nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 2.10: Summary
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[Slide 1] Welcome back to Unit 2 of our course. I guess it's now time to sum up what we did. 

[Slide 2] So let me just start by briefly reminding you what we have done in Unit 1, because that's kind of the backdrop that we started from, in Unit 2. So in Unit 1, we showed that you could write the current in this form, for an elastic or a Landauer resistor. And it's this difference of the two Fermi functions. And you can use, then this low bias approximation to get an expression for the conductance, current divided by voltage, which is like an average over energy of a conductance function. And that conductance function depends on the density of states and the time it takes for an electron to go through the channel. When then showed that you could start from here and get a general expression for the conductance that actually applies in both ballistic and diffusive limits. That is, if you had a ballistic conductor, that means the length is very small compared to a mean-free path, so you drop that. And then you have the ballistic conductance. If you had a long device, then the length is large compared to a mean-free path, and so you drop the mean-free path, and then you have a conductance that goes down with length of the channel. Just as Ohm's law requires. And this way of looking at it then, gives you a different expression for conductivity, which is what we call the new perspective. And to contrast that, with what you normally learn, starting from freshman physics or any textbook, usually is this Drude formula. So in this new perspective, the conductance is ballistic conductance per unit area times the mean-free path. Whereas in the Drude formula, it's like free electron density effective mass and this mean-free time. And one of the objectives we had, in this unit was to try to connect the two. And in order to do that, we had to introduce a specific model for these energy levels. 

[Slide 3] That is, in Unit 1, we just discussed everything in terms of a general density of states. We said, "Well, let's say we know the density of states. Then how, what should be the conductance? What should be the current? et cetera." Now what we did in this unit is we say that we adopted a model for density of states that's this energy band model. The band structure model, which is widely used for crystalline solids. And that's where it introduces this concept of effective mass and things like that, so that we can actually connect with the root formula. So we started in Lecture 2, with a, what's called an energy momentum relation. That is, in vacuum, it is well known that you can relate energy to momentum. That is energy is potential energy plus kinetic energy and kinetic energy is proportional to that momentum squared. So that's in vacuum. Now, what's not at all obvious is that, in a complicated object like a solid, you can still write the energy in terms of the momentum. You can still have this energy momentum relations, but with some caveats. This is one of the very early insights in solid state physics, when people realized that in crystalline solids, you could express electron energies as a function of momentum. But then, the energy momentum relation need not be parabolic, it could be highly non-parabolic. That is, E is not necessarily proportional to p squared. It could be a fairly complicated function. And the other important caveat is that you don't necessarily have just one band, you could have multiple bands. And of course, for this discussion, we're sticking to one band of that energy momentum relation, actually. Now, so that's what we introduced in Lecture 2. And then we had Lecture 3, where we introduced another important concept and that is this N of p, which tells you how many states are available, whose momentum is less than p. And there is a specific method, a specific scheme that people use to count states, that is how many states are available, using what's called this periodic boundary condition. And as I mentioned before, that the periodic boundary condition is not necessarily what the real solid obeys. But, the argument is that usually when you are talking of big solids, the actual boundary condition doesn't matter, so what you use is whatever is mathematically most convenient. Anyway, so this method of counting states, using periodic boundary conditions is widely used, and that's what leads to this function, here. And once you have these two inputs, you could combine it to get these radius quantities of interest, namely density of states, that's Lecture 4, number of modes, Lecture 5, electron density, that's number 6. And once you had these quantities, which enter the expression for conductivity, we were ready to connect it to the Drude formula. So that's what we did in Lecture 7. Then in the last two Lectures, 2.8 and 2.9, we introduced this new element into it, namely the third terminal, this gate voltage, which of course is a very important part of a transistor. 

