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So now we have learned what are band structures, and we have learned how we can construct a graphene

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lattice, and now we want to have a look at how this band structure, which was basically just a one

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dimensional example of nearly free electrons.

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How this band structure changes when we consider graphene.

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So in order to describe this band structure and to calculate it, we have to in principle, we have

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to solve the shooting equation for an enormous number of electrons that are in this crystal.

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This is not possible.

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Even the smartest people cannot do it because it just takes way too much time to solve it, and it's

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impossible for the fastest computer to do so.

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Physicists have developed several methods how we can simplify the problem and how we can solve it.

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So one of these methods is density functional theory, which is a bit more difficult than what we will

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be doing here.

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So I decided to pick what I think is the simplest, but also the nicest approach for solving this problem.

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And the cool thing is that I think it's even possible to solve it without knowing how the method really

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works and just accepting some initial thoughts and then thinking about how the latter's works and how

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the neighbors interact with each other.

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So the thing is, what we were doing in tight binding is we are looking basically at all of these atoms

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and say that the electrons can only be confined very closely to these atoms.

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And then, of course, they can jump from one atom to the other.

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So the electrons are always tightly bound to the atoms.

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And the only thing that we have to describe is the directions along which the electron can hop.

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And this is what we're going to do here.

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So what we're going to do is we have to define some so-called Hamilton matrix.

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So let me write this down, Hamilton matrix, and it will be a matrix.

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And the eigenvalues of this matrix will be the energies, and they will directly give us the band structure.

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So we just have to come up with the Hamilton matrix and then calculate the eigenvectors, the eigenvalues,

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which is also a nice exercise and something new that we will learn and then we can already have a look

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at the result.

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So the Hamilton matrix, I would call him and it will be a matrix, of course.

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So I define it as an array, and it turns out that this matrix is a two by two matrix in our case.

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And the reason is that we have two basis atoms.

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So if you would have a different type of letters where we have three bases atoms, then this would be

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a three by three matrix.

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So since we have this periodicity, we can just look at one of such unit cells where we have two of

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these atoms blue and red.

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And that is sufficient to just describe the behaviour of the electrons in this unit cell.

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And basically what we have to write here is so let me write it down like this first.

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So the first entry will be from blue to blue.

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Second entry will be from blue to red.

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Then we will have from red to blue and from red to tourettes.

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And this is all that we have to fill in.

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And if you look carefully, then you will see that if you if we just consider nearest neighbors and

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just the closest path an electron can hop to, we look at the red atom.

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Then the electron can only hope to one of these three neighboring blue atoms.

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So the hopping from rat to rat and also from blue to blue is zero because there is no closest path to

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these atoms.

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So our Hamiltonian just got way simpler than we initially thought because we only have these two elements

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and these will be the hopping paths from red to blue.

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So here we have three possibilities.

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And from blue to red, where you also have three possibilities.

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And the way this works and the way this is implemented is in terms of exponential functions.

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So I thought a bit about if I should really derive this for you and explain this to you, but I think

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this would really be too much and would lead too far.

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So I hope that you can just accept that these terms in the Hamilton Matrix will be these exponential

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functions and they will be the integer units.

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So imagine every integer imaginary unit I or in Python, we write one j times the dot product of the

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K vector.

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So basically K x komaki y with D Delta our position.

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So the basically the vector connecting these two atoms.

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And now we really have to write, what are these payoffs?

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And the first path would be from blue to red.

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And I first want to consider this path where it goes from here to here.

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So it would be zero comma, a zero.

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Then we have another term because we do not only have two hopping, two here, but also two here and

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two here.

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And this will of course, be the same form.

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So let me add to these three terms up, and now let me change and correct what I have written down here,

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because these things are, of course, different.

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So we just have to fix this one here and have to fix this one.

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And one of these paths will be along this direction, which will be on let me think it will be in the

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right direction minus a zero half.

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And then we have another one of these square with three of a two is along the X direction.

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So this again, something if you really want to figure it out, then post a video and have a careful

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look at the figure and then think about what are the vectors to the nearest neighbors.

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It may take a bit, but I think it's not that difficult to do.

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So we write a zero times and P Square Root three divided by two.

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And the Y coordinate change is minus A0 over two.

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And for the other vector, it's the same along the way direction and then the X direction.

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It's just a different sign.

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So we have now our three terms and the vectors are that connect to blue and red are zero A1.

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Three square root, three over two times a one and minus a one of a two and then here basically the

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same thing, but just with a different sign for the exchange.

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This would be this, this and that.

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Now, if we look at the hopping from red to blue, then we will see we have very similar hopping, but

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they are, of course, just in the opposite direction.

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So we have, for example, zero and minus zero.

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So I would just copy this and then just change the sign of all the hopping paths.

