nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/Epilogue
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[Slide 1] Welcome back to our course on quantum transport. As you know, we are just finishing up this course on EdX. And in future, this course, and along with the Part A, will all be available in self based format on nanoHUB-U. Guess it's now time for a little epilogue. 

[Slide 2] Now in Part B, as you know this is about quantum transport and as you might expect, quantum transport kind of starts with the Schrodinger equation for quantum mechanics, this E psi equals H psi where H is this matrix whose eigenvalues tell you these energy levels available for conduction. But as we discussed in Part A, Just having the mechanics is not enough to describe transport because transport involves mechanics as well as these entropy driven thermodynamic processes. And in the context of Semiclassical Transport, which we discussed in part A, you have to combine the Newton's Laws, straight mechanics, semiclassical mechanics, with the entropy driven processes which is this right hand side of the Bozeman equation. And this is the Bozeman Equation which is the centerpiece of all semi-classical transport. Now in quantum transport, I guess we replace this mechanics spot with Schrodinger equation, that's quantum mechanics. But then you have to add the entropy driven part. And this in this NEGF method enters through the self energy functions. We have this sigma 1, sigma 2, describing your two contacts. And the sigma's you know describing the interactions of the electron with the surroundings in the channel. And this leads to this additional terms in the Schrodinger equation. I guess it's a modified Schrodinger equation. And H has to be Hermitian in order to conserve probability but the sigma is actually non-Hermitian, which means because of the sigma, if you start out the system in a certain number of electrons, these electrons will actually want to leak out of the system. Probability will be conserved. So this will lead to outflow. But then there's also this inflow term because electrons, new electrons come in from the contacts. So the idea is if electrons here which go out into the contacts and new ones come in. And that's described by these additional terms. So how does this relate to the NEGF equations? 

[Slide 3] Which as I said is like the center piece in this quantum transport. Well what I'll show you in the next slide is how you can get from this what you call a one electron Schrodinger equation to the NEGF equations. And there's also this current operator which you need to calculate current but I won't be talking about this since we have talked about it in the actual course but this is just an epilogue view, kind of give you an overview and sum things up. So I'll focus on this. And the main point I wanted to make here is also that this is really the same as this standard NEGF equations that are usually derived using much more advanced techniques, following the original work from the 1960's, the Seminole work of 

[inaudible] and Keldysh and others. And the main thing have changed here is this notation, that is the standard literature uses G less and sigma less whereas we multiply by minus i and we call it Gn and sigma in. Because the Gn is the physical significance is it's like electron density. The sigma in is like inflow. Ok so how do you get from this equation to the NEGF equations? 

[Slide 4] Well the point is that if you start from this modified Schrodinger equation, you could write the psi in terms of this source and what goes in between. So this is obtained by straightforward matrix inversion from here. And what you get here, that's what you call the Retarded Green's Function. But this gives us psi in terms of S but it's inconvenient to work directly with the Schrodinger equation because when you have multiple sources, like s1 and s2, if you work like this and put in s1 plus s2, you get interference between them, which is not physical because these are incoherent sources. There are no interference terms experimentally. And it's more convenient then to actually work with The psi psi star, the psi psi dagger which is what we call Gn and SS dagger which is sigma in. So in that sense this is like, I guess psi psi star is interpreted as probability for individual electrons. But then when you have hundreds of electrons you could interpret it as the electron density. And similarly the source, its strength, is what you call sigma in. And where do you get this? Well just from here. psi equals GR s so psi psi dagger is GR s. And then GRS, dagger of the whole thing, which is s dagger, GA. GA being the conjugate transpose of GR. So that then gets us this NEGF equation. And as you can see it's a one line derivation literally. Ok and this is what I've been doing in my books usually for many years now but the standard derivation of this equation though Uses much more advanced techniques following the original work back in the 60's, as I mentioned. So the Keldysh paper, the title of the paper was Diagram technique for non-equilibrium processes. And what was considered there was, I guess, this dissipative processes that involve energy dissipation from the electrons into the surroundings. And to describe such processes, because it involves how energy from electron goes into many degrees of freedom, you need this many-body theory. And so what is used here is what's called this many-body perterbation theory. Now why wasn't this simple motivation done? Well that's because of this mindset in general that transport essentially involves many-body processes and you cannot talk about current flow without many-body processes. That was the general mindset. 

