nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L2.1: Introduction
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[Slide 1] Welcome to Unit Two of our course. This is the introductory lecture. 

[Slide 2] Now as you know, in general, what you are talking about in this course is this current flow in small devices, and basically, the way you view it is that there's a source and a drain, and electrons want to flow through this electronic highway with states that are available for transporting electrons. In part A of the course, we talked about semi classical transport where electrons are viewed sort of as particles. Whereas this course, part B, it's about quantum transport, and as you have discussed, in describing transport, you not only need these mechanics but also this what I call entropy driven processes. The kind of things that, I guess, these contacts represent. And what we have discussed so far in Unit One is this Schrodinger equation, E psi equals H psi, and we discussed how you can write the H in the form of matrices, which describe these, this electronic highway, the levels that are available for transport of electrons, but the point to note is whatever we have done so far, it just describes an isolated channel not transport really. 

[Slide 3] So if you think of a range of energies, just a small window of energy, dE, and with the number of states D0. This is the density of states, and you have some electrons n in there. The point is, it's an isolated channel. Nothing's coming in or going out as it stands and what you want to talk about in this unit and the next is how to include current flow, how to include this process of flow from the contacts. Specifically, the way you would think about it is if there N electrons in here that want to go out into the contact, and that's this nu one, which is the rate at which they want to go out. If the connection to the contact is strong, they want to go out rapidly. If it's weak, it will go out slowly. So the rate at which they go out into the contact, this is nu 1N and nu 2N the other hand, new electrons come in to the contacts, and that's this spot. So how much comes in, again, of course, depends on how strong the connection is but also depends on the density of states available inside the channel. So it's proportional to D0. So you write it as S1, D0, and as you might expect, this quantity which tells you how easily it goes out. This tells you what comes in. They are related because if you have a strongly coupled contacts, electrons will go out easily, but they'll also come in easily, and we'll see there's a relation between them. But in general, you can write then dN/dt, the rate at which the number of electrons inside here changes is equal to the rate at which they go out. That has a minus sign, because when it goes out it gives  you a negative dN/dt and wants to go down. Whereas the inflow, that's a positive sign, because it tends to build up, and one point that causes some confusion is people often say, "Don't you have to worry about exclusion principle, that well, what comes in doesn't that just depend on the states available but also on which of them are blocked because of the exclusion principle?" And I'd like to mention that for elastic processes where things are flowing without changing energy, the right way to think, and this isn't obvious really, but the right way to think is that electrons just flow out and new ones come in without any reference to the exclusion principle. So it's not like what comes and gets blocked. In other words, if all the states were completely full, let's say, then these states in here would also be filled, but it's not correct to think that the electrons in there just stay there. It's more like they are continually going out and new ones are coming in, and that's the picture we'll adopt. 

[Slide 4] Now the point we're trying to make, what we want to do here in this unit is figure out what to do with the Schrodinger equation which describes this isolated channel, how to modify it, so that this process of outflow and inflow are included. And first question, of course, is Schrodinger equation is in terms of a wave function. Here we are thinking of number of electrons. How are the two related? Well, N is the psi dagger psi. You know that the psi is this wave function whose square gives you the probability of finding an electron. So if we're dealing with one electron, then the right way to interpret psi star psi, the psi squared is what is the probability? You'll find it in a certain place. On the other hand, when you have lots of electrons, let's say 10,000 electrons, and there's 10% probability of finding an electron in some place, that means there'll be about 1000 electrons there. So in one electron terms psi square is like the probability, but when you have lots of electrons, the psi squared is gives you the number of electrons. So, and you'll notice that I also used a psi with a tilde in it. That's because, you see, in order to understand this flow process, it's important to go to the time dependent Schrodinger equation. What I introduced in Unit One was the time-independent Schrodinger equation, and by in large, you see, in this course, we are talking about steady-state flow not transient flow, and for this purpose, usually the time-independent Schrodinger equation is adequate, and that's what we'll be largely trying to do, but in order to understand current flow, we'll occasionally have to go back to this one, when we try to derive the basic equations that we'll be using in the first few lectures. The next four lectures we'll be doing that. I'll occasionally be going back to that one, and the connection between this and this is just that the psi tilde, the time-dependent one, is written as psi, exponential minus iEt over h bar, because if you take that form and substituted it in here, this equation will turn into that one, and the reason is there's a d/dt, and when you take time derivative of this exponential function, that's usually very simple, because time derivative simply means multiplying it by the exponent minus iE over h bar. And you see, the h bar cancels out i times minus, i is one, and you get E. So that's how this becomes E psi, and that exponential factor just cancels out on both sides. So generally, we'll be using this, but occasionally, we'll go back to this, and you should be clear about that connection. Okay, now here the question is then, as it stands, it describes an isolated channel, what do we do to get the outflow and inflow into the story, and what we'll show in the next few lectures, I guess, is that you can include the outflow by adding a term like this, and the inflow by adding a term like this, and where does this come from? Well, the right way obtain these terms would be from the boundary conditions. That is, you say when you have E psi equals H psi, that applies everywhere, but we are only interested in this region. So we apply certain boundary conditions, and from those boundary conditions, we can show that you get these additional terms. The boundary conditions kind of become equivalent to these additional terms, but that requires some discussion, which we won't get into right away. Actually, in the first few lectures of this course, will derive the basic equations, but we won't talk about how you could get the sigmas and S's from boundary condition. That we will try to develop later using different examples, because how you exactly get the sigma and the s, it depends a little bit of upon the problem at hand. So later in this unit, we'll do some 1D examples, and in the next unit, Unit Three, we'll be doing more advanced examples, but for the moment, what we focus on is that given this thing, these additional terms, how can they be related to outflow and inflow, and one important point I want to make in this context is, you'll notice in this simple picture we have two inflows, one from the left and one from the right, and yet, here, I had put only one inflow from the left and not the other one, and there's a subtle reason for that, why is not quite right to put both in here. 

