nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 3.10: Summary
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[Slide 1] Welcome back to Unit 3 of our course. I guess it is the last lecture. It is time to sum up. 

[Slide 2] As we may recall, we started from, I guess what you had done in the previous units, where it showed that the resistance has a part that is proportional to the length, and then there is this extra constant part, independent of the length. The part we call the interface resistance. And what we try to do in this unit is try to show why the interface resistance is associated with the interface essentially. How do we know that it is actually the interface and not somewhere else? And in the process, actually, we introduced a lot of very important and general concepts. And first thing we started by noting is that mentally we tend to associate resistance with heat loss. And that does not really give us this insight about why interface, why this ballistic resistance is associated with the interface. Because the heat loss is distributed spatially over an inelastic scattering length, and it can be all over the place. Largely in the contacts. But from that point of view, then, you'd say the resistance is not here or here. It is like everywhere. But then we all have an intuitive feeling that resistance is associated with obstacles. That is, if we put a little hole in here, the resistance should go up because electrons will have a tough time getting across. So if you want a criterion, something that lets you see where the obstacles are, then it is important to follow the voltage of the IR drop. Other than that heat which is I squared R. That was of course our first message, because that voltage drop corresponds to this intuition about resistance being associated with obstacles, even on an atomic scale. Now, the second important point was that what is voltage? And there, what I made a point of is that we should associate voltage with the electrochemical potential and not the electrostatic potential, because if you look at the electrochemical potential in a device like this, you see this drops at the two ends, which you could then associate with the resistances. Because this is like a series circuit. When a current flows, there is a voltage drop associated with each resistance. Current times resistance. So any time you see a voltage drop like this, you say there must be a corresponding resistance. Voltage drop here corresponds to that resistance. And this linear slope here, that is what corresponds to that channel. On the other hand, if you look at the electrostatic potentials, you see often it would be smeared out, and so it wouldn't have these-- this information that we talked about. It would tend to get smeared out by a screening length. More fundamentally, you see the electrostatic potential is not necessarily constant, even at equilibrium, because fundamentally what statistical mechanics, equilibrium, statistical mechanics requires is at equilibrium the electrochemical potential should be the same everywhere. So that d mu/dz should be zero. On the other hand, fundamentally electrostatic potentials don't have to be the same everywhere at equilibrium. And on the microscopic level, there is all kinds of electrostatic potentials, even at equilibrium. And so it is conceptually not very clear that you could-- conceptually confusing to associate that with current flow. SSlide 3] Now, in terms of the actual quantitative models that we use, that if you wanted to actually model how these potentials vary inside the device, what you would use is something like a Boltzmann equation for semi-classical transport, which is what this part of the course is about. Or if you are doing quantum transfer, you would use the equivalent which is this non-equilibrium Green's function, which is kind of like-- includes the quantum mechanics. And that is like that second course that comes after this, where we go into the quantum models. So these are what you might use to actually get quantitative results. So what I've shown here, for example, is a calculation of the electrochemical potential and the electrostatic potential, using this quantum formula, this NEGF. And you can see the electrochemical potential has these sharp drops, whereas the electrostatic potential tends to be more smoothed out. So these are the things that are used to benchmark. And the question I often get is, so we have a nice direct way of calculating things, so why bother with approximate concepts like electrochemical potentials or quasi-Fermi levels? Why not just calculate things rigorously and look at it? And that is why I think those are first-- I mean, when you work in a field, one thing you realize is that calculating something and understanding something are not quite the same thing. And in order to be creative, you kind of need physical pictures, or intuitive pictures. And that is where the qausi-Fermi levels help enormously. And I will try to summarize, give you a couple of examples in summing up. And that is how, when you see a new situation, long before you have calculated anything, you kind of have an intuitive feel for what to expect, and that is the value of the intuitive picture. On the other hand, it is very important to benchmark your intuition against the proper quantitative models and experiment. Because otherwise, of course, your intuitive picture will just lead you astray. 

