 Okay, so last I guess it was Monday right Monday afternoon we talked about this model for understanding current flow in small devices and as we discussed it's even useful for understanding current flow in larger devices as well and you have heard Professor Lundstrom and Professor Appenzeller last two days, the beautiful talks, they helped connect this way of thinking, this question kind of to real world experiments on graphene. So for example, Professor Lundstrom showed how you could use this to understand conductance measurements on graphene samples and you also heard Professor Appenzeller yesterday. Now what I'll be talking about today is a, we'll call a more sophisticated model, it's called this non-equilibrium green function model for transport and the basic equations I try to write down here, that's also what you have in your notes, one in your notebook if you look at it, these equations are right there and these equations are quite general in the sense that we have been using them for many different conductors over the years like we have been using it for say quantized conductance in two-dimensional conductors for silicon nanotransistors, Hall resistance, quantum Hall resistance, this magnetic tunnel junctions, spin torque, spin Hall effect, all kinds of things. So these are perfectly general that you could apply to all kinds of problems. What changes from one problem to the next is just how you write your H's and sigma's. So H is this Hamiltonian matrix that describes your conductor, sigma's, those are also matrices and they describe your connection to the contacts. So what's, so when you have a whole new problem like graphing, the question is how do you write down these matrices? That's all that's really different and a practical problem you might run into at some point is that the matrices are too big, right? Small problems, I mean there's no, you can easily do it on a laptop, once you have written down the matrices, you have the H's and sigma's, you find G, then you go to this equation, tells you this is the equivalent of electron density, if you want density of states, that's where you go etc. So these are things that I teach over a course which is on the nano hub that you can look up easily. There's a graduate course on this and an undergraduate course also actually. And I won't go into many of the details here because my purpose here is to give you an overview of things so that even if you haven't seen it at all, it'd still be useful. You'd get a good roadmap as to how this works. And especially of course I want to gear it towards how you apply it to graphing. And on Friday Professor Lundstrom will be talking about graphing p-n junctions. Now that's an example where you couldn't just use this formula. Why? Because this assumes there's a density of states that's kind of characteristic of your entire sample. So we are talking about a problem like this and the density of states is uniform across the thing. We are talking of a graphene sample. Now if you have applied some kind of a voltage that makes this side a little different from that. So there's a step in this potential. And so this side is say n-type graphene and that side is p-type graphene and you couldn't really use this. On the other hand you could use this formulation. You could use this to solve to find the conduction through a p-n junction like that. And that's something Professor Lundstrom will be talking about tomorrow. Now specific problems there may be easier or simpler ways of doing it. That's true. So, but what's nice about this NEGF method is how general it is that even if that you don't really have to make all kinds of assumptions in this. You can go ahead and almost blindly do it at first and see what you get. Of course, I highly recommend after that, you know, thinking seriously about what you're calculating. But the point is you don't really have to understand everything in order to apply it. Anyway, so right now then in this lecture I have kind of two objectives. One is I want to spend the next few minutes like 20 minutes or so trying to connect a little bit between this and this. Like last day we talked a lot about this. On Monday afternoon I tried to explain where this comes from, what it all means. I want to connect a little bit between them and show you the differences and give you a little bit of the flavor of where this comes from. Of course, without really going into any serious derivations because as I mentioned serious derivations these would be I mean a few hours lecture on in themselves. Now if you get back to this equation that I wrote here, let me write this in a slightly different way which actually has a very nice physical interpretation as well. In fact, last day after my lecture, remember talking to Professor Alam and he said that well, you know, what you said was fine but why didn't you give this other interpretation of things? I've always found that a whole lot more intuitive. So this is this other way of looking at it, which goes something like this. Let's take this entire quantity here and call it gamma divided by h bar. Now no particular reason why I put an h bar there but dimensionally the gamma is has the dimensions of energy and when you divide it by h bar this is like per unit time. Right? So this is like per unit time. So and you'll notice what is in here also has the same dimensions. You see velocity is centimeters per second. This is centimeters. So that's like per unit time and this is dimensionless. It's like per unit time. So I'll write it that. So what you get then is an equation that looks like this. Now the reason I say this as a simple interpretation is something like this. You could say that well, how do I get this equation? Well, let's look at this conductor. You have a certain density of states in here and gamma is the rate say at which electrons want to come in here, gamma over h bar. So and it's also the rate at which they want to go out into this contact. So I'm assuming for this discussion that the two contacts are equivalent but that's not necessary. You could have made it gamma 1 and gamma 2. But for this discussion, just assuming it to be equal. Now the point is then what you'd have written is that the current should be equal to if I were to write the current at contact 1, it would be something like q times gamma 1 over h bar times df1 minus n. Now what do I mean by that? It would be like n is the electron density inside in here and d times f1 is kind of what the electron is the electron density that the contact 1 would like it to have. Why is that? Because contact 1 wants to bring this channel into equilibrium with itself. And contact 1 has a Fermi function that's f1. So it really wants to fill up this entire density of states according to its own Fermi function. And if those two are equal, there would be no current. The contact would be happy. And so you'd expect that the current would be kind of proportional to that, the difference. And this proportionality, this constant here, that depends on how strongly it's connected to the contact. So you could put a gamma 1. And I guess for, I had assumed the gamma 1 and gamma 2 were equal, but that's not necessary. I mean, as I said, you could have carried a gamma 1 there. Now the I2 is q gamma over h bar times n minus df2. So df2, that's the electron density that contact 2 would like to see. But then it's actually a little different and so there's a current that flows. In that steady state, you could argue those two things have to be equal. The rate at which things come in, same rate, they flow out. And so if you equate those two, you'd get n is equal to d times f1 plus f2 over 2. It's just kind of intuitive that, you see, left contact wants to fill according to f1, right contact wants to fill according to f2. So it does something average. And if I had kept a gamma 1 and gamma 2, what I'd get is gamma 1 f1 plus gamma 2 f2 divided by gamma 1 plus gamma 2, that's it. Now if I take this and put it back into any one of these, after all those two are equal, I'll get this equation, that's it. So it's a very simple equation in a way. You know, you've got this density of states, current is proportional to f1 minus f2 and depends on the rate at which the electrons can come in and go out, that's the gamma. This gets divided by 2 because you have gamma over h bar and the gamma over h bar, like two conductances in series, so the net is like half of that, that's it. Now if you look at the definition of gamma then, what I wrote goes something like this, gamma over h bar is equal to vx over l times this lambda prime over l. So remember, this is the part we had for ballistic transport, this is the part that came because of all the back scattering. Let me put that in there. The question is, does this have a simple interpretation? Well, if we ignore this factor, then it's like we are talking of ballistic transport and so gamma, remember the way I defined gamma is the rate at which electrons want to come out or go into the other contact. And you'd think that if electrons flow freely through this contact, the time it takes for them to escape into any one of those contacts is equal