1 00:00:00,100 --> 00:00:05,206 So in this final part of the lecture we'll solve the Cooper problem, which is 2 00:00:05,206 --> 00:00:11,274 a problem of weakly attractive electrons near the thermos surface that form pairs, 3 00:00:11,274 --> 00:00:17,810 so-called Cooper pairs bound states of 2 electrons. 4 00:00:17,810 --> 00:00:22,226 And as we discussed these Cooper pairs, the, at low temperatures, they both 5 00:00:22,226 --> 00:00:26,386 condense into [INAUDIBLE] condensate should become superfluid, and this is 6 00:00:26,386 --> 00:00:31,364 this amazing state that we know as superconductor. 7 00:00:31,364 --> 00:00:34,724 I should mention that we will see actually the solution to the Cooper 8 00:00:34,724 --> 00:00:38,924 problem will involve essentially the same method we used in the first lecture this 9 00:00:38,924 --> 00:00:42,452 week in the context of single particles quantum mechanics, so it's going to be 10 00:00:42,452 --> 00:00:47,904 actually very simple. Technically, a very simple solution for a 11 00:00:47,904 --> 00:00:51,406 Nobel prize winning work anyway. So, it's not trivial but, you know, this 12 00:00:51,406 --> 00:00:55,206 is a very important work. So, here I'm actually showing the the 13 00:00:55,206 --> 00:00:58,640 title of the original paper by Leon Cooper. 14 00:00:58,640 --> 00:01:04,533 So and this work of Cooper eventually has progressed, has developed into a much 15 00:01:04,533 --> 00:01:09,503 more sophisticated theory, which is the Bardeen-Cooper-Schrieffer theory of 16 00:01:09,503 --> 00:01:15,132 superconductivity, also called the BCS theory. 17 00:01:15,132 --> 00:01:18,914 So, and this BCS theory has been the cornerstone of the theory of 18 00:01:18,914 --> 00:01:25,316 superconductivity for many, many years. And frankly, there is nothing better on 19 00:01:25,316 --> 00:01:30,768 the market in some sense. So, this BCS theory describes very well 20 00:01:30,768 --> 00:01:36,228 most conventional superconductors. And it provides some understanding also, 21 00:01:36,228 --> 00:01:39,900 in the physics of unconventional superconductors. 22 00:01:39,900 --> 00:01:43,695 Although they're sensuously a high temperatures super conductors for example 23 00:01:43,695 --> 00:01:47,30 although their understanding is not complete. 24 00:01:47,30 --> 00:01:51,432 now as I well I shown many times already excitations for various Nobel prizes and 25 00:01:51,432 --> 00:01:55,772 and that just caused briefly a number of Nobel prize winning works and they should 26 00:01:55,772 --> 00:02:01,238 mid, admit that. In most of these cases I had to sort of 27 00:02:01,238 --> 00:02:05,66 simplify things enormously when I was providing explanations, sort of dumb it 28 00:02:05,66 --> 00:02:09,260 down to some degree. So here, well I'm not going to discuss 29 00:02:09,260 --> 00:02:12,432 the BCS-theory, this is a very complicated theory, but the paper of 30 00:02:12,432 --> 00:02:17,453 Cooper is actually very, very simple. So, the original paper which started it 31 00:02:17,453 --> 00:02:20,410 all is very simple. And we will be able to understand 32 00:02:20,410 --> 00:02:23,940 essentially its main message and the derivation. 33 00:02:23,940 --> 00:02:28,660 Now but before going into this derivation let me discuss the origin of this 34 00:02:28,660 --> 00:02:32,863 attractive interaction. So I basically, w,hat I, what I want to 35 00:02:32,863 --> 00:02:37,259 solve again is a problem of electrons interacting via an attractive potential. 36 00:02:37,259 --> 00:02:41,600 But where is it exactly coming from? So this is actually very complicated and 37 00:02:41,600 --> 00:02:44,650 intriguing question and well, it sometimes the mystery of let's say high 38 00:02:44,650 --> 00:02:47,900 temperature of super conductivity is in that, we don't really know where it is 39 00:02:47,900 --> 00:02:52,920 sort glue this attractive interaction might be coming from. 40 00:02:52,920 --> 00:02:58,766 So the origin of electron interaction and conventional super conductors is also far 41 00:02:58,766 --> 00:03:03,750 from obvious actually. So, but there was a breakthrough paper 42 00:03:03,750 --> 00:03:08,761 back in the beginning of the 50s. so I'm showing you here the title of this 43 00:03:08,761 --> 00:03:13,420 paper, Superconductivity in Iso, of Isotopes of Mercury. 