 Online simulation and more for Nanotechnology. You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. Okay, so we're ready to get started. What I'm going to do is to give a very brief overview of what it is I'm going to be talking about in the next nine lectures that I'll be presenting. So there isn't a lot of material in this. It's just talking about what I'm going to talk about. Professor Dada's introduction was actually very good for me because the first half of his introduction and a lot of the questions that he was getting are things that we're going to be discussing in much more detail in the subsequent nine lectures here. So let me discuss what this is about. First of all, let me just mention everything on the nanohub is copyrighted, but it's copyrighted in a very generous way. So it means you can use the material, but there are some restrictions. You have to credit where you got it from. And you can't put it in a book, it can't be used for commercial purposes. So outside of those, there's some very generous copyright provisions that we encourage people to make use of this material. Okay, so the lectures that I'm going to be talking about, very much about the material. Professor Dada talked about the fundamental concepts of how you think about electron flow at the nanoscale. These 10 lectures are about how electrons flow in semiconductors, mostly a little bit in metals. And just by way of introduction and review. So you all know Ohm's law. We have a voltage source connected to a resistor current flows. And the IV characteristic is a straight line. Current is conductance times voltage, or voltage divided by resistance. And this has been known for a long, long time. So there are a couple of questions that we'll be thinking about here. Why is the characteristic linear? How large can V be and have the characteristics still be linear? But most importantly, what determines the magnitude of the conductance? When we have a hunk of semiconductor, a chunk of metal, or a very small conductor, like a nano-electronic device. There were questions about diffusive transport. So we'll be talking about both ballistic and diffusive. But just by way of review, think of an n-type semiconductor, large macroscopic sample, some length L, it's going to be a resistor. Some cross-sectional area, A. We hook a voltage source to it. Current flows, the current is carried by electrons. Let's see. So if we apply a positive voltage to the right side, we attract electrons. They try to flow through the resistor. But they undergo this random walk. The average distance between scattering events is much shorter than the resistor itself. There's a slight, there's a lot of random thermal motion. Electrons are moving in all directions, but there's a slight bias for them to move from the left to the right. They go out the contact, and because of the negative sign, that means that current flows in to the right contact. So this is going to be something that we have to be careful about later on. It's a circuit convention. Circuit people like to define the positive direction of current flow to be when the current flows into the contact. It's positive. Now if my x-axis moves from the, goes from the left to the right, a positive current is flowing in the minus x direction. There'll be some times where I'll have to be careful about that sign. Okay, so there's an electric field, and traditionally we think of this as all happening due to a force from the electric field. As Professor Dada mentioned, that's maybe not the best way to think about it more generally. But traditionally we think there's an electric field there, which is just a voltage across the resistor divided by the length. And the electric field points in the minus x direction, and that puts a force on electrons in the plus x direction, and that's what causes them to move. And they move at an average velocity because of all of this scattering. It's like a friction. The electrons don't just accelerate. They lose all of that momentum in the scattering events. And they end up moving at an average steady state velocity, which we call the drift velocity v sub d. And the drift velocity is minus mobility times the electric field because of the minus sign. So if we want to compute, basically I'm just reviewing the traditional textbook way that we treat this problem, and we'll be talking about how you treat it in a much different way in the course. So if I want to talk about the current, the current is charge, capital Q divided by the time it takes electrons to go across the device. And the amount of charge there is just the electron density, little n, times the volume of that resistor, which is the cross-sectional area, times its length, and then I have a Q for the charge on the electron. So that's the total charge. That total charge moves through the resistor in a time that's just determined by the length of the resistor divided by the average velocity that they're moving at. And the result then is that I can write current as electron density times charge times average drift velocity times area. And drift velocity is mu times the electric field. So we get this classic textbook expression for diffusive transport. Current is NQ mu times the ratio of cross-sectional area divided by length of the resistor. So we have current is proportional to voltage. And all of the constants out front are the conductance in units of one over own, and that's NQ mu times the ratio of area over length. And since we can make resistors with different dimensions, then the important parameter is this conductivity sigma, which is just NQ mu. Okay, so these are the traditional way we think about large resistors. Now if you go in the lab then and measure the IV characteristic of a resistor, you'll get something like this, if it's silicon. So you'll have a region where the current is literally proportional to the voltage. That's the low, people would call that the low field region, or the near equilibrium region, or the linear transport region. That's the region that Professor Dada talked about. And that's the region that all of the lectures and my set of lectures are about. If you continue to crank the voltage up, the current voltage characteristic will become nonlinear. Now in silicon, the high electric fields would accelerate the electrons. They would start to emit optical phonons. The velocity would slow down, and we would get the current to saturate. In other semiconductors, like three, five semiconductors, like gallium arsenide, we would get even non-monotonic IV characteristics due to carriers transferring between different