nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 4.7: Law of Equilbrium
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[Slide 1] Welcome back to Unit 4 of our course, and this is Lecture 7. 

[Slide 2] Now in the last lectures, last couple of lectures, I emphasized this very important property of all contacts that are in equilibrium, that it is easier to emit into a contact than to absorb from it, and that is what is, expressed in this property of the contact that we have talked about, and what we now like to talk about is consider a system that exchanges energy and electrons with this contact and comes to equilibrium with it, and the result we'd like to establish is that this system, which has energy levels E1, E2, et cetera, available to it, at equilibrium will be distributed amongst these various states according to this relation. So the probability that the system will be in a state I, having energy EI and number of electrons and I is given by this function, and the Z is a constant, and this constant has to be determined, such that all the probabilities add up to 1, and what I'd like to explain is how this result kind of leads into that one. That is a distribution of the system. This is a property of the contacts. This is kind of important because often I hear this statement that if you're talking about a small system, you should not use these probabilities, and that's not really true because it doesn't matter what the system is. It could be small. It could be big. It could be complicated. It could be simple. As long as it is in contact with a classical reservoir, a contact like this, this relation holds, because this is a consequence of a property of the contact. It's not a property of the system. So in order so see how you get from here to there, the way you think about it is consider two states, E1 and E2 of the system, whose difference is E, and the number of electrons is N1 and N2. So if the system is sitting in E2,N2, then what it has to do is absorb E,N from the contact, and go to E1,N1. On the other hand, if it's an E1,N1 then it has to emit E,N in order to two to E2,N2, and the reason is the way I've defined it, it's like E2 plus E is equal to E1. So you have to absorb. Here E1 minus E is equal to E2 so that's like emit. So there's this absorption process and the emission, and at equilibrium, those rates must be the same. So if P1 is the probability that the system is in E1,N1, then the rate at which it will got this way is P1 times Pr, and if P2 is the probability, it's here, then the rate at which it will go that way is P2 times P, and equilibrium, those two things, must be equal. 

[Slide 3] So you could now write P1 to P2 ratio as the ratio of P to Pr, this ratio of absorption to emission, which a property of the contact, and you use this contact property then to replace that, and you get this expression. All right, this just comes from the contact property that we have been talking about, and since E is E1 minus E2 and N is N1 minus N2, we could write it as the ratio of this to this, and now you'll notice this is exactly P1, and that's exactly P2. So you get this P1 over P2. So the point is that just by requiring that it's in equilibrium with this contact so that going right is the same as going left. The rates of going write and left, the ratio of P1 over P2 has to be equal to this, which is guaranteed if all the Ps are given by that expression. So we have naturally kind of taken a property of the contact and use it to derive the equilibrium distribution. See? And this is the central law of equilibrium. Central law of equilibrium statistical mechanics. Okay, and all of statistical mechanics, this is like the basic law, and everything else is derived from it, okay? So what I'll do next, in the next couple of slides, is try to explain how you get the Fermi function out of this, for example, but it's not just the Fermi function. The Bose function for photons or phonons, that also comes out of it, and it's in the lecture notes, though I won't be going into it. So I'll just illustrate how to use this with a few examples. But right now, the main point I wanted to make is that this really comes from a property of the contact, and in that sense, it's very general, irrespective of the details of the of the system. Okay, so how do you get Fermi function out of it? 

