1 00:00:00,060 --> 00:00:05,610 With previous lectures, we have considered a finite lattice of graphene, a so-called nano ribbon. 2 00:00:06,480 --> 00:00:11,730 And this is the corresponding band structure which gives you the relation of the energy of an electron 3 00:00:12,060 --> 00:00:14,190 with respect to the momentum. 4 00:00:15,150 --> 00:00:22,110 And we have seen that this pen structure is essentially the same as the other band structure from before 5 00:00:22,110 --> 00:00:24,210 of the infinitely extended graphene. 6 00:00:24,630 --> 00:00:27,690 It's just projected on a single plane. 7 00:00:28,380 --> 00:00:34,410 So the relation of energy to the momentum of the electrons is pretty similar in this case. 8 00:00:35,520 --> 00:00:43,050 But now we are going to apply a magnetic field perpendicular to the graphene, and we will see how this 9 00:00:43,050 --> 00:00:47,460 changes the band structure and the relation energy of K. 10 00:00:48,690 --> 00:00:51,270 So maybe a bit of background about this. 11 00:00:51,660 --> 00:00:57,900 So if you apply a very small magnetic field, then the band structure should not really change that 12 00:00:57,900 --> 00:00:58,290 much. 13 00:00:58,950 --> 00:01:05,489 And the only thing that will happen then in an experiment is that you will see that electrons that move 14 00:01:05,489 --> 00:01:11,640 through the graphene will be deflected along the perpendicular direction due to the Lorentz force. 15 00:01:12,180 --> 00:01:14,700 This effect is called the hall effect. 16 00:01:15,990 --> 00:01:21,930 And what happens if you increase the magnetic field to very large values of maybe a few? 17 00:01:21,960 --> 00:01:25,830 Tesla even is that the band structure will drastically change. 18 00:01:25,830 --> 00:01:29,250 All of a sudden, this is called Landau Quantization. 19 00:01:29,910 --> 00:01:31,320 We will see that here. 20 00:01:31,320 --> 00:01:35,400 We will find several very flat bands and very flat energy levels. 21 00:01:35,910 --> 00:01:43,440 And also in the experiments, they observe a quantized version of the hall effect, where all of this 22 00:01:43,500 --> 00:01:49,320 all of a sudden, the number of deflected electrons will increase in steps. 23 00:01:50,820 --> 00:01:57,180 So before I lose too many words about this, let us program and let us see what is going to happen here. 24 00:01:58,350 --> 00:02:03,540 So here we need even more physics backgrounds and to be honest, it's a bit difficult here. 25 00:02:03,660 --> 00:02:06,930 I just give you the essentials and then we are trying to program this. 26 00:02:08,580 --> 00:02:14,360 So what's happening here is we are applying a magnetic field along the Z direction, so out of plane. 27 00:02:15,300 --> 00:02:21,900 And maybe you know this from electrodynamics that we have to introduce now a vector potential which 28 00:02:22,410 --> 00:02:29,670 has fulfilled this relation to curl or the rotation of a vector potential must be equal to B. 29 00:02:30,630 --> 00:02:33,420 And so this is like some indirect definition of A. 30 00:02:33,780 --> 00:02:40,170 And there exist several choices for a which all give the same magnetic field and we really have to freedom 31 00:02:40,170 --> 00:02:41,940 to select one of these choices. 32 00:02:42,780 --> 00:02:49,800 And I have considered Tier two following choice minus B times y times the unit vector along the X direction. 33 00:02:51,120 --> 00:02:56,760 So this vector field only has an X component, and it has the value minus B times Y. 34 00:02:57,660 --> 00:03:03,720 And if you now calculate to curl and you will see that the magnetic field will be exactly zero zero 35 00:03:03,720 --> 00:03:06,600 B. So just a magnetic field along the z direction. 36 00:03:08,310 --> 00:03:14,730 Now how do we implement the magnetic field or the vector potential in the Hamilton matrix? 37 00:03:15,600 --> 00:03:20,040 This is done by so-called parallel substitution, but not really that important. 38 00:03:20,790 --> 00:03:26,400 The only thing that's going to happen is that we will get an additional factor for all the hopping terms. 