nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.1: Introduction
========================================
 

[Slide 1] Welcome to our course, the Fundamentals of Nanoelectronics. This is the first unit of the course entitled, New Perspective. So in this introductory lecture let me try to explain what this perspective is that we're talking about. 

[Slide 2] Now we're all familiar with the amazing advances of modern electronics. You all know how powerful laptops and Smartphones have become, and what has driven this process, what has made it possible is this continuing miniaturization of electronic devices. That is, what I have here is like a very simplified schematic of a transistor, it contains a channel that's a conductive region, with two big contacts, called the source and the drain. As I said, very simplified schematic, but the point I wanted to make is that over the years what has happened is this device has got smaller and smaller. Why? Because the smaller you make it the more you can pack into a single chip. So today's Smartphone or laptop has like a billion nano transistors in it. And if we look at one of those transistors, this active region is actually like a few hundred atoms across, today's transistors, okay? Now where the field will go from here on, that's hotly debated, people have opinions, all kinds of new devices people are looking at, but that's not what we'll talk about in this course because generally speaking we really don't know the answers there, we do not know what kind of devices will actually win out in the next 10 or 20 years. So this course is really about the fundamentals, the basic conceptual framework for how we think about small devices, because that, hopefully, won't change for many decades to come. Now how do you think about current flow in a device? Well, when you apply a voltage there's a current, and you usually define something called a resistance, which is voltage divided by current. If we take the inverse of that you call it conductance. And what is well known is that usually this conductance will depend on the cross section of area of your conductor and inversely proportionate to the length, the longer you make it, the smaller the conductance, higher the resistance. And there is this material parameter called conductivity, which depends on what material is used in that conductor. So some things, like copper, conduct extremely well, other things conduct poorly. So what determines the conductivity? Well, usually the way we think about conduction, the way you learn about this in, say, freshmen physics, is that the way electrons travel inside your conductor is through a diffusive process, what sometimes people refer to as a random walk. That an electron comes in from one contact, goes along for a while, hits something, gets turned around, hits something else, gets turned around again, and so on. And that's how it eventually gets from the source to the drain. And based on this picture you see something called the Drude formula. So there'd be this expression for conductivity that you learn, again, in any elementary course. That conductivity depends on this number of electrons, this electron density, the mean free time, how long an electron travels before it gets, before it hits something, and something called this effective mass. Now this is what we might call the old perspective, you see? So this is not what we'll be doing in this course, although I'll occasionally be referring to it in order to connect up with what you may have seen before, okay? The perspective we will have in this course, we actually start from what happens in small devices. So usually what happens is historically our understanding progressed from the top, down, right? Because 20 years ago devices were up here and no one had any idea what the resistance of a very short conductor would be, but today the answers are well known and what I'd like to argue is it makes good sense to actually start from this end. That is rather than start here and then try to project that understanding down here, what this course tells us is we start here, we start at this end, where the flow of electrons is more similar to what you might call ballistic. Ballistic means like a bullet, as if it goes through from source to drain, straight without hitting anything. And in this case the conductor is now established can be written as this fundamental constant. You see q is the charger in the electron, h is Planck's constant, so this is actually a fundamental constant, which has the dimensions of conductance, inverse of resistance, so actually this comes to about inverse of 25 kilo ohms. And that times quantity, called the number of modes, and we'll talk about what that means as we go along. So this is what you'd learn if you took a course in mesoscopic physics. So this result got established firmly, say, about 25 years ago, and now you'd learn about it in a mesoscopics course usually. But this would usually be viewed as a separate result for small devices, whereas, for big devices you still do the traditional thing. That's what the normal view is, whereas, what we'd like to argue is that for big devices, too, you could actually just start from here and show that the conductance can be written as the ballistic conductance times this factor. This factor where the lambda is the mean free path. And L is the length of the device. Mean free path means the distance that an electron travels in a mean free time, see, before it hits something. Now, you see, if you had a very short device, length is very short compared to a mean free path, then you see you can drop that. And this factor just becomes one, and so the conductance is equal to the ballistic conductance. On the other hand, if you had a very long device then you could ignore the mean free path down here in the denominator, and now you get a conductance that goes inversely as length. See, just as you expect for long devices, see? So this point of view actually naturally leads up to long devices, as well. And if you compared the two you can see that the conductivity should be equal to the ballistic conductance times the mean free path. So that's this new perspective we are talking about, where you start from this end, define something on the ballistic conductance, which depends on the number of modes, which is a material property. And then you say, well, that's the conductance you would measure if you had a real short section of that material, like a real short copper wire, a real short silicon wire, this is what you'd measure. But then when you make it longer then you would have the usual behavior described by conductivity, and what is that conductivity? Well, it's the ballistic conductance times the mean free path. So that's this new perspective. So how do we obtain this result? I mean this is what we'll be doing, actually, in this unit. 

