 you can follow along with this presentation using printed slides from the nano hub visit www.nano hub.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 yesterday the central concept I was trying to get across is this idea of a elastic resistor you know which has said when you first say it almost sounds it can be right even because the resistor has to dissipate heat and if it's elastic that means it doesn't exchange energy so how does it dissipate heat of course the central point is that it goes through the channel as shown on that right-hand side little figure there without losing any energy but the actual energy is all dissipated in the contacts that's the view and what this also tells you is that resistance is not necessarily associated with the heat dissipation per se that is you could have resistance in between because inside the channel let's say you have a obstacle of some sort that would increase the resistance but it's not necessarily dissipating any heat the dissipation is still in the contacts as far as the elastic resistor goes now this of course what has made it particularly interesting in recent years is that it seems to be a good representation of many nanoscale devices anyway but the whole idea sort of dates back to Landauer which is like 50 years ago when all this was not even experimentally relevant and he was thinking of it more as a conceptual tool for understanding transport because I think he kind of saw that transport there are these two types of phenomena involved the dynamical ones where that's this part where you do not exchange energy and the energy exchanging part the irreversible part where you're giving up heat and things like that and what makes transport complicated is these two things are intertwined and anytime you can separate it out it makes it much easier to understand what's happening and this elastic resistor let me just make one comment that you know one thing you could say that well real resistors long resistors especially if you have bias across them of course there's a lot of inelastic scattering inside the channel as well and this question is why does the why could we use the elastic resistor concept to be deriving results that are valid for big conductors because one of the points I tried to make was that if you look at those expressions we obtained from the elastic resistor point of view for the conductivity or those are standard results which are derived not for elastic resistors but for regular resistors using much more advanced formalisms using Boltzmann or Kubo formalisms that expression for conductivity as density of states times the diffusion coefficient for example see so those are really standard results but we are obtaining it with this elastic resistor model which shows that it really captures a lot of physics see and so one way to justify it would be to say well the real long resistor is so you can think of it like a series of little elastic resistors as if it goes elastically then loses some energy goes elastically then loses some energy and so on but what of course makes this thing very tricky is the point that you see if you have a little elastic section here we know that the resistance is rho L plus lambda over a and then you have another section we discussed and so this is L1 and this is L2 and if you just add them up in series you'll get the wrong result why because you'll be counting the lambda twice will be counting this interface resistance twice so the thing is that if you're thinking of it this way the point you have to be clear on is it this is why so important to sort out where all this resistance comes from really and the point to recognize is this is the part that is associated with all kinds of obstacles along the way inside the resistor which is why the longer it is the bigger it gets and then there's a constant part which is simply involved in getting in from the contacts into a resistor and getting out and that part of course does not happen in between because those in between contacts are not really there they're just conceptual things of thinking of it so that is the point you have to be careful on when you're applying these ideas and of course one new thing that comes out of it is just this thing that the resistance is conductivity times area divided by lambda plus L whereas what we usually do is have an L there so that is the important of course conceptual part but otherwise the expression for conductivity etc looks much the same way and don't mean to imply then though that there couldn't be all kinds of extra things that come in when inelastic scattering is actually distributed rather than in lumps like this so there could be other issues involved you see but elastic resistor is a very good starting point so in my mind it is almost like the hydrogen atom for transport it's like where you start that's the one and what I'll try to do in the next few lectures basically is try to show you how with elastic resistors you can understand all kinds of other things as well we'll talk about heat flow, spin, flow of spin all kinds of other things so far what you'll already have done is semi-classical transport this particular lecture we'll talk about quantum transport, quantum transport through an elastic resistor you see now everything I so one example I quoted yesterday where I said you know inelastic scattering can make an enormous difference well if you had one channel connected to the left one channel connect to the right and there was no inelastic scattering in between you wouldn't have any current on the other hand anytime you turn on inelastic scattering there will lots of current so obviously the lower one is not even an approximation for the upper one so not everything is captured of course there are other issues and as I said yesterday p-n junctions to some extent are a little bit like this any now so today this lecture then is about quantum transport and because everything we talked about yesterday it's called that would come under the category of semi-classical transport so in terms of the basic physics nothing we said yesterday would I mean everything we said is contained in the Boltzmann equation so whatever I said is consistent with what you'd get if you solve Boltzmann equation is just that you could see it I hope much more clearly because we were looking at an elastic resistor where there wasn't a lot of inelastic scattering in the channel whereas Boltzmann equation of course itself could handle all the inelastic scattering there's a clear prescription it's just that it wouldn't be as easy to see what happened that's all right so in that sense the corresponding thing to Boltzmann for quantum transport is this NEGF method which some of you are familiar with and some of you are not probably and I won't assume you're familiar with it but then the purpose of this lecture is not so much to teach you NEGF if you don't know it already it is more to tell you what it is about what exactly you're doing physically it's more about the concepts involved than about the mechanics of how you do it because one thing you realize in all this that I think Mark also mentioned this that you know calculating something and understanding something are kind of separate issues I mean one helps the other but then they're not identical really and so I'm not going to talk too much about the calculating part and there if you have questions maybe we could use the discussion session for some of that if you want right but the one question with quantum transport that often comes up is and that is what I kind of want to illustrate a little bit is that well for quantum mechanics one thing you have all heard of is a Schrodinger equation that describes the quantum mechanics of different materials I mean atoms solids etc so why exactly do we need an NEGF formulas why do we need something else couldn't we just go to Schrodinger equation for example then that is the