nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Scientific Overview ======================================== [Slide 1] Welcome to our course on the Fundamentals of Nanoelectronics. In this lecture I'll try to give you a feeling for the kinds of things that we'll be talking about. Now you're all familiar with the amazing things that today's smart phones can do and one of the things that makes this possible is that one of these devices has about a billion nanotransistors in it. Now how do you fit a billion nanotransistors into something that, say, 3 centimeters by 3 centimeters? We can do the arithmetic and we only need to do it approximately. You'll see you'll need to fit about 30,000 each way. You see? What that means is that each one of these here would have to be about a micron on the side. So micron by micron. So what I've shown here is a very simplified schematic of a transistor. What it consists of is this active region called a channel with two contacts, the source and the grain. And this is very simplified. I've left out a very key ingredient which is the third terminal called the gate. But the point is that this whole thing, this transistor, then should fit in to a region about a micron by micron if you want to have a billion of them in this chip. Now if the transistor takes up a micron then the thing is this active region is actually even smaller, like a factor of ten or more. So today's devices if we looked at the length of the channel, they're about a few tenths of a nanometer. A few tenths nanometers. And atomic distances are kind of down here. You see these length scales go by a factor of ten. So you can easily see that today's active regions have like 100 atoms. You see? Hundreds of atoms. Now -- And there's a lot of debate as to how long we can continue this way. Can we keep making things any smaller? But that's not what this course is about. We'll not be talking about specific devices or where the field is going. We'll be talking more about the general principles or the fundamentals of nanoelectronics. That is, how to think about current flow [Slide 2] in nanoscale devices because one of the things that's happened in the process of making devices smaller is the very nature of current flow is kind of has changed from what it used to be thirty years ago. You see thirty years ago the way you think about current flow, this is what experts usually carry in their head, is that an electron comes in from a contact like the source, goes along for a while, hits something, turns around, hits something else, turns around, and so on. That's what you call a diffusive random walk. And that's the motion, that kind of picture people think in terms of. And based on that you'd usually learn about current flow in, say, freshman physics courses or in any introductory course. You'd learn about this Ohm's law that resistance can be written as a resistivity which is a material property times the length divided by the cross sectional area. And then you'd learn about this Drude formula for the resistivity or its inverse is called the conductivity. So this is the kind of thing you'd normally start from. Now the other important thing that you learn in any mesoscopic physics course, that is if you took a course about small devices, then you'd learn that in small devices the current flow is often ballistic. That is like a bullet. And when you have ballistic transport then the resistance can be written as this quantum of resistance which is a fundamental constant like Planck's constant. That's the charge on an electron. And this h over q squared comes to about 25 kilomhs. And the resistance of a ballistic conductor is 25 kilomhs times 1 over M where M is this number of modes. And that is a very important concept we'll be talking about which has emerged from this field of nano electronics and mesoscopic physics. And usually you see these two are two very different things. This is kind of what you learn in a traditional device course or a traditional solid state physics course. This is what you learn in a nanoelectronics course or in a mesoscopic physics course. What I'd like to describe to you, though, is a viewpoint that these are all connected. That is the perspective I want to convey, this new perspective, is that in general you could write the resistance in this form. You see if you have a short device which means the length is much smaller than a mean-free path then you see you can drop that term because this is less than 0.01. Well, compared to 1 you can drop it. And then you have the ballistic resistance. On the other hand, if you have a long device, hundreds of mean-free paths long, then you see you can drop the 1 because 1 plus 100, well that's almost 100 anyway. So and then you have a resistance that's proportional to length and that then connects up to the Ohm's Law. In fact it gives you a whole different view of perspective on Ohm's Law. So the point I'm trying to make is then this is the new perspective, that what is the resistance of a device? You start from the ballistic end and say the ballistic conductor has this resistance and when you make it long you take the ballistic resistance and multiply it by that factor. That's the new perspective and what we'll talk about as how -- Where this comes from, how do you connect it to the Drude formula and things like that. [Slide 3] That's what we'll be doing actually in the first two units of this course. That is putting this discussion kind of on a solid footing, filling in all the details. In the third unit of this course we'll discuss actually a very important conceptual point and that's this. You see if you look at this resistance you notice there's two terms. One part that's proportional to the length, one part that's independent of length. Now the question is, "Physically where are these resistances located?" What I mean by that is you might think that the part that is proportional to the length, that should be associated with this black region. You see that's this -- You know that length is the length of this channel. So since it's proportional to the length of the channel you'd think this part would be associated with this middle whereas this constant is actually associated with the two interfaces. So the overall thing you could view almost as a series combination of two interface resistances and a channel resistance. Now this though is not at all obvious. This is the point that takes a lot of discussion and causes a lot of confusion. And the reason is that mentally people associate resistance with heating. You