[Slide 4] So that's something we introduced only in this last two lectures. Up to here, we are basically just talking of ordinary two terminal device. Here, we introduce this gate voltage, and once you introduce that, it also leads you naturally to this important part of the story, which is the potential energy inside the channel. That is, this gate voltage can be used to control this potential energy in the channel so that the band can be moved up, if the potential energy is positive, which means if the gate voltage is negative, or it could be moved down and that of course is the essence of transistor action. Transistor means the resistor whose resistance can be changed with that third terminal. And the way it is changed is, the gate voltage controls the potential energy in the channel and moves it up and down. So that's this U, which depends on the potential on the gate, depends on the potential on the drain and also depends on the change in the electron density inside the channel. That's this electron, electron charging. Electron, electron interaction. And this U has to be included when you actually calculate the current. That is, so far, our expression for current had G of E here, but now, you have to shift the conductance function also, this E minus U. And how do you find this U? Well, this tells you what U is, as a function of N and then we have another expression for N, as a function of U. Because N is the number of electrons, which depends on the density of states, and how it is occupied. So we have N as a function of U, U as a function of N and the two have to be in general, solved self consistently. You see, because you want the simultaneous solution of those two equations and that can be done in a straightforward way numerically. And in simple cases, you can even do it analytically. But one important thing is, all device simulation usually involves this self-consistent calculation. And one part of it is related to density of states, the part we've been focusing on, but the other part, is also essential and this is what kind of describes the electrostatics of the problem. And in many ways, the transistor is kind of an electrostatic device, where the most important part of its design is really its electrostatic design. Now, one important parameter we introduced here is this U0, which is what's often called the single electron charging energy. That is, it tells you how much the potential energy changes if we add one electron. Now usually in big devices, one electron doesn't make much of a difference. And if you add one electron, maybe the potential will change by one micro-electron volt. But of course, you still have to worry about this change, because usually, under operational conditions, you might change the number of electrons by like 10,000 or 100,000. So even if one of them changes it by a micro-volt, micro-electron volt, then 100,000 of them would change it by ten milli-volts or... so, ten milli-volts or one hundred milli-volts. So the actual change can be sizable. On the other hand, now it is there are small devices, very small devices where the single electron charging energy itself can be pretty big. So as a device gets small, of course, one electron makes a much bigger difference, in terms of what its potential is. And so the potential energy could be significantly affected and this could be big. And that leads to a whole new class of phenomena that we haven't even talked about. These are what are called the single-electron charging effects or Coulomb blockade. Those are important when this U zero is a relatively large number, you know, on the order kT or more. 

[Slide 5] Now, the other point I wanted to mention then is, that usually as I said, you could use this model, this model that we just described, to understand the operation of a transistor, you know, how the gate voltage controls it, how the current voltage characteristics, what they look like, and so on. But if you want to go beyond that, then what you have to take into account is that the potential inside the channel is not just a single number, like I mentioned, but actually could be varying spatially. That is the part that we have kind of completely ignored, whereas a more serious model of a device would have to include that. Now how do you include it? Well, instead of this equation, this little algebraic equation, you'd actually have a differential equation, that's this Poisson equation. So Poisson equation would describe this electrostatics. And sometimes you may want to have corrections to that Poisson equation based on this exchange and correlation, various quantum effects that people talk about. So people talk about corrections to this, but this is the basic electrostatic or the Hartree potential. That was what would replace our simple algebraic equation here. On the other hand, for the current and electron number, and so on, that is where usually you'd have something like this drift diffusion equation. There's an equation that says current is proportional to the slope of the electrochemical potential. And so we'd have an electrochemical potential inside the channel that's changing and you could calculate the current, this is continuity equation which says that in a one dimensional device, the current does not change spatially, it's constant. And all of these, of course, are based on the most general thing in semi-classical transport, which is this Boltzmann Transport Equation. That's usually the cornerstone of all semi-classical theory. So you might say, well, but all this of course, people have been doing for many years, long before nanoelectronics. So what new did we learn? As far as nanoelectronics goes? Well the key point, is of course, this L plus lambda part of it. It is usually thirty years ago, you'd have said conductance is sigma a over L. Whereas, what we know today is that as L turns to zero, actually the conductance turns to a constant number that's this ballistic conductance. So you need this L plus lambda. And if you invert this, you could write the resistance as a part that is proportional to the length, and a part that is constant. That's this ballistic resistance. So that we have done in the last two units, you know, in this unit and the previous one. What we haven't talked about though, is the origin of this ballistic resistance. That is, what we say is the resistance has a ballistic part and a part that's proportional to the length. And this is actually what comes out of the Boltzmann Transport Equation. So in that sense, it is all hidden in there. It's just that normally people don't worry about it. Normally, people are kind of almost unaware of this. Still, the modern developments in nanoelectronics. And what we understand today is that, in a channel like this, this part that is proportional to the length, you could view as the channel resistance, whereas the constant part, you could view as an interface resistance, which appears at the two interfaces. So when we go back and use the Drift Diffusion equation, like we always do, for device simulation, the important piece of new knowledge that needs to go in there is that you have to add the interface resistances. Now of course, if you're using the Boltzmann equation, and you're doing it correctly, these would come out automatically. If you're using the drift diffusion equation, you have to remember to add them. I mean, that is the important thing, that what everyone realizes is, that if you have a small conductor, or any conductor, you always have contact resistances or interface resistances at the end. What people didn't realize is that there's a fundamental limit to that. Usually you think that if I can improve the quality of my contacts, I can get rid of all contact resistance. What we know today is that that's not true. There is a fundamental limit, this ballistic resistance and so, your roadmaps of where you want to go, kind of devices you want to design in future, has to take those fundamental limits into account. Now the part we haven't talked about then, is that this extra ballistic resistance is actually associated with that interface. 

[Slide 6] And in order to understand that, we need to talk about how the potential changes inside the device. Because when you run a current, as you know, any time there's a resistance, there's a voltage drop across it. And so that's what we'll be doing in Unit 3, that is, what and where is the resistance or equivalently, what and where is the voltage.