nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L4.3. Rotating contacts
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[Slide 1] Welcome back to Unit 4 of our course on quantum transport, this is the third 

[Slide 2] lecture. So as you may remember in the last lecture we considered a device with just one point, one basis function, and including spin. Then the matrices that describe it are 2 by 2 matrices. and we assume the channel itself is non-magnetic. And so the Hamiltonian matrix describing the channel, that's like epsilon times i. So it's epsilon on the diagonal because up-and-down are both equivalent. But when we try to write the matrices describing the connection to the contacts, that's the sigmas and the gammas that's when we have to distinguish between up spins and down spins. One spin gets out easily, the other one doesn't. Now the question is how do you write the gamma for a magnet that's pointing in some other direction? In the last lecture, we argued that if the magnet is pointing in the opposite direction you could just reverse the alpha and beta. So instead of alpha beta you have beta alpha. Because the role of up-and-down spins are exactly reversed. But what would you do if you had a magnet pointing in some other strange direction? That's what we'll talk about in this lecture. And this is where it's important to understand this important distinction between spinors and vectors. And as I mention in the introduction, one good analogy for electron spin is the polarization of light, and polarization of light you visualize in terms of vectors. You learn in freshman physics that light has this electric field which defines its polarization, and if you had a polarizer, and an analyzer that are at 90 degrees then it will block the light. Whereas if they're parallel the light will flow easily. You could think in a similar way here these magnets are like the polarizers and the analyzers, and if they are parallel electrons flow easily. Lots of current, low resistance. If they are antiparallel, that's when it's blocked. High resistance, but note this important difference with light. With light if they were, if the polarizer and the analyzer were antiparallel that wouldn't block it. That is almost like being parallel. Light goes straight through. They are blocked  when they're 90 degrees. Ninety degrees is the orthogonal direction, but electron spins 90 degrees doesn't give a minimum. Its 180 degrees that gives a minimum. That's the-- those are your orthogonal directions, and that's why I guess, with light-- the polarization of light you understand in terms of vectors but with spins you need something else and that's what you call spinors. So we deal with the vector when you think of something pointing in some direction n defined by the angles theta and phi. You can write it, x, y and z components, as you know cosine theta is the z component, sin theta is the x-y component and then you could resolve it again and write sin theta cosine phi sin theta sin phi. That you learn in vector algebra. But if it's a spinor you visualize it the same way. You think of it as pointing in some direction but you do not write three real components for it. You write two complex components, an up component and a  down component. And what are these components. Well the components are cosine half theta and sin half theta. Exponential minors i phi half, exponential plus i phi half. Now this isn't meant to be obvious, and in the next lecture I'll try to justify this. But for the moment let's just accept this and we'll talk about the consequences of it. The main point to note is how you-- here  you have the theta and the phi and here you have half theta and half phi. With vectors, when you have something pointing in this direction, you can resolve it using this parallelogram law and write it as a superposition of this and this. With spinors, all the angles-- instead of theta its half theta. So in order to get 90 degrees you kind of need 180 degrees here to get orthogonal. And so a spinor pointing in this direction is resolved into this and this which are at 180 degrees as if you see, the angle got doubled and so what that means is a spin pointing in the plus z direction is actually a superposition of the spin in the plus x direction and minus x directions. Very counterintuitive. Very different from vectors. And so it takes some time to get used, and as we go along we'll mention in other non-intuitive issues. Now one of the non-intuitive things that we will address, actually right away, is this question of how would you write the components of a spinor or a vector that instead of pointing along plus n and is pointing along minus n in exactly the opposite direction. So these two quantities I'd written down before I denote as c and s. So c and s denote the components of something pointing this way. So how do I write the components of 