know, after all, how does a light bulb work? Well, it's basically a resistance. You run current through it, it gets hot, and that -- And it omits light. So anytime there's a resistance, you run current, it gets hot. And so you'd think that if I draw a picture like this we must expect that when I run current through it the heating in this region will be proportional to this resistance, the heating in that region would be proportional to that resistance, and so on. But that's not at all true. What's clear now is that in nanoscale conductors the heating is all over the place. In fact, the common paradigm people use often is that in a ballistic resistor actually all the heating occurs in the contacts. And not in the channel itself. So heating is usually all over the place. So based on that you couldn't have drawn a picture like this. So you might say, "Well, why do you want to draw a picture like that?" Let's just say the resistance is everywhere. Well, but that is something that goes against our intuitive feeling and that is this, that what determines the resistance of a structure like this, it is this channel region. Why? Because the contacts are big things. The channel is this narrow thing connecting them. That's what determines the resistance. You know, just like I'll often use this analogy of cars on a highway. You could have hundreds of lanes coming in, but if in the middle region you have only two lanes then it's those two lanes that control the traffic flow. You see? Something like that. Electrons -- The rate of flow is really controlled by the narrow region which means if I were to punch a hole in the middle of the channel the resistance would go up. No question. Just as you'd intuitively think, you know. Electrons would have a tough time getting around that. But does that mean that there will be a lot of heating right there? Not at all because heating means energy has to be transferred from the electron system to the solid. That is this collection of atoms. And so the atoms can start jiggling. That's what heat is. So that transfer has to occur. And if you have a hole like that then of course there's nothing there to transfer the energy to. So a hole would clearly increase the resistance, but doesn't necessarily involve any dissipation. So the point is that mentally if you want to draw a picture like this where the resistance corresponds to your intuition you have to go beyond this picture that's almost ingrained in our head, that resistance is associated with heating. Instead we have to associate it with voltage drop. That is how does the voltage change from left to right? Because when you run a current through a series circuit like this what you all know is the voltage drop in different regions is proportional to the resistance of that region. It's this I times R. But this brings up many subtle conceptual issues because -- And that's what the third unit is about, that what and where is the voltage. What is it that you should identify as the voltage and how does it actually change inside the structure. That's what we'll talk about here. Now once you accept that resistance is associated, not quite linked, intimately to the heating that it causes [Slide 4] that leads naturally to the viewpoint that we're using here which is we idealize this picture. You know, as I said, when a current flows some of the heating's in the channel. Some of the heating's outside. But the idealized picture that people use is something we call this elastic resistor. Or you could call it a Landauer resistor after Landauer who first proposed this as a way of thinking about current flow. The picture is if we think that electrons going from left to right here do not lose any energy at all, and all the energy dissipation occurs in the contacts. So that's the idealization. Right? So inside they may exchange momentum, but not energy. That is, as you know, just because an electron changes its direction of motion, momentum, doesn't mean the energy has changed. You could keep the same magnitude of momentum and still turn around. So inside the point is you assume that no energy is exchanged. Energy is all dissipated in the contacts. Now if you assume that, that's what we'd call the elastic resistor, it leads to a very clean description of current flow. And the reason it's very clean is that you see we have kind of separated two very different types of physics. You see physics has these two major branches which evolved independently. You know there was this mechanics which started with Newton's laws looking at the motion of planets, the frictionless motion. And then a few centuries later heat engines came along and people understood that heat was a form of energy and that's what was thermodynamics. And it took all of the nineteenth century to put this all together. You know, culminating in the work of Boltzmann. And what makes transport such a difficult subject, transport meaning this flow of electrons or any other entity, is that these two processes are all mixed up. This force driven things and what you might call the entropy driven things. And what makes this model particularly easy to understand is that the two processes are separated. And so you can take care of these entropy driven processes in a very elementary way. Indeed, so elementary that you often forget what that -- How profound a thing you are doing here. See? And this is what we use in most of the course. So in the fourth unit what we do is delve a little deeper in to these entropy driven things to give you a feeling for what makes entropy so different or heat so different from normal type of mechanics. That is, you see we all understand that if you put a battery in here, you take energy from the battery and heat up the contacts. That's -- And you say, "Okay. That's energy conservation." Now why can't you reverse it? Why can't you take energy from the surroundings and charge up your battery? Or light up your light bulb? We all know that doesn't happen. Right? So there is this very deep second law that forbids that. And the picture that kind of describes this is the fact that when you have heat generation what you're doing is taking energy from like one degree of freedom and spreading it out in to many. And that's something that happens spontaneously. What doesn't happen spontaneously is the reverse, taking heat out of many degrees of freedom and focusing it all in to one. And these are all very deep concepts of equilibrium statistical mechanics. See? Described by this quantity entropy. Entropy is a measure of how many states this has. And