1 00:00:00,220 --> 00:00:04,114 So in the second part of this segment, I'm going to elaborate on the weak 2 00:00:04,114 --> 00:00:08,272 localization phenomenon, and in particular, I will discuss in more detail 3 00:00:08,272 --> 00:00:12,041 the self-crossing trajectories that are, are responsible for it. 4 00:00:12,041 --> 00:00:17,644 And in particular, I will identify the physical circumstances under which they 5 00:00:17,644 --> 00:00:22,125 are actually important. And this discussion is going to be under 6 00:00:22,125 --> 00:00:27,150 the title I put here, Quantum corrections to diffusion, but before going back to 7 00:00:27,150 --> 00:00:32,250 these quantum corrections that we already mentioned in the previous video let me 8 00:00:32,250 --> 00:00:35,756 remind you what classical diffusion actually is. 9 00:00:35,757 --> 00:00:40,079 Actually the phenomena of diffusion you can observe at home by performing a 10 00:00:40,079 --> 00:00:45,910 rather, very simple experiment. If you put a droplet of dye in water and 11 00:00:45,910 --> 00:00:51,791 watch it spread out with time. So this evolution of this droplet, time 12 00:00:51,791 --> 00:00:55,158 evolution. Spreading out of this droplet is going to 13 00:00:55,158 --> 00:00:59,379 be described by the diffusion equation presented here with some diffusion 14 00:00:59,379 --> 00:01:02,970 coefficient D. So but the phenomenal diffusion the 15 00:01:02,970 --> 00:01:08,107 diffusion equation itself are of course much more general than this particular 16 00:01:08,108 --> 00:01:14,716 physical realization. And they happened whenever we have a large 17 00:01:14,716 --> 00:01:18,917 number of particles experiencing a random walk. 18 00:01:18,918 --> 00:01:23,754 So random is, if you can, you can think about it as particle which moves every 19 00:01:23,754 --> 00:01:28,362 time step it moves in a certain direction it can go up, it can go right, it can go 20 00:01:28,362 --> 00:01:31,975 up, it can go down, it can go left, up et cetera, et cetera. 21 00:01:31,976 --> 00:01:36,880 And so, at each moment of time there is a probability, a equal probability to go in 22 00:01:36,880 --> 00:01:41,300 any, in any direction. And so, if a lot of size particles are put 23 00:01:41,300 --> 00:01:44,540 together. They, if, and if you look at those 24 00:01:44,540 --> 00:01:48,753 particles from a large length scale. So, their density. 25 00:01:48,754 --> 00:01:52,184 The evolution of their density is going to be described by this equation. 26 00:01:52,185 --> 00:01:57,172 As with here rho is in general a, a function of a position and time. 27 00:01:57,173 --> 00:02:02,861 So, how it is relevant to the previous discussion about electrons moving in the 28 00:02:02,861 --> 00:02:06,735 metal? So there as we mentioned, so the disorders 29 00:02:06,735 --> 00:02:11,053 the imperfections in the metal play the role of scatterers. 30 00:02:11,053 --> 00:02:14,909 That's somehow randomized the electron trajectories. 31 00:02:14,909 --> 00:02:18,934 And so, we can think of what electrons experiencing this random walk. 32 00:02:18,934 --> 00:02:24,169 And since there are many, many such electrons in a piece of metal, so if we 33 00:02:24,169 --> 00:02:30,960 want to extract their density the density is going to follow the diffusion equation. 34 00:02:30,960 --> 00:02:35,825 So another interpretation of the diffusion equation is that this raw Is actually 35 00:02:35,825 --> 00:02:40,970 probability density of finding a particular a random working particle in a 36 00:02:40,970 --> 00:02:45,813 certain moment of time at a certain moment of time in a certain position in space. 37 00:02:45,813 --> 00:02:50,754 So there's probability interpretation of this equation if you want. 38 00:02:50,754 --> 00:02:54,324 Now another comment I would like to make about this equation is that actually in 39 00:02:54,324 --> 00:02:58,354 some sense, it looks similar to the Schrodinger equation in quantum mechanics. 