nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L4.8: Spin voltage
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[Slide 1] Welcome back to Unit 4 of our course on quantum transport. This is the eighth lecture. 

[Slide 2] Now in the last lecture we talked about this quantity Gn that you can calculate from the NEFG method and we discussed how from Gn you could extract information about the number of electrons and the net spins at a point. Now what we'll talk about in this lecture is how this information could be translated into this concept of electrochemical potentials or spin voltages, that is what would you measure if you had an external probe. And how if that probe is magnetic it would let you measure the spin component of this electrochemical potential, so that's what we'll be talking about. Now let me start by noting that just as Gn tells you the electron density, A tells you density of state. That is how many states there are. Whereas, Gn is how those states are occupied. And what we will assume is that in this material the density of states part is completely nonmagnetic, there is just as much up spins as down spins. And the only reason the device might have more spins of one type is because of how they are occupied. Now given this Gn and A I guess you could think off the occupation factor, which tells you how much the up spins are occupied and how much the down spins are occupied. So number of up spins is N plus Sz over 2 and the number of states is D over 2, so the fraction that is occupied on up spin states is N plus Sz over D. Similarly, the fraction of down spins states is N minus Sz over D. And this occupation factor the fraction that is occupied, that's what can be translated into this electrochemical potential. I think we discussed this in part A a little, but for low bias the fs translates linearly into a mu and so a lot of the time our discussion will be in terms of fs, but the idea is that can translate into an electrochemical potential and this is what a magnetic-- a probe would measure. When you come in with a weakly coupled probe what it does is it floats to whatever potential, whatever electrochemical potential it sees right below it, it comes to the same potential because I guess usually these are high impedance circuits, no current can flow. So no current flows means these two have to be at the same potential. Now let's first talk about how you could semi-classically calculate what this potential will be and then we'll do the full quantum thing. 

[Slide 3] Now semi-classically the way you can think is that where you have an up spin channel and a down spin channel and you have a probe and there's some effective conductance, this g, that connects the probe to the channel and we'll assume that the up spin and down spin have different conductances, g1 and g2. Now, of course, if this were a nonmagnetic probe, then those two would have been equal. Nonmagnetic means it doesn't discriminate up spins down spins, makes equal contacts to both. But here if it's magnetic, then the two can be different and so we'll put down two different symbols, g1 and g2. Now there's no net current here, so the way you can figure out what his potential f or mu would be is by saying that whatever current flows here plus whatever current flows here must be zero. In other words, no current flows out of this. So Kirchhoff's Law requires that this current and this current add up to zero. So what's this current, well it's proportional to the conductance times the potential difference and this one is again this conductance times the corresponding potential difference. Now if I set this plus this equal to zero, then with a little algebra I can solve for fp and that will give you fp equals this quantity in terms of this g1 and g2. Notice that if g1 and g2 are equal, then basically fp would have been half of f up plus half of f down, it'll just see the average. But if they are not equal, then it would be biased more towards the higher conductance as you would intuitively expect. Now instead of f up let me put in N plus Sz over D and for f down I can put down N minus Sz over D and then again with a little algebra, you'd get this expression. Now this first quantity N over D, that's like our usual occupation. You know, you've got certain density of states, how many electrons do you have? So that one you could call the occupation f with the subscript c is the charge. And the other end is Sz over D that tells you how many spins you have and that part is like a z component of f. That is what would translate into a spin potential sort of. And ordinarily you don't have this because you ordinarily wouldn't have any net spins, you just have a total amount of number of electrons. So this is the new concept that we're introducing, that's the spin voltage related to the spin voltage. And g1 minus g2 over g1 plus g2, that's what you usually define as polarization. That is how well this current problem discriminates between up and down, that's P. So what the probe measures then is the usual charge potential of c, but also the spin potential depending what its polarization is. Of course, if the polarization is zero, then all you measure is the charge potential, that's the normal probes. But with magnetic probes, you can have a significant P and then what you could be measuring would be this combination. And you could extract and the common way of extracting spin potential information is that you measure this fp with a magnet and then you reverse the magnet. Because if you reverse the magnet, then g2 and g1 get interchanged, so P becomes minus P. And so you measure fc plus P of z and fc minus P of c and by looking at the difference you can figure out what the spin potential is. So that's a standard way of measuring spin potentials, that is look at the potential with a magnetic probe, reverse the magnet, look at the potential again, take the difference. 