[Slide 4] Now, one example that actually we didn't talk about in this unit much, but let me just point out, is that you know the general approach you have been using is, that we talk of a small section in which we can ignore the inelastic scattering, so all the inelastic scattering is in the contacts, and so different energy channels are all independent. So for a particular energy, you could think of it as a series of resistors. If you look at another set of-- another energy-- you've got another set of resistors, another energy, another set, and so on. And what you have ignored, usually, is the inelastic scattering in our thinking. Now, what does inelastic scattering do? Well, conceptually what it tries to do is bring different energies into equilibrium. So what it does is, you could think of it as a kind of vertical resistor. One that connects one energy channel with another. You see, we have ignored these, but if you are to include inelastic scattering, pictorially what we would be doing is kind of including resistors that go between energy channels. See, and these potentials here, then, of course, they could be one set of electrochemical potentials that is appropriate for this energy, another set for that energy. Because, you see, the channel is out of equilibrium. So the distribution of electrons is not necessarily given by a Fermi function. And so the mu is not necessarily constant. But as long as you use a mu that is energy dependent, you can always represent any function in this form. And so it is just a convenient way of thinking of the Fermi function, you could say. So you could say, well, I've got different electrochemical potentials at this energy and at that energy, and in that case, of course, the inelastic processes will cause a flow. And if you think about it, the model that ignores these resistors might give you a certain current. What these resistors will do is actually give you more current in general. That requires a little thought, though. But ordinarily what would happen is that if you had contacts to all these energies, uniform contacts to all these energies, then what you would expect is that point should be at the same potential as that point, as that one. Because these are all kind of similar resistors. And if so, putting a vertical connection between two points like this makes no difference. You are connecting two points that were already at the same potential anyway. So in that case, inelastic scattering won't make much of a difference. And that is why usually for low bias, ordinary situations, as I say, neglecting inelastic scattering. This picture that we have talked about gives you the exact results. The exact results that you get solving Boltzmann equations. Now, on the other hand, if you had say a nano-transistor at high bias, where on the left-hand side of the source, the band edge is somewhere here, so you do not contact these channels, you only contact the top channel, while on the right-hand side you have a large positive potential, so the band edge is down here, and you're contacting all these channels. Now you see you do not expect that potential to be the same as that potential. And so inelastic scattering will make a difference. So, at high bias in general, inelastic scattering does make a difference. Will actually give you more current than what you'd have calculated otherwise. And similarly, you could have more complicated situations where on the left-hand side you connect to this energy, and on this side, you connect to a different energy. Actually pn-junctions are a little bit like that. On the inside, you connect to higher energy channels, the conduction band, on P side you connect to this, a different set of energies, and in that case, again, the inelastic processes will obviously make a big difference. In fact, if you neglect the inelastic processes, there won't even be any current flow because there won't be any path from left contact to right contact. It is these inelastic processes that make current flow possible in such a structure. So the point I am trying to make is using this concept of this electrochemical potential, you can have a pictorial view of whether you expect, say, inelastic processes to make a difference or not for example. And, of course, you have to check against actual calculations and against experiment. But it gives you a way of thinking about things. 

[Slide 5] Now, the general things that we talked about in this unit are this general idea of quasi-Fermi levels. So you see, what I just explained is how you can have a different electrochemical potential for different energy channels. That we didn't talk about much. But what we talked about a lot is this idea of quasi-Fermi levels. The idea that in a channel you could have a different potential for right-moving electrons and left-moving electrons. And that is a very useful concept in terms of understanding current flow in small devices. 

[Slide 6] We also talked about this idea of spin potentials. That is, in this whole field of spintronics, people talk about two different quasi-Fermi levels for up-spins and for down-spins. And materials with spin orbit with coupling, I tried to explain how mu plus gets associated with a mu up, and mu minus could get associated with a mu down. And in general, how you could actually have four quasi-Fermi levels for up-spin moving to the right, down-spin moving to the right, up-spin moving to the left, down-spin moving to the left. That is, in general, you could have this mu, this quasi-Fermi level for particular momentum and a particular spin direction. And all this then gives you a very useful physical picture in terms of thinking about how current flows on this scale. And all this rich physics you miss if you say that well then the quasi-Fermi levels are too complicated, we'll just think in terms of electrostatic potentials. Because this is the physical picture that most people use. And because, as I said, our intuition is usually guided by certain physical pictures. And this picture, what I am trying to explain is, is really not general enough. Not rich enough to capture all the new physics in nanoelectronic devices that are being discovered, and will be discovered in the coming years. Because as we move along, there will be a lot more developments I believe, in terms of how you make contacts, the kind of materials you look at, which give you exquisite control over different  quasi-Fermi levels. See, the way biological systems have this control over, say, the potential of sodium ions and potassium ions and so on, similarly we would have similar control over different groups of stakes, spins, right-moving, left-moving, induction band, valence band, all kinds of things, and you need this concept of quasi-Fermi levels in order to have a pictorial view of how current flows. So of course you always have the Boltzmann equation, or NEGF as your quantitative benchmarks. The quasi-Fermi level is a way of thinking that helps you understand or catalogue what you are seeing. 

[Slide 7] So with that, then, guess we can move on then to the next unit of the course where we will be taking a little deeper look at some of these concepts that we have kind of glossed over so far. You see, I made this point earlier that this viewpoint here, what makes it simple is that it separates out the entropy driven things from the force driven things. That is, the idea is, you assume that the channel has-- is described by pure mechanics, knowing elastic processes. Whereas all the entropy driven things, this thermodynamics is associated with the contacts, whereas in practice, of course, it is all mixed up. But what I tried to explain is you could idealize small units like this, and that gives you a lot of insight in terms of how it flows. But the part that we didn't get into at all, because we didn't talk in depth about what is it this entropy driven things mean, for example. And that is what we really want to talk about in this last unit of the course. See you then.