to the momentum
has changed drastically.
And these curves that we see here.
So maybe let me for a moment copy.
So a common piece so that we don't see them anymore.
So this year is called a band structure, and it is really important for describing materials.
And it allows us to determine if a material is, for example, a metal or an insulator.
And it's really, really different for different materials.
And in the following, we are going to consider graphene and we will calculate the actual band structure
of graphene.
And I think it's really a very interesting topic.
So to close the section, let me restore the initial parabolas so that you have this nice figure in
your notebook.
And yeah, I wrote it down here as a main effect of the potential of the band crossing split up.
But still, it's very important to realize that the periodicity has to be conserved, so it's still
periodic with respect to to pay your way.
So for here and here, the band structure is exactly the same.