nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L4.7. Spin density/current
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[Slide 1] Welcome back to Unit 4 of our course on quantum transport and this is the seventh lecture. 

[Slide 2] Now as you know, in this unit we have been illustrating the use of this NEGF method, which we developed in earlier units using examples from spin transport. And the spin transport involves usually the physics, involves magnetic contacts or special properties of the channel like spin orbit coupling. And we have talked about how to include these into the description of this H and the gammas or sigmas that describe the contracts. And one key point as I mentioned is when you include spin you have twice as many basis functions because every up-spin basis has a corresponding down-spin basis. And so compared to the spinless description, you have matrixes which are twice as big like 5 by 5 becomes 10 by 10. Now in this lecture what we'll focus on is the interpretation of the quantities that you can calculate from and NEGFs, specifically this Gn. That is given h and sigma or gamma, you can calculate current, you can calculate electron density, density of states, any quantity of interest. And what we'll talk about here is how you interpret that and specifically, we focus on this quantity. And what we'll show is that supposing we look at one point, which means two basis functions, then the Gn we'll get will be a 2 by 2 matrix. And what we'll show is that the elements of this matrix gives you the number of electrons N, but it also tells you the spin, the average spin, the net spin of the electrons. And spin is like it points in some direction, it has a z component and an x component and a y component. And what we'll talk about is, you know, why the Gn looks like this and how you can extract this information 

[Slide 3] from Gn. Okay, now just to remind you, a we have discussed earlier, if you have a spin pointing in the n direction, in this direction described as the vector n, then if you thought of it as a vector, it'sx, y, and z components would be given this cosine theta sine theta sin phi sine theta cosine phi, that's what you learn in vector algebra. With spins it has these two complex components, which are denoted by c and s where c is defined as given here and s is defined as given here. Now consider an electron in a state pointing along n, so that its components are giving us c and s. What is the corresponding Gn? That's this psi psi dagger. So the principle is simple we take c s multiply it by c star s star. And if we write it out you'll get this 2 by 2 matrix. You see this is 2 by 1, this is 1 by 2, so when you multiply you get 2 by 2. And these individual elements then you can do the multiplication and write it out. You will get cosine square half theta, cosine half theta, sin half theta and there will be phase factor because of this and same here and so on. Next what we could do is use this trig identities, a thing which you have used before, so you're probably getting used to it now. So write this cosine square half theta as half of 1 plus cosine theta or sin square half theta is half of 1 minus cosine theta. And this quantity is half of sin theta and same here. Now you'll notice that the elements that appear here, they are actually like the components of the corresponding vector. So for example, this cosine theta is actually the z complement of n. So you can write it as 1 plus n z. Similarly, the real part of this a sin theta, cosine phi, that's like the x component of n. The imaginary part is sin theta, sin phi, that's like the y component of n. So you can write these in terms of the components of that vector n. And this you could write it out in terms of r favor it 2 by 2 matrices, namely the identity matrix, the z component of the poly spin matrix, and then the x component of d and y component of d, poly spin matrices. This and this together give me 1 plus n x and 1 minus n z, that's the diagonal part. These two give me the off diagonal part and x minus i n y and n x plus i n y. So what you have then is i plus n z sigma z plus n x sigma x plus n y sigma y, so you could write it compactly as half of i plus sigma dot n okay. 

