nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L3.8. Inelastic Scattering
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[Slide 1] >> Welcome back to Unit 3 of our course on quantum transport. This is Lecture 8. 

[Slide 2] Now as you know in general when using this NEGF method, we have the H that describes the channel, and then we have the surroundings. There's the two contacts described by sigma 1, sigma 2. And then there is the sigma 0 which describes the interactions of the electrons inside the channel with their surroundings. Now in Unit 2, we had obtained expressions for this sigma 0 due to the interaction with the surroundings. However, at that time, we considered only elastic interactions. And also we didn't quite derive it, we kind of gave heuristic arguments why it works and we showed some examples back in this lecture. Now what I want to do in this lecture is firstly connect this with our more advanced view of sigma that we now have based on what we have done in this unit. So first I'd like to connect it, show how from this point of view you can get these results for elastic scattering. Like we considered before. And then we'd go on to discuss how you can extend it to in elastic processes so that you then have a way to write down these sigmas for inelastic interactions where energy's exchanged with the surroundings. Now in the last lecture, if you remember, from this sigma, you can get an expression for gamma. Where instead of the surface Green's function, you have the surface density of state or spectral function. And we related that to this lifetime and connected it with this Fermi's golden rule. Now we could use this same expression to connect up to these results that we had earlier. We could do it this way. You see, look at the sigma in, tau gn tau dagger. Well let's write it out in full matrix form. So sigma 0 in. The IJ component is given by tau im, Gn mn. And then tau dagger nj. Now and you have to sum over m and n. So this is just the usual rule for matrix multiplication. Now what you could do next is note that tau dagger nj is the same as tau jn complex conjugate because these taus are Hermitian matrices. So you could write it in this form and then this quantity here is what you could call D, Dim,jn. Of course, this D now has 4 indices. These tau, this was a matrix, we had two indices. This is what you might fall a fourth order tensor, imjn. And then you have sigma 0 ij is related to Gn mn through this fourth order tensor, Dim,jn. Now you might see, well, that doesn't quite look like this one because here your D had only two indices. Now that's because when you are doing this, we are considering a somewhat special situation. Whereas what we now have is a more general result. That is what you are considering then where scattering potentials, these taus, which have the property that they were diagonal. In other words, tau im was nonzero, only if i was equal to m. So you only had a tau ii. That's because we are considering the scattering potential which in position representation, that is when you use r as your basis, they look diagonal. So if you look this matrix out, the scattering potential, the matrix would only have potential elements on the diagonal. Not off diagonal. So that would correspond to this tau im being just tau ii. And in that case, what happens is Dim,jn is nonzero, only when m is equal to i and when n is equal to j. Because of the property of tau that we just discussed. And in that case, you get this Dii jj, which equaled in short hand form write as, ij. So this is kind of what we had done, except that I never spelled all this out, we wrote this down somewhat heuristically back in unit two. But now we actually have the full machinery to do it more formally, okay. 

[Slide 3] Now here what we have done is started with this expression of sigma in tau Gn tau dagger. And from that, you get this expression for sigma 0. Note that this expression applies whenever you take your channel and connect it to some contact. And this gn is the surface Green's function of that contact. But the way we are doing apply it to the scattering problem is as if the channel through the scatterer is connected back to itself. So that this gn, instead of being the Green's function of your contact is kind of same Green's function back again. Because scattering processes you could think of it as if the channel acts as its own contact. So electrons in the channel get connected or scattering into the channel itself, right. So that's why the small g suddenly became the big G. The big G which is associated with the channel. Now you could do the same again by starting from this gamma relation, tau a tau dagger. And what you'd get would be similarly gamma 0 here. Instead of a, you'd get the spectral function a. And net result is you would get an expression looking like this, gamma 0 equals DA. 