conceptual part I want to kind of get across this top so what we'll consider then is this elastic resistor and oh let me just mention this for those of you are meaning is that these are the equations of any GF that some of you as you know have seen it some of you haven't and the general scheme is the following that you have a channel that is represented by a Hamiltonian that H that's on the right hand side if you look that's this H this is the Hamiltonian whose eigenvalues give you the allowed energy levels that's the part you always learn in quantum mechanics usually and of course usually the way you see Schrodinger equation is like e psi is equal to H psi and that H is usually a differential operator but one of the things I think you learn is that you can convert it into a matrix and usually whenever you do a numerical calculation the differential equation is usually quite convenient for analytical solutions but many problems in quantum mechanics as you know don't really have analytical solutions in fact there's very few that have and when you solve it numerically most of the time you convert it into a matrix one former and the eigenvalues of that matrix give you the energies that's how usually it is anyway so that's that H so it's a matrix whose size depends on the size of your device of the channel essentially so for example if you think of the channel as say if you're using one of these real space you know the tight binding models and your channel has say 10 spatial points then H would be a 10 by 10 matrix for example now the part that is not as well appreciated is this connection to the contacts and as I said for the elastic resistor of course you see that is very important because as we discussed last day they all this it is this connection to the contacts that makes this current flow through this thing because something comes in and something goes out and that's the part which you normal quantum mechanics courses you don't quite hear much about you hear a lot about the H learn about the Hamiltonian all kinds of age how to find eigenvalues etc what you don't quite see much of is how to connect to the surroundings etc and and there I've written this sigma 1 which is the connection to the left contact sigma 2 which is the connection to the right contact and in between there's a sigma s which in general means all kinds of scattering processes inside the channel itself and as I said 20 30 years ago when your people were dealing with big devices big conductors nobody worried about the contacts so there's any gf formalism when it was first developed back in the 60s of course no one was thinking about contacts at all and they didn't have a sigma 1 or a sigma 2 and then in those days of course the physics of resistance was all in the sigma s it was all about the scattering you see whereas again in the 90s once this people started looking at small conductors that is where of course this elastic resistor is a very useful way of thinking about small conductors and then you as a starting point you can drop the sigma s you can just use a connection to the left contact a connection to the right contact sigma 1 and sigma 2 and that is of course in terms of understanding that's much easier because the theory of sigma s is a whole lot more complicated than the theory of sigma 1 or sigma 2 I mean this connection to the context that's a whole lot easier to understand and so again for the because of the history of this subject again most of the literature uses fairly advanced formalisms to come up with these equations on the other hand what I'll try to show you is how straight from the one electron Schrodinger equation you can see where it all comes from that's basically what I'll try to get across if you have to do a real calculation then of course you'd say well I'm interested in graphene how do I write down H next how do I write down the sigmas etc and that I won't go into because that some of you are familiar with and those of you are not I can't quite tell you in half an hour I mean it'll take longer so I won't really go into that part on how you write those things down but once you have those the left hand side gives you all the equations you need to calculate it because these are all matrices and once you have written the matrices the left hand side basically at the end of it tells you how to get current for example out of it given those matrices so first step is to write the matrices then this is it that's all okay so to understand this then let's start with the again the simplest elastic resistor which is a device with one level so I'll assume that in this contact f is equal to one what I mean by that is I'm assuming the chemical potential is up here and around this energy for the moment let's use f is one but of course f could be anything you just have to multiply finally by that f1 minus f2 but for this discussion let's assume this is one this is zero the moment and we're trying to calculate the current through that level and what I'll do is first we'll do what's on the left hand side which is like a simple semi classical picture you know like taking off electrons as particles go in and out and then we'll do the same problem with Schrodinger equation we'll say okay we're thinking of it as a wave what new things come out of it and what I'll do on the right hand side though basically the whole any gf method gives you a formal way of doing this with many levels and all that in a systematic way you don't have to worry about it anymore but in terms of understanding what you're doing this is a very good example to try you know one level how do I do the quantum mechanics of it so one level so let me try to write down the first let's do what's on the left that's this semi classical picture so there you'd say okay yeah I've got a level here the rate at which the number of electrons in that level changes if I put an electron in here it will want to escape out into the contacts depending on how well coupled it is and that's tell that's that second equal to minus nu 1 plus nu 2 times n nu 1 is the rate at which it wants to escape into the first contact nu 2 is the rate at which it wants to escape into the second contact so put something like this so this would be sort of like a lifetime so this could be like one every picosecond for example so that would be like 10 to the 12th per second some such number so this would be one and then that last term there that is like the rate at which electrons come in from the left contact because I've assumed this is one that's 0 so electrons only want to come in from here nothing's coming in from that side so I could write plus so what would be the steady-state occupation steady-state number of n well let's set that to 0 steady-state so you'd get n is equal to now how do I one way to fix the strength of this s1 is to say that well one thing I know is if there was no coupling to the right-hand side so let's say I detached this completely what that means is this was not there then what should have happened is this level should have just become full because it's connected to a contact which wants to keep it full you know the point I made last year the reason current flows is this one wants to fill it up that one wants to enter now if this was the only contact it was connected connected to then of course it would just get full so if nu2 isn't there then n should just be one so which sort of tells me that this ought to be equal to that's the other wise I won't even get the right answer for one contact so with that just if we that argument you can kind of set the strength of what's coming in so instead of s1 then so that tells you the steady-state number of electrons in there and after this you can calculate the current pretty easily if you want because what you could do is an interesting way of writing this you could write this in two parts so I'll write this as this s1 minus nu1n minus nu2n so just regroup