to this length of the channel divided by velocity, that looks perfectly reasonable, l over v. On the other hand, if you have a lot of scattering inside the channel, then it gets multiplied by lambda over l, so that the overall time is now proportional to l square. And that's because when you have lots of scattering, then as Professor Alam has actually discussed at length about this, that when you have diffusive transport, the time it takes to get through a length l is like proportional to l square. This is like the diffusion coefficient divided by l square. In fact, after this week, you might be wondering if it should be 1.93 or something. But the thing is in general, of course, ballistic transport goes as 1 over l, diffusive transport goes as 1 over l square. And what happens is this diffusion, this density of states of course is proportional to the size of the conductor when you get to big conductors. It's like the bigger you make it, more the states you have in a given energy range. There's a l there, ballistic transport, this one is goes as 1 over l. And so you get a current or a conductance that's independent of l. On the other hand, if you have diffusive transport, this goes as 1 over l square. And so you get w over l, ohm's law, etcetera. So that's how you would understand this. And this is kind of very intuitive, right? That just density of states times what you might call the escape rate, a whole different way of looking at it. In fact, actually when I teach the undergraduate course, I usually start with this one. Because the nice thing about this one is it works with density of states. You don't really have to understand EK diagrams or anything like that. You know, because one of the things you worry about when you get to small conductors is the conductor few atoms long, does it really even make sense to talk about an EK diagram, etcetera. And although in our thinking usually EK is very fundamental. And like graphene we are talking about, first thing we want to do is know what the band structure looks like. What's EK? What's this velocity d, dk, etcetera, right? Whereas when you think of it this way, you don't really have to know about EK diagrams. This could as well be amorphous silicon with as long as you know what its density of states is. So the only issue is what's the density of states? How easily does it get in and how easily does it get out? That's about it, right? Now the reason I bring this up here though is more because this helps connect to the NEGF equations more naturally. In the sense that from this point of view, you see you could write here d gamma 1, gamma 2. This is the density of states. This is how it couples to left contact. This is how it couples to right contact. And if you look at the NEGF equations, it's almost as if instead of having numbers like this, we have matrices like that. So instead of, so you have this Hamiltonian matrix whose eigenvalues give you all the energy levels. And the sigmas, those are basically telling you how easily they get in and out. Actually, these are also matrices. Now the only thing is H is always Hermitian. It has real eigenvalues. The sigmas are non-Hermitian. They have complex eigenvalues. And the imaginary part of that actually represents the rate at which they go in and out. So in that sense, if you actually look at the equations also, they appear somewhat intuitive if you look at this one. For example, how do you calculate current? You have an equation like this, trace of some matrix and then again some other matrix. And the thing is that this is like n, this gn, it's a matrix version of n. If you look at its diagonal elements, it will tell you the electron density everywhere. Similarly, A is like the density of states. If you, I feel like I wiped off that equation, but the one I had written there, if you look at that, what I had for the current was gamma times D times F minus M. That's what I had written before, you see. And what you now have is the matrix version of everything. That's it. So from that point of view, it kind of almost gives you a natural feeling for these things. All that happened was what used to be numbers became matrices. Now matrices, of course, are very important in terms of capturing all kinds of physics that you don't get through numbers. One is interference, that when you are adding positive numbers, if you have some amount of current through one level and you have a second level right around there, the two currents will add. So if you get one milliamp through one level, two levels, you'll get two milliamps. You couldn't get less. With matrices, though, in the right way, you could actually get less. They could be interfering the wrong way and all that. And under certain conditions, you do have quantum interference effects that work that way. And the thing is the matrices, of course, will capture all that physics. And without thinking, without, as I said, if you do it right, it's like, not that you have to understand everything and do things accordingly. It should all come out automatically. Now there's another piece of physics that's very important that comes out automatically from the quantum treatment. And that's what I'd call broadening. Now let me explain that point a little. Is this quantity here d gamma over 2h bar? As I mentioned, one of the nice things about this form of the equation rather than that one is that here I don't have to talk about EK diagrams. This could be amorphous silicon. You give me any density of states. Tell me how it gets in and out. We are done. Here I have to know EK relations to find velocity and stuff like that. Now, what if I applied this to a device with just one level? So I guess I could use this picture. So supposing we have a device where instead of having such a continuous set of levels, all we have is just one level there. Now if you looked at the current versus voltage of that one level device, it would look something like this. It would just step up at some point. Why would it step up? Well, because as I'm changing this voltage, bring this down. At some point, I'd be crossing this level, and that's when current will start flowing. As you know, in order to have current flow, you need to have states between mu1 and mu2. As long as this is above it, no current will flow. As soon as it crosses, it will flow. And so we'll have this nice sharp step. And what is the width of this step? Well, if you are doing this at high temperatures, it would be spread out. Why? Because Fermi functions are always spread out by a few kT. But if you are doing this at low temperatures, it will be very sharp. And the thing is, in principle, you might think that at low temperatures, this could be infinitely sharp. And if so, you'd think that the conductance would be infinitely large. Conductance meaning if I'm looking at DIDV. And that is wrong. What is now established quite clearly from an experimental point of view, something that was not known before, say, 1988 or so, but now is very clear, is that when you take a small conductor and measure its conductance, per mode, the maximum you'll get is this q square over h, which is like 25 kilo ohms. I mean, h over q square is 25 kilo ohms. And if you have two spins, you get half of that, 12.5 kilo ohms or so. Because two spins are in parallel. If you do it with carbon nanotubes, then you have two valleys as well. So it's more like six kilo ohms. But the point is, there's a maximum. It's not infinitely large. So how do you understand that? Because if you look at the density of states, here, it looks like a delta function. In principle, you might think this is so sharp that d times gamma could be extremely large at that point. But not really. What happens is, when you do a proper quantum treatment, the very act of connecting the level to contacts also broadens it out, you see? And that broadening is like uncertainty principle. That's the part that you would call the uncertainty principle. But of course, if you're using a wave point of view, like the Schrodinger equation, it comes naturally. You don't have to invoke it. So if you are using this set of equations, for example, that's based on the Schrodinger equation. So that will take care of it automatically. You would get this broadening naturally. And this is a very important thing that doesn't come out of a simple semi-classical viewpoint. In fact, last day, someone had asked me that, well, we have this dvx over 2l. And we obtain this factor from a purely semi-classical argument, namely, with just density of states times velocity we just counted current, et cetera. Now, when we evaluated it for a one-dimensional wire, we found that it came to 1 over h. And the one thing that might bother is where did the h come from suddenly? And h, of course, kind of suggests you must have invoked quantum mechanics somewhere along the way. And the way, actually, we invoked it here is that, you see, when evaluating the density of states, we said that you have some e k relationship, and the k's are separated by 2 pi over l. And that is kind of an uncertainty relationship. You see, this is in terms of k and l. You could turn it in terms of energy