nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L3.9. Can NEGF Include Everything?
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[Slide 1] Welcome back to Unit 3 of our course on quantum transport. And this is Lecture 9. 

[Slide 2] And as you know, in much of this course, Part A and Part B, I restricted our discussion to structures where the interactions within the channel are purely elastic. And I guess in Unit 2, we wrote down expressions for sigma 0 and sigma 0 in for elastic interactions. Now in the last lecture, we talked about how one could include inelastic processes through this function D which describes the strength of the inelastic interactions. And our discussion was entirely based on what you call the one-electron Schrodinger equation which describes-- which is in terms of this potential UR that describes the potential seen by one electron, right. And the D that we have here, it's like the average value of the matrix elements due to that potential. So, this is this UR and phis are the basis functions. So this tau im is like if I wrote this UR in the form of a matrix, what that would look like. And these Ds are these average values of these matrix elements here. And then notice, this is a matrix so it has two indices. Since I'm taking the product of two, the D has four indices. So, that's what you call a fourth-order tensor, right? Now the point is then, our entire discussion is in terms of this one-electron Schrodinger equation which is very different from what you'd see in much of the literature. Because much of the literature, the standard approach is based on this Many-Body Perturbation Theory, right, which we haven't discussed at all. And the power of the Many-Body Perturbation Theory is that it gives you very general expressions or a general method for writing down expressions for this sigma to higher and higher orders. That is what we have done here. It's kind of like a lowest order expansion. That is you're treating this interaction approximately. And this is like the lowest order, usually called the Born approximation. And guess what? We'd use this, the self-consistent Born approximation. But you could have a higher order term where instead of sigma being D times G, it would be like some constant times G squared or G cubed and things like that. And the Many-Body Perturbation Theory would give you a general way to write down these expressions to up to any order, but which we haven't gone into. Now, at the Born approximation level, you can write down the results more or less from the one electron Schrodinger equation itself. Although as we saw, there are subtle issues that you have to be careful about, things like why is this, Gp E minus h-bar omega plus Gn E plus h-bar omega? As I explained, the second term comes from this very subtle treatment of exclusion principle, right? So, those things, you have to be careful about but otherwise, you could write results more or less from the one-electron Schrodinger equation. Now, one important type of interaction that people are very interested in is electron-electron interactions because we have generally been treating electrons as non-interacting particles. But actually, of course, they are interacting, in a sense, each is a charged particle and the others feel its presence. So how would you include that? Well again, Many-Body Perturbation Theory gives you a systematic way of doing that. It's just that it's fairly complicated. It would involve higher order terms with multiple Gs in here. Now-- And that is something that one could use although I haven't seen too many concrete exact problems worked out that way. 

[Slide 3] Now what-- why don't we normally worry about electron-electron interaction? Well, the reason is that usually, we are interested in conductivity or conductance. And that depends on the momentum relaxation time. Now, when two electrons interact with each other, you see momentum is not relaxed over all, because one electron might pick up some momentum but there is this equal and opposite reaction. So, the other one loses that momentum. And so, in total, momentum doesn't change. And as a result, things like mobility conductivity to zero or to lowest order are not affected by the electron-electron interaction at all. So, does that mean it has no effect on transport? Well, not really. There are experiments which are very sensitive to this. And these are experiments which are phase sensitive, for example. That is, there is this classic experiment from 1985 where people measured the resistance of a ring. So, you have this ring and when an electron comes in from here, it can either go to the upper arm or the lower arm. And you measure this resistance as a function of magnetic field and what you find is a oscillatory magnetoresistance, that the resistance goes up and down. And our understanding is that, you see, if there's no magnetic field and if the two arms were identical, then the two parts like the double-slit experiment, the two parts would constructively interfere. But when you turn on the magnetic field due to vector potential, what we discussed, you know, a few lectures earlier, the vector potential gives rise to different phase terms for the two parts. And so, instead of interfering constructively, they could interfere destructively and thereby, the resistance would go up. And then again, it will become constructive and the resistance would go down. So, this has actually been experimentally observed. And as you might expect, an effect like this depends on the phase relaxation time, and not just on the momentum relaxation time. And the thing is, that when two electrons interact, I said, whatever momentum, one loses, the other picks up. Momentum is not relaxed but phase is lost. One electron sees this random potential due to another one and its phase gets messed up. So, phase relaxation is strongly affected by electron-electron interaction. And there are other experiments like conductance fluctuations, weak localization which depend on phase relaxation. And in general, such processes are usually very-- much more sensitive to temperature because when you go down to say, 10 Kelvin or below, often, momentum relaxation won't change very much. Mobility is usually would saturate around that temperature but phase relaxation continues to get better and get, you know, smaller and I guess, you would say, less and less phase relaxation as you go to lower temperatures. And so, a lot of the phase sensitive effects. In order to see them, you have to go way below 10 Kelvin. So, how do you include this? As I mentioned again, Many-Body Theory gives you a formal way of doing it except that it's fairly complicated. 