40 00:02:58,355 --> 00:03:04,000 So if we were to put in the measuring constant i here, which we don't have to 41 00:03:04,000 --> 00:03:09,200 put, but if we were imagine that some measuring time, so and so the coefficient 42 00:03:09,200 --> 00:03:12,534 D we would have let's say minus h squared over 2 m. 43 00:03:12,535 --> 00:03:18,078 Then this equation would have become a free [unknown] equation of a free particle 44 00:03:18,078 --> 00:03:22,056 of the wave function. So this is a purely mathematical analogy I 45 00:03:22,056 --> 00:03:26,612 should Could say, but this mathematical analogy in certain cases actually goes a 46 00:03:26,612 --> 00:03:31,034 long way, and allows us to calculate things that otherwise very, would have 47 00:03:31,034 --> 00:03:36,130 been very difficult to calculate. So, at this stage however, we don't want 48 00:03:36,130 --> 00:03:41,818 necessarily to involve any, this analogy. Instead let me focus on the question of 49 00:03:41,818 --> 00:03:46,814 relevance to the phenomenon of weak localization that I mentioned. 50 00:03:46,815 --> 00:03:51,655 Namely on the self-crossing trajectory. So here, I'm talking about self-crossing 51 00:03:51,655 --> 00:03:55,979 trajectories of some diffusive particle. So let's say, here actually, what I mean 52 00:03:55,979 --> 00:04:00,263 is that there is some small length scale at which particle diffuses around, but as 53 00:04:00,263 --> 00:04:04,484 a result, in a realization of this random walk, it may lead to this self-crossing 54 00:04:04,484 --> 00:04:07,648 trajectory. And so from the perspective of this 55 00:04:07,648 --> 00:04:10,633 crossing point, what it actually means is that. 56 00:04:10,634 --> 00:04:14,706 I'm asking the question of what is the probability for a particle to return to 57 00:04:14,706 --> 00:04:19,160 the same point where it started. So let's say I set an initial condition 58 00:04:19,160 --> 00:04:23,240 for my density at t equals 0 to be a delta function in space. 59 00:04:23,240 --> 00:04:28,319 A delta, very sharply localized density using a particular point. 60 00:04:28,320 --> 00:04:32,304 Now it turns out one can actually solve for exactly the diffusion equation with 61 00:04:32,304 --> 00:04:35,550 this initial condition is actually called the room function. 62 00:04:35,551 --> 00:04:40,917 In this fundamental solution the general solution to the diffusion equation is 63 00:04:40,917 --> 00:04:45,192 written here. So here again r denotes the coordinate in 64 00:04:45,192 --> 00:04:51,464 space at T star, D is the diffusion coefficient and the small d denotes the 65 00:04:51,464 --> 00:04:55,340 dimensionality of space. So it can be either one dimensional space 66 00:04:55,340 --> 00:04:57,683 or three dimensional space, or two dimensional space. 67 00:04:57,684 --> 00:05:03,580 And in order to, for us to find the probability to return to the origin. 68 00:05:03,580 --> 00:05:07,064 Let's say we have, we are looking for the probability of a self rising trajectory 69 00:05:07,064 --> 00:05:12,014 which starts from r equals zero. And goes back to r equals 0. 70 00:05:12,014 --> 00:05:15,906 So it means that we have to set r to 0 in this ex, expression or set expon, 71 00:05:15,906 --> 00:05:19,859 exponential to 1 and that's what we're going to get for the probability of 72 00:05:19,859 --> 00:05:23,730 self-intersection. So again, what this guy implies is that if 73 00:05:23,730 --> 00:05:28,830 we take random working particles, put it by hand in the origin and let it random 74 00:05:28,830 --> 00:05:31,581 walk. So this guy gives us the probability of 75 00:05:31,581 --> 00:05:36,169 its return to the vicinity of the probability density, more precisely the 76 00:05:36,169 --> 00:05:39,602 return to the vicinity of this point in the time sheet. 