nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L3.7. Golden Rule
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[Slide 1] Welcome back to Unit 3 of our course on quantum transport. This is Lecture 7. 

[Slide 2] Now as you know, in these last few lectures because we first developed this general way for calculating the self-energy. And then went through a couple of examples involving this graphene and magnetic fields on how to use this general method. Now in this lecture, we will do something different. What we will try to do is show how this result connects up to a totally different result that we haven't talked about but something that is widely used. If you have taken a course in Quantum Mechanics you've probably seen it. And people use this in the context of devices, in the context of analyzing all kinds of physics. And that is called this Fermi's golden rule. Now in the context of devices the way you usually apply it is that the way we think of conductors is that electrons are moving along and usually they have a certain velocity, certain momentum corresponding to a k. And then when they hit some kind of obstacle which could be a defect or an impurity. It sees a scattering potential due to that defect or impurity and as a result it's momentum changes. It turns in some direction k prime. And the rate at which electrons get scattered from k into k prime is given by this Fermi's golden rule. And the scattering rate which is like per second is given by this tau k k prime squared which is called the matrix element. And the way you evaluate it is by looking at-- by doing this integral. You take this potential, the scattering potential, you put in the-- one of the wave functions and you put in the other wave function the initial and the final wave functions. And then you do that integral either tau k comma k prime. And then there is this delta function which tells you that in ordered to be scattered from k into k prime the initial and final energies must be equal. And this is a result that is widely used. In fact, there is a generalization of this to time varying potentials like how electron gets scattered by things like phonons which we will come to a little later in this lecture itself. But let's start with the elastic case. Elastic meaning initial and final energies are equal. So as long as this potential is a static one the scattering is elastic. Now what does that have to do with self-energies? Well one thing we have discussed is this broadening function, gamma which is given by I guess you could call it the imaginary part of the self-energy. In a matrix sense it is like the anti-Hermitian part of this self-energy matrix. But this is this imaginary part, you could think of it as the imaginary part. And the point is that this gamma, you could view as like the lifetime of a state that is , what this gamma tells you is that if your electron is in a certain state, how long it stays there before it is scattered to another state. So that is what I mean by lifetime. And there is this Uncertainty Principle which says that the longer it stays in a state the sharper the level is. So in that sense the, breadth is smaller. So this is the breadth of a level, sort of and it is inversely related to how long it stays there. And so in that sense you can imagine this would have some connection to the scattering rate because the scattering rate is also a measure of how long an electron stays in a particular eigenstate k. And what we will show is indeed from this gamma you can get these standard golden rule expressions. Of course, we won't go into all the physics that you do using these expressions. I just wanted to connect up to this result because usually in standard quantum mechanics courses this result is derived in a very different way. And it is a very widely used result. And I just wanted to show that it connects up to what we have been doing here. Now for starters let me also note, again this expression that we have talked about that if in place of sigma you put in the tau g tau dagger and then take the conjugate transpose of sigma, put it in, you will get an expression looking like this. And this g as you know is the Surface Green's function. Whereas what appears here is this i g minus g dagger which is like the surface spectral function. Which means like the surface density of states. So this is the form we will be using: gamma is equal to tau a tau dagger. 

[Slide 3] Now how do you use this? Well the idea is you need a very different picture now. So far we have been doing everything in real space. But remember these are matrices and so you don't have to use this real space representation necessarily. You could work in terms of any basis functions. So supposing we are in these basis functions involving e to the power i k dot r type of states, what you call plain wave states. So that is this k. So the electron is in this certain state k which you think of as e to the power i k dot r. And it can scatter into all these other states which are kind of like a contact. Or if you are in one of these other states you can come in. So mentally these k's and k primes you could view. This has been your device, the place where the electron actually is. And all these other states are kind of giving you a continuum of contact states that it can scatter into. And you could still use the same formula. So you could go ahead and write: gamma for state k and we are interested only in the diagonal element. This is an eigenstate k. We are interested in the gamma-- the diagonal element of the gamma for that state. And that state has a definite energy so this E would be Ek, whatever the energy of state k is. And tau a tau dagger. And now I am just using the rules of matrix multiplication. You have k comma k here so the first index must be k, the last index must be k. And in between you have these k primes and what you are assuming is that the spectral function is also diagonal which is true if these are eigenstates. So you have this tau k k prime and then a k prime comma k prime tau k prime k. And sum over all k prime. So that is like the expansion of this assuming that a is diagonal and you are looking at the diagonal elements. Next, you note that dagger means conjugate/transpose so tau dagger k prime comma k is really tau k comma k prime conjugate there. And so what you have is tau k comma k prime and the same thing conjugate here. So that when you multiply this you would get the magnitude squared of tau. And this a, that is like the spectral function associated with the state k prime. And that is actually 2 pi times the density of states. And if you have a single eigenstate then its density of states if we assume is very sharp. Is delta E minus, well whatever the energy. So it is like delta E minus Ek prime and since we are interested in the energy associated with k, I put in Ek here. So instead of this a, we are putting in 2 pi delta, 2 pi times the density of states associated with state k prime. And this is tau squared and then I put in an h-bar here and a 1 over h-bar here. Why do I do that? Well because gamma is like h bar times the inverse lifetime so I kind of want to pull out an h-bar that way what is in here I can interpret as the inverse lifetime. So this then you could view as scattering rate. That you have a state, electronic state k and the rate at which it scatters out into various k prime states which are all these contact states is this S k comma k prime. And if you add up all those rates that tells you the total rate at which electrons scatter out of k. And so h-bar times that scattering rate is the broadening of the level k, gamma. So that is how you'd interpret it. 

