 Now, maybe I'm trying to think what would be the best order to go about this. So, maybe let me say a little bit about the spin orbit coupling and the role of applied B-fields, etc. Now, one point, if you remember, that we had in the contacts, I drew the density of states like this. And of course, all the interesting effects that we are talking about in spin valves, etc., come from the fact that you can do something a little different for one spin than for the other. Because usually, of course, they're all the same. You just multiply by 2 at the end. And here, of course, the channel, there was nothing special about it. It's just like any old channel. But the important thing was the contact had this, where magnets, and you could have parallel or anti-parallel as I mentioned. And here, when you look at the density of states, the thing is it is like one or two electron volts. And one of the points I sometimes stress is that, you know, that you have heard of the Zeeman effect, and that is that when you put a magnetic field, the energy levels of two spins, they're split. Split by an amount that is given by this Bohr magneton times the magnetic field. And as I said, you know, this every electron, we talked about the spin. But one of the things I didn't get into is that basically it is as if every electron is a magnet, in the sense that it has. And the way we know this, where this was all first detected, of course, is this famous Stern-Gerlach experiment. That was back in the early 1920s, 22 or so. And the experiment consists of the following. That is, you have a region with an inhomogeneous magnetic field. So magnetic field is increasing, let's say. And what was known is that any little magnet in an inhomogeneous field wants to bend in a particular direction. Because people say, well, the energy of a magnet in a magnetic field is proportional to the B. So if the B is changing, it is as if the energy is lower on one side than the other. And so it feels a force in the direction of the lower energy force. And the surprising thing, well, that electrons did bend, exactly. It's just that some bent upwards, some bent downwards. And they saw these two spots on the screen. In the original experiment, I think we're done with gold atoms. But later on, they did this with a hydrogen atom, which is very simple. And the effect is believed to be due to the electron. Of course, it's hard to do the experiment directly on the electron because the electron is being a charged particle. You also see this Lorentz force, which is separate from this one. Now you also see this qv cross B. So when they do this experiment, they do it with an atom so that basically it's neutral. So there's no qv cross B part to it. All that you have is this one. And from how much it deflects, you can say how big of a magnet that electron is. So basically every electron is like a little magnet. And how big is its magnetic moment? Well, that you know from this, deflection. And that's this Bohr magneton. And the Bohr magneton, the expression is qh bar over 2m. And if you put in the numbers, then these are all fundamental constants. I believe it comes to about 10 to the minus 23rd amps per meter. And I often think of it as a little current loop carrying 1 milliamp with 1 angstrom each side. I mean, no particular significance to those numbers. All I'm saying is if you had a little current loop going around like this and calculated its moment in a current times the area, you'd get 10 to the minus 23. But of course, I could easily have chosen something that was a little bigger, but the current was smaller. So no particular significance to those numbers. It's just the way I can remember what it is. So the bottom line is that's now an experimental fact. So if you ask anyone, how do we know electrons have spin? The bottom line is Stern-Gerlach experiment. It still goes back to this thing some 80 years ago. This is how you really know it's spin and that there are these two spins because there are two spots essentially on the screen. And when it was first discovered in a major surprise, I mean, change it, everybody's thinking over and over, okay. Now, the thing is that if you look at, so there is these two forces. One is this Lorentz force, which is totally separate. We are not talking about here. But if you're looking about this, what is the effect? How much energy is changed by a magnetic field? You put in the numbers, what you'll see is this. Again, this is MKS. So if I use magnetic field in MKS, like one Tesla, since this is 10 to the minus 23, what you get is about 10 to the minus 23 joules, which you realize is about 0.1 milli-electron volt. Because as you know, electron volt is like 10 to the minus 19th joules. So milli-electron volt would be minus 22 and then 1 tenth of that. So usually, if you put one Tesla field, which is a fairly big field, like 10 Tesla is quite big, 100 Tesla is really, really big. We are, but one Tesla is also fairly big and you get a small fraction. So usually at room temperature, 25 millivolts kT, you don't worry about it all that much. So the one question that always comes up immediately then is, how can these things have a separation of one or two electron volts? If you are trying to create this with a magnetic field, you need some unearthly field to have that, that you'll never get. So how does this happen? And this is what in the early, when people first start understanding magnetism, they realize that there was this enormous exchange field that is the reason you have magnets in a way. And what, that was another question someone asked me, so maybe this is a good time to say a little bit about it. And that is that, you see, every electron is a magnet, but then why isn't, I mean, every solid we have, silicon, gallium arsenide, they have lots of electrons in them. So why aren't they magnetic? Well, because normally the magnets are all random. And so, overall, you wouldn't get anything. But then, in the actual magnets, like iron, there is this enormous interaction among these magnets, which tends to line them up so that they all find it preferential to be pointing in the same direction. That's a low energy state, relatively. And in the beginning, I think in Feynman lectures, he says that, well, if you just thought of the interaction between two magnetic moments and say that, well, maybe it's the field of this that is trying to line it up, the magnetic field due to this pulls this together and so on, maybe that's what is happening. If you did that and calculate the corresponding transition temperature, the temperature below which it would become magnetic, you'll get something less than a Kelvin, very small. Whereas the fact is that magnets, of course, exist at room temperature. The Curie temperature is like 1,000 Kelvin. So obviously, whatever's the interaction that is keeping it together is something much stronger than what you would have expected from Maxwell's equations. And this is what people refer to as this exchange field. And it only acts when two magnets have overlapping wave functions. So when an electron is here and another one is there, and the wave functions do not overlap at all, then the magnetic force between them is whatever you learn from Maxwell's equations. You have two