by, you see, multiplying it by dE dk. As you know, dE dk is like h bar v. So this would be h bar v times this. And that's delta e. So the idea is you're saying you have one level. All the levels are spaced by that much. And that kind of limits the density of states that you get out of that calculation. And that is why, when you use an argument like this, you finally come up with a dvx over 2l. That's 1 over h. That's how the Planck's constant sort of comes in. Although it almost looks like you have not yet brought in Schrodinger equation, but the thing is that uncertainty principles are right there when you put in the periodic boundary condition. And the same here. Yes, like if you're doing it with one level, then you have to invoke this broadening amount if you want to get it right. Now, lots of times, of course, with big conductors, you have so many levels that you don't have to worry about the broadening of a single one. There's enough of them around that you get a broadening automatically. But for small conductors, that's very important. And that, again, is something that you come out of the Schrodinger equation naturally. Now, let me just say a few words about where these equations come from. As I said, this is related to the Schrodinger equation. But one thing that many people often ask me is, why not just use the Schrodinger equation directly? Why do we need these NEGF equations? Now, the Schrodinger equation, as you know, looks something like this. E psi equals H psi. So I'll write it this way. EI minus H times psi is equal to 0. So that's the equation you normally solve. That would be the Schrodinger equation. And this I is the identity matrix. H is your Hamiltonian matrix. So it's like E psi equals H psi. And when you solve that, of course, you usually think of the E's as all the eigenvalues of H. The thing is in a transport problem, what we are doing is an open problem. Where the electrons continually come in from the contact and go out. And in order to describe that, first thing is you need a source term on the right. What's the source term? It's like all the electrons that are coming in from the contact and continually hitting your device. That's the source term. But the important point is that when you hit it with a source term, it comes in and hits it, you also have to add some damping in here. There's something associated with it. These are the sigmas. This is the Hermitian Hamiltonian. And these are the sigmas, the non-Hermitian things that connect to the contact. So one way to think about it is almost like, as if originally we had an isolated LC circuit. And now that you're trying to drive it with a source of some kind, it also comes with a source resistance. And this resistance is like this one. And just as any resistance would broaden the resonance of an LC circuit, similarly, just this process of connection will broaden each level. And another, I mean, if you're a mechanical incline, you can think of it as a guitar. Like when you had this, when you didn't have these things, what you're trying to do is finding the resonance frequencies of a guitar string. Now the question is, if I try to excite that guitar string with a tuning fork of some kind, then what response will I get? Now, of course, if it exactly matches the guitar, let's say a guitar is tuned at 400 Hertz. And if I hit it exactly at 400 Hertz, of course I'll get a response. But even if it's like 401, I'll get something. So there'll be a broadening associated with it. And if it's a badly damped guitar, I mean, not the kind anyone would want. If it's badly damped, then it would be very broad. You know, you could hit it even at 350 and you'll get some response. And most of the time our devices are like that. They are, we have good contacts, which means they are really badly damped. In fact, one of the early things when people were making this transistor smaller, one of the things people always asked is, if I make this device less than a micron, shouldn't we start seeing all kinds of effects due to the particle in a box levels? Because anyone who has taken a course in quantum mechanics knows that when you make boxes that small, you get discrete levels. And the fact is that transistors went on getting smaller and smaller and now we are talking of 22 nanometers and no one sees much of particle in a box levels at all. Why is that? Well, because good transistors by definition have good contacts, they are badly damped. So all those levels are always broadened enough that it more or less looks continuous by the time you're done. So you don't really have a whole bunch of sharp levels like this. Each one sufficiently broadened that it becomes one big continuum, really. Anyway, but all of that, of course, goes in here automatically. It's like put in your sigmas. If you have bad contacts, the sigmas are weak, the levels won't be broadened as much. If you have good contacts, they'll be broadened, et cetera. So that's all taken care of. And this is the thing that NEGF allows you to do. That is that treat an open system because what you learn a lot about is the Schrodinger equation for an isolated system, finding the eigenvalues of H. But what this allows you to do is see how it is excited by a source term. And one of the things you can see is that the response of this system, psi is equal to Gs, where G is like the inverse of this matrix. And so that's what you call the Green's function. It's like you have hit it with a certain source. What response do I get inside? And it's equal to this Green's function times that. And that's this Green's function that you see here. And next thing you might wonder is how do I get the electron density out of this? And the electron density is something like psi, psi dagger. This you have to think through a little bit why I didn't write psi dagger psi. But to take the psi, psi dagger, that's like the electron density. And if you put it here, you'll get Gs, S dagger, G dagger. And so what is in here is like the strength of the source term. And if the source comes from contact one, it is proportional to the Fermi function in contact one and the degree of coupling to contact one. This gamma one is like the imaginary part of the sigma, the part that's responsible for the problem. And this is how, and this as I said, is like your electron density. And that's kind of what this equation tells you. GN equals G gamma one, G dagger, F1. But the point that I wanted to make here though is that why don't we use the Schrodinger equation directly? First point is, of course, we want to treat an open system. But that's fine, we'll say, okay, good, we'll use this modified Schrodinger equation. We're going to go with that. Well, the problem is that when you have multiple sources, like you have electrons coming from contact one and you have electrons coming from contact two. When you have multiple sources, you cannot add their wave functions. So if I take two sources and put a S1 plus S2 here, you know, this is from contact one, that's from contact two, then I'll effectively be adding the two sides. And that would be physically completely wrong. When you have multiple sources, what adds up is the psi star psi. And so what you really need is an equation for the psi star psi. And that's what the NEGF equations are. It kind of takes the psi star psi from contact one and the psi star psi from contact two and puts them together, adds the psi star sides. See, if you had added size, it would have been just completely wrong, that's fine. And with light, usually we are using Maxwell's equations, you know, which is, yes, please. I will tell you that the reference is from the source. But we don't have thoughts that, it's like we don't connect any leads to the material. We also have the reference, right? We don't have to get from that, but we work. So for the, it's like, we don't have thoughts from that, it's not come one. So his question is, if we do not connect to any leads, then you would have no, according to this, there would be zero source term and so there wouldn't be any electrons in sight, right? That's the point you're making. That if you didn't connect any leads from this point of view. The best way to think about it though is, yeah, actually if you don't connect it to anything, then what is in there sort of depends on the history of the material. You don't really know what would be in there. In the sense that if you, let's say, hit it with light, emptied out 200 electrons from it and then completely isolated, it would be that way. So in other words, that's then like the initial condition of the thing, right? And problems like that, sometimes people add a kind of initial condition term. In our context, the way I'd like to think about it is that supposing we think of a Gedanken experiment where we lower all these chemical potentials down here, we're gonna put a huge gate voltage. So completely empty. So in that sense, there is no source here because source depends on the Fermi function. And then it's true, you'd have depleted your conductor completely, right? And then as you raise them, you'll fill it up. Now one issue that often comes up is inside if you had a