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The previous lecture, we have derived the three band structure of graphene, so actually it's a 2D

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bend structure because, yeah, it's a 2D system, but so we have two different values or two directions

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for the vector components acts and.

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And then on the z axis, we have plotted the energy.

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So why is this special?

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Well, first of all, it's special because it's different from the electrons.

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But to be honest, that's not really that special because every material is different from free electrons

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and every material has different band structure because the electrons behave differently in different

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materials.

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So what is special is these points here because the bands really touch each other only in a singular

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point.

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And the thing is a material or a the electronic properties of a material are determined by the fact

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how these bands overlap or how they are.

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There is a gap between them.

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For example, if there would be a gap between these two bands, then it would be an insulator.

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If these bands would totally intersect or totally overlap, then it would be a metal and it would be

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a nice conductor for electrons.

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And here we have just these singular points, which make graphene very special.

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So it's like right at the edge between insulator and metal.

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And furthermore, these points are also very interesting because it is a linear bent crossing.

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So at these points, the energy depends linearly, at the momentum and at the K value.

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So we have learned for free electrons that it behaves quadratic li on P and quite tragically on K.

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And here we have a linear dependence which makes these electrons behave as if they would have lost their

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mass and if they would, as if they would behave relativistic li.

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So in relativistic physics, which is true when an electron moves really fast, for example, then the

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energy relation would look pretty difficult here.

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Actually, there is a small mistake that it's not plus here.

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It's like this.

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And if you want now have an object which has no mass, then this term would vanish.

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And then you would have to square root of p-square c squared, which is just Pizzi.

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So a mass less relativistic object moves or has a relation of the energy that is plus minus p times

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C.

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Or you could also say H R K Times C, and this is exactly what we say here for these points.

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So the electrons in graphene, when they have a certain momentum, which is corresponds to this value,

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then they behave as if they would have no mass, which is a very specific and very special property.

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So to see if this is really a linear bent crossing, let us change a bit to disband structure and modify

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the code to have a cut through these so-called Dirac points.

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So for this, we will just write Pulte dot plots.

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And now we leave basically the one of the two values constant and the and we only look at the other

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one.

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So we choose here K x two, p zero.

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And so we just look at the K y well, you.

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So I write K y list.

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And now we, of course, must select only a certain part of the list.

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So we take all the values here and for the other value we take basically the the middle point.

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So since we have here three hundred one, this would be one hundred fifty one.

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This would correspond exactly to the middle, so to equal to zero.

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But I want to make a general here, so I write K points divided by two, which is bad because we have

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an odd number.

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So I basically I use such an integer division, which would give us at the moment 150 and then plus

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one and the similar thing we do for the energy list.

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So I will just copy this and write energy list, and we need to have two of these commands because we

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want to plot to eigenvalues.

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So both bands basically and I think this should work already.

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Yes, here we are.

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Let's just add the plot label real quickly plotted on ex-Labour is k y and units of one of eight and

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do y label is the energy in our arbitrary units because we have not really specified the unit.

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And you see this is now a cut through this band structure.

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Going basically along the K-Y axis at the Value X equals zero.

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So we are crossing this point here and we are crossing this point here and we go along the valley in

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the middle, which goes until minus three.

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And this is exactly what we see here.

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And you see now, if we look carefully here, I mean, it looks a bit odd.

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This is because we did maybe not use enough of the points.

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But if you look here and you will see it's really a linear bend crossing, so you could check further

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by zooming into the figure just around this point.

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And by cranking really up the number of key points, what I think we can leave it as it is.

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And yeah, it looks good.

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I'm I'm happy with the plot.

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So we've seen really that the electrons in graphene can behave freely if we are here, because then

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the band structure is quadratic, like for free electrons.

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But if the value corresponds to these values here, then they behave as if they would have no mass and

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as if they would be relativistic.

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This is really a special property of graphene and one of the properties that led to the founders of

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graphene getting the Nobel Prize in physics.