nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 3.1: Introduction
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[Slide 1] Welcome back to our course. This is the third unit. It's called what and where is the voltage? Now if you're like me when you hear that title, you might wonder why are we talking about this? You know, we have been talking about resistance. We have been talking about current flow. Now why this question? What and where is the voltage? And that's what I'll try to explain, you see, in this introductory lecture, why it's a very important question, see. 

[Slide 2] Now what we have done so far -- I guess this was largely actually in Unit 1 itself that we talked about this channel with the voltage across it. And you can write the current for small voltages as the conductance times the voltage. And this conductances can be written in the form, as shown here, with the ballistic conductance G sub B. And the idea is when you have a very long channel, L is much greater than the mean-free path. And you have Ohm's law on the other end for a very short channel. The L is small compared to a mean-free path and the conductance becomes equal to the ballistic conductance. Okay. Now if we did this expression for conductance and just invert it, then you get an expression for resistance, and the resistance then looks like this ballistic resistance plus a term that's proportional to the length. So this expression just follows from here. Straight forward algebra. Now the question you could ask is now what's this ballistic resistance? What is it associated with? Because as I have tried to stress in the early days, that was one of the important conceptual issues, one of the sticking points, see. Where is this ballistic resistance? What does it signify? Now if you look at the part that's proportional to the length, you'd say well, that's proportional to the length; so we can associate it with the central region, the channel. The longer you make it the bigger it is. On the other hand, this constant part, what I'll try to show in this lecture is that constant part actually is associated with the two interfaces, and that's not meant to be obvious. Now why do we say that this constant part is associated with the interfaces? And here we have a major problem that bothers people because usually, you see, we associate resistance with heating because the way we think is, you know, think of a light bulb, is basically a resistor. When you run current through it, it heats up and that's why it emits light. In most people's minds then, the resistance is associated with I squared R, the heating. And the point I want to make is that actually if you tried to follow the heating, that's very misleading and not -- won't really get you what we are talking about here. So the point I want to make is that, you see, when you have small devices -- I made this point earlier -- that the heating is actually not in the channel at all, that is, in small devices a lot of the heating is in the contacts. In fact, for most of our discussion we have assumed that it is entirely in the contacts, but we have assumed that the channel is an elastic resistor or this Landauer resistor. So you might say well, if the heating is in the contacts and everywhere else, then why don't we just say that the resistance is everywhere? And if the heating is in the contacts, well, that's where the resistance is. Why do we say the resistance is in the channel and this much is at the interfaces? Well, you could adopt that view but it goes against something that we all feel instinctively. So it doesn't quite go with this intuitive feeling, and that is this. That if I gave this channel and punch a hole in the middle, we all know that the resistance will go up. You know, if it used to be 1 kilohms, I make a hole. It will probably will be 5 kilohms, maybe 10 kilohms. So in that sense the resistance is caused by the hole, but that doesn't mean that the heating associated with that resistance is occurring in the hole. Not at all, because hole really doesn't quite have the degrees of freedom needed to dissipate heat. You see, to dissipate heat what must happen is the energy must go from the electrons into the atoms and start them jiggling. You know, that's what heat is, this thermal energy. So that conversion has to take place. So if we put a hole here, the resistance will go up but it doesn't mean that the associated heating will occur there. It will occur somewhere else. So what we want then is a criterion for understanding where is the resistance, and that criterion can't be where is the heat. And what I'd like to argue is that criterion is that we should look at where is the voltage drop, that is, from elementary circuits, you know, that when you have a set of resistors like this and you run a current through it, the voltage drop across each section is proportional to the resistance. High resistance, a lot of voltage drops there. So the reason we say that this resistance is at the interfaces is because there's a voltage drop at the interfaces. 

