 You can follow along with this presentation using printed slides from the Nano Hub. Visit www.nanohub.org and download the PDF file containing the slides for this presentation. Print them out and turn each page when you hear the following sound. Enjoy the show. One part that probably many people asked me about, so let me say a few words about it. That has to do with this second quantization, that what does that mean and that's related to what you asked also. So, you see the thing you hear a lot is that how electrons, photons, all of these are both particles and waves and so in the particle picture, when you think of electrons or photons as particles, then you have an energy momentum relationship usually, that's how it, a particle has a certain relationship between energy and momentum which could say look something like this or if you are talking of the photons then it may look something like E is equal to C times P for example. So, when you think of as a particle there is an EP relationship and then comes the wave picture where you turn this into a differential equation, that's the wave equation that you have and corresponding to this there is this Schrodinger equation, so that's the wave equation and corresponding to this again, I guess you will have Maxwell's equation. So, for Maxwell's equation the way, I mean one form of it would be derivative, second derivative of E with respect to time is like C squared times second derivative of and by this E I mean electric field which I didn't write the C correctly, so in simplest form it would be something like this and you could say well if you start from this then you could just as here the energy momentum relationship gave you a wave equation by replacing E with I h bar d dt and P with d 2 dx 2, same you could say this became that. Thank you. Thank you. Almost proved that C equals 1 I guess. Thank you. So, now one way of interpreting the electric field of a photon is something like the wave function of the photon, I mean that's not how Maxwell wrote it down of course and Maxwell's time no one was thinking of you know the wave function in that game but you could view this as the wave function of single photons. So, for example the way you learn Maxwell's equations usually is that if you have two media let's say this is glass and this is air and we calculate that well if you have a steady stream of light coming in one watt and let's say half gets reflected and another half gets transmitted or whatever it is you can calculate knowing the dielectric constant there will be some reflection, some transmission etc. Now and this is of course intended for large beam of light lots of energy that's incident now what if you cut this down so that you are down at a single photon level one photon comes in and then something gets reflected something gets transmitted so point is it's not that half a photon will get reflected or half a photon will get transmitted the way you are supposed to interpret it is that the photon has half a chance of getting reflected and half a chance of getting transmitted but whatever happens I mean it will either be transmitted or reflected it won't break up that's the point so as soon as you go to that level that's when the probabilistic interpretation comes in and that of course in the beginning was a major philosophical jump for people to accept because the assumption always was that when I when there's no stochastic things involved when I know everything I should be able to predict what a photon should do and the point is what this tells you is you know everything about the light that came in that photon that was coming in but there's no way you can predict whether it will be transmitted or reflected all you can say is that if you do the experiment hundred times half the times it will be transmitted half the times it will be reflected okay so even with light you have that basic thing always you see and so one way of interpreting Maxwell's equations then is as if this E is like the wave function of the photon that would be one way of thinking about it as I said Maxwell then intended that way okay and with electrons say that's the way you think of psi usually that's supposed to give you the probability of things now the reason this is somewhat different from this is because we all I mean of course everyone realizes that you know electric fields are rather real in the sense you can measure them wave functions you can't really measure them and why is that well in Feynman lectures I think Feynman says at some point that well the difference is that because this electron these being Fermi particles usually whenever you have so let me draw this E k relationship so for electrons let's say we have this e versus p so the point is that in a given state here you can either put one electron in there or not have any electrons in this zero or one so for every state you can either have zero or no particles but you cannot put two electrons into it because of the exclusion principle on the other hand when you look at light and you plot omega p relationships like this and you take any of any particular frequency mode any mode there you could put any number of photons into it millions of them right and so it is as if you have a whole series you could go 0 1 2 and Feynman makes a point that you know when you have lots of them then you can all yeah when you can put so many of them into a mode they can also add up coherently to give you a huge electric field that's actually measurable which never happens with the electrons that's the way he put it and that's why finally when it comes to electrons it's only the size star side that's measurable and of course this is says in the context of trying to explain superconductivity where it says that when the superconductor is one case where electrons form Cooper pairs and then act like bosons so a lot of them add up in a way to give you a coherent wave function whose phase is actually measurable but that's a whole different thing but in this context then the point I'm trying to make is this step to this step that's what usually you refer to as first quantization and then from here there is this next step where which has both wave and particle properties and that's that second quantization so the idea is from this you get a E.K. relation and then the idea is that for every one of these it's like two states there possible there's a zero and you could either occupy this state or not occupy this state and so that entire so when you're and when you're using this picture I mean at this level of course you have to work with yeah these states and do the quantum mechanics of these states I mean that's the part where that's the second quantization that which allows you to handle and this is what we normally are not doing what we're normally doing is thinking of it in one particle terms trying to figure out what one electron does and then doing averages properly this is the point I tried to make earlier that do you add wave functions or do you add size star size etc. now one point that often comes up again these are the questions that people ask me is that this NEGF that I described to you the well doesn't this include all kinds of many electron interactions for example