39 00:03:27,420 --> 00:03:35,100 So additionally, to such terms here, like exponential function and then the hopping, we will get 40 00:03:35,100 --> 00:03:40,140 another factor which will have to shape exponential function times I. 41 00:03:40,170 --> 00:03:44,220 And then a line integral where we have to scale product of a. 42 00:03:44,670 --> 00:03:48,000 And here this will be along the direction of the hopping path. 43 00:03:49,590 --> 00:03:55,230 And this is a mathematical exercise that you may want to do or may not want to do. 44 00:03:55,500 --> 00:03:57,330 It's not really that important for this cause. 45 00:03:57,750 --> 00:04:05,640 When you solve this, you will see that for all of our toppings, the solution will be exponential function 46 00:04:05,640 --> 00:04:12,690 minus i.e. times Delta X times b times the average value of Y of the two atoms that correspond to the 47 00:04:12,690 --> 00:04:13,050 path. 48 00:04:14,220 --> 00:04:20,730 So basically, if we look at a hopping path, for example, here from purple to blue, then we have 49 00:04:20,730 --> 00:04:25,140 a certain Delta X and we have an average value of the Y coordinates. 50 00:04:25,770 --> 00:04:28,500 And this determines this additional factor. 51 00:04:30,290 --> 00:04:35,570 Also, we can see that are hopping along the y direction, for example, from blue to red, also from 52 00:04:35,570 --> 00:04:43,600 green to purple will not have an additional face factor because the Delta X, where is it here? 53 00:04:43,610 --> 00:04:45,980 The Delta X is zero in this case. 54 00:04:47,490 --> 00:04:55,950 This is because we have chosen our potential to be along the X direction and then it is perpendicular 55 00:04:55,950 --> 00:04:57,280 to the Y direction, of course. 56 00:04:57,300 --> 00:05:01,050 So this dot product here will be zero for these particular topics. 57 00:05:02,760 --> 00:05:06,690 So let's go ahead and implement this in our Hamilton matrix. 58 00:05:07,230 --> 00:05:14,880 So here I went to the top of our notebook here and just copied these three cells because we will want 59 00:05:14,880 --> 00:05:17,040 to calculate exactly the same thing. 60 00:05:17,850 --> 00:05:23,520 So you see, it gives the same results and we will just want to modify the hopping terms. 61 00:05:23,530 --> 00:05:27,900 So basically these off diagonal vectors here, that's the only thing we have to modify. 62 00:05:29,340 --> 00:05:37,530 So we have to multiply a face factor to half of the toppings. 63 00:05:37,860 --> 00:05:43,470 So all the toppings that have an X component or an X direction will have a modified topping. 64 00:05:43,920 --> 00:05:49,110 And the other ones which only go along y or minus Y, they will not have a modified hopping. 65 00:05:50,160 --> 00:05:54,810 And of course, as always, there are many ways how you can do this, how you can implement this. 66 00:05:55,350 --> 00:06:04,200 And I found it to be most easily when we divide these off diagonal this into three sub lists corresponding 67 00:06:04,200 --> 00:06:05,940 to the three individual toppings. 68 00:06:06,270 --> 00:06:13,650 So this one, that one and this one, because then we can address all three toppings individually. 69 00:06:14,850 --> 00:06:15,600 So let me do this. 70 00:06:16,230 --> 00:06:24,450 I will just write that the first off diagonal will be called off diagonal one eight and the command 71 00:06:24,450 --> 00:06:27,810 will be empty tile, then the array. 72 00:06:27,810 --> 00:06:31,530 And let me just make this all in one line so that we save a bit of space here. 73 00:06:32,010 --> 00:06:37,800 So we will then have just the first topping and then that's it. 74 00:06:38,100 --> 00:06:43,110 So we will to lead this one, though, after just be careful with the brackets. 75 00:06:43,860 --> 00:06:45,330 Um, let me see. 76 00:06:45,340 --> 00:06:48,350 I think this is not correct at the moment. 77 00:06:48,990 --> 00:06:51,540 Yes, so I have to close the array first. 78 00:06:54,060 --> 00:06:57,700 And then this one's for empty tile. 79 00:06:57,720 --> 00:06:58,050 Yes. 80 00:06:58,080 --> 00:06:58,470 OK. 81 00:06:58,830 --> 00:07:00,250 So I think now is correct. 82 00:07:01,260 --> 00:07:10,290 And then the last thing we can do for of diagonal 1b, which will be the other hopping, which will 83 00:07:10,290 --> 00:07:12,690 just have a modified X component. 