nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L4.10: Summary
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[Slide 1] Welcome back to Unit 4 of our course on quantum transport. This is the last lecture, so I guess it's time to sum up. 

[Slide 2] So, we started this unit, you know, coming from the previous units with this general NEGF method. That is, if you want to describe quantum transport through any device, you need this H which describes your channel, you need the sigma 1, sigma 2 which tell you the connection to the contacts, and you need the sigma 0 describing the interactions and then in principle you could use these equations. Given sigma 1, sigma 2, et cetera, you can calculate the current and usually if you want to include interactions then you have to do this self consistently. So, that we had seen coming into this unit. In this unit, what we were doing was no new equations as such. We're using the same principles, but to illustrate different class of problems involving the spin transport and my purpose was two-fold. One is because spin transport is a topic of great current interest. Lots of activity, lots of new discoveries, et cetera. And it provides a very good, I guess playground for using the quantum transport methods. But, I also wanted to include it because it-- in terms-- in looking at results and interpreting them, it gives you a deeper understanding of the nature of quantum transport and how it relates to the semi-classical picture. 

[Slide 3] Now, so basically if we look at our-- the lectures that we went through, the first step that I introduced is that when you're choosing your basis functions, in the spinless case, if you had, say, 20 basis functions, when you want to include spins, you'll need 40 because for every up spin basis there'll be a corresponding down spin basis. And that-- and after we have chosen this basis set, then the question is how to do we go-- how do write down these matrices. And we started by considering the contacts. The sigma 1 and sigma 2. And we showed how you could write the appropriate, I guess sigmas, including the spin for magnetic contacts. And this helped also introduce this very different nature of spinors compared to vectors. The point I tried to make is if it spins one of the confusing aspects is you visualize it just like any vector. Points in some direction. But, vectors you normally represent in terms of their-- in terms of three real components, the x, y, and z component. But, with spinors, you represent them in terms of two complex components and this is dictated by the physics, as I said. There is this very important difference that with vectors 90 degrees is orthogonal and that shows up in these experiments with polarizers and analyzers when you do it with light. Whereas with spins, it's 180 degrees that's orthogonal and that shows up in all the spintronic experiments with magnets. Now, having introduced this part, this conceptual aspect, this difference between vectors and spinors and how you treat the contacts, we then went on to talk about how you include spin-related effects into the Hamiltonian. That is, these days there's a lot of interest in materials with high spin-orbit coupling, which is a relativistic effect and so in these lectures we introduced this, again, very new concept to most of you and how this spin-orbit coupling effects get into the H and how that's represented in the Hamiltonian matrices how you put it into your tight-binding lattice models. Now, after that, I guess the last few lectures has been more about interpreting things and looking inside the device. That is, this Gn which is like the matrix electron density which lets you analyze the electron density and the spin density inside a device and that leads naturally to this concept of spin circuits. 

[Slide 4] So, just to remind you briefly what we discussed there is that, you see, when you have two n components, if your psi has these two n components, that is, you know, n points this way and then two because there's up and down, then the corresponding Gn matrix is this two-by-two-- is this two n-by-two n matrix and if you look just at the diagonal elements, that would give you the electron density associated with each basis function. And in a semi-classical theory, that's what you do. You just worry about the electron number, you don't worry about the off-diagonal elements. And often the question people ask is, well, what are those off-diagonal elements for. What information do they carry? This diagonal is easy to understand and by looking at spin, it gives you a little insight into what those off-diagonal elements are about because we don't quite look at all of the off-diagonal elements, we only look at these blocks. That is, we look at the off-diagonal element between each red and each blue at a particular point. And, as we saw, when you look that this block, that's when I guess the diagonal elements-- this tells you the number of up spins. This tells you the number of down spins. But, these off-diagonal things tell you this transverse spins. The spins in the transverse direction. And so each of these two-by-two blocks, I guess, has four independent quantities. This N, S. The N is the number of electrons. Spin is the--S is the spin density. And you could use that to define, I guess, a local charge voltage and a vector spin voltage. You see? Charge and spin, that's this four-component voltages and four-component currents. And these could be used to develop the spin circuit theory 

[Slide 5] that we talked about where you have conductance matrices that relate four-component voltages to four-component currents. And so when you have n points, spin circuit theory involves 4n elements. Each point you have four things. Whereas the full NEGF, on the other hand, involves these 4n  squared elements because it's two n-by-two n matrices. And as I mentioned in the context of spin, of course, this is a powerful approach that has been used in the field of spintronics, but to me this is also a paradigm for how you want to-- how you may want to go in future where if you want to include more quantum effects, more off-diagonal elements, you might want to define units that are bigger than two. Like, you could use an eight-component spin, for example, which would be described by 64-component voltages and currents and you'd have conductance matrices that would be 64 by 64. Of course that sounds complicated, but it's less complicated than the full NEGF which treats this entire thing as one big spin. Now, one other thing I really like about spin is that it also illustrates this relationship between the quantum and the classical. I guess you can kind of see it here that the quantum has these off-diagonal terms. 

