nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L4.2. Magnetic contacts
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[Slide 1] >> Welcome back to Unit 4 of our course on Quantum Transport. This is the second lecture. 

[Slide 2] Now as I mentioned in the introduction, a very important experimental result that actually one could say started the field of Spintronics, is the fact that if you've had a device with two magnetic contacts that were parallel, the resistance is lower than if they're antiparallel. And this is something that found widespread application in terms of reading information from magnetic discs. So the question is how would you include this in our NEGF model? Now the basic model, as you remember, for any channel with two contacts we need an H that describes the channel, you need the self energies that describe that two contacts, and once you have that, here we are talking of coherent transport. We're not worried about interactions within the channel. And so you can calculate this retarded Green's function from which you can get the transmission, which is like the conductance normalized to the quantum of conductors. Now so how would we write down these matrixes? Now consider in this discussion, so as you know in general you choose this lattice, or what you might call the basis functions. And if your lattice has say 10 points, then your matrixes are 10 by 10. And for this discussion let's assume the device only involves one basis function, one point. In that case the matrixes are just 1 by 1 so it's just a number. And that number is, let's say, epsilon. That's like a single energy level. I've written it as if it's a matrix by putting in these square brackets but it's really a 1 by 1 matrix which is a number. Now what about the sigmas? Well let's say the rate at which it goes out is given by this gamma 1. And then, as you know, if the gamma-- the rate at which it goes out gamma 1 then the corresponding sigma would be minus i over 2 times gamma 1. So that when we define i sigma minus sigma dagger, and this dagger is conjugate transpose, but since these are 1 by 1, transpose doesn't matter so it's just conjugate. And so when you do this what you get is like the negative imaginary part times 2. Anyway so for this gamma the corresponding sigma looks like this, and you're not worrying about any real part of sigma. Similarly for this gamma you have that one. And given this H in sigma you can write down the retarded Green's function. This is E. That's H. Next one is sigma 1. Next one is sigma 2. This is E minus H minus sigma 1 minus sigma 2 and then inverse. Now what's GA? That's this conjugate transpose of GR. And again transpose doesn't matter. It's just a number. So you want conjugate so plus i becomes minus i. That's it. And you take all that and you can put it in here. And again, Trace simply means this number because Trace is the sum of everything under diagonal. So you just have to multiply out those four quantities. And when you do that you'll get this. You see that gamma 1 is the gamma 1 here. That gamma 2 is that gamma 2. And GR times GA, that's like multiplying this with this, one is the conjugate of the other so you get the magnitude squared of that quantity, which is E minus epsilon squared plus gamma 1 plus gamma 2 over 2 squared. So that's the transmission. And usually what we do is we then multiply by 2 because we say well we all know that there are up spin levels and down spin levels and we have just been talking about one of them. So two of them will give you twice as much transmission, twice as much conductance or half as much resistance. Now for our present problem though we cannot multiply by 2. Why? Because the entire physics lies in these contacts making unequal contacts to the two types of spins. So you cannot just handle one spin and multiply the result by 2. 

[Slide 3] What you have to do is explicitly take into account the two channels, the two spins, the up and down. So now if you try to write the H instead of being just a number epsilon, you'll now have a 2 by 2 matrix. And we're assuming this is just an ordinary nonmagnetic channel. And so it's epsilon, epsilon both only diagonal. So these are equal. Diagonal matrix, both diagonal elements are the same. So that's like epsilon times the identity matrix. But when you write the gammas, that's when I guess you have to note that up and down should-- could have different gammas. Because as I mentioned earlier, the basic physics, I guess I mentioned this in the introductory lecture, the basic physics of these magnets is that one type of spin can go out easily compared to the other type of spin. And so we write gamma 1 up, gamma 1 down. Here you put the gamma 2 up, gamma 2 down. So each is now a 2 by 2 matrix. And once again we could do the same Algebra, multiply them all out and take the Trace. Except that here we don't really need matrixes and that's because all matrixes are diagonal. So basically all we are doing is treating two separate channels, an up channel and a down channel and adding them up. And so if you actually put in these matrixes, multiply it out, take the trace, what you'll get will look just like our old result except that there will be two terms, you see one from the 1,1 element and one from the 2,2 element, and when you take the Trace you'll be adding them. And for the 1,1 element you'll have something that looks just like this except for the subscript of u everywhere. And for the 2,2 element you'll have again looking something like this except for the subscript of d everywhere. So finally the answer will look something like this. 

