nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 3.8: Boltzmann Equation
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[Slide 1] Welcome to Unit 3 again, and this is Lecture 8. 

[Slide 2] And in this lecture, I'd like to introduce you to this Boltzmann equation, which actually is the centerpiece of all semi-classical transport, that is all results we are talking about have the basis. The basic equation is really Boltzmann equation. The way you justify it is usually starting from the Boltzmann equation. And, you know, everything we have talked, the diffusion equation, et cetera, if I had to justify it properly, this is where I would start. Now how come this isn't Lecture 1? Well, that's because this equation is somewhat challenging mathematically. You can see it's-- you know, it's a partial differential equation. And there are multiple variables. There's this z, which is the location, spatial location. And then there's pz, which is the momentum. And in general, of course, the more the independent variables you have, the more challenging it is, numerically and conceptually. And so that's why we didn't quite start from this like in Lecture 1 and then divide everything systematically. Instead, I tried to convey things more intuitively. But now in this lecture, what I'd like to do is provide you an outline of how you justify it directly from Boltzmann equation. Well, so first point is where does that equation come from? Now the first thing is, of course, you're all familiar with Newton's laws. That is how you describe the motion of a particle like electrons using Newton's laws. There's two equations. Velocity is this dz/dt. And force is the derivative of the momentum. Now in Boltzmann equation, we define something called this f, which is a function of z, pz, and t. What it tells you is the occupation of states with a certain momentum p at a location z at time t. Note that there is this important change in concept here. That is here the only independent variable is t. z and pz are dependent variables. Those are the location and momentum of a particular electron. Here, z, pz, and t are all independent variables. We stand at a particular z, at a particular pz, and ask how many electrons are there or what is the occupation of states around there? And the argument you can make is that this f z, pz, t should be the same as what I have written here because if you look back-- I mean if you look at the number of electrons at time t and try to relate it to what it looked like at time delta t ago, the point is delta t ago, anybody at the location z must have at z minus vz delta t because in time delta t, the electron moved by that distance. And so whatever it is at time t, it must have been at z minus vz delta t a little time ago. And the same with momentum, if the momentum right now is pz, it must have been that much at time delta t ago. So this statement then kind of connects this viewpoint, which you call -- could call a collective description, to the individual particle viewpoint here. Individual, this is how an individual electron moves. This is how the distribution evolves. That would be this connection. Now the next step is to take this right hand side, assume a small delta t, and do this Taylor series type expansion, which means to take that quantity on the right and write it as f z, pz, t minus the derivative of f with respect to time times delta t. Why? Because we are trying to -- so the -- what it is at t minus delta t must be what it was at t minus the derivative times delta t. Now this would be everything if t was the only variable involved. But then we have two other variables. And so for the z, we could write something like this, that the del f/del z times vz delta t. And similarly for the pz, you could write minus the del f/del pz Fz delta t. So note this very subtle argument here. First, there's this much be equal to that because of Newton's law. The second point is that this must be equal to that because of just by Taylor series expansion. So from here to here is a purely mathematical argument with no physical content, whereas this equal to this, that basically incorporates the physics of Newton's law. Now once you have this, you see you can cancel those two things. And what it tells you then is whatever's out here must be equal to zero. So that gives you an equation like this. And if you go to steady state, you could say, assuming everything's at steady state, nothing evolves with time, then you can drop that. So you'd get this equation. Now notice we have got the left hand side of the Boltzmann equation that I was trying to obtain. What about the right hand side? Well, the point is if you put it equal to zero, then what we have here is really equivalent to the Newton's laws. That is just a reformulation of Newton's law using these collective variables. But what Boltzmann equation does is more than that. I think I mentioned before that what Boltzmann does is takes Newton's laws and includes these entropic processes. Entropic-driven things are scattering into the whole picture. And that is why it can also describe irreversible phenomena because Newton's law is basically mechanics. To describe irreversible phenomena and devices that you have, you need this extra term, which is the scattering term. And that is what you generally write as the scattering operator acting on f. And of course, the precise nature of the scattering operator, that requires a lot of discussion in general. For our purpose, what we'll assume is a very simple form for this scattering operator. We'll assume that all the scattering does is it takes the f and tries to restore it to a local equilibrium f, which you call this f-bar. And so this f, f is equal to f-bar. Then the scattering processes do nothing. That is, if locally the distribution looks like its equilibrium, the scattering processes won't do anything, and it will go to zero. On the other hand, if you're away from equilibrium, then the scattering processes will give you a term that is proportional to how far from equilibrium you are. This is what's often called the relaxation time approximation. So this is the form of the Boltzmann equation that we could use. As I said, what we have done here is included the scattering term in this relaxation time approximation. 