[Slide 4] So for this, though, you have to think in a slightly different way. That is, usually we've been using what what's called this one-electron picture. Let say we have a one level there whose energy is epsilon, and we think of electrons coming into that level or electrons going out of that level. Now to apply this expression, though, we need to think in what's called Fock space, or this multi-electron picture. The way you think is there is one level, but the system can be in one of two states. That is, that level can be either empty or it can be full. So if it's empty, we call that the zero state, and the corresponding energy is zero. It it's full, we call that the one state and the corresponding energy is epsilon. So in order so use, then, this formula, we need this E minus mu N over kT. So here, E is 0, N is 0. So E minus mu N is 0. Here, E is epsilon, N is 1. So E minus mu N is like epsilon minus mu. So that's what we have written as x. Epsilon minus mu, divided by kT. So now we can write down the probabilities that the system will be in zero state or the system will be in the one state. See, it's just this e to the power minus x over Z or e to the power zero, which is 1 over Z. So those are the two probabilities. How do you determine Z? Well, all probabilities must add up to 1. So P0 plus P1 must be 1. So if you put in the values of P0 and P1, you get 1 plus E to the power minus x divided by Z, must be equal to one, and that immediately gives you a value for Z. So if you substitute that back in here, you get an expression for P0 and you get and expression for P1. So this tells you probability that the state is empty. The lower one guys you the probability that the state is full, and you could rewrite this with a little algebra in this form, and you'll notice this is exactly the Fermi function. You say, "What the Fermi function supposed to tell you?" It tells you in the one electron picture what is the probability that this state is occupied? So basically, it's P1, and you can see starting from here, we have now managed to get of Fermi function that you have been using all along. If you want to know the number of electrons, you'd say number of electrons, average number of electrons is 0 times probability you are in the 0 state. One times the probability you are in the 1 state, and so that's basically this, P1, and which is this Fermi function that we have been talking about. But in general, though, you see this is the more general expression. So we could use it for more complicated things. So let me give you an example. 

[Slide 5] Suppose we had a system with two one-electron levels. You know, like up-spin and down-spin. Now when you look at the Fock space fixed state, it's not just 0 and 1. There's two of them. Each one can be 0 or 1. So you have four possibilities. 0-0, both are empty. 0-1, empty, full, 1-0 is like full empty, 1-1, both are full. There's four possibilities. What are their energies? Well, 0 means energy is 0. If only one is occupied, the energy is epsilon. Doesn't matter if it's 0-1, or 1-0, since both have the same energy that epsilon. If both are occupied, well, then energy should be two epsilon, but then I've also added a possible interaction energy U0, because whenever two electrons are present, they would feel this Coulomb repulsion. So that's what we are calling this U0. What are the number of electrons? You see, to use this formula, of course, I need E and N. So what are the number of electrons? Well, here is 0. In these two states, it's 1. In that state, it's 2. So what is the exponent? This E minus mu N over kT. In this case, it's just 0. Both are 0. Here you get epsilon minus mu over kT. That's what you have defined as x. Here you have two epsilon minus 2 mu over kT. That's 2x, and then there's the U0 over kT. That's the y. So those are the exponents that we should be using. So based on that, you could write down the probabilities of all the states. P00 would be 1 over Z. P01,10, that would be E to the power minus x over Z. P11 would be E to the power minus 2x and E to minus y over Z. How would you find Z? Well the sum of all this must be equal to 1. So that's how you could do that. I won't go through the algebra, but one thing I wanted to point out is that if the charging energy is negligible. That is, if U0 is much less than kT, and that's usually the case in large conductors. That is, the interaction energy is very small, because you have a large conductor, and the electron is spread out. On the other hand, if you had a very small conductor, then there's one electron charging energy can be sizable. So this y can be big, and these are effects that people have observed in small conductors call single-electron charging effects or Coulomb blockade. But supposing this y is negligible? Then you see you should the on the answers you would expect from common sense. Namely, if I put y equal to 0, what you can show, and I'm not going through the algebra. You'd have to do this yourself. The P11 will be f square. What's P11? It's the probability that both are occupied. Now what's the probability, one of them is occupied? It's the Fermi function. Fermi function tells you the probability that a particular state is occupied. So what's the chance both are occupied? Well it's f squared. What's the chance one is occupied and the other is empty? f times 1 minus f, because 1 minus f is the probability of being unoccupied. Empty. What's the chance both are unoccupied? Well, it's 1 minus f squared, and if you take this plus 2 times that, because there's two states there, plus that, the answer is 1. Total probability is 1. So the point I'm trying to make is that this general method, if you apply it to the y equals 0 case, you'll get the sensible answers that you'd got out of Fermi function. On the other hand, of course, the power of this general thing is even when y is not small, that is, if you are in the single electron charging regime or Coulomb blockade, you can still use this to handle equilibrium problems. 