[Slide 6] Now, just to stress that, let me describe to you a short Gedanken experiment that one could do with present-day technology. So, think of a channel with two magnets pointing up like so. These are contacts to the channel and we apply a voltage across it. So, there's the negative voltages here, so electrons come in. And for this discussion, since it's a purely conceptual thing, I'll assume that our magnets are perfect, which means if I have red magnet, it only injects red spins. No blue spins. In other words, the polarization of the magnet is one. So, what it does is then fills up the channel with red spins and you think they would all be collected by another red magnet out here. And if you put a blue magnet there, nothing comes out because only red spins go through. OK. Now, the question is what would happen if we put a magnet in-between the two red magnets that's pointing sideways. So, it's neither up nor down, but in this direction. The answer is that of the 100-- if 100 electrons come in from here, half of them will go out there. Fifty. And the other half will continue beyond it. So, why do half go out? Well, the idea is that here it is injecting spins that are pointing this way and spinors have this property that you see. A spin pointing this way is actually a superposition of spins pointing to the left and spins pointing to the right. So, left and right. And so this is a superposition of the two and this is kind of like vectors, except that the angles are doubled. See, with the vector pointing this way, what we learn is this Parallelogram Law of Vectors that this a superposition of this and that. Here, I guess, instead of 45/45, it's like 90, 90/90, right? So, that's-- so all those hundred spins are in the superposition state and what this magnet does is kind of pulls out half of it because this one is kind of half like this. And this led to all kinds of conceptual problems or conceptual issues because you might say, well, 100 electrons come in, but-- I mean an electron either has to go in there or go on straight. You see, half an electron cannot come out here, as you know. And the answer is that that's why you require this probabilistic interpretation and that is that of these 100, if you looked at each electron, it would have a 50% chance of going this way and 50% chance of continuing. What that means is when I do it with 100 electrons, then on the average 50 will go this way and 50 will go on this way. In general, all the experiments we are talking about are steady state experiments averaged over time, done with many, many electrons. But, for an individual electron, the interpretation is probabilistic, which, as I said, raised many conceptual issues in the early days and even bother some people today because it means that the probability is kind of inherent. It's not the lack of knowledge. That is, we know that the spin points up, and yet we cannot tell whether a particular electron will come out here or go on this way. All we can tell is there's a 50% chance it will go this way and 50% chance it will go that way. Even though there's no lack of information, we know exactly what direction it points. Now, what happens after this? Well, if we look here, then all the right components have been taken out, so now only the left part is left. So, what does the left part do here? Well the left part itself is the superposition of up and down and so the up component is pulled out and the down component continues. Which means if we now look at these terminals, you'd see like 25 going in here and 25 going out there. So, if you look at this last magnet here. The thing is, originally no electron was coming out here, 0. Now I've put a magnet and I've kept everything the same, but we just put another magnet here and the net result is you have more electrons coming out here at this time. See? And that's hard to rationalize within our classical particle picture. You have to bring in all this probability and quantum mechanics. The quantum mechanical concepts. See? And if you take this magnet and let's say you rotate it so instead of being at 90 degrees, let's say we let it rotate along. So, if it was in the up direction in the-- pointing up, nothing would have happened. All 100 would have come out here and it would still have-- you'd still have got zero electrons coming out there. On the other hand, when it's here you get a lot coming out. And so if you calculate-- if you model this-- and this you could either with NEGF or with spin circuits. It's straightforward using the principles we have talked about. What you'd find is that if theta is 90 degrees, then you have a lot of current coming out. Right? There's 25% of the current is coming out here, 25% of what comes in. On the other hand, if the-- this magnet points along zero or 180-- or 360, I guess, the current is very low. So, by rotating this magnet, you can control the current that flows there. And things like this, I guess related ideas have been proposed actually for making spin transistors. And at this time, the way it stands I'd say this is not necessarily a very attractive way of making a transistor. And-- I mean here when you look at it, it looks like a big change in current. It's just that I assumed that all these magnets have perfect 100% polarization. But, as it stands, even the best magnets don't really have 100%, but that could change, of course, as we go along. So, when you take magnets with I guess less than 100% polarization, the modulation of current is less impressive. But, my purpose here is really not that I was trying to suggest a very hot new device idea. That's not quite the point here. The point here is that technology has progressed to a point where you can do subtle experiments like this that show the quantum nature of spin using solid state devices and present-day technology. You see, lot of these principles of spins were discovered like nearly 100 years ago. I mean since the Stern-Gerlach experiment which was 1922. And then there's a lot of experiments which involve electrons in vacuum. You see, that's how the basic theoretical framework was developed. And today technology has come to a point where you could do similar experiments instead of vacuum, I guess you can do it in solids using magnets. OK. And so that's one point I wanted to make and the other point I wanted to make is how important it is that you have contacts that can discriminate between different spins. So, you know, earlier I said I view spin as a general paradigm that one could be thinking of more sophisticated spins, but in order to do creative things with it, you also need the corresponding contacts. Contacts that would inject or detect certain types of spins preferentially. And what has made this field now so exciting is that, you know, with magnetic contacts and with all the spin-orbit materials, there are ways of generating and detecting these spins in solids. And I tried to give you a flavor for where the field is going. This is a rapidly developing field, so, you know, new things are getting discovered every year and so we'll see where it all goes. So, what I did in this unit then is kind of used this spin transport experiments with a dual purpose in mind. One was to introduce you to this new developments in this field, but also to show you-- show how you can use it with the NEGF method and how it gives you insight into the nature of quantum transport and what off-diagonal elements mean. 

[Slide 7] Well, with that, I guess we are at the end of our two-part course and what remains, I guess, is this short epilogue after this.