nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 3.9: Spin Voltages
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[Slide 1] Welcome back to Unit 3 of our course. This is Lecture 9. 

[Slide 2] Now, what I want to do in this lecture is to talk to you about a field of great current interest. This is this field of spintronics, where there has been enormous progress in the last ten, twenty years. And what I'll talk about is a very specific aspect in that field which relates very nicely to what we have been talking about in this unit of the course. As you remember, this unit was about this quasi-Fermi levels. That is, the idea that if you look inside the channel you have an average electrochemical potential, and then there's these two quasi-Fermi levels. One for right moving electrons, and one for left moving electrons. And the current is related to their-- difference between their two quasi-Fermi levels. Now, the field of spintronics these days, there's a lot of interest in something called topological insulators. And those of you who are familiar with the Hamiltonian, that's how it's written for these materials. But if you're not familiar, don't worry. The main thing this tells you is that if an electron has a momentum, k, and spin, sigma, then the sigma cross k is in the direction of the normal to the surface. So, if you're looking at the top surface, for example. For the top surface y is the normal. So what that means is sigma cross k would be in the plus y or minus y directions. And, so if you consider states moving to the right for which k is in the z direction, then the spin should be in the x direction. So that z cross x is in the y direction. So, that's what you could call the up-spin, let's say. And similarly there will be a down-spin, which will be associated with states moving to the left. In other words, in these materials, a state moving to the right also has a spin in a particular direction. This up-spin. And states moving to the left have a spin in the down-- down-spin. And that's, of course, very different from normal materials, where the state moving to the right could have either up-spin or down-spin. And same through objects moving to the left. Here you have this spin momentum locking. A particular momentum implies a particular direction of spin. So what that means is, I could write the current, which is usually this difference in quasi-Fermi levels, but I could write it as the difference between the quasi-Fermi levels of up and down. And this difference between up-spin and down-spin potentials, that's what's called the spin potential. And we have defined it with a factor of two, and that's just a matter of convention. So you could write this as-- the current as the spin potential times this ballistic conductance, and this factor of two that's just a matter of convention. And you could turn it around and write the spin potential as proportional to the current. So the point is that in this class of materials, any time you have a flow of current there'll be an associated spin potential. One little detail; You see, if you looked at the top surface, and you looked at the states, not all of them would have a momentum in the z direction. It would actually have an angular spread in the zx plane. So this is like the zx plane. This is the z direction, and some states will have a momentum that have an x component. But in each case, of course, the spin will be locked to a direction perpendicular to that. So when you add up all the spin potentials, and spin potential is kind of like a vector, so some potentials would be in this direction, some would be in that direction. So when you add them all up, you get an angular average. And that's this 2 over pi. But that's a detail. Main point is in this class of material, current flow gives you a spin potential. How could we measure it? 

[Slide 3] Well, we have talked about how probes measure potentials in the past. You know, where we had a mu plus and a mu minus, and the probe floated to some potential. Now we have a mu up and a mu down, and you could write the conductance associated with the up channel and the conductance associated with the down channel. And if you remember, the principal is the probe is a very high resistance thing, which does not draw any current. And so-- and this potential adjusts in such a way that the current flowing in here is exactly equal to the current flowing out here. That is, if we write these two currents, two inward currents, they add up to zero. And based on that, you could usually write the probe potential in terms of up and down. Now, ordinarily, what would happen is g-up would be equal to g-down, if you had an ordinary probe. And then mu p would just be half of these. Half of the sum. Technically the average of the two. But if you use a magnetic probe, let's say a magnetic that winds in the x direction. Then one of these conductances will be much bigger than the other. That's, again, another important development of the last twenty years in this field, which is the basis for a lot of useful devices. Which, where it is clearly established that a magnet in a certain point-- pointing in a certain direction has a relatively low resistance contact to spins in that direction. And so g-up could be much bigger than g-down. And then the actual probe potential would be, again, a weighted average of up and down, and the weighting depends on these two conductances. Up and down. Now, the important point is that if I can easily reverse the direction of a magnet, and if I reverse the direction of a magnet, then you see up and down get interchanged. And so, the probe potential should change. That is, instead of g-up here, I'll get g-down. Instead of g-down there, I'll get g-up. And so you'll have a different potential now. Because, previously, I'd say ninety percent of this and ten percent of that. If I reverse the magnet it'll be 10% of this and ninety percent of that. And so if you look at the difference in the probe potential, that is, with the magnet pointing in one direction, I get a certain potential. The magnet reversed, I get another-- different potential. If I look at the difference, the difference will be a measure of this spin potential. And multiplied by this quantity here, which depends on how different the two conductances are. See? And that is usually what's called a polarization of the magnet in this context. So, this is now a standard way actually of measuring spin potentials. That is, measure the potential with a magnetic probe, reverse the direction of the probe, and see how much it changes. If it's a non-magnetic probe, of course, you shouldn't get any difference here. Or, if there is no spin potential, then when you reverse the magnet, there'll be no difference in what you measure. But, if there's a spin potential, and you-- then you should measure the-- this difference. And so you could write, then, as the measured difference, what you're actually measuring, would be like what we had times this polarization of the magnet. That is how effective your probe is in measuring this quantity. So, this part is kind of the intrinsic spin potential. This kind of tells you what you'd actually measure with a probe. And it depends on the polarization of the magnet, and this other quantity I've introduced, that's like the property of the channel, which so far actually in our discussion is one. And what I'll next explain is how, in general, it could be less than one. When we get to that point. 