[Slide 3] Well, we start by saying that, well, in order to understand current flow, the first thing you need is the energy levels, the availability of energy levels inside the channel. But, as you know, the hydrogen atom there would be discrete levels, this 1S, 2S, et cetera. When you get to bigger molecules the energy levels get closer together, and when you get close to solid they're often very dense, you see? So that's what I've tried to show here, lots of states, and typically there's an energy range where you have certain states, some energy range where you don't have any states, and then again you have states and so on. So I'm just schematically showing what these energy levels might look like. And then you have a contact, and the idea that this contact has again lots of states, continuously distributed, and the details of the contact usually wouldn't matter too much. That's because when in terms of the flow of current, you see, the contact is a very big region, whereas, the channel is this narrow bottleneck that controls the flow. So usually the details of this doesn't matter too much. What controls the flow is this narrow region, okay? So in this contact you have all these states, and they're all filled up to a certain level, which you call the Fermi level of this electrochemical potential, which I'll show with the Mu. And the idea here is that electrons, as you know, left to itself they want to occupy the lowest energy state, but then they all cannot just go into the lowest energy because of the exclusion principle. So if you have a million electrons in here you have to have at least a million levels to accommodate them. So this Mu tells you how far you need to fill them up in order to accommodate all of them. Now if we go to the other contact there, also, you have a similar situation, except that everything is lowered by the amount of the applied voltage. It is because I have applied this positive voltage on the drain side, the positive voltage lowers all the energy levels by an amount equal to this q times V, because this is what you call an electron energy diagram, so positive voltage lowers the electron energy. So you have a similar picture then here, but then it is lowered by qV. Now why does current flow? Well, the way to think about it is that this contact wants to bring all these energy levels in the channel into equilibrium with itself. And in order to bring it into equilibrium it likes to keep these states filled because, after all, all states are filled up to here in the contact, so you'd like to have at equilibrium it would like to have the same chemical, electrochemical potential everywhere. In general, just as heat flow is driven by temperature, that is temperatures are different heat flows. Similarly, with electrons what drives the flow is this electrochemical potential. If they're equal then you have equilibrium. So this wants to fill them up. On the other hand, this contact you see has its Fermi level or electrochemical potential down here, so it really would like to keep those states empty. And so what it does is it pulls them out. Well, once it's pulled out, of course, this contact isn't very happy anymore, so it fills it up again. And then, again, this contact isn't happy, so it pulls it out again. And this just goes on forever, and that's why current flows, you see? New electrons come in and old ones go out, another one comes in, goes out, and then they go out into the circuit and flow into the battery, outside. So using this viewpoint then we will obtain an expression for the conductance, and we'll see how we do this. This will be in the second or third lecture as we go along, but the point is you can kind of see the end result is fairly intuitive. You see, the conductance depends on density of states, density of states around the Fermi energy. The density of states down here, it doesn't matter. What matters is the states up here, see? That's the density state, and t is the time it takes for an electron to get from left to right. So that's kind of what you might expect if you think of electrons as going down like cars down a highway. What is the total flux of cars across the highway? Well, it depends on how many lanes you have on your highway, that's the density of state, and it depends on the speed limit, how fast these electrons travel, how far the cars go, how fast the cars go. So we kind of see that that's intuitive, that's what will determine the conductance. But this approach, you see, is very different from the standard perspective, you know, the one that you usually would learn in freshmen physics, this Drude formula. And let me just briefly describe how that's done, just so you can see the difference. So usually the way it starts is by saying that electrons move because there's an electric field, the field of force, and so you have Newton's Laws, mass times acceleration, this