them a little bit why is that because when you regroup it you can kind of look at this this is like the current this is the rate at which electrons are coming in from contact one I see this is the rate at which it is kind of coming in this is the rate at which is going back out so the net current at contact one is this one and the net current at contact two is that one and steady-state they're equal of course whatever comes in goes out so when I'm trying to evaluate current the thing is this whole thing is zero but then this one or that one will give me the net current I could look at either one that's the point and the thing is this is important because when you this is a very good guide to what to do with the quantum thing also here it's pretty clear before we bring in all the other stuff so if you look at any one of them and of course this one has only one term so that's easier to do so what is the current then just nu2 times that yes right because the right contact I'm assuming we are at an energy such that the Fermi function is zero and so nothing is coming back and once it goes out it gets pulled out because that as I've always mentioned that this is the part that no one quite tells you explicitly but the point I make is ordinarily if electrons went from here to there sooner or later they would all come back again and there would be no net current current flows because as soon as you get there somebody pulls it out and whatever is left in here somebody fills it out that's why it keeps going forever and this is the part that is not in any Hamiltonian not anywhere you have to put it in and usually it is done rather surreptitiously and you kind of don't appreciate that point so yes so and this is zero otherwise in general what would have happened is this current would have been multiplied by f1 minus f2 that's what would have happened in general if you are two different for me if the f1 and f2 instead of as you I assumed one and zero but if this was say 0.5 and 0.3 at that energy then you would have to put that in that's the same argument we had I mean in all these elastic resistors at the end of the day we'll have different expressions for this g but basically it will always have this structure there is this f1 minus f2 because f1 is what's driving left to right right to left so eventually you'd have this also and the way I've done the current of course this is just per second you see if I really want amperes then I should of course multiply by q or minus q depending on how you're keeping track of what's the sign convention for your current but basically you'd have to multiply by a q right now if you remember yesterday when I tried to write down the current I said that if you had just one level the current would be q over t times f1 minus f2 and the idea was that this t I said is the time it takes to get from left to right this transfer time so in this model this thing is like 1 over t you could say that whatever I called transfer time yesterday I said well in a rate at which electrons are going is this t and so current would be q over t that's how I kind of started last day so in this context you could say 1 over t is nu1 nu2 over nu1 plus nu2 or you could turn this around and say t is equal to and that kind of makes sense because this is what you might call interface limited transport sort of it's like it doesn't make too much difference how long the electron takes inside because it's such a short device the real time it takes the time it takes to get from here to here is essentially limited by getting in and they're getting out and one of them you could say is the time associated is 1 over this rate like if this is 10 to the 12th per second then it's like a picosecond 1 over that and so total time is like the sum is a rate so this term here that's a rate it's dimensions is per second assuming there's no electron flow into the contact it's already 4 what can we now start off this way assuming u1 is 0 is it an electron, is it a channel? it won't go back to the input contact it can always get back in right because no from here you are continually so let's say you don't yet you want an equation that will work regardless of your f1s and f2s so for example supposing both contacts were empty then what should happen is you could start out with some electrons in the channel and it will just empty out so for example if this was 0 this will tell you that the number of electrons will die out with a time constant of so much that's the rate at which it will just empty out into the channels so in this model all I was saying is that the transfer time is just like the sum of the time associated with contact 1 and the time associated with contact 2 right and this you can call the kind of limit of interface limited transport that your conductance is just determined by interfaces what becomes the rate? how well connected your device is to the contacts so physically I would say if I put an electron in the channel how long do I have to wait before the extra electron is lost is it picosecond, microsecond, nanosecond that's the physical connection if you have to walk to other contacts to the rate then you would have to if you had multiple contacts here then I guess you could associate different times associated with going from 1 to 2 or 2 to 3 etc then you would kind of need a multi-terminal formula with different times associated with different points ok now let's do the quantum version of course you see that's the one I was trying to get at yes please no not in this model the way you're thinking I've got a channel once filling up once emptying it now the corresponding thing write a Schrodinger equation it would look something like this nice thing about the one level equation though is as I said in general the Schrodinger equation would have looked like i h bar d psi dT is equal to h psi where this h is the is a matrix the Hamiltonian matrix whose eigenvalues give you all the energy levels now in this case we are doing a one level thing there's only one energy level so basically it is this is just a number epsilon if you happen to have two levels then this would be a 2 by 2 matrix that would be a good place to start so this makes the Schrodinger equation particularly simple and this would be the time dependent Schrodinger equation and then there is a time independent Schrodinger equation which amounts to saying that if you write psi as this is time dependent and then there's a time independent one where you do this and then this will look like e psi equals epsilon psi that's the usual form now and then you say the only way you could have an electron inside is if its energy happen to be exactly equal to epsilon if you don't match that energy electron can be inside psi will be 0 that's basically it now this we cannot use for the problem as is because it is describing a isolated system that's the point this is just a level by itself what we have is this connection to the context things coming in so this part of it if you find the rate at which the number of electrons changes because the number would be like psi psi star so let's say we try to find d dt of psi psi star and you can actually show this very quickly that answer will be 0 you'll see why because this is like psi d psi star dt plus d psi dt times psi star and then you can use this d psi dt to replace this part so instead of this I could put in epsilon over i h bar psi and instead of this I could put in epsilon psi over minus i h bar I picked up the minus because I complex conjugated it and then you'll see the two terms just cancel each other out so basically if you are using this equation it tells you that the number of electrons in that level won't change d and dt is 0 that's it so if you use that that is equivalent to putting d and dt equals 0 but what we need is to put in things corresponding to this, this and that and in a way those are the sigmas in the energy of formalism