[Slide 4] Now what we want to do next is talk about how to generalize this for inelastic processes. Now what we did in the last lecture is we said we saw that. If you consider an energy E, then electrons due to interaction of phonons or other time-varying entities, it can either absorb an phonon and go up, that's these blue things. Or it can emit an phonon and come down. That's these red things. And one of the points I made something that's counterintuitive a little bit, is that it's much easier to go down than to come up. And that's where there's this N plus 1 and N. And so if the scatterers are actually an equilibrium, then the ratio of the two is given by exponential h-bar omega over kt. This you could actually check if you assume these Ns are phonons distributed according to say the Bose-Einstein statistics. And you could show that N plus 1 over N would be this exponential in here. But the main point here is that from a given energy, you could have either downward processes like so or upward processes. And one convenient way to do the bookkeeping which you'll be using is we'll write emission as positive omega and absorption as negative omega. So you could just think about downward processes. And that would be the positive omega. And similar arguments could be used for the upward processes, that would be that negative ones. Now what will show is that this expression that we had before, it will generalize into that expression. That is, if you have scatterers available with all kinds of frequencies omega, they could be described by this dh-bar omega, h-bar. 140 00:08:51,226 --> 00:08:53,356 It's a function of frequency. You know, certain phonons, you'd have certain strengths at different frequencies. And what will show is that this in scattering function at energy E depends on E plus h-bar omega, the Gn at E plus h-bar omega. This is, of course, different from what we had for the elastic case where everything was at the same energy. Inelastic, as you might expect, this energy, it depends on a different one. And this integral over h-bar omega runs from negative infinity to a positive infinity. The thinking is the positive part is the emission, negative part is the absorption. And we don't need to take care of or talk about each one separately, we can just focus on one of them. Say the positive one. And write down the expression. And then as long as you're integrating over all omega, it would cover both cases. Now the gamma 0 what will show is, and this is not intuitive and that's what I'll try to explain why. I'll try to give a little bit of an explanation where this comes from is we'll find instead of A of E, you'll get this Gn and Gp. Now Gp is something I have not used before. It's a new thing I'm introducing here. As I've mentioned before, A is like density of states, 2 pi times the density of states. Gn is electron density, 2 pi times electron density. And Gp is like the hole density, it's like the empty spot, 2 pi times that. So, in general, A of E would be like Gp plus Gm. And for elastic scattering, you don't have to look at each one individually because the sum is equal to the density of states. It's just that for inelastic scattering, instead of this, it's like one of them is getting shifted down, E minus h-bar omega, the other one is shifted up, E plus h-bar omega. And so I can't quite sum this up and write it as A at some energy, not in general at least. And so this part of it is kind of very non-intuitive, and I'd like to explain that a little further. 

