1 00:00:00,150 --> 00:00:06,630 So we've just learned how band structures arise and what they tell us to basically give us the relation 2 00:00:06,930 --> 00:00:10,470 of the energy and the momentum of the electrons. 3 00:00:11,040 --> 00:00:17,190 And we have seen that this can be quite different if you consider a periodic crystal and compare it 4 00:00:17,190 --> 00:00:18,330 to free electrons. 5 00:00:19,260 --> 00:00:25,050 And furthermore, it is also very different when you compare different materials. 6 00:00:25,550 --> 00:00:29,310 So this could be due to different elements in these materials. 7 00:00:29,310 --> 00:00:35,250 But in general, it can also be due to a different geometry and a different symmetry of the letters. 8 00:00:36,150 --> 00:00:39,600 And one particularly interesting material is graphene. 9 00:00:40,440 --> 00:00:44,270 And here you see a graphene lattice. 10 00:00:44,280 --> 00:00:49,710 And before we can continue to calculate the band structure of the graphene lattice and before we can 11 00:00:49,710 --> 00:00:55,320 explore what is so special about this band structure, we first have to understand the letters and have 12 00:00:55,320 --> 00:00:57,420 to construct a lattice ourself. 13 00:00:57,900 --> 00:00:59,670 And this is the purpose of this lecture. 14 00:01:00,150 --> 00:01:05,010 So at the end, we will be having this structure here. 15 00:01:05,010 --> 00:01:12,270 This lattice, which you can see, is periodic, and it consists out of two different species. 16 00:01:12,270 --> 00:01:12,930 So to see. 17 00:01:14,160 --> 00:01:19,200 So both of these circles blue and red, they are both C atoms. 18 00:01:20,070 --> 00:01:25,850 And you can see that when you look only at the blue atoms, they form a hexagonal lattice. 19 00:01:25,860 --> 00:01:27,480 So basically triangles. 20 00:01:28,260 --> 00:01:35,760 And you can also see that when you start from some blue atom here and you go periodically along this 21 00:01:35,760 --> 00:01:37,770 direction, there will always be an atom. 22 00:01:37,770 --> 00:01:40,680 And also when you go along this direction, this direction. 23 00:01:41,070 --> 00:01:44,160 And yeah, so you see the blue lattice is periodic. 24 00:01:44,520 --> 00:01:49,470 The red lattice is periodic, and both of them together are, of course, also periodic. 25 00:01:50,580 --> 00:01:54,750 So first of all, we must understand how can we construct such a lattice? 26 00:01:54,750 --> 00:02:00,480 And this is not just a practice to create a nice figure, but also to understand the physics behind 27 00:02:00,480 --> 00:02:01,860 it and to understand the methods. 28 00:02:03,090 --> 00:02:09,240 So what we're starting with is the letters constant and this is A0 zero. 29 00:02:09,389 --> 00:02:10,530 I call this here a zero. 30 00:02:10,770 --> 00:02:18,840 And it describes the distance between blue and red, and the unit here is angstrom, which is I can 31 00:02:18,840 --> 00:02:19,740 write it down here. 32 00:02:20,340 --> 00:02:30,060 The unit of zero is angstrom, which is 10 to the power of minus 10 meters. 33 00:02:31,860 --> 00:02:33,180 So it's pretty, pretty small. 34 00:02:33,870 --> 00:02:41,100 And the other parameter lattice length is square root of three times a zero, and this describes the 35 00:02:41,100 --> 00:02:44,400 distance between the atoms of the same species. 36 00:02:45,840 --> 00:02:51,310 So once again, when I say different species, I don't mean different elements. 37 00:02:51,330 --> 00:02:52,950 All of these are C atoms. 38 00:02:53,460 --> 00:02:57,420 I just mean that they belong to different supply to cease, basically. 39 00:02:58,380 --> 00:03:00,720 So we have now our description. 40 00:03:00,720 --> 00:03:05,160 So of course, we could have also chosen here one to make it more simple, but I decided to take the 41 00:03:05,160 --> 00:03:07,260 real experimental value of graphene. 42 00:03:08,190 --> 00:03:08,990 So we run this. 43 00:03:09,030 --> 00:03:15,750 Now we have to find a zero and the letters length parameter, and now we have to, of course, just 44 00:03:15,750 --> 00:03:22,980 make a scatterplot with all these coordinates and we could now go ahead and think about it geometrically 45 00:03:22,980 --> 00:03:26,640 and write a list of all the coordinates of all of these atoms. 46 00:03:27,240 --> 00:03:32,670 But of course, it is very it takes a long time and it's difficult and it's much easier to be clever 47 00:03:32,670 --> 00:03:36,420 about it and to construct them using some arrays. 48 00:03:37,560 --> 00:03:44,910 And for this, we first of all, have to define the lattice factors that describe the periodicity. 49 00:03:46,020 --> 00:03:49,110 And let me write this down here the letters, letters. 