The previous lecture, we have derived the three band structure of graphene, so actually it's a 2D
bend structure because, yeah, it's a 2D system, but so we have two different values or two directions
for the vector components acts and.
And then on the z axis, we have plotted the energy.
So why is this special?
Well, first of all, it's special because it's different from the electrons.
But to be honest, that's not really that special because every material is different from free electrons
and every material has different band structure because the electrons behave differently in different
materials.
So what is special is these points here because the bands really touch each other only in a singular
point.
And the thing is a material or a the electronic properties of a material are determined by the fact
how these bands overlap or how they are.
There is a gap between them.
For example, if there would be a gap between these two bands, then it would be an insulator.
If these bands would totally intersect or totally overlap, then it would be a metal and it would be
a nice conductor for electrons.
And here we have just these singular points, which make graphene very special.
So it's like right at the edge between insulator and metal.
And furthermore, these points are also very interesting because it is a linear bent crossing.
So at these points, the energy depends linearly, at the momentum and at the K value.
So we have learned for free electrons that it behaves quadratic li on P and quite tragically on K.
And here we have a linear dependence which makes these electrons behave as if they would have lost their
mass and if they would, as if they would behave relativistic li.
So in relativistic physics, which is true when an electron moves really fast, for example, then the
energy relation would look pretty difficult here.
Actually, there is a small mistake that it's not plus here.
It's like this.
And if you want now have an object which has no mass, then this term would vanish.
And then you would have to square root of p-square c squared, which is just Pizzi.
So a mass less relativistic object moves or has a relation of the energy that is plus minus p times
C.
Or you could also say H R K Times C, and this is exactly what we say here for these points.
So the electrons in graphene, when they have a certain momentum, which is corresponds to this value,
then they behave as if they would have no mass, which is a very specific and very special property.
So to see if this is really a linear bent crossing, let us change a bit to disband structure and modify
the code to have a cut through these so-called Dirac points.
So for this, we will just write Pulte dot plots.
And now we leave basically the one of the two values constant and the and we only look at the other
one.
So we choose here K x two, p zero.
And so we just look at the K y well, you.
So I write K y list.
And now we, of course, must select only a certain part of the list.
So we take all the values here and for the other value we take basically the the middle point.
So since we have here three hundred one, this would be one hundred fifty one.
This would correspond exactly to the middle, so to equal to zero.
But I want to make a general here, so I write K points divided by two, which is bad because we have
an odd number.
So I basically I use such an integer division, which would give us at the moment 150 and then plus
one and the similar thing we do for the energy list.
So I will just copy this and write energy list, and we need to have two of these commands because we
want to plot to eigenvalues.
So both bands basically and I think this should work already.
Yes, here we are.
Let's just add the plot label real quickly plotted on ex-Labour is k y and units of one of eight and
do y label is the energy in our arbitrary units because we have not really specified the unit.
And you see this is now a cut through this band structure.
Going basically along the K-Y axis at the Value X equals zero.
So we are crossing this point here and we are crossing this point here and we go along the valley in
the middle, which goes until minus three.
And this is exactly what we see here.
And you see now, if we look carefully here, I mean, it looks a bit odd.
This is because we did maybe not use enough of the points.
But if you look here and you will see it's really a linear bend crossing, so you could check further
by zooming into the figure just around this point.
And by cranking really up the number of key points, what I think we can leave it as it is.
And yeah, it looks good.
I'm I'm happy with the plot.
So we've seen really that the electrons in graphene can behave freely if we are here, because then
the band structure is quadratic, like for free electrons.
But if the value corresponds to these values here, then they behave as if they would have no mass and
as if they would be relativistic.
This is really a special property of graphene and one of the properties that led to the founders of
graphene getting the Nobel Prize in physics.