[Slide 5] Now supposing we have this situation and for our purpose now, ignore the blue processes. Because, as I said, we can just focus on the emission term, the red ones. And question is if I stand at some energy E, what will be the inflow? Well the inflow would be electrons from here wanting to get in there. And so if you have taken any course on devices or the solid state physics, you'd probably be used to saying, okay, the inflow depends on electron density at E plus h-bar omega times the whole density at E. Because of this exclusion principle that electron when gets in there, it needs to have an empty spot there. So you'd write n times p. Similarly, if you want to write the outflow, the rate at which it goes out, you'd have n of E times p times E minus h-bar omega. That electron needs to be here and the empty spot needs to be there, okay. And remember we are not considering the blue processes in this discussion. Just the red ones. Now as you said if we just reverse the sign of h-bar omega, you'd get the blue processes, okay. Now the point I want to make is this is what you'd see everywhere, but in this context, you actually need to add another term to both. So you add the same term to both, which means you're interested in the net flow. If you're interested in the difference, you wouldn't notice this. Because this just cancels that. And I'll try to explain where this comes from, but in the context of what we are doing, it's extremely important to add this additional thing. Now if you accept this and I'll come back to it in a minute, then when I, you see, add those two terms, what I get is n E plus h-bar omega is common. So that's what I'm writing as Gn, and Gn is this matrix version of the electron density. And then you have E plus n, that's like the density of state. So that's A of E. Here when I add them up, the N of E is common to both, so that's what I'm writing as Gn. But then you have this P at E minus h-bar omega, that's Gp. And N at E plus H bar omega, that's Gn. So you then get these forms for the inflow and the outflow. And we'll see what we do with that in a minute. But let me now rightly justify that shaded term in there. What exactly does it represent? Well, you see, what it represents is a process in which an electron from here is trying to get into this level, you know, n of E and this is also n of E plus h-bar omega. But is stopped from doing so because of the exclusion principle. An ordinarily, you might say well I tried to get in there, but I couldn't because of the exclusion principle, so nothing happened. But the correct quantum mechanical view of thinking about this is not that nothing happened, but as if it got in there, but an equal amount got returned. And so this process led to some broadening. Now let me explain this point a little better. Take a much simpler problem, supposing we have an energy level connected to a contact which is empty. And so electrons can flow out into the contact. And, as we know, that leads to a broadening of this level. No problems. Now what if this contact is completely full? Let's say it's filled with electrons which means any electron here can't get out because of the exclusion principle. So does that mean it's not broadened? That means you'd have an extremely sharp line? Not really. What's well established and I think there's no doubt about it is this level is still broadened. Because the way you should think about it is that electrons from here, not that they're blocked by the contact, but it's like they flow out and new ones flow in. So in that sense, coherence is lost. So processes like this where something got blocked is really should be viewed as not that nothing happened, but rather something went out and something else came, an equal amount came back in. And so if you're just trying to figure out what is the net flow, it doesn't matter, and so you can drop these. But when you're trying to calculate broadening, for example, you cannot ignore this. Because if you just believed in that term, then you might say, okay, since there are no holes, my contact is filled. No holes, so there was no outflow and, hence, no broadening. And the answer is no, that's not right. You, because even though you were blocked, there was an outflow and there is broadening, okay. 

[Slide 6] Now with that little justification for the additional term, we can now go back to our old result that we had and compare it to this outflow and inflow expressions that we had in unit two. You know, one of the things I spend some time discussing this idea that you can think of these inflow and outflow from the contacts as this sigma in times A and gamma times Gn. Now here also you'll notice you have something times A, so naturally you'd interpret that quantity as sigma 0 in. So this is the sigma N due to the interaction. And, similarly, here is something times Gn. So that naturally becomes the gamma 0. So these would be then the appropriate sigmas and gamma for inelastic processes. 

[Slide 7] So you could now fill it out. We have been kind of trying to keep the notations simple so we didn't write out in full detail the tensor nature of D and things like that. So you could fill it out and then it would look something like this. Sigma 0 in is equal to this D over 2 pi Gn like we have there. But then you have to, of course, integrate over all h-bar omega because phonons just don't have a single frequency. In general, they would have a complete spectrum. So that's this D h-bar omega. And then there is the additional tensor stuff I had mentioned. This ij here and an mn there, and this is this fourth order tensor in general. Which is this im,jn. Similarly, could take this one and generalize it, and you'd have this gamma 0 is related to this Gn plus Gp like I mentioned. So this is the general version. 

