1 00:00:00,510 --> 00:00:02,760 We're now up to the 18th lecture of the commencement, of course. 2 00:00:02,780 --> 00:00:08,339 And over the last couple of weeks, we have been building up our knowledge of how to understand electrons in solids. 3 00:00:08,340 --> 00:00:12,240 We know about the structure. We know about electrons and holes in our electron effect, the masses. 4 00:00:12,420 --> 00:00:14,280 We know about doping. And at this point, 5 00:00:14,280 --> 00:00:22,559 we actually have arrived at enough knowledge that we can actually think about putting together materials in order to make real devices, 6 00:00:22,560 --> 00:00:29,760 and in particular, semiconductor devices, semiconductor devices, 7 00:00:30,300 --> 00:00:36,330 which is in some way the crowning glory of the entire field of condensed matter physics. 8 00:00:36,330 --> 00:00:40,620 It is because we can make semiconductor devices that we can build things like computers, we can build cell phones, 9 00:00:40,620 --> 00:00:47,250 we can build iPads and all these other things that we take for granted and completely change our, you know, the way we live these days. 10 00:00:47,460 --> 00:00:56,850 It is all because we understand semiconductor physics extremely well. The good news is that it is totally unexamined, but that means that we can't. 11 00:00:56,880 --> 00:01:00,780 The bad news is that means we can't spend a whole lot of time on it. Even though it is an extremely interesting subject. 12 00:01:00,990 --> 00:01:08,130 You could spend the rest of your entire life studying semiconductor devices and building new ones, and probably some people in this room really well. 13 00:01:08,670 --> 00:01:12,719 The semiconductor industry is a huge industry that employs many, many physicists. 14 00:01:12,720 --> 00:01:15,870 So people will be thinking about semiconductor devices in the future. 15 00:01:15,870 --> 00:01:21,000 I promise you someone will. But for us, it is going to be limited to about 40 minutes. 16 00:01:21,360 --> 00:01:27,210 But we'll see how far we can get in 40 minutes. So the simplest semiconductor device we're going to discuss is the quantum wealth, 17 00:01:30,450 --> 00:01:37,620 which is nothing more than a a sandwich of one material inside another material. 18 00:01:37,620 --> 00:01:41,250 So imagine having a three layer material over here. 19 00:01:41,580 --> 00:01:48,330 It's over here. We'll put aluminium arsenide here is gallium arsenide and here's aluminium arsenide. 20 00:01:48,330 --> 00:01:52,610 So it's a thin layer of gallium on our side in between lemon mars and this is some 21 00:01:52,620 --> 00:01:57,540 position coordinate and we have actually studied this material before gallium arsenide 22 00:01:57,540 --> 00:02:02,339 it remember it is a it has zinc structures FCC with the basis aluminium arsenide 23 00:02:02,340 --> 00:02:06,090 is just like gallium arsenide except you replace the gallium with the aluminium. 24 00:02:06,090 --> 00:02:12,629 So it's also FCC with basis. And the reason they have chosen these these materials, aluminium arsenide, 25 00:02:12,630 --> 00:02:16,800 gallium arsenide is actually the there is a little bit of a gift of nature here. 26 00:02:17,250 --> 00:02:24,330 It turns out that the lattice constant the size of the in itself for gallium arsenide in aluminium outside is extremely, extremely precisely the same. 27 00:02:24,540 --> 00:02:29,280 It's just something that randomly occurs which makes these very desirable materials to work with. 28 00:02:29,550 --> 00:02:33,750 The reason that makes things very desirable is means that you can line up the atoms directly 29 00:02:33,750 --> 00:02:38,250 between the gallium arsenide and the aluminium arsenide and basically have just one giant crystal. 30 00:02:38,400 --> 00:02:43,170 And as you walk through the crystal, the only thing that changes is in going from this material to this material. 31 00:02:43,380 --> 00:02:49,800 All the aluminiums get secretly replaced with gallium, so it just looks like one big crystal of one material. 32 00:02:49,800 --> 00:02:54,720 But over here you have aluminium, here you have gallium, and then you back here and it's aluminium is again, 33 00:02:55,050 --> 00:03:04,020 the difference between the different materials is that the bandgap in aluminium arsenide is actually bigger than the bandgap in gallium arsenide. 34 00:03:04,020 --> 00:03:09,900 So you might have a band structure that looks like this. It's a big band gap out here, small band gap here, big band gap here. 35 00:03:10,260 --> 00:03:16,649 So if we plot the conduction, band energy conduction, it will be big here. 36 00:03:16,650 --> 00:03:23,100 Then it drops down in this region here, which is known as the quantum well, and then it comes back up over here. 37 00:03:23,100 --> 00:03:27,000 So this is the lunar arsenide, here's the gallium arsenide, here's the aluminium arsenide. 38 00:03:27,300 --> 00:03:34,920 So if you're an electron living in that conduction band, what you see is you basically see a well, some sort of confining potential, 39 00:03:34,920 --> 00:03:41,040 like a particle in a box, and you would form a wave function in that particle in a box in the usual way. 40 00:03:41,040 --> 00:03:45,809 And of course you can have exciton state wave functions also in that particle in a box. 41 00:03:45,810 --> 00:03:49,330 Also, the only difference between this and the kind of particle in the box problems 42 00:03:49,350 --> 00:03:53,370 that you solved in your quantum mechanics courses is that it's the M star, 43 00:03:53,370 --> 00:03:56,939 not m the effect a mass of the electron is important. 44 00:03:56,940 --> 00:04:02,370 The effect of massive electron in the conduction man is the important thing that shows up in in Schrödinger's equation, 45 00:04:02,610 --> 00:04:10,560 not the real mass of the electron. And the other thing that's different is that this V of x, this potential is actually due to the band structure. 46 00:04:11,310 --> 00:04:24,240 Due to band structure. So that is what's causing the potential to go up and down to the bottom of the conduction band is moving is moving up and down. 47 00:04:24,570 --> 00:04:29,370 Now, if you think about the Valence Band. It actually looks the opposite. 48 00:04:29,760 --> 00:04:37,740 The Valence Band E Valence, the top of the Valence Band starts low, goes up high in the middle, and then comes back down like this. 49 00:04:38,250 --> 00:04:45,540 And if you are a hole living in the Valence band and remember the holes try to go up instead of coming down like electrons try to come down. 50 00:04:45,780 --> 00:04:49,109 So in fact, a hole can get stuck in this or upside down. 51 00:04:49,110 --> 00:04:55,439 Well, and it would also form a particle in a box wave function inside are inside this 52 00:04:55,440 --> 00:04:58,860 quantum wall so you can trap both an electron up here and a hole down here. 53 00:04:58,860 --> 00:05:05,670 Now, this very simple structure is actually used quite frequently in in the semiconductor industry, very frequently for making lasers. 54 00:05:06,060 --> 00:05:14,790 What happens is you imagine putting an electron up here and a hole down here, and then an electron can fall into that hole while emitting a photon. 55 00:05:15,000 --> 00:05:15,690 And it comes out. 56 00:05:15,690 --> 00:05:22,830 And if you learn some laser physics last time, you understand how if you have transitions like this, you can build you can build lasers out of them. 57 00:05:23,040 --> 00:05:29,520 The thing that is neat about this structure is you can change the energy of the iron states by changing the width of the well. 58 00:05:29,820 --> 00:05:35,490 So if you make it a thinner layer in inside this of gallium arsenide, inside aluminium arsenide, 59 00:05:35,790 --> 00:05:41,220 then the the energy ion states change just because the energy I can say so is depend on the width of the box. 60 00:05:41,520 --> 00:05:46,440 And so you can tune the energy of the upcoming phonon by just changing the width of the well. 61 00:05:46,620 --> 00:05:50,310 And that's why this kind of device is frequently used in in Optoelectronics. 