nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L1,3: Differential to Matrix Equation
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[Slide 1] Welcome back to Unit One of our course and this is the third lecture. 

[Slide 2] Now, in the last lecture, we talked about this differential equation, Schrodinger equation, that is the starting point for describing energy levels in all kinds of materials. And what we want to talk about in this lecture is how you can turn this differential equation into a matrix equation, okay. So that the differential equation, for example, you have a function, this wave function, psi of r. In the matrix version, instead we have a column vector with numbers, psi one, psi two, psi three, et cetera. So how does a function become a column vector? Actually the basic principle is fairly simple. Think of a one D problem where instead of this vector r, we are talking of one dimension, Z. Now in a differential equation, the Z is a continuous video. We turn it into a matrix equation, we could discretize it, that is, look at a specific set of discrete points uniformly spaced along the Z axis. And the idea is that if you got those points close enough, then it would describe the solutions of the original differential equation fairly well. Now, once you have these discrete points, you can take this function and write down the values of that function at those points. So at the first point, psi one. Point two, psi two. Point N, psi N. And that's how you see a function becomes a set of numbers which then appear as a column vector in here. So in a differential equation, you'd be solving for that function. In this matrix equation, you are solving for that column vector. And the differential operator here, that becomes a matrix and how you convert or how you go from this operator to a matrix then there are definite rules for doing that. We'll talk about it a little bit. But there is, for example, in this context, if you were taking the Z axis and discretizing it, one way of going from the differential to the matrix, the operator to the matrix is this finite difference method. And then another one is this finite element method. And these are widely used, not only in quantum mechanics but in all branches of science and engineering. Now in the context of quantum mechanics, we could use that, for example, to find the energy levels of a hydrogen atom or for that matter, any atom. As you know, in an atom, the electron is held to the nucleus by this Coulomb potential and that's the potential that goes in there. Of course, if it's anything other than a hydrogen atom, then you also have to include the potential due to the other electrons which is what makes the calculation more elaborate. But once you have that potential, the point is it's a spherically symmetric potential and so although an atom is a three-dimensional object, the mathematical problem is effectively one dimensional. It's the radial coordinates so you need to find the function -- the wave function as a function of r, the radial coordinate. Now, and that can be done so you could use that quite effectively for finding the energy levels are in different atoms. Now, in general though, this is hard to use once you have a real three-dimensional problem. You see here, because of this spherical symmetry, it's kind of effectively one-dimensional but supposing you are talking of molecules or solids, then you see, you don't necessarily have a symmetry. You have to find, solve the problem really in three dimensions. And in three dimensions, you can see that if you need a hundred points along one direction, and another hundred point along the Y direction and another hundred along the Z direction, then overall the number of points inside that box will be one million. What that means is the corresponding matrix will be like a million by million matrix. And that could easily get out of hand. You know, hundred by hundred matrix, that's easy these days. But million by million, you know, that's challenging and you're still just solving a really small problem. Now, what is widely used is a slightly different method 