[Slide 5] So before 1990, that is before the advent of mesoscopic physics, the view was that the essential physics of current and flow and resistance is in dissipative processes. And the contacts and the elastic interactions, that's kind of an unimportant detail. Now what mesoscopic physics taught us is that contacts are important. And so before 1990 theoretical papers often didn't even draw the contacts. But these days usually people do. And so the feeling is, of course essential physics is still dissipation but then contacts are an additional detail. Whereas the perspective I have tried to convey is that the essential physics of resistance you can understand in terms of the contacts and elastic interactions. And the dissipation is kind of an additional detail. And this is very different from what you'd normally hear and so let me elaborate on that a little more. 

[Slide 6] So why is it that people think dissipation and many-body processes are kind of integral to this idea of current flow? Well that's because of something you may have heard in undergraduate that if when you turn on your switch and a light bulb goes on, if that one electron actually had to go from here to there, you would be waiting a long time for the light to turn on. But in practice is the light turns on immediately and that's because one electron doesn't go from here to there. What it does is it pushes the next electron and that one pushes the next one and that's how the signal travels. And so there's this feeling that current flow, or this conduction process, is inherently a many-body process. It's not about one electron going through but rather one thing pushing another and so on. But the point I want to make is that this process is actually modeled quit well in terms of a distributed RLC transmission line where you have the CDs, resistors and inductors and this capacitors in parallel. And usually this inductance has a magnetostatic component, the part that comes from magneto statics, and a kinetic competent, the part that comes from transport theory of the kind we are talking about. Similarly the capacitance has an electrostatic part and a, this quantum capacitance is related to transport or density of states type of properties. And if you look at the velocity, which is usually 1 over square root of LC. If we use the magnetic static L and the electrostatic C you'll get the velocity of electromagnetic waves. Whereas if you look at the transport once then it will give you the velocity of electrons. And usually what happens is the inductance is dominated by LM and the C is dominated by CE and so the signal travels at the velocity of light. In very small conductors that could reverse and you could have situations where signals travel must slower. But usually this is what you'd have. But the point I'm trying to make is all that is captured very well in this magnetostatic capacitance and electrostatic capacitance. And doesn't need to be part of how we calculate the resistance. The resistance that goes in there, that really doesn't involve many-body theory. You can understand it, at least to first starter, and you can understand it in relatively simple terms. 

[Slide 7] Now let me explain this point a little. This is something that has become clearer, I guess with the advent of Mesoscopic Physics and Nanoelectronics. Because one of the things Mesoscopic Physics taught us is that the ballistic conductor has a resistance, this RB, which can be associated with the two interfaces. And if in the middle of your channel you happen to have a scatterer, like a little hole in there, then the transmission of electrons from left to right is something less than 1, this T, and so the resistance is higher. It's the ballistic resistance divided by T. And this additional resistance is what you can associate with the channel. So this is well established. This is understood. You call it Landauer formula. But the part that causes confusion still is how do we take this and divide it up into a part that's associated with the interface and a part that's associated with the channel? Because the common wisdom is that of course if you want to locate where the resistance is, you should follow the heat because resistance gives rise to heating and so wherever there's a resistance there must be heat. But this is not correct. Because you see if we had a hole in the middle of the channel, it's obvious that the hole is what's causing the resistance, that's common sense. On the other hand, the heat that is associated with that resistance is also obviously not been dissipated in the hole because you see a hole cannot dissipate heat, heat meaning that the energy from the electrons has to go out into the lattice, start the atoms vibrating and jiggling around and that's what heat is. And the hole just doesn't have the degrees of freedom to dissipate your heat. So intuitively you can see that this hole causes the resistance but can't really dissipate the heat. So it makes sense to associate that resistance with the hole right there but then if you associate the resistance with heat, it's completely wrong because that heat is actually distributed all over the place. So how could you come up with a quantitative criterion that would allow you to locate where that resistance is? And in Part A what we argued is what you have to do is follow the voltage. This is a series circuit. The current is the same everywhere so wherever there's a resistance there's a voltage drop. But this leads to the question what is the voltage? Which voltage do we look at? And the answer is what we know, what device for this know very well and is well established in solid state physics, that what is important is the electrochemical potential and not the electrostatic potential. So in a macroscopic scale, that's usually well known. It's just that when it comes to small conductors, people are usually nervous about talking about the electrochemical potential. And that is where, what we argued in Part A, I think this was unit 3, That if you looked at how the electrochemical potential changes you'd see a profile like this. And indeed there would be a drop here that corresponds to this interface resistance, another one corresponding to that interface resistance and a drop here that corresponds to this channel resistance. But to understand this you have to invoke this concept of Quasi Fermi Levels. The idea that in a small conductor the right moving carriers can have a different electrochemical potential, or Quasi-fermi level, compared to the left moving electrons. So if you look at the right moving electrons, it has this profile, the left moving have this profile and the average is what this black represents. Now why does this profile drop suddenly at this scatterer? Because if you looked at that drop you might say well that looks like a major voltage drop and so there must be a power that's dissipated that's V times I. And the answer is no. This quasi-fermi level should not be viewed as an average energy of the carriers. It only tells you the degree of filling of the states. So the fact that it drops suddenly simply means that if we looked at the right moving states they would all be filled at this end but as you cross the scatterer they'll suddenly become relatively empty, that's all. Why is that? Well that's because of something we all know from common experience. If you're going down a highway and there's I guess a bottleneck somewhere, like a construction zone, then before the construction zone you have a traffic jam. You cross the construction zone, the roads all empty. That's basically what it's saying here. You have this electronic highway with a little construction zone here, this bottleneck, the states are all filled up right here, big traffic jam, right here all the states are relatively empty. That's it, see? But the point to note is this quasi-fermi level should not be associated with the average energy of the carriers. Because whenever you have a roadblock like this, if you looked carefully at the electron energy distribution you'd find that it is way out of equilibrium, which means electrons are hot. So what this hole does is it causes the electrons to get hot. And electrons eventually, of course, get rid of the heat to the surroundings but where that happens depends on the details of the problem. It could easily happen one micron away from here. So this was this concept that we talked about in Part A. And the question is could you capture this 