77 00:05:39,602 --> 00:05:43,263 So, now let us recall about what we were talking about this self crossing 78 00:05:43,263 --> 00:05:47,338 trajectories. So the context in which they appeared was 79 00:05:47,338 --> 00:05:53,036 the diffusion of electrons in the metal and we wanted to see again, we wanted to 80 00:05:53,036 --> 00:05:58,426 find the probability of where an electron could diffuse from an initial point and at 81 00:05:58,426 --> 00:06:04,294 some final point. And there are many different trajectories 82 00:06:04,294 --> 00:06:08,670 of the q that so and we focused on the quantum terms. 83 00:06:08,670 --> 00:06:14,231 The quantum interference terms were different trajectories where l1 not equal 84 00:06:14,231 --> 00:06:20,207 to l2 would interfere with one another and the interference term was this cosine of 85 00:06:20,207 --> 00:06:26,349 the difference in length between two given trajectories, times p fermi, the typical 86 00:06:26,349 --> 00:06:31,430 momentum of an electron divided by h bar. And the statement was, unless l1 was 87 00:06:31,430 --> 00:06:35,667 actually equal to l2 this interference term would kill each other. 88 00:06:35,668 --> 00:06:39,828 Or every [unknown] to zero, because you will get just a collection of positive and 89 00:06:39,828 --> 00:06:42,673 negative random numbers and on average you will get 0. 90 00:06:42,673 --> 00:06:47,716 But, for the special kinds of terms where you can go clockwise or counterclockwise 91 00:06:47,716 --> 00:06:52,336 around the certain trajectory, this delta l here would be 0 and loci of 0 is equal 92 00:06:52,336 --> 00:06:56,000 to 1 so we'll get prescribed contribution from these guys. 93 00:06:56,000 --> 00:07:00,270 And so what we're trying to figure out now is how probabilities for us to actually 94 00:07:00,270 --> 00:07:04,479 get the self crossing trajectory[UNKNOWN] and this is, is determined by this 95 00:07:04,479 --> 00:07:07,975 equation. But this guy tells us this probability 96 00:07:07,975 --> 00:07:12,687 density to return to, to an original point in the particular time t. 97 00:07:12,688 --> 00:07:17,102 While what're what we're interested in is any self-intersection, it doesn't have to 98 00:07:17,102 --> 00:07:19,869 a-, occur necessarily in this particular time t. 99 00:07:19,870 --> 00:07:24,088 It has to occur at some point. So in order to calculate the total 100 00:07:24,088 --> 00:07:30,022 probability which I will denote as P sub total, I have to integrate this expression 101 00:07:30,022 --> 00:07:33,590 over time, from some minimal, t to some maximal t. 102 00:07:33,590 --> 00:07:39,670 So in here, I should mention that I'm dropping the overall coefficients. 103 00:07:39,670 --> 00:07:46,946 I'm not keeping track of the coefficients. I'm just focusing on the most important 104 00:07:46,946 --> 00:07:50,015 truth. Now the minimal time here is the minimal 105 00:07:50,015 --> 00:07:54,890 time that the diffusion equation is able to resolve being an approximation it 106 00:07:54,890 --> 00:08:00,440 cannot describe the motion shorter in time than the consecutive collision let's say 107 00:08:00,440 --> 00:08:05,840 of the electron between two [unknown] so this guys of the order tau was actually 108 00:08:05,840 --> 00:08:09,183 introduced. In the beginning of the previous video, 109 00:08:09,183 --> 00:08:13,080 but I should mention that it's actually not important, what it actually is. 110 00:08:13,080 --> 00:08:17,121 It, it is only important that there is some cutoff. 111 00:08:17,121 --> 00:08:20,900 On the other hand, if there is no dephasing processes, if there is, you 112 00:08:20,900 --> 00:08:25,645 know, the phase, quantum mechanical phase is perfectly preserved, then there is no 113 00:08:25,645 --> 00:08:28,964 maximum cutoff, so this guy actually goes to infinity. 