44 00:03:13,420 --> 00:03:17,11 And to appreciate the importance of this paper, let me just remind you of the 45 00:03:17,11 --> 00:03:22,526 basic picture of, what a metal is. So, well, if you have a metallic system, 46 00:03:22,526 --> 00:03:27,82 well, assorted system, so the reason it's assorted is because, the ions form a 47 00:03:27,82 --> 00:03:31,879 crystal lattice. So they are positioned in space in a 48 00:03:31,879 --> 00:03:35,988 regular fashion. So I'm just drawing here for the sake of 49 00:03:35,988 --> 00:03:40,132 simplicity, as a square a lot as 2D but in relatives, it could be more 50 00:03:40,132 --> 00:03:45,805 complicated. Three crystal lattice And, you should 51 00:03:45,805 --> 00:03:52,196 think about having positive ions sitting here in these slides, and these line are 52 00:03:52,196 --> 00:03:56,952 slides. And these lattice is sort of elastic, 53 00:03:56,952 --> 00:04:01,488 well it's a recent object but it can have a oscillations, waves running through it, 54 00:04:01,488 --> 00:04:06,24 and we're going to discuss a little bit in the few, in a few weeks, and so these 55 00:04:06,24 --> 00:04:11,970 oscillations of the lattice are called phonons. 56 00:04:11,970 --> 00:04:15,890 Now electrons in the metal are moving moving around, they're free to move 57 00:04:15,890 --> 00:04:21,150 around, but they move around in the on the background of this lattice. 58 00:04:21,150 --> 00:04:24,490 So this red guys here are electrons. Okay? 59 00:04:24,490 --> 00:04:29,885 So this is pretty much what, well it's very [UNKNOWN] model, simplified, over 60 00:04:29,885 --> 00:04:36,814 simplified model of, metallic soil. Now, so this paper, what this paper has 61 00:04:36,814 --> 00:04:41,106 achieved is that it looked at different isotopes of mercury which is a 62 00:04:41,106 --> 00:04:46,644 superconductor. And let me remind you that the isotopes 63 00:04:46,644 --> 00:04:51,90 are essentially different versions of the same chemical elements. 64 00:04:51,90 --> 00:04:56,112 and basically everything is the same for different isotopes apart from the number 65 00:04:56,112 --> 00:05:01,608 of neutrons in each ion sitting on the lattice size, but basically. 66 00:05:01,608 --> 00:05:05,204 So different isotopes means that everything is the same, but the only 67 00:05:05,204 --> 00:05:09,710 difference is the mass of these objects on the lattice size. 68 00:05:09,710 --> 00:05:13,652 Which well now really shouldn't affect too much superconductivity because 69 00:05:13,652 --> 00:05:17,432 clearly superconductivity's coming from whatever electron, electrons are doing 70 00:05:17,432 --> 00:05:21,596 and it shouldn't have much to do with the lattice. 71 00:05:21,596 --> 00:05:23,714 Right? So because these guys are actually, well 72 00:05:23,714 --> 00:05:26,990 they can oscillate a little bit near their equilibrium positions, but they 73 00:05:26,990 --> 00:05:31,760 cannot conduct electricity. So as it turns out, and this is really 74 00:05:31,760 --> 00:05:35,595 the main message of the paper, that the transition temperature, in the 75 00:05:35,595 --> 00:05:39,666 superconductor, the temperature at which the resistivity drops exactly to zero, 76 00:05:39,666 --> 00:05:45,790 depended very strongly on the mass of these guys. 77 00:05:45,790 --> 00:05:52,80 So and the behavior the ions the lower was the transition temperatures. 78 00:05:52,80 --> 00:05:56,234 So here maybe it is hard to see so there are a few points so this has average mass 79 00:05:56,234 --> 00:06:00,245 number and this has transition temperature. 80 00:06:00,245 --> 00:06:05,105 So so this is basically let me write a tc and this is the mass of this ions on 81 00:06:05,105 --> 00:06:10,134 lattice size. Okay and so this was a very clear-cut 82 00:06:10,134 --> 00:06:15,237 trend and it was a smoking gun of that superconductivity had something to do 83 00:06:15,237 --> 00:06:22,720 With interactions between the electrons, and decrease the lattice. 