[Slide 4] So now what we want to do is do this problem with the full quantum mechanical way that is with the NEGF method. So now you'd say well the current of this probe is given by the sigma in A minus gamma Gn, this Trace. And since it's this high impedance probe no current flows, so that current must be zero. And what is sigma in? That's the part that you usually, you know, if this probe has a function f, an occupation function f and is fp times the gamma okay. So sigma in I've replaced with fp times gamma. So overall then it's like I have a channel described by a certain Gn matrix, which has information about number of electron spins, etcetera that is the density of states matrix and then this coupling to the contact, that's the gamma matrix which, as we saw earlier in this unit I think lecture three that you could write it in this form where this P is the polarization of that magnet. That's the difference between this coupling to the up spins and down spins. So you could write the gamma as this 2 by 2 matrix and then the way you figure out fp is it has to have the right value to make this current equal to zero. So this fp is a number, it's not a matrix, so you could take this out and write fp as Trace of gamma Gn divided by Trace of gamma A. So that would be the potential that the probe will float to. Now we can try to evaluate these matrices given, you know, that's gamma, that's Gn and that's A. 

[Slide 5] So let's try to do that. So first we can start with gamma A, so gamma times A. Now that's easy because this is an identity matrix, so it's basically matrix whys its just I plus sigma dot P. And then you have gamma times half D. So and I guess that's the 2 pi from here, that's why you have this gamma over pi times D. So this is the gamma A. Now you want the Trace of that. So trace I would say is very easy to see because it's I and various sigma matrices and as I had mentioned earlier, all the sigma matrices have a Trace of zero while i has a Trace of 2. And so when I take the Trace of this I just get 2 from here and then gamma over pi times D, so that's straightforward. Now let's look at the numerator. So that's Trace of gamma Gn  and gamma Gn means I have to multiply this with this. So it's I plus sigma dot P times NI plus sigma dot S. So when I multiply it out I get four terms. I times NI, that's NI, I time sigma dot S that's the sigma dot S, NI times sigma dot P that's N sigma dot P, and then there's the product of sigma dot P and sigma dot S, that's that one. So you've got these four terms. Now the last term here, this is where we could use this general identity I mentioned before, you know, when you have sigma dot v1 plus sigma dot v2 you can write as sigma dot P times sigma dot S is P dot S times I plus I sigma P cross S. So overall, then we have four terms, this, this, this, and then actually it's five terms now because we have three and this one has become two terms. And I want the Trace of this. Well, only ones that will give me any Traces the ones with the i because the sigmas all are Traceless, have zero Trace. So when I take the Trace of this basically I just get N from here and P dot S from there, so N plus P dot S and then the constants in front. The 2 come because the Trace of I is 2. So that's the Trace of gamma Gn and Trace of gamma A. Now we can take the ratio and we'll have this occupation the f for the probe or the-- I guess the electrochemical potential. So this ratio that's what's N plus P dot S over D. So this you can then write as N over D, that's like our usual fc and then the S over D that you can define as a factor fs and it's P dot fs. And you can compare it with what we had earlier using the simple semi-classical picture. What we had then was fc plus P fc. You see the semi-classical picture doesn't account for all the off diagonal stuff. So it is good as long as you are talking about spins in the z direction, but it's not good in terms of x and y. So as long as you are talking about spins in the z direction it gets you the right answers. And spins in the z direction and magnets in the z direction. So that's like the special case of this. By using NEGF you have the full matrices, so it has all the information about the spin and so we get that more general answer here. 

[Slide 6] And this you could translate into potentials as I had mentioned, so you could say okay, the potential and the probe is again the charge potential, the chemical potential inside P dot mus and mus is like the spin potential inside. And as I had mentioned earlier, a standard way of measuring the spin potential is measure mup with a magnet in some direction, switch the magnet around so that the P becomes minus P and then look at the difference because then the charge potential will cancel out and you'd get 2P dot mus. So this is then the standard way of using probes to figure out the spin potential. And there's a lot of interest in spin potentials these days because people have shown different ways of actually generating spin potential inside channels. 