84 00:07:15,180 --> 00:07:25,170 And of course, we can do the same thing for of diagnosi, which will just be eight times eight times 85 00:07:25,170 --> 00:07:26,070 k y. 86 00:07:28,140 --> 00:07:29,620 So like this? 87 00:07:29,640 --> 00:07:30,000 Yes. 88 00:07:31,200 --> 00:07:31,560 OK. 89 00:07:32,670 --> 00:07:33,360 And the same thing. 90 00:07:33,360 --> 00:07:39,000 We're also going to do for the other things which have just a different sign in the exponent. 91 00:07:39,000 --> 00:07:45,780 So let me copy this to two two, and that's change to sign here for the exponents. 92 00:07:47,610 --> 00:07:53,580 Now what we only have to do is we have to multiply here the modified hopping. 93 00:07:54,270 --> 00:07:59,760 So of course, only for those that have an X components or not for this one, but for this one and for 94 00:07:59,760 --> 00:08:00,210 this one. 95 00:08:01,110 --> 00:08:08,490 And for this, I will introduce an array of the modified hopping as well a remote, for example. 96 00:08:08,880 --> 00:08:09,330 Right. 97 00:08:10,200 --> 00:08:12,360 And here also Ara mods, right? 98 00:08:12,960 --> 00:08:17,700 And here our mod left and our mods left. 99 00:08:18,660 --> 00:08:21,120 And now we only must define what are these arrays. 100 00:08:21,240 --> 00:08:29,580 So Ara modified right equals and pre modified left equals. 101 00:08:30,270 --> 00:08:36,090 And the good thing is, when you look here, these these arrays, the modified hopping, they do not 102 00:08:36,090 --> 00:08:36,960 depend on art. 103 00:08:37,440 --> 00:08:38,669 So we are sorry. 104 00:08:38,669 --> 00:08:40,020 They do not depend on K. 105 00:08:40,020 --> 00:08:42,470 So we do not have to put them into the loop. 106 00:08:42,480 --> 00:08:44,730 We only have to calculate the ones in the beginning. 107 00:08:45,600 --> 00:08:48,240 And essentially they are just what is written down here. 108 00:08:50,240 --> 00:08:57,650 So let me start, it's, of course, in the narrator describing the hopping for all the individual hopping 109 00:08:57,650 --> 00:08:58,130 paths. 110 00:08:58,640 --> 00:09:01,670 So for example, this one, this one, this one, this one. 111 00:09:02,240 --> 00:09:06,380 But of course, subdivided, divided by left and right direction. 112 00:09:07,580 --> 00:09:12,620 So for a right direction, hopping this term will always be positive. 113 00:09:13,520 --> 00:09:17,060 And this one will just be the average value of the Y coordinates. 114 00:09:17,930 --> 00:09:24,770 So basically, what we have to consider here is this hopping, this hopping, this hopping, this hopping. 115 00:09:25,490 --> 00:09:30,980 And when you take the average value of the Y coordinates, you will see that these points will be equally 116 00:09:30,980 --> 00:09:31,550 spaced. 117 00:09:32,690 --> 00:09:38,240 And this is what we can use now to program this in a very simple way. 118 00:09:38,870 --> 00:09:42,440 We are just writing this in and p dot exponential. 119 00:09:46,000 --> 00:10:00,730 And it's yeah, it's difficult times, one j times and P as Q R T U three L for two 120 00:10:04,360 --> 00:10:05,560 times zero. 121 00:10:06,400 --> 00:10:10,720 And so this basically we have to define first. 122 00:10:10,990 --> 00:10:15,430 So the fields is equal to some number. 123 00:10:15,430 --> 00:10:18,250 Let's I don't know, maybe let's use zero point one. 124 00:10:19,060 --> 00:10:20,440 And so this is here to be feel. 125 00:10:20,450 --> 00:10:21,370 This is the I. 126 00:10:21,820 --> 00:10:30,820 And then we have square root three over two times a zero, which is, I think, the distance and the 127 00:10:30,820 --> 00:10:31,600 X direction. 128 00:10:33,710 --> 00:10:46,040 And then we must also provide an index I times and pitot square root three over two times letters length, 129 00:10:47,240 --> 00:10:54,050 which is basically describing this average value of Y for the different hopping paths. 130 00:10:54,380 --> 00:10:57,920 So the AI index will describe basically all the different toppings. 