84 00:06:22,720 --> 00:06:27,964 So a little later, people realized that what actually happens is that this these 85 00:06:27,964 --> 00:06:32,932 electrons, moving around, they whenever the electron passes through a region in a 86 00:06:32,932 --> 00:06:39,231 lattice, essentially because electron's negatively charged. 87 00:06:39,231 --> 00:06:43,787 And the ions sitting on lattice size are positively charged, so this electron 88 00:06:43,787 --> 00:06:49,580 polarizes the lattice, okay, so locally. And it takes some time for the lattice 89 00:06:49,580 --> 00:06:53,284 sort of to relax back. And then if a second electron comes in, 90 00:06:53,284 --> 00:06:56,612 so, let's say I have here now this guy goes away, and there's a second electron 91 00:06:56,612 --> 00:07:02,343 which comes in into the same, region. So, this, electron, this second electron 92 00:07:02,343 --> 00:07:06,843 is attracted in some sense to, to this region and, this, results in an effective 93 00:07:06,843 --> 00:07:13,60 correlation, effective, phonon-mediated attraction between the electrons. 94 00:07:13,60 --> 00:07:15,547 It's a very tricky business. So, I, if you don't understand it, well, 95 00:07:15,547 --> 00:07:17,925 don't be surprised. I don't understand it either. 96 00:07:17,925 --> 00:07:21,509 Attraction but that's how it works so, we'll have to you know just accept this 97 00:07:21,509 --> 00:07:24,594 fact. So that what happens is that electrons in 98 00:07:24,594 --> 00:07:28,626 some sense they exchange waves elastic waves which are called phonons and this 99 00:07:28,626 --> 00:07:33,620 leads to effective attraction. Well, and by the way for the model for 100 00:07:33,620 --> 00:07:37,841 the actual solution, I'm going to present the origin of the attraction does matter 101 00:07:37,841 --> 00:07:41,558 that much it will be just some attraction, some constant minus v not 102 00:07:41,558 --> 00:07:47,469 but, you know, it's good to know the physics behind it. 103 00:07:47,469 --> 00:07:51,757 For the actual compilation we'll need a specific mathematical model of this 104 00:07:51,757 --> 00:07:56,270 electron, electron attraction. And the true attraction the true 105 00:07:56,270 --> 00:07:59,130 interaction is actually quite complicated. 106 00:07:59,130 --> 00:08:02,784 But here I will present what I call a spherical cow model of the phonon 107 00:08:02,784 --> 00:08:08,728 mediated attraction between electrons. which I should say actually is used 108 00:08:08,728 --> 00:08:12,910 commonly even in research papers. So it works perfectly well. 109 00:08:12,910 --> 00:08:17,509 And even though we know how to write the true model which would involve all the 110 00:08:17,509 --> 00:08:22,345 complications of the theory. So the results, the outcomes, of the 111 00:08:22,345 --> 00:08:25,810 simplified model and the I'm not going to give a little more, or pretty close to 112 00:08:25,810 --> 00:08:31,40 each other, so there is really no reason for us to complicate things. 113 00:08:31,40 --> 00:08:35,371 In any case, so what we were talking about is that again, electrons moving 114 00:08:35,371 --> 00:08:39,631 around in a presence of a crystal lattice, they exchange, waves drawing 115 00:08:39,631 --> 00:08:46,205 through the lattice are these phonons. And so this diagram is an example of a 116 00:08:46,205 --> 00:08:50,950 fine metallic like diagram which shows this kind of a change. 117 00:08:50,950 --> 00:08:55,795 And in any process like this we must satisfy basic conservation laws 118 00:08:55,795 --> 00:09:03,170 particularly when must ensure that both energy and momentum are conserved. 119 00:09:03,170 --> 00:09:08,474 So and therefore what's important here in understanding the structure of this 120 00:09:08,474 --> 00:09:13,890 interaction is the typical energies involved in such processes. 121 00:09:13,890 --> 00:09:17,90 So what we know, what we've already discussed is the the typical energy of 122 00:09:17,90 --> 00:09:22,500 the electrons in the in the middle. So and this typical energy, I denoted as 123 00:09:22,500 --> 00:09:26,855 E sub F, the Fermi energy divided by the Boseman constant, sort of to convert into 124 00:09:26,855 --> 00:09:31,730 a more, familiar temperature units so the corresponding typical sort of temperature 125 00:09:31,730 --> 00:09:37,560 of electrons is going to be about 10,000 Kelvin. 