is called the drift velocity is equal to the force that an electron feels, q is the charge on an electron, F is the electric field. And then you say that, well, this is true in vacuum, but in a solid there's all this frictional force, kind of a viscous drag, which you can describe with a mean free time. So there's the momentum, divided by time, and this is like the frictional force. And at steady state nothing is changing with time, and so you can drop this term, and so you'll get an expression for the drift velocity. And this factor here is usually called the mobility, and that is something people often carry in their head, that when you talk about any material the first thing you ask is what's the mobility? That's the point. And then from this you usually calculate the current by saying current depends on electron density times the velocity, that may not be obvious if you haven't seen it before, and we'll discuss it later in a different a context anyway. But at this time I'm just going to give you an overall flavor of how one gets the Drude formula. So once you do that you get this current, it depends on the electric field, which determines the voltage, through this factor, and that's what's called the conductivity. But the point you'll note here is that the reason the electrons move in this viewpoint is because of the electric field. Now that causes a rather, an interesting problem, though, because if you think about it if electrons are driven by electric fields then you'd think that even these electrons down here should start moving when there's an electric field, and that is, of course, wrong. I mean no one claims that's true. Everyone knows that current flow is because of the electrons that are up here, close to the Fermi level. And electrons deep down, if they're deep down there's lots of states, and I've just shown the tip of the iceburg, usually there's lots of other states down here, the core electrons, but they claim no role at all  in current flow, see? That's well known, that's established. So the way usually people argue is that, well, you see the end that you use here is not the total number of electrons, it's the number of free electrons, okay? Why? Well, because the argument goes that the filled bands, all these bands are filled, and these filled bands do not conduct. And this is something I over the years have found quite difficult and non-intuitive to explain to someone, you see, why like filled bands do not conduct. And I think even last year in physics today, I remember reading a letter from a reader as to what exactly does this mean, where does this come from? So these, of course, raise a lot of questions, that when you think of electric field as driving the current it raises questions like this. On the other hand, the way we've described it, you see, it's pretty clear why these states do not contribute to current flow. Why? Because the reason we have current flow here is because this contact wants to fill it up, this contact wants to empty it, so this keeps putting electrons in, and this keeps pulling it out. What about the states here? Well, this contact wants to fill it up, this contact also wants to fill it up, and so it stays filled, end of story. So all these states are just filled, both contacts are happy, and nothing else happens, that's all, you see? So from this point of view it is perfectly clear why conduction takes place only in the narrow range of energies around the Fermi energy, okay? 

[Slide 4] Now from this point of view, as I said, we obtained this expression for conductance, which you will see leads naturally to the ballistic conductance that I talked about. And, in addition of course, not only do we get the ballistic conductance out of it, we will show that you can also use this viewpoint for diffusive transport, you see? And get an expression for conductivity. Now this is a different expression, it doesn't look like the Drude formula. Here we have like free electron density and things like that, whereas, here you have the density of states, density of states goes to the Fermi energy. And you'll have something called the diffusion coefficient. So this is the expression that will come out naturally from here. So is that a new expression? Well, no, this is actually well known. It is just that people normally don't carry it in their head. Why? Because to derive this expression normally requires advanced formulisms, like the Boltzmann formula or sometimes you refer to it as the Kubo formula. So it requires these advanced formulisms and so it's usually not in like chapter one of any book, it's deep down somewhere, it's chapter 10 or chapter 13 somewhere, and most people don't quite remember this. So what they usually carry in their heads is this one. But everyone agrees that this is really much more general, this is really what you should be using in general, you see? 

