nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 2.1: Introduction
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[Slide 1] Welcome back to our course on the Fundamentals on Nanoelectronics, this is Unit 2, the introductory lecture. 

[Slide 2] Now just to remind you what you were doing earlier we considered the small device and we wrote down the current, the expression for current and we then linearized it to get an expression for conductance and the conductance is like that, like this average, this energy average around the chemical potential of the conductance function which is a function of energy. What does the conductance function look like well it depends on this density of state and the time that it takes for the electron to get through and starting from that expression we obtain this relationship of how the conductivity is related to this ballistic conductance. That is what we call the new perspective, you see this new perspective where we write conductivity as ballistic conductance times this mean free path. We can contrast this with the old perspective or the one that you have probably seen in freshman physics, most text books start with something like this the Drude formula and here you'll notice the conductivity depends on what you call the free electron density and in Unit 1 we talked a little bit about the subtleties involved here and there's this effective mass and there's the mean free time. Now if you compare the two you'll notice this interesting correspondence though that there's these two things here, there's this n over m which is like independent of scattering processes it's like electron density effective mass, things like that. And correspondingly we have here this ballistic conductance which is kind of like that. And then there is another term which depends on scattering. This is the mean free time and that's the mean free path and the two are related of course because mean free path is the distance, average distance an electron travels in a mean free time. So in this new perspective then this ballistic conductance plays a pretty important role and this ballistic conductance, the expression for it as we discussed is this density of stage times the average velocity divided by the length and we also showed, or rather wrote it in the form of, in terms of what's called this number of modes. One of the important things you have learned in mesoscopic physics and nanoelectronics is that in really small devices that M is an integer so that ballistic conductance is actually quantized in terms of this conductances quantum, q squared over h. So these are the things we talked about in the last unit, now what we want to do in this unit is adopt a specific model for this density of states. As you can see the density of states plays a very central role in determining how well something conducts. As I said basically of course that material conducts if there is a large density of states around the equilibrium chemical potential of this Fermi level. But so far we have not talked much about this density of state, where does it come from, how do we model it and that's kind of the purpose of this unit is to talk about the specific this energy band models which give us this density of state and then try to compare it to the Drude formula and connect it up. 

[Slide 3] So what's this density of states? Well, I think in Unit 1 I had mentioned that the basic idea is that if you say had a hydrogen atom then you know you have discrete levels that are spaced by large energy like the spacing could be few electron volts between energy levels. Now as you get to larger things then the energy levels get closer together so you have lots and lots of allowed energy levels. In some relatively big conductors where you have hundreds of atoms it's convenient to talk in terms of what's called the density of states. So this axis is energy but density of states tells you in a given energy range, how many states you have, so it's the number of states per unit energy. Now how do you measure this? Well usually to one way is photoemissions experiments, hit it with light and see how much energy is needed to actually knock and electron out of the solid. So that's how you would experimentally measure it. Theoretically how you would model it, well the general models would all start from the Schrodinger equation, you see something that takes into account the wave nature of electrons. Now what we'll be doing in this unit though is a relatively simple model based on this idea of energy momentum relation. So you see we have these electrons in a solid and a solid is a fairly complicated object but one of the very important insights in solid state physics, an old insight like back in back from like 1930's is that electrons in a solid behave almost as if they are in vacuum but with a different energy momentum relation for a different mass. These things we'll talk more about but the point is that our starting point for this discussion and that's what we take up in the next lecture is what's called an energy momentum relation. Electrons in a solid, energy of electrons in a solid can be plotted as a function of the momentum. Note that this curve might look a little bit like that but it's really two different curves totally because here we are plotting energy versus momentum, momentum is a vector that can in general point in any direction. Here we have just drawn say, the momentum in a certain direction where it can be positive or negative and the energy is a function of that momentum. But is here what we have is energy and this axis is the density of states, number of states per unit energy which of course has to be positive. So there is no negative side to this graph it's all on this side. Superficially they look kind of similar but it's very different and one of our objectives is actually after we introduced this idea of energy momentum relation is to discuss in the next three lectures after that is how to translate that curve into that curve, it is how to start from energy momentum relation and get a density of states or this number of modes out of it, how to do that translation. That's kind of what the next three lectures are about. 

[Slide 4] Now another thing that we want to do here is connect up to this Drude formula, the one that I mentioned like what we might call the old perspective. And as I mentioned in Unit 1, the central difference between the two perspectives, here it is when you use this Drude formula the conductivity depends on the electron density and what's this electron density well the idea is if you have a certain chemical potential that tells you how far the levels are filled then this n is the number of electrons in that back below it and we talked about some of the subtleties involved like what happens when this electrochemical potential is up and near the top of the band for example. But the point is that Drude formula relates conductivity to the electron density whereas this ballistic conductance which we as we said plays a central role in our thinking depends on the density of states or the number of modes which is only at a given energy. I mean you have to average it over energy but at low temperatures it's just that energy and at higher temperatures there's an average involved. But philosophically a very different thing you see one is about a value at a certain energy, the other is about this integrated number of electrons. What we'd like to do in the next couple of lectures here is to connect up those two types, we'll talk about this electron density and why the new perspective that we're talking about usually agrees with what you get out of Drude formula also. But then there are times when they may not agree and of course I'd like to find out that what we are talking about is really much more general. That follows from the general Boltzmann equation. 

[Slide 5] Now, the next item that we'll talk about after this is a little bit about an actual transistor, that is as I mentioned before that one of the things that's driving the microelectronics industry is the ability to make devices smaller and smaller and the most important devices of course is nano transistor. A transistor is basically what we have shown here, you know there's a channel with two contacts you put a voltage, you get a current but I have kind of oversimplified because what we have left out a very significant thing I have left out is a third terminal here. Now this terminal actually is insulated from the channel so in that sense ideally no current should be flowing. Of course in practice there are some leakage currents but the point is in an ideal transistor there is no current flowing in this direction. All that this voltage does it to move the energy levels in the channel up and down, that is you see you have this chemical potential here and you have all these energy levels in the channel if you put a positive voltage it gets pulled down and if you put a negative voltage everything gets pulled up. Since conductivity depends on where the chemical potential is you see you can change the conductivity. That is if you raise it enough you can stop the current altogether if the chemical potential happens to get into that gap because there is two ways of thinking about it, one is chemical potential is fixed, we move the energy levels up and down or you could think the energy level is fixed, we move the chemical potential up and down. So essence of the transistor of course is to be able to control how well the channel conducts, it is called the resistivity in fact in the transistor you are changing that resistivity by several orders of magnitude. Through this gate voltage, so usually you'd have a current versus voltage relation looking something like this for a transistor where depending on the gate voltage you see, as you made the gate voltage more and more positive it will conduct better so you will get more and more current. So far usually of course we have been talking about this low bias regime that is when you have a relatively low voltage across the entire device so we are talking about this regime, that's what we call the low bias conductance and so we have been talking about this dI/dV around the origin. So what will we be doing in this last two lectures is first in one lecture we will talk about how the gate voltage controls this conductance, this low bias conductance and that will introduce a very important concept of quantum capacitance and then in this last one we'll talk a little bit about the shape of this current voltage characteristic. You'll notice these have this property of what we call saturation, it is as you increase the voltage the current wants to get saturated and not increase as fast anymore. That is a very important characteristic of transistors, that is ideally actually you'd want to saturate perfectly but that is proving more and more difficult to do as you go to smaller devices and that is something we'll talk a little bit about in the last lecture. So that's then broadly what we want to do in this unit and so we are now ready to move on 

[Slide 6] to the second lecture starting from this energy momentum relation.