[Slide 3] So the way we would look at that is we would say well, inside this contact the electrochemical potentials are located at say mu1 and mu2. These are the two points, two contacts, and they have applied voltage; so they are separated by the applied voltage, q times V. And ordinarily what you'd expect is, we'll say that the electrochemical potential varies linearly from one contact to the other. That's what's normally expected. What we'll show in course of this lecture, these lectures that's in this unit, what we'll show is that the actual voltage drop looks more like this black curve here, that is, it's linear but then doesn't go from mu1 to mu2, other there's a drop right at the interface and other drop right at this interface, and these are the drops that are associated with this ballistic resistance, the parts of the two interfaces, see. That's what we're discussing. Now one of the important questions that comes up that causes a lot of confusion of course is just is what is the meaning of voltage? Because what I drew here is what's called an electrochemical potential. And in our discussion that's what we have always been using, the electrochemical potential. On the other hand, most people if you ask them what is voltage, usually the answer you'll often get is the electrostatic potential And the reason is that this electrochemical potential is a deep subtle concept, you see. It's a concept from statistical mechanics and people feel uncomfortable about it, and they're often much more comfortable talking about something like electrostatic potential, which they feel everyone understands, you see, but this really isn't quite right, that is, if you ask people, they would say well, what's -- how do you describe current? And they would say well, current depends on the electric field and the electric field is proportional to the slope of this electrostatic potential energy. I guess, I'm writing U for electrostatic potential energy, and that's what often you hear from people. On the other hand, the correct answer, and I think all the experts will agree on that. The correct answer is that the current is proportional to the slope of the electrochemical potential and not the electrostatic potential. Now let me elaborate on this a little more. 

[Slide 4] Consider a structure, you know, a channel with the density of states, like I've shown here, and let's say that's the electrochemical potential. What does the electrochemical potential represent? Well, at low temperatures it will tell you the level up to which the states are all filled. And this is what goes into the Fermi function, as we have seen. Now supposing there's an electrostatic potential that varies across the channel. Well, then what you might expect is that the bottom of the band, that's the lowest energy point, would actually continually go down because there's an electrostatic potential, a positive electrostatic potential, meaning a negative energy; so the electronic energy goes down. And so if we drew the density of states plot, it would look something like this. So is there a current? Well, if you write the electron current as being proportional to this electric field, then I would say yes, there is a current, okay, but that's really not quite true. I mean, that's true only if the electrochemical potential also goes down, like I've shown here, only if this is parallel to this, and that's true often in homogeneous conductors. In homogeneous conductors the electron density is the same everywhere, and the electron density depends on this energy range between the mu and the bottom of the band. So in a homogeneous conductor often you'd have this situation, in which case this would be correct, but this is not true in general. You could have an inhomogeneous conductor, for example, where there's a lot more electrons on this side than on this side, and just at equilibrium you'd have a constant electrochemical potential with a high electron density on this side but a low electron density on that side. And does that mean there would be a current? Of course not. This is at equilibrium. There's no current in there. People would agree. But the way they would usually explain it is by saying that oh, you just -- we just had the drift current. You have to add the diffusion current to it, and the diffusion current is the slope of the electron density. And now there's a slope because it's smaller here than over there. Now what you can show is that the diffusion current can be rewritten in this form, that is, just as this is du/dz, you could write this as being proportional to the slope d/dz of mu minus U; mu minus U meaning that distance, you see, which is proportional to the electron density. So what you can show is that the diffusion current can be written this way so that if you add the two up, what you get is this expression. And this is the correct expression for the current. You see, it is proportional to the slope of the electrochemical potential. And this is fundamental. This is really the basic standard result, see. And if you just use the first term out of this and say that current is proportional to electrostatic potential, then you could be right but you could also be very wrong. You could be missing very important things, see. And they say well, is there any controversy about this? Not at all. I think all experts should agree that's the correct answer. And yet often people kind of avoid talking about mu, again, for the reason I mentioned before, that it's a difficult concept and especially under non-equilibrium conditions such as at equilibrium it's easy. You know, you have the same mu everywhere. It's kind of like temperature. At equilibrium the temperature everywhere is the same. But when you have heat flow when temperature is varying, that's when the concepts get more subtle. Here also mu is kind of like temperature for current flow, see. Equilibrium, same mu everywhere, but out of equilibrium how you define it does bring up subtle issues, and that's why often people try to avoid it. 