62 00:05:50,460 --> 00:05:54,570 Okay. All right. So this is a very simplest device that we ever need to think about. 63 00:05:54,570 --> 00:06:02,910 We don't have to worry about any opens. Now we're going to think about adding doping, which will enable us to make some much more complicated devices. 64 00:06:03,210 --> 00:06:06,330 So doping and although I said at the beginning of the lecture, 65 00:06:06,330 --> 00:06:14,069 the semiconductor devices is not examined of all doping is examined, all we are we discuss this at the end of last lecture. 66 00:06:14,070 --> 00:06:18,300 I'll remind you of some of the results that if we drop a semiconductor, 67 00:06:18,630 --> 00:06:25,800 the density of electrons in the conduction band is some constants which I want write down times an exponential factor, 68 00:06:26,220 --> 00:06:32,430 even minus beta e conduction minus mu. So basically the big part of this, 69 00:06:32,430 --> 00:06:39,419 the exponential factor tells you that you have to activate an electron up into the conduction band from the chemical potential. 70 00:06:39,420 --> 00:06:42,450 It's exponentially activated up into the conduction band. 71 00:06:43,350 --> 00:06:46,410 And then there's a couple of three factors out front which aren't that interesting. 72 00:06:46,890 --> 00:06:54,600 Similarly, and actually it should be clear that as you raise the chemical potential towards the conduction band, 73 00:06:54,600 --> 00:06:58,050 the density of electrons in the conduction band goes up. 74 00:06:58,650 --> 00:07:10,260 Similarly, if we want to know about the density of holes in the valence band P that some constants also even minus beta nu minus V. 75 00:07:10,830 --> 00:07:17,250 So similarly we have the holes being activated down into the valence band 76 00:07:17,250 --> 00:07:21,809 because remember it requires energy to push a hole down into the valence band. 77 00:07:21,810 --> 00:07:24,420 The holes want to bubble up and again it's activated. 78 00:07:24,570 --> 00:07:30,720 If you move the chemical potential down towards the valence band, then the density of holes goes up. 79 00:07:31,170 --> 00:07:32,910 The law of mass action. 80 00:07:34,170 --> 00:07:46,530 Mass Action tells us that the product of these two things and of t p of t equals is a couple of these these two sets of constants. 81 00:07:46,980 --> 00:07:54,840 Then this is the minus beta e conduction minus valence, and this is independent of the chemical potential. 82 00:07:55,560 --> 00:07:57,360 So what I want you to take away from this, 83 00:07:57,360 --> 00:08:05,460 these equations which have to do with the density of electrons and the conduction intensive holes in the valence band is that if we dope, 84 00:08:05,820 --> 00:08:11,160 if doped heavily with n, 85 00:08:12,270 --> 00:08:21,969 in other words at a lot of donors, then chemical potential goes up, up to the conduction band energy, 86 00:08:21,970 --> 00:08:30,270 it goes up towards the conduction band energy because we have to raise the number of electrons while keeping the product here fixed. 87 00:08:30,270 --> 00:08:32,940 And the only way that happens is by raising the chemical potential. 88 00:08:33,360 --> 00:08:45,060 Similarly, if we do not heavily, if doped heavily with P with acceptors adding holes to the valence band, 89 00:08:46,170 --> 00:08:55,830 then you goes down to the balance band energy that much good. 90 00:08:56,490 --> 00:09:01,230 That is everything that is examined of all of our device physics is these two two boards right here. 91 00:09:03,260 --> 00:09:08,270 So now what we can do is we can think about what happens if we put some dope ends in our semiconductor. 92 00:09:08,270 --> 00:09:16,160 And this brings us to our first interesting device, which is the Junction P and junction, 93 00:09:16,520 --> 00:09:22,310 which was surprisingly discovered or essentially discovered in 1874 by Ferdinand Brown. 94 00:09:22,670 --> 00:09:25,489 Ferdinand Brown actually had no idea what it was he discovered, 95 00:09:25,490 --> 00:09:30,230 but he was a clever enough physicist to realise that what he discovered was really interesting. 96 00:09:30,470 --> 00:09:36,830 In fact it became an extremely important thing. It was a little electronic widget that enabled the development of the radio. 97 00:09:36,830 --> 00:09:41,330 Without brands discovering of this device, we wouldn't have been able to build the radio. 98 00:09:41,330 --> 00:09:46,310 And of course, the radio is one of the inventions that really changed communications and changed the world in many ways. 99 00:09:46,610 --> 00:09:51,170 So Junction changed the world back in the 1800s or beginning of 1900s. 100 00:09:51,470 --> 00:09:59,000 But actually it was not really understood why the Junction behave the way it did for about 70 years after Brown's discovery. 101 00:09:59,300 --> 00:10:03,590 So let's see if we can understand what it was that Brown didn't understand. So, first of all, what is the structure? 102 00:10:04,010 --> 00:10:13,910 So you start with a big piece of semiconductor and you just up one side of it and you end up the other side and dope the other side. 103 00:10:14,390 --> 00:10:18,830 Now, when Braun actually built this, he took two pieces of materials and just stuck them together. 104 00:10:19,040 --> 00:10:19,999 In the modern era, 105 00:10:20,000 --> 00:10:25,850 you take one big crystal and you add some droppings over here and you add some dummies over here much more control than what Brown did, 106 00:10:26,270 --> 00:10:33,470 but generally the same idea. And so let us draw a picture of where the conduction band and the Valence band are. 107 00:10:33,830 --> 00:10:42,560 So the conduction band energy might be up here. The Valence band energy might be down here in Valence down here. 108 00:10:43,340 --> 00:10:48,559 And here's our division in the middle. Now, over on this side, we're doped. 109 00:10:48,560 --> 00:10:55,490 So that means we have a bunch of electrons up in the conduction band here and the chemical potential 110 00:10:55,760 --> 00:11:01,669 is way up here mu up here near the conduction band because we have doped it with the electrons. 111 00:11:01,670 --> 00:11:07,610 So the chemical potential went up towards the conduction band energy and now over down here we have a bunch of 112 00:11:07,610 --> 00:11:19,520 holes in the valence band and their chemical potential has been pulled down to near the valence band energy. 113 00:11:20,210 --> 00:11:27,320 So far, so good. All right. Now, the way it's drawn, this looks like you have negative charges on the left and positive charges on the right. 114 00:11:27,620 --> 00:11:31,579 That is not actually true because we had left out the charges on the nuclei. 115 00:11:31,580 --> 00:11:35,090 So the nuclei have charges are also nuclei nukes. 116 00:11:36,530 --> 00:11:42,079 We and they have compensating charges like this and competition charges over here. 117 00:11:42,080 --> 00:11:51,550 Because you remember when you add a for example, when you add a phosphorus to silicon, you add a extra nuclear charge, you also add an extra electron. 