nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L3.10 Summary
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[Slide 1] Welcome back to Unit 3 of our course on quantum transport. And this is the tenth lecture, so it's time to sum up what we have been doing. 

[Slide 2] Now, as you know, our interest is in describing current flow through nanoscale devices, where this channel has certain energy levels. The electrons are injected from a contact and picked up at another contact, giving rise to current flow. And one of the points of stress especially in Part A of this course, is that what make transport difficult is this mixing of two very different types of processes. What you might call the mechanical ones which are driven by force and the thermodynamics ones which are driven by entropy. And, so when you're trying to describe transport, it's not enough to have the mechanics, like Newton's Law in the-- for classical mechanics. You have to add these entropy driven processes to it. And in the semi-classical context, Boltzmann, showed us how to do that in the 19th century and that's Boltzmann. And this Boltzmann equation is the center piece of all semi-classical transport. And Part B of this course, what we are interested in this, Quantum Transport, where for the mechanics instead of Newton's Law as we have this Schrodinger equation. And what I'm trying to introduce to you is, the part about including these entropy driven processes to get this non-equilibrium green function methodology, NEGF. And so in Unit 1 of this course, we started by, just talking about this Schrodinger equation itself. And this is something you might have seen in other quantum mechanics courses that you have taken because that is described in all standard courses. What is non standard is the rest of it, although one thing we did in Unit 1 is introduce this tight-binding model, which you may or may not have seen earlier. Where we showed how you can write this H in the form of a matrix. But Unit 2, what we went on to is, 

[Slide 3] how to include the contacts. And there the first pint was to recognize that these contacts requires this additional terms, the sigma one and sigma two called the self energy and then there's a source term, electrons coming in, that's this s. And the resulting Schrodinger equation kind of looks a little different from what you normally see. Normally you just see that much but you these additional terms that you have to put in. And then I made the point that, with electrons, the observable quantity is this psi psi dagger. And so, it is convenient to work with equations where the basic quantities, the psi psi dagger. And these are the NEGF equations. And in Unit 2, I guess in lectures two to five, we obtained these equations along with an expression for this current operator on how to calculate the current. And this is the general expression for the current and for coherent transport that is, which is a sub set of the problems we want to consider. So coherent means, the only sigmas that are associated with the context, and you ignore all interactions within the channel. That's coherent transport. In general, of course, we are interested in non coherent transport, where you also include the interactions, which we'll be talking about, which you have also talked about actually but for coherent transport, you could simplify this equation into a different form that's convenient. So whether you have this current written in the form as proportional to f1 minus f2, and this is often called the transmission. But, I'd like to stress that, this only applies to coherent transport. You can not use it in general. It's this one that you should be using and more, more generally, you have this current operator which you could for all kinds of current. Anyway, so this, this all the things we did in Unit 2. 

[Slide 4] And another thing I introduce then was also the physical interpretation of different quantities that in order to keep your sanity and keep track of all these symbols. It is important to have a physical picture in mind as to what these things represent, like Gn is like electron density. This A or spectral functions is like the density of states. This is the in scattering, that's like the broadening and so on. Now in order to use these equations, of course, first you need the H which we did in Unit 1. But the next you need is the sigmas. And for the sigmas, in Unit 2, we talked about an elementary way of writing the sigma which applies to one dimensional problems. What we did in Unit 3 in Lectures 3 and 4 is introduced a more general expression for the sigma. Something you could use in general form, many, many types of problems. Where as this one is very special, applies only to the one D case. 

[Slide 5] And in Unit 2, we use this resonant tunneling as an example of how to use this method and what we showed in Unit 3 in the second lecture was how you could use this elementary method but for a more difficult problem, a two dimensional conductor. And this is a problem of great interest in its own right because as I've mention before, one of the seminal results of mesoscopic physics, is this quantized conductance. The idea that, when you take a narrow contuctor, as you make it wider and wider, the conductance doesn't increase linearly but rather, it increases in steps. And this a result that should come out naturally from the general NEGF method that we're talking about. And I always suggest this as a good example to try out to yourself because you have to do everything right in order to come out with the results as clean as that. You know, where you have nice quantization. This axis is the width of the conductor. This axis is the transmission at a given energy. And this a very good test example to try out because I always say, whenever you have a new method, first you should do a standard problem and make sure you get the standard results, before you do complicated unknown problems. Now, next we talked about this general method for getting the sigma and as an example of that, we said we could analyze ballistic conductors in graphene or in carbon nanotubes. So I guess, this result is for graphene. And again, if you had set up the method correctly, these should come out automatically. As I've often stressed, the part of this method is, you can do the calculation without necessarily understanding all aspects of the problem as long as you are competent and your setting it up properly. And then you can use your calculations to improve your understanding. You could say, well, why is this step at this energy. How do you understand that, so, that we can understand it. Use this to improve your overall understanding. Another example that we then talked about is this, hall effect, which involves, these transverse resistance. And it's calculated at a given energy like the-- where the electrochemical potential is at low temperatures. And this accesses the magnetic field. And at low magnetic fields, it kind of goes linearly, that's the classic hall effect. And at high energy, you have this amazing quantization. Now, which was again, another of the seminal results of mesoscopic physics that dates back to 1980. But if you have set you the problems correctly, these should come out automatically, right. So this is what we discuss in Lecture 5 of this unit, that was Lecture 6. Now, all of these problems, however, are examples of what you might call coherent transport. But the real power of the NEGF method is, 