right and this where I guess the answer is as many aspects to it so let me explain a little bit I guess Dr. Raja he talked a little bit about this the general NEGF formalism that how it actually the way you define this GN you know I wrote it here as just size I start I said that well you know take take one electron wave functions add up the size I start the more formal way of defining it would be the expectation value of certain second quantized operators etc. which is actually working in this picture you know which has both wave and particle picture all built into it actually working in that picture anyway so originally when this formalism was developed that's back in the 60s by a swinger Katanov and Bame and Keldish of course yeah so there's number of people who actually used develop different parts differently and I guess in the beginning I used to call it the Keldish Katanov Bame formalism and then I got letters from some of swinger students who are very unhappy that I left out swinger's name and and then I decided that you know I'm better off calling it something neutral like non-equilibrium green functions because there's so many people involved at different ways and and I get Bame was one of swinger students actually right anyway anyway so that's this general non-equilibrium green function theory and broadly of course as Dr. Raja pointed out that while equilibrium statistical mechanics is very well established non-equilibrium problems don't quite have such well defined methods yet by and large see and so this important thing here is the non-equilibrium green function so if you look at those equations up there the first equation up there that's what you would see in equilibrium green function theory also g equals e minus h minus sigma inverse that's the one that you wouldn't see there is let's say this third equation the g n equals g gamma g dagger etc you see that's got this two Fermi functions in it for example what I wrote yeah and this is the equation that first appeared in let's say Keldish's paper Keldish's 1965 paper there was an equation that yeah by the way I usually the notation I use is a little different from what's in the literature so I think where I would write it as g then all these things gamma 1 f 1 plus gamma 2 f 2 etc plus I think I had a sigma n right here so that's the equation that's this that's this third equation I had there and of course back in the 60s when Keldish Karanoff were developing this no one thought about contacts so of course they didn't have this it is really all about that because as I said when you're dealing with big conductors the entire physics of conduction is in those interactions that's the important thing right and so it was this one and the notation you'll see in the literature usually is it's not g n the right is g less so that's the notary that's the thing you'll see in Keldish's paper and the connection between these two is just that what I write as g n is actually minus i g less that's all and same here this is this minus i times this now why do I use this well because this one when you look at its diagonal elements it gives you the electron density so it has a very clear meaning physical whereas this one was motivated by as I think the name was this contour ordered theory so this less tells you what order you take a certain complex integral in the what order of times you use when evaluating a certain integral you see so that's where that less came from and so it wasn't quite so when you look at this one it's like the diagonal elements are not quite the electron density have to multiply by my minus i and all that so this is why I adopted this notation because when I was doing this of course I was trying to get all these I obtained all these results like I described to you this morning straight from the one electron Schrodinger equation because you see when it comes to those terms that involves the contacts that f1 and f2 those first two terms you don't really need many body perturbation theory to get those that part we can do exactly what I did in the morning you could do it a little better than that that's what's in my books you can look so the part where you might need many body perturbation theory would be to figure out how to write that down in a systematic way for all kinds of interactions to different different degrees of you know different terms in the perturbation series that's the part where you might need it and even there if you are talking about the elastic resistor where all the interactions are elastic then you can almost write it intuitively you don't really need it the real power of this is it allows you to handle much more complex interactions up to any order you want any degree of approximation and of course because in those days of course this wasn't even there this was the all-important thing and so that is why because of the history of the subject lot of the literature is always based on many body perturbation theory what that means of course is that if you do not know that you cannot quite get even get started there right now my other question that comes up here is that strong interactions as dr. rajah pointed out a strong interaction the strongly correlated systems that of course is a frontier subject in here we know where there is a lot that is not understood and a lot of interesting results and one question I often have from people is whether any gf since it's a many body method I mean people think of it that should actually include interactions to any order you want that's the feeling you see and what I and but what I've what I've always felt and what I have discovered looking at different problems is that that's not really true and I can give you a simple example of that actually and the reason I believe is that the way this you see the way these terms are calculated that's those contacts those are actually exact it's not a perturbation expansion or anything that's exact I mean whatever I did today we could take a little more time and do it properly and but that's all exact there's no approximation but anytime it comes to interactions whatever you do is approximate and you can do it and usually for most calculations you usually stop at the born approximation but you could go beyond that you know like condo effect that he mentioned actually it's a third-order term I mean the third-order perturbation yes yes yes I'd say whatever doing here one thing that you should include of course in the age is the potential from the Poisson equation right and that is basically is the heart return and as I say people make corrections to that one they include correlation and exchange and correlation corrections to that because the actual potential is always a little less but then there are things that this won't really cover and the simplest way I often point this out is by saying that you know perturbation expansion amounts to doing this I mean in in various forms it amounts to doing something like this and usually you say okay if you are not happy with x will go to x square if I'm not happy with x square will go to x cube good but does that work when x is equal to 2 not really that's it really doesn't matter how many orders you have gone up really doesn't get you there