nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.8: Angular Averaging
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[Slide 1] Welcome back to Unit 1 of our course, the New Perspective. We are now on Lecture 8. 

[Slide 2] Now what we did in the last 3 lectures, I guess, is started from this expression for conductance we had and applied it first to the ballistic limit, then to the diffusive limit, and then we said you could connect it all together and write G as sigma A over L plus, lambda. And okay, now notice there's a little mistake here. This should have been sigma A. See? And that's equal GB lambda. And as I mentioned, this is this new perspective we are talking about where the fundamental material parameter is this ballistic conductance, and the conductivity times area, which should have been here, is the mean-free is this ballistic conductance time the mean-free path, this lambda, which we defined as the ratio of this diffusion coefficient to the average velocity. So what we'll do in this lecture is we'll talk a little more about what these 2 quantities are, this D-bar and v-bar. You see the v-bar that entered the ballistic conductance, the D-bar that entered the conductivity, and this lambda, which is this ratio of D-bar to v-bar. Because this bar indicates there is this averaging involved, so let's start with the simplest case you see. That is a 1-D conductor. Seeing 1-D, and we are talking of remember elastic transports. We are talking of electrons with a certain energy. So a certain energy corresponds to a certain velocity. But the velocity could be either in the positive z direction or in the negative z direction. So it could have 1 of 2 values, plus v and minus v. No other value because, as I said, if you are considering electrons with a single energy, as you know in this elastic model, we can think of one energy at a time, and then do an integral over energy. So we're considering one energy, one particular value of velocity, but it can be plus or minus. Now so what is the average value? Well, average value of v-bar has got two possible values, plus v and minus v. So basically the average value is just v. What about the diffusion coefficient? Well, that's vz squared tau, again two value -- two possible values. Each is equal to v squared tau. So you can just write that. And then what we call lambda is this 2D over v, 2 D-bar over v-Bar. And when you put that in, you get 2v tau. Now the 2 might puzzle you a little bit because thinking about it, you see this v is the velocity. Tau is the mean-free time. So you say, well, velocity times mean-free time. That should be the mean-free path. Now why a 2 in here? And the thing is that this is the mean-free path for back scattering. So the way it is, is here's an electron going along, and every time it hits something, that's -- it turns it and it changes its direction a little bit. And tau tells you how often it hits something. Now the thing is that when it hits something, the direction changes. And we are assuming that the process is isotropic. So you are going this way. You might scatter to that direction or to that direction. But what we are really interested in this context is whether it has completely turned around, whether it has back scattered because as long as you're still going straight, the point is you are still headed in the right direction. You are still in a north-bound lane, so to speak. So of all the scattering processes, there will actually be half, which will kind of keep you moving in the same direction, and the other half, which will turn you around. And so the time, the average time for actually getting turned around is not tau, but 2 tau. And so this lambda, 2 tau times v, is -- you could call the mean-free path for back scattering. Now the exact factor that appears, that, as you will see, will be different in different dimensions because right now we are talking about a 1-D transport. Next we'll talk about 2-D, where you can go around in a circle. And then we'll talk about 3-D. And that exact numerical factor will change a little bit. 

[Slide 3] So let's do 2-D. So in 2 dimensions, of course, we have the z direction, which is the direction between source and drain, and then you have the electron velocity, which can be at any angle. So that's the angle theta. So how do you average it? You want the average velocity. So we want vz. So vz means the component along z, which is v cosine theta. So I put a v cosine theta there. That's the absolute value of v cosine theta. And we have to integrate from minus pi to plus pi. Now to get the average value, of course, you should also divide it by integral d theta. So one way to check is let's say the quantity you are averaging was not this, but something with a constant value, say 1. Then you say after the average, you should still get back 1, or if we're averaging, say, 2, you should get back 2. So if you divide it by the same thing here, but without that, that ensures that our constant thing will average to the constant value itself. So this is the average value then. And what you could then do is take out this absolute value kind of by integrating only from minus half pi to plus half pi because the idea is that this is the range where a cosine is positive. If you look at the range that we left out, that's where the cosine is negative. And in that range when you take the absolute value, of course, you would get back positive. And so what we could do is just integrate over minus half pi to plus half pi, or I guess you could have written a 2 here, which is like the -- you are multiplying what you get from here by 2 because the part you left out will give you exactly the same thing. But then you'd also have a 2 here. So I've just dropped the 2. And this is now a straightforward integral, and what you get is 2v over pi. So if we're in 2-D then, that's the average velocity you should use. Note this extra factor of 2 over pi. What about the diffusion coefficient? Well, now the average that is involved is vz square tau, that quantity. What's vz square? Well, that's v square cosine squared theta. So I have pulled the v square out of the integral and left the cosine square theta in there. So what you have to do is integrate d theta cosine square theta divided by d theta. And again, this is a straightforward integral. And the answer you'd get is v square tau divided by 2. 

