So before we conclude this lecture, I want to give you a short bonus lecture because we have already
seen that graphene and also the topological properties and the quantum hall effect both got a Nobel
Prize for physics.
It may be one of the next Nobel Prizes will be awarded to the person who really understands the twisted
bilayer of graphene, where you can see from some interesting patterns and we call this more' patterns
omolara lattices.
So we have you two graphene sheets orange and blue, and they are just slightly twisted with respect
to each other.
So you see here directly in the middle, the atoms are perfectly on top of each other.
But then due to the twist, they will slightly be displaced.
But of course, at some other point in space, they will once again be right on top of each other.
And this brings about these nice, superlative structures, especially if you zoom out.
I think you can see the bright spots here and the dark spots here and is just interesting from a geometrical
point of view, but also pretty interesting from a physics point of view.
Because these twisted by layers of graphene, they give rise to a very interesting phenomenon like superconductivity,
for example.
So let me show you how I created this figure.
So here I do not want to go with you through all the programming process once again because it takes
a lot of time.
And I think you will not really learn something new because it's pretty similar to what we did before
when we created, for example, this figure.
So basically, what we're doing is instead of defining only a single, a single subset or a single layer
with two subtle lattices, we will now define the coordinates for Subliners one and two folios one and
two.
So we will have sort of the force applied as this now and after I defined you a few variables, a few
constants.
I have just defined the letters vectors.
So as previously for software, this one and then we have here and the letters vectors for some letters
too, which are of course, rotated by the angle alpha.
This you can do by multiplying this with cosine alpha and sine alpha.
And as an example, the angle alpha here is something like two like two point six four degrees.
And then from analytical considerations, we can calculate what is the letters vector for the super
letters.
So basically the vector going from this red dot to this red dot.
And after we have done this, then we can also define some reciprocal letters vectors, which is not
so important here.
This is just for the visualization.
And then, of course, we can finally construct our coordinate lattices as we did before by just calculating
these linear combinations of these letters vectors off to letters one and off to some letters to.
And then we are basically finished.
We just have to plot this thing.
And actually, this command, you would already be sufficient.
So let me show you if you just write this, then we would already get our letters with the moiré structure,
which is not really that well visible here because the resolution is too low.
But what I did here is I added these this hexagon here for the unit cell and also these red dots for
the next centers of the next unit cells.
So as I said, this is a really a hot topic at the moment.
So if you're interested, it would be nice to look into this and maybe you can even come up with some
calculations.
But please be warned it's a bit more difficult to set up the Hamilton matrix for such a system and then
to solve it because the unit cells will be enormously large.
And also, we have to really figure out all the bond directions and all the neighbors by numerical methods.
So it will be way more difficult than what we did before.
But in principle, from what you have learned in the previous lectures, it should be possible to solve
such a problem as well, even though it will be much more time consuming and much more difficult.
So, yeah, as I said, this may be a project that will get the next or one of the next Nobel Prizes
in physics.
Because these memory structures and these twisted by layers, they have some very interesting properties
and they may solve large problems in physics.