localized state that is not quite connected to the contact, how does that ever get filled, right? And in a theory like this, if you just went at it blindly, if you left it empty, it would kind of stay empty. It wouldn't want to get filled. But the way, now if you did the complete theory, the one in which you have inelastic scattering, then what would happen is, let's say you have a localized states here, then the electron that comes in here will sooner or later make a transition into that level and then it will get filled up. So it'll eventually come to equilibrium with itself. But the equations I've written here is kind of a simplified version which does not have these inelastic processes in it. And so you're right that if you applied this formulation to a device like this, a localized state like this will kind of, if you fill it up initially, it'll be filled up. If you left it empty initially, it'll stay empty. Really. Was there another question? Yes. This is a steady state problem. We are thinking steady state. That's why I'm using the time-independent Schrodinger equation. It's not a given energy, right? Which if you're thinking in terms of light it would be like having the same frequency at the same frequency, right? But steady state at a given frequency, yeah. Yeah, so this current expression is in one sense a generalized expression for current and can it be used for also in coherent transport? Yes, actually this one can be. This one cannot be. This is special. This one you can use only when you have coherent, when you have elastic transport, actually. Not just coherent, it could be incoherent, but as long as it's elastic, you can do it this way. But in general, when you have lots of inelastic scattering or you have all kinds of non-equilibrium situations, you shouldn't use this, you should use this. Which is why in your notes that I, that you have, I put in this equation, I actually didn't write this equation at all. This is more to help connect with this one, right? But so ordinarily, I usually often don't use this. I kind of just go with that. Yes, please. What is the logic to add a, you know, size star, cyclical forces? Is it that the two wave functions are not in coherent? That the two wave functions are incoherent. So, right, with light, usually what happens is when you have two antennas, they have a definite phase relationship and you would add their electric fields. But if you have two light bulbs, then you're supposed to add powers. You shouldn't add the electric fields. With electrons, we never quite have the situation where it's like antennas, where it's coherent from two sources. Except you could say Josephson junction is kind of like that. In a Josephson junction, the current that you get is actually depends on the phase difference between the two. But that's a very special situation in superconductors. With normal electrons, there's no coherence between what comes in here and what comes in there. So it's like there's no definite phase relationship between them. So you're kind of average over all possible phases. So anyway, so this is the reason in my mind why you do not go directly to the Schrodinger equation. I cannot use that directly in the same way that when you take a course in electromagnetics, you do everything from Maxwell's equations. Although just as psi is like the wave function of electrons, similarly you could say electric field is like the wave function of a single photon. But it's just that there are coherent sources where you add electric fields due to photons. But with electrons it's like the psi star psi that you always want to add. Anyway, so that then is kind of a brief motivation for these equations. Why are we using the energy of equations? Well, these are the equations for the psi star psi. It kind of allows you to calculate electron density, density of states, all that directly. And I guess I try to motivate a little bit where this equation comes from, where this equation comes from. So you see where it, and if you really want to know the details, I'd say these are, this is all taught in this course, whose lectures are on the, on nanohub. Okay, so now I can go on to, yes. So can you please make more comment between these two kind of equations? I mean, so which parameter does scattering come from there? Scattering? For the generalized equation? Okay, I guess what I didn't do here, what I wrote here is the coherent version. Now, if you want the version with scattering, what happens is this, you have to put in a sigma s here. And that sigma s would enter here also. And here also there will be a third term. And the notes I handed out that they have there, those include these terms. And these terms are a little more difficult than the sigma one and sigma two, because these terms just are determined by the Fermi functions in the two contacts. Whereas for the scattering contact, I mean, you can almost think of it as a third contact, but the scattering contact doesn't have a well-defined Fermi function associated with it. And so it has to be kind of determined self-consistently. Now, in the afternoon's lab, the one that Shamiran will be running, I think he has a code which can do which can include these scattering terms. So it's like those lines are commented out, right? So you could either run it without it or run it with it. And you can see how to, so the example he will be going through is one where you calculate this transmission for a graphene sheet, like zigzag an armchair. So how you do that, that's basically what he'll be going through. And you can include scattering or leave it out. But the thing it won't do is talk about inelastic processes where actually energy changes because that kind of mixes up energy grids and makes it much little more complicated, the bookkeeping. Okay, so now I want to get onto the second part, which is about just the operational details. Because as I said, there's two parts to it. One is trying to appreciate where these equations come from. So you feel comfortable using them. And the second part is, okay, now that I have the equations, I mean, how do I apply it? How do I actually use it? So that's what I want to talk about now. So usually I recommend going through a sort of sequence of problems one by one to get familiar with this. And the first problem to do is just a one-dimensional chain. So if you have a one-dimensional chain, your Hamiltonian matrix might look something like this. You have something on the diagonal and you assume that you only couple to the nearest neighbors. You'd have a, usually that's negative, so I'm writing minus t, minus t. So this is a very common Hamiltonian people use with epsilon on the diagonal and minus t on the upper diagonal and the lower diagonal. That's a very common Hamiltonian to use. Now how would I choose these parameters? Well, if you believe that you have a single band with some effective mass, then the best way to choose it is again, choose it so that for a uniform wire, you'd have the same dispersion relations. So for a discrete lattice like this. And by the way, instead of writing these matrices, it's actually more convenient when you're thinking about it to write them schematically here like this. You know, it's like epsilon is the diagonal term. What you call the onsite term and this minus t, that's the coupling to the nearest element. Minus t, minus t, epsilon. So often in writing out this full matrix is kind of messy and vision. So getting from here to here, that itself is, when you go beyond one dimension, gets messy. But it's always relatively simple to draw this picture. So this is the picture that you should kind of have in your mind what you're doing. Now, so how do I choose these parameters? I'd say that if you had a uniform wire, it should give you the same ek relation that you believe is correct for the material you are trying to model. If you are taking this semi empirical viewpoint. So for example, this is a case where you'd expect e to be equal to e c plus say h bar square k square over two m. This is the effective mass. Now here, the corresponding thing, as I mentioned last day, the way you do it is, the expression goes like this, h of k plus, because in this case is equal to sum over m h n m e to the power i k dot r m minus r m. That's what you normally do. So this is the basic principle of band structure. But as long as each one is same, as long as periodic, you can find this. And here, these are all numbers. So if I just do this, I'll get the ek relation. In general, what could happen is you might have, I mean, each unit cell might have a basis. So these things could be a two by two matrix or a five by five matrix. And then when you do this, you might end up with a matrix. Whose eigenvalues you have to find for every k. That's the basic principle of band structure. Now when you apply that to this one, what you'll get is, e is equal to, you see, what I'm supposed to do is stand at some point n and then do a sum over all m's. And so when I do a sum here, I'll get three terms. One when m is equal to n, one when m is to the left of n and the other when m is to the right of n. So what I'll get is epsilon minus