nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/From Semiclassical to Quantum
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[Slide 1] Welcome back to our course on the Fundamentals of Nanoelectronics. Well, we are now at the end of Part A, and thank you for sticking with us, and what I'd like to do in this lecture is to give you a brief summary of this Part A and brief introduction to Part B. Now, as you remember, we started by saying that if you want to understand current flow through a device, the first thing is to understand this. Have a model for the density of states inside the channel. And next step is to locate the electrochemical potentials in the two contacts, which are separated by applied voltage. And why does current flow? Well, because the left contact wants to fill up these levels, and the right contact wants to empty them, and so electrons keep coming and keep getting pulled out. And this viewpoint, as I mentioned, is very different from what you normally learn in freshmen physics, where the thinking is that electrons are driven by electric fields. And when you think of electrons as being driven by electric fields, it's hard to understand why the electrons down here do not contribute to current flow. And, of course, there is no doubt that these electrons don't. It is generally accepted that current flow is due to only those states right around the electrochemical potential. Now from this viewpoint that we talked about in Part A, it is easy to see why these do not contribute to current flow. Well, it's because when you consider these levels, this contact wants to keep them filled, that contact wants to keep them filled, and so they just stay filled. Nothing else happens, whereas up here, this contact wants to fill it up, this contact wants to empty it. So from this point of view up in this expression for conductance, relating it to the density of states around the form energy, around this electrochemical potential, and the time that an electron spends inside the channel, 

[Slide 2] and this viewpoint then gave us an expression for the ballistic conductance right away. You obtain the ballistic conductance by noting that for ballistic transport, an electron just goes straight through like a bullet. And so the time it spends inside the channel is simply the length divided by the velocity, and that's, that gets us this expression for ballistic conductance. On the other hand, if you have diffusive conductance, so the electron goes back and forth, then it spends a lot more time inside the channel. Lot more time time means a lot of less conductance. And this factor by which the time increases is this one plus the length of the channel divided by the main free path. And based on this expression, you could obtain an expression for conductivity, which is this ballistic conductance per unit area times main free path, and that's kind of this new perspective. Gives you a different way of looking at conductivity. And the other thing that comes out of this discussion is that this density of state times velocity can be associated with this number of modes, and which is, again, a very important concept that has come out of nanoelectronics and mesoscopic physics. So all these things we covered in the first two units of the course. 

[Slide 3] And then we went on to a very important conceptual issue, namely, that when you look at the resistance that we just discussed, it has two parts - a constant part and a part that's proportional to the length. And what we argued is that the part that is proportional to the length can be associated with the channel, that's this black resistor, whereas the constant part can be associated with the two interfaces. So how can we do that? How can we separate it out and say, well, this resistance is here, and this, and this constant one is at the interfaces? That's actually a very difficult question. You see, that raises a lot of discussion primarily because, you see, mentally, we are all used to associating resistance with heating. And the nano device, what's well-known is that the heating is really everywhere, and, in fact, a lot of the heating actually occurs in the contacts. So if you are trying to locate where the heating occurs, that won't get you this picture at all. So you might say, well, then, why do you have this picture? Why not just say the resistance is everywhere? Well, but that goes against something that we all intuitively feel, namely, that the resistance is actually caused by the narrowest section of this structure, which is the channel. That there's this idea that when you have traffic flow, if you have ten lanes out here, the traffic flow is controlled by this narrow one-lane part in the middle. That's what controls the overall flow. And here also, then, it is this channel that controls the flow, and if you made a hole in the channel, the current would go down. The resistance would go up, but the heating associated with that wouldn't necessarily occur here. Because after all, a hole cannot dissipate heat. Heat dissipation requires that the electron give off energy to the atoms in the solid, and they start jiggling. You know, that's heat. So the great lesson of mesoscopic physics and nanoelectronics is that, sure, resistance is associated with heating, but that doesn't necessarily occur at the location of the resistance. Resistance is kind of located wherever momentum is lost, and if you want to locate where it is, what you have to do is look at how the voltage drops across the structure. And that raises another set of important conceptual questions about what does voltage mean, and what we discussed is how you should be looking at the electrochemical potential, and if you looked at the electrochemical potential in the channel which is otherwise ballistic except for a scatterer here, what you'd find is there would be a drop in the potential across that scatterer, and there would be small drops at the two ends corresponding to the interface resistances. And the other concept we introduced here was this quasi-femi levels. The idea that out of equilibrium you don't have just one single electrochemical potential, but you should rather think in terms of two separate electrochemical potentials, one for right-moving electrons and the other for left-moving electrons. And what we talked about earlier, this black one, that's really the average of the two. And, but the important mindset that one needs here that I feel, I hope I managed to convey is that when you see this drop in the electrochemical potential, you think that there must be a lot of heating going on here because people tend to associate electrochemical potential with energy. But the way you should think about it is the heating could be anywhere, really. Not there, but the way you think is this mu is simply a measure of the degree of filling of the levels. That is, what is the density of electrons in there. And the point is the reason you have such a sharp drop is simply because of something we're all familiar with. If we're going down a highway, and there's a construction zone in front, there's a lot of traffic right before the construction zone. Once you cross it, the road's empty, and that's exactly what this is. No, before the construction zone, there's lots of electrons. Once you cross it, the road is empty. So the, with this viewpoint, then, once you learn to disassociate resistance with the, from the dissipation. 