[Slide 6] that it also allows you to include interactions. You see this through the sigma zero. So the sigmas, if sigma one and sigma two due to the physical contacts. But then, electrons going through the channel interact with the surroundings and that can be described through these sigmas, the sigma zero, here. And how do you evaluate that? Well, you could use the same expression to come up with an expression for sigma 0 are sigma 0 in or the gamma. Because the way it works is, you see gamma is tau a tau dagger. A is suppose to be the surface spectral function or the surface density of states of the contact. But when it comes up to scattering processes like this, you could view it as if the conductor is its own contact. That is as if you have taken the conductor and connect it with replica of itself through the scattering potential that taus and so you could write the resulting gamma 0 as tau a tau dagger but the a is now the density of states of the conductor itself not of some other physical contact but it's the same a. And the tau and tau dagger then, that product is what we call the D. And it's a more complicated object than tau, in the sense that, these are matrixes with two indexes. Here when you look at the product, it has four indexes, it's a fourth order tensor. And this D then, describes this, the strength of the scattering process which is reflected in these taus. So this what you'd get if you had elastic scattering. And again as I pointed out, there's a lot of physics you could cover, considering elastic scattering processes, meaning ones where, no energies exchanged, and so, the energy, the sigma in at a particular energy depends on the electron density at that energy. On the other hand, what we did is, in Lecture 7 and 8, we talked about how you could extend this to include inelastic processes, where the in scattering at energy E, depends on the electron density at, E plus h bar omega. So that would be like an emission process. As if a phonon got emitted and the electron went from that energy to this one. And this integrals runs from negative to positive, and negative part takes of the absorption processes. Those are the minus h bar omega. 

[Slide 7] Now, one of the points I want to make here is, that everything we have done here is based on, this one-electron picture that is, the way we think of it is scattering processes give rise to a certain random potential that an electron feels which goes into this one-electron Schrodinger equation. And then from this you are-- you can get these taus and accordingly you can get Ds. Now, this one electron picture actually gets you the right results for elastic scattering. For inelastic scattering as I made a point, there are some subtle issues that you have to think about and that's related to this exclusion principle because normally, this is not a very obvious thing, right. And those are subtle issues we discussed when we talked about it in this-- I guess Lecture 8. Now the normal-- the standard way of obtaining these results is using this Many-body Perturbation Theory, which requires this advanced second quantized formalisms, which we haven't talked about, OK. And of course the part of these is in general, that it allows you to go to much higher orders. So for example given any specific interactions, you could write down higher order results that is, this is like the lowest order where sigma is proportional to G. You could write a next higher order which would have G squared, I mean two Gs' or three Gs' and so on. And this gives you a systematic way of writing all these interactions to any desired order. But for the lowest order which is usually called the Born Approximation, what you're using is a self consistent Born Approximation. In this order you can write the results more or less intuitively from the one-electron picture, though you have to be careful about these exclusion principle related subtle issues. 

[Slide 8] Now one important problem that often people worry about is this electron electron interaction, as I mentioned, this is a very subtle kind of interaction where momentum is not relaxed because what one momentum-- what one electron losses another picks up. But phase is destroyed and often standard conductivity on mobility is not affected by phase relaxation , its only affected by momentum relaxation. And so electron,-electron interactions do not have a direct effect. On the other hand, there are many other transport problems or experiments which are phase sensitive and so it is important to include phase relaxation without momentum relaxation. But we include electron-electron scattering from the first principles that's much more complicated. See, what you could do is adopt the phenomenological approach, and this is what we talked about in Lecture 9. Where we showed that if you had a wire with a scatterer, if you do that coherent calculation you get this oscillations in the potential-- the electrochemical potential around the semi classical value. And these coherent oscillations of course would usually be destroyed because of this loss of phase-- the phase relaxation and once you include this phase relaxation you indeed get a curve, this black curve where these oscillations have been suppressed. And how did you do this calculation? Well not by including electron-electron interaction from first principles rather by adopting a phenomenological approach that is we use the D which have this form and in this form it destroys phase but doesn't destroy momentum. On the other end if you use the D of this form, which is diagonal, it destroys momentum as well. And this is again reflected in this results. You see here, the oscillations are gone but this part is flat because this is a ballistic region with no resistance and so the slope is zero. On the other end once you include momentum relaxation even this part there's a potential drop and so there is a slope to it. 