50 00:03:52,350 --> 00:03:55,620 And since this is a two dimensional sample, we need two letters vectors. 51 00:03:56,430 --> 00:04:03,300 So and these letters vectors are supposed to span and to be able to construct the whole crystal. 52 00:04:04,230 --> 00:04:13,500 So I call them a one and a two and a one will be the lattice length parameter times a vector which I 53 00:04:13,500 --> 00:04:21,180 construct as an array and it will be one and zero. 54 00:04:22,800 --> 00:04:30,540 So basically just going along the X direction, a length of the parameter that is length. 55 00:04:31,380 --> 00:04:35,250 So this would be, for example, when we start here, we go along here. 56 00:04:36,330 --> 00:04:42,480 And so you see, if we take multiples of this lattice vector positive and negative, then we can construct 57 00:04:42,480 --> 00:04:45,000 all of these atoms that are on this line. 58 00:04:46,710 --> 00:04:51,930 So this would be the first lattice vector and then we need a second one, which allows us to describe 59 00:04:51,930 --> 00:04:52,680 all the others. 60 00:04:53,340 --> 00:04:56,240 So it should not be minus one zero. 61 00:04:56,250 --> 00:04:59,730 So it should be one of the others, and it really doesn't. 62 00:04:59,790 --> 00:05:00,800 Matter, which one we take. 63 00:05:01,070 --> 00:05:07,010 So I will just take a copy of this so that I don't have to write so much stuff here. 64 00:05:07,550 --> 00:05:17,900 But of course, we have to modify the vector now and it will be minus 0.5 and square root of three, 65 00:05:18,710 --> 00:05:20,030 divided by two. 66 00:05:21,200 --> 00:05:26,870 So if you calculate the norm of this, so the length basically, then you will see that this will have 67 00:05:26,870 --> 00:05:31,460 the length of what one would have exactly the same length as a one. 68 00:05:32,600 --> 00:05:35,690 So of course, maybe this looks a bit difficult now. 69 00:05:36,470 --> 00:05:40,580 So if you're really interested in the physics behind it and please post a video and think about it, 70 00:05:41,120 --> 00:05:45,530 why would this vector be minus one half and then square root three or four two? 71 00:05:45,740 --> 00:05:46,940 Well, actually, it's quite simple. 72 00:05:47,330 --> 00:05:50,270 It's because this vector here is of length one. 73 00:05:50,840 --> 00:05:51,950 Then this atom here. 74 00:05:51,950 --> 00:05:54,900 If you project it, it's right between these two atoms. 75 00:05:54,920 --> 00:05:57,620 So the X coordinate has to be minus 0.5. 76 00:05:58,100 --> 00:06:00,830 And then we know that the length of the vector. 77 00:06:01,160 --> 00:06:03,890 So basically, this length and this length are the same. 78 00:06:04,250 --> 00:06:09,410 So you just have to figure out what would be the value of the Y coordinate so that the length is then 79 00:06:09,410 --> 00:06:09,770 one. 80 00:06:10,850 --> 00:06:13,370 So if you're interested, please take your time and do it. 81 00:06:13,670 --> 00:06:18,510 But we're here for the numériques and for Python, and not so much for the physics. 82 00:06:18,510 --> 00:06:20,990 So I think it's OK if I just continue now. 83 00:06:21,920 --> 00:06:24,290 So we have constructed our two letters vectors. 84 00:06:24,890 --> 00:06:31,730 So once again, if we start from some random blue atom, we can now go along this direction or opposites 85 00:06:31,940 --> 00:06:34,640 and we can go along this direction and opposite. 86 00:06:35,330 --> 00:06:41,360 And we so to come to this atom, for example, we have to add both of these vectors. 87 00:06:41,360 --> 00:06:43,910 So basically we go first there and then there. 88 00:06:44,990 --> 00:06:51,830 So essentially, we can construct our whole letters by building a linear combination of these two vectors 89 00:06:51,830 --> 00:06:52,040 here. 90 00:06:52,600 --> 00:06:53,840 This is what we're going to do next. 91 00:06:55,340 --> 00:07:02,870 So now we are creating a list of coordinates, so I will write coordinates and I will call this coordinates 92 00:07:02,870 --> 00:07:10,160 eight because first of all, we will only consider the blue atoms, which we could label species eight. 93 00:07:10,910 --> 00:07:14,450 And I will show you a method how you can do this pretty easily. 94 00:07:14,780 --> 00:07:18,920 There are, of course, many other methods, but we are here to learn something new. 95 00:07:19,280 --> 00:07:25,160 So what I'm going to do is I will write a linear combination of both of these vectors as I set. 96 00:07:25,250 --> 00:07:31,100 So I write, for example, Pi Times a one plus J times a two. 97 00:07:32,120 --> 00:07:35,000 And now we have to specify what are these I and J. 98 00:07:35,360 --> 00:07:41,990 And they should just be some random integers, positive and negative as possible, also zero as possible 99 00:07:42,290 --> 00:07:43,580 so we can write, for example. 