126 00:09:37,560 --> 00:09:40,570 So this guys which are actually the main players. 127 00:09:40,570 --> 00:09:45,730 Of the theory have the energy, which you shall see with this temperature scale. 128 00:09:45,730 --> 00:09:51,650 Now it turns out that the phonons, it's not obvious from the previous discussion. 129 00:09:51,650 --> 00:09:55,682 It's not something we can get at, I'm just giving you an experimental fact, is 130 00:09:55,682 --> 00:09:59,273 that if the typical energy of the phonons, which I will call s h times I 131 00:09:59,273 --> 00:10:05,940 mean divide, so called divide frequency divided by the Boseman constant. 132 00:10:05,940 --> 00:10:10,100 So when convert it into temperatures, it's about 400 kelvin. 133 00:10:10,100 --> 00:10:13,14 A few hundred kelvin, so, which is two orders of magnitude smaller than the 134 00:10:13,14 --> 00:10:16,804 energy of the electron. So what we should think about is that, 135 00:10:16,804 --> 00:10:21,156 this phonon actually being exchanged by the electrons have energy which is much, 136 00:10:21,156 --> 00:10:25,337 much more than the electron energy itself. 137 00:10:25,337 --> 00:10:31,177 And, this, puts certain constrains on the possible, momentum and energies of the 138 00:10:31,177 --> 00:10:36,50 electrons that may experience sort of a process. 139 00:10:36,50 --> 00:10:40,610 And the simplest, sort of, version of the interaction which takes this into account 140 00:10:40,610 --> 00:10:45,592 is, presented here. So this is an equation for an 141 00:10:45,592 --> 00:10:50,272 interaction, momentum independent interaction between the two electrons V 142 00:10:50,272 --> 00:10:55,24 of p and what it tells us that the two electrons attract each other in this 143 00:10:55,24 --> 00:10:59,704 language if they are located in the narrow shell near the Fermi surface and 144 00:10:59,704 --> 00:11:08,490 the width of the shell is of the order of this phonon energy. 145 00:11:08,490 --> 00:11:11,955 So basically if, two electrons can exchange phonons, they do so and this 146 00:11:11,955 --> 00:11:15,365 results in the attraction which is manifested here through this negative 147 00:11:15,365 --> 00:11:20,190 coefficient. And V naught is some constant. 148 00:11:20,190 --> 00:11:25,230 And if two electrons are located well, beyond this narrow region So then they 149 00:11:25,230 --> 00:11:29,630 don't have a mechanism to attract each other. 150 00:11:29,630 --> 00:11:35,570 And their interaction, therefore, is 0. So if we look just at the upper part. 151 00:11:35,570 --> 00:11:40,628 So v of p is equal to minus v, 0. So this would correspond to, essentially, 152 00:11:40,628 --> 00:11:44,736 a local attraction real speed. So this is something that we actually 153 00:11:44,736 --> 00:11:48,262 saw. In, lecture number five for the delta 154 00:11:48,262 --> 00:11:51,800 potential. So this is the kind of problem we, 155 00:11:51,800 --> 00:11:56,806 discussed already. And, but, however, then, we were talking 156 00:11:56,806 --> 00:12:01,910 about the, one particle in a, in an quantum well. 157 00:12:01,910 --> 00:12:06,667 So now we're talking about two particles two excitations interacting with each 158 00:12:06,667 --> 00:12:11,559 other by this potential. But per, well put the results of the 159 00:12:11,559 --> 00:12:16,743 previous segment in this lecture we know that the two particle problem and quantum 160 00:12:16,743 --> 00:12:21,711 mechanics is to a large degree equivalent to a single particle quantumic angle 161 00:12:21,711 --> 00:12:25,959 problems and as we discussed in the previous video the only difference 162 00:12:25,959 --> 00:12:33,673 between the. single particle problem in potential v of 163 00:12:33,673 --> 00:12:38,710 r and 2 particle problem of 2 particles interacting by the same potential is that 164 00:12:38,710 --> 00:12:43,333 in the lecture we will get the reduced mass instead of the mass of eac, of each 165 00:12:43,333 --> 00:12:50,284 individual particle. And in the case of two identical 166 00:12:50,284 --> 00:12:53,680 electrons the reduced mass is equal to m r 2. 167 00:12:53,680 --> 00:12:56,530 So we, we have this simply this Schrodinger equation. 