[Slide 4] Now the next thing I want to do is make a little approximation which isn't really necessary. I just want to simplify the Algebra a little bit here. In general you don't need to make that assumption. And that is what we'll assume is that the Fermi level of the electrochemical potential is far below these energy levels. That is remember-- we assume that our channel has an up spin level and a down spin level, both with energy epsilon. And when you calculate this transmission what you want is the transmission at an energy equal to the Fermi energy. That's at low temperatures so that's this equilibrium electrochemical potential. And what you're assuming is that we can drop these two terms in the denominator. So when can we assume that? Well if this happens to be much less than that. So what does this represent? Well this represents the broadening of the level. You know any time you connect it to a contact, as we have discussed, it leads to a broadening and that's this broadening. And we're assuming that that is much less than the distance from the epsilon to the mu. So this is a case of what you might call a tunneling junction. So just one related fact I should point out is that these experiments with parallel and antiparallel contacts, I guess were originally done on materials like copper, which are good conductors, you see where the Fermi energy is actually in the middle of your energy levels. But nowadays a lot of the devices employ what are called magnetic tunnel junctions where this channel is actually an insulator and the electronic tunnels through it. So the approximation we are making here would apply to the tunnel junctions. But if you wanted to treat a good conductor like copper, then you should assume that this is right in the middle and then you should, to find the conductors, actually have to integrate over energy. And you could do that, it just takes a little more Algebra. And I won't go into it because our purpose here is more to explain how to write down these matrixes. Ok so anyway, so if you make this assumption then, then you see the denominators are both equal so I could put mu 0 minus epsilon squared, and the numerator I'll get gamma 1u, gamma 2u, plus gamma 1d, gamma 2d. So next then we note that if the contacts are parallel then when you look at the up spin channel it can say get out into the contacts at a rate say alpha. But if you look at a down spin channel the rate at which they can get out is much lower. Why? Because these are magnets that are pointing up. And so the density of states in these magnets, there is not enough density of states for down spins. And so you have this lower rate of getting out which we'll call beta. So beta is less than alpha. Now on the other hand when you look at the antiparallel channels, that's when you reverse that and we'll come to that in a minute. Now if you just put in these alphas and betas into here, you see what happens is, this is alpha, that's alpha so alpha squared plus beta times beta, that's beta squared. So the parallel transmission is like alpha squared plus beta squared divided by this denominator. Now we go to the antiparallel case. In the antiparallel case you see the up spin channel goes out here very easily, that's alpha. But has a tough time going out there so that's beta. This one has a tough time getting out on this side but can get out on this side much more easily so that's alpha. Now when you put those numbers in here you get alpha times beta plus beta times alpha so that's like 2 alpha beta. So what this tells you then is that the parallel transmission will be proportionate to alpha square plus beta square. And the antiparallel transmission is proportional to 2 alpha beta. And what we can show is that this is always bigger than that so that the parallel transmission is bigger than the antiparallel transmission and so the resistance is lower. 

[Slide 5] Now in devices people define something called the Tunneling magnetoresistance, so it is for tunneling devices. Tunneling magnetoresistance, TMR, and it's defined as the ratio of the antiparallel to the parallel resistance minus 1. Or you could think of it as the change in resistance, RAP minus Rp divided by Rp. So we can evaluate that by noting that resistance is inversly proportional to transmission. So RAP over Rp is like Tp over TAP. And if you put in the values we just calculated you'd get that. And the numerator then is alpha squared plus beta squared minus 2 alpha beta which is alpha minus beta squared. And if I multiply by 2 you have 4 alpha beta which is like the difference between those. And the reason I'm writing it this way is you can then divide through by alpha plus beta squared and you could write it in the form of 2P squared over 1 minus P squared, where P is this alpha minus beta over alpha plus beta. And this is a very standard result. That is this is something people use widely that-- what is P? Well that P tells you how good your contact is in terms of discriminating between up spins and down spins. You see that discrimination is reflected in the difference between alpha and beta so this P is a measure of that. Ideally what you'd like is beta to be 0, that is a contact should only let one type of spin get through and not the other type at all. And people are looking continually for magnets, for new materials which would actually do that. And this time usually it's never that good but over the years the polarization, this P, has improved significantly. And the way you relate the measured quantity, this TMR, to the P, is this 2P squared over 1 minus P squared. And you can see if P were 1 then actually of course you'd have an enormous TMR, that is the RAP would be way bigger than the Rp and that's exactly what you want for-- actually for applications. Anyway so the main point though as I said here was although as using this device as an example, my purpose was really just 

[Slide 6] to show how you write down the matrixes. And the point I tried to make is for magnetic contacts the first important piece of physics is that up spins get out more easily than down spins. And if you turn the magnet, let's say you reverse it, then of course the beta and the alpha are reversed. And that seems like an intuitive argument. What is not clear from this argument though is how would we handle a magnet that's pointing not up or down but somewhere in between? And that requires us to  get into the subtle issues of magnets which we'll take up in the next lecture.