nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.5: Ballistic (B) Conductance
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[Slide 1] Welcome back to the first unit of our course, The New Perspective. This is Lecture 5 and we'll talking about the ballistic Conductors. 

[Slide 2] Now what we did in the last lecture is we showed that the current divided by voltage is given by this average of a conductance function, it's average over energy. And the conductance function itself, at any given energy, depends on the density of states of that energy and the time it takes for an electron to get from left to right. Now in general, I won't be writing this integral over and over. We'll talk about the conductance at a given energy. And it's understood that if the temperature is low enough, then you can just look at the value of this function at the Fermi energy or electrochemical potential. On the other hand, if the temperature is relatively high and this conductance is varying strongly with energy, you'd actually have to do that integral. But for our discussion right now, I won't be writing this integral explicitly anymore. We'll just assume we're talking about this at some energy. So what about the ballistic conductance? Well, ballistic means that electrons go through the channel like a bullet. And so it's straightforward to write down the time it will take to get from left to right. It's just the length of the channel divided by the average velocity, it's velocity of the electron. So if you combine those two things, you get an expression for the ballistic conductance. I just put this time from here into this expression, and that's what you'd get. Okay, now how does the ballistic conductance depend on the dimensions of the channel? Well, the first thing to note is this density of state is proportional to the volume of the channel. You see, usually in large devices people don't talk about density of states per unit energy. They talk about the density of states per unit energy, per unit volume, in which case it's a constant value for a particular material; whereas here we have defined the density of states as just per unit energy, not per unit volume. And so our D depends on biggest proportional to the volume. Okay? Now a thing to note then, is that this density of states is proportional to A times L, whereas the time itself depends on L. And so when you divide it, what you get depends just on the area. Okay? So you if plotted it, it would look something like this. And this was observed, I guess, back in around 1969 and it's often called the Sharvin Resistance. The point to note here is that this ballistic conductance is independent of the length. You could take a ballistic Conductor, make it twice as big. It's still the same ballistic conductance, which is very different from normal conductors where if you make something twice as long the resistance is twice as much, conductance is half as much. Okay? Now as I said, this was discovered around 1969, 1970. 

[Slide 3] But about 20 years later, something else came along which actually had a major impact on this field, which actually one could say was one of the first things that started this whole field of mesoscopic physics and this way of looking at things. What this experiment showed was that if you go to small ballistic Conductors, ones where the area is relatively small, then instead of seeing a conductance that goes linearly with area, you'd see something that goes in steps. It's like this quantum of conductors that I'll talk about in a minute times an integer, okay? Now how do you understand that? Well, this is something we'll talk about more in the next unit. But let me just get ahead of myself for a minute and tell you a little bit about it. If you take a 1 dimensional wire, what you can show is that the time it takes for an electron to get from left to right, this t, and this D is the density of states, this quantity 2t over D is equal to Planck's constant h. And if you take that number and put it into this expression for conductance, you'd get the ballistic conductance as q squared over h. And this is what is known as the quantum of conductance. And it's the inverse of resistance. So this quantity is like inverse of 25 kilo-ohms. Now how do you understand this? Well, 1 way to think about it is consider a device with, say, 1 level. And usually when we think of energy levels, we think of sharp energy levels, you know, things that have a very well defined energy. Now in practice, of course, energy levels are not sharp. It's actually broadened out somewhat. And the fundamental limit on that broadening is given by this uncertainty principle. That is if an electron stays in that level for a length of time t, then the minimum broadening is h over t. That's like this, as I said, this uncertainty principle. And if the level is spread out over Delta E, that means the density of states is like inverse of Delta E. See? And so from this uncertainly relation, you can kind of see why t over D might be proportional to h. Of course you won't get a numerical factor out of a hand waving argument like that, but you can kind of see why this has a fundamental limit there. And with this, you can then rationalize this ballistic conductance as being q squared over h. And this is something we'll go into in more detail actually in the second unit, where we'll do it properly. But from this point of view, you see 1 dimensional conductor gives you a ballistic conductance of q squared over h. And the real conductor you could view as lots of 1 D conductors in parallel, which you could call the conducting channels. So if you have M of them, the conductance would be q squared over h times M. And that's exactly what this experiment shows. It shows the ballistic conductance, which is some fundamental unit, this quantum of conductance times an integer. And one thing I should mention though, that actually this conductance here is normalized not to q squared over h, but to 2q squared over h. And the reason for that is energy levels usually, normal materials, come in degenerate pairs. That is 2 that have the exactly the same energy. There's an up spin and a down spin. So for a 1 dimensional channel, the conductance would have been q squared over h. But usually even in a 1 dimensional channel, you'd have two states that are up spin and down spin. And so it will be 2q squared over h. And then if you have M channels, you'd have 2q squared over h times M. Anyway, but as I'd mentioned here I've kind of gotten ahead of myself. This is what we'll talk about in the next unit. 

[Slide 4] For the moment let's just go back to the old result, which is the ballistic conductance is given by this product of density of states and the velocity. And this is the Sharvin Resistance. And this is the average velocity. So what do you mean by average? Well, the idea is that, in this channel, we are interested in the velocity along the z direction. That is the direction from the source to the drain. But in general, when you look at electrons inside they're going every direction. Right? Now, of course, if we just take the average of vz, the answer would have been zero because you have vz in every direction. But here for ballistic transport, what we are saying is we are interested in the average velocity of carriers going from left to right. And because you have applied a voltage, there are no electrons going from right to left. That's a point we'll come back to later. But so this is what the average velocity is. It's the average of the absolute value of vz. It is only the average of those that are going to the right. On the other hand, those that are going to left of course would have a negative of that. And if you took them all together, the answer would have been zero. But here, this is what we're interested in. 

[Slide 5] So that's the ballistic conductance. We are now ready to go on to talk about the Diffusive conductance. Thank you.