nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L2.4: NEGF Equations
========================================
 Slide 1] Welcome back to unit 2 of our course. This is the fourth lecture. 

[Slide 2] Now what we did in the last lecture was to develop this quantum model, which includes inflow and outflow. So, at the level of the Schrodinger equation you have this additional term, the sigmas and the S's. And what we showed was they give rise to these inflow that's S1 dagger AS1. S2 dagger AS2 and the outflow of this side dagger gamma psi dagger gamma 2 psi. And this gamma is this antihermitian part of sigma A that's like a density of state related to the greens function. So, these were all the concepts that we introduced in the last lecture. And in this lecture what we do, want to do is put it in the form of the regular and NEGF equations, which will be the standard form that we'll be using and referring to later. But before we get started, just wanted to remind you of something I mentioned in the introduction and that is that when you look at this number of electrons I said you could look at psi dagger psi. And psi is let's say the column vector with 3 components. If you let's say have a lattice with 3. I mean in practice of course it may be 10 or 100 or 1000. But, illustrate the point. Just think of the lattice with 3 points and then psi has 3 components and the psi dagger is being those 3 things congregate. And it's transposed so it's a row vector. Now when you multiply the 2, what you get is 1 by 3 times 3 by 1. Of course for matrix multiplication those 2 terms must match and what you get is a 1 by 1, which means we just get a number when you multiply the 2. Now you'll notice here we have psi dagger gamma psi. The gamma is a matrix. But the psi dagger gamma psi that when you put a gamma in the middle we'll be putting in a 3 by 3 here. That's fine. But when you multiply it you still get 1 by 1. So, the point is that's still a number, single number, the 1 by 1 when you multiply it all out. Now on the other hand if you look at psi, psi dagger then the order is reversed. And now when you multiply it, it's 3 by 1 times 1 by 3. And so what you get is a 3 by 3 matrix, which we call Gn. And I guess the 2 Pi else here couple of words about it later. But basically this is, this Gn is like a matrix version of this number of electrons. And here I guess I get 1 number which is like the number of electrons. Here I get a matrix, but if I add up the terms on the diagonal then I will get the number. So, the number of electrons is like the trace of this matrix. And other matrix will use in their NEGF equations is the similar quantity but related to the source term S. So, SS dagger is what we will call sigma in and over 2 Pi. And that again will be a 3 by 3 matrix in a problem like this. 

[Slide 3] Now, now let's look at our expression for current. As I said if you looked at contact 1 you have an inflow and an outflow. So, if you write it out the current will be, this is the inflow, that's the outflow. And each one of these is actually a number as we just discussed. So, if it's a number I could always call it the trace. Of course not usually trace means you add up all the diagonal elements. But if you have just a 1 by 1 matrix, which is a number, then the trace is basically that number itself. So, this is a number and I could write it as trace. Of course it kind of looks silly why I'm calling it a trace, but you'll see that in a minute. But let's do that and then I know this rule that once I have traces and I'm taking a trace I can always move these things around as long as it preserves the cyclic order. So, what I could do is, here the psi dagger instead of being up front here, I could put it at the end. So, you have a psi dagger up there. Similarly here it's 1 dagger instead of being here I could move it there. Now notice that now the trace actually makes sense. What I mean is if we think about it if the psi was a 3 by 1 column vector then this is actually the 3 by 3 matrix. That's a 3 by 3 matrix. So, the thing we have here is actually the 3 by 3 matrix. And so it makes since to take it's trace. See here as I said it was a number the trace was kind of a cosmetic thing that almost looks silly. But, the reason I did that was so I could move it around. And once you have that you see I could replace psi, psi dagger with Gn over 2 Pi. SS dagger with sigma n over 2 Pi. What happened to the 2 Pi? Well I had this h bar down here, 2 Pi h bar is h. So, now it has become h over here. Another thing to notice, that this quantity here is like the flow of electrons per second. But if you actually want charge current or coulumbs per seconds then you need to multiply by the q as I have done here. So, that's this expression for the current. So, now you see from this picture that we had from the last lecture I could turn it around and write it like this. You know you have this channel inside in which you have Gn, which is the matrix representing the number of electrons. I have A which is the matrix presenting the density of states. And the rate at which things come in depends on the trace of the sigma in times A. The outflow is the trace of gamma times Gn. So, physically these quantities are just like what appear in our semi classical model and you can see the correspondence. You see A is like density of states. Gn is like number of electrons. The gamma is like the nu, this outflow. And the capitol S is inflow is like the sigma in. This picture is important to keep in your mind because as we do this NEGF equations there will be so many symbols that unless you have a good physical picture it's hard to keep track of them. Okay? So, that's why I spend so much time trying to get just this physical picture in terms of inflow and outflow clear. Okay? 