[Slide 7] So, so far we have talked about magnetic contacts, but another way of doing that is actually with ordinary contacts, but using materials that have spin orbit coupling because with ordinary contacts what happens is when you run a current you have states going to the right that are filled-- much more filled compared to the states going to the left. And so think as we had discussed earlier both in part A and also earlier in this course in part B, that when current flows the right moving states are more occupied than the left moving states. And in a material with spin orbit what happens is if you look at the right moving states usually you'd expect there would be 100 up spins, 100 down spins, but in spin orbit materials there's more up spins than down spins because of this spin momentum coupling that we have talked about in a couple of lectures earlier. And so mu plus greater than mu minus translates to having more up spins than down spins because states moving to the right are more up than down, whereas states moving to the left are more down than up okay. So this is something I think we had briefly mentioned in part A, I think in unit 3 of part A, we had briefly mentioned the spin voltages in materials with high spin orbit coupling, but the point is that that is now a good way that has been established as experimentally is a good way to establish spin potentials. One way to establish spin potentials is to use magnetic contacts. Another way is use channels with high spin orbit coupling and then you can use ordinary contacts. But no matter how you establish it, of course, how do you measure it, well use a magnetic probe that will be the straightforward way to measure spin potentials. 

[Slide 8] Another class of experiments that's again had a lot of interest these days, I mean actually for a long time that are done with spin, is non-local measurements. So what do you mean by non-local, well the idea is let's say you have a material here and you run a current at one end of the material, so current flows here. Now it's called non-local because although the current is flowing here you're measuring a spin potential somewhere out here, which is outside the current path. So ordinarily you see if a current flows here, we expect there shouldn't be any action out here. But spin potentials can actually be measured outside the current path and that's again a standard experiment that's been done for over 30 years now. It has been, you know, tried in many different materials under many different conditions. And the basic idea is that when you run a current here because the resistance of up spins and down spins is different at this interface, so when you look inside the potential for up spins is different from the potential for down spins inside here. But then it doesn't stay that way, I mean they are separated but that's not an equilibrium situation, they you want to come back together. So as you move away on this side they'll gradually come back together, but it takes a while. In others words, it might take like a 10th of a micron or a micron for the two potentials to come back together, depends on this spin flip length. You know, the length that it takes for spins to get equilibrated. And so out here the two are separated, if you go many microns away they'll all join up. But if within this distance, this spin flip length, you put another probe, a magnetic probe, then you could measure this spin potential difference and that's what you call a non-local measurement. So you couldn't do this 1 millimeter away, but you could do this like a 10th of a micron away from the actual current path. And when you put a magnet here to measure this potential, if you rotate the magnet, so you know this potential, this up-and-down are related to this magnet. So up means pointing this way, down means pointing the opposite way and if we rotate this magnet the potential that this magnet will float to will have this P dot mus. So P is the direction of the magnet, mus is the spin potential, which is in the direction of this magnet. So basically this P dot mus is like the dot product of this and that, which you'll have this but basically it'll be like cosine theta. And so you should see a potential that'll be oscillating as you rotate the magnet. So usually the magnet is rotated I guess in the plain, but it could be rotated in this plain. Usually it's harder to take it out of plane like this, but it's in plain you would rotate it and you'd see this oscillation in the potential. And such measurements have been made. In fact, another way to do this measurement is instead of rotating the magnet you could make the spins rotate as they propagate. Because here you have lots of spins in this direction, but if you put a magnetic field here then you would have this Hanle effect, which is that you have spins pointing like this, anytime there is a magnetic field the spin wants to rotate around that magnetic field. So if you put a magnetic field in this direction for example, the spins would rotate in the plane. And that again would give you these oscillations which people have observed. The more recent experiments are where you have rotated spins not with a B field, but with a gate voltage that changes the spin orbit coupling. That is spin orbit coupling as we discussed gives an effective B field, which depends on the direction the electrons are flowing and if you can control the spin orbit coefficient, then you can again change the degree by which the spins rotate in getting from one place to another and that would again show up as an oscillation in these potential. And those things have all been observed and there's a lot of experiments on these lines, which I'm not going into because this lecture is really more about how spin potentials are measured using magnetic probes and this general concept of the spin voltage. 

[Slide 9] And the main point I'm trying to point out at the end is that there are many experiments now, many ways of generating and generating these spin potentials. It could involve magnets or it could involve spin orbit coupling, etcetera. Now how would you model these things? Well what we discussed in this unit is how you could write down the H and how you could write down the sigmas. So for an experiment like this you might need three sigmas for the three contacts for example. But one important point that we haven't talked about is that the spin flip interactions, as I said here you have some excess spins which gradually gets reduced as they flow away. And if you want to include that in a model, then you'd have to include spin flip interactions through the sigma zero type of thing. And that we haven't talked about and that's relatively hard to do, not so much conceptually as numerically because if you have a long device all these matrices tend to get big. And so as a practical matter, it is harder to treat this entire big device within a single NEGF framework, although in principle we could do that. Now what we'll be talking about in the next lecture is a method that's widely used, that's being finding use in spintronics that involves this concept of spin circuits and we'll talk