nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L3.1. Introduction
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[Slide 1] Welcome to Unit 3 of our course on quantum transport. Now this unit three is a continuation of Unit 2. 

[Slide 2] Now if you remember what we did in Unit 2, this is the outline for Unit 2. What we did there is we talked about how to include these contacts into the description of quantum transport. That is what you learn in traditional quantum mechanics, about the Hamiltonian H, which describes an isolated channel. That's what we did in unit one. But unit two was about how to include contacts into this description. And that comes in through these sigmas. So the first few lectures of unit two we developed the equations of quantum transport. This Green's function and then this correlation function, which is like the electron density. And A called the spectral function which does like the density of states. These are all matrices. And if you want the current, then you had a separate equation for calculating the current at any contact. And for coherent transport there was a special version that you could use. But in general, this is the version that you want to use. This is what describes the current and the different contacts. So this is kind of what we developed in the first four lectures in unit two. We then went on to look at a few examples. Because once you have the equations of course, you have to apply it to different examples in order to get a feeling for how it works. But what we used are what you might call one-dimensional examples involving conductors that are one-dimensional. And what we want to do in this unit is look at more examples. See? And these example were selected in a way that they have some interesting physics also. Although I will be using it mainly to illustrate how to use this method of quantum transport that we are discussing. So that's what we'll be doing in this unit. So in this brief introduction then, let me try to give you a quick outline 

[Slide 3] of the things we'll be going through. So as I said, we want to go beyond 1D examples. And so what we'll do is we'll start with an interesting problem, which you'll hear in many contexts. This quantum point contact. This is a very seminal experiment about 30 years ago, where people observed this quantized conductance, which has now become very much a part of thinking about nanoelectronics. And I'll use this example to illustrate how one might use the overall quantum transport method that we are talking about. And this requires us to go beyond 1D and do a 2D problem. 2D, so you could start with a 2D lattice. Lattice where the diagonal elements of epsilon as usual, and the t's are the connection to the nearest neighbors. So in unit two we had looked at 1D examples like that. Here we'll start with a 2D example. And we'll talk about how from  by looking at this example you could get this result of conductance quantization, this idea that when you have a ballistic conductor, the conductance comes in units of Q squared over H. How that comes out naturally from this method. The next thing we'll do is we'll talk about a more general way of calculating this sigmas. That is for the quantum point contact, using this model, one advantage is that you can calculate the self-energy in a relatively elementary way by extending what we learned in unit two for 1D examples. So it is a relatively simple extension. Whereas for more general problems, in order to calculate sigma, you need this general formulation of self-energy. And that's what we'll be doing in the two lectures after that. That's where we'll obtain this general expression. That the self-energy is given by this tau g tau dagger. What's tau? Well tau is a matrix that describes the coupling of your conductor to the contact. That's this tau. And this g as you will see is what is called the surface Green's function of the contact. And sigma in general is given by this expression. And this is a general expression that actually works in any problem. Whereas what we'll do in 3.2 here,would be a rather special method that doesn't generally always work. And I'll explain when it works and when it doesn't. Now to illustrate this new method then, 

[Slide 4] we'll look at a few problems. For example, if you want to describe graphene, well you don't have a square lattice where the carbon atoms are arranged in this hexagonal lattice. That is a problem where you could use this general expression. So we'll be looking at problems like that. And we'll also look at problems involving magnetic fields where even if you have a square lattice and you want to include magnetic fields to find the self-energy, you should use this general method. So these are two examples. Two examples which are of interest of themselves. The physics is interesting. But what I'll try to show is how this physics comes out automatically from the general formulaism. Which you could use to model. So one of the powerful things about this method that we are discussing here is that you can use it almost blindly without necessarily understanding the physics of the problems we are talking about. That is there is a general way you can write down the H's. You can write down the sigmas and then use the equations we talked about. And you can calculate current. You can calculate electron density. You can calculate density of states, et cetera. So you can extract all of this information without necessarily understanding the problem very well. And that I always stress is kind of a strength and a weakness. Strength because you can get started without understanding everything. Of course it's a weakness if you stop there, in the sense if you don't spend the time to actually understand things. Because what's really powerful about this method is you can calculate a lot without understanding deeply. Which means you can use it as an aid to understanding. So we'll use these two examples then to extract some physics and we'll see how the numerical results you can understand in terms of simple models. Now what we'll do after that is we'll talk about a related topic that is in many courses on quantum mechanics you will hear about this Fermi's golden rule, which is widely used. And which is usually derived in a different way. What I'll try to point out in this lecture is that this general expression for self-energy could be viewed as a generalized version of this Fermi's golden rule. And so once you understand this, you could use it to look at many other kinds of problems as well where normally people use this golden rule. So that's what we'll be doing in this lecture. 

[Slide 5] Now then in the following lecture, that's this eight, what we'll be looking at is this scattering processes. That is, as you know, this sigma one and sigma two, these represent the two contacts. But in general, you also have to include the interaction with the surroundings. And that's what this sigma zero is about. Now what we did in unit two is we looked at elastic interactions. That was in lecture 2.9. And we showed that in that case you can write down the sigma zeroes in a relatively simple way, where sigma zero is this D tensor times the Green's function. And sigma zero N is the D tensor times the correlation function. This simple prescription works very well for elastic processes. And that's what we discussed in unit two. What we want to do here in this unit is talk about the more general case of inelastic processes. As well as processes where you're interacting with a system that is not necessarily in equilibrium. So in general, the expressions for sigma zero and sigma zero in will be more complicated than this. And that's what we'll talk about in this lecture so that you can apply it to more difficult problems. Although many problems, there's a wide variety of problems that you can handle without getting into this part, using just elastic scattering. But to complete the story I'd like to tell you about how you do the more general case of inelastic scattering. 

[Slide 6] And finally we'll be talking about this self-consistent potential. That is one thing I've mentioned before, though I haven't stressed it too much, is that in general, of course electrons are interacting particles. And you are normally thinking of electrons as if they are non-interacting. Because you try to take care of the interactions through a self-consistent potential. This is this mean field picture that one electron feels some average potential due to all the other electrons. And this interaction can be represented by a potential U, which has to be calculated self-consistently. Now in part A, when we were doing semi-classical transport, I think we spent a little time talking about this. Now in the context of quantum transport also, in general, especially when you are far from equilibrium, it is important to take this self-consistent potential into account. And we'll discuss in this ninth lecture. We'll talk about how you could include this and how it is usually done. And I'll also point out that there are certain cases where the standard cases may not work very well. And this is the regime of what's called single electron charging effect or coulomb blockade. Where you could use the same equations, but you'd have to figure out a different way of including these interactions. Because if you do it using the standard way, then you could be missing these single electron charging effects. So that's what I'd like to talk about in this lecture. Because often people think that this NEGF is a general many-body technique and as such it includes everything. And what I want to point out here is that if we just use the standard prescriptions, there are cases where it may not really include everything. And so you have to be clever and creative in order to include these effects as they come up in different context. 

[Slide 7] So that's then a brief outline of what we plan to go through. So with that introduction then, let's get started with the first topic, which is these quantum point contacts where we'll be talking about a 2D lattice and we will be discussing how you can calculate the sigmas in order to handle transport through a two-dimensional conductor. And we'll be focusing on sigma one and sigma two. We won't be worrying about the sigma zero or the U, which we'll bring up only in the latter half of this unit. Thank you.