[Slide 4] So from this point of view then we have again this expression for this scattering rate which looks just like the standard golden rule expression. So the point I was trying to make is how from what we have discussed you could get the standard golden rule expression. Also if you want this matrix element then the method for finding the matrix element is you do this integral over volume. You put in the scattering potential and then you put in the initial and final wave functions. This is something that we discussed in unit 1 in lecture 1.3. We haven't used this very much because for our tight binding model usually we have been obtaining the-- these parameters of these-- of the tight binding model by comparing with experiment. We actually haven't been using this first principles relation. But if you look at this lecture, I think I introduced this basic relation which you could use in this context. If you wanted to evaluate it you'd say, "Okay the initial and final wave functions are displaying waves, normalized to certain volume." And then if you multiply it out you'd get that this tau, the matrix element is kind of like the Fourier transform of the scattering potential at the value beta where beta is the difference between the initial and final wave-vectors, etcetera. So this part I am not going into in any detail. These are things you will see in many courses in many different contexts and I am just trying to connect it up to our NEGF result. 

[Slide 5] So we will actually use a more general version of this that is not k's and k primes. But arbitrary eigenstates m and m prime. So often in a solid you have this model where the eigenstates are plane waves but that is not the most general thing. So in general the eigenstates could look like-- could be different like in a hydrogen atom the eigenstates would be-- would not be plane waves, they would be localized things. So whatever they are the point is the golden rule applies in general. That the scattering rate from a state m to m prime is given by something like this: Where this matrix element is given by this wave function of m, wave function of m prime and the scattering potential. Now the version that you often see used involves time varying potentials. So if you have an impurity or a defect that is like a scattering-- that is like a static scattering potential. On the other hand when you have phonons which represent the giggling of the atoms. Then what an electron sees is more like something that is time varying. And in that case the result, the Fermi's golden rule looks something like this. It has one term that looks like this. Where this N of omega is like the number of phonons that are present with a certain frequency omega. And this tau that is the matrix element due to 1 phonon. So if you had just 1 phonon whatever potential you'd have got gives rise to a certain tau. You could calculate that and then if you have N phonons the scattering rate would be N times bigger. And this delta function here, note that what it ensures is that the initial and final states have the same energy here because delta function is non-zero only when the argument is zero. Which means when Em is equal to Em prime. But in this case Em is not equal to Em prime. Rather Em plus h-bar omega is equal to Em prime. And you interpret that as saying, "Well what the electron did was it was in state m and it absorbed the phonon of energy h-bar omega and went to the state Em prime". So this is what you might call the absorption term. So usually Em plus h-bar omega equals Em prime. And in that case the argument is zero so the delta function gives you a non-zero result. There is also an emission term that goes with it. Here it is Em minus h-bar omega is equal to Em prime. And the non-obvious result is that emission rates usually are larger than the absorption rates. This is absorption. Emission is proportional to N plus 1. And usually we call that stimulated emission and spontaneous emission. So this is of course a non-trivial result. I think we may have talked about it a little bit in part A in unit 4 when we were talking about second law in equilibrium statistics. But as I said, we are-- this is not meant to be obvious at all. Now let me first quickly justify why you might think that in the presence of a time varying potential the energy would change. 