t to the power ikxa minus t e to the power ikxa. So one of them will be minus. So you'll get these three terms. And if you add them all up, you'll get epsilon minus 2t cosine kxa. Now how do I make this look like that? The answer is that for small k, you can make it look that way by using this typical Taylor series expansion for cosines. And what you get then is instead of cosine, you put 1 minus k squared a squared over 2 and then if you match it with that, you'll figure out what epsilon to use and what t to use. Now this is a method I like in general because it's very general that you could have a much more complicated dispersion relation. But the first step is try to write it as a sum of various terms with different exponentials like this. And then you can just read off what to use. In other words, you see here what I did was I said I know epsilon and t and so what is the ek relation? So I got this and then I got this and then I got that. I could have gone the other way. Dispersion relation. So first I'm going to write it in terms of sines and cosines so I can turn it into exponentials. I'll come here and then I'll just read it off because this one doesn't have any exponential factor with it, that must be the diagonal term. This one has a e to the power minus ikx so that must be what goes to the left. And this has got a e to the power plus ikx must be what goes to the right. Now this is a principle I've seen generalized to much more complicated problems. I mean, or we are doing say graphene with a particular mode, all kinds of things. You can generalize this. I mean here it looks kind of trivial but this is a principle you could generalize to much more complicated things. So the whole idea then is you want to choose your parameters so it gives you the correct ek relationships for a uniform wire. But after that of course what you do is you apply it to the problem where you have some potential on it. So for example you want to treat a p-n junction left and right there is a potential difference. Now once you have applied a potential of course there is no ek relation anymore. The thing is the ek relation is just for choosing the parameters to use but then you would apply it to problems where the potential is actually changing and what you hope is it's changing slowly enough that this is still okay. That's usually the spirit of all effective mass equations. As I said, the usual way you do effective mass equations is you go from here instead of k you put minus i ddx that's the usual way you do effective mass equations. That's kind of what people have been doing in the 60s since the 60s and this what I just described to you could view as a discrete version of that same approach. What you do is take your ek relation first write it as exponentials then pick off your parameters. Good. So first problem then to do is this one. Now if you do this question is what sigmas should we use. Now this is where the simplest sigma to use here would be something like this the minus i eta some imaginary number and then zeroes everywhere. So this is sigma one. So what does that tell you? It's like from point one electrons can easily escape and so that's represented by this. This tells you how easily it can escape from point number one and the rest of it's all zero. Similarly if I want to write sigma two I would write sigma two is like zeroes and then i eta down here the rest are all zeroes. Why is it down there? Well because the second contact connects to this side is to the right hand side so that would be one simple I guess a minus i so this would be one simple example to go. Now if you do that what you would get is something like this when you look at the energy versus transmission you see this 1d wire as you know will have some discrete levels you might use periodic boundary conditions or something to get them whatever they are and because of this i eta each one will have broadened somewhat. So what you would get is some transmission looking like that. Now you might say well that's not very satisfactory because after all you know I have a 1d wire I can think of all these different levels and if it's a short enough wire I should get a transmission of one through each one of them. So I really want to get something like this you know so where these limits are given by you know it's epsilon minus 2t cosine ka so when ka equals 0 it's epsilon minus 2t when ka equals pi it's epsilon plus 2t and the transmission should be just one all the way in between. You'll never get more than one but what you generally get is something rounded like this the question is how would you get that now this is what you would get if you follow if you do something else that is instead of this sigma 1 being minus i eta if you choose it properly meaning this requires more discussion why this is what we use and what you should use is minus t e to the power i ka where for any given energy you find the corresponding k using this relationship so for any given energy you find the k and put it in here and that's what you use and if you use that because what it reflects is the fact that different electrons at different energies can actually go out into the contact at different rates depending on their velocity faster ones get out faster slower ones get out slower whereas if you put a single number here independent of energy that's like saying everyone gets out at the same energy and then what happens is certain energies are kind of well matched and certain energies aren't but if you use that one then it all fits nicely and it actually get a transmission of one across the entire band and this is again a very good problem to go through because unless you do everything right you usually don't get something nice and clean like that where the transmission is exactly one all the way across and then the next problem I usually recommend doing is the following and that is the 2d version of that so you have entire 2d lattice like this and here again what we want to do is have some onsite number epsilon some connection to the surroundings say minus t and this one as I said visualizing here is relatively easy what might confuse you is then how do I translate this into a matrix because when I try to write a matrix it's like point one is connected to this this this and this and it's not just you know it's not just to from one you're not just connected to two and three but you might be connected to say 10 and 11 and so keeping all that straight then gets you have to do the bookkeeping in a way you're comfortable with now what I have found most useful usually in this context is call this entire column something say alpha so if you're on this side you've got say 10 points then alpha is a 10 by 10 matrix so conceptually think of it as if this entire column is alpha this column is again same thing alpha same thing alpha and then the coupling between this and this that's what I call beta so any two dimensional conductor like this two or three dimensions you could visualize as as if it's a 1d chain with alphas connected by betas and the thing is the alphas and betas themselves are now matrices matrices whose size depends on the width of this thing okay and that's the way I found it easiest to do the bookkeeping and now if you want to write the Hamiltonian it looks like alphas down the diagonals betas on the upper diagonal and down here in this problem actually it's also beta but in general going this way and going this way may not be quite the same and graphene is an example where they are not the same and but in general what will happen is if this is beta this will want to be beta dagger yeah but you have to be careful to write the right thing there that's all okay so this is how you can set up the H now as far as finding the sigmas that's where again you can either do the simple thing put just I eta that's like well nobody knows exactly what the contact is it's not a great contact anyway so we'll just put something there and try to adjust it so that you get the right current levels or you could say now we have got really good contacts you really want to simulate the situation where you have graphene connect or this conductor connected nicely to the contact where there's very little reflections at the contact then there is a way of doing something that's the equivalent of this but that again takes some discussion as to how you find that the basic rule is that the sigma is given by beta g beta dagger where this is alpha and this is connected by beta where this g is obtained by solving a matrix equation like this g inverse is equal to we'll write this nicely here g inverse is equal to EI minus alpha minus beta g beta dagger that's all so you know the alphas and betas so you start with some guess for g evaluate this quantity and keep iterating that's the point that you have to start from some guess it doesn't quite match so you correct it a little bit you have to solve it iteratively now you could have used this in one dimension in that case alphas and betas are just numbers and this becomes a quadratic equation you can solve just analytically and that would have given you the answer I gave you before so there's