magnets, you can figure out exactly what it is. But if the wave functions overlap, then there is a much stronger and more complicated force, for which, I mean, I can say the words, but necessarily wouldn't really be convincing where it comes from. But the point is this is there, and this is what leads to magnetism. And you can see easily that the fact that this is split by a volt immediately tells you, you see, 0.1 millivolts required 1 tesla, so 1 volt would require like 10,000 tesla. So effectively, that exchange field, that one electron is feeling due to the rest of it, is like 10,000 tesla. It's that high. It's an enormous, right? And to some extent, I mentioned earlier about the spin torque, the fact that now you can inject spins into a magnet. So supposing you have this magnet here, pointing in this direction, and you inject a lot of spins from here. Let's say looking like this, then this magnet now feels an enormous force due to this extra spins that you have injected. Now you might say, well, I'm not injecting all that many. Maybe one in a million or so. But that's fine. You see, ordinarily it would have been feeling this 10 to the fourth tesla, but even if, let's say, you had a spin that was like one in a million, one in a million at, I mean, normally you have spins there that's like one on every atom, let's say. But what you have done is created an imbalance that's like one part in a million. Well, then instead of 10 to the fourth tesla, you'd have 10 to the minus two tesla, which is 100 Gauss, and that's enough to turn a magnet. That's the kind of field you need to turn a magnet, basically. You see? So now you might say, well, you already had 10 to the fourth tesla in there, and you add a 10, and you do some non-equilibrium thing, and get a 10 to the minus two in there. How does it matter? It's a minor difference. But it's just that the magnet normally doesn't feel anything due to this. Because as long as you have a magnet and you put something in the same direction, it just ignores it. It's anything off its axis that exerts a torque on it that actually wants to make it go round. So the big 10 to the fourth tesla, it has kind of adjusted to it, makes no difference to it. It's in the same direction as the magnet doesn't matter. But this little extra one is at some angle, which is why, of course, it's very important to have these two magnets have a little bit of an angle to it. So when they build these devices, they almost deliberately actually build in a little angle. Because if they were exactly the same, you wouldn't get as much of an effect. You'd still get some because of thermal fluctuations, because it's never quite straight, so it jiggles around, and so it feels some torque. And they say that if it's a little hotter, it's easier to turn it. So when the inject spins, you'll have a lower threshold current if it's a little hotter, et cetera. But then just to get it to turn it better with less current, they deliberately build in a little bit of an angle. Because if it's exactly in the same direction, nothing would happen. That's important. Anyway, so in terms of the magnitudes of these things, then what I wanted to be clear on is how different this exchange field is compared to our usual magnetic fields. So it's a very focused thing in the sense you only get it if wave functions overlap, very focused thing locally, but magnitude is much bigger than what you normally expect. So just a little bit of imbalance in spins, and it exerts a sizable thing. Because, you know, ordinarily you'd think it would be very hard to be turning a magnet with just hitting it with these little spins, because a magnet is, you know, so many spins together. But of course that is true. I mean, you couldn't do anything to this magnet if it was micron or even 10th of a micron thick. The ones that are getting turned are like a few monolayers. These are nanomagnets in the thickness. As far as ADA is concerned, they're fairly big. It's the thickness that is. Okay, so in terms of the effect of applied field on devices, this was a question that lots of times you don't necessarily have to have the field on when you're doing this. Because what you could do is at first start with parallel magnets and then turn up the field and get it to turn. And then after it has turned, then do your measurement again. Not like you have to leave the field on while making the measurement. So you just have to turn this thing. But even then, there are many people who say that if you just see a magnetor resistance, that's not a convincing enough proof that you have these effects that you're talking about. You should look for something like the Hanley effect or things like that. Because they say that, well, you know, just the fact that you have a magnet there means you'll be all having all kinds of fringing fields. And then if the resistance changes a little bit, when you turn it, who knows it's not some complicated Hall effect because of the field that changed, et cetera. Especially if you're talking of, you know, a fraction of a percent, a very small thing. No small things can happen for multiple reasons, right? So you. I mean, what are you talking about? I mean, using speed torque, you are trying to switch a nanometer. Now say, have a nanometer dimension 100 nanometer, 60 nanometer, thickness say 20 nanometer. So what's, I mean, do you have an estimation of what kind of field you require to switch the nanometer? Where do I come from? Yeah, thickness 20 nanometer sounds high to me for switching with spin torque. My guess is we do have to have more like two nanometers. What I've seen. The thickness. Area is fine. Area is fine because it doesn't matter if it's big because you're hitting the whole thing with spins, right? So point is for every little unit cell, you have a spin coming in. That's fine. So it's area big doesn't matter. Yeah. No, usually then you're using something where the two parameters people talk about, magnet, this is m versus h. So they talk about the coercive field. What is this field you need to switch it from one side to another and then the complete total magnetization, saturation magnetization. Those are the two numbers you usually get. And where you think about this is that, as if there's this energy barrier. So when you're here, this is it. When you're here, it's like this. And this barrier is actually the product of hc and m saturation. And that barrier kind of determines the stability of this thing. And so for a nanomagnet to be, and actually the barrier depends on volume. So in that sense it is true that as you cut something down, your, the barrier is going down. But as long as the barrier is safe about one electron volt, the magnet is, say for one electron volt, I think the estimate is the magnet should be stable for about 10 years. What I mean by that is if you put it here, then for about 10 years you are sure it will be there. But on the other hand, if instead of one electron volt, this was like 0.1 electron volts, then you couldn't be that sure for 10 years. I mean, then maybe a month. And then when you get it lower and lower, of course it gets worse. At some point you are not even sure for the next second. Why is it, there is a question because, I mean, we are actually doing a