[Slide 4] That leads kind of naturally to this overall model that we're using, this Landauer resistor or the elastic resistor. That is, in our entire description, what we're assuming is that in the channel, the electron doesn't exchange energy with the surroundings. That is no heating occurs here, and the heating occurs at the two ends. And the point to note is that this process of heat dissipation is fundamentally different from the usual reversible processes that we are used to. That is, these heat dissipation is driven by entropy, and this is the concept we talked about in the fourth unit. The idea that heat dissipation kind of, although it is basically energy, but it involves the process of taking energy from a single degree of freedom, and dissipating it in many, many degrees of freedom. And that is what makes the process this irreversible in the sense that you can go from one degree of freedom to many degrees of freedom, but it's hard to take energy from many degrees of freedom and put it back here. That is why in a structure like this, you could connect a battery, take energy from it, and heat up your contacts, but you cannot take heat, energy from the contacts and charge up your battery, although in an energy conservation wise, there's no problem there. You're taking energy from the contacts and charging up your battery, but the second law doesn't allow that because that would involve kind of going backwards like this from many degrees of freedom into one. And what makes transport, the description of transport relatively complicated is that these two types of processes are all mixed up whereas by using this paradigm, this idealization of separating it out, we are able to create all these entropy-driven processes in a very elementary way, and that leads to this expression for current that we talk about in Unit 1 itself. 

[Slide 5] You see. Now one question you might ask is that what we just discussed might be applicable to small devices where there is actually experimental evidence that a lot of the heating occurs in the contacts, but how about big devices where obviously a lot of heating does occur in the channel itself. And that is where I argued in Part A, in what we just discussed, that you can think of long resistors as a series of short elastic resistors. And this viewpoint gives you results that are, that agree with what you get from the rigorous theory for low bias, and for high bias, in general, it provides a good approximate physical picture, and so what is this vigorous theory. Well, the rigorous theory for semi-classical transport, which is what we were discussing in Part A. Namely, the way to think of electrons as particles. The rigorous theory is this Boltzmann equation, after Boltzmann who kind of combined the mechanics of Newton with this entropy-driven processes from thermodynamics and came up with this Boltzmann transport equation which is the centerpiece of all semi-classical transport theory. In all of the results we've discussed, they come, they're all consistent with the Boltzmann transport equation, but what we do is use this simple picture to give you a pictorial understanding of things, because in general, knowing how to solve an equation and understanding the physics are two different things, and it's very important to have an intuitive picture that is correct. You know, that actually corresponds with the math. And that is what we have tried to do in all of this Part A. You know, convey this physical picture that goes with this Boltzmann equation. And what we do in Part B after this is the quantum version of that. So instead of Newtonian mechanics, we now have Schrodinger equation as quantum mechanics, but, of course, that's not enough. You still need the entropy-driven processes, and when you put that together, you have this non-equilibrium Green function method, and this is what we do in Part B in quantum transport. So there we start with the Schrodinger equation, which looks something like this. I guess even if you haven't taken a course in quantum mechanics, you have probably seen this. I've seen this on lots of t-shirts. So you're probably familiar with that, but the point to note, again, is if you're trying to describe current flow, this is not enough. This just describes an isolated channel. You have to worry about how to include the entropic processes. 