[Slide 3] and that's called this idea of using what are called "basis functions." So what we do is this wave function that you're trying to find, you write it as a linear combination of a set of basis functions. What are these basis functions? Well, they can be anything in principle. Certain choices may work better than others. But the point is, that they are known functions. They're your favorite functions, whatever they are. It could be four-year CD's or it could be localized orbitals, all kinds of things but the point is these are known. The unknown is your wave function and you're expressing the unknown as a linear combination of known things, you see? And once you do that, there is a definite prescription for turning this differential equation into a matrix equation and you can write down these matrices that appear here by doing a set of integrals. That is, if you want the mn component of the H matrix, you integrate over volume your basis function m and then the operator, that's this differential operator acting on the basis function, n. And since these are known functions, you can evaluate these integrals and usually that's the time-consuming process. You do it numerically but you can do it because you know these functions. And once you've evaluated those integrals and you have these matrices, then you see, you could solve the matrix equation itself, okay. And there are all these first principles methods which, you know, it'll do just that actually. And there's a widely-used software called this "Gaussian" which uses Gaussian basis functions because that makes the evaluation of the integrals easier. Now, we won't really be going into that, on how you do evaluate these integrals from first principles. What we'll be talking about is the semi-empirical methods where you get these matrix elements by comparing with experiment. And well, let me explain a little further. So consider this problem with the hydrogen molecule. You know, two nucleus, two positive nucleus separated by a certain distance and this problem doesn't have spherical symmetry because you've got the H here, and the H here, and this direction is special. So you cannot just solve the problem in the radial coordinate. So ordinarily, it would have been, you would have required a huge matrix. You'd actually have to solve the three-dimensional problem. But using the method of basis functions, what you could do is use two basis functions. One corresponding to what the wave function would have looked like just for a single hydrogen atom. You know, the one S wave function corresponding to one atom, whatever that was, this exponential E to the power of minus R over A zero. That's what it looks like. So that's a known function. You use that as one of your basis functions. And then another one here, identical but shifted over. That's your second basis function. And given those two functions, you can evaluate those integrals and come out with this matrix. And you can then obtain the energy eigenvalues by solving this matrix equation and you'll see right away, of course, the enormous simplification. If you had been discretizing the lattice directly, you'd have a million-by-million matrix. Here, we just have a two-by-two matrix. And of course, question you'll ask is how accurate is it? And the answer is yeah, it can be fairly accurate if you do it right, you see. Now how do you solve this equation? Well, multiply from the left by the inverse of this matrix. That way, from the left of this, you take this out because when you multiply by the inverse, you just get the identity. You can drop that. And on this side, you are left with the inverse times that. In two-by-two matrix, with a little algebra, you can multiply it out and you'd have this E psi equals that matrix times psi, a standard eigenvalue equation, okay. And if it has a big matrix, you could have gone to a computer. This is just a two-by-two. You can write it down analytically. It's quite simple especially because the diagonal elements are equal. Because when the diagonal elements are equal, the eigenvalues are simply the diagonal element plus or minus the magnitude of the off diagonal one. So if you do that, you'll get these two eigenvalues. And how would you find those eigenvalues? Well, that's what you could do by comparing with experiment. 

[Slide 4] Now, so here you can understand these two approaches that I mentioned, the first principles approach and the semi-empirical approach. In the first principles approach, you have started from the Schrodinger equation with the correct differential operator in there and done these integrals and that's how you'd have gotten those numbers, the s and the epsilon and the t. In a semi-empirical approach, you'd say, "Well, let's look at certain known experimental quantities which depend on these energy levels." So for example, like the ionization energy, what it takes to knock one electron out of a hydrogen molecule or perhaps the binding energy of a hydrogen molecule. So use certain well-established experiments to extract these parameters. Well, but if you are taking it from experiment, then what do you do with it after that? I mean, usually the purpose is to explain experiments. But if you are using the experiments to get your parameters, what you can do then is of course, use it to explain other experiments like you are -- if you are interested in the conductance of a hydrogen molecule like someone puts contacts across a hydrogen molecule and people have done that actually. And they want to model the conductance. Oh good, you could start with this two-by-two matrix and then talk about how to get the contacts on it. So bottom line, use your favorite experiments, well-established experiments to get your parameters. That's the semi-empirical method and then use it to explain other experiments. 