118 00:11:51,560 --> 00:11:54,740 The whole thing is neutral. The electron goes off into the conduction band. 119 00:11:54,740 --> 00:11:57,950 But you left a positive nuclear charge behind as well. 120 00:11:58,250 --> 00:12:05,569 So everything here is electrically neutral. No electric fields, electron negative charge is compensated by the nuclear positive charge, 121 00:12:05,570 --> 00:12:11,420 and the whole positive charge is compensated by the R by the acceptor negative charge here. 122 00:12:11,660 --> 00:12:20,210 So far so good. Okay, now here we have electrons up here in the conduction band and we have holes here in the valence band. 123 00:12:20,420 --> 00:12:28,040 And in fact, this electron, if it falls down to the Valence Band annihilating its hole, he can let off energy. 124 00:12:28,220 --> 00:12:30,860 He can mirror photon, for example, and lower the energy. 125 00:12:31,160 --> 00:12:39,350 So what happens is that the electron here are now a little bit of a razor transform, annihilates the whole electron, 126 00:12:39,650 --> 00:12:45,500 annihilates the hole, and you end up with a recently annihilate one more here, annihilate one more. 127 00:12:45,860 --> 00:12:52,970 And we end up with this region in the middle from here to here, which is known as the depletion region. 128 00:12:53,420 --> 00:13:03,980 Depletion region where there are no electrons and no holes, there's no mobile charge carriers anymore. 129 00:13:04,130 --> 00:13:08,540 You only have the static nuclear charges left. Nothing moving around. 130 00:13:08,900 --> 00:13:17,000 So you might think. This annihilation will keep going because the electrons are up at high energy and the holes are down at low energy. 131 00:13:17,270 --> 00:13:22,460 So we can just keep annihilating electrons with holes, keep emitting photons, and lower and lower and lower the energy. 132 00:13:22,670 --> 00:13:26,209 But that is not true. In fact, it does not keep going. 133 00:13:26,210 --> 00:13:29,960 And the reason it does not keep going is because we are leaving a net charge behind. 134 00:13:29,960 --> 00:13:37,070 This region now is charged. This is net positive charge because of the doping and this is net negative charge, net minus charge. 135 00:13:37,070 --> 00:13:42,290 So as we start annihilating, we leave actually a charged region here and a charge region here. 136 00:13:42,770 --> 00:13:51,139 And that means we have an electric field against an electric field, points from the minus to the positive and minus divisor positive. 137 00:13:51,140 --> 00:13:55,520 Negative. Which way? I see the direction of a positive charge is the negative. 138 00:13:58,110 --> 00:14:01,860 Points this way. The electric field goes from positive. 139 00:14:01,860 --> 00:14:10,979 No test charge goes towards the negative. Yes. Okay. Got it. Right. So so there's an electric field here that we're building up. 140 00:14:10,980 --> 00:14:15,780 And as we annihilate more more charges here, we end up with more and more electric field. 141 00:14:15,990 --> 00:14:20,550 And that means if we want to take one more electron here and annihilate one more hole here, 142 00:14:20,700 --> 00:14:30,240 we first must we must climb up a potential hill and electrostatic potential hill to get past this electric field to reach the hole here. 143 00:14:30,480 --> 00:14:37,020 At some point, this electrostatic potential hill becomes too big and we can't get over it and we can't have any more annihilation. 144 00:14:37,440 --> 00:14:42,120 So the depletion with depletion. 145 00:14:42,270 --> 00:14:46,430 Depletion with. Mission with. 146 00:14:48,400 --> 00:14:56,520 Is that by selling the gap, energy gap, the amount of energy you would you would gain by annihilating one more energy, 147 00:14:56,850 --> 00:15:01,680 with one more electron, with more whole, it has to equal two times Delta PHI. 148 00:15:01,950 --> 00:15:07,950 The amount of energy you have to put into it to get the electron over the electrostatic 149 00:15:07,950 --> 00:15:14,250 hill that you're fighting against to move the electron past the discharge region. 150 00:15:14,340 --> 00:15:17,850 Is that clear? Roughly someone? 151 00:15:17,860 --> 00:15:29,130 Yes. Okay, good. At least someone said yes. So this this potential here is just the integral of EDL from well, across the system. 152 00:15:30,740 --> 00:15:36,260 All right. So so that generally we we do not believe that this depletion with is going to be infinitely big. 153 00:15:36,260 --> 00:15:43,430 It's going to stop at some region. But this is not a good way to draw what goes on because this looks a little it looks a little strange. 154 00:15:43,430 --> 00:15:49,339 It looks it looks incorrect. I mean, it looks like the electron here is at higher energy and the whole is at lower energy. 155 00:15:49,340 --> 00:15:53,149 So it kind of looks like the electrons and still want to annihilate the hole. 156 00:15:53,150 --> 00:15:55,820 But we know that there's an electric field that's making it difficult. 157 00:15:55,820 --> 00:16:02,780 So we what we want to do is we want to redraw this picture in a way that will make it obvious that the electron does not want to come in, 158 00:16:02,780 --> 00:16:10,190 annihilate one more hole. So what we do is we draw things in terms of the so-called electrochemical potential. 159 00:16:10,730 --> 00:16:23,080 Electrochemical potential. Potential, which is the combination of the regular chemical potential and the electrostatic potential. 160 00:16:23,530 --> 00:16:25,270 Okay. So we're going to plot that instead, 161 00:16:25,480 --> 00:16:34,480 which is going to combine together the so the band structure diagram as well as the Coulomb effects of the space charge that we left behind. 162 00:16:34,840 --> 00:16:39,910 So over on this side, draw the same picture we had before. 163 00:16:40,270 --> 00:16:45,820 So we have the conduction band of the energy up here, the electrochemical potential input here. 164 00:16:45,850 --> 00:16:49,120 So this is a new minus EFI over here. 165 00:16:49,390 --> 00:16:52,390 We have the valence band energy over here in Valence. 166 00:16:52,760 --> 00:16:59,680 Now over here on the other side, we are going to line up the electrochemical potential, exactly the same. 167 00:17:00,310 --> 00:17:06,520 But here the valence band energy will be here in Valence and the conduction band energy will be up here. 168 00:17:07,150 --> 00:17:10,510 The conduction and these get connected together like this. 169 00:17:12,180 --> 00:17:15,870 Okay. That's. Those are supposed to be parallel lines. Not very parallel. 170 00:17:16,130 --> 00:17:22,510 Uh, improve it. The improvement rises up a little bit more, so they look very vaguely parallel. 171 00:17:23,370 --> 00:17:27,010 So these are supposed to be. Okay. 172 00:17:27,400 --> 00:17:32,770 Good. So all I have done is I've taken that picture up there and I have twisted it. 173 00:17:33,000 --> 00:17:37,750 I have of bent it so as to account for the additional electrostatic potential. 174 00:17:38,200 --> 00:17:44,770 So now the on the left, in the right, the electrochemical potential lines up with the electric chemical potential over here. 175 00:17:44,980 --> 00:17:51,430 And what that tells us is that there is no net driving force for moving an electron from this side to this side, 176 00:17:51,640 --> 00:17:53,500 that there is a change in the chemical potential, 177 00:17:53,770 --> 00:18:01,030 but there is are equal change in the electrostatic potential that compensated the left hand side and the right hand side are completely lined up. 178 00:18:01,030 --> 00:18:06,099 As far as the electron is concerned, it does not want to move to the left. The electron didn't want to go left, don't want to go to the right either. 179 00:18:06,100 --> 00:18:09,580 It just wants to say stay put. So this is the equilibrium situation. 180 00:18:09,970 --> 00:18:13,600 Is that sort of clear how this has been drawn? Okay. All right. 