nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L1,2: Wave Equation
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[Slide 1] Welcome back to Unit One of our course. This is the second lecture on the wave equation. 

[Slide 2] Now, in the introductory lecture, I had mentioned that in modeling this hydrogen atom, if we postulate that electrons have this wave property so that it needs to fit into this circumference requiring that the circumference is equal to an integer times the wave, de Broglie wavelength, then you could actually get results that match the experimental observations quite well. And what the Schrodinger equation showed was when you provided a solid mathematical basis for this, it showed that using this equation, you could get the same results actually and plus some additional things. For example, it showed that you could have multiple wave functions which give you the same energy and these you could interpret as multiple energy levels with the same energy, degenerate energy levels are the same. And same here, the s, p, d levels and so on. Now, how do you, where do you get these levels and the wave functions? Well, that's by solving this differential equation which we will not be getting into. One of the things I often say is that the differential equations, of course, how to come up with a solution, that's difficult. But if someone gives you a solution, you can usually check if he's right or not. You don't have to take their word for it because if someone says, "Okay, here's the wave function for this level and it has an energy of so and so." Okay, good. You can always put it in and take derivatives. It will take you a while but after you've done all that, you can see what energy you get and whether it satisfies that equation or not. But so you could check out some of these, you know, with the wave functions which are all available. You can easily look them up, the wave functions in a hydrogen atom. But we won't be going into that. But the point is, of course, what it doesn't quite give you is this intuitive feeling for you know, why you have discrete levels because that, at the end of the day, is really based on this idea that when you have waves and they have to fit in then it gives rise to discrete levels. And so what we'd like to do is talk about in this lecture, to talk about some simpler examples where and which will also introduce a very important concept of this dispersion relation. And these simple examples are easily solved and these are solutions that you should be generally familiar with and will also show how these simple models show why when you can confine, you get discrete levels. Now, in this problem, when you apply it to the hydrogen atom, of course, you are confining the wave with the Coulomb potential. That is, you have a nucleus here which pulls the electrons towards it. So you have a potential one over r that looks something like this. And the electrons wants to get stuck in here. So this is kind of a complicated potential that keeps it localized in that region. And the discrete levels basically come from this confinement of the electron. That's the general principle. And what we'll do is look at that principle in a context where the math is a little simpler. 

[Slide 3] So the problem we'll look at -- so instead of this actual equation with a complicated potential, let's think of a problem where you have a constant potential there, okay, the U zero. Now, if you have a constant potential, then what I claim is you can easily write down the solution to that differential equation. Because in general, whenever you have a differential equation with constant coefficients, you can write the solutions in the form of exponentials or what are sometimes called plane waves. That's E to the power of i, kxx, kyy, kzz. How do we know that? Well, you don't have to take my word for it, even if you haven't taken a course in differential equations because you can check. You can easily check that this works that if I take this wave function, put it back into the differential equation, it satisfies that equation. How will I show? Well, the thing is that del squared that you see here, that's what called a Laplacian operator. It's the sum of the second derivatives with respect to x, y and z. Now how do you take the second derivative of this? Well, when you have an exponential, that's one of the beauties of that function is taking derivatives is real easy. That is, if you take the first derivative of this function with respect to x, it is equivalent to multiplying it by ikx, you know, whatever is in the exponent. So taking first derivative is like multiplying with ikx. Taking second derivative, well, that's like multiplying by ikx twice which is like minus kx square. So this differential operator acting on that function essentially is like multiplying it with the appropriate numbers. And so you can easily write down -- I mean, perform these derivatives, that is, the two derivatives with respect to x gives you minus kx square, the minus take off that minus over there so you get h bar square over two m, kx square. And then ky square and then kz square, okay. And so overall then, you get E psi equals that number times psi. And you could cancel out the psi's and you get an expression for the energy, E is equal to this, nu zero plus this function of k. And this is what's called a dispersion relation, the Ek relation. Okay, then for waves, you know, homogeneous medium, that's usually a very fundamental property that people look for. And you'll notice that when you think of electrons as particles -- now in the first course, we talked about the energy momentum relation and usually in the wave viewpoint, the Ek relation will look just like the energy momentum relation except that instead of momentum, you have h bar k, okay. Now this dispersion relation then is what we'll try to use in the next slide to try to explain why when you confine the wave, you get discrete levels. Because as it stands, it looks like energy is continuous. I could choose any k I want and I get any energy I want. But as soon as I confine it, we'll see what happens. But before I go on, let me just point out one little notational thing and that is that this quantity here, you see, is often written as exponential I k dot r. That's because, you see, r is this vector x in the x direction, y in the y direction, z in the z direction. Those are the unit vectors. Whereas the vector k is like kx in the x direction, et cetera. And when you take k dot r, when you take the dot product of two vectors, what you do is you multiply the x components, add it to multiply the y components, multiply the z components and add them all up. So that's kxx plus kyy plus kzz. And that's exactly what we have here, okay? So you'll often see this notation. 