that's the h h doesn't give you any changes then there's a sigma 1, there's a sigma 2 and then there's a driving term that's basically where you're headed now how do you put that in? the answer, the way it goes is that if I put in epsilon minus say i gamma over 2 supposing I do that so this is of course a real number but now I'm adding an imaginary number now what you can show easily is that in that case the d dt of psi psi star will actually be, it won't exactly cancel out because this one, the imaginary one that one actually will find what you get from here and what you get from here will add instead of canceling out and the net result is that at the end of the day you'll find you'll get minus gamma psi psi star which is a little bit of algebra you can check it out so finally you'll get this so adding something like this here is sort of equivalent to adding a gamma times n you know that's the kind of thing we're trying to add here to get this effect so the simplest way to get it into Schrodinger equation is to put a minus i gamma and you'll note that you have to if you want to actually this will be gamma over h bar because the way I've done this gamma has the dimensions of energy because it's epsilon is energy that's also energy whereas what I get here of course has the dimensions of per second just like nu and energy divided by h bar is of course per second because energy is like h nu divided by h that's like per second so the point is that in order to get this nu 1 plus nu 2 here what you need to do with the Schrodinger equation in order to get the equivalent of that is put a gamma 1 and a gamma 2 and what are gamma 1 and gamma 2 well those things divided by h bar so gamma 1 or gamma 2 divided by h bar should be equal to what was nu 1 or nu 2 in the semi-classical model so this was the rate at which electrons escaped into contact 1 or contact 2 what you should do is that's what why the 2 there you're rough basically the idea is this that when you look at the solution to this psi looks like e to the power minus gamma t over h bar over 2 because of the 2 that I put in so usually what happens is psi is e to the power minus i epsilon t over h bar now because there's a i there the corresponding solution looks like this now what you want is the probability to decay as gamma t over h bar and probability is the square of this so when you do psi star that takes off the 2 so in order for the probability to decay with gamma the wave function has to decay as gamma over 2 because probability is psi square that's about it if I put anything real then d d t of d n d t would be 0 d d t of psi psi star would remain 0 because putting anything real there is just like changing the epsilon a little bit that makes no difference to the basic thing and the reason you see that if it is anything real then the solution looks like this and the magnitude squared of that is always 1 it's independent of time nothing changes that's basically it but by putting this in you add a solution that looks like this and then the magnitude is actually changing with time and it represents the fact that you have connected this system to the contacts and things are going out into the contacts that's about it so now the corresponding time independent Schrodinger equation if I were writing it would look something like this where it writes gamma as gamma 1 plus gamma 2 and there's one more thing I need and that is the source term remember what I've done is kind of changed the Schrodinger equation to get this effect in it in the sense that if I take that equation and find d n d t it will look like this it will match this semi-classical picture I have in mind and then there is the additional source term I would need to put in let me put this once now then you can write the solution easily now one of the important things that comes out of the quantum way of doing it compared to the classical picture is what's called broadening and that's the point I wanted to illustrate and that is see in this problem that we are talking about that I've got a level here which is at epsilon classically the way we think only way an electron could get from left to right through this would be if its energy matched that epsilon exactly if it was a little bit off one way or the other as long as it's all elastic it wouldn't be able to get through but quantum mechanically it doesn't quite have to match I mean e could be a little bit different from epsilon and you'd still have a wave function inside in response to your source so your epsilon's right here e is a little bit off doesn't quite match that's fine you could still get a response to it of course when e is equal to epsilon this is zero and so the response is a maximum that's fine but it doesn't have to match exactly and so what you'd find is that if you looked at the electron if it is not exactly on this a little bit off then it will broaden it out somewhat effectively so this connection to the contacts is giving you a broadening and this comes out of the quantum treatment it is a wave effect in a way this whole thing the point I wanted to make here also is the following that if you take this one level system and you used our semi classical picture the one that we discussed last day and let's say we calculated the current versus voltage and for this discussion let's assume is zero temperature so we've got this and what I do is I at zero voltage mu2 is here and then I pull it down so gradually I'm pulling the drain down and I have this perfect electrostatic so this one doesn't move down I don't want all that complication I'm just pulling this down then what will happen is at first no current will slow as soon as I cross this current will start flowing so basically you'd have something looking like this so if you looked at DIDV that's the conductance so supposing someone measures this and then they looked at the conductance right around here you see the conductance would look like it has this enormous peak question is how big is that peak and of course one point I tried to get across yesterday we discussed is how there is this maximum conductance you see we wrote the conductance as this q square over h times m with the idea that m tells you how many channels you have and if you have one channel you should be q square over h many people measure two q square over h because there's always the two spin channels but that's the best you could get if it was ballistic conduction and all that now what is a little bothersome when you look at this is it looks like there's no upper limit to that peak that could be anything it could be way up something and because this is a very sharp level as you're conducting through it you can show that the maximum current you'll get will be something like q gamma over h bar gamma 1 gamma 2 over gamma 1 plus gamma 2 etc so it will be something like this and if the transition takes place infinitely sharp then of course the conductance is infinite you see but the fact is experimentally people have now measured this even in a hydrogen molecule something where this is a pretty good model where you're really conducting through one level so this is not just the academic thing we're doing this represents what happens in a hydrogen molecule and when they make good contacts to it and they measure the conductance they do get two q square over h I mean something on that order so what did we miss here what we missed here is this broadening that goes with it the point is that that level won't be this sharp once you connect to it properly from the contacts once you connect from the contacts it will broaden out and it will broaden out by an amount of about gamma so that this transition won't be as sharp as I drew it but will actually be spread out over a few gamma and so when you look at the slope the conductance will still be q square over h that's the point this will be spread out over gamma over h bar so when you look at