[Slide 6] And that's where, I guess what we show, is we introduce, this is what we'll do in Part B. We'll introduce these new things called the cell energy functions. The sigmas. The sigma1 denoting the connection to the source contact. Sigma2 denoting the connection to the drain contact, and the sigma0, which is the connection to just the surroundings. I mean, there's no physical contact there, but when the electrons going through the channel, it is interacting with all the surroundings, its atoms there, and that itself needs to be discussed, described again through a separate function like this. So these are the things that we'll talk about 

[inaudible] in the next course, and one of the examples we'll look at is this basic physics that we talked about already, namely, if you had a channel with a barrier, what is the resistance of different sections of this. What is this interface resistance, and what is this resistance associated with the barrier? Now, in Part A, we talked about this in semi-classical terms. You know, in terms of the cars on a highway. How there's a drop right there, right at the barrier. That's this red curve. This is the curve that you could rationalize in terms of, as I say, the cars on the highway. But now, of course, we're doing the quantum theory. We are thinking of electrons as waves, and when we do that, you'll see, you'll get this black curve, whose average is kind of like what we had before, but it oscillates around it. Now why does it oscillate? Well, because we are thinking of electrons as waves now. That's how we are modeling it, and waves show interference. They show standing waves. You know, when you have two components, one going to the right, one going to the left, there is interference between them. So all that is reflected in this theory. Now, why don't we normally worry about it? How come everything we discussed in Part A is usually quite good, and that's what you normally observe experimentally. The reason is that this is what you might call a coherent theory, coherent meaning you're assuming that the waves don't interact with anything else. In practice, when an electron wave is going through the solid, the solid is not a rigid thing sitting there. All these atoms we're jiggling around. I mean, that's this thermal energy. They're all vibrating around their equilibrium position, which means as far as the electron goes, it kind of as if it's going through a fog with a kind of almost a thunderstorm going on. And that destroys its phase. And so in practice, you often don't see these oscillations unless you are at real low temperatures. So what you might often see is something like this where the oscillations are much smaller, and there's even a slope here, it's not flat because in going through this fog, the electron loses momentum, and that gives it a resistance. And there's, but there's still a remnant of some oscillations. That's the kind of thing people experimentally see. And the point is this NEGF method allows you to include that kind of thing into this theory. That's what we'll do in Part B, of course. Through the sigma0. Through this interaction with the surroundings. The surroundings that are in the jiggling around and destroying the phase of the electron. And here the way I've drawn it, you can see there's a lot of resistance in this part because momentum is being lost. On the other hand, there are all kinds of scattering processes that destroy the phase of the electron but do not affect its momentum. Now that, too, could be included into the NEGF theory, and in that case, when you calculate how the potential drops, you'd see something like this. Again, there's a little bit of oscillations, but as far as, there's no slope here. Because there's no momentum loss, there is no corresponding resistance either. And one point, I want to stress that in all these calculations, we actually haven't included any dissipation inside the channel. The sigma0 just involves all kinds of interactions but without actual energy exchange. Whereas the usual view, of course, takes this dissipation, dissipative part as kind of the fundamental thing about resistance, and so it starts from there. And that makes the usual theory, usual approach much harder because describing dissipative processes involves many body theory, and this many body perturbation theory, which takes few semesters to master, but the point I want to make is this entire, all this physics, you know, about where the resistance is, where the voltage drop is, all this we are getting without including any dissipative processes inside the channel at all. 