181 00:18:14,530 --> 00:18:20,520 As a terrible picture, isn't it? Okay. But this this device here, this is basically the physics of the junction. 182 00:18:20,800 --> 00:18:27,459 And even as it stands, just like that, without doing anything further, it's an interesting device because it's basically a solar cell. 183 00:18:27,460 --> 00:18:30,760 A solar or photoelectric cell. Photoelectric cell. 184 00:18:34,940 --> 00:18:40,460 So. And the idea of this is that you put in photons. 185 00:18:40,820 --> 00:18:44,630 Put in photons, get our current. 186 00:18:45,650 --> 00:18:59,680 Get current. And this idea of putting in photons and getting our current is a $20 billion industry right now with people trying to build for 187 00:18:59,680 --> 00:19:06,049 ourselves that are going to replace our fossil fuels and potentially save our world from global warming and everything else. 188 00:19:06,050 --> 00:19:11,350 So I encourage all of you to go into the semiconductor industry and try to figure out how we're going to make solar cells more 189 00:19:11,350 --> 00:19:16,510 efficient so that we can get all of our power from these things instead of from burning coal and oil and things like that. 190 00:19:16,810 --> 00:19:24,220 At any rate. Let us try to figure out why it is that this device has this property that if you put in light, you'll get out current. 191 00:19:24,820 --> 00:19:32,650 So if you put in light into this into this picture, you can excite an electron from the valence band up to the conduction band. 192 00:19:32,680 --> 00:19:39,040 So you leave a hole behind and you excite an electron up to the conduction band from the valence band. 193 00:19:39,340 --> 00:19:44,100 Now, in this depletion region, there's an electric field, there's a strong electric field in this depletion region. 194 00:19:44,130 --> 00:19:47,350 You can see it in this picture by the curvature of these bands. 195 00:19:47,650 --> 00:19:53,710 And what happens is the electron wants to it is pushed by the electric field in this direction because it wants to go down to lower energy. 196 00:19:53,950 --> 00:20:01,450 And the hole is pushed in this direction, wants to go up to higher energy because holes always try to go up in the in the band structure. 197 00:20:02,350 --> 00:20:09,400 And either way, this is always current. Going to the right is either positive, current, positive charge going to the right. 198 00:20:09,700 --> 00:20:13,390 This is plus going to the right or negative charge going to the left. 199 00:20:13,720 --> 00:20:18,220 And both of those are current in the same direction. So just by putting in a photon here. 200 00:20:18,520 --> 00:20:23,060 Photon. Which creates this exhortation. 201 00:20:23,390 --> 00:20:27,950 You managed to create current. 202 00:20:28,490 --> 00:20:33,500 So this is the, you know, the the physics is going to save us from global warming one of these days. 203 00:20:33,710 --> 00:20:37,310 So please, you know, go out into the world and figure out how to build this better. 204 00:20:37,580 --> 00:20:42,320 You know, they're still not quite as cheap as burning fossil fuels that people aren't willing to use them yet. 205 00:20:42,530 --> 00:20:46,070 But maybe someday we will. Okay. 206 00:20:47,240 --> 00:20:53,629 This is not what first Brown discovered in 1874, but he discovered with something very with the same device. 207 00:20:53,630 --> 00:21:02,120 But he discovered another effect that these things have, which is maybe equally interesting, which is the effect of rectification. 208 00:21:05,230 --> 00:21:15,070 Which means net current flows one way low, one way, one way, but not the other. 209 00:21:22,580 --> 00:21:24,560 And if you're a good with your circuits, 210 00:21:25,130 --> 00:21:32,180 you might be able to figure out why it is they're having something with this property which rectifies enables you to build a radio, 211 00:21:32,390 --> 00:21:37,760 which is what he realised very quickly that if you have something that rectifies allowing content to flow one way but but not the other. 212 00:21:37,940 --> 00:21:41,710 You can build a radio with it. So. Why? 213 00:21:41,890 --> 00:21:47,020 Why is this? Why does it have that physique? So let's take this device and let us imagine. 214 00:21:48,280 --> 00:21:50,050 Well, let me draw the same picture again, 215 00:21:50,260 --> 00:21:55,510 except what I am going to do now is I'm going to put a voltage across this picture to try to get current to flow. 216 00:21:55,780 --> 00:22:05,230 So let's do that. So exactly the same picture you see here is the electrochemical potential meu minus e ify on the left. 217 00:22:05,560 --> 00:22:09,910 And then down here is the valence band, Energy Valence. 218 00:22:10,510 --> 00:22:16,089 And then over here, instead of lining up the electrochemical potentials exactly on the left and the right, 219 00:22:16,090 --> 00:22:20,350 I'm going to shift the chemical potential down to the electrochemical potential down here. 220 00:22:20,350 --> 00:22:25,209 So this is if it was lined up. Exactly. It would be exactly the same lined up here. 221 00:22:25,210 --> 00:22:31,150 And now I'm going to shift it downwards by easy here, by some energy. 222 00:22:31,150 --> 00:22:37,270 So this is the electrochemical potential on the right. So now because I'm putting a voltage across the whole thing. 223 00:22:37,720 --> 00:22:48,070 So then we have to put in the valence band energy against new minus EFI here, it's here and valence band energy is here. 224 00:22:48,430 --> 00:22:57,760 Valence and then the conduction band energy will be here and then in the depletion region have electric field which connects these together like that. 225 00:22:58,280 --> 00:23:01,510 Okay. So is it clear what I have drawn here? I've taken exactly the same picture. 226 00:23:01,750 --> 00:23:07,300 I've just applied a voltage across the to to shift this chemical potential down to here. 227 00:23:08,290 --> 00:23:17,850 And. All right. So now now that I have applied this this voltage, how much current flows? 228 00:23:17,860 --> 00:23:21,939 Well, there are several ways you can get current. One way to get current with this process. 229 00:23:21,940 --> 00:23:30,400 One is that an electron in the Valence band could get excited up to the conduction band and then eventually find its way down the hill. 230 00:23:31,120 --> 00:23:36,130 So let's call that process one. And process two is sort of the reverse of that, 231 00:23:36,460 --> 00:23:42,130 which is the hole up in the conduction band could get excited down into the Valence Band because remember, 232 00:23:42,130 --> 00:23:47,080 it requires positive energy to push a hole down into valence band and then find its way up the hill. 233 00:23:47,470 --> 00:23:56,830 So process one and two, one and two. This either an electron going to the left or a hole going to the right. 234 00:23:57,370 --> 00:24:02,710 And either way, its rate is either the minus beta e gap, 235 00:24:03,190 --> 00:24:09,219 because the limiting step is to get the electrons up into the conduction band or get the hole down into the valence band. 236 00:24:09,220 --> 00:24:12,370 And that costs the gap energy. So it's activated. Okay. 237 00:24:13,010 --> 00:24:18,430 So that is one thing that can happen that can cause current. But another thing that can happen that can cause current is you have some number of 238 00:24:18,430 --> 00:24:22,810 electrons up here in the conduction band already and they can just climb up the hill. 239 00:24:23,950 --> 00:24:30,460 Let's call it three or we have some holes here in the Valence Band and they can just climb down the hill, let's call it four. 240 00:24:31,450 --> 00:24:40,000 So three and four, three and four, we have electrons going to the right or a holes going to the left, 241 00:24:40,900 --> 00:24:46,120 and they happen at a rate of either the minus E gap minus EV. 242 00:24:47,110 --> 00:24:51,400 Why is that? Well, the height of this hill from here to here. 243 00:24:52,640 --> 00:25:03,870 Is Gab minus eve. If we had not applied this way, it would be exactly the same gap energy that we would need to climb up the hill. 244 00:25:03,890 --> 00:25:12,020 But now we have reduced that hill. We reduce the size of the hill by ev by applying the voltage across the system. 