phone simulation. In the phone simulation, we have found, I haven't done the theoretical model, but we have found that it takes, take that one nanomagnet with two nanometers, the thermal fluctuation and you have a nanomagnet is after pressure of 160 or 170. It takes place, I mean, with a thermal magnet. Oh, I see. Okay, so this I'll have to, but a lot will depend on whether the parameters you used what barrier this would correspond to. Because they say that they, you know, the time, the lifetime is dependent exponentially on that barrier and with some constant in front. And so what I'm not sure is what you used for that barrier height. But devices with nanomagnets in them, you know, one or two nanometers in them are, of course, for logic whether it can be used, that's a whole separate issue. But for memory, I think that's being used, you know, these are things people are actively working on, looks like those products will happen, right? For memory, from what I've heard. So the point as we're getting to here is that many people would say that, well, you know, GMR is not a good enough experiment because if you see just a fraction of a percent change in resistance, who knows what caused it? Maybe it's some complicated hall effect. But the way you can convince me that it's really got something to do with spins and spins how they turn has to, you have to do a Hanley experiment. Now the Hanley experiment would work something like this. And that is, you see, I mentioned that was the end of my talk this morning that you can have parallel and anti-parallel, but you can also have things that are in between. So in general, the current you would get would do something like this. This is parallel, that's anti-parallel, et cetera. But then if the magnet were in between, you'd see some in-between signal, right? And I mentioned this is this cosine square half theta, half being this important thing, distinction from photons. Now, if instead of turning the magnet, you had a way of turning the electron itself, it gives you exactly the same effect. So what I mean is you could keep the magnet fixed. They're both, you know, just parallel or anti-parallel, whatever it is. But then the electron that comes in, if as it goes along it turns, let's say, then by the time it gets here, question is what angle is it in? And that's what will determine how much of a current you'll get, right? And that's, and so in a hand-leaf effect, that's effectively what you're doing. You're putting a magnetic field and the general understanding is that any time you have a magnetic field, if you had a spin in a particular direction, it would want to precess around it. You know, that would come out of, that's what you expect for, you know, classically also in a magnetic field, magnetic moments, what they're supposed to do. And it would come out of Schrodinger equation or any GF anywhere, if you do the quantum mechanics, right? That would come out, and it would precess around it. And that's what the hand-leaf effect is based on. So it's like, okay, I inject spins this way and I put a magnetic field this way, let's say. Then what will happen is it will precess. And the current should be oscillating like this. Now usually, I think there are experiments by Applebaum which shows many, many oscillations, you know, like maybe five or 10 periods. But most of them usually just show barely one oscillation or so. Because after that, usually, again, fake coherence issues come in. So you don't quite see this clean theorist's curve like this, what you see is more something that dies out. Or maybe an oscillation or two. But in silicon, there are these experiments by Applebaum which show like five or 10 oscillations. Magnetic field. And spin orbit is kind of related in the sense that the simplest way to think about the spin orbit coupling is this relativistic effect because of which a moving electron kind of, the electric field itself looks like an effective magnetic field. That's all they say. So if you put an electric field in this direction and an electron is moving this way, then the effective field is like k cross e. So the effective magnetic field. So if you had a, so if you can change this electric field, it would be like changing the magnetic field and you should see similar oscillations. And there is a paper last year which reports experiments showing this. I think they show a couple of oscillations. It's done this. And of course, again, when you see, these are always small effects when you start out. And whenever you have small effects, you have to worry about is it really why you think it is or is it due to something else? And one of the convincing things they have there is that if the injected spins are in a direction perpendicular to the b field, this effective b field, then you see the effect. Whereas if you turn these magnets so that the injected spins are in the same direction as the b field, in which case they shouldn't turn because if we feel this way and injected this way, nothing will happen, then the oscillations go away. So they show that control experiment that if you're in the same direction, you don't see anything. So they do the experiment with the magnets in a particular direction. And then with the magnets turned in the direction of the b field itself. And so but then, you know, this is only one experiment. So one needs many more experiments and checks and all that to be sure what happens. So that is one, now actually the spin orbit coefficient itself though, by the way, I shouldn't have written equal to because there is like a constant in front of me. There are other constants that I have not put down. Now, the spin orbit coefficient itself is known from many other experiments actually, which I have not gone into this, what I call the Shubnikov-Dihas oscillations from which you can measure this coefficient and how strong it is. So that is known independently from other experiments. But here it has a direct effect on the current in this structure. Now, and the spin orbit coupling, as I said, is variable in different materials and silicon, it's much smaller than gallium arsenide and which is smaller than indium arsenide. Indium arsenide is the one that was used for many of these experiments. So if you're looking for strong spin orbit coupling, that's where you usually go to indium arsenide. If you're trying to avoid it, then silicon is good place. Usually, okay, so I think. Now, regarding this morning, the question you had about this screening of time, I guess you are saying that the electron feels some effective potential with all that included in it. The important part of what I was trying to say this morning with that device was that usually, so if you haven't started with an anti-parallel device and in an anti-parallel device, let's say you have these reds coming in from here and the blues coming in from here. And ordinarily, of course, because this only has blues, this only has reds. I said, you know, think of the perfect device, you have perfect contacts, nothing in between. And ordinarily, how much goes this way or how much goes this way would be determined by this F1 minus F2. And that's why conductance, current, everything depends on that. Whereas, here I was trying to give you an example