[Slide 7] So we'll also look at other interesting problems, you know, like, one that we talked about I believe in Unit 2 of Part A. We discussed how the conductance, the ballistic conductance would be linearly proportional to the width in a semi-classical view. That is, if you think of electrons as particles, then the ballistic conductance would be this density of state times the velocity, and you'd get this linear increase. And that was observed right around 1970 or so, but twenty years later, people looked at smaller conductors, and that is when they saw this quantized conductance. And that comes from the wave nature of electrons. Now we gave some elementary explanation for why this happens, and that was I believe in Unit 2 of Part A, but in Part B, of course, we have the full quantum theory, and this will just come out automatically. So once you understand this, this will just follow naturally from there. See, because it includes the wave nature of electrons. Another problem we'll talk about, something we didn't discuss at all in Part A, namely the Hall effect. So that's a very well-known thing, back from the 1880's I believe. That when you apply a magnetic field, there is this, what's called the transverse resistance. Transverse resistance means there's a voltage in this direction proportional to the current flow in this direction, and that transverse resistance is supposed to increase linearly with magnetic field, and there's a simple semi-classical view of that. But what was discovered in 1980 was a very striking result, namely, that at high magnetic fields, this transverse resistance is actually quantized, and has this very accurate values, you know, out to, like, six or seven places in decimal. And so accurate that the NIST, this National Institute of Science and Technology, they use this as a standard of resistance. So this is something, again, an example we'll talk about using the NEGF method, and you'll see how it comes out naturally. 

[Slide 8] Another example we'll talk about, I guess in the final unit of this course, of the Part B is the spin transport because that's a very active area of research that has developed a lot in the last two decades. And one example is kind of, like, what we discussed in Unit 4 of the, of Part A. We say that think of an anti-parallel spin valve where you have a magnet that's pointing downwards, connecting to down spins, and another magnet pointing upwards, connecting to up spins. And no current flows ordinarily because, you know, up, you know, these up spins come in, but can't get out, and down spins can get out, but cannot come in. But then if you put a magnet that's pointing sideways in between, you can increase the current. Now the picture of spin that we used in Part A was this elementary semi-classical picture that you have two types of spins, up and down, red and blue. But what that doesn't let you think about is, OK, we have red and blue, but how do I think about something that's pointing sideways? And the view that comes from this quantum nature of spins is that something that's pointing up is actually a super position of something that's pointing left and something that's pointing right. And it's a little bit like vectors, but not quite. You know, with vectors, you know that something pointing at 45 degrees is a super position of x and y. Something in the middle. Here, it's, like, something pointing up is a super position of left and right. So it's a little bit like vectors but something different. You call these spinors. OK. And the result is that, you know, we, that you can understand why the presence of a magnet like this, for example, would increase the current. It's because the, these up electrons that are injected have a left component and a right component. The right component gets pulled out by this magnet. What remains is the left. And that left component, then, again, has a little bit of up and a little bit of down, and the down part can flow out. So there's a current that flows, and all this is very counterintuitive. Not at all obvious. You see, and that's what makes spin kind of mysterious in a way, and you can see one of Feymann's lectures I think in Volume Three where he talks about interesting experiments like this, although not in the context of solid state devices, but in the context of, like, Stern-Gerlach experiments in vacuum. And, of course, what is new today is that people can actually build solid state devices where some of these principles are actually relevant, and the NEGF method that we'll be talking about in this Part B shows you how to model all this, and how to think creatively about new devices that make use of these principles. 

[Slide 9] So with that, then, let me just briefly mention that what we finished is Part A. That's based on this semi-classical picture of electrons, that things of electrons as particles, which gives you a very good understanding of many devices. Many types of nanodevices, but then there are also all kinds of other devices on the horizon which involve the wave nature of electrons, and that require this quantum model. But underlying it all, of course, what we are trying to do use this idea of this Landauer resistor or this elastic resistor to give a simple physical picture that goes with the math, and the simple physical picture, of course, the simplicity comes from separating out the mechanics from the thermodynamics and getting a deeper understanding of these entropy-driven processes. And we are, we convey all this assuming only we have minimum amount of pre-requisites. You know, what you have seen, what you have used in Part A, but Part B requires familiarity with matrix algebra, that's for the quantum mechanics, but other than that, we don't really assume any prior acquaintance with either quantum mechanics or statistical mechanics, and our past experience, I guess we did this about three years ago, has been very positive. People generally liked this way. Based on this pre-requisite, they were able to follow what we were doing. And based on their feedback, what we have done is reorganized this course a little bit, and also the text to that goes with it. And the text, of course, covers both Part A and Part B, and what we just finished, then, is Part A, and I hope you liked it and will join us for Part B. Thank you.