[Slide 4] Now let's look at a couple of examples to make sure we understand this. So for example, if you have a state that is pointing along the plus z direction. So we'll confine ourselves to states in the z x plane, so that we can take the phi to be zero, so there's no phase factor here, that's zero so that drops out. So if theta is zero, that's in the z direction. So you get cosine 0, that's 1 sine 0, that's 0. So the wave function is like one zero. That's a spin pointing in the z direction. So when you take Gn, that's like psi psi dagger, so 1 0 times 1 0, that gives you 1 0 0 0. Now consider something along the negative z direction and you can show in that case the wave function would be like 0 1 0 1, so when you multiply it out you would get 0 0 0 1. So that's a spin in the negative z direction. Now what about a spin that points along the x direction. Okay, then you can show that your wave function would be something like 1 over square root of 2 1 1. Why is that, well phi is 0, but now theta is 90 degrees, so this is like cosine 45, which is 1 over root 2. This is sin 45 1 over root 2. So that's the wave function. And so you have to multiply it by the psi psi dagger, so the dagger is 1 1 and of course, you still have this divide by square root of 2. So when you multiply it out you get half 1 1 1 1. So in this case you see you have both diagonal elements are 1 1, it kind of suggests that as is, you know, this was an up spin electron, so this was 1, the rest is 0. This is down spin, this is 1, rest is 0. And the x spin pointing along x is like a little bit of plus c and a little of minus c and so that's why it's like half here and half here. But it goes more than that, you will notice I also have an off diagonal element. Now to appreciate this subtle point consider what you mean by unpolarized electrons. That is normally you see you have normal numbers of up and down, that's what you might call unpolarized electrons. Now if we had unpolarized electrons, then the corresponding g n would be like 50 percent of this plus 50 percent of that and that would look something like this. You see this is 1 rest 0, this is 0s, that's 1. So if we take 50 percent of this and add 50 percent of that you get this. So that's the Gn corresponding to unpolarized electrons. And here it's hard to write down a wave function you see because if you say well, you know, unpolarized electrons are like equal amounts of plus c and minus z and so I'll write the wave function like this, well that would imply a Gn of something looking like that. And that would actually physically not represent unpolarized electrons, it would represent ex-polarized electrons. So there is a big difference you see x is equal amounts of plus c and minus c, but unpolarized means I guess incoherent mixture of 50 percent of this and 50 percent of that. The difference is you if I put an analyzer in the plus c direction, so if you had unpolarized light 50 percent of-- not light I guess electrons, 50 percent will get through. Similarly, if we looked at minus c another 50 percent would get through. The same with plus x, but in this case if you put an analyzer in the x direction even then actually only 50 percent would get through. What you can show is this is 50 plus z 50 percent minus z, but it's also 50 percent x, 50 minus x. Those are all basically the same thing. And here if you put an analyzer in the x direction 100 percent would get through. This may not be obvious from just what we discussed, but the important point I'm trying to make is these off diagonal elements are extremely important. And if you leave out the off diagonal what you get is this random mixture of plus c and minus c, whereas it's this one that shows you it's not a random mixture, it's a coherent super position of plus c n minus. Now this is how you'd interpret the wave function of the Gn for a single electron. If you have lots of electrons, then you can just add them all up. So instead of i plus sigma dot n, you'd have n i plus sigma dot s where n is the total number of electrons and s is the total spin. So the idea is that for single electron you interpret these inn terms of probability, that you have got an electron, if I measure the spin in a certain direction what is the probability that the answer will be positive, but I get it, get a spin in that direction. So that's let's say 50 50. But if I have millions of electrons what it means is half a million are up, half a million are down. But a lot of these subtleties are involved in this off diagonal elements, which  xs minus isy part of it. 

