nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.6: Diffusive (D) Conductance
========================================
 

[Slide 1] Welcome back to Unit 1 of our course, the new perspective. We'll now talk about the diffusive conductors. In the last unit, last lecture rather, not unit. 

[Slide 2] The last lecture we talked about the ballistic conductance. Of course, we have this general expression for conductance, and what we did was say that for ballistic transport, the time is length divided by average velocity, and that's how we got this expression for the ballistic conductance. For diffusive conductance, we'll do something similar. We'll argue that when an electron follows this diffusive path through a conductor, that is, it doesn't go like a bullet. What it does it hit something, goes off in another angle, hit something else, and eventually gets to the other end. Then the time it takes to cross a length L is proportional to L squared and can be written as L squared divided by two times what's called the diffusion coefficient. See. Now this we'll talk a little more about it in the next slide, but if you accept this, then you can combine it with our general expression for conductance, and now you will get an overall conductance that will be proportional to area divided by length. You see, for the ballistic case we just got area. Now we get area divided by length. That's because the time, instead of being proportional to length, is now proportional to length squared. And so this is just like what you expect for big conductors. Conductance is area divided by length times what you call the conductivity. So you can identify that with the conductivity. Now as I mentioned earlier, this expression for conductivity is standard, it's known, but not as well-known as the Drude formula, which usually you see in freshmen physics texts, and, but this really is more general, alright. And you can see how easily it comes out from this approach. We had a general expression for conductance, and time is proportional to L squared. That comes from the, this general theory of random walks and Brownian motion and if you combine the two, you get that. OK. Now what I'd like to talk about next is this expression for the time that we are using here. 

[Slide 3] I'll just to justify this a little bit, and so that's this time that's proportional to L square over 2D. Now one way to think about it is to use this basic expression that I had introduced in one of the earlier lectures when I say that the number of electrons in the channel is related to the flux of electrons per second times the time that each electron spends in the channel. Now this is a, this takes a little thinking, and as I mentioned earlier, there's an analogy that I've often used to justify it. If you remember that if you have a graduate program with, say, a hundred students graduating every year, and each student, they spend five years in the program, then the number of students in your program will be 500. Hey, that's not a completely obvious thing. You'd have to think through it a little bit, but it's well known. It's something you would, might have encountered before, and what we have here is the electronic version of that same argument. OK, now based on this, then, you'd say, OK, let's try to plot the, what's the number of the electrons inside this channel. So what I've plotted here is electrons per length per unit energy inside this channel, and for ballistic transport, you see, let's first do the ballistic case, and then we'll do the diffusive case. For ballistic transport, you see, we can say that all the states that are carrying current from source to drain will all be occupied if the, let's say the Fermi function in this contact at this energy happens to be one, which means if all the states here are filled, then all the states in the channel will also be filled because they're all coming from here whereas the states that are carrying electrons the other way will all want to be empty if in the channel, if in the contact the states are empty. Now you could, again, understand by thinking of, in terms of cars on a highway that is supposing we have the sources, let's say one city. Let's say somewhere in the south, and this is another city somewhere in the north. So you have this northbound lanes, and you have a southbound lanes. Now let's say something big is happening in this northern city. So that lots of people are trying to get from this source to the drain. Which is kind of the equivalent of having f1 equals 1. That is at this energy, there's lots of electrons waiting to get on the highway. On this side, there's no electron trying to come back. Well, what did you expect to see on the highway? What you'd expect to see is all the northbound lanes would be just completely packed, like a traffic jam or mostly, all bumper to bumper. You look at the southbound lanes, they're all completely empty. So that's exactly what you'd expect to see. Half the states completely full, other half completely empty. So what would be the number of electrons? It's, like, D over 2 is like the density of states per unit energy. Yeah divided by 2, well, because it's half the state. It's only the northbound ones. And then I divided by length to get the electrons per unit length. So that's D over 2L, you see. So if you look at the number of electrons inside, in this channel, it will be this electron density times the length. That's the total number of electrons. So you'd write q as D over 2. What about the time? Well, the time for ballistic transport is L over v. And what this expression tells you is if you want current, you just take the q and divide it by the t, and that's how you'd get an expression for the current. Now let's go on to diffusive transport. What we'll do in this case is we'll kind of use the relation in reverse. That is, you see, for ballistic transport, it was easy to write down the time. For diffusive transport, it's not quite so easy. So instead what we'll do is we'll try to write the current using the standard diffusion equation, and then try to deduce the time from there. So what, so first let us try to see how the distribution of electrons will look like. So what's different now is, you see, for ballistic transport, all the northbound lanes were filled, all the southbound lanes were empty. For diffusive transport, it's, like, people, these electrons get onto the northbound lanes, but then they hit something and turn around and come back. So at this end, all the northbound lanes are filled, but then, by the time you get through your channel, the number of electrons has trickled down to some very small value at the other end. Whereas if we look at the southbound lanes, they're all empty here because no one's getting onto it. There's no one from the drain trying to get on, but then it gradually builds up. Why? Because the electrons that turned around from the northbound lanes are actually going backwards. See. See, if you actually solve this properly, and this is something we'll do in the, a later unit, like, the unit three of this course where we'll actually talk about the differential equations properly. But you can kind of see that all the northbound lanes will be very full at this end and get relatively empty at this end whereas the southbound lanes also similar. It's kind of empty at this end, but will gradually fill up at this end. So what's the number of electrons then on the road? If you look at this entire road, and look at what's the number of electrons, here, you had a certain number, D over 2 for northbound lanes, certain number, D over 2, rather, 0 for the southbound lanes. Here, north and south are equally occupied, but each one is only half as much because instead of a rectangle it's, like, a triangle. So when you look at the area, you get D over 4, but then it's D over 4 for northbound lanes, D over 4 for southbound lanes. And so when you add it up, the charge is still the same, D over 2. OK. What about the current? Well, that's where we invoke the diffusion equation because the diffusion equation says the current in this diffusive transport is given by this diffusion coefficient times the derivative of the electron density. So what's the electron density? Well, that's, like, D over L at this end. We've got southbound and northbound. So D over 2L for each one. So it's, like, D over L. That's the electron density, and then dn and dz. So D over l goes to 0 there, and this length is L. So it's D over L divided by L. So this is diffusion coefficient. Carries over like so, and the dn  dz is like D over L divided by L because there's a straight line. So the slope is this much, which is D over L. It's like D over 2L for each of the two types of two lanes, the northbound and the southbound, and this much is L. So it gives you this. Now you can combine the two, and you'd get the time, which is Q over I, and it will be what I had written down before, this L square over 2D. So if you accept the diffusion equation, you can kind of see how you'd get a diffusive time that is L squared over 2D bar. Now where does the diffusion equation come from? Now that's kind of a detour that I don't want to get into because it's something that's very well known, and one of the nice references which gives a very readable description of all this is this book on random walks. It's a short book actually, and what it shows is this diffusion equation, where it comes from, and the fact that the diffusion coefficient is given by the average value of v squared tao. I mean, v component of v along the z direction. 

[Slide 4] So this is then our basic result that what we had obtained in the last lecture was a ballistic conductance where we, what was involved was the average velocity, which is the average value of the absolute value of vz, and now what we have done is looked at diffusive transport, and shown that you obtained the standard result for the conductivity in terms of this diffusion coefficient and the density of states and the diffusion coefficient is given by v squared, vz squared tao. So we are now ready to connect up the two and obtain a general expression, and which is what we'll do in the next lecture. Thank you.