this is what kind of thing we talk about in unit four of the course. See? And we also connect it to this Landauer's Principle that you may have heard of. So actually Landauer was a very deep thinker and somewhat ahead of his time. And the two major contributions of his that have inspired a lot of people, that you see a lot of discussion today, even today, is this idea of this elastic resistor which is at the heart of this whole transport picture that we are talking about, and then there's this Landauer's Principle which has to do with information. And these are usually two different things in the sense that people who are familiar with one are often not familiar with the other. Two totally different conferences, for example. But they both have to do, though, with this. With a deeper understanding of what entropy driven processes are about and what that means. And this is the kind of thing we try to convey in that last unit. [Slide 5] Okay? Now getting back to what we are doing then, you say that, "Okay. I understand that you have got this nice ideal picture." And this picture actually even describes nano devices quite well because there's good experimental evidence that in really short devices most of the heating occurs in the contacts, and that people say that is why they don't burn up. Because if all the heating was in the actual nano conductor, a little nano conductor couldn't get rid of all that heat and would burn up. But because it's in the contacts it doesn't burn up. So there's good evidence that nano devices kind of approach at least this ideal. But you might say, "Well, but what use is that when I'm trying to think of a big device?" And the answer is that when you have a big device you can think of it somewhat approximately as a series connection of lots of little devices. That is you have a little elastic resistor like this in series with another one in series with another one. And what we'll show you is that this picture actually gives you all the standard results that are normally obtained from rigorous theory for low bias. That is for when the voltage is relatively small this low bias transport, it gives you all the standard answers, this picture. And when you have high bias it gives you a general approximate physical picture. Now I might say, "Well, what is this rigorous theory that we are talking about?" [Slide 6] Well, for semiclassical transport, that is for when you're thinking of electrons as particles, then the starting point is Newton's laws, but that's mechanics, and to that you have to add to this entropy driven processes in order to have a transport theory. And that's this Boltzmann transport equation. This is a picture of Boltzmann. You see? With his contribution about entropy, this S equals k lambda. Now so this is -- This Boltzmann equation is what? Is kind of this centerpiece of all semiclassical transport theory. And whatever we are talking about you could deduce rigorously from here. See? This just gives you a nice approximate picture for it. And if you wanted to do quantum transport where you think of electrons as waves, then you'd have a similar -- You'd need a similar thing. Your starting point would be the Schrodinger equation which is quantum mechanics, but then you have to add the entropy driven processes to it and what you get is what we call this Non-Equilibrium Green's Function method. And we have these two separate parts to the overall course. This is just part A. This is about semiclassical transport where our benchmark theory is Boltzmann equation. Whereas for quantum transport our benchmark theory would be this NEGF. Now what you might wonder is, okay, you have an exact rigorous theory. Why do we need an approximate picture? [Slide 7] And this where I think Feynman expressed it very nicely. What he said is that, you see, even if you understand all the equations, mathematically, it doesn't mean you understand the physics. That is it doesn't mean you can intuitively think about things and be creative with it. In order to be creative you need a physical understanding or an intuitive picture. You see? And what you have to do, of course, is make sure that this intuitive picture corresponds with reality. Now we all use this unknowingly in the sense that even though we know that the Boltzmann equation is what describes things and so if you have to calculate something you'd go to a computer and maybe ask it to solve the Boltzmann equation. But when you're thinking about it usually you have a picture in your mind based on Drude's formula. You have a picture that current is driven by electric fields. And as I'll try to argue in this course, that picture can be very misleading. You see? These days there's a lot of interest in all kinds of devices like spin transport devices where you have one spin flowing this way, another spin flowing this way. And if you think that electrons are driven by electric fields it's very hard to understand why one flows one way and the other flows the other way. So what we are trying to convey to you here is a different physical picture. You see? But of course this physical picture is based in and on the Boltzmann equation. [Slide 8] Okay? So with that then we are kind of ready to move on. You see? As I said, there's four units in this course and this basic concept, this part A, is based on the semiclassical picture. There's a part B that comes later which is based on -- Which introduces the quantum approach to it. But whatever we do here, though, because of this new approach, one thing it allows us to do is even though we are talking about some of the deepest concepts in statistical mechanics, see we are able to do this with a minimal amount of prerequisites. The kind of prerequisites that all students in science and engineering have. And when we first did this, you see, I always wanted that. Could we really convey all these things with so few prerequisites? And I've been very encouraged by the response we got on our first offering about three years ago. And when no one complained they couldn't follow. Okay? And they were very happy with the general presentation. And so -- But they gave me a lot of feedback on things that could be reorganized, could be done better. And we have taken all that and reorganized it now. And the text that goes with it we have also rearranged the text somewhat. And this new version, this second edition, will be available to you. I guess in this course you can download copies. Thank you and I guess it's time to get started then with unit one which is this new perspective. Thank you.