168 00:12:56,530 --> 00:13:00,235 And here I write it first in real space sort of but keeping in mind that there 169 00:13:00,235 --> 00:13:04,460 are in fact additional constraints on the interaction. 170 00:13:04,460 --> 00:13:08,816 So if we do if we follow the same route that we did in lecture number five and do 171 00:13:08,816 --> 00:13:14,666 Fourier transform. So, we can write this equation in this 172 00:13:14,666 --> 00:13:20,630 form, so delta of r emptying on psi of r picks up psi of of 0. 173 00:13:20,630 --> 00:13:24,310 And so here I'm going to have this integral over all momentum. 174 00:13:24,310 --> 00:13:28,730 But basically a way to introduce constraints on the potential which we 175 00:13:28,730 --> 00:13:33,82 just discussed would be to limit the integration or momentum here by only 176 00:13:33,82 --> 00:13:37,910 momenta that appear in the vicinity of the Fermi surface, you know, so that the 177 00:13:37,910 --> 00:13:46,203 possibility of phonon exchange exists. And so this equation essentially, these 2 178 00:13:46,203 --> 00:13:50,490 equations with the appropriate sort of caveats. 179 00:13:50,490 --> 00:13:54,434 About where this reaction is possible is exactly the Cooper pairing problem, and 180 00:13:54,434 --> 00:13:58,784 as you can see it's essentially identical mathematically to the kinds of problems 181 00:13:58,784 --> 00:14:05,10 we saw in lecture number five in the single particle of quantum mechanics. 182 00:14:05,10 --> 00:14:09,630 And so, well we can we can follow the same route but at at this stage we may be 183 00:14:09,630 --> 00:14:14,506 a little bit surprised by this resemblance. 184 00:14:14,506 --> 00:14:18,598 Because, I mentioned that this, eventually the result of this calculation 185 00:14:18,598 --> 00:14:23,905 is going to be the appearance of a bound state between the two electrons. 186 00:14:23,905 --> 00:14:28,461 But, as we discussed, weak attraction in, three dimensions, and this is a 187 00:14:28,461 --> 00:14:33,330 3-dimensional problem, does not result in a bound state. 188 00:14:33,330 --> 00:14:36,792 So delta function potentially 3D does not have a bound state. 189 00:14:36,792 --> 00:14:41,622 Uh,however, an interesting thing that happens here is that because electrons 190 00:14:41,622 --> 00:14:46,592 really play a role in, in this, pairing and this interaction, with for-nodes, 191 00:14:46,592 --> 00:14:52,190 they exist in the vicinity of this firmer surface. 192 00:14:52,190 --> 00:14:56,192 And this surface is two dimensional. This, gives rise, in a sense, to 193 00:14:56,192 --> 00:15:01,244 reduction of dimensionality. Although, in real space, we started with 194 00:15:01,244 --> 00:15:06,488 a three dimensional, system, so this effectively what you are dealing with, 195 00:15:06,488 --> 00:15:13,72 with here has used dimension. And this gives rise to the appearance of 196 00:15:13,72 --> 00:15:19,526 a bond state in a very unexpected way. So, to see this we essentially have to 197 00:15:19,526 --> 00:15:27,382 repeat exactly the same steps as we did in parts 3 and 4 of lecture number 5. 198 00:15:27,382 --> 00:15:31,870 So, so the only difference here is due to the fact that this energy in the left 199 00:15:31,870 --> 00:15:38,570 hand, in the right-hand side, if we want to find, really consists of two parts. 200 00:15:38,570 --> 00:15:42,350 So this is the energy of the electrons that already have a finite energy at the 201 00:15:42,350 --> 00:15:46,592 at the Fermi surface. So remember is that we're talking here 202 00:15:46,592 --> 00:15:51,80 about these excitations that have the energy of two E-Fermi, and then there is 203 00:15:51,80 --> 00:15:56,960 an energy due to the interaction which we want to negative. 204 00:15:56,960 --> 00:16:00,672 So if this guy delta is negative, it means that we can have former bonds 205 00:16:00,672 --> 00:16:05,30 stayed, and by doing so lowering, lower the energy. 206 00:16:05,30 --> 00:16:09,400 So and well this fact actually makes all the difference. 