an analytical solution there but in general if these are a 2 by 2 matrix or a 200 by 200 matrix then you actually have to solve this matrix equation iteratively so that's done in this code that I guess Shomiran will be discussing with you that's done there so this is then the general method that is you'd write down the h go on and this is a good problem to do because when you have a two-dimensional conductor one of the very standard results that you should try to get is that as you make the conductor wider or if you have a fixed width and as you change the energy the transmission should actually go up in steps as a function of energy and why does it go up in steps because conceptually you can think of a two-dimensional conductor as an entire set of 1D conductors in parallel that is those transverse modes you can conceptually decouple that as a lot of independent modes all in parallel and the first one starts conducting at this energy and you get this the next one its EK relation the bottom is a little higher so it starts conducting here and then the next one starts conducting here and so when you add it all up you get the steps and this of course was one of the seminal experiments back in 1988 that kind of gradually I guess inspired everyone to start using the Landauer point of view because this idea that I guess experimentally this was observed as a conductance quantization that as you change the gate voltage the conductance changed in steps and that is something that the Landauer point of view explained in a natural way and so that's what I guess inspired a lot of people to then start using this way of thinking about conductance but in this context of course I say that this is one of the good test problems to go through always because as I said you don't get nice steps like this by accident unless you have done a lot of things right so this is always a very good example to go through now once you have this example then you are ready for things like graphene so what would graphene look like well as you know last day we drew this so again depending on whether we are doing armchair or zigzag suppose you could have current flowing this way or current flowing this way now if you are having current flowing this way for example you might want to take this block and call it your alpha take next block call that your alpha same thing and then there's the beta that's connecting them right so that's how you set it up and then after that of course the Hamiltonian is just alphas on the diagonal betas on the upper diagonal beta daggers down here same thing but on the other hand if your current flows in this direction then you would want to take your alphas this way so you take a slice this way call that your alpha and then you have to write your alphas accordingly and here again the betas are such that it's a little different the beta and beta dagger are kind of a little bit different and that's why you have to be careful as to which way you put that when you are finding the self energy in this direction you have to use beta g beta dagger but then this beta should be that one the one that appears on the upper diagonal whereas when you are looking for the contact on this side it's as if the beta and the beta dagger exchange roles and all that again has to be done right so that you actually get nice steps in the transmission so that again is a very good test whether you have done it right or not otherwise you have to connect the sigma one and the sigma two correctly on both sides if you get them interchanged then effectively you would have put major reflections at both interfaces because it would be like you connected the wrong thing there but here again as I said this is if you want to have good contact so electrons go reflection obviously into the contact it's like you're travelling along graphene and when it gets into the contact also it flows out smoothly lots of times experiments you are trying to simulate might have bad contacts where electrons don't really get out that easily people are not really observing quantized conductance at all in that case it may not be necessary to do all this in great detail you could just have a little i eta and get started in that question that sometimes comes up is the following that is if you are doing a p-n junction say for example so how would you set this up basically it would be alphas and betas like this and then one would put a potential so on this side all the diagonal elements would be lifted up appropriately or lifted, moved down and this would be moved up appropriately to reflect the potential change and then you can calculate transmission from left to right and you would have the transmission for a p-n junction if you have set it up right it should all come out from there and the only practical problem then is just the size of the thing how big can you handle really and this is where we are looking for this is something that we have been looking for good ways of handling bigger problems now the only one that seems to that's relatively easy to appreciate in this context is the following and that is if you can think of it as bunch of modes what I mean by that is if you can mentally decompose this graphene sheet into an entire set of parallel modes like this so that you can then do current flow in each one separately then of course instead of having a big 200 by 200 matrix you have I guess 100 2 by 2 matrices for example you have much smaller things to do which would be much faster besides it would be easily parallelizable anyway so this decomposition when it works it's very nice and of course if you do not have any scattering processes if you are not trying to model any scattering processes that actually take you from one mode to the other then the point is let's say you have current flow in this direction and you are in this direction I guess that's the y direction then as you know people usually assume periodic boundary conditions as long as you have a big conductor because in all of solid state physics you are always assuming periodic boundary conditions not because anything is periodic but because you think that well it's a big conductor so what you do at the edges doesn't really matter so we do assume it's periodic because it's automatically the simplest thing because then anything going in then this k's are what you call good quantum numbers that if you have an electron with a certain k it doesn't hit the edge and get converted into anything it just goes around and comes back so it's an eigenstate of the system even of the finite system so this periodic boundary condition you could use if you have wide graphene sheets and then you'd have a separate one for each mode like this and now you could ask that well how do I write down the Hamiltonian for one of these modes you know each mode as Sakmin described that day and you went through the CNT bands each mode is obtained from the graphene relation by putting k y b equal to some multiple of 2 pi let's say if you are doing the periodic boundary conditions as you know so we have a certain k y b and we want to describe transport along x question is how would I choose my Hamiltonian to describe it and this is where I'd say again choose it according to this principle that is look at the h of k for that k y and then just read off what should be used so for example let me just show this and then I can we can sum up this thing last day I'd said that for graphene h of k looks something like this there's a minus t out front don't worry about that so supposing this is the h of k as you know for graphene this is the one that you solved to obtain the band structure of graphene now what we are trying to figure out is what to do about a particular mode in graphene because we are conceptually separating this graphene into a whole bunch of separate decoupled modes and we are trying to model the current flow through this one and this mode has a specific value of k y b whatever that is so what that means is for this discussion this cosine k y b is just a number so let me call it say some number c so now what happens is I could write this in the following way 0 1 1 0 plus 0 2 c 0 0 e to the power i k x a plus 0 2 c 0 0 e to the power minus i k x a so what I did was I took that and wrote it as a sum of three things now why am I writing it this way because remember my objective is to take this h of k write it as a sum of exponential and then you see I can read off all these couplings right away so what's the onsite this so when I'm trying to model this every one of these wires like the site consists of two things like the a and b sites because each site has like two basis functions there's two of them there and then there's two of them here and then there's two of them here and the onsite is a 2 by 2 matrix it's 0 1 1 0 and the connection to the right is 0 2 c 0 0 and the connection to the left is this that's it similarly if you want you know modes in the other direction then what you do is you set a specific value for k x and then try to separate it out as e to the power i k y b and e to the power minus i k y b and you can read it off what to use you see a very powerful