[Slide 5] Now, this is a hydrogen molecule. In general, we are talking about solids. So for example, a very popular solid these days, one on which there's a lot of work is graphene and what it represents is this one-dimensional layer. It's and in graphite, you see the layers are very weakly coupled to each other. So there's -- and what I understand is experimentalists can actually peel off single layers just using a Scotch tape, see? And there has been a lot of work on this material in the last decade. And what it represents then is this two-dimensional layer of carbon atoms arranged in a hexagonal lattice. So you want to find its energy levels. Well, in the method of basis functions, first question is, what basis function should we use? And the answer is well, consider these two carbon atoms right here and for each carbon atom, if we are to choose a basis function, what would we choose? Well, it would be the atomic orbitals. Well, what are the atomic orbitals? Well, carbon has an atomic number of six. It means it has six electrons. So if you have, if you look at the energy levels, there's the one s level and incidentally, I'm sure you have heard this before that these levels usually come in pairs. So if there's an up-spin level, there's also a down-spin level. It's only in the last unit of this course that we'll talk explicitly about spin. But for the moment, what I just like to state is that levels always come in pairs. So although I'm drawing one, there's actually two levels there and because of the exclusion principle, one level holds only one electron. So since there's two levels there, it can hold two electrons. And then there's the four left over because carbon has six electrons so the other four have to go into the n equals two levels. Now n equals two, you have a s level, the two s and then you have the two P levels so three of them, two P's. So there's four levels here which could accommodate eight electrons in principle but it holds only four electrons in the carbon atom. Now here's the other carbon atom which again is exactly the same situation. And now when you bring them together like in a bond like this, you're going to bring them together, the energy levels get somewhat rearranged. Now the one S level doesn't play any major role because these are what you call the core electrons. These electrons are strongly tied to this nucleus. They don't even know that there's another carbon atom there. They're just tightly around one nucleus. So those usually, you don't worry about when trying to model current flow. So you take that out, usually and out here, you could say that, well, I've got four levels here, and four levels here. When you bring them together, the levels will rearrange. Just as in a hydrogen molecule, we had two levels which rearranged into an upper level and a lower level. Here also, it will rearrange into a set of four levels up here and a set of four levels down here. And including spin, that's eight levels so these levels down here, can accommodate all the electrons. So all eight electrons go in there. And roughly speaking, of course, that's essentially the origin of binding like why does it want to stick together? Because the general principle is by sticking together, the energy is lowered. Why is the energy lowered? Well, as long as they were apart, the four electrons were kind of here and four electrons here. When you bring them together, you have these composite levels which hold all eight electrons. So this is the origin of  covalent bonding, roughly speaking, and it works best when you have these half-filled shells, that is, you had these levels which could, in principle, hold eight electrons but have only four. And so when you bring them together, this lower half can accommodate all your electrons. The upper half is completely empty because if, let's say, all eight were filled here and this also had all the eight filled then when you bring them in, you'd have to put eight here and put eight up there and you wouldn't gain much by way of energy. But because it's half-filled, you can take everything and just put them all in here and overall, the energy gets lowered. So you call these the bonding levels and call these the anti-bonding levels. Now where is the Fermi energy or the equilibrium electrochemical potential? Well, at equilibrium again, these levels accommodate all your electrons so the Fermi energy, as you know, separates the filled levels from the empty levels so it would be somewhere in between. And what that means is that as far as current flow is concerned, what we're really interested in is the energy levels right around the Fermi energy. And this is where one enormous simplification that we have for graphene is that in terms of these levels, the ones closest to the Fermi energy, you can describe them fairly well in terms of the Pz orbital. You see, in general, we need to use four basis functions per atom, you know, S and the three P's. But if we are only interested in describing those levels properly, you can use just the Pz so just one basis function per atom. So when modeling this then, we'll have one basis function here, one basis function here and then just like the hydrogen molecule, we had a one s basis function here and a one s there. Here, you'd use the two Pz and the two Pz. Z being actually the direction perpendicular to the plane, see? And so just one basis function per atom. Now, you might say, well, what's so -- why is that any simpler than the high -- I mean, what's so special about it? After all, in the hydrogen molecule also, you have that. Yeah, but the difference is now although you got only one basis function per atom, but overall, it's still a huge solid. What that means is the matrix is not two-by-two. It's a huge matrix. So you got this one basis function here and you got one parameter that probably describes the matrix element in H corresponding one. So you can write that entire matrix in terms of one or two parameters. But you have this huge matrix so how do you find its eigenvalues because you need those eigenvalues in order to compare with experiment. And this is what we'll be talking about in the next few lectures, the idea that because we are usually dealing with periodic solids, you can take advantage of the periodicity to write down the energy eigenvalues of an infinite matrix analytically on paper without ever going to a computer. 

[Slide 6] So that's what we'll be talking about what is generally called as principle of band structure, that is, what it basically involves is how to write down the energy eigenvalues of an infinite matrix analytically by taking advantage of the periodic nature of the matrix. So that's what we'll be doing in the next few lectures. Thank you.