114 00:08:28,965 --> 00:08:35,470 And therefore, the integral that we actually have is an integral from sum tau 115 00:08:35,470 --> 00:08:39,033 to infinity of d, t over t to the power of d over 2. 116 00:08:39,033 --> 00:08:41,637 And this integral we can now easily calculate. 117 00:08:41,638 --> 00:08:48,216 And what is extremely important, what is crucial Is that this integral is finite 118 00:08:48,216 --> 00:08:55,272 if, we are in 3-dimensional space in which we live, but it's infinite, it's equal to 119 00:08:55,272 --> 00:08:59,359 infinity, if we are in one or two dimensions. 120 00:08:59,359 --> 00:09:04,480 So physically it implies the following so that if we start let's say, imagine you 121 00:09:04,480 --> 00:09:09,172 are in a spaceship and you start, you're sort of flying around in completely random 122 00:09:09,172 --> 00:09:13,932 directions in 3 dimensions, so this result tells us that there are going, you're not 123 00:09:13,932 --> 00:09:17,710 going to, you're going to be lost. You're never going to return to the point 124 00:09:17,710 --> 00:09:21,258 where you started from. On the other hand, if you start walking 125 00:09:21,258 --> 00:09:26,718 around in a city, in a town, in a city you don't know, and you, you walk randomly 126 00:09:26,718 --> 00:09:32,256 long enough, you're actually guaranteed to return to the vicinity of the point where 127 00:09:32,256 --> 00:09:35,447 you started. From the point of view of the electrons, 128 00:09:35,447 --> 00:09:39,429 so this result implies I think we have a motion of electrons in a. 129 00:09:39,430 --> 00:09:43,252 One and two dimensions. So there will be an infinite number of 130 00:09:43,252 --> 00:09:47,400 self-rising trajectories and the probability of self rising is going to be 131 00:09:47,400 --> 00:09:50,265 equal to 1. So this term is actually going to matter 132 00:09:50,265 --> 00:09:54,666 enormously at low temperatures. And they're going to blow up and lead to 133 00:09:54,666 --> 00:09:59,312 localization in, in low dimensions and this is actually what happens. 134 00:09:59,312 --> 00:10:04,431 And so if we now actually think about the physical consequences, of course, again, 135 00:10:04,431 --> 00:10:09,426 so this, in principle everything I'm talking about here is rather complicated 136 00:10:09,426 --> 00:10:12,562 theory, so if we were to study it in full details. 137 00:10:12,563 --> 00:10:17,780 But, I just basically want to, maybe I will leave this result here. 138 00:10:17,780 --> 00:10:21,940 The totally probability of self-intersection is really the key. 139 00:10:21,940 --> 00:10:25,870 The reason for this video is just to give you this result essentially. 140 00:10:25,870 --> 00:10:30,493 What it tells us in conjunction with the path integral argument and some iterative 141 00:10:30,493 --> 00:10:33,488 arguments. Is that in low dimensions the electrons 142 00:10:33,488 --> 00:10:35,730 are going to get localized at lower temperatures. 143 00:10:35,730 --> 00:10:38,510 They're going to tend to return to the same point. 144 00:10:38,510 --> 00:10:42,890 And this localization is going to be consequence of the, wave nature of 145 00:10:42,890 --> 00:10:45,810 particles. From the experimental point of view, what 146 00:10:45,810 --> 00:10:50,026 it actually menas is that If you perform a measurement of, let's say, resistance, 147 00:10:50,026 --> 00:10:53,160 which is 1 over conductivity, as a function of temperature. 148 00:10:53,160 --> 00:10:58,030 So at high temperature we're, very with [unknown] disturb the phase. 149 00:10:58,030 --> 00:11:02,320 So this is quantum interference phenomena. Basically unimportant whether or not the 150 00:11:02,320 --> 00:11:05,817 trajectories cross or not doesn't, doesn't play any role. 151 00:11:05,818 --> 00:11:09,429 So you will have some resistance, and it will flow actually decrease. 