[Slide 8] in a quantum treatment, this little semiclassical picture? Because the plots I drew, the semiclassical picture, that's what you get out of Bozeman equation. Now in NEGF what we discussed, I guess in this part, I think this was unit 2 we went through this example, that if you do a coherent treatment, which is if you only take into account the two contacts and ignore any interactions inside, then you'll get this black curve. So the red is the semiclassical, what you get out of Bozeman. The black is the coherent NEGF treatment. That's what you'd get here. Now in practice you wouldn't usually see all these oscillations. Why? Because in a real solid there's all this random potentials, as the electron is going through there's all these atoms jiggling around, which electron fields in the form of this random scattering potential, in which it destroys this phase. And NEGF allows you to put that into the model through the sigma 0. And if you put that in, the sigma 0, when you include that, Then you'll get a plot that looks like this. If you include sigma 0 such that there's only phase relaxation, then as you can see these oscillations have died out, the black looks almost like the red but with some ruminant fluctuations around it. Now if you include interactions that also relax momentum and not just phase, then you'll get a plot looking like this where the oscillations have died out but in addition there's a slope here. Why? Because the momentum relaxation gives you a distributed resistance. Because of the distributed resistance there's a slope in the voltage. And all this physics can be included by choosing this D carefully. So this interaction sigmas are related to the G's through this D, which has a simple physical meaning. This is actually a tenser with which what it represents is the mean square value of the scattering potential. See because the dimensions of energy squared, if this is one electron volt, this would be one electron volt squared. And by choosing the structure of this denser carefully, you can get phase relaxation or you can get momentum and phase relaxation or one thing we're working on these days is spin relaxation. So you could incorporate all that into this model. But the point is none of this includes any dissipative processes inside the channel. In other words what you're really assuming is all the dissipation occurs in the contacts. And that's what usually makes this calculation relatively simple. And the point I'm trying to make is even though dissipation is not included explicitly, the physics of resistance is all coming in very nicely. You know a hole barrier, this barrier which represents the hole, does give you a drop in the voltage giving a resistance, as you'd expect. All that physics is in accord  with intuition. So would dissipative processes make a difference? The answer is it could, 

