nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 2.6: Electron Density (n)
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[Slide 1]  Welcome back to Unit 2 of our course. 

[Slide 2] Now this is Lecture 6 and in this lecture what we'll try to do is talk about this electron density. That is so far you see we haven't quite brought in the electron density into the discussion very much. Because when we talk about the ballistic conductance, conductivity, the expressions we had involved density of states, it involved number of modes, but didn't quite involve this electron density. And what's electron density-- well the way you can think about it is if you had a band with a density of states like I've shown here, and if your electrochemical potential was somewhere here, then at zero temperature what would happen is all the states below the electrochemical potential, that's mu, below that would be occupied. So how many electrons would we have? Well it would basically be whatever many states you have below mu. So and what does n tell us, this n? It tells us how many states are actually there whose momentum is less than p. Whereas what we want to know is how many states are occupied. And of course all states below mu is occupied. So what you could do is actually use that function to obtain the electron density. Let me explain a little further. So this is the total number of states, now if you want electron density then you usually talk about number of electrons per unit length, or per unit area or per unit volume. So what we could do is take that capital N and divide by L in 1-D, divide by WL in 2-D, or divide by AL in 3-D. And so you'd have this function which you could call electron density, and it's exactly that function except that I have left out the L and the WL and the AL, and so this is what you're left with. And from this how could I find out the number of this electron density with a given electrochemical potential? Well first step would be get your energy momentum relation so you can convert the p into an E. That will then give you kind of electron density as a function of energy and then at zero Kelvin, of course all the state below E equal mu are filled. So what does this function tell you? Well it tells you the electron density you'd have if all states were filled up to E, because this tells you the total number of states per unit volume that you have whose energy is less than E. So at zero Kelvin, if I put E equals mu, I'll get the correct number of electrons. The electron density, I guess. Now, so this is fairly straightforward if you think about it at zero temperature. In that case the function we have been discussing, this comes back from, from Lecture 3, that itself will give you the electron density. Now what we want to talk about is how do you do this when T is not equal to zero Kelvin. That when it is at reasonable temperature, not zero. 

[Slide 3] Because then as you know the states are not occupied, we cannot quite say that the say that the state below mu are occupied and above mu are empty. They are occupied according to this Fermi function. And the Fermi function well below mu is one, well above mu is zero and it changes in between over an energy range of the order of kT. That's something we talked about early in Unit 1 if you remember, in fact, this picture of the Fermi function, this is the equilibrium Fermi function, guess I should have written f0, f zero, here, this equilibrium Fermi function this I've taken from one of the Unit 1 lectures. Now if you wanted to calculate the number of electrons what you should do then is N0, number of electrons would be like D, density of states, times f0. Density of state tells you how many states you have per unit energy, f0 tells you the Fermi function, tells you if the state is occupied or not. So if you want the number of electrons then you should take the number of states and multiply it by, whether its occupied or not. Now what's D? Well that's the derivative of capital N. So you have dN/dE times f0. Now what you could do is a bit of algebra, you can integrate this by parts. That is you have two functions here, N and f0, the derivative is on the first function, but there is this trick of integrating by parts whereby the derivative instead of being on the first function is transferred over to the second function. So if you do that you'll have first what you call this surface term, that is N times f0, whose values at the two limits you need, minus infinity and plus infinity and then you have the other term where us see the derivative is now gone from N and instead it appears on the f0 and there is a minus sign associated with it. So, here we had dnd times f0, now we have n times minus df/dE. Now, this term is zero. Why is that? Well, look at the two limits. Consider first the positive infinity. Well positive infinity means very large energy, that's where you can see the Fermi function is zero, so f0 is zero, so N f0 is also zero. Well what about minus infinity? Well that means I'm at very low energies. Now of course f0 is not zero, its one. So I can't ignore f0, but the point is N, the N of E that function is zero down here because it starts at the bottom of the band, that's where it starts growing. So, this quantity then is zero at both limits, doesn't matter what it is in between. What is needed here is the value at the two limits and they're both zero, so you drop it. And so you're left with that. And this kind of you can visualize, if you have a function n of E whose derivative is that function, D of E, so this is the integrated version of that, and the thing is how do you find the number of electrons, you can take D multiplied by Fermi function integrate or you can take N multiplied by the derivative of the Fermi function and integrate and those are the same thing. This is mathematically the same result as this, you see. And here the point is that when you go to very low temperatures, then of course this df/dE kind of becomes like the delta function right at that energy. So when you integrate you basically get the value of n at that energy, which is what we talked about in the last slide. That at zero temperature the electron density is simply given by the value of N of E, at mu. Whereas if you are at non-zero temperatures what you need to do is average this N of E over a certain range of energies determined by this df/dE. 