[Slide 5] It goes something like this. You see, supposing we put in both. Then when you solve this equation, you'll get psi in terms of S1 plus S2. So you'll get psi is equal to S1 plus S2 times whatever. We'll see what it is later but you could write it this way, but you really interested in is. psi dagger psi. Okay? Now which then becomes S1 plus S2 dagger times S1 plus S2. Now the thing is, when you multiply it out, you get four terms. Now two of these correspond to the S1 and S2. This is the strength of the source in contact one, strength of the source and contact two, but you got these two other spurious term, which involves this cross product of left and right, which kind of represents sort of interference between left contact and right contact, which is never observed experimentally with normal electrons. And so, if you include terms like this you are getting unphysical answers. So these are really never there. In real systems, whenever you have multiple sources, you never see interference between them, and in that sense, I often made this analogy with light, that with light, if you look at a thermal lightbulb, and would you have two light bulbs, you never get interference between them. And electrons are kind of like thermal sources. Now with light, there's also coherent sources or lasers, for example, where you can get interference between two different sources. With normal electrons there's nothing quite like that. And so, bottom line is, we need a theory which works directly with these products like psi dagger psi rather than with psi, because if you're working with an equation like this, it is not quite right to add multiple sources there, because then your theory will give you interference terms. 

[Slide 6] So instead, what we do is try to develop a set of equations which applies to these products, and those basically are the NEGF equations of nonequilibrium Green function equations that we'll be talking about. Now here, again, I like to make a distinction between psi dagger psi and psi psi dagger. These are not quite the same thing. If these were numbers, of course, they would have been the same thing. A times B is just B times A. but these are actually column vectors. Like, if you had a, say a lattice with three points, for example, then psi would be, like, a column vector with three things, and the psi dagger would be this thing, conjugate transpose. That's what the dagger stands for, conjugate transpose, and so you get psi1 star, psi2 star, and psi3 star, and when you multiply the two, you just get a number. See, and the reason is that this is like, I guess, 1 by 3. This is like a 3 by 1, and when you multiply it, you just get one number. The psi1 star psi1 plus psi2 star psi2, plus psi3 star psi3, now by contrast, you see, if you look at psi psi dagger, then the two are reversed. So you now have a 3 by 1 multiplying a 1 by 3, and what you get is a 3 by 3 matrix. So psi1, psi1, star goes there. Psi1, psi2 star, that goes there. Psi1, psi3 star. That goes there. So you actually have a 3 x 3 matrix, and whereas this quantity here is just a number. How could you get this number out of that matrix? Well, you can just add up the diagonal elements, and that's what is generally called, taking the trace of the matrix. So the trace of psi dagger psi and the trace of psi psi dagger, they are both the same, but the basic thing, this is a 3 x 3 matrix. That's just a number. And anyway, so this is the quantity that we call Gn. It's kind of like this n, but then it's this matrix version of it, right? So it's like the matrix that represents the number of electrons. And there's another quantity we'll introduce, which is the strength of the source, like, ss dagger. That's what you call sigma in, telling you what comes in. And what the NEGF equations represent is an equation that connects this to that. So rather than work directly with psi and s, you work with these, and the reason is that when you have multiple electrons, you cannot quite superimpose their wave functions, but you can superimpose their psi psi dagger. 

[Slide 7] Now how do you get an equation in terms of the psi psi dagger and ss dagger? Well, first step is, you see, if you have an equation like this. E psi equals H psi plus  sigma psi plus s, you could take all these three terms on one side and write it in this form. This I is an identity matrix of the same size as H and sigma, and E is just a number of energy. So you have this. So given this, you could write psi in terms of s by taking the inverse of this quantity. That's what's called a retarded Green's function, GR, and psi equal to that Green's function times s. Now if you want psi dagger, well, you have to take the dagger of this quantity, and if you remember from linear algebra, when you take the dagger of this, it gets turned around. So you first have the letter s dagger and then you have the GR dagger, and the GR dagger is what's called GA, the advanced Green's function. So now what we want is psi psi dagger. So what's psi psi dagger? Well, it will be like GRs, and then s dagger GA, and this psi psi dagger then is what we call Gn ss dagger is what we call sigma in, and I've also put in this notation that's also used a lot in the literature where what I am calling Gn is often written as minus iG less, and what I'm calling Sigma in is written as minus i sigma less. Okay, now from here then we get this equation Gn equals GR sigma n, GA. Or in the other notation G less equals G sigma less, GR sigma less, GA. And these are the basic NEGF equations. This equation for the retarded Green's function and this equation for this Gn, which is, like, the number of electrons. The operator that represents the number of electrons, and we'd like to say a few words then about this alternative method that you see a lot in the literature. 