but and there are examples which are kind of in that category and in the context of transport I felt that for example coulomb blockade that's an example which ordinarily you might think that well you could just take any gf and treat the electron-electron interactions a little better don't just don't just stop at heart rate let's put in a few more things but you still won't get the right all the right features of coulomb blockade you know you might be able to manage to get certain things but you won't get a lot of other things and those are things I have often mentioned in different context so for example one of the examples we discussed was that if you had one level say and you connected it to contact let's say the mu is the chemical potential is right there and you can put a small voltage across it and question is will it conduct right and usually you'd have said that well you have connected it to the context let's say it's broadened out something like this and as I said always up spin and down spin are degenerate so usually have two levels they usually have these two levels here and you could calculate the potential self consistently from heart rate and all that and let's say you try to calculate what the conductance is and if you have say one electron inside here then you think that the Fermi level is somewhere here it's half filled and you got two levels half filled so basically one electron now the thing is that if the interaction is strong enough it won't stay that way what will happen is one of these levels will sink down and one of these levels will float up so instead of both of them up spin and down spin both being right there and thereby giving you a lot of density of states to conduct what will happen is your one will go down one will go up and so you actually won't conduct very well now why is that the simplest way people think about it is they say well you know there is this self interaction correction that is the idea is that no electron feels any potential due to itself so what happens is let's say this the way I have drawn this this has one electron in it and because you have one electron in it what has happened is the other level has floated way up why because it feels a potential due to that you know you have to include this potential due to this one so it has floated up by what's called the single electron charging energy but you say well but then why didn't this one float up well because it doesn't feel anything due to itself that's all it only feels something due to that one but then that one's way up there it's empty there's nothing in there anyway you see so if you had included this point in your thinking that no level feels anything due to itself but only feels something due to all the other ones then the solution you would have come to would not have looked like this where both levels were degenerate you know up and down in the absence of magnetic field are supposed to be degenerate so you wouldn't have come to this situation where it is degenerate but which one goes down and one goes up and to me this would be a bottom up view of the mark transition essentially you see the mark transition that Dr. Rajya talked about this would be my bottom up view of the mark transition the simplest mark transition in a way right so what I'm saying is yeah oh yeah what I said was that let's say it is down what I'm trying to say is that that is a self-consistent situation what could happen is they it could have gone the other way also yeah yes please feel free to you know speak up yeah please try another way but why it has to be done so let me consider a comment for a body of just one side there is certain level and a certain interaction view okay and I'll write it in the following form so this is just one level which is decoupled from the leaves completely this model can be exactly solved okay in the site places there are only four states which are all of one is zero occupied state one is up down and the fourth one is up top that has two electrons single electron and zero okay now you want to enforce one electron occupancy on this state right now since you know all the states you also know all the energies of this because you can just take this Hamiltonian and apply to this operator we'll get an energy zero here we'll get an energy epsilon d again epsilon d and 2 epsilon d plus u okay now what you can do is you can find the partition function exactly and then find out what is the average occupancy and right now if you enforce that sum over n sigma equal to 1 what you will get is this epsilon d equal to minus u by 2 okay if u is large which is what the case we are considering here then epsilon d will be very much below the level that is the case so this is a question whether so it is almost like if I have my chemical it's an open system so you could say well if I have my new somewhere what is the n that's one way of thinking but the other way is if you say that the n is fixed then question is what should it look like and then it is kind of forced to flow flow down because one needs to be filled right you need to have one electron in there if you put your epsilon d equal to minus u by 2 right then this is energy zero this is zero and these two are way down so the transitions between the singly occupied state and the zero occupied state are equally probable as compared to the singly occupied state and the w occupied state so on an average the negative fluctuations of the occupation over are equal to the positive fluctuations of occupation over so the average occupancy is one so this is a regime that you see in the beginning yes please yeah but those would all be one electron arguments so isn't it that probably isn't wouldn't help yeah here the point I'm trying to make is that when you have strong interactions strong so that this one electron charging energy is large and we can talk about in a minute how large it needs to be then a simple Poisson equation would give you wrong things or even if you corrected it you wouldn't get the right things but one way you can get almost the right things is if you enforce this self interaction correction if you kind of build into this this thought that one only feels the potential due to the other except that that's really not a general solution but if you do a Google search on self interaction correction you'll find in a hundreds of papers people who have tried in a decent where many different schemes how to make this how to use this correction and these are all to be corrections to the because what you'd really like to do is not go to a second quantized picture because that is harder to do in a bigger system especially you'd like to have this one electron picture the idea that one electron sees some potential due to all the rest right that's what this you is supposed to be what average potential it did feels due to other things and question is can you get the right physics within that picture I mean to the extent it was you call it a mean field here that's what the mean field is it's the mean field due to all the other electrons and at the least what you need to do is have this self interaction correction in that mean field but even that won't get you everything and one of the things I've discovered