[Slide 5] So how did we get this very general result so easily? Because, as I said, usually it takes advanced formulism to get there? And the way we get there is by using this concept of elastic resistor or you could call it a Landauer resistor, after Ralph Landauer, who kind of pioneered this way of thinking, you know, long before it became experimentally relevant. But in small devices it seems clear that often when an electron goes from left to right, you know, when it goes from left to right there's always some heat dissipated, as you know. And usually when you run currents things will get hot, you know, that's how heaters work, that's how lightbulbs work, but that heat in small conductors often most of the heat will actually be dissipated in the contact rather than in the channel itself. And there's good experimental evidence that some of these very small conductors, if all that heat were actually dissipated here it would have burned up because this is a small thing, it can't get rid of all that heat, but it doesn't burn up because the heat is actually in the contacts, which are big things that can get rid of the heat, you see? So this is what I'd call an elastic resistor, you see elastic in the sense that the main channel, the channel that actually determines the resistance, as I said, the resistance is determined by this channel, which is the narrow bottleneck part, it determines the resistance, but the heating associated with that resistance occurs in the contacts. And for ballistic transport that kind of sounds natural, you know, after all when a bullet goes through a medium, the medium doesn't get heated up, what gets heated up is whatever the bullet hits, so it's kind of similar. Now this kind of a device, this elastic resistor is relatively simple to understand and model, and the reason is that there's a clear separation between these two types of processes. The one type, which you could call mechanics, and the other type, which you could call thermodynamics. Let me explain, see, in physics these are two very distinct branches that develop independently and separately. So mechanics started with Newton, understanding planetary motion, the frictionless motion of planets. Several centuries later heat engines came along, people realized that heat was a form of energy, and that's led to this science of thermodynamics. And these are very different things. In mechanics everything is driven by forces. In thermodynamics things are driven by something a little more abstract that people have, it is harder to grasp, this entropy, see? But what makes devices very difficult to understand is that usually these two things are all mixed up, see? And to describe devices, usually the starting point is for semiclassical trans pictures is this Boltzmann transport equation because late in the 19th Century, you see Boltzmann put together this basic Newton's Laws for Mechanics with these entropic forces and developed what's called the Boltzmann transport equation, which is now widely used, it's like the cornerstone of all semiclassical transport theory, people use it for neutron reactors, it's used for electronics and devices and so on. And if you want quantum transport you'd have to use something that's like a quantum version of that, instead of Newton it's Schrodinger's equation, to which you have to add entropic forces and that gets to what you call this non-equilibrium green function method, okay? But in this course we stick more to the semiclassical picture. So this is the Boltzmann equation that we'll normally use, you see, to describe devices. And when you talk of small devices, though, because the mechanics is separated from the thermodynamics it makes it much easier. You'll see that the results come out relatively easily, in fact, so easily that you don't even realize that you're doing something profound, see? So, well, I get it, for small devices you have this simple picture, but how do you handle big devices? Didn't you just say you'd get me this expression for conductivity that applies for big devices, you know, this conductivity for diffusive transport? Well, when it comes to big devices the way we think about it is as if it's lots of little nano devices in series, so mentally, conceptually think of it as electron goes elastically from one point to another, loses some energy, goes again elastically, again loses some energy. So this is an approximate picture because in reality, of course, these black things on the red, things are all mixed up. But conceptually you can picture it as if it's a series of such things, you see? And what we show is this is the approach that we are using to get results for big devices. Now you say, well, but doesn't that mean this part has to be ballistic, so how are you getting the diffusive part into the story? Right? Because, as I said, usually we have this ballistic transport, but then to get conductivity we're getting it for diffusive transport, but the point I would like to make is that this part, the transport doesn't need to be ballistic, it just needs to be elastic, you see? The difference is elastic means there is no exchange of energy, the electrons do not give up any energy to the surroundings, do not heat up the lattice. Ballistic means there is no loss of momentum, but you could have electrons knocking around, like this, losing lots of momentum, but not losing any energy, so all these could be elastic processes conceptually. And that's the picture we use to get the results for diffusive transport. It is elastic over one of these small units, but not necessarily ballistic. And what we'll see is the results we get agree exactly with what you'd get from a more rigorous theory, which are well known from the Boltzmann equation for low bias, and for high bias this still gives you a nice physical picture on how to think about it, and that's something we'll talk about in the next unit of the course a little bit. 

[Slide 6] So, with that introduction then, I think we are ready to get started. So the next topic we'll talk about, get started by discussing these two key concepts that you need in order to discuss current flow. First is the density of states, as I mentioned, and the second is the Fermi function. Thank you.