[Slide 3] something pointing the other way? Now if it's a vector, actually the answer would be very simple. You'd say well if I have a vector pointing this way. Anything pointing the other way will just add a minus sign in front of it. So this vector and this vector, one is just a negative of the other, but that's not true of spinors. spinors as I pointed out before, this 180 degrees is actually an orthogonal direction. It's not just the negative of the other one. Orthogonal means it cannot be expressed in terms of the last one. So to find the appropriate components we have to go through it more formally. So formally what we can say is that when you go from plus n to minus n what you're doing is you're turning theta into pi minus theta. Instead of being this angle, you're taking this and turning it up to here so that the effective angle is pi minus theta. And instead of phi you're writing pi plus phi which kind of rotates it to the appropriate direction. So you can think this through but if you put those n. If you do those substitutions in the vector components you'll get what you would expect intuitively. Namely the x will become minus x, y will become minus y, z will become minus z. You can check that. But  with the spinor when we do that. What would happen is instead of theta, we would put pi minus theta instead of phi. We would put pi plus phi. Same here, and now you see cosine pi over 2 minus theta over 2. That's actually sin theta over 2. Similarly sin pi minus theta over 2 that's like cosine theta over 2. And exponential minus i pi over 2,  and here you have exponential plus i pi over 2. So what I  have done is taken i pi over 2 outside and put an exponential minus i pi over here. So just that substitution into this then gives you the appropriate expression for the components of minus n. And one point I want to make here is when you write the components of a spinor, this overall quantity here does not matter. So if you put exponential i pi over 2,  or in any phase factor out here doesn't matter. What matters is this relative phase between them. So we could write the minus n as just this which is what I've written here. And this part and the exponential i pi is a minus one. So that's that. So, and the phase factor in front that really doesn't matter you could put anything in front. And as I said, this is something we will justify a little better in the next lecture. But one point you might say, well if you just can add anything, doesn't that cause big problems? I mean after all when you're how do you interpret experiments? You know doesn't sign matter? And the answer is you see, with the polarization of light describe an electric field. The sign matters because electric field is a measurable quantity. And if you say I can easily put a minus sign in front, that would make no sense. But with electron spin you see, there's the wave function that's psi. But the observable quantity is psi psi dagger. And so it doesn't matter what phase factor you put in front. The psi is different, but the psi psi dagger isn't. And so when you're interpreting experiments that shouldn't matter what you put in front. Now notice what I've written here, you could write as this is really what I defined as c, but complex conjugated. And what I wrote here as s complex conjugated gives you  that. And the point to note is this is not just negative of that. So you couldn't express this as just some multiple of this. This is a totally independent thing. In fact what you can  show is this and this are actually orthogonal, in the sense that if you multiply this by the conjugate transpose of that you'd actually get zero. They are orthogonal things. 

[Slide 4] Now the next thing then, given this result, the plus n and the minus n. How would we use it to write down gamma 2? And there the argument goes like this. That you see for a magnet pointing in the plus z direction? We have written gamma 1 in this form. Now if we used plus n and minus n as our basis functions then the matrix for gamma 2 would also look like that. In other words gamma 1 looks like this because our basis functions are plus z and minus z. This magnet is identical but it points instead of pointing along z it points along n. So if I use plus n and minus n as my basis functions then of course it should look just like that. There should be no difference. But of course what I'd like to know is what does gamma 2 look like in the old basis and plus z and minus z. And the way I can do that is I can take this and do a basis transformation and this is where if you rusty on basis transformations, you may want to look it up. You know something you've seen in linear algebra. And the way it works basis transformations is you put a matrix here, U, whose columns tell you what the old basis which is plus n and minus n. What plus n is in terms of the new basis plus z and minus z. Which as we discussed is c and s. Similarly you take minus n express it in terms of Z and minus z. That's just minus s star and c star. So this is the basis transformation matrix and the way you do this transformation the similarity transformation is put U here and you put a U dagger there. And this is the conjugate transpose of this one. It also represents like what the plus z is in terms of plus n and minus n. Or minus z is in terms of plus n minus n. So how do I remember what to put on this side and what to put on that side? Well that's where I've often pointed out that you see when you're multiplying matrices it is the rows that multiply into columns. So if you have plus n minus n here this also must be plus n minus n. Similarly plus n minus n this must be plus n minus n. On the other end if I had taken this thing and put it here then you see what, I'd have plus z and minus z over here which wouldn't match that and so that would be wrong. So the way you can line it up is by remembering that rows must match the columns, rows must match the following columns. Now once you have written this then, that tells you how to write down the gamma 2. And whatever we will do after this will be just algebra. You know multiplying them out, just printing out the result a little bit, that's all. So you multiply those two things usual rule of multiplication matrix multiplication is c times alpha minus s star times zero. So that's c alpha, c times zero minus s star times beta, that's minus s star beta and so on. So you can write the two-by-two. Next you can take this and multiply with that and that's when you get this matrix c alpha times c star that's cc star alpha minus s star beta times minus s plus ss star beta, etc. So this is then the gamma 2. So you have done the multiplication what we will now do is make 