the gamma I mean qv would be gamma so gamma over q so it would be spread out over that range and so when you take that and divide it by this you'll get about q square over h so finally the point is even in this one level thing although the thinking is now totally different you still get that same still get this upper limit on the conductance but you really need wave mechanics to get this right to get this broadening into the story now again lots of times it doesn't matter because you have many levels which are very close together if you spread them each all out whether you spread them out or not doesn't really matter too much so lots of times you don't worry about it the semi-classical picture is fine but it's the only the quantum picture that will give you this broadening now one way to set the strength of s would be again sort of like what we did before but here we said well I know when u2 is 0 what n should be etc so what I could say is I'll set the strength of s in the Schrodinger one by saying that after I integrate this over all energy so supposing I find n by integrating over all energy then what that will look like is so what I've done is this is psi so if I multiply by psi star that tells me that that's like the electron density but then I have to integrate over all energy and when I integrated over all energy what I get that should match whatever I got classically that would be one way to set the strength of what this is now in practice the way you do it of course usually is you start with an infinite system and then you calculate and then you eliminate the outside and calculate what this strength is that's how you get it in the regular NEGF method the way you normally do it what I'm trying to do is get you kind of bypass a lot of those details and roughly tell you how to see the answers because what I'm really trying to get across is the concepts behind it not necessarily the details of the method itself so this quantity then you could write as s1 s1 star over 2 pi so what I did was I pulled out the s1 s1 star and divided by gamma over 2 pi and put a gamma over 2 pi there now why am I doing that well because this is this Lorentzian function whose integral is 1 that's why I'm writing it that's all so and then I can say well this should be equal to what I normally what I expected to be and that's how I could kind of figure out what the strength of that source term should be to match but the important point that this method gets across is the idea that n or if I want to calculate current how would we do it if you remember we had these two ways of doing it I could either do s1 minus nu1n or I could look at nu2n and the easier one is of course that so here also same thing if I wanted current I'd put a nu2 here which would be like gamma 2 over h bar put that in and then you'd have almost the same thing but now with a gamma 2 over h bar here that's all so that's how you do the quantum treatment of this one level problem would proceed but the important thing is just Schrodinger equation isn't enough to connect it to the contacts when you connect it to the contacts it kind of automatically gives you this broadening that usually the Schrodinger equation we're used to e psi equals h psi that kind of gives you the resonant frequencies of an isolated system sort of like a guitar string what are its resonant frequencies but now what we're trying to do is excite this guitar string from elsewhere and the way you excite it you don't have to match the resonant frequency exactly it could be a little off and it would still vibrate in fact usually these things are so well damped that you could be well off and still be vibrating actually and so that's what we have kind of done we took this and added the damping terms to it now the one other point that I want to make here and that is the difference between this quantum treatment and the classical treatment and that has to do with you see this was the simple classical treatment and then for the quantum thing what I did was I said let's write it this way we have a this was the basic Schrodinger equation that's the damping term the connection to the contacts and this is the source here also we did exactly the same and in this discussion we assume that electrons only come in from the left and not from the we assume this is one that's zero now what if we want to include the f1 and f2 into our discussion so I've got some f1 here and I say we'd like to include all that what I said was we'll assume one the other is zero and at the end of the day you can always throw in the f1 minus f2 we kind of know what that part should be but if you wanted to include it here actually it's not very difficult in the sense that all you'd have to do is you'd have to you know I argued this one is actually same as new one but you could just add a new one f1 plus new two f2 so what I mean by that is this is s1 and you could add a s2 like this and if you went through the algebra you'd get all the right answers with the f1 minus f2 in place no problem at all but the point I'm trying to drive home is you can't do the same thing with the quantum one because with the quantum one we say okay now I've got two sources on two sides so maybe I'll put a f1 here and f2 there and of course we realize that these are kind of like square roots of that one so actually you probably have to put a square root of f1 and a square root of f2 if you did that let's say you did that but the point I'm trying to make is after that when you calculate psi and you calculate psi psi star you'll get a bunch of unphysical terms why? because you see the psi then will look like s1 square root of f1 plus s2 square root of f2 divided by e minus epsilon plus i gamma over 2 like we had before we had only this one before now we have two terms good but then when I take psi psi star I'm squaring this and when I square this I don't get two terms I get four terms I get something that will look like s1 f1 something that will look like s2 f2 and those of course would be just like the classical thing and then there would be stuff that are non-classical there would be things like this and those really should not be there no experimental evidence for anything like this with ordinary contacts at least with superconductive contacts yes I mean in a way Josephson effect is kind of like an interference between two contacts but normal contacts none of this the argument is what happens is what comes in from contact 1 and what comes in from contact 2 have no phase correlation so it's like because these are complex numbers but they have a random phase that changes with time that's the way to think so that this quantity like this has a phase that is randomly changing with time and as long as you're looking at time average things you never would see any of this whereas these are of course real numbers so you can average for a very long time you'd still see it but here you have in mind so that's why you need to drop all this but this is the problem with trying to do quantum transport straight from Schrodinger equation problem meaning if you have multiple sources I cannot just put them in here and calculate because you know when we use Maxwell's equations if you have multiple antennas that's fine they all go into the same Maxwell's equation but then multiple antennas usually interfere you know two separate antennas would interfere two lasers would interfere but here it is like two incoherent things they're not supposed to interfere so you cannot just add them in and so anybody who uses this of course the way they think about it is they're supposed to do one source at a time so you do this one get your answers weight it by F1 take this one weight it do your calculation weight it by F2 weight it up so as long as you keep that in your mind you're fine but you have to remember to do that in general now what the energy of method does is then it says that well the fundamental