nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/L1.1: Schrodinger Equation
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[Slide 1] Welcome to our course on the fundamentals of Nano electronics. This is Part B, Quantum Transport and this is Unit One of Part B which is about this Schrodinger equation. And so, this is the introductory lecture. 

[Slide 2] As you know, in this course, our defining picture has been this one which is a channel through which current flows and they have a source and a drain and one thing I stressed is the current flow depends on these energy levels in the channel, the density of states that's available. And one of the important points I tried to make is that what generally makes transport complicated is that this process of electrons getting from one contact to the other involves two types of processes, the mechanical ones which are like force-driven and then the ones that are entropy-driven. And in transport theory, it took a lot of work to combine these two into a single view point and that's this Boltzmann equation which combines Newtonian mechanics with the entropy-driven processes. And in Part A of this course, we were talking about what you might call the semi-classical transport where you view electrons as particles and what we showed was that this new perspective that is inspired by the developments in Nano electronics. It gives you a simple way of looking at transport by sort of separating out these force-driven processes from the entropy-driven processes. And in this part, we'll do much the same but for quantum transport. And of course, in quantum transport, Newton's laws get replaced by Schrodinger equation which takes into account the wave nature of electrons. And what we'll do in this unit is we'll start by talking about the Schrodinger equation which you have probably seen this somewhere. It looks like E psi equals H psi where H is the differential operator. And this H, as we'll show, you can convert from the differential operator into a matrix whose eigenvalues give you these energy levels. And what we'll be talking about in this unit, of the course is just this part, namely, the Schrodinger equation, how to write that matrix H which gives us the eigenvalues. In the next unit and after that, we'll talk about the entire process of transport. We'll talk about this non-equilibrium green function method. But right now, we're just talking about this part of the problem. 

[Slide 3] Now the Schrodinger equation, we started, I guess he wrote, Schrodinger wrote down this equation like 1925 or so and what we're trying to do at that time was trying to understand the experimental observation on the simplest element, the hydrogen atom. And the idea is that you have a nucleus with a positive charge. To keep it general, I put plus Z times q where Z is the atomic number. Of course, for hydrogen, that's just one. And then you have this negative charge, the electron that is we picture as going around it, kind of like planets around the sun. And you would say that the force that the electron feels due to the nucleus that's given by this Coulomb force goes as one over r squared. And that should be equal to the centripetal force, this MV square over R. That's the velocity of the the electron. And you could turn that, solve that to write the velocity in terms of the radius which tells you that if you had an electron with a large velocity, it would have a small radius, okay. Now, you could calculate the energy of the electron. It would be given by two parts, the potential energy, that's the potential energy due to that positive nucleus and this is the usual one over R potential of a point charged. And then there's the kinetic energy, half mv square but since v square is related to this, we could write the -- we could eliminate the V and write everything in that form. But the basic point then is that you think the electron could have any velocity and depending on its velocity, you'd have a certain radius where it would orbit. And depending on that radius, it would have a certain energy. And this could vary continuously. You could have faster, higher velocities, smaller radius and different energies and so on. Now what was experimentally known was that the hydrogen atom, the electrons appear to have discrete energies. Well, how do they know it? Well, the most straightforward experiment would be to look at the light that is emitted by a gas of like hot atoms because when you heat it up, electrons get excited to the higher energy levels and when they jump down, they emit light. I mean, the same way a light bulb works. You heat it up, it emits light. And the light that came out of a hydrogen atom, it appeared to have discrete frequencies that corresponded to specific energy differences. And the point was then to try to understand why the frequencies were discrete. Now what was noticed around 1915 or so, is that, if you postulate that electrons are waves and said that the electron cannot orbit in any radius it wants, it only has to choose those radii for which its wavelength, you know this is the de Broglie wavelength of a particle whose velocity is v. This is h divided by the momentum. If that wavelength is an -- if the circumference is an integer multiple of that wavelength because the idea is it's a wave so it must kind of fit into that circle. And so the circumference must be some integer multiple of the wavelength. So if you postulate that, then you say you could combine those two things and come up with allowed values of radius, that is, previously, it was like the electron could have any velocity, any radius but now what you find is that only specific values would fit. And so that gives you specific values of the radius rn and this quantity here is what is known as the Bohr radius. And it's about half an angstrom or so. Half an angstrom means like 1/20th of a nanometer. And if you take this rn and put it back into the expression for energy, what you'd find is you're using the same expression but now r cannot have any old value. It only is allowed to have certain specific values and when you put that in, you'd find that the energy goes as one over n square. And the simplest argument actually gave these results which matched the experiment quite well. So it's clear that there was a whole lot of truth here. And this is roughly what's known as the Bohr model of the atom. And note that this whole thing, of course, physically hinges on this idea that an electron has these wave properties and the wavelength must fit into this circumference. Now the point was then to put this insight onto a solid mathematical footing. And that's exactly what the Schrodinger equation does. 

[Slide 4] So in the Schrodinger equation, I guess here, this is just a heuristic guess, what the Schrodinger equation does is it actually gives you a wave equation. There's a wave function, psi. And it says E psi; energy  psi is equal to this differential operator times psi. And this potential, this function here is the potential energy due to the nucleus. And what we say is that in order to be the allowed orbits are described by those wave functions which satisfy this equation. And when you do that, you would indeed get discrete energy levels, that is, if you put in a wave function like this, you might find, well, if you just put in any old wave function, this equation would not be satisfied. But for certain specific wave functions, you would find that it would be satisfied and there would be a specific value of energy that would work. And so what -- and this can be done analytically. This is what Schrodinger showed is that indeed, there were wave functions, specific wave functions for which you could find these energy levels which matched exactly what we just got from the heuristic argument. What is more, it also showed that you could have other forms of the wave function which give you multiple energy levels. So in our old argument, it was like those one here, one here, one here whereas here, what you found was it could be like multiple wave functions that would give you the same energy. So something looking like this would give you that energy. Something looking like this might also give you the same energy. So that you then interpret as there are multiple levels having the same energy and these are what are called the s levels and the P levels. So there is for then equals two, there is one s level and there's this three p levels. n equals three, there's this s, p and d levels, one, three and five of them. So this is the kind of thing that's you probably see in some freshman course. You know, this kind of revolutionized our understanding of atoms and what was shown as that from this point of view, you could understand the periodic table and so on. So that gave people, of course, a lot of confidence in this equation. And this since then, has become the starting point for understanding energy levels in any material. 