207 00:16:09,400 --> 00:16:13,365 Actually this is how mathematically the fact that we're dealing with this 208 00:16:13,365 --> 00:16:19,775 manifest itself in this equation. Since the remaining calculation follows 209 00:16:19,775 --> 00:16:24,350 almost 1 to 1, that in lecture number 5, and I also present this particular 210 00:16:24,350 --> 00:16:28,775 calculation, all the details in additional materials, so you can read 211 00:16:28,775 --> 00:16:34,122 through this. So I'm not going to repeat this again in 212 00:16:34,122 --> 00:16:37,139 this video. So what I'm going to do, I'm going to 213 00:16:37,139 --> 00:16:41,43 just present the final result for this Our key parameter delta which again is 214 00:16:41,43 --> 00:16:44,703 the energy of the bounced state of two electrons that form sort of a large 215 00:16:44,703 --> 00:16:50,430 molecule, if you want, which is called Gutterberg here. 216 00:16:50,430 --> 00:16:55,638 So and this delta is is going to be of the water of minus h of energy so this is 217 00:16:55,638 --> 00:17:03,890 phonon energy, typical phonon energy that we discussed in the previous slide. 218 00:17:03,890 --> 00:17:07,402 But the key thing here is that there is multiplied by an. 219 00:17:07,402 --> 00:17:11,489 Exponential of minus one, some constant, I'll call it n naught, and define it in a 220 00:17:11,489 --> 00:17:17,298 second, times v naught. So where n naught is equal to m, p fermi, 221 00:17:17,298 --> 00:17:25,910 so this is this threshold momentum, divided by 4 pi squared, hq. 222 00:17:25,910 --> 00:17:29,15 So this is just constant, but what I want to emphasize, well, first of all, there 223 00:17:29,15 --> 00:17:33,64 is a solution, which is great. So the electrons indeed,can find a way to 224 00:17:33,64 --> 00:17:36,424 lower their energy by forming these [UNKNOWN] that eventually form the 225 00:17:36,424 --> 00:17:41,641 super-conductor. but what is not so great is that these 226 00:17:41,641 --> 00:17:50,800 small interaction energy appears in the exponential in, minus 1 over d naught. 227 00:17:50,800 --> 00:17:55,675 So if you notice, small then, well the, energy of the bound state, so 228 00:17:55,675 --> 00:18:02,630 called,[UNKNOWN], is exponentially small and this is exactly what happens. 229 00:18:02,630 --> 00:18:09,630 And those of you followed, all segments in lecture number five can recognize. 230 00:18:09,630 --> 00:18:13,506 This result is a result that we have seen, actually, in the context of 231 00:18:13,506 --> 00:18:19,614 two-dimensional, single particle problem. And this reflects this reduction in 232 00:18:19,614 --> 00:18:24,546 dimensionality that I mentioned before. So, the last two things I'm going to 233 00:18:24,546 --> 00:18:27,900 mention in the end is. First of all, this result. 234 00:18:27,900 --> 00:18:31,853 For delta is a function of not is a very unusual function which we've already seen 235 00:18:31,853 --> 00:18:37,390 but, this function is special because it doesn't have a Taylor expansion. 236 00:18:37,390 --> 00:18:41,95 So there is no way one can approach this result by doing so called perturbation 237 00:18:41,95 --> 00:18:44,620 theory. And this may be one of the reasons why it 238 00:18:44,620 --> 00:18:49,84 took so long for people to figure out the heat of super conductivity, because there 239 00:18:49,84 --> 00:18:54,460 was no way to approach, it sort of in a conservative way. 240 00:18:54,460 --> 00:18:58,118 And, the last thing I am going to mention, is that well, if we did not have 241 00:18:58,118 --> 00:19:01,900 this exponential, if we were to image that this delta was the order of this 242 00:19:01,900 --> 00:19:06,880 phonon energy. This would have actually implied that we 243 00:19:06,880 --> 00:19:12,358 can get super conductivity at very high temperatures up to room temperatures. 244 00:19:12,358 --> 00:19:17,390 So, this would have been great we would have the ability to transport electricity 245 00:19:17,390 --> 00:19:21,696 with no losses. So, this is really the grand challenge in 246 00:19:21,696 --> 00:19:26,755 the field of how to go beyond this coupling this so called weak coupling. 247 00:19:26,755 --> 00:19:30,590 Bearing model, so how to first of all describe this situation where the 248 00:19:30,590 --> 00:19:34,815 interaction is so strong and how to get rid of this exponentially this is a sort 249 00:19:34,815 --> 00:19:40,784 of practical problem that there's no solution at the moment.