[Slide 6] So let me just describe to you a problem that has been extensively investigated in the last 20-30 years is the following. That if you have a small structure, a quantum dot with two levels, and let us say I have a way of moving the electrochemical potential. So in practice, you might have a gate electrode that moves the energy levels, and the mu stays fixed, but conceptually, it's a little easier to think as if the levels are fixed, and you move the mu. So the way you think is when the mu is way down here, both states are empty, so you have zero electrons. If the mu is way up there, both states are filled, so you have 2 electrons, and the question is how you get from 0 to 2, and you might think that as you raise mu, once you cross epsilon, that's when both states should get filled. So you might have thought that you should go from 0 directly to 2, but in the single-electron charging regime what happens is it doesn't go directly to 2. It kind of goes through another step, 1, and then goes to 2. So question is where does this step occur, and where does that step occur? And those are answers that will come naturally out of this general equilibrium law. So let me explain. So for this purpose, it's convenient to think as if you are at low temperatures, because low temperatures means temperature is 0. So this quantity up here is very big, and in that case, you see, there's a very simple rule, and that is that when you have small kT, the system wants to go to whatever state has the minimum value of that quantity. So if this is 100 for one of them and 200 for another, it will go to the 100, and that's not obvious though, because you might say, "Well, if it's 100 for one and 200 for another, shouldn't the probabilities be like E to the power minus 100 divided by Z and E to the power minus 200 divided by Z?" I'd say yes, but then doesn't that look like both are 0? But the point is both cannot be 0 because probabilities must add up to 1. So how do you find the Z? Well, you'd say that some of the two must be 1, and that would give you Z equals the sum of those two things that are both very small, but the thing is, this is a lot bigger than that. So Z is approximately equal to the first one. So what that means is that P1 is approximately 1, and P2 is approximately 0. So that's the basic rule I was trying to say. That once you get to this regime where these are big numbers, because kT is small, the system just has high probability of being in those states which have the minimum value of E minus mu naught, and you could use this idea to try to understand how this curve, how you get this curve. The way you can think is when mu is way down here, mu is a large negative number. What that means is x is a big number. So if x is big, then the minimum is 0. So the system goes to 00, because remember, the system is looking for whichever state has the minimum value in that column, and if x is large, it wants to be there. But then as you raise mu, there will come a point when the x will become smaller and smaller and eventually it will become 0, and then less than 0, and once that happens, the system will go here. So when will that happen? Well, whenever x becomes equal to 0, because x was kind of big, and then it got smaller and smaller. Once it's equal to 0, after that, it'll become even less and that's when the system will go to 01, or 10 states. Now if you keep making x smaller, then you see, this will keep getting smaller, but the thing is, this will get smaller at a even faster rate, because there's 2 in there, and so at some point, this will become the minimum, and at that point, when x is equal to 2x plus y, that's when the system, instead of being here, will go to there, and that's where you'll have the second transition, and with a little algebra, you can show that that occurs at mu equals epsilon plus U0. So in other words, there's this one electron that walks in when mu crosses epsilon, but the second electron walks in at a higher value of mu when it's epsilon plus U0. In the multi-electron picture in this Fock space, the way you think is the system starts out here. As you raise mu, it goes there, and as you raise mu further, it goes there. In the one electron picture, of course, the way you think is as you raise mu, one level gets filled, and because it gets filled, you can say that the level kind of floats up by U0, because it feels the repulsion due to that one filled level, and then the mu has to go even further to fill up that second level. That would be the way you could rationalize it, but the right way to do it is really from the multi-electron picture, and these are all relevant to a whole wide class of phenomena that have been investigated in the last 20-30 years. But that's kind of a detour from what we are talking about here. Main point I wanted to establish is this general law of equilibrium, and the idea that the Fermi function is kind of a special case from it. But this is the general law that you should start from that describes all equilibrium systems and it's really a consequence of this property of the contacts that we talked about. 

[Slide 7] What we want to do next is go back to this concept of entropy and try to connect it to information, and that's what we'll be doing in the next two lectures. Thank you.