valleys. We could get some very complex effects. All of those are beyond the scope of what we're going to be talking about today. They're interesting and important effects in devices, but they're not the ones that we're going to be talking about in these sets of lectures. So we exclude high field transport. We focus on the region where current is linearly proportional to voltage. And there are a lot of interesting things to say about that regime. And it's also an important region for all devices. Now over the past 20 years or so, there's been remarkable progress in the science of nano-electronics. People have learned how to attach electrodes to molecules and measure their IV characteristics. And what you'll find is if you apply a small voltage across a small molecule, the current is proportional to the voltage. It has some conductance. And this is the point that Professor Dada made. If you're starting point in thinking about what's the conductance of a molecule is nq mu times a divided by length, that's not a very good starting point. It's very difficult to know how to apply that conventional formula for large structures to a molecule or to a small device. Even more, we'll find that the resistance doesn't, it's not a continuous function of the dimensions. It's discreet and quantized as we increase the cross sectional area. We'll increase the conductance in steps, not continuously. So there are some interesting things to go on there which are now well established experimental facts. They were of interest primarily to the physics community 15 or 20 years ago. But electronic devices have gotten so small that they're more and more relevant for mainstream device technology. And even in transistors, we begin to think about these kinds of things. So there are lots of different kinds of carrier transport. The focus of these lectures is a near equilibrium transport. There's this whole field of, if you apply very large voltages, the current becomes a nonlinear function of the voltage. That's called high field transport. People also refer to it as hot carrier transport. Because the carriers have much more energy than they did in equilibrium. If you measure that energy by an electron temperature, the electron temperature is much higher than the lattice temperature. Now, there are even more interesting effects that we won't be talking about. When you go, people many years ago measured the velocity of electrons at a high electric field in bulk silicon. What you find when you go into small devices is things get even more complicated. There is no one-to-one mapping between the electric field and the current. I mean, there it depends on the history of the electron. How long did it spend in this region of electric field? Where is it coming from? What energy did it gain in the electric field upstream? And there are very interesting effects that occur in small devices, things like velocity overshoot that become very important in small devices that are even more complicated. And people use sophisticated simulation programs like Monte Carlo simulation programs, hydrodynamic balance equations to treat those. Those are all very interesting and important effects too. But none of which I'm going to be discussing in these lectures. And then there's quantum transport. In the second half of Professor Dada's lecture, he talked about the non-equilibrium greens function approach. Items one through three, we think about the electron as a particle. It's a semi-classical particle and we don't have to think about it as a quantum mechanical entity. But the other thing that's happened over recent years is that devices that become so small that the wave nature of the electrons becomes very important. Now for a long time, we've worried about electrons tunneling through thin layers, but even more importantly, now we're thinking of as electrons move from the source to the drain and transistor, quantum mechanical reflections and tunneling and various quantum effects are becoming more and more important. So one of the things that you're seeing here recently is that mainstream semiconductor manufacturers are beginning to acquire computer-aided design tools that treat quantum transport because channels have become so short that it's necessary to do that. In many of these summer schools, we spent a lot of time on quantum transport. In this particular one, we don't spend very much time on quantum transport. Professor Dada gave you an introduction and we have an evening session where you'll see some real industrial strength simulation capability that's being done here at Purdue and Professor Clemix Group. So for those of you that are interested in that topic, I encourage you to make sure that you attend that evening session. And then there's another field that we did a little bit of discussion at the previous summer school about this whole idea of transporting random and disordered media, where there are grain boundaries and polycrystallins or amorphous materials. This is another topic that we think the concepts that are applied here can be applied to those problems too, but that's still very much for us a research topic where the methods haven't fully been worked out. And it's a topic that is of more and more importance in electronic devices because there's more and more interest in artificially structuring materials at the nano scale and very often there's highly random nature to those structures and trying to understand the nature of transport and those structures is becoming more and more important. But, so just so you know what we will be talking about and what we won't, we're going to focus on neuro-equilibrium transport because it is really the starting point for all of these others, you know? A good solid understanding of neuro-equilibrium transport is what you begin with and then you start adding complexity in. All right, so that's point one, that's why we're starting it. It's also extremely useful. So if you're working experimentalists or growing semiconductor materials or producing graphene or looking at the properties of topological insulators, sort of the first thing you do is to go in the lab and do measurements of their neuro-equilibrium transport properties. You try to understand something about transport by understanding their mobilities or their conductivity, diffusion coefficients, thermoelectric properties. So it's extremely useful in characterizing the properties of the materials that you're working with. And the third point is that in almost in any device that I know of, there are always low-field regions that end up being very important in