1 00:00:00,060 --> 00:00:05,880 So before we conclude this lecture, I want to give you a short bonus lecture because we have already 2 00:00:05,880 --> 00:00:13,110 seen that graphene and also the topological properties and the quantum hall effect both got a Nobel 3 00:00:13,110 --> 00:00:14,340 Prize for physics. 4 00:00:15,060 --> 00:00:21,870 It may be one of the next Nobel Prizes will be awarded to the person who really understands the twisted 5 00:00:21,870 --> 00:00:29,600 bilayer of graphene, where you can see from some interesting patterns and we call this more' patterns 6 00:00:29,610 --> 00:00:30,870 omolara lattices. 7 00:00:31,980 --> 00:00:38,760 So we have you two graphene sheets orange and blue, and they are just slightly twisted with respect 8 00:00:38,760 --> 00:00:39,390 to each other. 9 00:00:40,050 --> 00:00:45,090 So you see here directly in the middle, the atoms are perfectly on top of each other. 10 00:00:45,270 --> 00:00:48,750 But then due to the twist, they will slightly be displaced. 11 00:00:49,290 --> 00:00:53,880 But of course, at some other point in space, they will once again be right on top of each other. 12 00:00:54,780 --> 00:00:59,160 And this brings about these nice, superlative structures, especially if you zoom out. 13 00:00:59,160 --> 00:01:05,910 I think you can see the bright spots here and the dark spots here and is just interesting from a geometrical 14 00:01:06,090 --> 00:01:11,010 point of view, but also pretty interesting from a physics point of view. 15 00:01:11,010 --> 00:01:17,790 Because these twisted by layers of graphene, they give rise to a very interesting phenomenon like superconductivity, 16 00:01:17,790 --> 00:01:18,540 for example. 17 00:01:19,650 --> 00:01:22,950 So let me show you how I created this figure. 18 00:01:23,520 --> 00:01:30,900 So here I do not want to go with you through all the programming process once again because it takes 19 00:01:30,900 --> 00:01:31,470 a lot of time. 20 00:01:31,470 --> 00:01:36,990 And I think you will not really learn something new because it's pretty similar to what we did before 21 00:01:36,990 --> 00:01:38,760 when we created, for example, this figure. 22 00:01:39,990 --> 00:01:48,090 So basically, what we're doing is instead of defining only a single, a single subset or a single layer 23 00:01:48,090 --> 00:01:54,480 with two subtle lattices, we will now define the coordinates for Subliners one and two folios one and 24 00:01:54,480 --> 00:01:54,810 two. 25 00:01:55,500 --> 00:02:02,370 So we will have sort of the force applied as this now and after I defined you a few variables, a few 26 00:02:02,370 --> 00:02:03,030 constants. 27 00:02:03,960 --> 00:02:06,420 I have just defined the letters vectors. 28 00:02:06,840 --> 00:02:13,230 So as previously for software, this one and then we have here and the letters vectors for some letters 29 00:02:13,230 --> 00:02:17,340 too, which are of course, rotated by the angle alpha. 30 00:02:17,610 --> 00:02:21,990 This you can do by multiplying this with cosine alpha and sine alpha. 31 00:02:22,890 --> 00:02:28,590 And as an example, the angle alpha here is something like two like two point six four degrees. 32 00:02:29,970 --> 00:02:38,010 And then from analytical considerations, we can calculate what is the letters vector for the super 33 00:02:38,010 --> 00:02:38,460 letters. 34 00:02:39,120 --> 00:02:43,680 So basically the vector going from this red dot to this red dot. 35 00:02:45,360 --> 00:02:50,160 And after we have done this, then we can also define some reciprocal letters vectors, which is not 36 00:02:50,160 --> 00:02:51,170 so important here. 37 00:02:51,180 --> 00:02:52,830 This is just for the visualization. 38 00:02:53,460 --> 00:03:02,100 And then, of course, we can finally construct our coordinate lattices as we did before by just calculating 39 00:03:02,100 --> 00:03:08,310 these linear combinations of these letters vectors off to letters one and off to some letters to. 40 00:03:10,020 --> 00:03:12,350 And then we are basically finished. 41 00:03:12,360 --> 00:03:13,980 We just have to plot this thing. 42 00:03:14,700 --> 00:03:18,100 And actually, this command, you would already be sufficient. 43 00:03:18,100 --> 00:03:25,440 So let me show you if you just write this, then we would already get our letters with the moiré structure, 44 00:03:25,770 --> 00:03:29,370 which is not really that well visible here because the resolution is too low. 45 00:03:31,350 --> 00:03:38,130 But what I did here is I added these this hexagon here for the unit cell and also these red dots for 46 00:03:38,130 --> 00:03:40,620 the next centers of the next unit cells. 47 00:03:41,880 --> 00:03:44,690 So as I said, this is a really a hot topic at the moment. 48 00:03:44,700 --> 00:03:52,830 So if you're interested, it would be nice to look into this and maybe you can even come up with some 49 00:03:52,830 --> 00:03:53,610 calculations. 50 00:03:53,970 --> 00:04:00,420 But please be warned it's a bit more difficult to set up the Hamilton matrix for such a system and then 51 00:04:00,420 --> 00:04:04,050 to solve it because the unit cells will be enormously large. 52 00:04:04,470 --> 00:04:11,900 And also, we have to really figure out all the bond directions and all the neighbors by numerical methods. 53 00:04:11,970 --> 00:04:14,940 So it will be way more difficult than what we did before. 54 00:04:15,900 --> 00:04:21,540 But in principle, from what you have learned in the previous lectures, it should be possible to solve 55 00:04:21,540 --> 00:04:26,670 such a problem as well, even though it will be much more time consuming and much more difficult. 56 00:04:28,050 --> 00:04:36,780 So, yeah, as I said, this may be a project that will get the next or one of the next Nobel Prizes 57 00:04:36,780 --> 00:04:37,320 in physics. 58 00:04:37,620 --> 00:04:44,310 Because these memory structures and these twisted by layers, they have some very interesting properties 59 00:04:44,310 --> 00:04:47,370 and they may solve large problems in physics.