thing then is that we can add up from many different things is not the Psi itself but the Psi Psi star and so it works directly with the Psi Psi star and try to give you an equation for it so it would be things like Psi Psi star or Psi Psi dagger as what we call the correlation function so what that means is of course in this example Psi was just one number and Psi star again another number so when you multiply it's just one number but more generally what would have happened is this Psi could be let's say two component so it tells you the wave function at 0.1 wave function at 0.2 and Psi dagger then would also be two component and when you multiply the two you'll get a 2 by 2 matrix the diagonal elements which is like so let me actually write this so this is Psi 1 this is Psi 2 this is Psi 1 star this is Psi 2 star so when I multiply it out you'll get things like this Psi 1 Psi 1 star Psi 2 Psi 2 star Psi 1 Psi 2 star Psi 2 Psi 1 star so this is the basic quantity this correlation function that you usually calculate in an NEGF formalism and if you look at its diagonal elements it is basically giving you the electron density it is telling you how many electrons at 0.1 how many electrons at 0.2 but then it has some additional information this part of it also in it and many times you don't need it necessarily and you could just look at the diagonal elements it will give you the same density and this is in general true that in the quantum formalism whatever you think semi-classically as say electron density so if I had two points I'd have to tell you the electron density here and the electron density here so I'd have something like N1 N2 in the quantum way of thinking what used to be two things kind of become four things because N1 N2 and then there's some more that's usually what you deal with and this is a quantity that if you find for one source and then you find it for another source you can just add them all up I mean they're all you don't have to do it separately they're all additive so the way the quantum equations would look like think so let me just quickly give you an idea about the first equation on top and the third equation the electron density equation for example so as I said I'm not going to go through this very much just want to give you a flavor of where this came from because this is just the generalized matrix version of the simple thing I was doing you see that's it and the way it works would be something like this that basic Schrodinger equation looks like EI minus H times psi equals 0 that's the usual Schrodinger equation you would see in textbooks then we say well now you have to connect it to the two contacts one contact gives you a sigma 1 another contact gives you a sigma 2 two contacts and these are complex numbers in general and the imaginary part of it is this gamma the broadening so usually you'd write gamma 1 is equal to I sigma 1 minus sigma 1 dagger in the imaginary part you do something like this and the other thing is I now need a source term over here S1 now from here I could write psi is equal to Green's function times S1 where this Green's function by the way nothing to do with conductance yesterday I was using G for conductance and this G has nothing to do with conductance then you just get one number right so that would basically just give you the sum of the two but you really want this full information when you calculate Gn so what I mean by that is this is an important point this is a 2 by 1 this is a 2 by 1 so when I multiply the two things I get a 2 by 2 but if you multiply the other way you'll get a 1 by 1 you just get a number which will actually be just your total number of electrons in the whole thing and if that is all you want that's what you'd calculate but in the NEGF formalism usually what you are calculating is this full correlation matrix so that equation up there this second equation that is really tells you this this is what you are looking at and the way you get the Gn then would be something like this psi psi dagger is equal to G S1 S1 dagger G dagger so psi is equal to GS1 and what is G? it's the inverse of that that's the green function of the system it's the inverse of this matrix and that's the first equation up there G is equal to EI minus H minus sigma's inverse that's it so it tells you the response of the system if I hit it with a source S what wave function is created inside that's the response that's the green function of the system so how much the wave is high in the quantum picture it's not a solution we are doing a one electron wave function so in a way what I did is the special case of when H is just a one by one you know this can be a matrix of generally 10 by 10 100 by 100 whatever it is and what we just went through is the case when this was one by one that's all there is a number of operators at all I mean if you could interpret it as a second quantized operator et cetera but that's kind of not necessary here, not beyond it how did you write it? because it's our last line how did I write what? a picture how did you get the second? there's sort of an equation oh this one? yeah no all I am saying is we are motivated by the semi-classical picture to do it and then I am saying let's take the standard Schrodinger equation and modify it by adding a few terms so that it has the physics of the left one right so the gamma one and the gamma two will give you the new one and the new two and then you need a source term and how do you find the strength of the source term? well the best way is to kind of again match the results you know in this context if you did it from more from first principles then you would get it automatically but here if you are quickly trying to get the answer go to the limits where you know the answers and say that well the source has to be so much in order to give you sensible answers epsilon minus i gamma plus gamma 2 over 2 this is your having modified equation yeah the way I would say the epsilon is the h the gamma one is the sigma one gamma 2 is the sigma 2 right when you add f1 and f2 all that gives you an extra whatever you can approach yeah the point I was trying to make is here you should not be adding f1 and f2 as it is because while on the left hand one you could easily add you could put in the f1 and f2 here and it would all be done and you would get a current that it depends on f1 minus f2 all done no problem at all if you add f1 and f2 you do not get the right physics at all so here you should not be doing it on the other hand you can do it in NEGF NEGF and that is why I guess I would say that why we need NEGF rather than do Schrodinger is that in Schrodinger you have to be very careful about these tricky points that different sources are incoherent well the NEGF kind of handles that nicely that is what it is I just need a few applications so on the right hand side equation to the diagram there what is Kami-Rama one and what is in the Schrodinger plus so these sigma1 and sigma1 is really this thing this is sigma2 that is this one and when I write a plus I mean dagger that is conjugate transpose so for one if this was a number this would be like taking a negative imaginary part of sigma so basically if this was minus i gamma over 2 that would be gamma now in the matrix sense of course what you are looking at is what you might call the anti hermitian component of sigma1 in the matrix sense so these are all matrix generalizations of this simple this as I said won't get clear in this time if you haven't seen it already in this room I think there are some of you who have seen it already whom I can't add much to probably in this direction and the others I am just trying to get across the conceptual picture behind it rather than the details of it what is the significance for the after maybe tomorrow when I talk about spin that is one very good example where you can appreciate what those terms are about so I will bring that up