131 00:10:58,190 --> 00:11:02,090 So this one, this one, this one, this one, this one, and so on. 132 00:11:03,350 --> 00:11:08,030 OK, so now of course, we have to once again define what the value of AI is. 133 00:11:08,510 --> 00:11:13,520 So I think I have to close the exponential function first. 134 00:11:16,180 --> 00:11:16,800 Let's see. 135 00:11:17,230 --> 00:11:19,570 No, I think, yeah, this one was incorrect. 136 00:11:19,930 --> 00:11:32,050 Yeah, now I have to write for I in range zero comma atoms, and now the only thing that we must modify 137 00:11:32,050 --> 00:11:39,730 is that only every second topping and every second atom will have a hopping to the right. 138 00:11:40,450 --> 00:11:46,450 So it will actually not be every I, but it will be every second I. 139 00:11:47,620 --> 00:11:48,040 Where is it? 140 00:11:48,070 --> 00:11:48,520 Here it is. 141 00:11:49,690 --> 00:11:52,690 So what we can write down instead is. 142 00:11:56,640 --> 00:11:57,150 Like this? 143 00:11:59,710 --> 00:12:06,100 And once again, since it's the off diagonal, we just go from the first to the second last element. 144 00:12:07,570 --> 00:12:13,510 So maybe this is a bit complicated here, but I think if you think about it very carefully, you will 145 00:12:13,510 --> 00:12:13,890 manage. 146 00:12:13,900 --> 00:12:15,460 But maybe it's a good idea. 147 00:12:15,610 --> 00:12:22,450 If you didn't understand to just post a video, look at the figure once again and think about how can 148 00:12:22,450 --> 00:12:28,780 we express these modified hopping factors here for the different topics? 149 00:12:30,130 --> 00:12:36,460 So now if we look at the left hopping, then we will have essentially the same thing just with a different 150 00:12:36,460 --> 00:12:36,820 sign. 151 00:12:37,870 --> 00:12:39,580 So let me add a different sign here. 152 00:12:39,820 --> 00:12:46,510 So no, we almost at the end we have defined the race for the modified hopping along the right and along 153 00:12:46,510 --> 00:12:46,990 the left. 154 00:12:47,590 --> 00:12:55,510 And basically, what we have done here with this trick is we have basically modified is hoping for all 155 00:12:55,510 --> 00:12:56,440 the different toppings. 156 00:12:56,980 --> 00:13:04,720 But here's the important thing is that for these terms, we use this and Tyler commands where actually 157 00:13:04,720 --> 00:13:09,100 I just realize now that I deleted accidentally too much. 158 00:13:09,550 --> 00:13:11,170 So I forgot. 159 00:13:11,960 --> 00:13:16,360 Um, let me see and pray. 160 00:13:16,840 --> 00:13:18,250 Yeah, I forgot the zero here. 161 00:13:18,820 --> 00:13:21,680 I deleted this one because this is the important point. 162 00:13:22,210 --> 00:13:25,120 Every second term in this rate is zero. 163 00:13:25,480 --> 00:13:28,850 So it doesn't matter if we multiply this one here. 164 00:13:28,870 --> 00:13:31,210 So this is a bit of a trick to make it a bit more easy. 165 00:13:32,050 --> 00:13:35,440 So let's restore the zero here that I deleted accidentally. 166 00:13:35,920 --> 00:13:38,830 And I think also here. 167 00:13:39,010 --> 00:13:40,600 Yeah, exactly. 168 00:13:42,670 --> 00:13:48,580 And here, here and here, this is exactly as we did before. 169 00:13:49,390 --> 00:13:52,330 I just accidentally deleted it when I updated these terms. 170 00:13:53,290 --> 00:13:59,410 So now we have our off diagonal terms, and this means we only have to update the Hamiltonian. 171 00:14:00,970 --> 00:14:02,890 So we right here one a 172 00:14:05,950 --> 00:14:07,540 one b one C. 173 00:14:10,320 --> 00:14:19,080 And the same, of course, four of diagonal to a b c. 174 00:14:22,020 --> 00:14:29,250 And I hope that now everything should work because we are then solving and adding it to the energy list 175 00:14:30,750 --> 00:14:33,870 and we reshape and let's have a look at the plots. 176 00:14:34,620 --> 00:14:35,910 Yes, that looks really good. 177 00:14:37,050 --> 00:14:40,530 So yeah, it seems that the beef was already quite large. 178 00:14:40,560 --> 00:14:42,360 Let me go to a very small value here. 179 00:14:42,960 --> 00:14:48,750 You run that again because first we have to check that for very small beef you values, we can restore 180 00:14:48,750 --> 00:14:52,260 the old result and actually change this one. 181 00:14:52,260 --> 00:14:55,050 You also to 160 so that we can compare it better. 