[Slide 4] Now the next thing that you wanted to show quickly is that, and I say that Gn is like n and A is like density of state. So, what you might expect is, if I take the trace of Gn that will tell me the number of electrons per unit energy in a given in an energy range. And if I integrated over energy then I should get the total number of electrons. And that you might expect based on psi, psi dagger and this 2 Pi part effect kind of comes from this Fourier transform type of relationship. Mainly you see if you had a psi in time squared integral there's this Parseval's theorem that is in quantum mechanics energy and time alike Fourier transform variables. So, when you go from one to the other there is this Parseval's theorem of the same, which relates the integral dT psi D squared to integral dE psi E square. And there's usually the 2 Pi. So, that's kind of the origin of the 2 Pi there. Doesn't matter too much as you'll see, I mean kind of cancels out. But, you have to remember when trying to find the number of electrons from Gn, you have to remember to divide by 2 Pi. Now similarly when you look density of state you expect the same thing, that if you have A and you integrate the trace over all energy and A of course we defined as GR minus GA so it looks like that. But what it should be equal to is the total number of states. That means if I had started from an h matrix, that let's say was 100 by 100, then as you know we'd see it has 100 eigenvalue that are 100 states. And after adding all the sigmas and everything else when you calculate A and you take the trace integral over energy then this answer should be 100. Okay? And that's not obvious at all but we can show it. But I won't go into it in the lecture. But the main thing then to keep in mind as you try to navigate through all these symbols is that this A is like density of state, Gn is like number of electrons. Gamma is the rate at which things leak out. And sigma in is the rate at which things want to, the source term that electrons want to come in. Okay? 

[Slide 5] Now once you have that picture you'll remember from our lecture 2 on semi classical model I argued that there has to be a basic relationship between S and nu. You see the S that comes in and the nu that tells you the rate at which they flow out. If one is big the other must be big also. If it's easy to flow out it must be easy to flow in. And the way we argue it is, think of a special situation where you have only 1 contact connected. And in that case the number of electrons to the number of states ratio must be equal to the Fermi function in the contact. That's the basic law of equilibrium statistical mechanics. That if in your contact the certain energy range, that's a 50% of the states that occupied. And then if you have a channel come in contact with that at equilibrium then the channel also states will only be 50% occupied. So, that requirement kind of leads to this relation between S1 and nu1. And from the analogy you'd say, well that suggests that the sigma in in the quantum picture from contact 1 should be gamma 1 times F1. Similarly for contact 2. Sigma in2 should be gamma 2 times F2. 

[Slide 6] Now we can move on to the NEGF basic NEGF equation that I think we had discussed in the introduction. That what we want to do is start from the basic, this modified Schrodinger equation in terms of psi and s and try to get the NEGF equation in terms of psi, psi dagger and ss dagger. That is in terms of Gn and sigma in. And the way you do that is, first you take this, invert it to right psi as greens function times s. So, that psi dagger is s dagger times the advanced greens function. I see the retarded Green's function defined here. Advanced 1 is its conjugate transpose. So, then when you put in look at psi, psi dagger, you got GRs and then S1 dagger GA. This is what you call Gn over 2 Pi. This is what you call sigma in over 2 Pi. You'll notice that 2 Pi is just canceled out. So, you get that, this NEGF equation, which relates the psi, psi dagger to the ss dagger. The important thing is that in the Schrodinger equation it would be kind of tricky if you put in 2 sources, because then you would be getting interference between those sources. But in the NEGF equation here you can add sources. So, you can say, what is sigma in if you have multiple sources? Well just take the sigma in due to the contact one. Sigma in two due to the contact 2 add them up. And as we just discussed this one is gamma times F, gamma 1F1 that's gamma 2F2. 

[Slide 7] Now one interesting check to make, and this gives us a very important identity that you should be familiar with. It's that supposing is these are general equations on the non-equilibrium conditions. But, supposing we have an equilibrium situation, which means the 2 contacts have exactly the same Fermi function. F1 and F2 both equal to F0. Well in that case sigma in would be gamma 1 plus gamma 2 times F0. And the sum of the 2 that's what we have called gamma. So at equilibrium this is gamma times F0. Well let's put that into this equation. So, you get GR gamma GA times F0. Remember these are all matrix's, where F0 is just a Fermi function. It's a function of energy. So, at the given energy it's just a number, 0.5, 0.4, whatever it is. But as these things are all matrixes. Now the point is that equilibrium of course what we expect is that the number of electrons would be equal to the density of stage times the Fermi function. That as I mentioned is this basic law of equilibrium statistical mechanics. But if you have a channel in equilibrium with a contact then the given energy range, you know the fractional states that are occupied. The fractional states that are occupied must be the same. So, what should happen is this should be equal to A times F0. But, the question is, is that true? Is A equal to GR gamma GA? Well, it's not obvious that this is guaranteed, because the way we define A was IGR minus GA. That doesn't look the same as that. What I'll show next with a little algebra is that this is indeed identically equal to that. So, how do we show that? Well first thing is we start from the definition of GR. We take the inverse of both sides. So, we can write GR inverse as EI minus H minus sigma. Next we take the conjugate transpose of GR inverse, which is EI minus H minus sigma dagger. With here also we are taking the conjugate transpose. I remains I, H is Hermitian so it remains H. But sigma becomes sigma dagger. And the point of  all this, that here is inverse and then dagger. But in matrix algebra you've probably seen, if not you can work it out. But instead inverse and dagger you can reverse the order. You could have the dagger and then the inverse. And then the GR dagger would be GA inverse. So, GA inverse is equal to that. Next what we can do is subtract from here this one. So, GA inverse minus GR inverse is equal to this minus this, which is sigma minus sigma dagger. And if you remember we had the definition of gamma as I times sigma minus sigma dagger. So, sigma minus sigma dagger is minus I gamma. Now what you can do is multiply from the left with GR and multiply from the right with GA. So, you have GA inverse when you multiply from the right with GA it goes away. It becomes identity. And then multiply by GR so you get GR. The second term the GR, GR inverse is just identity and multiplied by GA becomes this GA. And here your minus I gamma, then you have the GR on the left, GA on the right. So, you have this expression. And that then leads to this expression for A. A is I times GR minus GA. When you multiply by I the minus I drops out. So, you get GR gamma GA. And interestingly is GR gamma GA, but it could have been GA gamma GR just as well because instead of multiplying from the left with GR and from the right with GA I could have done just the reverse. I could have multiplied from the left with GA and right with GR. And then I would have got that. So, that just for completeness this is an identity that is often very handy. But in this context of course the main reason I was doing it was to just that the NEGF equation does give you the correct equilibrium result when all the Fermi functions are the same, because GR gamma GA is indeed equal to A. So, when you apply to an equilibrium problem it gives you the expected results that the number operator here, the Gn, the matrix that represents the number is equal to the matrix that represents the density of state times the Fermi function. Okay. 