is some things that actually work for equilibrium problems where it's kind of close to equilibrium won't really work when you put a bigger bias and try to do non equilibrium problems so for example when you look at the current current versus voltage with two with one level I had said earlier how it steps up like this we discussed this that if you had a mu here and you pulled it down at some point you'd have a cold current through this well if you have two levels what do you get that's the question now that with two levels you think well it would just be double of that well but because of this coulomb blockade effect if you're really in this regime when you zero is large the point is you'll actually see two steps and now the fact that you get two steps you can get it by putting in this self interaction correction that you one doesn't feel the other what you won't get with that though is the steps are not equal this is two-thirds and that's one-third that won't come out of that simple picture so this is an example where something when you're close to equilibrium might work quite well but when it's a non equilibrium effect you may not get it at all because as you know one of the very powerful theories in terms of handling many body interactions this correction is the DFT which has worked very well and widely used but most of the applications have been to equilibrium problems you know when you're talking about the equilibrium structure and all that where but basically of course it tells you how to correct the heart rate potential so as to include a lot of these many body effects and does it very well but it close to equilibrium works very well but out of equilibrium so whenever I talk to many of the people working in molecular electronics and they tell me you know we have this version of DFT which works and I tell them well can you get this two-thirds and one-third thing out of it is that oh sure well we'll do that next time I meet have you done it not really I can't so I've never seen a actual mean field treatment that will get you that two-thirds and one-third you see yeah well because we are still calculating steady state so I think you don't really need the time dependent you know this is a important interesting important point that these greens functions you know are like R R prime T T prime and this T T prime is T you can always transform it to T minus T prime and T plus T prime over 2 and time dependent DFT of course is working with that one but I think you need more some self energy that's energy dependent which is this part so GW or you think things like that anyway that's a whole different story but but the main point I wanted to I thought you know this is something I should mention and because what I feel is that any GF you know works very well as long as this mean field picture works and so the question you ask is when does the mean field picture work in things like this the basic point is that this single electron charging energy that means here I have a channel when I put one electron in how much does its potential change into the now if it's a very big conductor then of course one electron changes things by a micro volt who cares it's not really doesn't matter but when you have something really small then adding one electron makes an enormous difference right and the smaller it gets the more important it is and so so for example if you are talking about a hydrogen atom you know one thing we all know is a hydrogen atom has these two 1s levels which is this is the vacuum level this is like minus 13.6 electron volt and there's an electron in there now and you draw these two levels so it looks like there is a level empty where you could put an electron in not really if you actually try to put an electron into a hydrogen atom that energy is somewhere here way up there because the single electron charging energy is like 10 volts on something as small as a hydrogen atom one electron that there will be 10 volts so so the smaller it gets of course this u0 gets bigger and bigger so as long as it is much less than kt you are fine you're fine meaning you're okay using our standard which is why a lot of the coulomb blockade effects in small conductors where first observed at 4 Kelvin 10 Kelvin low temperatures you know to go to low temperatures to see it even now lots of times that's where you see that but there's a second thing and that is that let's say it's bigger than kt so that that this one is not satisfied there is a second thing that also helps the mean field picture and that is the gamma because the point is as I said as soon as you connect it to contacts these things broaden out so and what you can see is of course is that if that broadening were bigger than u0 then you wouldn't even have noticed all this you'd be fine respect is this is very close to the interacting doesn't it where if you look at it in terms of the boundary then the boundary goes and seek to the power minus u by gamma and gamma is very large and you buy gamma and then the pk would not really matter on that absolutely yes yes yes so the point make it so the limit we are usually like to operate in where you can safely use any gf without any special issues is when you is less than or of order gamma gamma being this connection to contacts how easily electrons flow in the absence of any self-interaction if like the bias which I apply is very large so in that case also you cannot take it as a small perturbation so this perturbation theory should not hold in that case yeah except that the way we are doing it I do not view that the bias is being treated as a perturbation though in any of this right but what drives things this is the point I try to make what I like about the non-equilibrium method is you're really driving it with those Fermi functions the f1 f2 that's where you are feeding in electrons etc and the electric field just goes in the Hamiltonian as is I mean you're not there's no perturbation theory about it so if it's a big field here you can include that the main in practice you could worry about whether the model Hamiltonian you're using whether that is those are often limited in validity in terms of their energy scale you know you use some ek relation that works over this range of energies and if you are injecting electrons all over right so that's a limitation from the Hamiltonian point of view you should be that would be my feeling it is this electron electron interaction that turns it into an unmanageable problem and that's the central unsolved problem in all of condensed matter really that there's almost hardly any exact solutions to anything every stage that is almost like the frontier thing and the high TC superconductor generators is like now what 20 years old more than that right in 1986 right okay so we're talking 24 years and still no no solution right okay let me come to that right okay so and this one as you can see kind of brings in this particle aspect into it so as long without this it is almost like electrons are behaving like waves you see and as I said as long as you are looking at just the average flow you know the point I made earlier that usually like if you have air and glass and your light coming in one watt comes in half what