method that I've seen in terms of how you go from any desired e k relationship to a discrete Hamiltonian that gets you the same e k relation so that you can use it for doing other calculations for non-uniform materials but of course if you want to handle general problems where electrons get scattered from one mode to the other then there is no simple way like this you really have to treat that entire two-dimensional sheet and then it becomes a practical problem is just how to handle large matrices really and so just to sum up then the one point I wanted to stress is that you as you look into this field of graphene and you look at the literature you hear about all kinds of strange properties that graphene has see all kinds of issues like you know this electron hole symmetry bedding phase all kinds of other things that you hear about and one question that people ask me is that well but if you use NEGF does it include all that or do I have to take this and do something special with it and my answer is no really everything I've seen so far that's all included in here the only thing that NEGF doesn't include is that here when you are including electron-electron interactions it has to be done self-consistently through a potential U which you know you might take from Poisson equation or put in corrections for exchange etc that's there but other than that and that's included in this model through what you call a mean field that's the U and there can be of course all kinds of interesting effects that cannot be captured within such a mean field theory it's just that to date I haven't seen any experimental results that show anything that cannot be understood within this field theory so I'd say whatever has been seen so far and the various things that people are talking about in this context are all described by this NEGF equations really so for example one interesting result in this quantum Hall plateaus is that all semiconductors as you know have this Hall resistance is quantized in graphene it turns out that the alternate ones are missing and the reason for that of course there's lots of papers discussing all this but one of my favorite stories is one of the students in my class he didn't know anything about graphene at that time I didn't know much either and he just used this NEGF for calculating it and he came back to me and said I'm doing something wrong I keep missing the alternate plateaus I don't know I must be doing something wrong and then we looked into the literature and found you should have be having only the odd ones so the point is that all of this is really included in there and this kind of tells you the power of the method is that of course that you know it's like how much you can calculate without understanding anything and that's of course also the danger but the real way you can use it effectively is if you use it to calculate but then try to use it to improve your own understanding so in a way it's like do it yourself quantum mechanics I mean use it to calculate what it is and then use that to improve your own intuition understanding about all this thank you so very good question the question is that I said when you're doing graphene you can use this periodic boundary conditions and the question was then how do you do zigzag graphene where you have edge state in general I'd say when you have small conductors anything small meaning the width is small you really should not be using periodic boundary condition because the real boundary condition then matters so when I said use periodic boundary condition it is in the same spirit that you use it in all of solid state physics not that it is real but because we believe that the edges because the whatever property we are calculating does not depend on the surface conditions or the boundary so we use whatever is simplest so that applies usually to large graphene conductors you know when the width is fairly large now graphene is probably the only material where actually you can even have a periodic boundary condition in the sense that carbon nanotube has it so whatever you calculate with periodic boundary conditions would be relevant to carbon nanotubes but if you have real graphene nano ribbons with edge states and all that then this wouldn't do you'd have to do something else because then the simple decomposition of kyb equals this that's not good enough because the states actually involve both plus ky and minus ky it may be possible to decompose it into modes that are not like 2 by 2s but maybe 4 by 4s but those are things I haven't actually tried out so I don't know but there may be ways of decomposing it in that case for the exchange interactions usually what happens is so the question is about this electron-electron interaction what we put in here and the u that goes in here this is a u that depends on the electron density so in general when you are using any gf the u will have to be calculated self-consistently with the gn so when you have obtained the electron density whatever u you put in will depend on that right now the simplest thing that people use is of course the Hartree approximation which is the what you'd get out of which is equivalent to using Poisson equation so you take this electron density put it in Poisson equation whatever you get that's the Hartree but then in what is believed is also that Hartree kind of always overestimates things that is because of this exchange interaction you can all actually electrons don't interact quite that much it's something less it always subtracts from that now the simplest thing that people put in is some version of the local density approximation which is this correction term that is proportional to the power one third now this is where anything you are doing is usually approximate and people who work on this I guess everyone has their favorite thing and I'd say anything you use would have to be justified by benchmarking against some experiment although I don't think there is any a priori way of saying that this is much better than everything else except that in density functional theory that has been very successful over a wide range of problems except that density functional theory is usually applied mostly to equilibrium problems not enough so is there a way that we can use density functional theory to get the parameters that we could then put into idea for the exchange interaction for the exchange interaction so here based on density functional theory one might choose something that would be appropriate so those usually I think there are various expressions in the literature that people use okay now to consider temperature temperature usually enters the question was about temperature how you include temperature and one way temperature of course enters easily is through the Fermi functions just to the contact in the real world the other way it enters is because there is a lot more scattering in the in the inside the conductor because all the atoms are jiggling around a lot more so there is a lot more inelastic scattering a lot more elastic scattering all that so now that is the part that I suppose you would have have to calibrate against what people believe in terms of the I mean you would have to use the model with the scattering in it and then use deformation potentials to estimate that scattering potential right so so temperature also the temperature much so that means it is not just simple to add temperature in the Fermi so so yeah to the one the Matsubara green functions that you are referring to but that has to do with the equilibrium green function theory no but in the non-equilibrium model I would say that it is already in here right I do not need it separately that way that Matsubara green function that is a way of doing the Fermi function sum right in that complex plane right but that is used widely in the equilibrium green function theory right but this is this non-equilibrium green function theory where I would say because as soon as you apply a voltage I would say effectively you are already at very high temperatures anyway right if you put half a volt across something so I think you are really dealing with it at a different level here you do not need to do any you do not need to use any different greens functions really so the Matsubara greens function I think you would see more in the context of the equilibrium green function theory so the question is then it is a non-equilibrium system then how am I using the Fermi functions right yes I do not think it is quite to the temperature in the Fermi very good so the question was that how am I using a Fermi function here for example since it is a non-equilibrium problem and that is a very important conceptual point to be absolutely clear on and that is the Fermi function refers to the contacts so the whole idea is that whatever we are considering the graphene that sits here and in order to drive current through it you have to have two large contacts two large contacts which maintain two different chemical potentials so you have got two Fermi functions not one Fermi you have got two Fermi functions right now if you actually calculate the occupation of levels inside here