245 00:25:12,590 --> 00:25:20,840 So the total current total j j equals well is going to be a combination of these two things. 246 00:25:20,840 --> 00:25:27,870 I call it A1 to either the minus beta E gap minus a34. 247 00:25:27,890 --> 00:25:34,730 Those are just coefficients either minus beta B gap minus any which. 248 00:25:35,650 --> 00:25:37,729 Okay. Then we need to know what these coefficients are. 249 00:25:37,730 --> 00:25:43,190 But actually there's sort of a trick for that that the current when there zero voltages should be zero. 250 00:25:43,460 --> 00:25:47,120 If you do not apply any voltage in the system, you should not get any current. 251 00:25:47,390 --> 00:25:50,390 So that means a one to an 84 should be equal. 252 00:25:50,720 --> 00:26:03,980 So j is proportional to each the minus beta e gap factor that out times are either the EV or Katie t that is beta minus one. 253 00:26:05,890 --> 00:26:10,920 Okay. So this is what's known as the diode equation. Diode equation. 254 00:26:13,330 --> 00:26:22,210 The injunction or the Rectifier is sometimes known as a diode and we can plot its behaviour here. 255 00:26:22,990 --> 00:26:32,640 So we have. Current voltage on this axis and current on this axis and is exponential in. 256 00:26:34,790 --> 00:26:41,089 In voltage. So this is this is zero current here and this is negative current here. 257 00:26:41,090 --> 00:26:45,230 So maybe I should have drawn it like the zero is here and negative is here. 258 00:26:45,560 --> 00:26:54,590 So with positive voltage it goes up exponentially, whereas negative voltage it goes to a constant which is extremely small constant. 259 00:26:54,950 --> 00:27:02,780 So this is frequently drawn figuratively as being straight up like this and straight flat like this, 260 00:27:03,350 --> 00:27:11,110 which the APR APR say it allows current to flow in one direction and allows no current to flow in the other direction. 261 00:27:11,120 --> 00:27:14,389 Frequently we draw in a circuit diagram. 262 00:27:14,390 --> 00:27:21,140 We draw a diode like this diode, which means current flows only in one direction. 263 00:27:22,340 --> 00:27:26,840 It's pretty neat device to put into your circuits when you're designing circuits. 264 00:27:29,120 --> 00:27:33,500 So this was what Vernon Brown discovered experimentally, more or less, 265 00:27:34,250 --> 00:27:39,980 and it was not understood until the 1940s when people really developed a good understanding of semiconductor physics. 266 00:27:40,790 --> 00:27:47,420 And by that time when they finally understood semiconductor physics, they realised they could do a lot more with it than just building diodes. 267 00:27:47,660 --> 00:27:52,879 And this enabled them to start working on the next big invention that changed the world, 268 00:27:52,880 --> 00:28:01,760 which is the transistor invented in 1947 by the Bell Labs team of John Bardeen, Walter Bratton and William Shockley. 269 00:28:02,030 --> 00:28:06,019 Bardeen is definitely a name that everyone should know because he won two Nobel Prizes, 270 00:28:06,020 --> 00:28:09,349 one for the transistor, one for the theory of Superconductivity. 271 00:28:09,350 --> 00:28:12,140 A very bright but pretty much unknown person. 272 00:28:14,080 --> 00:28:24,190 And this device, the transistor is the elementary that the smallest piece, the elementary piece, the building block of every computer ever made. 273 00:28:24,220 --> 00:28:28,720 Now it is in your average cell phone or laptop or iPad. 274 00:28:28,930 --> 00:28:33,340 There are billions of these transistors sitting on, literally billions of them in there. 275 00:28:33,520 --> 00:28:37,780 It is the elementary switch or elementary amplifier that enables everything to work. 276 00:28:38,920 --> 00:28:44,500 So transistors come in many different varieties. The one we're going to discuss is actually known as the MOSFET. 277 00:28:45,070 --> 00:28:56,680 MOSFET, which stands for a metal oxide oxide semiconductor field effect transistor. 278 00:29:02,760 --> 00:29:07,320 And the star. And I explain what that means as we as we get to it. 279 00:29:07,620 --> 00:29:13,230 Although I should comment that the original transistor that they built in 1947 was not of this design. 280 00:29:13,500 --> 00:29:18,960 They were aware that they could build this design, but for various reasons they decided that was not the one they were going to try to build first. 281 00:29:19,230 --> 00:29:22,340 This is the type of transistor that is most frequently in circuits now. 282 00:29:22,350 --> 00:29:27,120 So that's why explaining that one is also easier to understand, except nicer to talk about. 283 00:29:28,020 --> 00:29:36,120 The way it works is fairly simple device. You take a big piece of a semiconductor and you drop it and it has a surface like this. 284 00:29:36,940 --> 00:29:41,040 You can do it backwards, you can open if you want it to, but let's imagine it's doped. 285 00:29:41,490 --> 00:29:46,200 And then we take two small regions and we end up them, make it a little bigger, 286 00:29:47,040 --> 00:29:54,090 this region here and up this region here, and then we'll end up this region here, this and this is all doped. 287 00:29:55,280 --> 00:30:00,860 Okay. So what we have now is we have a surface with an end to end and a big bulk with p doped. 288 00:30:01,070 --> 00:30:09,470 We'll put contacts here on the wires coming in. This will call the source source wire and this is called the drain wire drain. 289 00:30:10,310 --> 00:30:12,980 And then we try to run current from the source to the drain. 290 00:30:13,370 --> 00:30:19,250 Now if you look at this structure here, you'll realise that current is not going to flow from the source to the drain. 291 00:30:19,430 --> 00:30:23,500 You'll have a depletion region at the interface between an MP. 292 00:30:23,510 --> 00:30:26,750 There's an MP junction there and PNG junction here. 293 00:30:28,080 --> 00:30:34,570 And one of the PM junctions wants the current to only flow one way and the other one wants it to flow only the other way. 294 00:30:34,590 --> 00:30:41,040 You can kind of think of it figuratively as being, you know, two back to back diodes pointing in opposite directions. 295 00:30:41,460 --> 00:30:45,810 So one of them only allows the current to flow to the left, the other one only allows the current to flow the right. 296 00:30:46,020 --> 00:30:48,420 So between the two of them, you can't get any current through it at all. 297 00:30:49,100 --> 00:30:52,920 Okay, so this looks like a boring device, but now we add the last part of the device. 298 00:30:54,030 --> 00:30:57,360 We put a big piece of metal on top like this, which is known as the gate. 299 00:30:59,490 --> 00:31:02,610 That metal metal gate. 300 00:31:03,120 --> 00:31:07,920 And then we attach it to the source via a little bit of battery here. 301 00:31:08,340 --> 00:31:15,630 So voltage plus battery like that, attach the metal gate and then we put an insulator in between here, which is usually an oxide layer. 302 00:31:17,010 --> 00:31:21,240 And this is what gives the name of the device metal oxide semiconductor. 303 00:31:21,250 --> 00:31:26,130 So the gate is metal, the oxide is in the middle and the semiconductor is on the bottom. 