where ordinarily no current would flow because red comes in, can't get out, blues come in, can't get out. But the only way they can flow is by flipping their spin. So if a red came in from here, flipped spins and went out, then it would be able to. And the point was that if you put lots of, if you had lots of red impurities here, but no blue impurities, then what happens is when a red comes in from here and it's looking for an impurity to flip off of, it finds nothing. It's all red because this interaction is red blue becomes blue red, that's the exchange interaction. So red electron, blue impurity becomes blue electron, red impurity, that's the basic interaction. So it is as if you have stopped anything from going this way because red can't find a blue. On the other hand, the blues have lots of reds so they can go this way. So there is this, you're built in something that allows you to go one way but not the other, right? And that's what leads to these anomalous things. And that is something that wouldn't come out of this equation. You see what I tried to write down was a different equation that you need almost to handle things like this. And I wanted to kind of make that point that I've said that for the elastic resistor, you know how all this understanding gets much simpler. In a way this is still an elastic resistor because this interaction is elastic, no energy is exchanged. So it is really what makes this kind of different is that it is interacting with something that has a degree of freedom and so has entropy and other things associated with it. That's what kind of makes it different, that you are scattering off of something which is out of equilibrium, of course, it is all red. So although the basic device you can understand very easily how it works, how you can go one way but not the other, but then the implications of it that how general is this, when can this happen, all the things you can learn from it, that's what really makes it interesting, you see? And that, of course, you can think about it. I think the details I've written up in various places. So you can look at that if you're interested, yes. Since spin-up and spin-down states are different energies, so when you space flip, will that be still elastic, elastic? Because energy is changed, right? Now elastic in the sense, what happens is, so supposing I have an, so I have this red electrons, so this is energy density of states. And energy, so this is the red electrons, blue electrons, and so I have a red electron sitting here. And it interacts off of an impurity which happens to be blue. So this is the blue impurity. I'm sorry, I guess I should have drawn it. I've been using, now when this scatters off of that, the thing is it goes to exactly the same energy because in this discussion impurities, whether it is red or blue have exactly the same energy, no difference. It's exactly the same energy whether it's up or down. So no energy is exchanged at all. In general, interactions involve both. And the reason I was doing this with a, where you are assuming no energy exchanges, it brings out the importance of order and entropy and things like that. Because it is almost, what it means is, everyone understands that you can have a device and if you shine light on it, it will give you energy that you can run a light bulb with. That will be a photo cell, for example. So then you say, well, where did the energy come from? Just came from your light. And the point that you sometimes see is that, well, energy can also come from entropy in a way. And the thing is, this thing is not providing you with any energy at all, really. The energy is coming from the contacts as I was trying to make the point that if you look at energy conservation, it's coming from the contact. It's just that normally you're not allowed to do that. You're not allowed to take energy from this random reservoir around you and do something useful with it. That's the second law. Here you are being able to do it because you are taking something ordered and turning it into something disordered. It's the picture that I had, you know that. As long as it's all red, it has got a very low entropy because it can be that way only one way, whereas what's on the left is like completely disordered. There are many, many ways of having it. So what drives it from right to left is the fact that you're going from one state to thousands of possible states. That is really what drives it. And this is the thing that is difficult to put into any mechanical descriptions of things in terms of forces, et cetera. It's a whole different kind of thing. As I said, in physics, these two branches develop separately. The understanding of mechanics and the understanding of these entropy-driven things. Essentially. And putting it all together was this major achievement of the 19th century. So, and this one is kind of irreversible. In the sense the right will go to the left, but the left won't come back to the right. Left, you know, left to itself won't come back to the right. And yet if you're just writing a Hamiltonian and trying to do Schrodinger equation out of it, anything that goes one way will go back the other way too. And this is something I've seen in all kinds of other contexts also. So for example, one question that sometimes comes up here. If you have a conductor and you connect it to a contact, question is, we usually say that when an electron comes in from here, it just flows out and there's hardly any reflection. So electrons don't turn around, it just goes through. Now, and that is true. That is why you have this quantized conductance that you see when you make good contacts to something. Because the quantized conductance is perfectly ballistic. Anything that comes in here goes out. No reflections, et cetera. But if you just did a, like, solve Schrodinger equation in 1D, you know, and you put in something like this with a big something that broadens out. And you solve Schrodinger equation for a wave. What you'd find is there's usually a fairly sizable reflection often. And that's because anytime you're doing mechanics, it's like you have one mode. Whenever you go out there, also, you're kind of still in one mode. It may be some combination of things, but degree of freedom hasn't increased. On the other hand, the reason you don't have much reflection, if you just think classically it's really very simple, it's this. You're coming in with, you know, it's like you're coming in one lane. And when you go out here, it's like hundreds of lanes. So this electron can get reflected in one way, but can get transmitted in 100 ways. But when you do a quantum mechanical simulation of this, that's not necessarily happening. It is not getting this, because mechanics will never increase this entropy. I mean, it won't take one thing and distribute it among 10 things, as long as you're doing pure mechanics. And so what I've often seen is you might calculate a lot of reflection, but once you put some scattering into the model, then you'll find out it transmits fine. You get much better transmission, actually, with some scattering into it. So this is the part that I say that, well, that this entropic thing, just because you have solved Schrodinger equation very carefully, doesn't