nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 2.3: Counting States
========================================
 >> 

[Slide 1] Welcome back to the Unit 2 of our course, the energy band model. This is the third lecture. 

[Slide 2] And what we did in our last lecture was we talked about this energy momentum relation, and what we want to do is translate a given energy momentum relation into a density of states or number of modes. And an important step in the process is this idea of counting states, and that's what this lecture is about. So the basic idea is that when you have this energy momentum relation, see, usually you see -- you could say that electron could have any momentum. And if it can have any momentum, then correspondingly, it could have any energy. And in that case, it's not clear how to count the density of states. It's like a continuum. But the argument you make is that once you have an electron in a solid with a certain finite size, then not all momenta are allowed. Only certain momenta are allowed. And so, you have these discrete values of momentum that are allowed inside the solid. And corresponding to each discrete value, of course there are allowed energies. So if this is one allowed momentum, then there's a corresponding allowed energy here, which then gives you an energy level. And similarly at that momentum, there's another allowed energy level in this band, which will give you an energy right there. And so, this discrete set of momenta kind of translate into a discrete set of energies, which we then translate into this density of states or number of modes. And what we'll be doing in this lecture, though, we'll be trying to count this discretization in momentum. That is, what we'll ask is that given a certain maximum value of momentum, p, how many allowed states are there whose momenta is less than that. So we define this function N of p, which tells us, as I said, the number of states within in a circle of radius p. I mean circle if it's two dimensions, sphere if it's three dimensions, and just a line if it is one dimension. In fact, we'll do all those three cases, the three dimensions. But once you have this N of p, that's of course not the end of the story because what we really want to do eventually is to get to density of states and number of modes. And the way it works is that given this N of p, and since we know the energy momentum relation, you could kind of eliminate the p and get an N of E, which tells you how many states have an energy less than some value, let's say. And from that you can deduce the density of states. Density of states is like the derivative of that in the sense N tells you all the states up to a given energy, E, and then we ask, if I increase this energy a little bit, how many extra states do I get? That kind of tells you the density of states at that energy. So this is what we'll be doing later in the subsequent lectures. Right now, though, this lecture is entirely about finding that function, N of p. So let's get down to it then. 

[Slide 3] How do we discretize the momentum? Well, that's where you have to bring in the wave nature of electrons, which means basically you need quantum mechanics. But as I mentioned in this course, I'm not trying to -- I'm trying not to get deep into quantum mechanics. We'll do it in an elementary way. And that is based on this idea of a de Broglie wavelength, which I am sure you have seen before in freshman physics. The de Broglie wavelength is related to h divided by the momentum. So what -- then what we argue is if you have a solid with a given length, L, then only those momenta are allowed. So an electron cannot have any momentum. It can only have certain values p sub n such that the corresponding wavelength is h over pn and the length of the solid is an integer number times the wavelength. So the condition you're imposing is that an electron with a certain momentum has a certain wavelength. And that wavelength must kind of fit into this box of length, L. Now if you accept that, we could turn that around and say, well, pn must be an integer times h over L. And that, of course, immediately then tells you what this spacing is. The allowed values of p are simply discrete multiples of h over L. So if you had a big box, L would be large, and so these allowed values would be awfully close together. If you had a small box, L would be small, and the allowed values would be further apart. And that is why, of course, as you make a box bigger and bigger, you get more and more states. You know we have discussed before that usually density of states usually depends on the size of the solid. You make something twice as much, you get twice as many states. Here in this model, the way it works is you make something twice as big. The state allowed values get twice as dense. Now I want to make a couple of comments about this momentum and k because here we have talked about an energy momentum relation. Often people talk about the Ek relation. And this momentum and k are related by just p equals h bar k. And they kind of embody this particle property and the wave property. That is, when you think of electron as a particular, it has this momentum. When you think of it as a wave, it has a wavelength. And k is like 2 pi divided by the wavelength. So there's a wave property, the k. And of course, this wave particle duality says that this particle property has to be related to the wave property through this relation and this h bar times 2 pi over wavelength. And this h bar, that's actually Planck's constant h divided by 2 pi. And so, 2 pi times h bar is just the Planck's constant. And so, p is equal to h divided by the wavelength, which, of course, you could turn around and write wavelength equals h over p. So these are connected things. Now one point I wanted to mention also is that we found this quantization condition in terms of the momentum. Often in the literature or in books, you'll see it in terms of the k. And these are just equivalent things because p is h bar k. And what you can convince yourself is if p is an integer times h over L, then k would be kind of that divided by h bar. And since h divided by h bar is 2 pi, it says that the allowed values of k will be integer times 2 pi over L. And that's a formula often seen in books and in the literature. But in terms of momentum then, the allowed values are spaced by h over L. And so now you can easily find this function that we talked about. You see we were trying to write the function N of p, which tells me how many states are allowed, how many momentum states are allowed whose momentum is less than some value p. Well, in 1-D, if you want all these states going from minus p to plus p, that means the range is 2p. And they are spaced by h over L. So you divide 2p by h over L. So this would be N of p in one dimension. Now what we'll do next is extend that to two dimensions and three dimensions, and then we are kind of done with this lecture. 