[Slide 4] Now, what you could do though is adopt the phenomenological approach and that's something we already discussed in Unit 2. I just want to remind you. That you see, if you consider this 1D wire with a scatterer in the middle, and you look at how the occupation of stage varies inside the channel, and that occupation is kind of like the electrochemical potential, so basically, it tells you how the voltage drops across the channel. And semiclassically, you'd expect drops at the contacts because of the contact resistance and then a drop in the middle because of the scatterer. And when you do a coherent NEGF simulation, you get this black curve, the one with the oscillations because of the standing waves. In practice, you may not see this because of phase relaxation due to electron-electron interaction. How do you include that in the model? Well, what we did back in this lecture, what I showed you was results that-- and then now, you can see the black curve, the oscillations are almost gone but otherwise, it looks just like this semiclassical curve. You see? There is a little oscillation right around there but-- and these-- then, this model captures this idea of phase relaxation and how did we do this? Well, we use this with m equal to i, n equal to j and this Dij had this structure. It was all ones. And the property of this kind of a D matrix is that it relaxes phase but does not relax momentum. On the other hand, if you use a D of this type, you see where it's not all ones but just ones on the diagonal. Then, it relaxes momentum as well. And so now, when you look at this plot, the oscillations are diminished as you might expect, but there's an additional slope to it. And that slope is because there's momentum relaxation. Momentum relaxation gives you this additional resistance and hence, a additional potential drop. So my point is, this is an example where I'd say electron-electron interaction is important in the sense that it gives phase relaxation. But, rather than do it from first principles, it may be much more practical, much simpler to do it phenomenologically. We could say that, well I know from experiment that you have a certain amount of phase relaxation and a certain amount of momentum relaxation. So, let's just choose our Ds in order to be compatible with that. 

[Slide 5] Now, there is another place where you have to include electron-electron interaction, in fact, this, I would say is a much more important thing that you almost do routinely. And that is that in a device, whenever you're running current, usually the electron density inside when you apply enough bias could change from the equilibrium value. And so, the conductor could actually get charged. And that would, of course, change the potential in the conductor. And that needs to be included through the self-consistent potential U. This is something I stress a lot or at least a little bit in Part A of the course, but we haven't raised it in Part B. But this is something that's generally well-known. I mean, even when you do semiclassical device analysis, you always-- along with your transport equation like drift diffusion, you have Poisson equation and you kind of solve them self-consistently. So here also, you would need a Poisson equation. And I guess what I've  written here, you could view as a version of the Poisson equation. So, Poisson equation would be like del squared U depends on n. Whereas the integral form of that would be U is equal to n times the potential due to a point charge. So, this is kind of the solution to that if you do not have any conductors around. If you have other conductors around with boundary conditions then it's more convenient to solve Poisson equation. And how do you get n? Well, you could take Gn. That's the electron density. Look at its diagonal element and then integrate over all energy. And what I've not shown here is that, of course, what you are really interested in is how much the electron density changed from the equilibrium value? And so, this is like the total electron density. You should subtract out the equilibrium value from it, and the same in the Poisson equation. Now, this kind of a thing, you could call it Poisson equation or you could call it the Hartree approximation. This, as I said, is a routine, should always be included in transport in general and especially if you are moving away from low bias. Now, in small conductors in addition, there's this exchange corrections that people say that, well, actually the potential you calculate from Poisson equation is a little too large. The actual potential that an electron fields, due to the other electrons, is somewhat less. You see? That's why there's a negative sign there. Why is it less? Well, the physical idea is that the other electrons always move to avoid this electron. That's the process that you-- it's hard to describe within a one electron picture but in the many-body picture, it would be like other electrons would avoid it for two reasons. One is, electrons that have the same spin would avoid it because of the exclusion principle, because you cannot have two electrons at the same spin at the same place. On the other hand, if there are-- and that's what this correction is about, the exchange correction. But in addition, electrons which have opposite spins are all electrons would avoid it just because of the Coulomb interaction. And that's this correlation part of it. So in general, people have worked hard trying to find the proper form for this correction, something that would accurately include this exchange in correlation corrections. And some of it is based on Many-Body Perturbation Theory. Some of it goes beyond that uses other methods to estimate what this is. So in general, this is a big area of research to find the appropriate self-consistent potential to put into this-- put into our model so that it would mimic, so that it would accurately describe this electron-electron interactions. And note that this correction actually goes to the Hamiltonian and not to the sigmas. It's not about a phase correction. It's like a change in the potential that in electron fields, actually. Now, there's one point I wanted to mention here though, that often, you might think that since this is a general technique, any problem you could-- we could always attack using this method. And I'd like to mention that that's not quite true because there are many methods that could require non-perturbative approaches, for example. 