let us say so let us say I have got these two different Fermi functions something here and something here now as we discussed inside if you do the simple ballistic theory then what you get inside will be some super position of the two Fermi functions that will be the simple view mentioned so if you plot that here what it would look like is something like this so this is what if you applied a simple ballistic type of theory without including any interactions if you just did that you would get something that really doesn't look like a Fermi function at all inside it is hopelessly out of equilibrium this is badly out of equilibrium if you think about it and it makes sense in the sense that down here both contacts want to fill up your level so it is filled up here both contacts want to empty it so it is empty in between this contact wants to fill it up that wants to keep it empty so it is half filled perfectly simple right now once you put in scattering processes what scattering processes do is they don't like this badly you know hopelessly out of equilibrium distribution they would like to turn this into a Fermi distribution of like this so all that scattering processes will do inside is they will work very hard to turn this into that now the way it would get into the non-equilibrium green function theory is through this so if you put enough of that with inelastic scattering and with inelastic scattering actually the equations are a little more complicated than this what I have written here is assuming it is all elastic but if you include inelastic scattering then actually the reason it gets complicated is as long as it is elastic every energy is an independent calculation I mean that is like what you would call this embarrassingly parallel calculations in that sense every energy can go to a separate computer essentially but as soon as you put inelastic scattering all the energies are coupled together so you have to keep track of all of that and the book keeping everything is a little harder but the NEGF formulation tells you how to do it in principle and if you put all that in then you indeed relax this distribution towards it and how depending on how strongly you put in you might get more of something that looks more like this or like that in other words NEGF if you put inelastic scattering it will continuously take you from this to this now the strength of that inelastic process that is temperature dependent if you are including phonons because the sigma s you know it is kind of like d times g it is done self consistently I mean this d has nothing to a density of states I think this is actually a tensor d i j k l and that tensor depends on the number of phonons emission processes would be n plus 1 absorption processes would be n etc and I think in my 2005 book in chapter 10 all those equations are all there it is just that we have not used it extensively there is a few examples in that book that show what you get out of this but and if you include that you will get all the processes there you would get all the inelastic processes you would get the temperature dependence of scattering processes in there but this is a very different approach from the equilibrium green function theory because in equilibrium green function theory what you are really doing is you are really not approaching the non-equilibrium problem at all you are using the fluctuation dissipation theorem and you say that conductance is proportional to the equilibrium noise and then you calculate the equilibrium noise so it is a very different philosophy although both are green function theories and the non-equilibrium one connects very nicely to this whole and our point of view and everything does that answer your question now if we talked about specific problems then I think we can discuss how I would do it you know specific problems that you would do that formula when you get a result from this should somewhere connect to this problem yeah but that is is there any specific case where you can use this and say towards equilibrium but still in non-equilibrium and through a fluctuation you may not but the Matsubara is only for again linear response for equilibrium theory it is the equilibrium theory for linear response and there I would say but what is the problem that one would do with it like for example I have it is very hard usually to from that formalism to get this quantized conductance because when you have a conductor like this and it is connected to two contacts and you want to find the conductance then what happens is usually you are using this fluctuation dissipation theorem so the conductance is proportional to noise now the thing is if you want to get the quantized conductance out of that theory then you have to consider the noise that arises from connecting to the contacts namely you see I take this material it has a certain noise I mean if it is isolated let's say there would be no noise but because you have connected it to the contacts continually go in and come out if you take that noise and connect find the corresponding conductance you would get the quantized conductance but it is a lot of work but most of the time of course when people are using it they are not even considering contacts it is like pre mesoscopic days when people calculated conductance you see before 1988 the pre mesoscopic days when people talked about resistance and conductance all the physics of conductance was right here right I mean before 1990 no theorist even drew the contacts I mean experimentalists knew about it of course but no theorist ever drew a contact that kind of started around 1990 ok and so in a way what is simplifying all this is the point that in small conduct as most of the physics is here and and this is a minor detail whereas till 1988 as I said and all through the 60s, 70s all the physics of conductance was here and this was a detail nobody even thought about ok and so most of the literature on Maksubar and Green's function is about this one and so quantized conductance won't come out of it this quantized conductance is something you see only in a ballistic conductor when this is all gone and all your fluctuations are because of that so you can do it but again it's a whole different thing so it's hard to find a good problem that is done both ways that we can actually compare but this method we have used in also all kinds of problems where you know heat exchange is seriously involved that is the point I often make is that usually if you had a material with say two levels it's a device with two levels and I make a contact across it two contacts across it and I look at the current here the thing is if you have a lot of inelastic scattering like you are worried about you won't see any effect in the terminals at all why because anytime you raise an electron from here to here you got an electron there and you got a hole left behind this electron has half a chance of going to the left half a chance of going to the right what's left behind here has a half a chance of being filled up from here half a chance of being filled up from there so externally there will be no net current at all in a circular because of the symmetry this point that Professor Alam has been making that a good solar cell you almost have to have a way of collecting it at one now the same device though if you can connect it in such a way that there is no connection here let's say so that from this level it only connects here because there is a lot of density of states here and it only collects here because there is a lot of density of states here then you see anytime there is an inelastic process you will immediately have an current in the external circuit so what I say always is that if you really have a good theory of inelastic scattering apply it to this device and then we can actually compare and all that even as you are applying it to this device there is no difference between a good theory and a bad theory they will all give you zero in various ways that's about it we really should do this device and in a way this is the essence of any solar cell and all kinds of thermoelectricity everything you really want to you know connect well to one side and connect this one well to the other side Sir, limit to the amount of voltage In principle, yes it's just that as the voltage goes higher and higher all kinds of inelastic processes are much more important and so at least the simple version where you are neglecting inelastic processes that's not a good idea the more the voltage you put but lot of the nanoscale devices we are usually talking half a volt that should be right actually even at half a volt probably in transistors if you want to do it right and that's one of the things we have been talking about that how to have a model that where you include all the inelastic processes and could be applied to a transistor with inelastic processes whereas lot of the work on transistors it's like you want to figure out what's the maximum on current that you could get and you say well let's assume it's ballistic it won't get any better than that see even that isn't very impressive with inelastic processes thank you