nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.9: Drude Formula
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[Slide 1] Welcome back to our Unit 1 of this course, the new perspective. We're now on the 9th lecture. Now, we've more or less developed this new perspective. 

[Slide 2] And, what we want to do in this lecture is tell you a little bit about how this new perspective connects to the standard Drude formula that I've mentioned a few times now, and which is what you normally have seen in freshman physics. And, for this discussion, I'm assuming a density of states picture looking something like what I've shown here. You have a band of energies here. Then, you have some gap, a small gap here. And then, you have another band of energies here. And, you might think of this as conduction band and valence band if you're familiar with those terms. And, let's say your electrochemical potential is somewhere here for the moment, so states are filled up to here. Now, how do you find the conductance? Well, the perspective that we have talked about, the conductance is given by this energy average of this conductance function. And, the conductance function is given by q squared D/2t. Okay? And, what we saw is that for diffusive transport, this leads to an expression for the conductivity that we talked about. Now, the point I want to emphasize is that this involves energy average and the averaging is done with this derivative of the Fermi function. See. And what does that derivative look like? Well, it's got a peak right around the electrochemical potential. The width of this is on the order of a couple of kT, a few kT. So, the point is that the conductance is determined entirely by the property of electrons in a narrow energy range right around mu. Okay. And what is down here doesn't matter at all. Current flow is determined by electrons only and this small energy window. Now, if we look at the Drude formula, we have something called this electron density. And often, you might say well there's lots of electrons. You should probably be including everything. But, that's not true. The way the Drude formula is applied it says that this n is really the free electrons because filled bands do not conduct. So, these are all filled bands that do not conduct at all. So, what you should use is only the free electrons which go from here to here and stop right there. You don't count all this. Okay? And, if there's a clean gap, I guess, you can kind of understand that. Though, I've always found this is one of the hardest things to come to terms with when you first learn about these things. And, in materials like graphine, there's actually no clear gap either. You see, this kind of ends where that one starts. There's no gap actually. But still, you should only count up to here, not below that. See? That's well known. But, the point is this is a very different result from this one. Okay? Now, what you might think is, what would happen, let's say the Fermi energy instead of being here like we had talked about was actually down here? Well, from our point of view, all that happens is, the df dE moves down here. So, when I'm looking at the conductors, this density of states, instead of looking at the density of states up here, I should be looking at the density of states down here. That's all. Nothing's changed. Same formula, just look at the density of states wherever your chemical potential happens to be. But, if you want to use the Drude formula, then, of course, you might think you want to use the electrons in here. But not at all. That's not what you should be using. What you should be using is the holes, the empty states up here. So, if you're near the top of a band, you should look at this empty states. And, as I said, this is one of the things that takes a lot of time to get used to, actually, when you first learn about current flow. But, if you adopt this point of view, then I'd say there's no confusion at all. You always use this formula, the df dE factor is either here or here depending on where the chemical potential is. 

[Slide 3] Okay? Now, let's go a little more into this. So, let's take the case where the Fermi energy or this chemical potential is up here. The point is that the expression we obtained for conductivity would involve df dE whereas, if you look at the Drude formula, it involves a number of electrons. And the number of electrons is obtained by taking the density of states, multiplying it by the Fermi function because Fermi function says how many of the available states are actually occupied by electrons. So, if you actually want to calculate n and if you want to calculate this area here, what you have to do is state the density of states multiplied by the Fermi function which tells you if it's occupied or not and then do the integral. So, if you wrote this out it would look something like this. And, the first thing to note is here we have df dE here you have pure f, just the Fermi function. And these are two very different functions, you see, those two functions because, if you plot it out, this one looks like sharp peaked function right along the electrochemical potential. That one, as you have discussed is one below the chemical potential and is 0 above. So, this is like the derivative of that. So, it's a very different function. And, what I've shown here is what you people generally refer to as a degenerate conductor. This word degenerate is used in many different contexts. In this context, degenerate conductor means that the Fermi energy is kind of way into the band somewhere so that when the kT, this rest of this curve this kT is much less than the distance from the Fermi energy to the bottom of the band. So, that's what you'd normally call a degenerate conductor. Now, in that case, you can see, there's a very big difference between this function and that function. And, it may not be at all obvious that an expression like this would give you the same result as an expression like this. Okay. And, in the next unit of this course, actually, we'll talk more about this. We'll try to show under what conditions this and this actually end up giving you the same answer. But, for the moment, what I want to tell you a little bit more about is non-degenerate conductors. That is people who deal with semiconductors, you see, often use this non-degenerate approximation And, for non-degenerate conductors it's relatively easy to see the connection between this and that. So, what's a non-degenerate conductor? Well, it means that the chemical potential, instead of being up here, is actually down somewhere here below the band. So, if it's below the band, that means, of course, these functions are also moved down. 

