nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 2.4: Density of States
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[Slide 1] Welcome back to Unit 2 of our course, and this Lecture 4. 

[Slide 2] Now what we did in the last lecture is we obtained, or we discussed this rule for counting states, you see, using this periodic boundary condition. And obtained this function called N of p. What does N of p represent? Well, it tells you how many states are allowed, which have a momentum less than p. So as if, in momentum space, you consider something like a circle or a sphere; I guess a circle in two dimensions, sphere in three dimensions, and look at how many values of momentum in here are actually allowed, using this periodic boundary conditions. So that's what we did in that last lecture. What we want to do is now, in this lecture, takes this function and deduce the density of states from it. Which is the quantity that we actually want? And as I mentioned once before, the way you do that is, you'll need an energy momentum relation. And here we are assuming isotropic energy momentum, which means momentum is the same-- I mean, energy is the same, for momentum in all directions, only depends on the magnitude of the momentum. So that's this E of p. When I write p, I mean the magnitude of the momentum. And we could combine those two together and that is a function of E. What does that represent? Well it represents all, how many states you have whose energy is less than E. That's this N of E. And once you have that, you can take its derivative and that will give you the density of states. Why is that? That requires a little bit of thinking, because the way you should think about this is, N of E, tells us how many states we have in this blue region, assuming that the blue region corresponds to a certain value, E. Now, let's increase E a little bit, from E to E plus dE. So corresponding that the momentum's increased a little bit, and so we have a slightly bigger circle. And so when we go from E to E plus dE, there is an increased number of states available. And what you want to know is, how much it increased, because that's what represents the density of states at that energy. So density of states is given by the increase in N for a given increase in E. So that's how we are going to calculate the density of states. So now let's get onto a specific example. 

[Slide 3] So as I said, we have a function called N of p and that function, in deducing that function, we didn't have to worry about any energy momentum relation; it's a perfectly general argument, but the result depends on the dimensions, which I'll represent with a small d. And so we write N as momentum to the power d, because in 1-D, it's proportional to d to the power 1, 2-D is d squared, 3-D is d cubed. And there's a constant in front, which I wrote as A. And I guess when I look at it, I realize that's an unfortunate choice, because I'd earlier been using A to represent the cross-sectional area. Here, I should have called, given it a different name, like K or something else. What I mean is, pick some unimportant constant, because in this discussion, most important thing is how it varies with p. And whatever the constant in front, I called it A. Now what about the energy momentum relation? Well, in general, you might have say a bottom of the band Ec and then it varies with p in a certain way, and let's write it as E of p equals Ec plus again, some unimportant constant, and then p to the power alpha. As we discussed before, often one deals with a parabolic bands in which case alpha is 2, that is energy goes as p squared. But here, let's keep it general, we'll call it p to the power alpha. Now what you want to do is combine those two things, to get the expression for N as a function of E. And this requires a little bit of algebra, and what you'd get is this expression. That is, you see, p to the power alpha is like E minus Ec. And what we want, N is p to the power d. So you kind of have to take E minus Ec and raise it to the power d over alpha, and that will give you N. So this is a bit of algebra. I think you can try this out. From this and this, you can eliminate p to get this expression. Well, and now we are done. We want the density of states, so we just take the derivative of that. So when you take the derivative, as you know, when you take the derivative of say, to the power d, or d over alpha, what you'd get is this quantity, to the power d over alpha minus 1. Because whenever you take derivative, it subtracts 1 from the exponent. So that's the general expression for the density of states. Note that it depends on two things: d, which is the number of dimensions, alpha which tells you the energy momentum relation, how energy varies with momentum. So let's now look at a specific example. 

[Slide 4] Let's assume alpha is equal to 2. That is the, as I said, the most common case, that's this parabolic band. And if you put alpha equals 2, and consider three dimensions, that means d is 3. So the exponent here becomes 3 divided by 2, which is one and a half minus 1, so that's like a half. And so, the density of states looks like E minus Ec to the power half. Which is the square root of E minus Ec. And this is the most common form that you'd see for density of states is this parabolic band, assuming a bulk 3-D semiconductor, for example. And the density of states goes as this, square root of the energy. Now, the answer will change though, if we consider say, two dimensions. What do we get if we have two dimensions? Well, now it's 2 divided by 2 which is 1 minus 1, that's zero, which means d is like E minus Ec to the power zero; it's like a constant. And so when you plot it, you look-- it will look something like this. Of course, below Ec there is no states, so here it's zero, but once you cross Ec, it's a constant. That's this 2-D result. And the other interesting result that you will also see a lot is 1-D, that's what you sometimes call a quantum wire, like if you had a single, 1-D wire. In the 1-D case, you see it is 1 divided by 2 minus 1, so that's like a minus a half. And so, it's like 1 divided by the square root. And because it's 1 divided by a square root, it has this singularity right near the band edge, wants to go up. And these are things that have been experimentally observed, and people can measure the density of states on different materials and they have seen this, actually. This 3-D, the 2-D, the 1-D version. Note that, these results kind of depend on both the dimensions and this alpha. So for example, if you had a material with alpha equals 1, that is where energy momentum relation, instead of being parabolic was linear, and there's actually a material that's of great interest these days called graphene, where that's true. Where the common assumption is that the energy is linear with momentum. So all the results will change, because you see, now alpha is equal to 1, and you'd have to use that. And of course, in general energy momentum relation doesn't have to look like a polynomial, it doesn't need to have this polynomial form that I mentioned. You could have things like, that look like this. You know I earlier once mentioned that energy momentum could have this, quote unquote relativistic form. You know, where energy is the square root of some constant squared plus some p squared. Well, in that case, of course, you couldn't use that formula. You'd have to take this E and that N, find N of E and then take its derivative. And it would require some algebra to get there, but the principle is very straightforward. Take the N, take the E, eliminate p, get N of E and then take its derivative. 

[Slide 5] So I think we're done here, what we're trying to do. Mainly, start from this rule for counting states that we got from the third lecture, combine it with the energy momentum relation, to get the density of states. That's what we talked about. And what we'll now talk about next, is how you get the number