nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L4.9: Spin circuits ======================================== >> [Slide 1] Welcome back to Unit 4 of our course on quantum transport. This is the ninth lecture. [Slide 2] Now, in the last lecture, we talked about these general different configurations that are now being investigated experimentally. The local configuration where you have these spin-related effects in the current path and also non-local configurations where the spin-related spin voltage is outside the current path, for example. Now, in any case, all of these could be, in general, treated within the NEGF method. The first step being setting up this dual basis functions with the up and down spins and then writing down these matrices, as we have discussed. Now, a very common device is this magnetic tunnel junction, which looks basically like this. Two magnetic contacts and this channel is actually a very thin insulator through which electrons can tunnel. So, like a couple of nanometers. And a model like this should describe a magnetic tunnel junction quite well, although, you know, the H we have discussed is this effective mass model and depending on the nature of the insulator, one may need to go beyond that. Now, there's another class of devices, the spin valves which actually were used a lot before, but magnetic tunnel junctions usually give much bigger magnetoresistance and so, for practical applications, a lot of the applications use MTJ's. Now, the difference with spin valves is and for spin valves this channel is actually a conductor. Something like, say, copper. Now, structure like that, if you, I guess analyze using the NEGF method, depending on the length of this region-- remember in the MTJ this is a very short region, like a couple of nanometers. For spin valves, this could be a fairly long regions, tens of nanometers, say, or 100 nanometers. And depending on the situation, you could get all kind of interference effects. That is NEGF could give you all kind of interference effects that are actually not observed because-- I think I mentioned this before that in general at reasonable temperatures there is this strong electron-electron interactions which often destroys a lot of the-- like a phase-related interference effects. And so, to get rid of those effects so you can actually compare with experiment, it's important to include the interactions. The sigma 0. And same in this structure. You see, you could model it with NEGF, there's three contacts, so you'd need three sigmas, but then you also need to include the interactions and these interactions, as I said, could be the phase-breaking type. The type that destroys phase and so destroys the interference effect and some of it could also be spin-flipping types where you have a spin voltage here, but as you go away from it, the spins get equilibrated by spin-flip processes and so out here you may not have much spin-flip left. And if your NEGF model is to capture all that, that must be in sigma 0. Now, what I'll talk about in this lecture is a different approach that's used-- that's finding use in this field of spintronics. Different meaning I'll try to explain the relation with the NEGF method and how it works. [Slide 3] Now, the idea is that, well, as you know, in the NEGF method we have these matrices and if you go to any point here, I've got these two basis functions. At point 1, it's up, down, up, down, and so on. And one thing we discussed earlier is that if you looked at the diagonal blocks, these two-by-two blocks, then from that matrix, the Gn matrix, that block, you can extract information about the number of electrons and the number of spins. Now, in NEGF calculation, of course, doesn't just give you these blocks. It gives you this entire matrix and all these off-diagonal elements, those have the phase information. The part that leads to interference. And, of course, the purpose of these interactions, I mean what these interactions do is destroys this-- these off-diagonal elements. In fact, if the interactions were strong enough, if spin-flip processes were strong enough, it would even destroy some of the transverse spins, the spin-related effects. And in that case you could use a completely semi-classical approach that focuses just on the diagonal elements. That's kind of what a semi-classical approach does relative to the quantum approach, that in an appropriate basis it uses just the diagonal elements and those you could interpret in this context as the number of electrons in each of these associated with each of the basis functions and then you could write semi-classical equations like diffusion equation in terms of those diagonal elements. Whereas what NEGF does is, of course, takes this entire thing. All the off-diagonal things. That's what makes it harder to implement as well, numerically, because it's got a lot more information and the thing to note is that what interactions does-- interactions do is to destroy this additional information in a way. So, if you'll put in the right amount of it, you know, you'd-- these things would get reduced. And with enough of it, you'd reduce to the semi-classical picture. Now, spin information is a little