[Slide 9] Now, another case where people use the-- people need to calculate these potentials is the self consistent potential due to charging effects. And this is again part of with standard device analysis that you have to include the potential U due to the change in the electron density. And usually that's calculated to the Poisson Equation or what you might call the Hartree approximation. But in general for small conductors it's often important to include this corrections for exchange and correlation. Its many-body effects and again the way you try to calculate these terms is using this Many-body Perturbation Theory. Although, there are, approaches where you go beyond that, do something different. But the point is this is of course is a vast field in itself. And in order to follow that you'd need to understand this, many-body formalism. But one point I want to make is that people often think that this is general so you should be able to handle anything, the Many-body Perturbation Theory. But what I would like to mention is that there are many problems where this Perturbation Approach is not the best, it doesn't give you the best results. You really should use other methods non Perturbative Approaches. And one example of this is this coulomb blockage phenomena, single electron charging effects. Which are observed and which is a vast field of research on its own and these require non perturbative methods in general. 

[Slide 10] Now over all then the point I wanted to make is that these basic NEGF equations on this current operator in this course we obtain them form the One-Electron Schrodinger equation using this one-electron approach. And whereas commonly they are obtained from this Many-body Perturbation Theory and these equations then, you could solve to obtain the current for a given applied voltage once you know H sigma 1 and sigma 2. This is what you might call coherent transport. Where the sigmas are associated with the contacts. In general, though you need to include interactions, this is this U the charging energy then the sigma zeros which could describe elastic processes or inelastic processes and these have to be calculated self consistently in the sense that they're dependent on the greens function and the electron density matrix inside this. And so it's like you should make some assumption calculate this feed it back again recalculate and keep doing this until the whole thing converges. And in general in device analysis, people always solve the transport physics self consistently with the Poisson Equation and this is a more generalized version of that you could say. And how do you write down these-- what are these self energy equations? Well in some cases you can get it directly from the one electron approach especially if it's elastic. If it is inelastic, then there are subtle issues like, as I mentioned involving the exclusion principle and of course the formal way of getting those results would be from this Many-body Perturbation Theory. But you can kind of-- once you know the results you can kind of rationalize and understand where it comes from. There are other problems where you actually need this Many-body Perturbation Theory in order to write down these self energy functions. But then of course as I mention there could be problems that require non Perturbative Techniques and so you might have to go beyond this. But one last thing to mention here then is, that you see 

[Slide 11] in general these equations are obtained from Many-body Perturbation Theory and so in peoples' mind, this whole approach is kind of tied to the Many-body Perturbation Theory and so people view NEGF as an esoteric tool which is accessible only to those who are well versed in this formalism. But the point I wanted to make is that there are many problems of interest these days where you can-- where it do not really need to calculate the sigmas using this because the kind of sigmas that captured the physics can be obtained more or less from the one electron approach. And so, this part, I feel should be much more widely known it should be part of all graduate and even undergraduate curriculum, see. And this is the-- in that case you see you'd be able to do all kinds of problems that are of current interest for example and this isn't happening because in people's minds again this equation and everything is kind of tied to this very advanced formalism. Whereas, ideally what should happen is, see-- this should be something that should be widely known and the part that needs the advanced formalism would be certain types of scattering processes. So for example the Boltzmann equation I had mentioned is the center piece of all semi classical transport and the Boltzmann equation has something called the scattering operator. Now, how do you calculate this scattering operator? Well those methods have evolved, they often use this Fermi's golden rule. You know, something that was not even known in Boltzmann's time. But the Boltzmann equation you know, still used, the scattering operator, how you evaluate it, that has continued to evolved, the same here. These are the basic equations, how you calculate these sigmas, how you calculate this self energy, that will evolve and as I said some methods might even require non perturbative techniques, et cetera. 

[Slide 12] Well, with that done, I guess we can close up this unit and we'd end with one more unit in this course about Spin Transport which gives you a very good example of great current interest where you can use the formalism that we talked about the NEGF method, you know, which I usually don't refer to as NEGF of method but rather say that well its just a basic method for contacting Schrodinger, for adding contacts to the Schrodinger equation. And in the next unit I guess we'll be presenting examples involving Spin which I believe give you a lot more-- lot of insight into the nature of quantum processes and how they differ from the semi-classical picture. Thank you.