[Slide 5] it look a little prettier. So first thing we do is for the c's and s's we put in the actual-- what it represents so cc  star that's just cosine squared half theta. Ss star that's sin squared half theta. Cs star you see that is this times the conjugate of that that's like cosine theta over 2 sin theta over 2. Which is half sin theta. And now you notice the phase factors don't cancel because s star has exponential minus i/phi over 2. So when I multiply the two I get exponential minus i phi. Similarly here I get the conjugate of that and here again you could just put in the numbers and you'll get that. So this comes that. Next step is do a little more algebra, and you can show that this quantity is actually half alpha plus beta plus half alpha minus beta times cosine theta. Whereas this one is half alpha plus beta minus that quantity. So here what I am using is again this trigonometric identities which are the cosine squared half theta plus sin squared half theta is one and cosine squared half theta minus sin squared half theta is cosine theta. So making use of those identities I get from here to here. Next what I'll do is 

[Slide 6] we will take this and break it up into four matrices. Separate matrices. So what you can do is first of those is alpha plus beta over 2 times the identity matrix because you'll notice there's a alpha plus beta over 2 here and alpha plus beta over 2 there. So we have taken care of that. Next you note this alpha minus beta over 2 cosine theta and a minus alpha minus beta over 2 cosine theta. So that I could write as alpha minus beta over 2 cosine theta and then plus one and minus one. Plus one for this minus one with this. So these two added together then take care of the diagonal elements. The off-diagonal is still zero. For the off-diagonal what we note is, you first write the real part of this. The real part of this is alpha minus beta over 2 sin theta cosine phi. And you'll notice that real part is the same for this one as well and so we have 1,1. So this then gives me the real part of the off-diagonals and when you look at the imaginary part that's when you have alpha minus beta over 2 sin theta and then e to the power minus i phi is cosine phi minus i sin phi  so you have the sin phi here. And your minus i plus i. So this matrix then can be written as the sum of four matrices each one with its own component own coefficient in front this actually is generally true you could take any matrix and write it in terms of, any two by two 

[Slide 7] matrix, and write it in terms of these four matrices. Now, this is the identity matrix and these three matrices are also known-- I mean they are given a name and this is called the z component  of the Pauli spin matrix sigma. And nz that's of course the z component of n. Which is cosine theta. So we have alpha minus beta over 2 that consine theta is nz and this then is that Pauli spin matrix with z component. Similarly this is the Pauli matrix x and sin theta cosine phi that's just the x component of n. So you could write n x sigma x. This is the Pauli spin matrix y, sigma y, and this is then the y component of ny sigma y. So you see you have i then you have nz sigma z plus nx sigma x plus ny  sigma y which you could write as sigma dot n. Because you remember when you have dot product of two vectors like a dot b it means ax bx plus ay by plus az bz. Similarly, sigma dot n denotes sigma x nx plus sigma y ny plus sigma z nz. And what you could do is combine those two and write it in this form. You see you could put the alpha plus beta over 2 outside so then you'd have i and here you'd have alpha minus beta divided by alpha plus beta which if you remember in the last lecture mentioned you usually define as the polarization of the magnet. It tells you how effective a magnet is in terms of discriminating between the two spins. So the final result then is that the gamma 2 looks like alpha plus beta over 2, that's half the sum of the diagonal element times 

[Slide 8] this quantity i, this P sigma dot n. So that's the basic result then of this lecture and in getting there though, I introduced these Pauli spin matrices and also I introduced this very non-intuitive concept of spinors that many of you haven't seen before. And as I pointed out, when a spinor points in some direction defined by theta and phi it's components are given by the c and s. And if it's pointing in the negative n direction the components are not minus c and minus s, but rather it's minus s star and c star. And all these non-intuitive issues I guess this I just stated without any justification and in the next lecture what I'll try to do is