[Slide 4] So that's kind of a summary of what we just talked about. What I want to show next is how from here, we can understand this discretization of energy levels for a simple one d problem. So and this is again, if you took a course in quantum mechanics, this is like the first problem you'd go through, something -- it's called a particle in a box problem. That's supposing you have an electron confined to a box whose length is L which means in this region, the potential is constant, flat. But then as soon as you get to the edges, you have this very high potential which keeps the electron inside. So we have a one D problem. So you could say, well, and since U is equal to U zero meaning a constant, it means I should be able to use the solution I just found. And if I plot that, it would look like a parabola. This is E versus k. At k equals zero, its U zero. And then increases as kz square. That's the parabola here, okay. Now, when I want to use this to understand the energy levels here though, I cannot quite use this form of the wave function, this E to the power of ikzz. Why? Because you see the wave function in this problem has to fit into this box. Now, it has to go to zero at the ends, you know, where you have these infinite walls. And the function like E to power of ikzz is never zero. It is exponential, this imaginary quantity. So it has the same magnitude everywhere. It doesn't go to zero any place. So if just a function like this would not describe something that has to go to zero at the ends, so what you have to do is superpose a plus kz and a minus kz because you have these two solutions, plus kz and minus kz which have the same energy. And in general, if you had two solutions with the same energy, any linear combination of those is also a solution. So you could take something like this and how do I choose my A and B? Well, I know that at z equal to zero, the wave function has to go to zero. All right, z equals zero, this is one. That's also one. So if it has to go to zero, then B must be equal to minus A. So this has to look something like this and this with a little trig identity, you know, is the sine function. And as you know, sine function goes to zero at z equal to zero. But you see, the function also has to go to zero at the other end, at z equals L. That requires that kzL be equal to n pi. Why? Because the sine function is zero, every time this integrant is a multiple of pi. So kz then cannot just have any old value. But it must be an integer multiple of pi over L where L is the size of the box. So what that means is previously, what we had was this dispersion relation of E versus k and k was a continuous variable and which means E was also continuous. But now because we have put it in a box of length L, the kz has become quantized. It has to be the multiple of n pi over L. And correspondingly, the energies in that also now have discrete values. So in this simple model, you can kind of see how by confining something, you get discrete levels. And of course, the smaller the box, the further apart these points will be. Why? Because the L in the denominator here, the small box, the allowed values of kz will be much further apart whereas in a big box, they will tend to be much closer together. 

[Slide 5] Now, so this is this general principle that as I mentioned before, that waves when confined show these resonant frequencies. And one example of that which we are very -- which most of us are very familiar with is the guitar string. That if you had a guitar string pinned between two points, now ordinarily in a string, you could have an acoustic wave of any frequency. But once it's pinned, only certain frequencies are allowed. And what determines those frequencies, well, in a guitar, for an acoustic wave, the frequencies are linear function of this k. And once you confine it, only certain values of k are allowed and correspondingly certain values of frequency. And for the acoustic wave, this is a straight line which means the allowed values of frequency are like one, two, three which you call the fundamental and the harmonics and so on. And if you want, in a guitar, if you want to make the frequency go higher, what you will do is make the string shorter. And if you want to go lower, you'd make the string longer. So same math that we're using for the particle in a box with electrons. But then, of course, the physics is very different. Here, we are talking about acoustic waves, you know, something very concrete we all understand like a guitar string. And here, the wave equation involves something is that displacement of the string that what you are describing is how far the string is displaced from its mean position. In the electronic case, we are talking about a wave function which is a little more abstract and as it, and this wave function is not something that's usually directly measured. What you tend to measure is psi, psi-star, that is, psi, psi-star is what tells you how the electron probability is distributed. That is for a given electron, what is the probability of finding it in different places? Usually, we'll be talking about thousands of electrons so you could interpret psi, psi-star essentially as the electron density in the sense that for one electron, if the chance of finding it here is point one, then at 1000 electrons, it means there will be about a hundred electrons put there. So psi, psi-star is what you could think of as electron density. But the point is usually psi is not the directly observed quantity. 

[Slide 6] It's the psi, psi-star. Now in our actual problem, as you know, what we're interested in is solids which have this periodic arrangement of atoms. And so, when you look at the microscopic or the atomistic potentials, it would look something like this. You'd have positive charges arranged periodically and then at the ends, you'd have these walls which keep the electrons confined inside this solid. Now you'd have electrons whose energies are way down here and they get stuck on these atoms. So they're really confined. And they will usually have discrete levels, you know, which are spaced by many electron volts. These are the core levels which usually we don't talk much about in the transport, in what we are interested in. On the other hand, there will be electrons whose energies are up here and they are much less confined in the sense that they can spread out over the entire solid and for them, the energy levels are relatively continuous because they are much more delocalized. They are less confined. And often, if you are interested only in these energy levels, you use something called an effective mass equation. In this equation, you don't quite put down a potential for this nucleus, for all these atoms. You ignore that completely. So one of the seminal insights in solid state physics was that an electron in a periodic crystal behaves almost as if it's in vacuum but with a different effective mass, different mass. And that's why I've put a M zero here and an M to indicate that this is a different mass from this one. So this one also looks like a Schrodinger equation but it's really what you would call an effective mass equation and sometimes people use this because it's much simpler than this. Because here, you do not need to include all the atomistic potentials. So if you put in a potential in the effective mass equation, what it really represents is all the additional potentials compared to the ideal periodic lattice, because the periodic lattice has already been taken into account as part of the m. So we'll often see that. So that was a very quick introduction then to this wave equation and these basic concepts, namely dispersion relation and the idea of that whenever you confine a wave, you tend to get discrete levels. 

[Slide 7] Now with that, we are ready to move on to the next topic, which is how you convert the differential equation into a matrix equation. And the rest of the lectures will be about the matrix equation. Thank you.