nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L1.10: Summary
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[Slide 1] Welcome back to the last lecture now of unit one of our course. 

[Slide 2] I guess it's time to sum up. Well as, you know, when we're talking about current flow or transport, it usually involves these two types of processes. And so, in part A of this course we were talking of semiclassical transport using semiclassical mechanics, there's particle mechanics and then there's the entropy-driven processes. And when you put that together, you have the Boltzmann equation. So for part A are standard kind of starting point in thinking is usually this Boltzmann equation and what we try to convey is that using this idea that in nanodevices you can think into by separating out the two types of processes you can get a simplified view, simplified way of looking at things. And that viewpoint is what we are also continuing in this part into quantum transport. Of course, instead of Newton's laws, we now have the Schrodinger equation that describes the wave nature of electrons. And that's this E psi equals H psi. And H is what describes the energy levels in the channel. And here again, we have to start from Schrodinger but then that's not enough. We have to add these entropy-driven processes to the story and that's this non-equilibrium Green function method, which is what we'll be going on in the next unit. In this unit, what we have really talked about is just this, the mechanics part, the Schrodinger equation part. 

[Slide 3] So what we showed was, you see if you're interested in a solid like this. Again, the starting point would be the Schrodinger equation, which is first written down in the context of understanding the spectra of hydrogen atoms. But since then has been applied to-- in all atoms, molecules, solids, everywhere, the starting point is again the Schrodinger equation. Of course, in a hydrogen atom, the potential that appears is only the potential due the nucleus whereas in any solid you have to include a component due to all the other electrons, which is what makes the calculations much more difficult and elaborate. Now, a method that is widely used in this context for solving the Schrodinger equation is this idea of basis functions. That is you take this quantity, the weight function that you're looking for and write it as a linear combination of some well-- some known functions. What do you use for the known functions? Well, people use all kinds of things. What we are using here, what we are generally talking about in this course is what you might call atomic wave functions. That is, if you looked at one of the atoms, what do wave functions would have look like. Of course, in general, for every atom, it wouldn't be just one wave function there would be a few of them. There could be a 2s, 2p, etcetera. There will be multiple wave functions at the same atom. And the thing is, you take all these bases functions. One, if you look localize around each atom. And overall, if you have N basis functions, then you can turn this Schrodinger equation into a matrix equation where the matrix is N by N. And in the first principles method, of course, you would calculate the elements of this matrix first principles and there's a definite prescription on how to do that. In the semi-empirical method, which is what we are doing. The idea is to write it in terms of a few parameters and then obtain those parameters by comparing with a few standard experiments. And once you have this, then, you can go on to describe all kinds of other experiments. And there, what we spend most of the time in this unit talking about is how you can write down the eigenvalues of a huge matrix. This is a huge matrix because after all these basis functions, you have a few basis functions per atom, per unit cell. And let's say you have thousands of these, then, this is a huge matrix. But the great-- The basic lesson that I was trying to convey in this unit and I don't want to spend a lot of time talking about is that because it's periodic, you can actually write down all the eigenvalues without going to a computer. You don't have to solve this entire huge top matrix, you know, thousands-- thousand by thousand. You can just write it in terms of a small matrix. And what determines the size of the matrix? Well it's the number of basis functions per unit cell. So, we took the case of graphene where you could do it with two basis functions per unit cell. In general, you might need much more. You know graphene is a particularly nice example because it's very relevant to research and the nice thing is that this two-by-two description actually gives you fairly good results. You know, based on which people actually write professional papers, OK. On the other hand, in many other cases to get to good reliable results, you would need to use many more basis functions per atom. 