goes back what goes through question is do I need to worry about the quantum nature of photons to calculate this and I say that well as long as you're trying to figure out the steady state flow of things you can just use Maxwell's equations you're fine the point where you start worrying about the particle nature of the photon is when you worry about noise for example so if you really cut this down to one photon then of course it's probabilistic one photon comes in there's half a chance it will be reflected half a chance it will be transmitted and in practice of course it will either be transmitted or reflected not that half will go one way or the other and that would then you'd have to if you wanted to calculate the noise associated with all that then you have to take the particle nature into account right and in the beginning what Dr. Raja was talking about involved the noise due to electrons he talks about short noise etc and there of course again you have the exact same issue that when an electron goes through it's like if you're just calculating the steady flow of electrons that how many electrons cross through per second you don't really have to worry about this aspect but what you're now asking is what's the noise right and then you have to bring in the particle aspect and which is which then requires usually the way it is done is with that second quantized formalism right Landauer had a relatively physical way of describing it right relatively non-mathematical way have the physical description to I think right anyway so but otherwise usually when you are looking at steady-state problems most of the time I'd say that electron is really behaving mostly like a wave not a the particle aspect is not there in this discussion very much as it stands right so in that case why is it that the semi-classical description works so well what I mean by that is you know yesterday we talked about the semi-classical picture where you think of electrons as particles essentially why is that working so well that's because of usually all interference effects just wash out more or less at room temperature especially as I said there could be interference effects and some have actually been observed you know in the last 20 years but by and large they tend to wash out if there are many levels the right the other thing was the broadening right so much of that usually that is why usually the semi-classical picture work and so whenever people say the well you know we'd like to set up an energy F to do this problem I always say the well you know that makes it a little harder because as you know instead of electron density you have the matrix etc it gets a little harder but is it really necessary and often the motivation comes from well you know I have a the device there's a part where I have a Schottky barrier where there's tunneling involved and I don't quite know how to put it into Boltzmann okay then maybe you could do any GF for that part because any GF for a big thing is relatively harder to do so you always have to evaluate whether the energy F is really why in that thing there are two steps in there yeah this is where the cleanest way I know is yeah those that would be a bit of a diversion diversion meaning I need to do some background discussion and that is when people discuss Coulomb blockade usually in that limit rather than use the one electron picture they usually go to a two electron picture or multi electron picture what that means is they say that okay I have got this two levels in here so instead of thinking of in terms of this one electron thing we'd say that this system can be in one of four states which is what Dr. Raja wrote here those are the four states so you could be in a state where it is zero zero meaning both states are empty or you could be in a state where it is zero one or you could be in a state where it's one zero or you could be in a state when both are occupied and this has say energy zero this has an energy epsilon that says that was the notation epsilon D and this would be two epsilon D plus and then this is that well when an electron wants to cut so the way current flows of course the basic thing is still the same an electron has to come in and go out see if it is in the zero zero state it's like an electron comes in it will go up to zero one if an electron comes in with that energy and then it goes out again and so what happens is the reason you get this current is when you have the electrons have enough energy to be doing this they do this now the second step comes in when the electrons even have enough energy to be doing that to take you from zero one to one one so the point is so when you actually start calculating currents you use this picture this is the full second quantized picture right the way you are thinking now is not in terms of one electron trying to find a level and then getting out and all that but more like the entire two system is in this whole thing is one big system which has happens to have four levels so yeah so this would be the second quantized so when you have very strong interactions then that is when you is much greater than gamma that is when we have a again a well-defined method for doing this that you can solve it this way and the only problem with this one is a practical one namely that you see you have two levels so the number of states here was four if you happen to have n levels the number of allowed states will be 2 to the power n why because you see every one of them can either be occupied or empty and so if you have ten ones then you can think of two to the power ten states each one's zero or one it's like a binary binary number essentially right and so it gets out of hand quickly that's a but that's a practical problem no conceptual issues at this at this end on the other hand when you is of the order of gamma or less then you can use any gf like we discussed there's no particular problem there the middle region is where it's not at all clear why well what's wrong with this one why can't I use in the middle region well because in this picture is hard to get this broadening into the picture in the in this many-body picture in a any gf gives you a nice way of putting the broadening in the one particle picture I have never seen a good way of putting a broadening into this picture in any reasonable way and I've actually taught a lot about this I've not quite seen anything there and so there is no clean way of that you know of the first special problems people have found out special techniques but there is no general way that takes you smoothly from here to there that I know of no general way and in a way as you say the high TC superconductor is kind of in between you know if there was a general way of course you would have solved that problem in a way because this is the band conduction limit this is the more transition limit and in between is I guess the condo effect is somewhere in between actually condo effect the way when you see those condo peaks that you see which comes about I mean there's an extra there's a level here there's an extra peak here that condo peak that you see is kind of when you is in a somewhere in this transition region so it's the transition that's not very clear at