[Slide 5] Now, for example, as I said, Schrodinger wrote down his equation in 1925. In the next 30 years or so, people had more or less applied it to all the atoms in the periodic table. So if you can see this, that axis is the atomic number. So the point at the left end is the hydrogen. That's what Schrodinger had shown that the one S level had an energy of on this scale, one, because the energy has normalized to that quantity. So that's the one s level of hydrogen. But as you go down the periodic table, the energy goes deeper and deeper. Deeper and deeper because the nucleus has bigger positive charges and so it holds on to the electron more tightly. But then, there are other levels that come in, one s, two s, three s, et cetera. And you'll notice, when you're at this end of the periodic table, these are what we call the core electrons which have a very low energy, tightly bound here. Whereas there are also electrons up here which are not as tightly bound and how do you measure all this experimentally? Well, the standard method is this photoemission. Photoemission meaning you hit it with light and see what energy of light is needed, H nu, to knock the electron out of the atom into the surrounding vacuum, surrounding atmosphere. So and that's how, as I said, by 1960, people had more or less applied it to the entire periodic table and it all worked very well. You can see how well the experiment and the theory correspond. One point I should make is that you might think that doing the rest of the periodic table couldn't be too hard compared to the hydrogen atom because after all, all we need to do is change the value of Z. It's just the atomic number that's changing. Well, not really. Actually, as soon as you go from hydrogen to helium, the problem gets a whole lot harder. And the reason is that you see this potential, what I've written down is the potential an electron feels due to the positive nucleus. And that's enough for hydrogen. But in any other atom, you also have to include a potential due to the other electrons like if you consider just helium, it has two electrons. So when you are trying to find the energy levels corresponding to that electron then what does it take to knock an electron out of this atom? You have to include a potential here that this one feels due to the other ones. So for example in hydrogen, as I mentioned, it takes 13.6 EV to knock an electron from here out into the vacuum. So you might think that in helium, it would take four times that because the atomic number is two so two squared, that's like 52 electron volts. In practice, it only takes about 25 electron volts to knock an electron out of helium. Why is that 30 volt difference? Well, that's because there is this other electron in helium which is kind of repelling this electron and telling it to get out and so, it's much easier to get it out, okay? So -- but all this was included in the theory and this was done, 

[Slide 6] as I said, by 1960 or so. Now the problem we are interested in is of course, more complicated than atoms. We're interested in solids. And largely, periodic solids where you have this arrangement of atoms. Now here, the common method that is used to solve the Schrodinger equation is to use this concept of basis functions which we'll talk about in a couple of lectures later. And in this -- when you use this, what you can do is convert this differential equation into a matrix equation. And these basis functions here, kind of what we'll be using, you could -- there are many different types that people use. But what we'll be using are these atomic wave functions like if this was an atom, we know what the wave functions would have looked like. And these are what you use for as basis functions to expand to write r, the one we are looking for. So in this solid, what we are looking for, we write it as some number times this plus some number times that plus some number times that. So it's a linear combination of these. And as I mentioned, when you do that, you get a -- you can turn this differential equation into a matrix equation and this matrix equation then is what you solve because there are very powerful matrix methods for finding the eigenvalues of matrices like this. And this matrix, there's a theory that tells you how to write down the elements of that matrix given the differential operator. And we'll talk a little bit about it but not much because we won't really be using or trying to evaluate the matrix from first principles because in general, in this field, 

[Slide 7] you would see two types of approaches. One is what's called a first principles approach where you start from the Schrodinger equation, do these integrals and come up with these matrices. And there are commercial software which does things like that. For example, there's this Gaussian which uses Gaussian basis functions to do these problems. On the other hand, what we'll be using is what's called the semi-empirical methods where we say that well, we have a nice periodic solid and we could try to express these matrix elements in terms of a few basic parameters which are determined by the spacing of the atoms and nature of the atoms. And these parameters instead of calculating them from first principles, you could fit it to a few known experiments. And once you have those parameters, you can go on and model much more difficult experiments. But with a few basic experiments, you could try to fit that. And that's the approach we'll be using. But in order -- and in order to do that though, what we need to do is kind of identify these parameters and figure out that in terms of these parameters, how -- what the eigenvalues of that matrix will look like that we can actually compare with the experiment. So much of this unit, we'll be talking about this part of the story which is generally called this "band structure of a solid," that is, given a periodic solid, how do we write down the eigenvalues without necessarily going to a computer because often you might think you have a solid with say millions of atoms. So we are talking about a huge matrix because the size of the matrix depends on the size of the basis, number of basis functions. And that's then like millions. So that would be a huge thing. But there is this very well-developed theory which takes advantage of the periodic nature of solids to write down the eigenvalues without actually solving this million-by-million matrix. 

[Slide 8] And that is what we'll largely be talking about in this unit. So what we'll talk about in the next two units then is I'll introduce this wave equation and how you convert it to a matrix equation. But after that, it will be about how to write down the eigenvalues of the matrix equation. Thank you.