[Slide 6] Let me explain a little what I mean by that. You see, we could write this U, kind of symbolically as U0 times N. And remembering, of course, the N is the delta N compared to the equilibrium. So, the potential that you see is due to the potential due to one electron times the number of electrons. So, this quantity here is what's called a single electron charging energy. That you have this conductor, if I put one electron there, how much does the potential energy change? And you could estimate that  just by thinking of the conductor as a little sphere whose radius is R. Then, this is something you have probably seen in freshman physics, is that the potential, when you put one electron on that sphere, the potential changes by q divided by 4 pi epsilon R. And so, the potential energy is q squared divided by that. So, if you use an R of 1 nanometer, the corresponding energy would be 1.6 eV. Of course, I've used the vacuum permittivity. In a solid, the permittivity is larger, it could be 4 or 10. And that would make the U0 smaller. Now, usually what happens is, normal devices, U0 is a very small number but U is not negligible because the change in the number of electrons is big. So, U0 may be just say 1 microvolt but you might-- but then, delta N could be tens of thousands. And so, you would still have a sizable potential to include. But there-- you wouldn't have single-electron effects. Single-electron effects arise when  you are in a transport regime where the single-electron charging energy is much bigger than these other energies of interests like kT and broadening. And in that case, you see effects experimentally. This Coulomb blockade is a vast field of research which if you just did a straightforward Poisson equation type theory, you would miss completely. And in the-- I won't go into it but in my notes, it describes some simple examples which you can see. Ordinarily, you don't worry about it because U0 is usually relatively small. kT at room temperature is 25 millivolts. Often, if you are talking of transistors, you have good contacts, electrons flow easily and so, when electrons flow easily, it also means the broadening gamma is large. And so, you normally don't go into this regime. On the other hand, as I said, there's a vast field of research working on conductors which are actually in this regime. And my point here simply is that you might think that this regime you could describe using Many-Body Perturbation Theory, but that's not necessarily true. You might need  these non-perturbative methods. Now, what do I mean by that? Well, roughly speaking, you could view perturbation theory as an expansion of this kind, that you need the inverse of 1 minus x. You need the inverse of something like E minus H minus US or UR, this random potential. You need the inverse. And what you do is make an expansion like this. And then, you look at different terms in the expansion and Many-Body Perturbation Theory tells you how to write down each term. And this is fine as long as x is less than 1. But once x gets to get bigger than 1, you see, this doesn't make much sense. And so, a perturbative approach is not necessarily always applicable. When you have strong interactions, you need non-perturbative methods or at least, you have to work in such a way that you're expanding around some other parameter which happens to be small. So, you have to choose your x carefully, that is you always go to some standard state and then expand around it. And you have to choose those things carefully. So, in general, the point I'm trying to make is that although this is a very powerful method that has been applied to many different problems, one should-- it doesn't mean it covers everything necessarily. 

[Slide 7] So, to sum up then. I guess we have these basic equations of the NEGF method that we have discussed. And the question is, how do you calculate these interactions, the sigma 0s? And, what we-- in the past, we have talked about elastic interactions. Now, we have talked about how to include inelastic interactions but within the Born approximation. And many applications can be addressed within what we have discussed. And all of our discussion is based on the one-electron approach. And the standard approach using Many-Body Perturbation Theory, it gives you systematic ways of writing down some of these sigmas in four different types of interactions. But, one should be careful. It doesn't necessarily cover everything because there are many problems that may require a non-perturbative approach. So with that, I guess, it's time to sum up. And that's what we'll do in the next lecture. Thank you.