necessarily mean you have got it all right. And sometimes you may not, maybe making it worse in a way, because you're bringing in things that are really not happening in the real world. And it's a similar issue about this. I think someone asked me about what does it take to make a good spin contact. And this is where I guess there's been a lot of work. I'm not sure if there's a consensus on the understanding of things, but my view of it is something like that when you look inside the channel, let's say you've got 10 lanes, and then you have this contact, and of course what makes the two spins different is that you have different number of lanes in the contact. So let us say you've got 100 lanes here for the minority spin, and maybe 10,000 lanes for the majority spin. And what you're really trying to do is inject more of one spin than the other. And the point is that if I look here, of course it makes no difference, because you're finally going to feed it into 10 lanes. So whether you have 100 or 10,000 doesn't matter. You keep them full anyway, no matter what. So the trick is then to put enough of a barrier here so that it gets harder for both of them to fill it. But then after some time, this guy won't be able to fill it anymore very well, but this will still be able to, because 10,000 lanes even working not so well can still fill it. And then you'll start seeing a difference around here. That's my understanding of this. And again, these are things if you're just doing a pure quantum mechanical simulation with same number of lanes, et cetera, you may not necessarily get it out of that. You see? So a lot of things from common sense looks like that's what should happen. If you did a simple classical treatment, you'll get it, but the full quantum mechanics might miss it. You see? The full quote unquote full quantum mechanics. Yeah. Of full meaning, yeah, let me qualify that. It should be full quantum mechanics if it means including defacing processes, including scattering processes, then it should get it. Because one of the points I tried to make is that with any GF, you can go from the fully coherent quantum limit to the fully incoherent quantum limit. And in the incoherent quantum limit, it would cover, it would be essentially equivalent to Boltzmann. It's just that often people don't put in enough scattering to be doing that. I mean they do not put in any scattering usually because the coherent calculation is a whole lot easier to do. So that's what you normally do. So you ignore all that. And then you're missing a lot of this. Okay, now let's see where we, now regarding speed, I suppose, to depend on the scheme one has in mind. So the way I see it is, the way we are talking of spin transport, it's basically carried by electrons. So whatever it is that's carrying the charge, I mean, and limits the speed of that, that's the same thing with the spin too. So usually transistors are limited by transit times. Here also it would be similar. Now it's just that the question of how you use it, and this is where I've made the point that if you really want to use spin, then one should go away from the paradigm of having the information in the form of charge. What I mean by that is in a regular CMOS, what you do is charge one capacitor, the information is in the capacitor, and you charge this capacitor according to the information on this capacitor. And that energy, of course, comes from here. And usually when you talk about spin transistors, the idea is that well, as far as outside is concerned, it's still the same. It's still driving capacitors, et cetera. What you do is replace this part somehow. Something to do with spin, you do something here. And when you do that, usually any advantage you may get with spin is very quickly lost. Because as you try to measure, because of this polarization, as I said, even if you had one millivolt signal inside, by the time you measure it, it's 0.1 millivolts, usually because the polarization of the magnets is probably 110%, et cetera. So normally whenever you do that, you lose a lot of this thing. On the other hand, if you could do it directly so that the information is not as a capacitor, but as a magnet, and you put a voltage here, and what that does is it inject spins, and then at this end, another magnet gets turned according to that spin. So then you can be changing the state of this magnet according to the state of this magnet, and then again use that information to go to the next one, et cetera. But then advantages are always in the spin domain. You're never really trying to measure the voltage, convert it, reconvert, go back, et cetera. That's the idea. So these things, of course, are very much, we'll have to see where all this goes after you've looked at all the various aspects. I think someone asked me this question about spin transistors. But yeah. That's a big question, I support that, but the spin way, the speed of the spin, you just talked about the speed of the spin propagation, which is just the electrons. But the spin wave, which you had on your earlier slide, which was the red and blue propagate, remember the spin wave transformer? Yeah. The speed of that up and down spin. Right. We had the side channel. Oh, yeah, but there was no. There was up spin, it was propagating. Oh, yes. That was the, got it, got it. And the gradient of the potential. This one, right? That's what that's a fuse. Right, right, right. Yes, but it is still basically electrons that are going upwards, let's say. So for example, in that picture, it is, I didn't mean to imply that anything special was that it was a collective effect of any kind. It is still basically electrons. And what would happen is the red electrons and the blue electrons had two different chemical potentials near the channel, and it gradually dies out towards the end. And so the reds go out, get spin flipped because of various mechanisms in that semiconductor, and then just come back. Come back again, right? That was the picture there. And so, of course, there are magnetic materials where there are things like spin waves, and I think people have been considering what it would take to use spin waves like that in magnetic materials to propagate information. But that is separate. Here I'm just talking of transmission of this information through copper, silicon, any semiconductor or metal, really. And there I'd say it is just like any other. Now the thing is that if you're going to use it to turn a magnet, then the other issue that comes up is how long does it take to turn a magnet? Because to me, that is one of the most important issues here that usually in the past spins and magnets have been kind of two totally different fields. You know, I've often talked to people who are often surprised to hear that they're even related, you see? Because pintronics and magnetics seem like two totally different fields, you see? And what has now happened is, I'd say it has become one big field. And because magnets, especially when you scale it down, when you go to big magnets, what happens is a lot of the forces involved are things like domain walls, magnetostatics, et cetera. So it sounds like a totally different ball game. But once you get down to small things, so basically it's a material in