[Slide 4] So let me say a few words about what you need to handle silicon for example. You see, silicon is something we didn't address in this course-- in this unit so far. Because then we won't be talking about this very much either. But let me just take a slide to give you a flavor of what it would be involved if you wanted to do silicon. You know a very common semiconductor. Silicon as, you know, has atomic number of 14. So if you look at the atomic levels, there would be the 1s level which will accommodate two electrons, again, because of the two spins as we have mentioned. There'll be 2s and 2p that's four levels which accommodate eight electrons. So that's the eight. So 8 and 2, 10 and that leaves another four electrons for the n equals 3. And when you go to n equals 3 you have again the s state and P state. So if we look at that four which could accommodate eight but then there's only four of them. And then there are higher levels. There's the 3d, there's five of them there. And then there's 4s above it and so on. That's all. So when you are trying to write down the basis functions of the units-- in a unit cell, what you would use is the basis functions for 1 atom and then you ask, well how many atoms in a unit cell. Well, these common semiconductors all have what's called diamond lattice. And the diamond lattice is like an FCC latice. FCC is face-centered cubic. What that means is, you have squares like this with atoms every corners of the squares and one atom at the center of each face. So if you turn around, every face has one atom at the center. So that's what you called face-centered cubic lattice. You could easily Google that and look up the pictures on the web if you can not see this very well. But all these diamond lattice then, the weight is different from the FCC is that here, I'm showing just one atom at this point. But, in all these common semiconductors like silicon, you'd have two silicon atoms at each point. So it would be one here and another one a little bit off on this side. So the unit cell is actually two silicon atoms. You know, just as in graphene, the unit cell was two carbon atoms. So here you got two silicon atoms, OK. So with these two silicon atoms then, you'd have something like this and again to get a good description then, how many basis functions would we used per unit cell. Well, usually as we discussed, you don't use the core electron. These core states at all. So leave that out. So you got 1 and 3/4 here, 1 and 3/4 so you could use eight of them and usually people also include the spin explicitly into the description. So it's 16. So, lot of the bond structure of silicon, the minimal amount people would use is 16 basis functions per unit cell. What that means is when you plot it out, it will look like a complicated diagram. There would be 16 branches to it. Remember in graphene we had two and so we had two branches. Now they'll be 16, some overlapping, so often, you know, people call it the spaghetti diagram. You see all these lines going everywhere, OK. And it takes, you know, takes sometime get yourself oriented and see which bonds are coming from where. And all that we didn't talk about in this unit at all. I'm just trying to give you a flavor of what the bond structure looks like for real solids, you know, other than graphene. So, the minimal amount is 16 basis functions. But then, people use more basis functions to get better descriptions. Sometimes people use the 3d levels, sometimes people use the 4s levels or both. And that will again give you many more basis functions and so many more lines to the bond structure. And the other question that you could ask is what's the Brillouin zone look like? What's the reciprocal lattice? Well, again, the basic unit cell of course has this face-centered cubic structure as I mentioned and you can show that the reciprocal lattice of a face-centered cubic structure is the body-centered cubic structure. Body-centered means you have the square, the cube as before, but the additional atom is in there center of the cube and not on the center of the face. And then that's the reciprocal lattice and so if you want the Brillouin zone, so I'm bringing it here so you can see it, that's the Brillouin zone. So, you kind of look at those reciprocal lattice vector and draw by sectors. And when you draw all those by sectors, you'd get this complicated geometric shape here and that's the Brillouin zone. So, all these details of course take time to get use to and we'll not be talking about it in this course. What we did in this course instead is we talked about the basic principle starting from simple 1d solids 

[Slide 5] and worked up to the, I guess, the highest level of complexity we reached was this graphene. And in graphene again, it's two silicon atoms per unit cell. We now have two carbon atoms per unit cell. And carbon has a lower atomic number, it's like 6 and so it's only the 1 n equals 1 and the n equals 2 levels. And then of course the n equals 1, these being the core levels you usually don't include. So, you have the n equals 2 levels. But here you might say, well, that's-- the situation is just like silicon because after all, you got this four here and four here, so you'll need eight basis functions per unit cell. And that is kind of true if you want to describe all these levels but the normal simplification in graphene that you have is that when it comes to current flow, again, that ones that matter at these levels, because those are the closest to the Fermi energy which is in the middle. And those can be described largely in terms of the Pz orbitals. What that means is, you could just use one basis function per atom, which means two basis functions per unit cell, and still get a reasonably good description of the energy levels of graphene. And that is kind of what we built up to this course and we talked about how you can analyze it. Now, one thing I should mention that you can get these energy levels of a uniform sheet of graphene assuming periodic boundary conditions just from this 2 by 2 matrix. Of course when we do current flow in small structures, then you can not use the bond structure anymore because it's not an infinite sheet with periodic boundary conditions. Then of course, you have to do the real structure and if the real structure has a hundred atoms, you'd have a, you know, a matrix with hundred basis functions. So it would be 100 by 100 matrix. No question, right. Right now, all we are trying to do is figure out what are the right parameters to use for this t for example, the one that connects two neighboring carbon atoms. So, what we want to know is the eigenvalues of this infinite sheet of graphene and compare it to experiment and based on that, extract a suitable value of t. So, that later on, when we talk about current flow 