more robust than the usual phase information and these days there's a lot of experiments which show spin-related effects, but they don't have these other interferences. And in that case, you see you could capture the physics with something that goes between the full NEGF and the full semi-classical. Full semi-classical is you stick to the diagonal. Full NEGF is keep the entire matrix. And the spin circuit takes these two-by-two blocks, keeps the off-diagonal information here so you know about the transverse spins and all that, but then it just stick to the blocks. Now, that's a definite simplification from the full NEGF. [Slide 4] So, let me try to give you a feeling for how that works. So, the idea is consider this device like a spin valve device. Two magnets with a channel in-between and the NEGF would have this H and the sigma 1, sigma 2, and these interactions, sigma 0. Now, once you threw away those off-diagonal things, conceptually you could break it up into three pieces. One is this piece which is this magnet-channel interface. Then there is the bulk of the channel, and then there is another magnet-channel interface. Now, this you could represent like a diffusion equation or, in this context, you could reduce it to a circuit element. But, it's not the usual circuit element you learn in elementary circuit theory. That's because, you see, in order to have all the information there, at any note, at any point, if you look at the voltage, it is not a single number. It's got its charge component and it also has a spin component. OK? The three spin component, so it's actually a four component voltage. Similarly, if you calculate the current, there would be these two-- the charge current and this full three components of the spin current. So, there'd be four spin currents and so when I write a conductance matrix-- conductance element there, it's not just a number like you learn in circuit theory. It's actually a four-by-four matrix, one that relates the four components of the voltage to the four components of the current. There's another subtlety involved and that is when you look at this you'd say, well, this is like a series circuit. Right? In circuits you learned this idea that in a series circuit the current everywhere is the same. But, that's not quite true about the spin currents because charge current, of course, if one milliamp comes in, one milliamp has to get out there. But, spin currents are not conserved, necessarily. You could have a certain spin current come in, but which through various spin-flip processes got destroyed in-between. And so, for a proper description that captures the physics, in addition to the series elements you kind of need a shunt element here. And a shunt element over there. And same with the actual channel. You need these shunt elements. So, with a-- but with a circuit like this, and I-- you should remember, of course, a circuit is kind of within quotation marks because it's not the usual circuit. It's a circuit with four component voltages and four component currents and that's what I'd call the spin circuit. But, with that, you could then model this whole thing, whole structure, and the advantage of it is you have now kept the spin-related information. That's this diagonal two-by-two blocks. But you have just thrown away all the other off-diagonal information, which in many experiments usually not relevant. I mean it's like you don't see it because dephasing processes destroy it anyway. And in NEGF, to have that come out right, you'd have to include all the dephasing processes. [Slide 5] Now, just to give you a feeling for what the circuit elements look like, let's first, I guess forget this part and look at this contact. Now, if you look at the series element, again, it's-- as I said, this conductance matrix is a four-by-four matrix. There's four components. There's the c that's the charge component, that's the usual one. And then there's the z and then there's the transverse ones, x and y. C, z, x, and y. And you'll notice here the non-zero elements are up in this corner. So, how do we obtain this and this G zero? That's a constant times this 1, P, P, 1. Now, where does that come from? Well, you could kind of think of it this way. If you had used up and down as your basis functions, then we know we have seen before that the up has a different conductance from the down, g1 and g2. You know, that's what makes these magnetic contacts-- that's what gives these magnetic contacts spin-related properties, right? The ability to discriminate between spins. It's this difference between g1 and g2. Now, the c and z, that's kind of like-- that requires a basis transformation because, you see, the-- as we discussed earlier, the up component is like n plus sz over 2 and n minus sz over 2. Similarly, if we turn that around, the charge component is like up plus down over 2. The spin component is like up minus down over 2. So, if you make that transformation, then you see you'd have this quantity will transform into something like this. So, this is what getting us straight forward basis transformation. That is, you were in the up/down basis, now you're going into this charge-spin basis. So, c and z and this transforms