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So I've copied this, and now I just have to change here to sign and here and here and also here and

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here.

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OK, so this would be the Hamilton matrix for the key vector, which we have to find to be K, X and

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Y.

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So of course, what we must do now is we must create such a plot where we have on the x axis the key

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vector, which in our case will have two components.

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So we must actually make a three d plot where we have at the X and the Y axis, the K X and the K Y

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values and then A the z axis.

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We have the corresponding energy.

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So we basically must loop over all the cakes and all the K y values and then calculate the eigenvalues

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of the Hamiltonian, stored them in some memory or list and then make the 3D plots.

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So this is what we're going to do next.

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So I start with paid K X list, which is, as always, a little space function.

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And here I decided to go from minus PI.

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So of course, you know, already the solution.

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So I know what's the proper choice for the boundaries.

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But if you're just starting, you could.

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You could have also just started with going from minus one to one, and then you would have realized,

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Oh, I cannot see everything.

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I have to make the boundaries a bit larger.

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And yeah, if you do mathematics, you can show that these are the appropriate boundaries.

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And for the number of K points are two key points because we will use this later as well.

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And we will define key points to be maybe let's just use 50 for the beginning or 51 and the why list.

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For the moment at least, we will take the same as the K X list.

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All right.

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Yeah.

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So of course, now I get an error because here I have written K and K-Y, which are not defined yet,

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but we will fix this in a second because we will right now for K X in cakes list.

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So we loop over all the values in the list.

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And we do, of course, the same for K-Y.

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So I could just copy K Y K Y list.

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We define the Hamiltonian.

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And now we must calculate the eigenvalues.

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So I write here Energy's R.D. eigenvalues, or you could also say the vice versa, the eigenvalues are

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the energies.

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So we will write.

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A first rate to and I will explain to you and p dot linear algebra dot igen values of a permission matrix.

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So this is just a detail, I think not so important here.

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The fact is that this this matrix always fulfills some mathematical property.

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It is a her mission matrix and therefore we can here, right?

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It could call a routine to calculate the eigenvalues for a her mission matrix, which makes the calculation

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faster.

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So this would give us 90 eigenvalue, and of course, we have to store them somewhere.

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So I will write and I will create an empty list energy list,

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and I will add these values here to the list.

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So energy list dot append and then we store these and you see, when I run, no, there is no error.

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So, yeah, let's see if it worked.

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Let's have a look at energy list.

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And it gives us a list of arrays, which is a bit odd, but we can fix this later on.

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But the important thing is that we have now many arrays and all of them have two values corresponding

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to the two eigenvalues of this matrix.

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So I think this worked pretty well.

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So what we can do now is we can basically, first of all, turn this into an hey.

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We could have also defined that here already as an array, but then we would have to use a different

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syntax here.

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So not really a big deal.

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We just write and pray and then we can update our energy list.

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And also, I will create a new K, X and K list.

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Which is an empty dot, the mesh grid out of the old K, X and K-Y lists.

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So you will see in a second what this does.

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Maybe you can imagine already.

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So when I write here tax list before I run this command, then I see that this is just a list of all

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the values.

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Basically, these 51 points.

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But if I now run this command first and then call K X list, then you see it's a very large raise,

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even that large that it doesn't plot here.

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So we have fifty one points here, fifty one points here and fifty one of these lists in total.

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So it is really a meshed grid that we can now use for our plotting.

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So one thing that I want to or that they have to do before is I have to further fix the shape of the

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energy list.

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So this is because it doesn't have the correct syntax yet.

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And first, when I when I proposed this notebook, I tried, of course, just with this energy list

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and I found that it didn't work because it has to be wrong for shape.

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So this is something that's sometimes a bit annoying.

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I find in Python that you have to reshape these areas just to make sure the syntax is correct.

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But yeah, it is what it is.

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So what we have to do is we have to write and reshape of energy list.

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Then we have to define how we want to reshape it.

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And basically, I leaf the data are the same, but they just change the ways the brackets are set and

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the correct way to do it is like this from a to.

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So now have a look.

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This is our old energy list and this is our new one.

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So now the shape of the array fits exactly the shape of our attacks and Qi wireless that we have updated.

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So it looks correct so we can override the old energy list.

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So by the way, now it's probably a good idea since we have overwritten some of these variables to just

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run all three of these cells once more so that we make sure that everything runs properly and is updated.

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And now we can finally go to the plot.

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So we need a three plot.

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So has always been just first ride projection, equal 3D.

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And we give it a name lot, not 3D.

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And now we can go a hat basically and just plots three dots.

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And here I want to show you a surface plot.

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So I tested several types of plots before, but I find two surface plots the nicest here.

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So let's use plot underscores surface.

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You can, of course, experiment here and try different options and look at the plot in different ways.

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But just for this cause, what I will show you here is the surface plot.