[Slide 9] depending on the problem at hand. So the way I think is at a given energy you could think of it as these resistors in series. There's the interface resistance and there's the channel resistance and you can draw it mu like so. But then there are many, many energy channels. So each energy channel you could view like this. Now if we looked at the mu, usually at each energy, if we looked at how the mu varies, in principal it could be different for the different energies. Which means in the most general case you should think of it as if the mu has an energy dependence. If you are trying to, this mu represents this filling of the states. But the point is that itself could be different for different energies and you might need a different mu to represent it. Now what inelastic processes do is they give you couplings between different energy levels. And so they cause what you might call vertical flow, they cause current flow from one energy channel into the other. But then if you have uniformed contacts and if all the energy channels conduct relatively equally, then what you might expect is that the potential at these points and different energy channels are almost the same. And in that case adding an inelastic process will not change the overall resistance that you calculate because there'll be hardly any vertical flow. And that's why usually for low bias and uniform materials, if you ignore the dissipation, that is if you ignore these connections and just say all dissipation occurs in the contacts, the resistance you calculate is right, is correct. It matches what you get from the rigorous theory. But this doesn't necessarily have to be the case, For example even this problem that we talked about with the barrier in between, if you look carefully you see these energies don't conduct as well as those energies because of the barrier. And so what electrons will tend to do, if you include an inelastic process into the model, is you'll rise up in energy, try to travel here and then come back, which means on this side it would actually absorb heat from the lattice thereby cooling the solids so this side could get cooled while this side gets heated. And this would be kind of like the Peltier effect, the thermoelectric effect. And these are things that would come out of a detailed NEGF model provided you include the inelastic processes. And of course if you don't include the inelastic processes you'll miss all that and overall resistance you calculate would also be a little bit off because you wouldn't have the vertical flow in your model. Now you may have non-uniform contacts. See now so far we have talked about uniform contacts But supposing you had non-uniform contacts where on this side you connect to this energy level but on this side you connect to many energy levels. So this is kind of what you have in a transistor when you add high bias because on the drain side you've applied a positive potential and you've lowered the conduction bandage. Well then again you might expect a lot of vertical flow, which could affect the overall current and the resistance. The most striking example is one where one contact connects here and one contact connects there, like in a P-N junction where you could have one contact that connects to the conduction band, another one that connects to the valence band. And then if you don't have vertical flow, you don't have any current flow at all. And only current that you actually get is because of the vertical flow, in a P-N junction what you would call recombination generation processes. So the bottom line is that there are many cases which you have talked about where you can calculate resistance correctly without including this vertical flow but that's not necessarily true, right? There are many problems that do require it. 

[Slide 10] But the main point I'm really trying to make is that in general there is that mindset that current flow and resistance requires many-body processes. And so in order to get started, in order to understand where these equations come from you first have to master many-body theory. So NEGF is viewed as this esoteric tool which is only accessible after you have mastered many-body theory. And I've quoted from the preface of a recent book, an excellent book, by the way, but which kind of says what the common perception is about NEGF. And the point I have tried to make is well if you just consider elastic interactions then you can use relatively straightforward one electron theory to get the same equations. And even the dissipative processes, if you stick to the lowest approximation, that's the born approximation, then you can get the correct answers out of one electron theory but you have to be careful about these exclusion principle factors. And so you could get started with one-electron theory. Learn these equations. Learn how to use them. 

[Slide 11] And then as needed for different problems, figure out how to get the required sigma. And in this sense think of the Boltzmann equation, as I said, Boltzmann equation has these scattering terms on the right. How do you calculate the scattering terms? Well those, the methods have evolved. You often use this Fermi's Golden Rule, which came long after Boltzmann. And similarly here we'd expect that as we go along, as we look at different problems, we'll come up with new better ways of calculating these sigmas or this self consistent potential system U, that you need. New methods will come up and some of these might actually need non-perturbative methods for strong interactions, for example. But those we can evolve as we go along. But in order to get started it's not like you need to learn many-body theory right away. And this part of the problem where you are neglecting inelastic processes completely, that itself is relevant to many current problems in nanoelectronics. 