which the spins have a certain interactions among them which tend to keep them lined up, right? And people have good descriptions of how that works. And what people have done is taken the standard equations that are used for modeling magnets and then figuring out how to handle spin currents. You know, something that five years ago wasn't there or 10 years ago no one had done. But in the last 10 years, people have figured out how to describe how a magnet is driven by spin currents, for example. So those are things, right? So this is a very important development, but then you might say that, well, magnets are big things, you know, it takes a long time to switch one of them. So wouldn't it be slow? And the thing is, again, that is because our experience is with relatively big magnets. Big magnets meaning ones that have millions of spins in them. And then it does take longer to switch. On the other hand, if you, a magnet can be stable with even a thousand spins in it. You know, there's the things people have built and so on. And then they can be switched a whole lot faster and with a whole lot less energy. But those are, again, things that need to be looked at to be sure, yeah. So what you're going to see is when you're talking about the spin logic, spinless logic, and the chart-based logic that we have talked, and the chart-based logic, we are talking of the scaling down of the, I mean, so it can go through, but in the spinless body system, we really want to divide. Then if we are to keep two nanobytes very close together of the contacts, then there will be an interaction between them. So you cannot go very close, right? So that's a problem within the system here. One more thing, so how do you see that? Yeah, this is the part I do not know enough about. But what I am told is, again, as you go to small magnets, that interaction is relatively small compared to the interaction due to the spin-driven things. So in general, people who build these things, they feel that the spin torque provides a much better route to scaling things down. And it could overcome what happens due to the stray fields themselves, physical distance between them. But this is something I have not thought carefully about to be sure, really. Okay, about this polarization, I suppose this is a matter of choice to some extent, in the sense that you could have defined magnetor resistance in different ways. So the way it is motivated is this magnetor resistance is, I think, parallel conductance over anti-parallel conductance minus 1. So it is this. So you look at how much the conductance changes and divide it by the anti-parallel conductance. Now, why anti-parallel? You could have used parallel, for example. But when you do anti-parallel, then, of course, in the best of situations, this could be 0. So you could actually get 500% magnetor resistance, for example. On the other hand, if you had defined it as the sum here, for example, then, of course, you would never be more than 100%. But what you see in the literature, you know, you have reports of 500% magnetor resistance. But that's because the definition is like this, which could well exceed 1. And similarly, what you call polarization, again, I guess the motivation is it should be 0 when the two R's are equal. That means you're not discriminating between two spins at all. That's what should be 0. And on the other hand, if one of them is 0, then it's like 100%. So that's how you're kind of choosing what it is. And I think this is the standard that people use. And then the magnetor resistance comes out as p square over 1 minus p square. I mean, the model that I went through in the morning, I think that's what it would give you. Yeah, I'm forgetting what the spin alignment was about. Yes, please. So electrons would really go through a chain of red and find themselves with one another. Right, so, right, right. So what we're discussing in this picture, it's what I call this one electron picture. So always the thinking is one electron feels some average potential due to everything else. And there is no collective effect involved in this. So different electrons don't need to have any particular coherence amongst themselves or anything. It's basically still a one electron effect, really. I'd say that every electron moves in some mean field, overall. So nothing beyond mean field, right? No effect beyond a mean field is how we think about this. Right, and? Right, for instance, you have to need to from the Coulombian interaction. This is what we talked about, electron transport. We used something to see. Which is the Poisson equation type of thing. Oh, so you're saying you could have a similar thing here. Right, yeah, if the similar thing would be due to magnetostatics, and that is probably a whole lot weaker. The part that you worry about a little bit may be the exchange interactions which could be bigger if they came close enough. Now in a magnet, of course, that is all important. That the way you say it, the reason you have a magnet is people say that if you put a magnetic field, then, of course, up and down will separate, and you'll have a lot more of one of them than the other. And you can show that the magnetization will be tan hyperbolic mu BB over 2KT. Tan hyperbolic comes from this ratio is this exponential. And then if you write down how many are up and how many are down, you'd actually get that. And the way you explain magnets, why magnets form, again, this is in Feynman lectures, is that if I put an external field, this is the overall magnetization I'd get. But as the magnetization forms, it provides an effective field, almost like Poisson equation. And that is the part you put in here with lambda m. And so you could have a correction due to that as if there's an internal field of some kind. And then you show that if lambda is big enough, then compared to this temperature, if lambda is big enough, then even when this is zero, you could have a solution. And so that is why it kind of by itself becomes a magnet. So this would be in the sense of a self-concent mean field, almost. This is the mean field theory of ferromagnetism. And so they say you plot the left-hand side is a straight line. The right-hand side is something like this. And depending on the parameters, you might have a non-zero solution. That's how you show it easily. But in any of these descriptions, in the ordinary semiconductors, no one seems to worry about it. Because in any case, the spin densities involved are like one in a million. Which means if you have one here at a particular atom, the next one is after a million atoms or so. Relatively not dense, not very dense really. In magnets, you worry about such things. And so the equations people use to describe the magnets, the LLG equation, there is, of course, an internal field. This spin wave function coherence, that's also an interesting point that you see. Interference, as I mentioned before, one of the distinctions I made that how when you do this quantum calculation, you have the correlation function, which is a 2 by 2 matrix. And the diagonal elements have a sensible meaning. It's like, how many here, how many there. And then there's this coherence thing and between the two wave function. So in the case of up spin and down spin, as