nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L2.10: Summary
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[Slide 1] Welcome back to Unit 2 of our course. And I guess it is now time to sum up what we did in this unit. So, in the last unit, that's Unit 1, we talked about the Schrodinger equation. This E Psi equals H Psi, where H is this matrix whose eigen values give you the energy level. The energy levels that define this electronic highway, the highway along with electrons from source to drain. And our purpose in this unit was to add the, this flow process, the contacts. That's why we call this contacting Schrodinger. The process that involves this outflow of electrons and inflow of electrons from the contacts. 

[Slide 2] And what we talked about was how you can modify the Schrodinger equation and add these additional terms, the sigma1, sigma2 and the s1 and the s2. And so that they correspond to this inflow and outflow. Now, one important point, that I tried to make at the outset is that at the Schrodinger equation level, it is not quite right to add multiple sources, because if you do that, you'll get interference between them, which is unphysical. Because in general, electrons are like thermal sources. That is, with light, as you know, there are thermal sources where the power adds up, but there's also coherent sources, like lasers, where the electric field adds up. But when it comes to electrons, it's all normal electrons, it's all thermal. So it's the Psi star Psi that needs to be added, not the Psi. And so what we need is equations that relate the Psi Psi dagger directly to the ss dagger, rather than Schrodinger equation which relates psi to s. And once you have equations like this, then multiple sources can be added up. And based on these new quantities that we introduce, this Gn, which is like the number matrix, sigma in, which describes these source terms, we obtain a flow diagram. So it is kind of like the simple particle picture that we had, except that all these quantities have become matrices. So here we had say, N electrons. Here you have a Gn matrix, and if you take the trace of that matrix, you'd get the number of electrons. Similarly, you have this nu which tells you the rate at which electrons leak out into the contact. Here there's a gamma matrix divided by h, that's like the nu. So there is these analogies. A which is like density of states. Gn like number. Gamma like nu, and sigma in is like the source, source strength. And this, I guess correspondence is useful to keep in your mind, as you try to navigate your way through all these new symbols and concepts that we are introducing. 

[Slide 3] Now there's a little detour, which I want to mention because it might come in handy depending on what problems you are working on, and that is this idea of a current operator. This is, in general, in this course, we'll be talking about charge currents and for charge currents, you see what we have written down here is the inflow and outflow of electrons. So how many electrons per second come in? So if you look at the units of what I have written down here, it would be like in a given energy range, it would be per second. Now, if you wanted charge current, you'd just multiply it by the charge on an electron and you'd be done. On the other hand, you might be interested in the flow of some more complicated quantity, like spin. Something that itself is represented by a matrix, let's say Y, and you want the current of Y. So in that case, it would be nice to have a current operator so that you could multiply it by Y and take its trace. And of course, if Y is not a matrix, just a number, then you could pull out the Y, and just take the trace of the operator itself. And in that case, you would get back what we had here. So what I showed was that in general, you could write a current operator such that this quantity, if you take its trace, you get that. But then, if you wanted the flow of some other quantity, some more complex quantity Y, then you might want to work directly with these. 

[Slide 4] But much of this course though, we'll be talking of charge currents, and there, what we have is this basic set of equations that you could apply to any problem of interest. This is what we developed in the first half of this unit. And in order to apply these equations, what you need is a sigma and a sigma in and the antihermitian part of sigma is the gamma. And if the contact involved is in equilibrium and described by a Fermi function, then there is a basic relationship between the in scattering and the, this outflow of the gamma. In other words, sigma in must be equal to gamma times f. And this is a fundamental requirement based on equilibrium statistical mechanics. And so, what that means is, for every contact what you really need to know is sigma. If you have sigma, you have the gamma because it's the antihermitian part. And if you have the gamma, you multiply by f and you have the sigma in. So there's only thing you need to find out, which is the sigmas. Now, but this argument though, that sigma equals gamma times f, only applies if the contact itself is in equilibrium. But if you look at this dephasing contact, you know, which as I mentioned is extremely important because usually electrons suffer this dephasing processes as it goes through the device which often destroys a lot of interference effects. And if you want to model that, model the effect of this random potential that the electrons face when going through a device, then you need this dephasing contact, the sigma zero and sigma zero in. And those are not described by any Fermi function ordinarily. Although there are models people use where you assume that it's also described by a Fermi function. But we will not be doing that, and what I prefer is one where we don't make that assumption. But what you have to be sure then is, that you choose these in such a way that there is no net current at that contact. And that means the inflow should be equal to the outflow, because at the physical contact you could have 100 per second coming and 90 per second leave, and there would be a net flow of 10 per second. At the dephasing contact, if there is 100 per second coming in, 100 must leave, otherwise we wouldn't have steady state. You'd be losing electrons. 

