nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 3.3: Quasi-Fermi Levels (QFL's)
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[Slide 1] Welcome back to Unit 3 of our course and this is Lecture 3. 

[Slide 2] Now what we did in the last lecture was to introduce this new boundary condition that for the electrochemical potential that we could use to solve the diffusion equation and get the correct answers once that actually includes the ballistic resistance and as we have shown that this new boundary condition kind of amounts to having these extra resistances at the interface. Now what we want to do next is try to show where this boundary condition comes from except that we need to do that in two steps and we take the first step in this lecture. That is in this lecture what we'll do is we'll introduce this concept of quasi-Fermi levels. The idea that there are two quasi-Fermi levels one for right moving carriers and one for left moving carriers and we'll show that the correct boundary conditions for these quasi-Fermi levels is that the mu plus. The one for right moving carriers at z equals zero is mu1. That is this mu plus has to connect up to mu1 here. Whereas when you look at mu minus it has to connect up to mu2 at this end and what we usually call mu, what we have been calling the electrochemical potential is actually the average of the two. So what we derived in the last lecture was the appropriate boundary condition for this average. And what we'll do in this lecture is we'll try to obtain the correct boundary conditions for these two quasi-Fermi levels and then in the coming lecture we'll connect it up. We'll show that this implies this how to get from here to here that's what we'll do in the next lecture and thereby complete the story. But right now then what we're trying to do is explain what's this mu plus and mu minus and why these are the appropriate boundary conditions to use. Now for this purpose it's useful to start from the ballistic end. That is this is the picture of course for any channel which could be very short or it could be many mean-free paths long but let's state the case of the ballistic limit which means where the length of the channel is much less than a mean-free path. In that case you see this middle region, this channel resistance that's a very small quantity because L is much less than lambda and so the entire resistance is just an interface resistance. So in that case what you would expect is that the way the chemical potential will vary is inside the channel it will hardly change. Why? Because that resistance is small. This change in potential kind of corresponds to the resistance. 

[Slide 3] So what you expect is this will be flat inside and the question is why is it halfway in between? Now what we like to argue is that actually there are these two separate quasi-Fermi levels and if you looked at the quasi-Fermi level for right moving electrons it will be up here connecting up with mu1 and if you looked at the quasi-Fermi level for left moving electrons it would be down here lining up with mu2. The question is why is that? Now to understand this let's think a little bit about what this electrochemical potential represents. As you know it comes from the Fermi function. The Fermi function has this property that if you go to energies high above mu it's zero. If you go to energies way below mu it's 1. So any state below mu, way below mu is completely filled. It's a hundred percent chance of being filled. Fermi function is 1. Up here hundred percent chance it's empty. The Fermi function is zero. And at low temperatures this transition is actually quite abrupt. That is the energy range over which this change takes place that's on the order of a few kT, so when temperature is small you have what's called a brick wall type of thing where it goes abruptly from 1 to zero and so when you're thinking about things it's often easier to think in the zero temperature limit because that's still approximately correct even when the temperature isn't really zero. So when I draw electrochemical potential up here what I really mean is that all the states below it are completely filled in this contact. Similarly, when I draw the electrochemical potential in this contact here it means that all the states below it are completely filled down there. 

[Slide 4] But here the states are empty, so if you consider this energy range here between mu1 and mu2 then if you looked at the Fermi function it would be 1 in this contact and zero in this contact. One here because it's below mu, zero because it's above mu so here then we have a situation where in this energy range there's lots of electrons filling up these states, whereas these states are just completely empty and what we'd like to argue is that as a result all the right moving states would be completely filled and all the left moving states would be completely empty. Now for this I've often used the analogy that expresses the situation quite well. It's like think of two cities and I use a local example since I live in this small city called Lafayette and it's to the south of Chicago. And then there's a highway in between and as with any highway there's northbound lanes and then there's southbound lanes. So now think of the situation where something big is happening in Chicago so a lot of cars at the Lafayette end are trying to get on the highway. They all want to go to Chicago and not surprisingly there's no one trying to come back. So at the Chicago end it's all kind of empty. That kind of illustrates the situation here. F1 equals 1. F2 equals zero. So what would you expect if you looked at the occupation of the lanes on the highway which is like the occupation of the states in the channel? What would you expect? Well, what you would expect is that all the northbound lanes would become completely filled and all the southbound lanes would be complete empty. And that's exactly what would happen here too. All the right moving states, that's kind of like the northbound lanes, they're all completely filled, filled up to here. That's mu plus and all these states are completely empty. That's the mu minus. So this is the situation when you have ballistic transport, that is length is much less than a mean-free path. In this context what it means is that when an electron, when this car gets on the highway it has no way of turning around. It's ballistics so it has to keep going straight until it gets out at the other end whereas, in practice if you had the diffusive conductor, you see that would correspond to a situation where cars could turn around and come back. So in that case what you would can expect is on the northbound lanes you start out with bumper to bumper traffic but then as you go along it kinds of thins out. Why? Because some cars turn around and go back. Similarly, when you look at the southbound lanes it all starts out being empty but then it gradually builds up. It builds up because cars from here have actually turned around. 

[Slide 5] So again, in terms of this quasi-Fermi levels what you expect is something like this that the quasi-Fermi levels for right hand electrons start out at mu1 but as you go through the channel it gradually goes down. And if you look at the quasi-Fermi level for left moving electrons it again starts out down here but then gradually builds up. So the boundary condition then that you expect for this electrochemical potential or this quasi-Fermi levels, this mu plus and mu minus is that mu plus at this end should be equal to mu1. In other words, if states are filled here they will want to be filled inside as well and if states are empty here they'll also want to be empty right there. Now the key point here of course is that when you think of mu then think of it as something that tells you of the degree of filling of the states not as something that signifies the average energy of the electrons because you see if the average energy were actually going down then there must be some dissipation and as I've mentioned before on this scale of things dissipation usually happens somewhere else, mostly in the contacts. So when I draw this mu going down like this I don't imply that energy is being dissipated down here necessarily at all. All that it says is that the number of electrons, density of electrons will keep going down and that we expect just from common sense, I mean I just based on what I said. If you had a highway with lots of cars trying to get from south to north the northbound lanes would be completely filled and then if cars got fed up and started turning around then what would happen is the filling would gradually go down and the other side would go up. So the basic boundary condition then we expect for these two quasi-Fermi levels the first point is that you expect there to be two quasi Fermi levels, two levels that describe the occupation of the northbound lanes and the southbound lanes. They don't have the same electrochemical potential and the second is that the boundary conditions from these Quasi-Fermi levels is mu plus at z equals zero is mu1. Mu minus at z equals L is mu2. 

[Slide 6] So with that then we are ready to move on to the next step which is these are the quasi-Fermi level boundary conditions. The question is how do we connect the boundary conditions we talked about in the last lecture which is for the average of them? That is this electrochemical potential. So what we'll show then in the next lecture is how what we just obtained implies what we had deduced earlier. Thank you.