[Slide 3] Now so this equation then, if we wanted to go further, I want to include -- I want to introduce this idea of quasi-Fermi levels. So you could take the f, this -- which could have any form in general, but write it in the form of a Fermi function, but with these quasi-Fermi levels, mu. That is at equilibrium, actually, the f would be given by a Fermi function, where that's a constant. But it, now out of equilibrium, let's assume that this can be written in this form, which is -- looks like a Fermi function, but the mu is different at different locations. And the mu is different for different momenta. Now if we use this, if we accept the -- if you start from this form, then you can write the derivative, the del f/del z, in terms of del mu/del z with this df0/dE in front. And this, of course, is -- again, involves exactly the same derivatives and Taylor series expansion that we have been using since Unit 1. Similarly, you could write the del f/del pz in this form. And so you could then take the Boltzmann equation, which is in terms of f, and write a similar looking equation in terms of the mu. So just as f was dependent on z and pz and E, but here the mu depends only on z and pz. And that's because as far as the E part is concerned, we have assumed a particular form. The one nice thing about this form is it also immediately shows you that at equilibrium, the equilibrium solution is given by mu equal to a constant. That is, if mu is constant, then as you can see, this equation is automatically satisfied because d mu dz is zero. D mu, d pz is zero. And the mu-bar is equal to mu. There's no scattering processes. That's all zero. So this is what you'd call the equilibrium solution. And the Fermi function, of course, is the equilibrium solution. That's what equilibrium statistical mechanics is, that at equilibrium, it's the same chemical potential everywhere. But what we're interested, of course, in is the non-equilibrium solution. And under non-equilibrium conditions, not all mu's are equal. In general, every z and pz can have a different electrochemical potential, or quasi-Fermi level associated with it. Now when does this work? Now usually in devices, if the applied voltage is small compared to kT, compared to the thermal and broadening, then usually this assumption, that the -- that out of equilibrium, f looks like a Fermi function with varying mu, that's usually a good assumption. Now if the applied voltage is larger, then that's not necessarily true. The distribution in energy may not look like a Fermi function. And I guess in the earlier lectures, I think when I talked about quasi-Fermi levels, when you talked about Landauer formula, I had mentioned that, that in general, f may not quite look like that. In which case, you should define the mu just so that it gives you the right number of electrons because in all our discussion, that's what really matters. In which case, you might just go from f to mu, not with all these derivatives and all, but with the idea that any change in f is related to the change in mu through qV. That's kind of what I -- what we did when we were discussing the Landauer formula, for example. But either way, the bottom line is you could go from this Boltzmann equation in terms of f to an equation in terms of the mu. And main thing you're gaining from this is you're taking the energy variable out of the discussion. So you can just talk about the mu directly. Now the next point is a little more subtle, and that is that under many conditions, you can ignore this middle term because this is the term that actually involves the force due to the field. Usually this f, that's the force that an electron fields. And that is where the electric fielder often entered the Boltzmann equation. Now when can you drop this? The argument goes something like this, that at equilibrium, of course, the derivatives are both zero. And when you go out of equilibrium, these things are like first order. But if the electric field is also zero at equilibrium, then you see this term is kind of second order. That is when you apply your voltage, you are applying an electric field, so you get a first order term there. And that's also another first order term, so the total thing is like a second order term. And so you can drop this. Now this wouldn't be true though, if let's say there was a built-in field in some region for example, like we discussed. Or if there was an applied magnetic field as in the discussion of Hall effect. And in the notes you can look at the treatment of the Hall effect for example that I have there using the Boltzmann Equation. But for this discussion then, let's assume again, drop that term. Then you can write the rest of it in this form. vz del mu/del z is equal to minus, mu minus mu-bar over tau. Notice that in general then, mu at the given location can be different for different momenta. Now, this is where you could simplify a little further. by saying we'll only keep track of two mus. These two quasi-Fermi levels. One for forward moving electrons, those which are vz greater than zero and one for backward moving electrons, those with vz less than zero. So you could write for those which are forward moving, I call that mu plus. And those which are backward moving, I call it mu minus. Now the mu plus for those, the velocity is positive and so the velocity is equal to the absolute value of the velocity. Well, as for mu minus, it has the negative velocity. So you could write it as the negative of the absolute value of velocity. And so these would be then the two equations. One for mu plus and one for mu minus. Now, to complete the story we need to discuss what is this mu average that is appearing here? And the point is that these are, what it represents is what the scattering processes do to the system. That is what the scattering processes try to do is take this mu plus and this mu minus and collapse them in to a single mu. Because you see this, having two different quasi-Fermi levels, one for plus and one for minus, that's a non-equilibrium situation. Scattering processes don't like non-equilibrium situations. You see their mission is to restore equilibrium. So what they try to do is collapse the mu plus and mu minus into a single value mu-bar. And so this average mu, this mu-bar would be this average of the two. That's where they'll be trying to collapse it. 