[Slide 5] Now one of the things that often people do is that they say, well, let's not get into this question of what is this mu? That's electrochemical potential inside the channel because the channel is out of equilibrium. Let's instead talk about the electrostatic potential. Now what we'll discuss again in this unit in a later lecture is that if you actually looked at the electrostatic potential, it kind of looks like this. It's a smoothed out version. It doesn't quite show this, discontinuities, you know, the ones that I say that are associated with the ballistic resistance. It smoothes it all out, which means looking at the electrostatic potential you don't quite get that part of the physics at all, see. So for example, this is a calculation. This is an actual quantitative calculation based on quantum transport formulism. And you can see, if you look at the electrochemical potential, there's a clear discontinuity. If you look at the electrostatic potentials, that's much more smoothed out. Now of course, to talk about electrochemical potential inside you have to be careful on how you define it. And in this paper that was done based on quantum transport. What we'll do in course of these lectures is we'll talk about how you define it in the semiclassical context in terms of the Boltzmann equation. 

[Slide 6] And what we'll show then is that this will introduce using the Boltzmann equation, this whole concept of Quasi-Fermi levels. This idea is when you're out of equilibrium, you don't quite have a single electrochemical potential, like I've shown here. More generally, you could have multiple electrochemical potentials. But in this context it's as if the right-moving carriers have different electrochemical potentials from the left-moving carriers, see, and these are what you call Quasi-Fermi levels. And when I draw a single electrochemical potential in the middle of this black one, what I really mean is the average of the two. And these are of course important subtle concepts that we'll be talking about in course of these lectures in this unit. Now this is something that was -- I've described in my book and like 20 years ago. But usually when I teach this, I often don't stress these points too much because in the past I felt well, these were difficult conceptual things but probably don't matter in terms of real calculations. But nowadays there's been enormous progress in the field of spintronics, which actually have made all these issues much more important. And that is why in teaching this course I decided to actually have a well-defined unit that talks just about this point namely, this what and where is the voltage. Specifically, for example, there is a lot of interest these days in something called topological insulators, and in these materials you have what's called spin-momentum locking. So what's this? Well, the idea is that electrons which are moving to the right, that is, which have a momentum in some to the right also have their spins in the upward direction. So if you're moving right, the spin is up. If you're moving left, the spin is down, you see. So that's what you call spin-momentum locking. Because ordinary conductors if they're moving right, you could have either up spin or down spin. If you're moving left, you could have up or down. But in these materials those two things are locked together. And so what that means is that what I've shown here as this difference between the Quasi-Fermi levels for plus and minus, which, as I said, occurs in any material. Whenever there's a current flow the Quasi-Fermi level for right-moving carriers is different from that for left-moving carriers. But what happens in these materials is that difference also shows up as a spin potential, that is, as a difference in the Quasi-Fermi levels for up spin and down spin. And this can actually be measured as well with a probe, that is, if you put a probe here, you know, it's like a scanning probe with which you look at the potential, the electrochemical potential right there, ordinarily what you'd measure is the average of the two. So you just get this black curve. But if you use a magnetic probe, then depending on the direction of the magnet you could be measuring the spin potential, that is, if the magnet like points into the paper, like I've shown here, then you'd be measuring, say, up. If its point out of the paper, you'd be measuring down. So what that means is that these Quasi-Fermi levels differences can actually be measured and have been measured, see. And there's also other structures where people have engineered it. I mean, there's been enormous progress in this field of spintronics in the last 10, 20 years. And one of the things people have shown also is you could have situations where, when you look in a certain region, the up spin potential goes down as you go to the right. Whereas, the down spin potential goes up as you go to the right. So the result is current, of course, depends on the slope of this electrochemical potential. What that means is up spins diffuse to the right; down spins diffuse to the left. Now something like this is very hard to understand if you're thinking in terms of electrostatic potentials, because after all there's only one electrostatic potential. It either slopes to the right or it slopes to the left. And in terms of that it's very hard to understand how one type of carrier could be going right and the other type of carrier could be going left. And what I believe is in the coming years there will be more and more of such examples in nano-electronic devices which will make it very important to really understand what potential means, what it is really about. And this idea of Quasi-Fermi levels and the difference species can have different Quasi-Fermi levels, see. 

[Slide 7] Well, with that introduction then, we're now ready to move on to Lecture 2 where the first thing I'll try to show is how this additional, this ballistic resistance that we talked about, how we can get that out of the standard diffusion equation by modifying the boundary condition. That's what we'll start with the next lecture. Thank you.