152 00:11:09,430 --> 00:11:13,704 But then, at some point, it will actually shoot up and become an insulator. 153 00:11:13,704 --> 00:11:18,748 Now, in this qualitative curve, so this part of the curve belongs to the weak 154 00:11:18,748 --> 00:11:23,082 localization phenomenon that we have been discussing. 155 00:11:23,082 --> 00:11:29,093 And this part is the classical due to conducitivity which goes back to more than 156 00:11:29,093 --> 00:11:32,301 100 years ago. So, and the prediction is that in one and 157 00:11:32,301 --> 00:11:36,522 two dimensions, per this argument, these quantum interference terms are going to 158 00:11:36,522 --> 00:11:40,428 take over, and they're going to get an insulator, while in three dimensions, 159 00:11:40,428 --> 00:11:44,712 actually, what's expected is they're going to remain metal, and the resistivity is 160 00:11:44,712 --> 00:11:47,569 going to saturate in 3D at some finite value. 161 00:11:47,570 --> 00:11:52,577 And this is indeed what happens. So, and there do exist quasi one 162 00:11:52,577 --> 00:11:58,401 dimensional and quasi two dimensional [unknown] let's say nano wires and so 163 00:11:58,401 --> 00:12:04,498 called semi conductor heterostructures, two dimensional fields where this is 164 00:12:04,498 --> 00:12:07,640 observed Zero. So I should emphasize again that the full 165 00:12:07,640 --> 00:12:11,160 theory of weak localization is an extremely complicated theory, and we just 166 00:12:11,160 --> 00:12:14,926 got a flavor of it. So hopefully but it's not the full story 167 00:12:14,926 --> 00:12:20,029 and interestingly enough, what I told you about here actually this weak 168 00:12:20,029 --> 00:12:24,109 localization, is a precursor to a different phenomenon. 169 00:12:24,110 --> 00:12:30,143 Which is actually called strong localization for which Phil Anderson, a 170 00:12:30,143 --> 00:12:35,012 professor Emeritus at Princeton got his Nobel Prize in 1977. 171 00:12:35,012 --> 00:12:38,668 So we obviously don't have time now to discuss in any detail this very 172 00:12:38,668 --> 00:12:43,276 interesting and complicated theory of Phil Anderson, but let me just mention that 173 00:12:43,276 --> 00:12:47,500 next week, we're going to go back to, to centered quantum mechanics and one problem 174 00:12:47,500 --> 00:12:50,505 we're going to solve is a particle in a potential way out. 175 00:12:50,505 --> 00:12:55,620 And in this problem we, we're going to see that the particle when put in a potential 176 00:12:55,620 --> 00:13:00,516 well gets localized by the well with some discrete energy levels not continuum of 177 00:13:00,516 --> 00:13:05,340 energy levels but discrete levels and so the answers of the theory of Anderson if 178 00:13:05,340 --> 00:13:09,660 it can be summarized in a few seconds is that in a random landscape of such 179 00:13:09,660 --> 00:13:12,690 potential wells, so this is let's say v of r. 180 00:13:12,690 --> 00:13:17,305 So the energy levels that electrons are going to acquire are not going to match 181 00:13:17,305 --> 00:13:20,270 each other. And so they're not going to be able to 182 00:13:20,270 --> 00:13:24,370 move up from well to well because of the energy conservation. 183 00:13:24,370 --> 00:13:28,518 And this would completely suppress the conductivity and this will lead to the 184 00:13:28,518 --> 00:13:32,631 strong localization. So that's the only thing I'm going to say. 185 00:13:32,631 --> 00:13:37,593 And next week we're going to study actually we're going to study simpler 186 00:13:37,593 --> 00:13:42,662 phenomena using the Schoeringer formulation in particular the appearance 187 00:13:42,662 --> 00:13:44,338 of discrete energy wells.