[Slide 6] in graphene, we could use that t. And what we showed I guess in these lectures is that because you can use just two basis functions per unit cell, you're going to extract these eigenvalues just from a 2 by 2 matrix. And this 2 by 2 matrix, the eigenvalues are given by this epsilon plus minus the magnitude of h zero. And there again, you can simplify this expression somewhat, which is commonly done by just looking at the valleys. That means the points where this h zero is a minimum, because that's where the energy levels will be closest to the Fermi energy. These valleys when you look at in k-space, you see them kind of spreading in this hexagonal pattern, here, OK. And often, you might think that as 6 valleys but the point we're trying to make is, it actually-- it's only two valleys. And for that, you have to consider this Brillouin zone. And around each one of those valleys, you can make this approximation so that E just looks like it's proportional to k. And that is of course one of the very important properties of graphene that distinguishes it from a common semiconductors. Common semiconductors E is proportionate to k squared, h-bar squared k squared over 2m. Here, around those values, you can write it as is, it's Ek times k because this is the magnitude of k, OK. And if you construct this reciprocal lattice, what you see is those six valleys are actually 6 1/3 valleys and so you have this two effective valleys. And that as I have mentioned before has a very direct experimental consequence. Namely, the ballistic conductance usually is 25 divided by 2 kilohm, that 12 and a half kilohms because of the two spins. In graphene and carbon nano tubes, it's more like 6 kilohms because there's also the two valleys, because all these valleys and spins are like-- conceptually like conductors all in parallel, which reduces the resistance by that much. And also, because E is proportional to k, we have this simple relation here. We could extract the velocity. What is the velocity of an electron in graphene? Well, it is this-- that usually velocity is proportional to k, but here, because E is proportional to k squared, and velocity being derivative of E with respect to k is usually proportional to k. Here, E is proportional to k and so the velocity is a constant number. It doesn't matter what the k of electrons is, and this comes as a surprise because, you know, ordinarily, we think that velocity is like H bar k. I mean, rather momentum is like H bar k, and so, more the k, more the momentum. And it if there's momentum, you'll think there's more velocity. Well not really, as far as not in general. In this context, yes, momentum is H bar K but the velocity is independent of that. It's a constant number. How much is that? Well, you can put in the values. a, that's very well known because that's basically this distance. As you know, these are the-- this is the arrangement of atoms in graphene. So you have a carbon atom here and a carbon atom here that bond length is known to be, I guess, about 1.4 angstroms, 0.14 nanometers. And so a is like one and a half times that, so that's the 0.2 nanometers. And that leaves only one other parameter t, I mean, H bar, that's, I mean, that Planck's constant, I mean that's well known, that's a universal constant. So when you put that in, if you put t equals 3 eV, you'll get about 10 to the 6th meters per second. And in graphene, what's experimentally known is that velocity is about 8 times 10 to the 5th meters per second I think. I mean, in this order and depending on what velocity you choose, you can adjust your t. So that kind of illustrates this principle of the semi-empirical method because the idea is we are turning Schrodinger equation into a matrix equation. And that matrix has only one parameter, this microscopic parameter, the t. How do I choose it? Well, I choose it by making sure that I match certain standard experimental results. So in this case, I could say, I used the appropriate t so that I get the right velocity, OK. And it just depends on that. So with that introduction then, this unit was intended to introduce to you the Schrodinger equation but more importantly, the matrix version of it and how you get the parameters of that matrix. 

[Slide 7] And now, we are ready to move on to talking about current flow or transport problems. And as I mentioned before, transport problems is not enough just to have quantum mechanics. You kind of need to have this entropy-driven processes, in other words, to start with, even if there's no interactions in the channel, you still have these contacts where all these entropy-driven processes work. And so what we need is to figure out how to take the Schrodinger equation which describes an isolated channel and attach context to it. And so, that's why we call it contacting Schrodinger and that's what we'll be talking about for the next two units of this course. Thank you.