[Slide 4] Now for the 3-D case, which is the most complicated, I guess, mathematically. Here the way you think of it is you have these 2 angles, 1 angle, which is with respect to the z axis, and then you project it down to the xy plane and look at the angle with the x axis. So there's these 2 angles, theta and phi, which define a particular direction in 3-D. Now if you're not familiar with that, you know, it takes a little time to get used to this. So what I'll do is I'll say -- refer you to standard texts, you know, where people find, say, the area, the surface area of a sphere. As you know, if you had a sphere whose radius is r, the surface area is 4 pi r squared. How do you find that? Well, usually what you do is do this integral. And as I said, if you haven't seen this before, then this is not meant to be obvious at all. But I'm just saying this is how people normally find the area of a sphere. This is how you define the solid angle, is d phi and sine theta d theta. And then when you integrate all this, you get 4 pi. You get 2 pi from here and another 2 from there. That's how you get 4 pi. And that's how you get the area of -- surface area of a sphere, 4 pi r squared. Now what we need, though, is just this part because that kind of tells us how to integrate over angle. In 2-D, it was simple. It was just integral d theta. But in 3-D, you need this solid angle here. So you have to do this integral. And so when you write down the average velocity, you have to take v cosine theta like before, because we want vz, and vz is v cosine theta. So that's the v cosine theta. But you have to average over this solid angle. And again, the same issue, what you want to do is this cosine theta is positive for parts of it and negative for parts of it. What you want to do is just keep the parts that are positive because the negative will give you exactly the same thing. So if you take it out from both numerator and denominator, you are fine. Now you can do this integral, and that will give you half of phi. Now if you want the diffusion coefficient, again same story, now it's v square cosine square theta times tau. The v square tau is pulled out. So you have cosine square theta, which has to be integrated over the solid angle. And this will give you v square tau over 3. So we have kind of done -- gone through this thing then for 1-D, 2-D, and 3-D. 

[Slide 5] So if you collect all those results together, you see, what you would have is average value of velocity. In 1-D, it's just v, 2-D is 2. There is a 2 over pi, 3-D there is a factor of half. For diffusion coefficient, actually the result is quite simple. It's v square tau, 1-D is 1, 2-D is half, 3-D is 1/3. So it's like 1 divided by the number of dimensions. So -- in fact, there is actually a simple argument that can be used to get that result without doing all these integrals. But I won't go into that. I wanted to do everything in a straightforward, parallel way. So you get these averages directly. And now if you look at the way we define lambda as 2 times the diffusion coefficient over velocity, you can see what you will get. When you divide this by this, which is v square tau divided by v, that's what gets you the v tau. And the numerical factor in each case, you can try out. It's like 2 times this divided by that. So it's -- for 1-D, you get 2, like we had discussed before. That's this mean-free path for back scattering that I had mentioned, 2 times this is 1, divided by 2 over pi. That's what gives you pi over 2, 2 times this is 2/3 divided by 1/2. That's what gives you the 4/3. So the mean-free path the way we have defined it, the one that leads to this simple expression as L plus lambda, that mean free path then is v times tau, which is what you normally might think as mean-free path, but times a certain factor that depends on what the dimensionality of your conductor. It's always bigger than 1. That's because what we are really talking about is this mean-free path for back scattering. And so tau is the mean free time for scattering, but then only a small fraction -- only a fraction of all the scattering processes involve back scattering, where you actually turn around. And so the factor here is always bigger than 1. In 1-D, it's 2. in 2-D, it's 1/2 pi, which is like 1.6 or so. And here in 3-D, it's 4/3, which is like 1.3. So what we'll do next is I guess this kind of completes this new perspective that we are talking about. And what I want to do in the next lecture is talk a little bit about the standard perspective, the Drude formula, and how this relates to that. Thank you.