all that and it's not clear how to handle that in general it's open questions and people have special things for special problem but it not a general thing that I can throw up and say okay just go do this that that's not it it's not that simple any questions related to this is this clear because this is important to know the when you can use it and when you cannot and so usually when people see coulomb blockade at room temperature it means that they have contacts that are very weakly coupled so that these gammas are really small so electrons don't get in and out easily so for example in a MOS transistor you'd normally not get into that as far as source and drain is concerned because by definition you are trying to make good contacts there you are not going to put a bad contact in your source or drain on the other hand if you are having electrons jump into the gate that part of it could be involved something in this regime because there it's of course a bad contact really getting in and out is kind of in the coulomb blockade regime of it so anything you wanted to add yes often I'm not seeing that you you have a comment yes please go ahead I'll give it over to you so it's like having one single important part, but with two opportunities over there, two levels which are available. It's actually equivalent to the massively covered quantum power. And in that, that problem has been taught in a really exact way, only very recently. And it's not being yet considered, I think, out of the blue. Yeah, often I'd have said, NEGF probably wouldn't add much to this. Because NEGF, the real power of it, is like this connecting to the contacts, the leads, right? Transport, right? When transport is involved in connecting to the leads, the fact that it has the broadening automatically and correctly and all that, that's the real power of it under non-equilibrium conditions. And of course it's non-equilibrium part of it, right? Because to me it is the quantum version of Boltzmann equation. And even Boltzmann equation applies to dilute gases when you do not have very strong interactions. When you have strong interactions, even Boltzmann, you know, there are all these hierarchy of equations that people talk about with BBGKY hierarchy. And same here. So NEGF, in my mind, is kind of like the equivalent of Boltzmann, right? So whenever this quantum dynamics is important, whether it's tunneling, interference, etc., but one particle picture is fine, then I think it's relatively straightforward to use, then the answers are clear. Conceptually, what has been the challenge of putting the contacts? I mean, putting the contacts. If you're putting broadening properly, I mean, if I were to do it intuitively, they have to go wrong, for example. Yeah, in the one electron shot? Yeah, if I take a physics course, quantum mechanics course, I cannot do this, right, immediately. Usually you do not do the contact part, right, that's true. Yeah, somehow that contact problem, yeah, I always feel it's a very important one, but it's never quite been addressed much. And partly because contacts involve current flow, and historically, people always associate resistance with the sigma s, not with this elastic resistor that I'm talking about. And this is where I feel that in view of the developments in nanoelectronics, one should revise this point of view, and this should be kind of where all transport theory should start. Sir, in equilibrium, one can use it very easily. Out of equilibrium, yes, sir, but in equilibrium, getting the broadening is very easy. Okay, so the way you can do it is, consider, again, a single level shot, right, which is coupled to a conduction path. So it's a broad conduction path, which is in contact with a lead, which is in contact with a single level, and you're modeling it using a certain hybridization parameter v. So that's how some of it is. It's a single particle picture, absolutely, right. So you have one level epsilon d, and you have a v which takes you from d to c, and c to d back, right. So your Hamiltonian is basically like this, epsilon k, epsilon d, v and v. Okay, that's your Hamiltonian. You just diagonalize this, you get two branches, okay. So, and what you see there are actually the two branches. Sir, there's actually an example in your book about these two levels that they hybridize, right. So, but using non-equilibrium, it suddenly becomes much harder, and you create it like this. Sir, in the Landau description, for example, the best thing about it is that the electron interactions are not taken into account, but the bias, you can put in as high as you want, right. The bias is taken completely non-perturbitantly. And the problem is that if you want to include both, I'm talking about electron interactions or just non-equilibrium. So, non-equilibrium. Non-equilibrium, right. So, what you now have is this v, which is the hybridization, it can be called vh, and now you have a potential bias as well, right. Now, adding these two both together seems to be making the problem difficult. And maybe the idea is that once we see, in many problems, what happens is, taking the causal effect, the quantum scale, as I mentioned, goes as this, right. So, you cannot do a perturbation in the hybridization term, right. If you do a perturbation about v equal to 0, you will not get a series. There is no Taylor expansion, because it's, you can see it's an essential singularity, right. So, the point is that the problem that we are trying to do is very similar to this, because we are trying to do, let's say you want to do a perturbation about this contacts, about the hybridization, then you will not get a Taylor series, you will not be able to stop any of that. In fact, there is no Taylor series for the hybridization. And I presume the bias is also very strongly influenced by that. That is probably what the problem is. But any equilibrium, I would say, the problem can be done very strongly. So, the, here, let's say this, the top one is this equilibrium green function. So, and if you read a standard many-body theory, like, say, Fetter and Wallach it will be all about the equilibrium green function. It should be all about that, right. That's what it is. Whereas the second, that one, the GN, that equation is, as I said, that's the thing that first appeared in Kaldeshe's 1965 paper, and it's not as widely known, or as widely used, actually. And I'd say it is in the device context that it's used a lot more. Because, here, because usually in our problems we are always dealing with non-equilibrium things. Most of the stuff. And the standard approach often that was adopted there for quantum transfer was this Kubo formalism which kind of doesn't quite tackle the non-equilibrium problem. Instead, it maps it into an equilibrium problem. In other words, it says that the fluctuation dissipation theorem. Linear response. So, it says that the DIDV is proportional to the equilibrium noise. And then you can use equilibrium stat mech to calculate the noise, right. So, here. But the thing is it only gives you linear response right around equilibrium. And this is where I have felt that a lot of the interesting things in future probably will be far from equilibrium things. Especially, if you think