[Slide 6] So the way you can justify it is by saying, "Well, you know, the Schrodinger Equation looks something like this. So when you have a scattering potential, you have a term here that involves the scattering potential times the wave function. And let us say the wave function looks like this where the electron is in a state whose energy is Em. Now the scattering potential is cosine omega t which you could write as the sum of 2 exponentials. So when you multiply these 2 you get a time term that looks like Em plus h-bar omega. You see that is minus i Em and minus omega and when you add it up it looks like this. So the result of multiplying with this term is to give you something whose energy is Em plus h-bar omega. And that is like the absorption term. On the other hand, the result of multiplying this with the other term is to get a term that looks like Em minus h-bar omega and you could call it emission. So in simple 1 electron picture that is how you sort of justify an absorption and an emission term. You see the cosine has a positive exponential and a negative exponential. One of them gives you the absorption; the other gives you the emission. Of course, what it doesn't explain though is why one is N and the other is N plus 1 because after all, cosine is a-- I mean, this potential is a real function. So these 2 must necessarily have the same amplitude. And so if these are the same, 1 must be the complex conjugate of the other. So if they are the same amplitude you'd expect both rates to be equal. Now one way to understand this unequal thing is by adopting a different picture. 

[Slide 7] So let me explain that a little. So our picture right now is if your electron is in the state m then it can absorb a phonon and go to a state m prime. Or it can emit a phonon and go to a state m prime-- a different state m prime. And the rates are proportional to N plus 1 and N. Now, earlier we talked about this elastic scattering when the final and initial energies were the same. What I'd like to explain is how you can adopt a different picture in which you can start from here and actually get that expression. So the way you do that is you have to start with a somewhat different picture than what you are used to. It goes like this: so these are the electronic energy levels m, m prime, etcetera. Associated with having say N phonons present. But the way you should think is that electrons and phonons are like one big system. So electron is in this state and the phonons are in state N. whereas when you emit a phonon you go to a state where you have 1 more phonon. So you go to the N plus 1 phonon state. Now the energy levels of the N plus 1 phonon state are moved up by h-bar omega. Why? Because here we are trying to draw the total energy so here you had only N phonons. Here you got 1 more phonon, so the total energy is increased by h-bar omega. Similarly you have an N minus 1 phonon state where the total energy is 1 phonon less. And now the point is that when an electron happens to be in state m with N phonons present, it could scatter either this way which would be emission or it could scatter this way which would be absorption. In one case, go from N phonons to N plus 1 phonons. In the other case you go from N to N minus 1 because you have absorbed 1 and here you have emitted 1. So the point I wanted to make is that you need adopt a picture like this then you see absorption and emission also look like elastic processes. Elastic meaning where the initial energy is the same as the final energy whereas when you adopt a picture like this, then initial and final energies are different. But in this picture they are the same. And so you can actually come up with this result starting from this picture using the golden rule we had for elastic processes. 

[Slide 8] So let me show how. Consider this red process that is the emission term. But now what will happen is starting from our original result we have tau m m prime squared but you also need another element here which is the matrix element from going from N to N plus 1. And when you look at the argument of the delta function, instead of Em, you have Em plus N h-bar omega, why? Because this state is associated with N phonons. On the other hand, this state is associated with N plus 1 phonons so you have Em prime plus N plus 1 h-bar omega. And now you can cancel the N's so that you have this Em minus Em prime minus h-bar omega. Now, we will come to that in a minute. Now if you look at the blue term that is where you have to look at the matrix element between N and N minus 1. And now again, initial energy is the same but the final energy has an N minus 1 to it. And now if you do this algebra I mentioned the first one becomes Em minus Em prime minus h-bar omega. The lower one becomes Em minus Em prime plus h-bar omega just like the absorption term and the emission term. And these parts of the matrix element that go from N to N plus 1 and N to N minus 1, of course, to talk about it in depth we'd need to talk more about these phonon states and what they look like. But in this context let me just mention the basic idea is that this quantity is proportional to whichever is bigger. So when you go from N to N plus 1, it is N plus 1. When you go from N to N minus 1, it is N. So whichever is bigger that's it. So if you accept that then you see what you have accomplished. You started from this elastic expression and got the inelastic expression, okay? 

[Slide 9] So in closing, what we tried to do in this lecture then is start from results that we have been talking about in this unit in general about this general expression for self-energy and broadening and connected it up to the Fermi's golden rule for elastic processes and for inelastic processes. So what we will do next in lecture 8 is talk about how these inelastic processes can be included into the general NEGF method. Remember, so far we have usually not talked about inelastic processes. As we mentioned, the physics of elastic resistance is much simpler. And we talked about how you can describe transport assuming that electrons go through the channel without exchanging energy. What I'd like to do in the next lecture is explain how you could include such processes into the NEGF method so that you'd have the full method in front of you. Thank you.