So we've just learned how band structures arise and what they tell us to basically give us the relation
of the energy and the momentum of the electrons.
And we have seen that this can be quite different if you consider a periodic crystal and compare it
to free electrons.
And furthermore, it is also very different when you compare different materials.
So this could be due to different elements in these materials.
But in general, it can also be due to a different geometry and a different symmetry of the letters.
And one particularly interesting material is graphene.
And here you see a graphene lattice.
And before we can continue to calculate the band structure of the graphene lattice and before we can
explore what is so special about this band structure, we first have to understand the letters and have
to construct a lattice ourself.
And this is the purpose of this lecture.
So at the end, we will be having this structure here.
This lattice, which you can see, is periodic, and it consists out of two different species.
So to see.
So both of these circles blue and red, they are both C atoms.
And you can see that when you look only at the blue atoms, they form a hexagonal lattice.
So basically triangles.
And you can also see that when you start from some blue atom here and you go periodically along this
direction, there will always be an atom.
And also when you go along this direction, this direction.
And yeah, so you see the blue lattice is periodic.
The red lattice is periodic, and both of them together are, of course, also periodic.
So first of all, we must understand how can we construct such a lattice?
And this is not just a practice to create a nice figure, but also to understand the physics behind
it and to understand the methods.
So what we're starting with is the letters constant and this is A0 zero.
I call this here a zero.
And it describes the distance between blue and red, and the unit here is angstrom, which is I can
write it down here.
The unit of zero is angstrom, which is 10 to the power of minus 10 meters.
So it's pretty, pretty small.
And the other parameter lattice length is square root of three times a zero, and this describes the
distance between the atoms of the same species.
So once again, when I say different species, I don't mean different elements.
All of these are C atoms.
I just mean that they belong to different supply to cease, basically.
So we have now our description.
So of course, we could have also chosen here one to make it more simple, but I decided to take the
real experimental value of graphene.
So we run this.
Now we have to find a zero and the letters length parameter, and now we have to, of course, just
make a scatterplot with all these coordinates and we could now go ahead and think about it geometrically
and write a list of all the coordinates of all of these atoms.
But of course, it is very it takes a long time and it's difficult and it's much easier to be clever
about it and to construct them using some arrays.
And for this, we first of all, have to define the lattice factors that describe the periodicity.
And let me write this down here the letters, letters.
And since this is a two dimensional sample, we need two letters vectors.
So and these letters vectors are supposed to span and to be able to construct the whole crystal.
So I call them a one and a two and a one will be the lattice length parameter times a vector which I
construct as an array and it will be one and zero.
So basically just going along the X direction, a length of the parameter that is length.
So this would be, for example, when we start here, we go along here.
And so you see, if we take multiples of this lattice vector positive and negative, then we can construct
all of these atoms that are on this line.
So this would be the first lattice vector and then we need a second one, which allows us to describe
all the others.
So it should not be minus one zero.
So it should be one of the others, and it really doesn't.
Matter, which one we take.
So I will just take a copy of this so that I don't have to write so much stuff here.
But of course, we have to modify the vector now and it will be minus 0.5 and square root of three,
divided by two.
So if you calculate the norm of this, so the length basically, then you will see that this will have
the length of what one would have exactly the same length as a one.
So of course, maybe this looks a bit difficult now.
So if you're really interested in the physics behind it and please post a video and think about it,
why would this vector be minus one half and then square root three or four two?
Well, actually, it's quite simple.
It's because this vector here is of length one.
Then this atom here.
If you project it, it's right between these two atoms.
So the X coordinate has to be minus 0.5.
And then we know that the length of the vector.
So basically, this length and this length are the same.
So you just have to figure out what would be the value of the Y coordinate so that the length is then
one.
So if you're interested, please take your time and do it.
But we're here for the numériques and for Python, and not so much for the physics.
So I think it's OK if I just continue now.
So we have constructed our two letters vectors.
So once again, if we start from some random blue atom, we can now go along this direction or opposites
and we can go along this direction and opposite.
And we so to come to this atom, for example, we have to add both of these vectors.
So basically we go first there and then there.
So essentially, we can construct our whole letters by building a linear combination of these two vectors
here.
This is what we're going to do next.
So now we are creating a list of coordinates, so I will write coordinates and I will call this coordinates
eight because first of all, we will only consider the blue atoms, which we could label species eight.
And I will show you a method how you can do this pretty easily.
There are, of course, many other methods, but we are here to learn something new.
So what I'm going to do is I will write a linear combination of both of these vectors as I set.
So I write, for example, Pi Times a one plus J times a two.
And now we have to specify what are these I and J.
And they should just be some random integers, positive and negative as possible, also zero as possible
so we can write, for example.
For I in range
and no, I wrote just write and max and minus and max and and max and the same for J.
So we just copy this for J and Max and Max.
And of course, we have to define and Max and I will choose here for.
All right.
So now we have our list of coordinates.
We can have a quick look, coordinates eight is an array with X and Y component.
So of course, for the plot, we know that we must have two hours first.
One only the X coordinates second one, only the Y coordinates.
So we are going to transpose this and now we have it.
And basically, we can now say X one comma x and also X1 comma y one is this one.
And now we can use a scatterplot, plot scatter and then X1 comma y one.
And you see, we have now the blue dots from here.
So of course, we can now make the plots look a bit nicer.
For example, we could now write you size 100s and color blue.
And also, if you look carefully, you will see that this plot might be a bit distorted.
I'm not 100 percent sure if it really is, but to make sure that it's not distorted, we can write something
like plot dots, axes and then set aspects to equal plans for the data elements.
So, yeah, I'm not sure if it really changed anything, but I think now it's we are on the safe side
that we really have the correct geometry of our lattice.
So now what we are missing here is the RET atoms.
And that's the special thing about graphene.
We cannot just add another vector here and change the linear combination because there's no way with
these lattice factors to get from blue to red.
What we must do is we must construct a totally new set of coordinates, so coordinate B, which will
also be this linear combination.
Because yeah, going from red to red is the same as going from blue to blue.
But the red atoms are shifted and they are shifted by some direction and the y direction, as you can
see here.
And so what we do right here is we are adding here another vector, which will be zero comma a zero
because we have to find a zero previously.
Here it is to be the nearest neighbor distance.
So now let's see what happens if we at these points here, so I have to copy both of these commands
and have to say X2 Y2 coordinates, b transpose and Tier two two.
And here I will write, read and then you can see we now have our graphene letters.
So since we have two of these lists of coordinates, you could also see that we just have to take two
of these atoms, which are right above each other, red and blue.
And then we take both of these and periodically shift them by the lattice vectors and then we can construct
our whole crystal.
So you see, there are always these two pairs of atoms.
And that's the nature of graphene having two basis atoms.
And the first one is the blue one and the other one is the red one.
So in this lecture, we have not only constructed a nice figure, a nice sketch of graphene, but we
have also learned already a lot about graphene.
It has two basis atoms.
We have determined the letters vectors that describe the periodicity, and this will help us much in
the following to calculate the bend structure.