304 00:31:26,700 --> 00:31:32,040 The purpose of the oxide is just to make a small insulating layer so the metal doesn't touch the semiconductor. 305 00:31:32,550 --> 00:31:37,860 You can make it of anything you want. It's just typically made of oxides, which are very nice insulators and they're easy to build. 306 00:31:38,130 --> 00:31:40,380 You can make out of any insulator you wanted to put there. 307 00:31:40,380 --> 00:31:43,890 You could make it a diamond, you can make it, you know, it could be just vacuum if you wanted. 308 00:31:43,890 --> 00:31:47,220 But the point is you have to have the metal not touching the semiconductor. 309 00:31:47,370 --> 00:31:51,209 So it's usually called metal oxide semiconductor. In fact, the gate does not have to be metal either. 310 00:31:51,210 --> 00:31:55,260 It just has to be conducting. So it can be whatever you want as long as it's conducting. 311 00:31:55,500 --> 00:31:59,910 At any rate, it's called metal oxide semiconductor because it's almost always made of metal oxide and semiconductor. 312 00:32:00,360 --> 00:32:04,969 All right. So what's the purpose of this metal? This metal gate, you charge it up. 313 00:32:04,970 --> 00:32:14,600 So it has plus charges on it. And this plus charges act like a capacitor and they're going to attract negative charges on the semiconductor here. 314 00:32:15,170 --> 00:32:21,410 Now, this region is semiconductor, so we're basically just charging up a capacitor plate between the metal and the semiconductor. 315 00:32:21,710 --> 00:32:27,230 This region here on the top of the semiconductor, it used to be p do not even have holes there. 316 00:32:27,470 --> 00:32:30,230 But now we have attracted a lot of electrons into that region. 317 00:32:30,470 --> 00:32:36,350 So instead of being p doped, it is now majorly doped because we have we just attracted electrons there. 318 00:32:36,350 --> 00:32:42,560 So now it's n so instead of having something that looks like that picture I had originally, it now looks more like this. 319 00:32:43,370 --> 00:32:49,190 This whole region here all the way from here to here is negative, is n, is n, 320 00:32:49,220 --> 00:32:54,770 doped, has has electrons in it, and the depletion region sort of makes a big. 321 00:32:56,800 --> 00:33:04,780 Sort of environment like this. Now, here you can get current directly from this end region to this end region via this electron channel. 322 00:33:05,050 --> 00:33:12,880 So you've added electrons to this layer, so the electrons can travel directly from the source to the drain without having to go through a junction. 323 00:33:12,910 --> 00:33:16,660 That's the point. That good? Happy with that? 324 00:33:16,750 --> 00:33:21,850 All right, good. So the characteristics of such a device look more or less like this, 325 00:33:23,170 --> 00:33:29,320 that if you plot the current versus the voltage, this is the voltage between the source and the drain. 326 00:33:29,620 --> 00:33:37,900 This is the current that flows from the source to the drain. If you put no gate voltage on V8 equals zero, you get almost no current flowing. 327 00:33:38,230 --> 00:33:44,320 But if you put a little bit of gate voltage on the gate equals say two volts equals two volts. 328 00:33:44,560 --> 00:33:53,170 You all of a sudden get a lot of source drain current. So this enables you to take a small voltage on the gate and control a very large current. 329 00:33:53,560 --> 00:33:57,690 So this is basically why this device is so useful. 330 00:33:57,700 --> 00:34:01,900 It acts as an amplifier. It can amplify a very small signal on the gate, 331 00:34:02,140 --> 00:34:06,920 and then you can do interesting things with it because now it's become a big current that you can work with. 332 00:34:08,050 --> 00:34:13,270 The elementary circuit, that element that builds all the computers in the modern age. 333 00:34:13,720 --> 00:34:20,950 All right. Well, I hope many of you will go on and study more semiconductor device physics more than these last 35 minutes. 334 00:34:21,640 --> 00:34:23,700 But this is all we have time for in this course. 335 00:34:23,710 --> 00:34:29,470 Unfortunately, we have to move on to the the final segment of our course and the final segment of a course. 336 00:34:29,710 --> 00:34:37,000 Is there leaving all this stuff behind us? And we move on to study magnetism, magnetism. 337 00:34:37,660 --> 00:34:40,569 So take a deep breath and we move on to magnetism. 338 00:34:40,570 --> 00:34:47,230 And and the first question one probably wants to ask is, why are we studying magnetism and why are you studying magnetism now? 339 00:34:47,980 --> 00:34:52,629 The whole course, pretty much from day one, all the way up to lecture 18 halfway through, 340 00:34:52,630 --> 00:34:58,959 which is right now we've really been studying waves in solids one way or the other, 341 00:34:58,960 --> 00:35:02,320 whether they're vibrational waves or whether they're electron waves or the x ray waves. 342 00:35:02,530 --> 00:35:06,940 We really have just been studying waves in solids in one way or the other, and magnetism is a little bit different. 343 00:35:06,970 --> 00:35:12,630 We're not really studying waves and solids when we're studying magnetism. So why is it that we're studying magnetism? 344 00:35:12,970 --> 00:35:19,660 There's a lot of reasons why we want to study magnetism. The first reason why we want to study magnetism is it certainly is an effect that has 345 00:35:19,660 --> 00:35:24,580 been known about since antiquity and the records of people knowing about magnetism. 346 00:35:24,580 --> 00:35:27,639 1000 B.C. in Greece, thousand B.C. in China. 347 00:35:27,640 --> 00:35:32,080 They knew how to build compasses and well, probably predates that by by thousands of years. 348 00:35:32,080 --> 00:35:35,980 Probably they were cavemen who knew about magnetism. You stuck notes on the refrigerator. 349 00:35:37,270 --> 00:35:40,460 The theory anyway. But. 350 00:35:40,870 --> 00:35:46,090 But despite the fact that this effect was was known about thousands and thousands years ago, 351 00:35:46,270 --> 00:35:50,230 there is absolutely no understanding of it whatsoever until people understood quantum mechanics. 352 00:35:50,470 --> 00:35:58,570 And that is one thing that makes this very interesting. In fact, there's a theorem by Baur and Van Nguyen, known as the Bois Van Buren Theorem. 353 00:35:58,990 --> 00:36:03,880 Niels Bohr I think everyone knows, but Enrico Van Leone was one of his students. 354 00:36:05,020 --> 00:36:18,100 We Win Theorem. And their content and the more value and theorem is basically no magnetism without quantum sim without quantum. 355 00:36:21,350 --> 00:36:25,760 Nothing in classical physics can ever give you magnetism in any sense. 356 00:36:26,090 --> 00:36:33,410 So that is kind of an interesting statement. So it tells us that, you know, that magnetism is a really good place to go to understand, 357 00:36:34,250 --> 00:36:37,549 you know, the macroscopic ramifications of microscopic magnetism. 358 00:36:37,550 --> 00:36:40,040 So that's one of the reasons why we want to study it. 359 00:36:40,280 --> 00:36:46,100 Another reason we want to study magnetism is because it is a rather spectacular failure of everything we've been doing so far. 360 00:36:46,370 --> 00:36:52,770 That magnetism, the band theory we've been studying so far will always predict that you have the same number of ups, 361 00:36:52,790 --> 00:36:57,050 spin electrons as down spin electrons, whereas in magnetism it's just not true. 362 00:36:57,080 --> 00:37:00,620 You have magnetic substances where we have a different number of spin up electrons and spin 363 00:37:00,620 --> 00:37:04,519 down electrons is completely outside of band theory and it's completely not predicted. 