of future directions as involving say connections to biology which in my mind is a far from equilibrium problem all kinds of things. I'd say one really needs good ways of handling far from equilibrium problems. In fact, there are some problems are inherently out of equilibrium. You cannot read the problem in equilibrium. It is a non-equilibrium problem. I guess you could measure specific heat and things like that. Non-equilibrium. And think, I mean, oh, you'd have a lot of, you think it won't be measurable. Yeah, stat mech itself would break down, right. Because stat mech has been built around the thermodynamic level. So, the fluctuations are assumed wherever you do a canonical ensemble or partition function analysis. Where the n is assumed to rotate. Otherwise, you would get fluctuations to the rotor or the pulverotent. You wouldn't get extensive quantities that the entropy would not be. Yeah, what I'm wondering is with the time average, you can get around it. You don't have the ensemble average. The percentages are not equal to the ensemble average. So, that, the whole periodic hypothesis means that. So, when you say that, so if you look at the resistance of this something very small, you're saying the fluctuations will be very large. Even after, so the thing that, I think with, that kind of came into mesoscopic physics in the late 80s was that previous, prior to that, everyone always calculated ensemble averages. And then they said, well now we should calculate sample specific things. Right, and so, as I mentioned before, on a single, if you measure conductance versus Vg in some of these FETs at low temperatures, it's supposed to go up, but then there is all these fluctuations, which are all sample specific. For a particular sample, you'd get this. If you move it around, you'll get something else. Right? So, and these can be calculated. Of course, the idea is that if you knew exactly how the impurities were distributed, you'd probably be able to. But then you don't quite know. So, what you can put in is some random thing and you'll see something of the right size. And then, what was, one of the interesting things that was proved there was the size of these fluctuations would be this, of the order of q square over h, etc. Right? I want to make any n. What is the order of n you call it? One over the one? Yeah, there is usually the opposite number and the difference happens. But then, as you go along with the other systems, what you should be considering is the 1 by unit. That's what we're considering. And that should be compared to the average value. So, there is, I mean, that cost of averaging is specific to different systems. But I would guess of the order of 100 other items. So, that looks like getting strong. Because most of the amount of scale, I mean, now people are trying too much. One dimension is one thing, but the other dimension is pretty large. If you take a big dimension, you have two dimensions, which might come in a long and you have a lot of a lot of problem to manage it in the central area. So, in that case, I think it's big. Yeah, yeah, yeah. To these systems, even when these systems which are large, they are not very much similar to the average. So, even the average time of these systems is not really something wrong. Because in, using average time of these systems that can equate the time average and the average time of the system. The next step. So, in this example, you'd say, this one is an ensemble average, which is what you'd get if you took lots of samples and average day. While this is a time average thing, of course, I mean, because you are sitting there measuring it over time anyway. And this is what you're actually measuring the time average thing. But if you took hundreds of these samples and average the answer, you probably have got that one. Right? And any ensemble average, you'd get that. That's the, that's the point you're making. I mean, there are violations of fluctuation. This is the same theorem. It basically comes from the fact that the fluctuations are not described by the dissipation. Right. So, I guess a very good question is there, whether this is a fluctuation by that is, fluctuations are not supposed to be reproducible. What I said was a very important point was this is something that you could go down every day and measure exactly this. That is what made this a very important observation then. That, yeah. You could go back the next day and you still measure exactly the same thing. Any other questions related to this? Yes, please. I didn't hear you very well. Yeah, please. Speak up. The movement of the item chain. So how can we define velocity of a phonon? How can we define that? Velocity of a phonon. How do you define what? So I think it's a collective variation of all atoms. So how can we define the phonon? Oh. So usually people calculate this omega versus k, right, of phonons in a given solid. Right? And if you look down here the slope is this sound velocity. If you look at this slope. Right? And that's what we usually is the sound that we're used to like in a solid. Right? That would be a sound. And here of course the wavelengths involved are very long. And then when you go up this all day. Now, so velocity is the slope of this. Now in the, so what moves is that vibration, I suppose. So just like sound. If you tap it at one end something propagates. So question is at what speed does that propagate? And that's supposed to be this d omega dk. So for example, there is this thing called optical phonons which usually hardly move. And the reason is that it is just a localized vibration usually. So let's say in gallium arsenide is this gallium and arsenide just moving there. And the next unit cell doesn't necessarily even know about it. So you could do something at this end, nothing will propagate. And that's basically why it's almost flat. But the velocity would be this slope. And when you come here it doesn't move quite so fast. That's how you think. And with the ek relations also with electrons as I mentioned the general definition of velocity is this dE dp. Basically d omega dk in the same idea, of course. I mean, the problem is that you have to look at the problem of scratching the dE dp object. So what you're saying is there is a delocalized object. Now how do you displace the propagation of this delocalized object? I mean it's not a local object as a particle. Then you could really measure the distance and the time it covers and then you find the velocity. But if you have a delocalized object which is like a symbol of any particle. Right? Now what I'm going to say is suppose you consider a free electron. Right? A free electron is always studying the equation. What do you get? You get a plane there. Now a plane in that situation can you really find where it is? You cannot. Right? But you can define a velocity. Yeah. That is what they define in terms of the group velocity. Right? That is the argument you go through that if you had like I said if there's a plane wave here the wave function of course is e to the power ikx and that's everywhere. So if it's everywhere you can't tell what its velocity is. So you kind of have to define a wave packet one place in x and then look at it at 9 o'clock