100 00:07:44,570 --> 00:07:47,060 For I in range 101 00:07:49,970 --> 00:07:59,660 and no, I wrote just write and max and minus and max and and max and the same for J. 102 00:07:59,870 --> 00:08:05,600 So we just copy this for J and Max and Max. 103 00:08:06,080 --> 00:08:10,580 And of course, we have to define and Max and I will choose here for. 104 00:08:12,500 --> 00:08:12,950 All right. 105 00:08:12,950 --> 00:08:15,220 So now we have our list of coordinates. 106 00:08:15,230 --> 00:08:21,500 We can have a quick look, coordinates eight is an array with X and Y component. 107 00:08:22,580 --> 00:08:27,710 So of course, for the plot, we know that we must have two hours first. 108 00:08:27,710 --> 00:08:30,800 One only the X coordinates second one, only the Y coordinates. 109 00:08:31,250 --> 00:08:34,580 So we are going to transpose this and now we have it. 110 00:08:35,419 --> 00:08:45,470 And basically, we can now say X one comma x and also X1 comma y one is this one. 111 00:08:46,730 --> 00:08:55,790 And now we can use a scatterplot, plot scatter and then X1 comma y one. 112 00:08:56,960 --> 00:08:59,870 And you see, we have now the blue dots from here. 113 00:09:01,100 --> 00:09:05,600 So of course, we can now make the plots look a bit nicer. 114 00:09:05,990 --> 00:09:11,630 For example, we could now write you size 100s and color blue. 115 00:09:14,180 --> 00:09:21,200 And also, if you look carefully, you will see that this plot might be a bit distorted. 116 00:09:21,210 --> 00:09:27,230 I'm not 100 percent sure if it really is, but to make sure that it's not distorted, we can write something 117 00:09:27,230 --> 00:09:42,440 like plot dots, axes and then set aspects to equal plans for the data elements. 118 00:09:43,790 --> 00:09:48,410 So, yeah, I'm not sure if it really changed anything, but I think now it's we are on the safe side 119 00:09:48,680 --> 00:09:51,560 that we really have the correct geometry of our lattice. 120 00:09:53,220 --> 00:09:56,640 So now what we are missing here is the RET atoms. 121 00:09:57,450 --> 00:09:59,250 And that's the special thing about graphene. 122 00:09:59,580 --> 00:10:09,120 We cannot just add another vector here and change the linear combination because there's no way with 123 00:10:09,120 --> 00:10:11,550 these lattice factors to get from blue to red. 124 00:10:12,240 --> 00:10:22,860 What we must do is we must construct a totally new set of coordinates, so coordinate B, which will 125 00:10:22,860 --> 00:10:25,440 also be this linear combination. 126 00:10:25,440 --> 00:10:29,670 Because yeah, going from red to red is the same as going from blue to blue. 127 00:10:30,270 --> 00:10:37,080 But the red atoms are shifted and they are shifted by some direction and the y direction, as you can 128 00:10:37,080 --> 00:10:37,650 see here. 129 00:10:38,610 --> 00:10:46,980 And so what we do right here is we are adding here another vector, which will be zero comma a zero 130 00:10:47,820 --> 00:10:52,590 because we have to find a zero previously. 131 00:10:53,010 --> 00:10:57,240 Here it is to be the nearest neighbor distance. 132 00:11:00,190 --> 00:11:06,250 So now let's see what happens if we at these points here, so I have to copy both of these commands 133 00:11:06,250 --> 00:11:13,720 and have to say X2 Y2 coordinates, b transpose and Tier two two. 134 00:11:14,200 --> 00:11:21,190 And here I will write, read and then you can see we now have our graphene letters. 135 00:11:22,810 --> 00:11:30,280 So since we have two of these lists of coordinates, you could also see that we just have to take two 136 00:11:30,280 --> 00:11:33,910 of these atoms, which are right above each other, red and blue. 137 00:11:34,360 --> 00:11:43,600 And then we take both of these and periodically shift them by the lattice vectors and then we can construct 138 00:11:43,600 --> 00:11:44,920 our whole crystal. 139 00:11:45,640 --> 00:11:48,670 So you see, there are always these two pairs of atoms. 140 00:11:49,360 --> 00:11:53,290 And that's the nature of graphene having two basis atoms. 141 00:11:54,040 --> 00:11:57,280 And the first one is the blue one and the other one is the red one. 142 00:11:58,450 --> 00:12:04,960 So in this lecture, we have not only constructed a nice figure, a nice sketch of graphene, but we 143 00:12:04,960 --> 00:12:07,750 have also learned already a lot about graphene. 144 00:12:08,140 --> 00:12:09,970 It has two basis atoms. 145 00:12:10,270 --> 00:12:17,530 We have determined the letters vectors that describe the periodicity, and this will help us much in 146 00:12:17,530 --> 00:12:19,870 the following to calculate the bend structure.