tomorrow a lot of times you are just looking at the electron density in which case you just pick up the diagonal terms in fact if you don't have to calculate the off diagonal terms computationally often that's good because there is so much less that you have to store and all that and it makes things much easier actually if you can just ignore the rest but the equations themselves of course give you the whole thing though the equations I had up there and what I sorry what I was trying to explain to you was that what I am trying what I got to you was that first equation the green function what that means is like e i minus h and then this sigma's inverse and why is that important well because that times psi is equal to the source and so psi is equal to inverse of this times the source and the inverse of this is what you call the greens function that's it that's the significance of the top one and what I am trying to get across now is the third one that the g n equation where that came from yeah in this context I would say the motivation is just the one level thing where if this was minus i gamma over 2 this would be gamma you see what I mean that if that gamma in the one level problem the gamma represented the in this one level problem the gamma represents the rate at which it leaks out so if it leaks out at a certain rate then the corresponding self-energy has to be minus i times half of that self-energy meaning what we put into here and in general if you want a leaking out described by some kind of a matrix what should be here should obey that relation so this is so I am saying it is of course hard to go from a one number to a matrix it is from the matrix to one number is a unique process from one number is hard to deduce what the matrix version would look like but this is just to motivate it that's all I am not really deriving it here but what I want to show you very quickly is how do you got the third equation right and that is as I said psi equals gs so when you take s dagger is gs s dagger g dagger and this thing is the strength of the source term and here in the one level case sorry in the one level case what you get is the strength of the source term is proportional to gamma which kind of makes sense that if you couple something strongly if the contact is coupled strongly the corresponding source from the source is also stronger so gamma is the rate at which it goes in and out to source and in general also this is true that this is proportional to gamma 1 and so the expression you get then is gn is equal to g gamma 1 g dagger so just from here if you accept that the strength of the source is gamma you will get that and my point is that here all the different sources like f1 f2 etc they can be superposed so what I mean is I was trying to make the point that you cannot just take s1 multiplied by square root of f1 s2 multiplied by square root of f2 you couldn't do that but here what you can do is you can multiply this by f1 and then have a g gamma 2 g dagger multiply that by f2 etc and you are done that's the and that's like the third equation there actually gn if you look at it oops sorry I guess I moved off the third equation therefore the electron density gn is g gamma 2 g dagger f2 plus g gamma 1 g dagger f1 okay so the point I am trying to make is these are the general equations of NEGF and if you are trying to seriously learn NEGF I would say there are two parts to it one is understanding where these equations come from and the second is how to use those equations and those are almost disjoint you can do that in any order almost right and because in terms of how to use it it is almost like well once you know how to write the matrices you can go ahead and use it and I said you know that's the power and the danger of NEGF that you know it's kind of scary how much you can calculate without understanding anything and but nice thing about it is that once you know how to implement it if you do it right it doesn't take too much thing you don't have to understand everything before you start looking at real calculations and so you can actually use the calculations to improve your understanding of things well you know what would happen here well let's try it out almost use it in that mode so let me just spend the next few minutes showing you two examples two examples or maybe one example depending on the time of things that would come out of this formalism as I said I didn't really derive it for those who haven't seen it before I mean there's no way we could do it you know it could sink in and this time it takes much longer than this to see all this properly but what I tried to motivate is exactly the question I always get from people is well if I want to do quantum transport why can't I start with Schrodinger equation why do I need NEGF and the point is just this yeah you could start with Schrodinger equation in a way but then you have to figure out how to put in the contacts because what you learn in all your quantum mechanics courses is lot about the Hamiltonian not much about the sigmas about this contact part of it and the simplest thing is the real contacts which as I said usually in transport theory in the past nobody even worried about sigma 1 and sigma 2 historically when people are dealing with big conductors the physics of resistance was all in the sigma s and that's a whole lot harder to write down actually the sigma 1 and sigma 2 those are relatively simple so as examples one of the examples I often use is this one that's called that I've shown here two problems that's done here one is a wire so you have this channel sorry and somewhere in here there is a big obstacle and that's that delta function in the middle you know which is a big potential thing that whenever electrons hit it they tend to get reflected you know so they don't quite transmit through it very well but they transmit somewhat and so you calculate the conductance function or this transmission that we talked about normalized conductance as a function of energy and what you get is the blue curve in there that is as you go higher in energy you can get through the barrier a little better so the transmission or the conductance improves so that's that but what is striking is when you put in two scatterers there so you got two of them and then you use this formalism and of course this is based on Schrodinger equation which affects in it and so on and so now you find that at certain energies you actually get peaks that's that red curve you see you're looking at the conductance at certain energies you have this very sharp peaks which almost show perfect conduction you see this is a wave property this is something that would never come out of a standard of any particle picture you know particle picture is like a highway you know you have a construction zone sometimes people have getting through it and you do not alleviate the traffic problem by building another construction zone in front of it but when it comes to waves that's exactly almost what's happening you have a barrier and you put another one at the right energy it actually helps you get through why because the reflection from the two interfere and if the wavelength is right it will cancel out but in particle terms there's no way one can get through two barriers better than it can get through one under certain conditions they may be equal but it couldn't be better than that so that could come so if you're at an energy like C you can see you have perfect transmission through the two barriers and if you're at a place like D then it is much less than what you expect yes sir so this is a wave property that would actually come out of this kind of calculation and this wouldn't be coming out of the discussion we had yesterday the semi-classical transport where I said well if you had two barrier where we had a distributed thing so anytime there is when you had a distributed scattering process what happened