[Slide 8] So to finish up then what I like to point out is one more thing. And that is, we just obtained the expressions for sigma in and gamma. Now what about the sigma? Because you know, usually we start from sigma and then we get gamma by doing i sigma minus sigma dagger. And so sigma was tau g tau dagger. And so the gamma becomes tau a tau dagger. And from that point of view, you'd have got gamma 0 is equal to D times A and A is Gp plus Gn. But as we just discussed, the process, there is the subtle argument about exclusion principles that one has to go through so that you don't get this A, rather you get Gp at E minus h-bar omega. And Gn at E plus h-bar omega. So what that means is we have a-- what you have obtained is this expression which is a little different from what we had before. But the question is what's the corresponding sigma? Because given a sigma, it's straight forward to go to gamma. Into gamma i minus sigma minus sigma dagger. But given gamma, how do you go back to sigma? And the point is it won't be anything as simple as saying that, oh, just put in the Green's function at E plus h-bar omega or E minus h-bar omega. What you have to note is something like this. That the sigma, the imaginary part of the sigma, that must be the gamma. So that's what I've written here. So if you did i sigma minus sigma dagger, you'd get gamma as long as this h and gamma, these are both Hermitian matrices. So what I can show is that if you write sigma in this form, then gamma 0 would be equal to i sigma 0 minus sigma 0 dagger. But question is what is this part? What you might call like the real part of the self energy. Because no matter what h 0 is, this form of sigma will lead to the gamma that we have obtained. So you can choose any arbitrary h 0 and this would be consistent with that in the sense that if you use that relation from this one, you'd get back that one. So how do we determine h 0? Now that's where usually you invoke a very different principle unrelated to all this. And that goes something like this. So algebraically, let me just multiply this by 2i and write it as 2i sigma 0 is equal to gamma 0 plus 2i h0. Now the point is that sigmas in general just have the following form. Gamma 0 convolved with this function. So this is the symbol for convolution. So gamma 0 convolved with a delta function. In fact, any function convolved with the delta function gives back the same function. So that first term is really just another way of writing gamma 0. But the second term, this one, it needs to have this form that is gamma 0 convolved with i over pi E. And any function is it convolved with that, that has a name, it's called the Hilbert transform. So generally this is a, I guess, a general principle that for response functions like sigma, the imaginary and real parts are Hilbert transforms of each other. Now where does that come from? Well this is a convolution. Now if you went to the time domain, what it would look like is this function, the Fourier transform looks like a unit step function. So what I've written here is like the Fourier  transform of a step function. Which is 0 for negative t and has value of 1 for positive t. And when you Fourier  transform, of course, convolution becomes an ordinary product. So the point is that this form ensures that the Fourier transform of sigma 0 looks like some function, which is the Fourier transform of this times the step function. So what it ensures is that this function will always be 0 for negative t. And that is a property that it needs to have for physical reasons. Because the sigma is like this Green's function which is the response of the system. And the response is causal in the sense nothing happens till you hit it at t equals 0. So when Fourier transformed, this function needs to be 0 for t less than 0. And by choosing this part to be of this form, we ensure that. Because the Fourier transform of this is the causal, this unit step function. So and this is a principal that occurs in many different contexts. So it's nothing to do as such with quantum mechanics. We call this the Kramers-Kronig relations. That idea that any response function that something that has to be causal, causal meaning which has to be 0 for t less than 0. Any such function, the real and imaginary parts must be Hilbert transforms of each other. Bottom line, then, when you have inelastic scattering, we have an expression for gamma. But to write sigma, of course, the imaginary for this part is easy. But to write this part, the Hermitian part of sigma, that's where you have to invoke this causality principle related to this Hilbert transform idea. 

[Slide 9] So to sum up, then, what we showed in this lecture, what we talked about is how to write down in general sigma zeros for inelastic processes. These inelastic processes are described by certain potentials, these taus, the scattering potential. So if it's an phonon, this would be the potential that one phonon causes the-- what an electron sees you do one phonon. So those would be this potentials, and the Ds are like that, these are random potentials due to the phonons, so this is like the ensemble average of that quantity. So that's the D, that's what goes in there. So that gives us sigma 0 related to Gn. Gamma 0, this broadening is related to this Gp and Gn. And if you want the sigma 0, then it has got the gamma 0, but then the other part, the real, the Hermitian part of sigma is given by this Hilbert transform. And this D, of course, is what describes the properties of the scatterers. And, in general, you'd have this absorption and emission. Emission is the positive frequencies, absorption is the negative frequencies. And if the scatterers are in equilibrium, then the ratio of the two would be given by exponential h-bar omega over kt, okay. So in the next lecture, then, we'll talk a little more about how interactions are included in NEGF and what the limitations are. Thank you.