controlling the performance of the devices. So one of the most challenging problems in scaling MOSFETs these days is the parasitic resistance of the source in the drain. So these are neuro-equilibrium transport problems. So wow, it's, to a lot of us, it's very interesting to think about what's happening in the intrinsic channel, what's controlling the performance of the device, maybe a neuro-equilibrium transport problem. And Professor Dada alluded to this, so I don't need to say much more. Neuro-equilibrium transport is something, I learned it when I took a course as a PhD student in the late 70s from a textbook that was written in the 60s. It's an old field that has been around for a long, long time. What more can you say about it? Oh, it turns out you actually can say a lot more about it. And the work in the last 15 to 20 years in mesoscopic physics where people have learned how current flows at very small scales has really given us a much different way of looking at this traditional problem. Now we think it's much more physically transparent. When you read the traditional approaches, there's a lot of mathematics in solving the Boltzmann equation that just tends to obscure, at least for me, the physical content. The underlying physical content is actually quite simple. The mathematics of this approach is also much simpler. Even though it'll turn out, in the end you'll get the same results when you look in the diffusive regime. The advantage is you can go to the ballistic regime too, just as easily. And so it's more broadly applicable to a wider range of problems. So that's why we're doing it. Now what lectures are we going to have? Like I said, you're going to hear a lot from me. I hope you'll hang in there with me. In lecture two this afternoon, at the end of the day, just before our reception out front, I'll go through the general model that we're going to use. This is the model that Professor Dada discussed. I'll just go through it one more time. It probably doesn't hurt to see it. We'll go through it step by step. It'll look a little bit different because it's just the way that I do it, but it's really, I learned it all from Professor Dada. So it's all of the same concepts. And that will be the starting point for all of the other lectures. We'll apply that not only to electrons, but eventually to phonons. So then lecture three, let's see what. All right, so basically that's what lecture two is all about. And I'm also, the foundations for all of the lectures are discussed in Professor Dada's lecture notes and lessons from nanoelectronics. So in terms of the fundamental concepts that underlie all of what I'm going to be talking about, I would refer you to those. We're going to be focusing a little more on how you take those concepts, apply them to real problems, analyze experimental results from devices, and questions like that. So lecture three then we'll talk about resistance. How do we compute the resistance of a small structure? And how do we get the right answer in the diffusive limit so we get all the answers that people have known from the 50s or before? But how do we get the right answers in the ballistic limit and what new things happen there? Okay, and I'm primarily, you know, you can do this in 1D for a nanowire. You can do it in 3D for a bulk chunk of material. You can do it in 2D for transport in a plane like an inversion layer of a MOSFET. Many of the problems that we're interested in these days are 2D. Channels of transistors, graphene, things like that. So I'm going to be doing most of the work in 2D just because we don't have time to work at all and you'd fall asleep if I did everything in 1D and then 2D and then 3D. So I'll do it in 2D primarily but one of the nice advantages of the technique is you can easily switch between dimensions. So we'll try to understand these traditional expressions. Where do they come from? Under what conditions can we use them? And we'll just mention how you can extend these concepts to other dimensions. Now we can also treat not just the conductance but we can treat thermoelectric effects. So I've got two lectures on thermoelectric effects. So we're going to spend some time on that. So as Professor Dada mentioned, a lot of the questions that came up during his presentation are things that we're going to be discussing in these lectures. If you apply voltage difference, current flows. If the two contacts are at different temperatures, current flows. When current flows, it also transports heat. So all of those effects can come out of this general picture too. And this is a condition, sometimes when you read this in books it gets highly mathematical. So I'm going to do this in two parts. The first part I'm going to try to do it mostly from physical approaches and explain what's going on. And what kind of effects are we going to be looking at? We'll take this semiconductor and what's been known for a long time is that if the right contact is hotter than the left contact, that if you put a high impedance voltmeter across this slab, you'll measure a voltage. And the voltage will be proportional to the temperature difference. If the temperature difference is small, that's what we mean by near equilibrium transport. And the constant of proportionality is called the CBEC coefficient and it has units of volts per Kelvin. And it's negative for n-type conduction and positive for p-type conduction. So in this case you'd measure a positive voltage if I have a n-type semiconductor. People also sometimes call it the thermal power. Thermal power, CBEC coefficient. And it was discovered a long, long time ago. It's the basis for thermal couples. And more recently people are interested in using this for solid state cooling and power generation applications like that. So we'll try to understand where this CBEC coefficient comes from. Now the other half of this problem is that if we run a current source through this resistor, heat will flow. And what you'll observe is that heat will be absorbed from the ambient at one of the contacts. So it'll get cold at one end. And heat will be dissipated at the other contact. So this is called the Peltier effect. And the coefficient pi is called the Peltier coefficient. It turns out that there is a really close connection between the CBEC effect and the Peltier effect. They're not actually two separate effects. They're related by this relation pi is equal to t times s which is called the Kelvin relation which has also been known for a long time. So this was also discovered a long time ago but continues to be an important effect to understand. So as