182 00:14:56,880 --> 00:15:02,580 So this is the original result for 160 atoms and zero magnetic fields. 183 00:15:02,850 --> 00:15:07,650 And when we have just a small magnetic fields, then we will get a very similar results. 184 00:15:08,740 --> 00:15:10,960 So now let's see what happens if we increase the fields. 185 00:15:13,750 --> 00:15:16,420 So here only very small changes visible. 186 00:15:16,840 --> 00:15:23,050 So this would not really, I would say, would not correspond to any land or quantization, and no quantum 187 00:15:23,050 --> 00:15:24,430 hall effect would be measurable. 188 00:15:24,910 --> 00:15:27,520 But you see already a bit of a split up here. 189 00:15:28,490 --> 00:15:35,280 So now if we keep on increasing the value, then you see, oh no, it gets really interesting. 190 00:15:35,290 --> 00:15:41,860 So this blog area here totally the forms and especially in the middle, you see very flat levels and 191 00:15:41,860 --> 00:15:48,370 these are these so-called Landau levels to form when you apply large magnetic fields and that give rise 192 00:15:48,370 --> 00:15:49,840 to the quantum hall effect. 193 00:15:51,040 --> 00:15:53,470 So now let's go back to our value of zero point one. 194 00:15:54,820 --> 00:15:57,940 And here you see, the quantization is enormously. 195 00:15:58,030 --> 00:16:04,720 So you have these totally flat levels here, here, here, here and then in between, you have a few 196 00:16:04,720 --> 00:16:06,550 states that bridge these gaps. 197 00:16:07,270 --> 00:16:13,810 And these are in fact the so-called edge states that give rise to this quantum hall effect. 198 00:16:15,340 --> 00:16:22,060 So what we can say is, and I wrote this down here and the headline that we have a quantum hall effect, 199 00:16:22,060 --> 00:16:26,020 which basically means we have a metallic behavior due to etched channels. 200 00:16:26,950 --> 00:16:33,370 So it turns out that if you would only have these flat Landau levels here, our sample would be an insulator 201 00:16:33,370 --> 00:16:37,380 because there are these gaps and the electrons cannot bridge these gaps. 202 00:16:37,390 --> 00:16:40,210 So a current cannot really flow in the sample. 203 00:16:41,140 --> 00:16:46,690 And in fact, this is really what's happening in the inner part of the sample if we apply a large magnetic 204 00:16:46,690 --> 00:16:47,050 field. 205 00:16:48,850 --> 00:16:59,740 However, in the arch layers so pretty close to to these atoms here at the bottom end to top the electron 206 00:16:59,740 --> 00:17:05,770 behavior changes, and they will form such edge states that bridge the gaps and they will give rise 207 00:17:05,770 --> 00:17:11,260 to this very interesting conductivity phenomenon called quantum hall effect. 208 00:17:12,190 --> 00:17:18,369 And in fact, this phenomenon has to be analyzed, of course, in much more detail to really be able 209 00:17:18,369 --> 00:17:19,240 to understand it. 210 00:17:19,599 --> 00:17:25,839 But I think even this lecture was very difficult because I had to introduce these modified toppings, 211 00:17:25,839 --> 00:17:28,000 and it was quite difficult to program these. 212 00:17:28,420 --> 00:17:34,960 And so I would say we stop at this point with having the band structure, with the Landau levels and 213 00:17:34,960 --> 00:17:37,240 with the edge states bridging the gaps. 214 00:17:38,260 --> 00:17:44,800 So if we would now continue to analyze these edge states in terms of their so-called topological properties, 215 00:17:45,250 --> 00:17:52,780 we would at some point recover the results of these very clever men here, which got the Nobel Prize 216 00:17:52,780 --> 00:18:00,250 in 2016 for their theoretical discoveries of the topological phase transitions and topical topological 217 00:18:00,250 --> 00:18:01,360 phases of matter. 218 00:18:02,200 --> 00:18:06,790 So they did not only do this for graphene, they did this in general for the quantum hall effect and 219 00:18:06,790 --> 00:18:08,020 also for other systems. 220 00:18:08,500 --> 00:18:12,970 But here we have basically combined the quantum hall effect and graphene.