and then look at it at 10 o'clock again and see how far it got. Right? That's right. So if at 9 o'clock you want to localize it here you have to take a whole bunch of k's and put them together. So according to some k you put a whole bunch of them together that's how you localize it and one of the exercises they go through is that an approximate description of how fast it gets is given by this group velocity which is d omega dk. In the wave picture that's why they say the group velocity is d omega dk whereas in the particle picture this definition Hamilton's equation of motion when the way Hamilton wrote his equations it was this dx dT is supposed to go as dE dP and then dP dT is supposed to go as dE dx. So when you think of it as particles that's the Hamilton's equations of motion whereas when you think of it as waves this one is basically identified with group velocity well yeah well this one is like Newton's law you know it says that momentum is equal to the force that's it and this is where yesterday I made the point that you know it is dP dT that's equal to the force not mdV dT because depending on if you are trying to use non-standard EP relationships then you have to be careful which one you use etc but these are the general equations that would work regardless of the energy momentum relationship to one question I think someone asking me is whether that if you have an electric field in some region so that let's say we draw the potential a sloping down so this is X and the potential is sloping down that's since there's an electric field then the P keeps increasing and question was whether the way I was describing the elastic resistor that the electron is going through the fixed energy does that include this fact that the K is that the momentum is increasing and the point is actually does because the way you think about it here is that you have some EK relation here and down here it's like the same thing kind of went down so if you are fixed energy you just end up with a bigger P that's it you see so the fact that the P increases that's equivalent to saying the energy stays the same and the potential energy went down because total energy is constant the kinetic and that's kind of part of the same story it don't really need to include that separately this I just remember because someone had asked me whether this was included or not and so this statement that dp dt equals du dx can be proved just from this idea that the total energy stays constant I think there was one question regarding the thermoelectric effect I don't know I guess we have a few minutes so if there was anything else related to the second talk on the thermoelectric because we more or less focused on the quantum transport part and of course there were lots of issues to clear up so I think it's good and someone I think said a note and this I was not aware of this that actually two there are metals which have different sebeck coefficients the point I had made if you have a mu here and the density of states is up here then you would have one kind of sebeck coefficient on the other hand if you had a density of states here like this you'd have a different sebeck coefficient because of the point what we discussed and that I said kind of explains it all comes from this f1 minus f2 the fact that a hot thing looks like this a cold thing looks like this and here f1 minus f2 drives electrons this way here f1 minus f2 drives electrons the other way and that's why there's this difference and I guess the question was then how is it that there are metals which have different signs for this sebeck coefficient because all metals of course this is a p-type semiconductor but there are metals which are metals usually would be like a conduction band I suppose so usually there's it'll be in the middle of a density of states somewhere and this I've not thought about but my first guess would be that there is a decreasing density of states locally at that energy probably so what I mean by that is the sign of the sebeck coefficient all depends on if you draw the mu whether you have more states above or below that's the issue really and if locally around there happens to be an opposite slope it could change the sign that I see but this is something I haven't looked into so I'm not sure so my experience in this has been that if you look at resistivity then it does not really depend that much on the detailed k-dependent structure but if you look at the thermo-electric power or the thermo-electric effect of the all-coefficient most of the onsite coefficients basically the L1 to L1 and so on L1 1 is very simple doesn't depend on the detailed structure but if you look at all the others then they depend on the detailed construction and the full-blur zone how the k-dependence is for e k the e versus k-vector that actually comes into picture because you find out something from your transport and that really matters though with thermo-power I'd say that you know that is usually the standard expression for evaluating that coefficient the one on the right and one on the left for evaluating conductance that's the L1 1 the second coefficient is like the L1 2 which involves this e minus mu and so to me I think the important thing is whether what it is above mu compared to what it is below mu so basically it's a question of whether these energies conduct better or these energies conduct better and that's what determines as far as thermo-power goes Hall effect I feel is really this band structure I mean that EK relationship and all because and that is why I feel that in amorphous materials Hall effect is so poorly understood while thermo-power is relatively well understood I think whether it divides into like higher unoccupied part is more occupied part which part conducts better right right because above it electrons will be driven in one direction below it they will be driven in another direction balance is determined by which one which side conducts better that to me is the essence of all the thermo-electric effects really one other point that said that if you have an energy level here that an electron is conducting through and this is mu 1 this is mu 2 then effectively the electron takes that much energy from this contact and dumps it into that contact so this actually gets cooled take energy from the contact and I said that was kind of the origin of this peltier effect in a way and here of course I assumed that this contact has no special structure in the density of states so that when a current flows it flows symmetrically right around here that's the assumption here but of course in any real material it is always biased a little bit one way or the other so the current here could be flowing a little higher up and the entire peltier effect is because you have one material where it flows say a little say 1 millivolt higher and somewhere else where it moves say 10 millivolt higher and then in order to adjust that region must either cool or heat according to whether it went up or down so peltier effect in that sense what you measure is kind of the difference between what you have on the left and what you have on the right and for our discussion I assumed coefficient or 0 peltier coefficient so you got just that