was the occupation of the positive going things went down gradually and as we said the transmission goes as lambda over L plus lambda now that is all semi-classical result of course that is when you neglect interference but once you include interference that's not what will happen if you have these two barriers it is not like they would necessarily behave in that semi-classical way so this is one thing that quantum formalism gets you now what if you include say many scatterers so again when you have many scatterers of course the semi-classical picture would give you this, this is what we know with many scatterers, there would be places where it would suddenly go down a little bit you know this is localized scatterers yesterday we had distributed scatterers but with localized scatterers it would go down in steps etc and at the end of it if you use that classical thinking you get that red curve and again a smooth curve no sharp function if you know what it is for one scatterer you can figure out what it should be for six scatterers but quantum mechanically of course the answers depend on how those six scatterers happen to be placed now question is in the real world would you see this and the answer is most of the time at least at room temperature what you really see is the classical one you wouldn't see the quantum I mean and it's not because there's anything wrong with the wave picture it is just that usually there's a lot of incoherent processes a lot of deep phasing going on I mean says sure you have one scatterer here and another one here and we are saying they will interfere but the thing is you have at room temperature all these atoms are jiggling around so when an electron goes through this it does not see a fixed potential it sort of sees a big jiggling thing and so its phase is what it sees at this instant is different from what it sees a picosecond later or another picosecond later and anything we are measuring of course is averaged over nanoseconds at least probably micro or milliseconds and so at the end of the day you do not see much so in any GF you could put in a sigma s the scattering term in the right way and it would actually get you the classical curve so this is one thing I'd say is a good again a good homework problem to go through actually that the quantum one the one I wrote there that one only has a sigma 1 and a sigma 2 but no scattering sigma but by putting in enough of a scattering sigma you could turn the quantum curve into a classical curve and that way any GF allows you to bridge that entire regime from the pure quantum calculation to what you'd expect out of Boltzmann the classical curve is essentially what you'd get out of Boltzmann so this is a good example and those quantum effects though at low temperatures are sometimes seen because I said that it always averages out over time now that's kind of true but in big conductors there is also what people call ensemble averaging which means they say that well if you have a big conductor like this it's kind of like as if multiple conductors and if this one has one kind of interference that one has a different one and what you're measuring is an average so 30 years ago of course everybody assumed that anything you measure is some kind of an ensemble average and it's no point calculating sample specific things but what was found is that when you started looking at small conductors you could often see sample specific things so for example with certain small conductors you could measure the conductance versus gate voltage so let's say with a gate voltage I change the carrier density and what is supposed to happen is it's supposed to go up because carrier density is going up but with small conductors at low temperatures they would see something like this I mean average would go up but there will be a lot of these conductance fluctuations and the interesting thing was you think well that's just noise but the thing is it's not noise in the sense it's reproducible you go back the next day it's still exactly the same pattern so it's not noise in that sense and the other interesting thing they found was in those days I remember they would take this conductor bring it up to room temperature you see the measurement is being done at 4 Kelvin or lower but then they bring it up to room temperature and it's a different pattern something else and the understanding is that what happened in the process of bringing it to room temperature and taking it back is that the impurity is moved around because this pattern of course depends on exactly how these scatterers are arranged you see if they if this one is not here but another but say it moves away by a fifth of a wavelength then of course you'll have a different pattern all together but as long as you keep it at 4 Kelvin you could measure day after day and still be seeing the same thing so these were experiments like in the early 90s which actually established this reality of this quantum interference in the sense it is measurable on the other hand it's true that at room temperature much of this doesn't play any real role and there's this entire field of course of localization which is related to that so you know that if you take scattering in that quantum one there's no scattering what is the scattering center let me make that let me qualify that when I say scattering what I mean is that when we model this there's an H and then there's a sigma now these scatterers I'm putting into the H so in the Hamiltonian itself we have certain potentials now to destroy that interference what you have to do is connect this to a sigma s in the sense that those would be like incoherent processes that actually change the phase relationships and all that so there is a big difference between putting something in a sigma and putting something in a H so when I say there is no scattering in the quantum calculation there is a sigma s the sigma s that kind of serves to deface things and destroy phase information and all that but as long as you put in a specific potential in H there's no defacing it's a coherent thing it will give you interference as everything that's what that quantum calculation is about so this is a good example to go through if you are trying to learn any GF because basically you would have to it's a one dimensional problem how to write the H, how to write the sigma s etc it's a good one dimensional problem and if you come to this then I'd say the next step usually is I recommend is trying the Hall effect that's a nice two dimensional problem to do but it's harder because you know in two dimensions then you have to learn how to write the H, how to write the sigma s etc but what you you can use any GF to actually calculate the Hall voltage as a function of the magnetic field and it will you know show you this classic results that you know as you know one of the striking results that has been observed in two dimensional conductors is that this Hall resistance which starts out linearly with magnetic field but at high magnetic fields shows these very exact quantizations and so on okay that's about since it's almost 10 o'clock let me stop here and we'll continue in the next lecture go ahead for the contacts in a yeah I guess I'd usually say that the rate at which electron escapes let's say is 10 to the 12th per picose 10 to the 12th per second the corresponding gamma would be a millivolt or so kT is 25 millivolt at room temperature but when you're actually doing this one dimensional wires what you want to have is perfect coupling to the contacts and I think those gammas are bigger than the one millivolt I think is not a perfect coupling to the contact so there I think what needs to happen is it kind of matches the tight binding hopping parameter so what I mean is usually in a tight binding model there is this T0 which couples different levels and I think the gamma is of the same order as anyway so we'll continue at 1030