into this where what you have on the diagonal is the sum of the two. What you have off-diagonal is the difference of the two. And this you could write in the form I have written here because you could take the G zero out and the ratio of this delta G to G0, that's what's called the polarization that we have discussed before. So, that's how you can kind of understand. You see this top two-by-two part of the conductance matrix. Now, the bottom part here, that's the transverse components. You'll notice it's zero. So, what that basically says is if I have a magnet pointing in this direction, in some direction like this, you could have spins in this direction flow in, but any spin that is transverse is killed very quickly. It doesn't-- cannot propagate through this very well. So, if you had spins in the transverse direction come in here, it wouldn't be able to get through and that's reflected in this high conduct-- this low conductance. That is it doesn't let it flow at all. And that's this experimental fact that in a metal-- in a magnet what happens is any time there's a spin that is in a transverse direction, it is-- it dies out within a monolayer or two. Very soon. And the reason is something like this. Supposing you had a whole bunch of spins pointed this way, you know, each one traveling in a different direction, as you know. Electrons usually are traveling in multiple directions. Now, in a magnet, yes, there is this internal magnetic field which makes this spin precess and so each one of them is precessing, but when you come out here they have all precessed by different amounts because depending on the direction the electron is traveling, one might spend one picosecond getting here. Another might spend half a picosecond getting there. Another might spend two picoseconds getting there and so they will have precessed by different amounts. And so by the time you travel a little bit into it, they've all randomized completely. They all started out this way, but by the time they're here, they're in every direction. So, bottom line is experimental fact within a magnet, transverse spins die out very quickly, cannot propagate and that's reflected in this zero conductance here. Now, if you look at the shunt part, then this reflects the fact you'll notice there is something-- non-zero things here. What that says is that if you had transverse spins coming from here, they wouldn't be able to get into the magnet, but they would disappear and the disappear is the part that goes into this shunt, goes off. That's this shunt conductor and these play a very important role, these mixing conductances that are now being investigated and play an important role in many current experiments which we are not going into. [Slide 6] Now, the next thing you could ask is what about this magnet. Well, that magnet is pointing in some other direction. So, here we assume the magnet was in the z direction and so c and z were kind of together and x and y was the transverse part. So, here also you see you could right these matrices assuming this was the z direction, but then of course you have to rotate the magnet, and that you can do again with a unitary transformation. A transformation that-- let's say you want to rotate the magnet in the z x plane, then there is the standard rotation matrix going from z to-- if you rotate by theta, that's this z x part of it that transforms z into x and x into z. And the charge part remains fixed and the y part remains fixed because you're kind of rotating around the y-axis. Similarly, for any rotations you could figure out the appropriate matrix and then do a basis transformation. [Slide 7] Now, if you put that together, so you have these conductance matrixes here, these conductance matrices here, and based on that you could now calculate what is the resistance one would measure between those two terminals using this what you might call the spin circuit theory. You'd get results that match experiments very well. See? And this is-- these are things that have been over the last decade or so. And so you can capture a lot of this physics then that is actually observed which involve this spin in a non-trivial way. What you-- you could capture it with the spin circuit theory. See what you are plotting here is the resistance as a function of the angle between the two magnets and when the magnet angle is zero, that's the parallel case. The-- when the angle is 180 degrees, that's the anti-parallel case. Those are the two special cases, parallel and anti-parallel, but then there is all this other stuff in-between. See? And, as I've mentioned before, going from parallel to anti-parallel, the resistance becomes high. This is a little bit like this polarizer and analyzer I mentioned with light, except that with spins you have to go 180 degrees to get a minimum flow, which means a maximum resistance. Whereas with light you'd have to turn the analyzer by 90 degrees. Here you have to go 180 degrees. But, the point I wanted to make is that if you want-- if you're using NEGF, then you see to get computable results, you need to include the dephasing processes in the channel because otherwise whatever you calculate will tend to have a lot of irrelevant