[Slide 8] So, then to sum up then what we did here was develop these NEGF of equations. See which is terms of, you know you have a channel described by H and you have the sigma 1 and the sigma 1 in. Sigma 2, sigma 2 in. That describes the connection to the 2 contacts. And in terms of these you can interpret it term as this inflow and outflow. And the basic NEGF equations are given here. And of course in navigating through all these symbols it helps to keep this analogy in mind. This analogy of the simple semi classical picture. And one last point  is we have generally been talking about 2 contacts, sigma 1 and sigma 2. But a very important part of this NEGF of formalism, this way of doing it, is it also allows you to include what you might call conceptual contacts. That is if there is interactions of the electrons with the surroundings in the channel so that it loses momentum or loses spin, etc. Those could be represented also with sigma 0 and sigma 0 in. Kind of like these, sigma 1 and sigma 2. And those just have to be added on otherwise the same basic picture would still work. But the one important difference to keep in mind in this context is that when it comes to sigma in 1 and 2 they are related to this gamma 1 and gamma 2. So, what that means is when it comes to the contacts, when you look here it looks like you've got 2 things, sigma 1 and sigma 1N. You know one related to this outflow the other related to the inflow. But, you don't really need to figure out both independently, because if you know sigma 1 you can find out gamma 1, multiply by the fermi function and you would have the in sigma. Same with contact 2. But when it comes to this conceptual contact the one that represents interactions there this argument doesn't work, because that contact is not necessarily in equilibrium with anything. That this is a conceptual contact. It just represents this interaction that as you view interaction as if the electron went out into some contact and came back in. And this is the picture  due to Buttiker. And in fact Buttiker's model what you do is model it with a real contact as if you had 1, 2, and 3. But in the NEGF method which I consider more powerful and more general, you right it does the sigma 0 and sigma 0 in. But you keep in mind that this is not a contact in the sense there is no Fermi function associated with it. And so you cannot write the in term easily in terms of gamma and F. So, you, I couldn't write sigma and 0 equals gamma 0F0. So, what you can do, that's what we'll discuss I think, I think in the ninth lecture when we talk of dephasing. Main point to know is that for the dephasing contact, this one, you have to ensure that whatever you do to figure out your sigma 0 and sigma they satisfy this relation. Why? Because at any contact you see, that's the inflow and that's the outflow. In these contacts this doesn't need to balance that because these are physical contacts where actual current is flowing. So, you could have 10 electrons per second come in and 5 leave. But, in this conceptual contact it's floating. Electrons have no place to go in the sense there's no real contact there. And so whatever the inflow must exactly balance the outflow. And there is no net current because if they're unequal there's no place for that net current to flow. 

[Slide 9] So, that then kind of summarizes what we did here. You see this idea of a channel described by H and the connection to the contacts. The physical contacts and the conceptual one contact described by the sigmas and there is the sigma and there is the sigma in. One that represents an outflow, the other that represents the inflow. And you can have this picture in your mind of the expression for outflow and inflow. And as I mentioned a very important thing to keep in mind is this Gn is just like number of electrons, A is like the density of states. And this kind of summarizes the equations we'll need. So, what we'll be doing after this in lecture 6 through 9 is applying these 2 different 1D examples. This would be like that starting point. And in the next unit, unit 3 we'll be looking at more difficult examples because these basic equations are all right here. Now the next lecture is a bit of a detour in the sense. We'll talk about what's called this current operator. It is here we have focused entirely on charge currents because that's the quantity of primary interest. But these days there's a lot of interest in other, the current of other entities like spin for example. And so I would like to say a few words about a general current operator that allows you to calculate the flow of any quantity. So, that's what we'll be doing in the fifth lecture. But after that we'll be looking at examples that apply these equations. Thank you.