I mentioned, you could interpret that as how many up electrons you have, how many down electrons you have, and what is the coherence between up and down. And in general, there is always these off diagonal elements, even in ordinary transport from one point to another, for example. And why don't we worry about it? Because usually there's all these defacing processes that more or less destroy it. And one interesting point is that, yeah, and usually this is something we don't have to worry about at room temperature much at all. And the reason is, I mean, the biggest defacing actually comes from electron-electron interactions. Namely the idea being that one electron, as it moves along, due to all the other electrons, of course, it sees some average potential, which is the Poisson. But then there is a fluctuating thing on top of that. Because what it feels due to the other electrons is also fluctuating on a picosecond scale. And so that more or less takes care of the phase of the whatever phase the electron had, because of that fluctuating potential on the picosecond scale. So this electron-electron interaction is the biggest source of defacing. And something you often overlook because it doesn't affect your mobility or current or things like that at all. Because electron-electron interaction, one electron loses momentum, another one picks it up. So as a whole, the current isn't affected. So lots of times people say, well, you know, I'm doing this experiment at 10 Kelvin. And, you know, the mobility doesn't increase anymore. The phonons are all frozen out. So there should be no defacing. Well, not really. As any experiment that is sensitive to phase, 10 Kelvin isn't cold enough usually. So for example, one of the classic experiments that kind of almost launched this field of mesoscopic physics was where people measured the resistance of a ring, a ring-shaped conductor. This was 1985. Ring-shaped conductor, and you measured the current. And as you change the magnetic field through this, it changes the phase of this arm with respect to that arm. And you'd see an oscillation in the resistance. So this was demonstrated. But the point is, you'd say, well, this then is a very good measure of what is the phase coherence length. You know, the fact that you don't see it at room temperature, why? Well, because the phase information is completely lost. So how cold do you make it? Often you might have thought, well, 10 Kelvin should be good enough. After all, below 10 Kelvin, mobility doesn't increase. Not really. I mean, 10 Kelvin, you wouldn't see a thing. You have to go to less than a Kelvin to be seeing most of this. So electron-electron interactions is very much a source of all these phase decoherence. Now the interesting thing about the spin is that that information seems much longer-lived. So what I mean is, you know, I said that the fact that people see these hand-leaf oscillations, that means, you know, this electron is turning and all that, that is related to this off-diagonal term in this description. And that seems to be observable at fairly high temperatures, comparatively. And so in spite of all this scattering is going on, the physical picture is if an electron comes in this way, its actual electronic state may be scattered all over the place. But the point is it doesn't lose memory of its spin direction at all. Because most of these, all these various scattering processes, it's never quite affecting your spin at all. If you're in this direction, it doesn't change you this way. Because to do that, you need a magnetic impurity or something of that sort. And so this is a much more robust piece of information. So even though the basic state is getting scattered all over the place. So you might say, well, the phase is all lost. But then whatever that is, this information isn't really lost at all. And sometimes I say that, well, if you're thinking in terms of a ring, it's like if you have a big fluctuating potential here of some kind, then it will destroy interference. But if you have an identically correlated potential on this arm as well, you'd still see the interference. In others, you destroy interference because of difference in the way you affect the two paths. If you affect it the same way in both paths, then it is not affected. So it is that most of the things that destroy phase kind of affect both spins the same way. You know, you could kind of say, and that is where the spin information isn't quite destroyed by that. It's almost like taking this whole thing and jiggling. That's fine, as long as both arms jiggle the same way, it makes no difference. Which is why, of course, when you do interference experiments, you try to keep the two paths as close as possible. If they're far away, then there'll be a lot more difference between them. As an experiment, you try to keep them as close as possible. Anyway, so about this linear response issue, the main point I think have is that the point I just tried to make earlier, that when you do this Taylor series expansion and come up with this del f, 0, del e things, because you do this and then multiply by qv, and it seems then that you could only apply it to voltages that are much less than kT. And the point I tried to make earlier is it's kind of true, but then this model, you're supposed to use it with the idea that the real thing is quite long. And it is as if you're doing your theory on a small piece here, a small coherent piece here, a small inelastic piece here, and then you have a contact, then you have a contact. So you're calculating this coefficient for something that is actually much shorter than the real thing. That's the thing. So I said that, well, we discussed this, and I said, OK, you have calculated a resistance. But then calculating it, you use the Taylor series expansion. That only works for 25 millivolt. And but then the resistor I measure, it's linear up to 1 volt. I mean, any resistance you measure, any resistance like a 10 kilo ohm resistor, it will be 10 kilo ohms, you know, up to volts. And the reason the justification is that you're really calculating the resistance of something whose length is a whole lot less than your resistance, the one you're measuring. And the real thing is supposed to be a series combination of all this. So that even if you put 1 volt here, what you're putting here is whole lot less. So this really makes sense to talk about linear response if I am working at a higher voltage. So a very important question then would come up is whatever you are measuring, let's say current versus voltage, let's say it looks like this. And when people use a linear response theory, they're doing this. And if you ask the theorist, he's of course doing this for V tending to zero. A totally separate question you have to ask him is that up to what voltage is this supposed to hold? And if you are actually in your device, you're going way beyond that, then you ought to be using a nonlinear theory, of course, whatever it requires. But often, at least for this thermoelectric materials, I believe within the range of operation, they are fairly linear. And I think in that field, they make a distinction between thermoelectrics and thermionics, right? Where thermoelectrics, they use the linear response, but