364 00:37:04,520 --> 00:37:09,230 So magnetism is a way to understand with the limitations of what we've already been doing. 365 00:37:09,590 --> 00:37:16,190 The final reason, which is maybe not so obvious, is that magnetism is a really great place to study. 366 00:37:16,190 --> 00:37:23,239 Phase transitions between phases of matter so everyone's familiar with in phase transitions between ice and water, water and steam. 367 00:37:23,240 --> 00:37:26,420 And you've probably studied these things in your thermodynamics classes, 368 00:37:26,750 --> 00:37:30,800 but when one really gets down to understanding the details of these phase transitions, 369 00:37:31,010 --> 00:37:37,250 most of what we know about phase transitions comes from studying magnets, where things are much better controlled and much better understood. 370 00:37:37,430 --> 00:37:40,340 And much of what we know about the melting of ice into water, 371 00:37:40,550 --> 00:37:46,880 we actually know because we studied magnets and then we translated our knowledge into understanding ice and water. 372 00:37:47,180 --> 00:37:52,339 That is not an obvious statement. We will make some statements about phase transitions in this course. 373 00:37:52,340 --> 00:37:55,100 If you go on to study other courses, you'll learn a lot more about it. 374 00:37:55,520 --> 00:38:04,610 Okay, so let's start by defining a couple of types of magnetism, types of magnetism that we're going to be working on of magnetism, 375 00:38:05,600 --> 00:38:08,060 or at least some of the types of magnetism we're going to need to know about. 376 00:38:09,420 --> 00:38:15,860 Usually what we do is we apply a magnetic field to some object and we measure groups, 377 00:38:16,370 --> 00:38:21,860 we measure a magnetisation m and there's a proportion if there's a proportionality constant, 378 00:38:22,250 --> 00:38:30,680 we call the proportionality chi over new night where chi is known as the susceptibility or magnetic susceptibility susceptibility. 379 00:38:30,680 --> 00:38:35,210 I'm going to spell that wrong as you see ability susceptibility. 380 00:38:38,090 --> 00:38:50,780 And Moonshot is the usual magnetic permeability, the thing that's permeability, the thing that's on your data sheet and your exams. 381 00:38:51,260 --> 00:38:58,190 So given that as a definition, there is a couple of kinds of magnetism that you can have. 382 00:38:58,190 --> 00:39:06,950 One is power magnetism, which means or a paramagnetic is any material with greater than zero. 383 00:39:07,310 --> 00:39:13,460 In other words, you apply a magnetic field and you get a magnetic moment in the same direction as a field that you applied, 384 00:39:13,730 --> 00:39:17,960 which seems to make some amount of sense. And we have know a couple of examples of that. 385 00:39:17,990 --> 00:39:21,080 Example example one is a free spin. 386 00:39:22,610 --> 00:39:25,910 Probably studied this in statistical mechanics last year. 387 00:39:27,050 --> 00:39:33,280 If you have a free stance or is sitting in outer space, if you apply a magnetic field to it, its energy is lower. 388 00:39:33,290 --> 00:39:36,529 If the magnetic moment points in the same direction as magnetic field. 389 00:39:36,530 --> 00:39:40,520 So the spin will try to flip over and point in the same direction as the magnetic field. 390 00:39:40,850 --> 00:39:43,190 Hence the free spin is a power magnet. 391 00:39:44,300 --> 00:39:54,050 Example to which we studied earlier in this term is a Fermi Gas or electron gas which is studying the Sommerfeld model. 392 00:39:54,050 --> 00:39:58,280 And we found Pauli power magnetism, power magnetism. 393 00:39:59,930 --> 00:40:06,230 And there was some calculation which went along with that and it is basically exactly the same physics. 394 00:40:06,440 --> 00:40:13,010 You have a bunch of electrons, you apply magnetic field. The moments of the electrons want to align with the magnetic field to lower that energy, 395 00:40:13,160 --> 00:40:18,140 but they can't all flip over because of the exclusion principle, so only a few of them flip over. 396 00:40:18,320 --> 00:40:23,240 But nonetheless the magnetisation that develops is in the same direction as the magnetic field that's been applied. 397 00:40:23,240 --> 00:40:31,160 So this is a power magnet, albeit a weaker one, than the free spin, because only some of the electrons will flip over in the case of a Fermi Gas. 398 00:40:33,740 --> 00:40:41,500 We'll see more about those in a while. So another type of magnetism which is common is dear magnetism. 399 00:40:44,260 --> 00:40:46,540 Which basically means Kai is less than zero. 400 00:40:46,750 --> 00:40:54,670 In other words, when you apply a magnetic field, the magnetic moment wants to point in the opposite direction as the magnetic field. 401 00:40:54,680 --> 00:40:56,920 Now, we may not know any examples of that immediately, 402 00:40:57,190 --> 00:41:07,299 but there is something that's qualitatively similar that we do now qualitatively like lenses law, 403 00:41:07,300 --> 00:41:12,100 which you may remember from first year in lenses law, 404 00:41:12,100 --> 00:41:16,510 to remind you if you have a loop of wire, if you apply a magnetic field to the loop of wire, 405 00:41:16,690 --> 00:41:20,500 currents will flow in that loop of wire to oppose the magnetic field. 406 00:41:20,680 --> 00:41:25,930 So you're developing a magnetic moment in the opposite direction of the change in magnetic field. 407 00:41:27,360 --> 00:41:30,450 Our loop of wire, however, is not a dire magnet. 408 00:41:30,840 --> 00:41:34,200 The reason for this is because that will develop a magnetic moment. 409 00:41:34,380 --> 00:41:42,030 But eventually the the current in the wire wall will decay and go down to zero and you'll have no magnetic moment left in the wire. 410 00:41:42,210 --> 00:41:50,610 So, in fact, it's not magnetic at all. It has no magnetic moment unless the wire is a superconductor which has no resistance or current never decay, 411 00:41:50,610 --> 00:41:56,700 and the current will keep flowing around forever and ever and ever. And and the magnetic moment would stay there forever and ever. 412 00:41:56,700 --> 00:41:59,730 And that's a real die magnet. We do not study superconductors this year. 413 00:41:59,940 --> 00:42:04,889 It's just nice to know that. That you could do this with a superconducting, get a real die magnet anyway, 414 00:42:04,890 --> 00:42:10,440 that the physics of magnetism is actually quite similar qualitatively to the physics of lenses law. 415 00:42:10,680 --> 00:42:14,450 But when you add quantum mechanics to it, you kind of, you know, 416 00:42:14,460 --> 00:42:21,270 in some interpretation you can think of these currents as being being stuck there forever and always opposing the magnetic field. 417 00:42:21,270 --> 00:42:28,500 Even when you when you do it with lenses like you have to keep changing the magnetic field to get the current to keep flowing in a die a magnet. 418 00:42:28,710 --> 00:42:32,130 As long as the magnetic field stays on the current, the current should keep flowing. 419 00:42:32,430 --> 00:42:41,070 So we'll see how that happens in a bit. The third case that we need to know about is the case of a ferromagnetic, 420 00:42:45,360 --> 00:42:56,190 which is when magnetisation is not equal to zero, even when B equals zero, something like iron. 421 00:42:57,430 --> 00:43:03,730 Has this property, the thing that sticks to your refrigerator. This is the thing that we usually think of as a magnet when we think about magnets. 422 00:43:04,060 --> 00:43:10,000 So a good place to start. Our study of magnetism is one atom at a time. 423 00:43:10,240 --> 00:43:20,890 So we'll start with atomic magnetism, atomic magnetism, and we'll build up to magnetism of real materials. 