[Slide 4] So, so far I talked about these topological insulators, where you have this perfect spin momentum locking. That is, if anybody-- if a state is going to the right, it has a spin in a particular direction. Whereas you have these non-magnetic materials, the ordinary materials, where usually if you're going to the right, you could have a up-spin or a down-spin. Now, in general, though, this logical-- lot of materials which have spin-orbit coupling. And in general anytime you have spin-orbit coupling you have a situation like this. Where you have right moving states, left moving states. You have up states, and you have down states. So in general, there's this four combinations. Right moving up, right moving down, left moving up, left moving down. Now, usually what you expect, this is generally referred to a time reversal symmetry. That if on a highways you have fifty lanes going to the right, there should also be fifty lanes going to the left. Of course, they don't have to be filled with cars, equally, when they're out of equilibrium. One side could be more filled than the other. But as far as the lanes are concerned there should be-- if there's fifty to the right, there should be fifty to the left. Now, that's not really the most general situation. In materials with spin-orbit coupling, which have this called time reversal symmetry, if there's fifty lanes-- fifty up-spin lanes going to the right, there should be fifty down-spin lanes going to the left. Similarly, if there is forty down-spin lanes going to the right, there should be forty up-spin lanes going to the left. See, now there is two must have the same number of lanes, these two must have the same number of lanes. But fundamentally there is no reason why those two have to be equal. And depending on the strength of this spin-orbit turn, M and N could be widely different. Of course, normal materials and where M is equal to N, that's what we are generally used to. But M can be very different from N, and of course, an extreme example is this topical insulators where N is actually equal to zero. So, right moving means up-spin, left moving means down-spin. Because N is zero. But more generally N is not zero, and you could define this polarization as M minus N over M plus N. That's the p that enters here. So you'd still have a spin potential, as long as M is different from N, but it would be somewhat smaller because of this factor. Now these are details that's in the notes. I'm not going into it, because as I said, this is a little bit of a detour from the main topic. But main point that I really wanted to get across in this unit is this concept of quasi-Fermi levels. 

[Slide 5] The idea that different classes of states could be occupied differently, and hence you could have different electrochemical potentials or quasi-Fermi levels. So, much of this unit we have been talking about the difference between right moving states and left moving states. And how they can have different quasi-Fermi levels. In the field of spintronics, it has become quite common to talk about the spin potential, the difference in the quasi-Fermi levels for up spins and down spins. And, as we move into new materials, especially with spin-orbit coupling, and we talk about ballistic conduction and so on, more generally you see we should have four quasi-Fermi levels, because there are four types of states. There's up-spin, down-spin, right moving, left moving. So there's four combination, and each one could have a different quasi-Fermi level. And this is, of course, very hard to describe. If you take the attitude that, quasi-Fermi levels are complicated things, confusing things, let's just stick to the electrostatic potentials. Well if you just stick to electrostatic potentials, you are missing a lot of rich physics. That is the real miss-- I've made other reasons why electrochem-- electrostatic potentials will not be very useful, because of how the they smear things out. I've made that point before. But right now the main point I want to drive home is as the field of nanoelectric devices develops, and you look at newer and newer devices, with all kinds of contacts and materials, with which you are able to control different types of states. You'll get into situations where there'll be multiple quasi-Fermi levels. Just as in biological systems you have different potentials for potassium ions and sodium ions. So you'd be in a much-- you'd have this rich physics coming out of different types of quasi-Fermi levels, and in order to understand all that, it is really important to get beyond this mindset of using electrostatic potentials to understand current flow. It is really this quasi-Fermi levels that are more fundamental and more central. 

[Slide 6] So that sort of brings us to the end of the topics we wanted to talk about in this unit. And now it's time to sum up, and that's our next lecture.