in thermionics, they never use the linear response theory. They use something different. Although the basic equations I'd say are very similar. It's just that one has this f1 minus f2, the other, you have to do the Taylor series expansion in a way, right? And you could always use the other one for thermoelectrics as well if you want it. Yeah, this electron-phonon interaction, that last question then. Yeah, I guess there are these standard ways people put it in. Now the way I usually picture it is, there is a fluctuating potential. So the question you have to ask is that for a phonon, what is it that an electron feels due to the phonon, right? And if it's an acoustic phonon, you use a deformation potential or so, which is known from experiment, that for a given amount of strain, what happens? And then you calculate, well, for a, so at room temperature, you know from the Bose function how many phonons you have. It's n, which is given by one over e to the power h bar omega over kt minus one. So at room temperature, n for h bar omega much less than kt, that would come to kt over h bar omega. And based on that many phonons, like for one phonon, you can figure out how much of a potential an electron would feel that's based on this deformation potentiality. And then you can estimate what the size of that fluctuation would be. So then you're saying that, okay, this is the fluctuating potential. And then depending on what model you're using, whether it's a classical one or a quantum one, it enters in different ways, right? So you'd have to look at specifics to see how you put that in. But I think there are many examples where I think people have done this in different ways, okay? But the way I visualize it again is that one electron feels, you know, there is a fluctuation that it is seeing. And it's a time dependent thing. So it is fluctuating at a certain scale. Now the other thing with surface roughness scattering that people worry about is the correlation length of the roughness potential, et cetera. So in one electron term, the important thing is to figure out. So it's like from an electron's point of view, you're seeing some fluctuating potential like this. And the point is what is the size of that fluctuation that you can expect? So for example in nanowires and all, just the width fluctuations could be giving you something that you should include, et cetera. So all these, of course, depending on specific problems, you'd have to work out. Okay, so I think yeah, that more or less covers it. So yeah, I mean, thank you for your attention and any feedback you provide is very useful. As Mark mentioned earlier, we were trying to put these lecture notes together in a form that would be published by World Scientific. It's supposed to do it sometime this year. And any feedback you provide, I think it will help us improve it. And this should be available, I guess, by early next year probably. Oh, sorry if I missed it. Yeah, please go ahead. The effect of? Oh, yes, yes, okay. Let me just say a few words about that. Yeah, so the point, you know, the usual Schrodinger equation would look something like E psi equals H psi. And H is always supposed to be Hermitian because it's, and the way you justify it is that eigenvalues should be real. So this has real eigenvalues. And so when you look at the eigensolutions, they look like e to the power minus epsilon t over h bar, but these are the eigenvalues of H. That's how usually you're done. Now, when you add a sigma to it, of course, this is not Hermitian. In fact, the anti-Hermitian part is what is responsible for letting electrons go in and out. And so when you actually calculate its eigenvalues, they are complex. And so along with this, there will be a complex part. I mean, there'll be a part that is real, right? And there's a real part. So this is also there. And then, so that's why the sigma has two parts, a Hermitian and an anti-Hermitian. The Hermitian part just shifts your energy a little bit. So it was here. So when you connect a system to a contact, it has two effects. One is just that the level gets moved a little bit. The other is that this imaginary thing part, which tells you that an electron here won't stay there forever. It'll escape as fast whenever it finds a way out, right? That's what tells you, that's what this is. Now, so that's how the eigenvalues are included. Now, if you are trying to use this in the way the eigenfunctions are often used in standard quantum mechanics, that is where you have to be a little careful because when this is Hermitian, all the eigenvectors are orthogonal. When this is not, I think that's not guaranteed. And so there is this special way people do it with bi-orthanormal functions, et cetera. So if you're trying to use all that, you have to be careful. Because a lot of the theory of quantum mechanics is built around, you know, Hermitian make research. As soon as you bring this in, there are other issues to be that you have to worry about in that context, okay? Okay, did I miss anything? Yeah. This one, the spin wave function overlap. Could you repeat the question? Yeah, this might depend a lot on the details of things that I'm not sure of. Yeah, go ahead. I was just wondering, is it on the order of this few nanometers or can it be as big as the 10 nanometers? Yeah, ordinarily when people talk of overlaps, especially with, say, if you had an impurity where the wave function is very localized and the electron wave function is relatively spread and the electron wave function is relatively spread out, then, of course, overlap is limited to the length of this wave function. The part that makes your question a little harder is because if you're talking of two electrons and they both have delocalized wave function all over, then I'm not sure exactly what's the best way to think about that, right, in terms of overlap. Because physically, I know that anything more than a 10 or 20 nanometers apart should not be feeling like that. 20 nanometers apart should not be feeling any special potential. Usually, as I said, what people do is they use an effective mean field that one electron feels due to everything else, right, which comes from various things. And so far, I don't, I mean, there is this exchange correlation correction that people include, okay, which could be viewed as what you're saying, right? That one electron, what does it feel due to the other ones? And usually they say that every time, the point I made that what potential does one electron feel, you calculate something from Poisson equation, but actual potential is always a little less what you put in, this exchange correlation correction. And the physical picture people have is that this one electron kind of digs a hole around it. So although there's a certain average density elsewhere, right around it, people move out of the way. And so the result is there's always this exchange hole around it, which makes the actual potential, it feels a little less than what you might have thought from Poisson. That's the picture people have usually. So probably those kinds of issues would be involved, but in the thinking, the way you're including it is as part of that mean field that it feels, right? Okay, so thank you very much and... Thank you.