[Slide 5] Now, in terms of how you obtain the sigma, we went through a 1D example. We took a case of a 1D wire, and assumed this incident wave, the reflected wave and a transmitted wave, and we showed how, using simple arguments and applying these boundary conditions, boundary condition meaning that at this contact, you assume there's only an outgoing wave. So by applying these boundary conditions, you can figure out what the appropriate sigma is. And so this is the example that we went through, and once you have obtained the sigma of course, you can get the sigma in using the relation that we talked about. And this is sigma that you can use for any 1D problem. The result is a very simple one. But you see, if you had a 1D wire, diagonal elements are epsilon, the coupling is t, well then the sigma basically looks like zeros everywhere except for te to the power ika, at the point where it is connected. So if it is sigma 2, the t to the power ika appears at this end. If it is sigma 1, then the t to the power ika appears at that end. Now the other example we went through is one that involved the dephasing contact. 

[Slide 6] And this is where we took this example of a wire with a barrier. Which you know, our simple view would be there's an interface resistance and then there's a resistance due to the barrier. This is a problem that we discussed at length, I guess in part a of this course. And the question is, now that we have a quantum model, what do we get? And if you use a coherent theory, then you get all these oscillations which indicate quantum interference effects. So that's what you'd get in a coherent theory. On the other hand, if you use an incoherent theory, which means, like if you use a particle picture, semi-classical, then you get this red curve. Now question is, how do you interpolate between them? And the point I want to make is NEGF allows you to interpolate between them. Which is do a full quantum theory, but include dephasing processes. And how do you include the dephasing processes? Well I say one way of doing it is with this choice, that a sigma zero ij is equal to Dij times GR . And sigma's zero in is Dij times Gn. And as long as this D is a symmetric matrix, electrons will be conserved. So you won't run into any fundamental problems. But how you choose the D will determine whether you get phase relaxation or whether you get momentum relaxation or both. So here, for example, particular choice of D gave pure phase relaxation. Whereas, another choice of D gave momentum, as well as phase relaxation here. So you can include all kinds of different physics through the choice of D, that's my point. And usually the approach is that you treat the D as something to be obtained from microscopic theory, because basically, what the D represents is the RMS potential that is, there is this random potential inside your channel, and what the Dij represents is the correlation between the potential at Point I and the potential at Point G. And ordinarily you might start from a microscopic theory of the scattering mechanism, like phonons, and try to figure out what that correlation is. But the approach we are taking is, we are not obtaining the D from microscopic theory. Instead we are saying, let us a use a D that gives us a consistent model, in the sense you don't lose electrons, but then we choose the D so it gives us the correct relaxation processes. It is in a given system, you usually know what kind of momentum relaxation time there is. You know that from mobility. Or you might know what the phase relaxation times are. And so you choose the D accordingly so that it describes your system correctly. And and the NEGF method gives you a general way of including such relaxation processes into the model. And this is where it's different from a pure coherent theory, where you would necessarily have things like this, including I guess we should show interference effects. 

[Slide 7] So that kind of brings us to this end of this unit. You see what we have done in the first half of this unit is set up these basic equations, the basic NEGF equations that you could use to analyze any device, including quantum effects. And to apply to a particular problem, what you need is the sigma. And how do you obtain the sigma? Well, in the second half of this unit, we saw a few simple 1D examples. In the next unit, what we'll look at is more advanced examples. That is we'll look at two-dimensional devices, we'll look at graphene, we'll look at devices with I guess, magnetic fields, and the last unit of this course, we'll look at devices involving spin transport. Thank you.