[Slide 4] But I'd like to make a couple of comments before we move on. And that is that remember the essence of this whole counting scheme, the way we are counting states is to say that length has to be an integer number times the wavelength. And this is what's often called the periodic boundary condition. And you might wonder where does that come from? Why does the wavelength have to fit into this length, L. And what I'd like to point out is that this actually kind of assumes that your solid is kind of like a ring. You see? Not a linear solid like you normally think of solids, but more like as if it's in a form of a ring because if it's in the form of a ring, then you could argue that when you have a wave that goes around this ring, it has to -- when it comes back, it needs to connect up to where it's started. Otherwise, there would be discontinuity there. And so you could argue that the overall wavelength must always fit into the length. And this is what you might call this -- I guess what you might call a poor man's quantum mechanics. That is, you are trying to get this quantum effect into this discussion in an elementary way. And this is sort of what people did before Schrodinger's equation came along. For example, if you are familiar with the Bohr model of the atom, the Bohr quantization condition, that also you could interpret in this way, as if you are saying that the wavelength has to fit into the circumference of that circle. But if you're not familiar with that, don't worry. The point I wanted to make here is that this boundary condition called the periodic boundary condition basically says that the integer number of wavelengths must fit into this solid. And one thing that might bother you is, well, I kind of can justify it if I had a ring. But I really don't have a ring. I usually have a solid looking like this, for example. And the answer is that justification people use is that if you talking of large solids, and you're trying to find the density of states in a large solid, usually the condition at the surfaces does not matter. So since the exact boundary conditions do not matter, what we do is use whatever boundary conditions make our life mathematically the simplest. And that's this periodic boundary condition. That is, if you actually knew the exact boundary conditions, you know the whole details would be much more complicated, and if you wanted to include them, whereas, here the argument is physical. You are saying, well, I know from experience that the real boundary conditions don't matter if I'm trying to find the density of states in the bulk of a large solid. Since they don't matter, I won't bother about what it actually is and won't -- and we'll just use an idealized boundary condition. And that is widely used in solid state physics, you see, for counting states. Of course, you might say, well, what happens in small things? Then you can do -- and you're absolutely right. In small things, periodic boundary conditions may or may not work very well. In fact, interestingly, there is one material where you can actually study it in a form that obeys the period boundary conditions. You see usually you will seldom have something in the form of a circle. But you see there is this material called graphene, which is like a single layer of carbon atoms taken from the surface of graphite, you know which is one of -- a material that's widely investigated. You know if you did a Google search, you'd get lots of papers about it, very exciting research area. Now this graphene is like a sheet of atoms. But it's also found in the form of a ring, which you call a nanotube. And one thing you could ask is, is the density of states in a graphene different from the density of states in a nanotube. And the answer is if it is a large sheet of graphene or a large nanotube, then they're almost the same. On the other hand, when you get to small things, you do see important differences. In graphene, in fact, there are edge states for certain orientations and so on. So bottom line is periodic boundary condition is widely used, works very well for large solids. In small solids, though, you may need to be careful. And there could be other things that this won't quite cover. So with that bit of a detour then, let's get back to what we're trying to do, namely find this function N of p using periodic boundary conditions. And we have already done that for 1-D. 

[Slide 5] What we now want to do is extend that to 2-D and 3-D. And that's fairly straightforward actually because if you're in two dimensions, then you see what you need to do is consider a circle. And within that circle, you have these states, and L is the length in one direction. W is the length in the other direction. So you have all these states which are spaced by h over L in one direction and H over W in another direction. So each state kind of occupies an area that's like h over L times h over W. And of course, the area of the circle of radius p is pi p square. And so N of p is area of the circle divided by that. And you can probably guess what the answer would be for three dimensions. Instead of the area of a circle, you have the volume of a sphere. And you can divide it by h over L and then again h over W1, h over W2. You've got these two widths. And the product we have written as area. So you could collect all this together and write a single expression for N of p in these three cases, the three dimensions, 1-D, 2-D, and 3-D. And the N of p is proportional to p to the power number of dimensions. That is, in one dimension, it's proportional to p to the power of 1, two dimensions it's p to the power of 2, p square, three dimensions is p cubed, p to the power of 3. So that's what we have written out here as p over h to the power D, and then the other factors are all in here. So this is the function that we'll now use in the coming lectures to get onto what we really want 

[Slide 6] to find, which is, say, density of states and number of modes. Note that this comes simply from the idea that corresponding to a momentum, there is a de Broglie wavelength, and it needs to fit into the box. And it doesn't make any use of the energy momentum relation at all. So this is a general method for counting states based on periodic boundary conditions that's widely used in solid state physics. And what we'll now do is couple it with the energy momentum relation from Lecture 2 and try to get at the density of states. That's the next lecture.