[Slide 8] That is, what we have done here, what we'll try to do in this course is start from this one electron Schrodinger equation and get these basic NEGF equations, and this is the way I've been doing this for a long time, it's like described in my books, but the standard way that is widely used is based the seminal work of Keldysh, and other people, actually, back in the 60s, and that work is based on what's called Many-Body Perturbation Theory. The reason is, you see, before the advent of nano electronics and mesoscopic physics, the general idea was that resistance, or current flow, involves dissipation, and so it is very important to include dissipative processes, and dissipative processes, of course, involve many body interactions. Because dissipation means the energy in the electron goes out into the phonons and by interaction with the surroundings, and that requires this many body theory, this many body perturbation theory, and that is how these equations were originally obtained, a lot of literature follows that. Whereas what we are doing is following what we have learned from nano electronics and mesoscopic physics, that the dissipation is kind of incidental to current flow and resistance. Much of the physics of resistance can be understood without including the dissipation, but the point I'd like to make, though, is it's not like we ignore interaction completely, because if you ignore interaction completely, then you don't quite need this NEGF method necessarily. You could use this what's called scattering theory, but what we do, what I like about NEGF is it allows us to include interactions, and the only thing is, these interactions, we assume, do not lead to any exchange of energy, but as we'll see, they have an enormous amount of physics in there, these interactions, and by ignoring this energy loss processes, this energy dissipation, what it allows us to do is get the same equations. Note that these are exactly the same equations that are obtained from the many-body method. So it's the same equations, but it's done in a much more elementary way from one electron Schrodinger equation. 

[Slide 9] And just to give you one example, consider a wire with a scatterer in between, a barrier of some kind, let's say a defect in there, or an impurity. If you have taken part A of this course, you might remember that this kind of is like three resistors in series. The resister at this interface, another one at this interface, and then the middle one, and so if you looked at how the electrochemical potential or the occupation of states, how it varies through the device, you would expect a drop at one end, a drop around the scatterer, and another one at the other end. Now if you do a coherent theory, where you ignore all interactions using quantum transport methods, what you'd get, you'd see these oscillations, and the reason you see oscillations is because related to standing waves that are familiar with waves. But in practice you often don't see such pronounced oscillations, and the reason is that real devices are not quite clean things like this. There's a lot of random scattering going on, which destroys a lot of the phase information, and this you can include into your theory by adding the sigma 0 and the sigma 0 in. In addition to the 1 and 2, which represent the contacts, and when you include this in, you'll see the oscillations get a whole lot smaller. What's more, you'll notice there's a slope here. Why the slope? Well, because the interactions that we included also gave rise to moment relaxation, and so they gave rise to resistance, because the point is, resistance comes from relaxation of momentum. Now if we want to include only phase relaxation and not momentum relaxation. That is, processes that destroy the phase of the electron but not its momentum, and these are very common. For example, a very common interaction is electron-electron interaction. That is one electron interacts with the other ones around it, and this destroys its phase, but it doesn't destroy momentum, because whatever momentum one electron loses, the other ones pick up. So as a collection, they kind of keep their momentum amongst themselves. So if you include that, the momentum relaxation, you get something like this, but you can also choose these sigmas in such a way that it gives only phase relaxation, and in that case you get something without that slope. But because of phase relaxation, of course, those oscillations have gotten much smaller, but there is no slope because it's pure phase relaxation. So the point I'm trying to make is this NEGF method that we'll be talking about, allows you to put all this physics into the theory, but through the sigma 0. Something that a normal, coherent theory doesn't allow you to do, because coherent theory is just based on 1 and 2. But we are not including dissipative processes at all, and that's what, I guess, keeps the overall framework simpler, because dissipative processes generally require deeper discussion these  many-body interactions, and we'll do a little bit of that at the end of unit three, but by and large, what we'll be talking about, we ignore dissipative processes. 

[Slide 10] So with that long introduction then, we're ready to get onto it, get onto the real stuff, and what I should point out is, the next four lectures here are kind of about developing these equations. You know, the basic idea that you modify the Schrodinger equation, and then you have the NEGF equations, which relate bilinear products, et cetera. That's what we're doing in these next four lecture, and after that, what we do is look at examples, and to some extent, the two are disjoined. In the sense here is what you see it how these equations come about. Here you see how to apply it to simple 1D examples. Whereas, in Unit Three, what we do is more advanced examples. So you may want to kind of look at this quickly. Get on with it. Look at the examples, and then, if necessary, come back and look at these four lectures. Okay, with that then, we are ready to get on with 2.2. Thank you.