[Slide 4] So if you now take that, these two equations, and you make use of this relation, you can put this together and you'll get d mu plus dz equals d mu minus dz is equal to the separation of quasi-Fermi levels for plus and minus state divided by 2 vz tau. And this is the mean-free path. And so you could write this, this is one over the mean-free path, and if you remember back from, I think it was Lecture 3 or 4, that we've talked about the relation between the separation between quasi-Fermi levels and the current. That is the current was like the ballistic conductance times the quasi-Fermi level separation. So I could replace this in terms of the current. And so I could now turn this around and write the current in this form, I is equal to this ballistic conductance times mean-free path d mu dz-- d mu plus dz or d mu minus dz. And this is kind of like looking like that diffusion equation that we have been using. If you notice that this GB lambda is actually conductivity times the area. So basically then, current is proportional to the slope of the mu plus or the slope of the mu minus. 

[Slide 5] So you could use this-- you could say that we can now solve this equation using the boundary conditions at the two ends. You see? You could say that mu plus at one end has to be mu1. Mu minus at the other end has to be this mu2, and for the current to be constant it must value the linearly, etc. So this is the equation that one could solve to get the variation of the two quasi-Fermi levels that we talked about. On the other hand, you could say that let's not bring in these two equations for two quasi-Fermi levels. Rather, let's just use a average because look, I could take this equation and add to it that equation and get a single equation for the average potential mu plus, plus mu minus divided by 2, and it would look exactly the same. So that's the normal diffusion equation. And as we discussed earlier in this unit, you could solve this equation, but with the boundary conditions that we discussed. I think it was back in Lecture 3 or 4, that the mu is equal to mu1 minus q i RB over 2. That was the new boundary condition. That is what people tend to do is solve that equation, but forgetting the second term. And so they tend to get mu, but put it equal to mu1 on the left and mu2 on the right. But so all of this then follows mathematically from the Boltzmann equation, with appropriate assumptions, which can be, I guess, scrutinized and discussed. But so as I said, if I were to do all this in a proper way, then you could have started from the Boltzmann equation and derived things step by step, whereas what I prefer to do, though, is usually you know start with intuitive description, that how things are generally working, and then tie it together mathematically, which is what I tried to do in this lecture. 

[Slide 6] Now so this unit then, then the next thing we'll do is we'll talk about this topic of great current interest. There's a lot of work nowadays on spin potentials. And there is a lot of activity. And of course, the reason I am talking about it in this context is because it relates in a natural way to the concepts we have talked about here, the mu plus and the mu minus, because in certain materials, the mu plus can translate into a quasi-Fermi level for up-spins, and mu minus can translate into a quasi-Fermi level for down-spins. But that's the next lecture.