I said, I'll discuss this in two pieces. The first piece I'll try to keep the mathematics to a minimum. The second piece I'll try to show how, if you just take the model that we established and that we used for electrical conduction, how you can just mathematically go through and derive all of the standard coefficients that when you read papers and see the thermoelectric, the equations for thermoelectric parameters, we'll see that we get all of the standard results. Okay, now, a lot of whether we're ballistic or diffusive depends on whether the conductor is long or short compared to the mean free path, the average distance between scattering. Now we're not going to go deeply into how you calculate scattering rates and mean free paths but in this lecture I'll try to give you some sense as to how do you do those calculations? What are the general properties of ionized impurity scattering and phonon scattering? What does this mean free path really mean? You know, I always tell students, there's things like thermal velocity, you calculate the thermal velocity, but I know of at least three different ways to define thermal velocity. There are different ways to define mean free path. When you read in a paper, somebody quotes a mean free path, you have to be very careful that you understand what they mean by that mean free path. In this approach that we're using, it's a specially defined mean free path, we call the mean free path for back scattering. And it makes everything work out in this land hour approach. And we have to be careful that we use the correct mean free path. There are various numerical factors. You might look in a paper and they'll quote a mean free path and you might try to calculate it using what you learn here and you'll find you're off by a factor of pi over two. So we'll explain where all that comes from. Little details that you have to be careful about. Now the traditional way of doing these problems has been to solve the Boltzmann transport equation. And A, first of all, you should know what the Boltzmann transport equation is and how people solve that. And B, sometimes it's a more convenient way to solve a particular problem. At least that's what I find. One of the ways it's especially nice for it is computing what happens when you apply a magnetic field. When you want to do something like a Hall Effect measurement. So in the first half of this lecture, I'll discuss how you solve the Boltzmann equation and show you how all of the results you get from that are the same results we've seen in previous lectures. And in the second half of the lecture we'll add a magnetic field and show where the Hall Effect comes from. Because that's a very important near equilibrium transport problem. Now lecture eight is a little bit about how you do measurements. So one of the reasons that people are so interested in near equilibrium transport is it tells you a lot about the properties of the material that you've grown or that you're working with. So a lot of people spend a lot of time in the lab doing measurements, trying to extract mobilities, feedback coefficients, things like that. So we'll go through some of the fundamentals of measurement considerations and talk about some standard things like four probe techniques, transmission line methods, banderpaw techniques, how you do Hall Effect measurements just to acquaint you with how all of these things work. Now these lectures are primarily a set of lectures about electron transport, near equilibrium electron transport. But as Professor Dada mentioned in response to a question, very similar concepts can be applied to phonons. So the purpose of this lecture is just to show how you take the same general model and apply it to heat transport, how you understand the thermal conductivity of a material and we'll show that you get the standard results that you'll find in textbooks, relatively simply by doing the calculations in this way. And then the final lecture, I'll be exhausted by then but I hope you're still with me. It's sort of a case study. In most of these lectures, when I work things out and show you how they'll work out, I'll be assuming a standard semiconductor with a parabolic density of states and an effective mass, make everything simple. There's no reason to do that. In fact, in a lot of our calculations now we have to use a full numerically computed band structure. Techniques can be applied to that. But sometimes students get the idea that the effective mass results get so embedded in your consciousness that frequently you want to use them when you shouldn't. So the purpose of this lecture is really not to talk about graphene, although there's a lot of interest in graphene these days. It is really to show you how you take a material that has a much different electronic dispersion for which it's not even clear if there is an effective mass or how you would define it and to show that that really doesn't matter at all. That you can work out everything that we've worked out for parabolic bands with effective masses just as easily for a new material like this and understand the results that you read in papers. Okay, so the objectives are just to introduce you to this field. This is sort of if you work on electronic materials and devices, you have to understand your equilibrium transport. So the objective is to introduce you to this field using this bottom up approach. We want to acquaint you with some key results that you should know about. Things like what is the quantum of conductance all about? What is ballistic transport all about? How do the standard measurement techniques that everybody use work? So when you read papers, you're familiar with all of that. And I can't possibly say everything that you need to know for your career but hopefully we can give you a starting point that there's enough that you know that you can get started. When you encounter a research problem or you have to go beyond what we've covered here, you can understand, okay, I know where I'm beginning from and you can figure out where you need to go to get the additional details to solve that problem that you're interested in. Now if you're interested in more information, as I said, just Google electronics from the bottom up. You find a lot of information there. I have a course that I teach and the whole course is online. It's going to be taught again this fall and I'll revise it but the version from a couple of years ago is online. So a lot of the things that I won't have time to go through in detail from time to time during the lecture is I'll refer you to a lecture on my course. Okay.