[Slide 4] And you'll notice here, you see, that how similar it is in general with the way we think of conductance or all the quantities we've been talking about. They also involve this average over energy in the same way using this df/dE function. So what we could do then is, this is this expression we have for N of p, if we take electron states per unit volume you get the small n of p and then you can combine it with the energy momentum relation to get an n of E and then if you average it over energy you'll get n zero and this is in spirit much like what we had been talking about. Conductance for example. So conductance also involves this average and usually we have been talking about this function and usually I have not been carrying around this integral everywhere. But G of e is like the conductance at a given energy and its understood that any time you want actual values you should multiply it by df/dE and average, like I told here. And then we're talking about this G of e that's determined by this ballistic conductance at the mean free path and the ballistic conductance itself has this expression of density of states times velocity or the number of modes. These are all functions of energy, but finally of course if you want a measurable quantity like the conductance at a given temperature you will have to average. So same here now. We now have a quantity n of E here which tells you the electron density and the understanding is at zero temperature electron density is just whatever n happens to be at E equal mu. At non zero temperatures you have to average. Now before finishing up this section I'd like to point out one little subtlety with respect to bands that run downwards. That is-- this discussion is fairly clear and straightforward like what we just went through, when the band runs upwards like this. That there is a bottom of the band and energies are available above it and there's nothing below. 

[Slide 5] On the other hand think of a band that is running downwards. Where there an upper value that's often written as Ev for the Valance band and the energy momentum relation runs downwards. As I mentioned that's quite common in solids. That's what you usually have in the valance band. And now if you have a electrochemical potential mu, then the point I'd like to make is that this N of e that we discussed will actually give you the number of states above mu and not the states below mu. That is it will essentially give you what you call the number of holes, that is the empty states, that is N0 is actually integral of density of states times a Fermi function but rather its density of states times one minus the Fermi function. Fermi function tells you whether a state is occupied. One minus f tells you if it's empty. So, this quantity what I'd like to show you is, this is what our N of E will give you. Let's say we define it this way, then for density of states then we write dN/dE except that now we need a minus sign, why is that? Well because you see now when you talk about N of E what it represents is all the states whose energy is higher than E, because momentum is zero right here. N of p tells you how many states have a certain, N of p tells you how many states have are available below a certain value of p, and that corresponds to how many states are available above a certain energy E. You see. So you have an N of E that is actually increasing as you go down in energy. What that means is density of states is kind of like the negative of its derivative. So that's why I wrote a minus dN/dE, and now you can do this integration by parts again. And you'll get these two terms. N times one minus f0 and then there will be another term which will be N times df0/dE. So this is exactly the same integration by parts that we did in the last slide. But one very important point I want to make is that this N times one minus f of zero is a quantity that vanishes at both ends, so you can actually ignore it. On the other hand if I had started with f0 instead of one minus f0, then I would have got N f0 over here. And N f0 would not be zero at one of the limits. Which means that I wouldn't be able to ignore this term. So whatever I did here only works if I use one minus f0, doesn't work if we use f0, because as I said, when you go to very low energies, you see, f0 is one and N is still increasing you see. So N is not zero, not like the previous case where I could say that when there are very low energies N is zero so n f0 is zero. So here N f0 is not equal to zero, but N times one minus f0 is. Because down here f0 is almost one, so one minus f0 is zero. So I can drop that term. So finally of course what you are getting then is that this quantity, our capital N, averaged over energy here will actually give you the number of holes in a band that is running downwards, rather that upwards. And that I had mentioned in the context of Drude formula earlier 

[Slide 6] that in Drude formula when you have q squared n tao over m, that n for valance band really should be interpreted as the number of the empty states. Anyway, so I guess what we have showed here is that this n of p that comes simply from counting states, no energy momentum relation involved. That n of p you could combine it with an energy momentum relation to get n of E and that would be like the electron density which you can average over energy to get this measureable electron density. So, that brings us, kind of introduces this electron density into our discussion and what we really want to do after this to finish up this discussion is connect this new perspective to the old perspective or the Drude formula. You see the Drude formula had conductivity expressed in terms of electron density. Whereas the new perspective we have the ballistic conductance and as I discussed this kind of, there is this correspondence. The role of electron density is played by the ballistic conductance and then you have the mean free time and the mean free path and in order to connect to these we needed an expression for electron density so that we can compare and that's what we have, what we obtained in this lecture. What we'll do in the next lecture is connect the two.