424 00:43:21,160 --> 00:43:35,740 And I should put it in this caveat that we always will we will ignore ignore the magnetism of nuclei, nuclear nuclei all together. 425 00:43:35,980 --> 00:43:45,460 And the reason why we ignore the nuclei is that if you think about the magnetic moment of some object, it's given by the born magnets on where to. 426 00:43:46,960 --> 00:43:52,600 Of the object for an electron. This is roughly the the up to the factor. 427 00:43:52,600 --> 00:43:54,220 This is the magnetic moment of an electron. 428 00:43:54,430 --> 00:44:03,790 It is similar for a nuclear on a proton or a neutron, but the mass that enters in here is bigger by a factor of a thousand or 2000 or something. 429 00:44:04,030 --> 00:44:11,829 So that the magnetic moment that you get from from a nucleus is much, much, much, much, much smaller than the magnetic moment you get from electrons. 430 00:44:11,830 --> 00:44:15,820 And that is why we're usually not concerned with the magnetism of nuclei. 431 00:44:16,060 --> 00:44:16,870 It is there. 432 00:44:16,900 --> 00:44:23,860 It sometimes turns out to be important, especially if you're thinking about things in extremely low temperatures or if you're looking at very, 433 00:44:23,860 --> 00:44:28,000 very subtle and fine effects if you have to worry about the magnetism on nuclei. 434 00:44:28,210 --> 00:44:33,490 But for us, we are only going to be worried about the magnetism of the electrons because it's just much bigger. 435 00:44:35,230 --> 00:44:39,129 So we're going to ignore the nuclei and then we have to start the way. 436 00:44:39,130 --> 00:44:43,600 We always start by writing down a Hamiltonian for the electrons in the atom. 437 00:44:43,900 --> 00:44:49,090 So I write the Hamiltonian Hamiltonian, and it is conventional to use sort of a script for the Hamiltonian, 438 00:44:49,090 --> 00:44:54,490 so you don't get it confused with the magnetic induction H you know, being H and new and that kind of stuff. 439 00:44:54,730 --> 00:45:01,900 So we use a script H for the Hamiltonian and we'll write it in this hopefully ways that you've seen before. 440 00:45:01,900 --> 00:45:11,830 So there's a a kinetic term, P plus A squared over two M where A is the vector potential double cross A is B, 441 00:45:11,830 --> 00:45:16,960 whatever magnetic field I decide to apply to this system, then there's going to be. 442 00:45:17,680 --> 00:45:20,860 So this is just the kinetic energy, then there's going to be an attraction to the nucleus. 443 00:45:21,090 --> 00:45:35,469 V vrr And the last term we have seen before g you b b that sigma and this is the Zeeman term, zeeman arm two ns maybe one. 444 00:45:35,470 --> 00:45:42,400 N Anyone now want one and I give them on term is my notes. 445 00:45:43,030 --> 00:45:46,390 My notes say my notes. Don't say it. Okay? I don't know. All right. 446 00:45:47,650 --> 00:45:52,900 So this is the term that tells you that the magnetic moment of the electron is lower energy. 447 00:45:52,900 --> 00:46:00,190 If the electron points in the same direction at the magnetic moment, points in the same direction as magnetic field, this gets very confusing. 448 00:46:00,190 --> 00:46:06,489 The signs get very confusing here. Remember that the magnetic moment of an electron spin points in the opposite 449 00:46:06,490 --> 00:46:09,820 direction of its magnetic moment because the charge in the electron is negative. 450 00:46:10,870 --> 00:46:16,020 So it looks as if the energy would be lower here when the so here, 451 00:46:16,030 --> 00:46:20,950 here the energy is lower if sigma is in the opposite direction of B, so it's pointing in the opposite direction. 452 00:46:20,950 --> 00:46:25,590 So it's negative. But if sigma is in the opposite direction is B, the magnetic moment is actually the opposite direction. 453 00:46:25,600 --> 00:46:29,409 Sigma So the magnetic moment is pointing in the same direction is b minus signs. 454 00:46:29,410 --> 00:46:37,990 Ah, you know the data. Me okay, maybe is the more magnetised for the electron energy is the g factor which will usually take to be just two. 455 00:46:38,560 --> 00:46:42,760 All right, so what do we do with this? 456 00:46:42,760 --> 00:46:48,660 Well, we're really would like to treat the magnetic field a little bit as a perturbation if we could serve. 457 00:46:49,160 --> 00:46:54,610 You know, we'll start by rewriting this Hamiltonian as in the following way, 458 00:46:54,610 --> 00:47:04,360 h equals P squared over two M plus V of R and that is just h zero or HB equals zero. 459 00:47:05,380 --> 00:47:11,820 Separate out that term first. Then we have a couple terms left over that we didn't write yet. 460 00:47:11,820 --> 00:47:18,720 So maybe that's writers here plus plus all this stuff we're going to write next. 461 00:47:19,050 --> 00:47:29,220 So one is the cross term with the vector potentially over 2mp, a plus eight P, and then there's another term, 462 00:47:29,220 --> 00:47:40,740 the vector potential squared over 2ma squared, and then we have the same on term left over g u b b that sigma. 463 00:47:41,930 --> 00:47:47,819 And then we have to do a little bit of work here. And one piece of work we need to do is we need to simplify this term. 464 00:47:47,820 --> 00:47:54,950 And the way we're going to simplify that is by choosing a gauge. Choose gauge for the vector potential. 465 00:47:55,340 --> 00:48:04,159 And the gauge I'm going to choose is that the vector potential is one half the cross R And if you choose it like that, 466 00:48:04,160 --> 00:48:13,520 this, this does give you del cross a equals B, if you choose it like this, then actually commute through with each other. 467 00:48:13,520 --> 00:48:23,570 And it's maybe not so obvious to see if that is true. But it is true because that X component of A contains the y component of R, 468 00:48:23,780 --> 00:48:30,919 so the x component of a can commute with the with the x component component of p and those are the ones that are multiplied by each other. 469 00:48:30,920 --> 00:48:43,730 So it doesn't matter which order you put them in. So choosing this gauge, we can then we write this as E over to M to a dot p, 470 00:48:44,360 --> 00:48:51,889 so those two right through each other and then we'll write in our expression for a while. 471 00:48:51,890 --> 00:48:57,650 We cancel the two with the one half and we get B cross R dotted with P. 472 00:48:58,430 --> 00:49:07,190 We use the triple product identity to rewrite that as E over to M the dot R across p, 473 00:49:07,640 --> 00:49:13,640 r across p and then just absorbing a factor of h bar into the R across p will 474 00:49:13,640 --> 00:49:18,709 recognise this thing as in the angular momentum and put the power up here. 475 00:49:18,710 --> 00:49:32,600 So we get new B and D dot l where L is the orbital angular momentum equals orbital angular momentum. 476 00:49:38,290 --> 00:49:44,140 So putting this all together, we end up with the final Hamiltonian that we're going to work with. 477 00:49:44,680 --> 00:49:50,710 The Hamiltonian is the Hamiltonian at the equals zero plus couple terms. 478 00:49:51,280 --> 00:50:03,070 You b be dotted with the orbital angular momentum plus g times the spin angular momentum plus this one more term they were left out. 479 00:50:03,790 --> 00:50:15,259 E squared over to m e squared. And I guess maybe we should stop there and pick up next time with this Hamiltonian 480 00:50:15,260 --> 00:50:18,520 and will analyse the effect of the magnetic field on this Hamiltonian. Okay. 481 00:50:18,790 --> 00:50:24,250 See you tomorrow. Don't forget the meeting, meaning right after the lecture. 482 00:50:25,160 --> 00:50:30,640 And then also the hands like this afternoon. 5 p.m. searching for extrasolar planets or something. 483 00:50:30,770 --> 00:50:31,810 Okay. By.