interferences. Interferences meaning where electron goes there, comes back, and interferes with the incident wave, giving rise to standing waves, et cetera. And which would not be observed experimentally if only, you know, because of the dephasing processes and you'd have to include all that. Whereas in the spin circuit theory, you're kind of dropping all that at the outset. You're saying, well, we know experimentally those are not important. Leave that out. [Slide 8] Now, if you-- in the spin circuit, you could also include the effect of the channel. You see what I did here was I said the channel is just short. We're assuming this is a short channel, doesn't play any important role. On the other hand, if you had a long channel, then the dynamics of electrons inside the channel could be included through these conductance matrices that you use for the channel itself. And I won't go into how you do that because that, again, depends on the nature of the channel. Depends on those details. As I've said before, the channels could be, you know, the normal type. Could be magnetic. Could be-- could involve spin-orbit coupling. And each one has these different properties as we have seen. And depending on what kind of material we are talking about, there may be different formalisms, different approaches, more appropriate for calculating this. For example, for normal materials and magnetic materials, the spin diffusion equation in some form could be used. This is like diffusion equation, but then keeping track of spins. Keeping track of the longitudinal spins, actually. That could be used for these classes of materials. This one might require more the quantum approach and so-- and these are all, I guess, topics of current research where people are working on this, but the nice thing also about the spin circuit method is, you see, with NEGF this whole thing is one big problem. And if you-- you cannot solve this problem without solving that one because it's one big thing. Whereas with spin circuit, it's like, OK, this is-- this describes the magnet-channel interface done. That's this one-- that's the other one. The other interface. This is the channel itself and you could solve it kind of one by one and use different approaches for different things. So, everything doesn't need to be NEGF. This part could involve some other approach more appropriate [Slide 9] to the material at hand. And when you use the spin circuit, of course the biggest advantage is just in terms of computational simplicity. That is because if you have n up-spin basis functions and n down-spin ones, so overall then your matrices are like two n by two n. So, when you use full NEGF, you have this 4n squared elements whereas when you use spin circuit you kind of use four elements per point, so overall you have 4n elements. And this I usually also point out that with spins what you are doing is treating the quantum effects or these off-diagonals effects for each point separately. You know the two basis functions for each point, but in principle we could define a bigger spin that involves say four of these different points, you know, which could be either four points spatially or it could be just other basis functions. S orbitals, p orbitals at a particular point. The point is the spin circuit treats these two component spins. In principle, you could take, say, eight of them and define an eight component spin. You see? And if you recall spin has eight components, then you see to represent it in a spin circuit you'll now need 64 things. When you had two component spins, you needed four things, which you interpreted as charge and the three components of the vector, the spin vector. With eight components, there'll be 64 component things. So, you'd have voltages and currents that would be 64 component. You see? But it would still be simpler than the full NEGF. So, instead of doing a full NEGF on the whole thing, you could break it up into little pieces and each piece would be represented by a voltage and a current, but which are 64 components each. You see? But, that's for future. I don't know of concrete examples where people have done that, really. But, in my mind, the spin is just a nice paradigm. Nice two-component paradigm that gives you a lot of insight into how these things work. In fact, this full NEGF method you could say treats the entire device as one large spin. So, it's like if you had 100 points and 100 up spin and 100 down spin, it's almost like you had this giant spin with 200 components. You see? And NEGF is treating this entire thing as one spin, but what you could do is break it up into little pieces and that gives a paradigm going forward because with NEGF, what everyone recognizes is it's a very powerful method that lets you do all kinds of problems and in principle it's straightforward, but when you try to implement it in practice, very soon it gets out of hand. And so ways to do approximate calculations that capture the physics are very-- will be very important in the coming years as people look at different types of quantum transport problems. [Slide 11] With that, then I guess we are near the end of this unit and in the next lecture I'll try to sum up briefly.