[Slide 4] So, it's right around where the chemical potential is. Now, in this case, as you can see, all the conduction, of course, is somewhere here, and so the relevant energy is somewhere here not in this region. In this region, the Fermi function is big but there are no states to conduct. So, the only states that matter are up here and that's where what you have is the tail of the Fermi function and you have the tail of the df dE. And, the point is that the tail of the two functions actually look very similar. Why is that? Because, when you're looking at the tail, then you see that exponent the e minus mu/kT, you know the energy is way up here. So, that's a big number. So, it is exponential of a large number. Okay, because you're looking at this region. And, because it's a large, exponential of a large positive number, now, that's also a large number and so you can drop the one. And, if you drop the one you get exponential minus e minus mu/kT. So, the Fermi function then becomes this exponential function which you often call the Boltzmann distribution. That's this one. And that's what you call the non-degenerate approximation. Now, the point is that the Fermi function is an exponential. Then you see there's not much difference between f and df dE because one very nice property of exponentials is that whether you take derivative or integrate, exponential stays an exponential. You see. And so, what you can show easily is the derivative of that with respect to e is really just f divided by kT. So now, there is this function and that function is essentially the same shape. It's just scaled by this kT. That's all. So, in that case, when you look down here, it seems like well the function we have and the Drude formula don't look all that different anymore. And, after all it's a q squared and a density of states, you have a q squared and a density of states. Here you a Fermi function, here you have a derivative of the Fermi function. But then, we agreed that in this limit the Fermi function and this derivative are much the same. So, they're kind of looking similar. So, let's try to sort out which terms are different. You know, so, if you look here you see that this term the tao/m, that doesn't appear anywhere here, so, lets pull that out. So, when you look here, the term that's different is this d bar. And, also, instead of f we have df dE, but then the df dE is like f divided by kT which is why I've divided by kT. So, you could say that this expression and that expression are essentially the same except for a factor of d bar/kT over here and a factor of tao/m over there. Now, let's multiply both by q and then you might recognize this q tao/m depending on your background. I think I brought this up once in an earlier lecture. This is a quantity that's often called a mobility. In fact this is the quantity that people carry in their head. Anytime you have a new material that's with lots of mobility, that's mobility. But here what you have is the diffusion coefficient. But again, if you have taken a course on semiconductors, or in some other context you might have heard of this Einstein relation. It says that diffusion coefficient times q/kT is mobility. And that's again true only for non-degenerate conductors. Not true in general. In general, there's a correction factor. But, for non-degenerate conductors, that and that are the same thing. So, basically, then, the expression we are talking about, the one that I said is the more general expression is essentially what you would've got out of a Drude formula as well. But, you also see this interesting subtleties that you have to be careful with about when you use the Drude formula. So here, we talked about this conduction through the electrons in the conduction band. 

[Slide 5] Now, what if you had electrons in the valence band down here? Then, as we just discussed, you actually need the number of holes. Now, why is that? Well, one way to understand it is that you see now we are down here and we're looking at this non-degenerate case. Since the mu is here, you see, the del f del E doesn't look anything like f at all, if you think about it, because here I argued that both the del f del E and f are essentially exponentials. So, del f del E is equal to f/kT and that's what I said, well that's where you can, where the two expressions all become the same. But, look down here, this one is still an exponential just like that one. But, this one is not 0 at all. It's now up at 1. It's a constant. So, you could obviously not equate that to that anymore down here. But, the point is that, when you're down here, it is 1 minus f that you can approximate with an exponential. This you'd have to convince yourself you'd take f write out 1 minus f and then assume that this e minus mu/kT is this large negative number. So, if you have that, so the e's we are talking about are somewhere here, mu is here. So, e minus mu is a negative number. So, what you can show is you have this exponential of a large negative number is 1 minus 0. And, this approximation then is, can be used. So, in other words, you don't have the Boltzmann approximation for f anymore. What you have is the Boltzmann approximation for 1 minus f. And so, in this energy range, you can replace del f del E with 1 minus f. So, what that means is, if you're trying to use the Drude formula, then, you would associate this n, the number of free electrons with the number of empty states or number of holes as we had mentioned earlier. But, the method I really wanted to get across though is, this Drude formula is what we learned in freshman physics, what you carry in your head and so on. But, what we obtain straightforwardly in this course so far, see this is the expression that you normally don't see because it requires advanced formulisims you use to get here. That's why, normally, it comes much later in any course. And, not like what you have in chapter 1. But, here, this is what kind of came naturally because of our new perspective and this is really much more general. Whereas, when you use something with the Drude formula you have to be careful about all kinds of subtleties like this as to this what exactly are the free electrons, the idea that sometimes you should be counting electrons, sometimes you should be counting holes. And, this non-degenerate approximation is where you see a nice clear connection. What we'll do in the next unit of this course, we'll talk about things in general, you know, where you don't use the non-degenerate approximation. But the basic message is that what we obtain in this course here, this expression, that's really the general one. And, when you use this one, you don't need all these subtle arguments. See. This is general. 

[Slide 6] Okay. With that, then, we are now ready to sum up what we did in this unit of the course. Thank you.