[Slide 5] Now next point is given that Gn let's say looks like this, how would you extract the information we want because this is a 2 by 2 complex matrix. What I can relate to easily is things like n and s. That is what is the number of electrons, what is the total amount of spin. So given this matrix how will I extract those quantities? So there's a mathematical and there's a formal way of doing this. Of course, you could look in here and say, okay this is n plus sZ, this is n minus sz, so I'll add these two up and that should give me n because the sz will cancel. And formally you could say oh what you should do is take the trace of Gn. Trace means add up the diagonal elements. So you can see if you add it up you'll get n. Actually there's a more formal way of seeing that and that is, you see you will note that you can write this as n i plus sigma x sx, sigma y sy plus sigma z s c. Now when I take a trace of this matrix, you'll notice that all the three poly matrices have zero trace. You see if you add up the diagonal elements that's0, add up the diagonal that's0, add up the diagonal here that's also zero, and the only one which doesn't have a zero trace is the identity matrix. There the trace is two. So when I take the trace of this, trace of i gives me two, which cancels the half and I get n. And the rest don't give me anything okay. So in this case, as I said, you could have seen it from here right away, but this formal thing actually helps you to also see how you could extract information about sx for example. So supposing I want to know to know what is the spin in the x direction? What I could do is multiply Gn by sigma x. So what would happen is I'm multiplying from the right with sigma x, so you get sigma x times i, that's sigma x. Here I get sigma x times sigma x, which is sigma x squared and the thing is sigma x squared is actually i. And this you can see from this general relation that I've stated earlier, you know, that is obeyed for any arbitrary vectors v1 and v2. So if we choose v1 and v2 to be the unit vectors in the x direction, then you see this becomes sigma x, that's also sigma x, so you get sigma x squared. X cross x is 0, so that drops out. X dot x is 1, so you get i okay. So you can show that sigma x squared equals i. Another thing you can check out is that if you have sigma y times sigma x, which means we choose this to be the y vector, unit vector along y, this is the unit vector along x, then y dot x that drops out, y cross x that's minus c and so this becomes minus I sigma z. And if you too look at sigma z sigma x that would be like putting a unit z vector here, a unit x vector there and then you will get plus i sigma y and that's what I've used here. You see, I'm multiplying by sigma x, so I've got sigma x squared, that's i. sigma y sigma x, that's minus i sigma z. And then sigma z sigma x, that's plus i sigma y. So when I multiply by sigma x, we get this. Now how does that help us? Well now you see 

[inaudible] the trace, I only get sx. Why, because all these other sigmas have 0 trace and the trace of i is 2 which cancels the half and so you get s x, so you get that. And what you can show is the same works for the other components also. Multiply by sigma y, trace it you'll get sy. Multiply 

[Slide 6] by sigma z, trace it you'll get sz. So in general, you can multiply by sigma in any particular direction i and trace it and you'll get the spin in that direction. And in general, if you want to write the spin in a particular direction n, some special vector n, that's like you have to take s dot n. So what you should do is multiply Gn over 2 by with sigma dot n and then take the trace, so if n isx you multiply by sigma x and is y you multiply by sigma y and so on. So this would be the general way then of finding the spin in any particular direction once you have calculated the Gn. Now just to connect up with want you normally learn in quantum mechanics, I'd like to point out that you could actually interpret this as the spin operator. That is you see Gn over 2 phi is like psi psi dagger, as we discussed before and when you are taking trace, you can move these things around cyclically. Namely trace of a b c is the same as trace of b c a, as long as you move it in the right order you are fine. So here also what we can do is I can take the psi from here and move it to the end and then you can see what you are taking is the trace of psi dagger, this quantity and psi and that's the usual prescription you learn in quantum mechanics. That given certain wave function the way you find the expectation value of some quantity, which is represented by an operator is by taking this case of psi dagger, the operator times psi. So this you could interpret as this spin operator 

[Slide 7] in the direction n. Okay, now this is for if you wanted the spin or the number-- if you wanted the number of electrons of course you just take the trace of Gn. And you have the number and if want the spin you do this. Similarly, you could calculate currents if you wanted. That is, earlier we talked about the current operator if you multiply by sigma dot n, trace it, then you'll get the spin current 

[inaudible] direction and the spin current is a vector. And note that current is a vector usually because it can have a direction, but here we're talking about current in a particular direction. And it's still a vector because it could be carrying a spin in a particular direction. So what I mean we're talking about current in this direction and this vector represents the direction of the spin that is being carried right not the direction in which the current is flowing, that's this factor. Now here we said that this is a 2 by 2 matrix if you look at this one point. In general, a device might have many more points, you know, I've shown three here, it might have 20 points or so. But then the way you should look at it is if you looked at the Gn matrix here, it would be like a 6 by 6, but what you're looking at is the diagonal blocks. So this first block tells you the n and s here at this point, n 2 s to tell you what it is here and 3 s 3 tells you what it is there and so on. 

[Slide 8] So you can calculate it anywhere inside your device. So what we'll do next is, so what we have talked about is how from the Gn you can get this n and s at a particular point in your device and what we'll talk about next is how that then translates into this concept of voltages or the electrochemical potential, the usual potential and the spin potential of the spin voltage. That's what we'll do in the next lecture. Thank you.