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So we need a list.

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OK?

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Why list?

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And then the energy list?

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And I think maybe this should work already, let's check it does not work because the the argument must

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be two dimensional, and I think the problem here is that we have to eigenvalues.

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So we are trying to plot two values for every kxan k y, which is something that cannot so easily be

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done.

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So we must extract.

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You're only the first eigenvalue.

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And the correct syntax to do this is take all the values in the first level, the second level and then

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the third level, which is take the first value.

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Now you can see this is our band.

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So when I want to add the second band as well, I can write like this.

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Now we have two bands and it looks correct, so I know the band structure of graphene, so it looks

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correct already, maybe a bit ugly, but looks good.

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Now what we can do is, or what we should do also is to add some labels, of course, club 3D.

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So you should always add labels, especially if you're going to show the figures to someone else.

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So we can right here.

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Okay?

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X is actually in units of one over a year.

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Then we have my label like this.

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And the Z label will be the energy.

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So we did not really specify the unit for the energy, and this is also a bit difficult because I would

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have to get the correct value from the literature.

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So I would say we just keep it as it is and say energy in some arbitrary units.

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OK, so now we have these descriptions here, and what we can also do is we can change the color.

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For example, I mean, this looks already quite good, but we could use a different color, for example,

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a color map.

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And let's try Klimek winter.

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Sorry, the syntax was bit off here.

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So we are like this.

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And so you see, the lower band has now a different color and the upper band, we can also change to

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bottom.

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So now the color looks a bit different.

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So I don't know if it looks nicer, but at least it allows us now to tell a bit better the actual value

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here.

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Because you see the higher devalued, the brighter the color gets.

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And so it is a bit easier now to understand what's going on here.

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Also, we can change the view because it's not really possible to look inside here.

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So that's a bit unfortunate, but we can fix this by writing for 3D view in it.

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And then you have two numbers which basically correspond to two angles in space, and I have tested

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before that.

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This one looks quite nice because now you can really see, Oh, there is a gap here.

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OK, nice.

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We didn't see this before.

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So now let me just add some more options.

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You will see in a second what these do.

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Of course, they are totally optional.

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You can also play with them and try different things.

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But I tried before, and I find the plot looks quite good if we use these options here.

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So in a second, you will see what they do.

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Of course, we have to apply them to both of these plots.

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And now you see it.

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We will have a closer look at the figure in a second.

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But before we do this, let me increase also the number of key points here.

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So that's the whole plot looks more smoothly.

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So here you have to be careful, of course, because the higher the number you take, the more memory

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it will take and also the longer it will take to do the calculation.

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So especially if your computer is a bit weak, please don't go too high with the number here.

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Otherwise it will crash.

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And yeah, I take 301.

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So you see, it takes already a few seconds.

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Now it's finished to calculate.

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And now, of course, we have to rerun everything.

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And also the plot takes no, a few seconds.

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But now we see what's going on here.

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I think now it looks really nice because we have this transparent band structure where we have the line

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width of the grid, which is really narrow.

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And yeah, we can really see what is going on here.

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Also, the last thing that I want to show you here is that we can export this so we can write Pulte

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safe, think as we did before and we give it a name.

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All right.

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Band structure, graphene dots, PMG.

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And if I run this, the picture will be very blurry, even more blurry than this one here.

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So it's a good idea to increase the resolution.

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And for this, you can use the DPI option Typekit 300 and you have to rerun wait a few seconds.

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And once it's finished, you will find the file and structure.

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Dot underscore graphene dot PMG in the directory where this notebook is saved.

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So I will be back in a second.

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So by the way, I just noticed that I had a small typo in the name of the file, which I have just corrected.

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No, it's really band structure on the school, graphene, the peg.

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And this is what we get as a result.

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And I think it looks really cool.

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And you can see now how different values for the cave vector relate to the energy.

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And so the discussion is really exactly the same as before for the one dimensional case we have now

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the relation, how is the energy of the electrons related to the momentum?

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Or you could also say to the velocity of the electrons, but the momentum is better to say.

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It's more correct, actually.

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So if the electron is moving, say along the the X direction, then it will have a finite K X value

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and then you can see how the energy changes.

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And I think it's pretty interesting because if the momentum of the electron is small, then it will

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be somewhere close to zero in the momentum space.

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And you see close to zero, you have this typical minimum shape here, and this is actually very similar

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to the free electron parabola.

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So it will depend linearly on the connector.

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So it means for small momentum, the electron will behave as it would be freely.

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So like free electrons, but when it has a larger momentum and then you see that there can be quite

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strong modifications here.

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And especially interesting are these points here with the bands cross in just a single point, and this

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is something very unique and very special about graphene.

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And I will discuss this and we will explore this a bit more in detail in the next lecture.