[Slide 12] And let me just give you one example before I wind up. And that is this problem that we talked about in unit 4 of this course, there's a  spin transport, that one of the major developments in the last few decades has been, is magnetic contacts which can inject spins preferentially. And for this discussion let's assume we have a perfect red magnet which only injects and detects up spins, the red spins and a blue magnet which can only inject or detect down spins. And so if 100 electrons come in here per second, they'll flood this channel. None of them can go out here so there's no current here. All of them will go out there, all 100 of them. And out here you won't get any. So this would be the situation with these four magnets. Now question is what happens if this magnet, instead of pointing down is actually pointing sideways? Then the answer is of these 100 electrons that come in per second, 50 of them go out here, half of them. Why? Well because an up spin is a super position of a left and the right. And this is very non-intuitive at first. What you are probably more familiar with is for photons where you think of polarization as a vector, then you're used to decomposing a vector like this into two components in 45 degrees on either side. Whereas here it's like 90 degrees on either side and it requires a whole different algebra, the spinors, ok? And the whole super position principle is, of course, at the heart of all the mysteries of quantum mechanics that have bothered so many people, the idea that one electron, even though we know everything about it. We know it's an up spin electron and yet we cannot predict whether it will go out there or not. It only has a 50 percent probability of going out. Of course if we do the experiment with hundreds of electrons then you can say that half of them will go out, which is what we are doing, steady state electrons, lots of electrons. Well what happens on this side? Well of the 50, again you can view the 50 here which are pointing to the left here, Those are again super positions of down and up. And so half of them, 25, will go out of this up magnet. And the other 25 will go down the channel And come out at the downright. So overall you see what happened was I turned this magnet, instead of pointing down I pointed it sideways and I have no current here and now I have lots of current there. And in principle you could work out how this number will change as we change the direction of this magnet from up, as it rotated down, And you'd get a curve looking something like this. But if it is either up or down, you don't get much current. But you get lots of current once it's sideways, ok? And this is the basis for, I guess close to some of the ideas for some of the ideas for spin transistors that you see in the literature. And of course whether this is useful as a device, that remains to be seen. But right now I'm just trying to make the point that there's all kinds of things involving subtle quantum mechanics which are now technologically accessible with solid state devices. And all of this you could model using this NEGF Theory that we discussed. Where you have an H that describes your channel, the sigmas that describe these contacts. And if you wanted to include spin scattering you could do that also with the sigma 0. And one of the interesting things people are looking at in this context of spin transport is this idea of spin circuits. Because one of the major challenges with NEGF is that although in principle you can handle any device, but the fact that you are keeping track of all these off diagonal elements makes it gradually, I guess big devices become numerically intractable. Whereas in a semiclassical treatment you only keep the diagonal elements. So if there are N diagonal elements, that's kind of like N squared off diagonal elements. Now, but all those off diagonal elements are usually not important. Dephasing processes destroy many of them. So the challenge is how to keep the relevant off diagonal elements and not get bogged down by the irrelevant ones. And spin circuits is kind of a method that allows you to do that in the context of spins. That is it keeps the off diagonal elements relevant to spin but ignores everything else. And this might well be a good paradigm that goes beyond spin. Because spin has these two components whose correlations you are worried about. In general you might have a 20 component spin and you might want to keep the relevant off diagonal element. But that's all for future work. We'll see where all that goes. The main point here though is that NEGF, even without all these many body interactions, there's plenty of interesting subtle physics that one can explore. 

[Slide 13] Where do you go from here? Well, as I mentioned, if you want to, I guess, deepen your understanding you should go through detailed numerical examples. We have only gone through a few simple ones. In my books and on my website I have a lot of these math lab codes that you can use that you can go through, look at the results. In the context of spintronics, you need to know more about magnets, which we did not talk about so that's magnet dynamics. That's usually described with a LLG equation. But that's not so much for this transport as for spintronics because these days magnetics is kind of intimately tied with spintronics. In general, of course, transport theory involves both mechanics and the statistical mechanics. And also many-body theory, second quantization, which we haven't touched much at all. And even here I've only tried to convey to you the basics without going into depth. And you could deepen your understanding with all these, with many of the traditional courses available both in physics and in chemistry. And on NANOHUB as we go along we might try to develop more courses that cover these aspects in depth. 

[Slide 14] But the main thing then in this course that I tried to convey is these NEGF equations are incredibly powerful. They tell you how to add contacts to the Schrodinger equation and analyze real devices. But unfortunately the common view is these NEGF equations and current flow in general involves many-body theory. So unless you master many body theory you can't even get started. Whereas the point I'm trying to convey is lot of the essentials you can get from one electron theory and this really should be a part of your graduate and even the undergraduate curriculum. And it is with this in mind that we developed this set of courses as Part A and Part B. where we assume a minimal amount of prerequisites, you see the kind that all students in science and engineering normally have. Now the point I'd also like to stress is that this approach and what we do in this course, this one electron approach, it's not just for people who lack background. Even if you have very good background, you understand many-body theory, I'd say that this one electron picture helps bring clarity to this discussion. In the field of spintronics, for example, 

[Slide 15] I know that there are many separate issues people are wrestling with. You know how do you handle equilibrium spin currents? Because one thing about spin currents is they can be nonzero even at equilibrium so how do you interpret all that? How do you subtract out things and look at transport processes, etcetera? And all these discussions can be clarified. You can see the essential points much clearer if you have this simple one electron picture. 

[Slide 16] And this is where I often quote from Feynman lectures where he makes this find very clearly that even when you have all the equations doesn't mean that you have understood it and can use it creatively. To use it creatively you use this, physical understanding. And that is kind of what we are really tried to convey in this set of courses. That is with a minimal amount of requisites, 

[Slide 17] how to understand the basic concepts in quantum transport. And I guess in the earlier course we talked about semiclassical transport. And as I said, we're just finishing this course on EdX but in future it will be available in self paste format on nanoHUB-U, both parts. And our objective here really is to convey the essentials, the physics, without being bogged down by a lot of-- without the need for a lot of sophisticated background, which you can pick up as you go along. And that's why we have tried to keep the prerequisites down to a minimum. Well it's been a real pleasure actually interacting with many of you online and I hope you also got something out of it.