nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 4.5: Second Law
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[Slide 1] Welcome back to Unit 4 of our course. This is Lecture 5. 

[Slide 2] Now, in the last two lectures, we have talked about two very important thermoelectric effects involving this interconversion of heat and electricity. And also in the first lecture, I mentioned that, although heat is a form of energy, it is very different from other forms. In the sense it's this random form of energy. And so, when everything's at the same temperature, you cannot just take heat from your surroundings and convert it into electricity. You need a temperature difference in order to be able to do that. And there is this very deep law, the second law of thermodynamics, which governs all these processes. In the sense that, it doesn't matter how ingenious a device you design, you cannot violate the requirements of this second law. And that's what we want to talk about in this lecture. And for this purpose we define another general setup. See we have a contact one. Electrochemical potential, mu1. Temperature, T1. Electrons coming in, N1, which flow out into this channel. There's this channel, energy, E1. Energy, E2, coming in from contact two. Similarly, there's this contact two where these electrons, N2, come in, which flow into the channel. Now, the arrows that are shown are all reference directions. In the sense that, if ten electrons come in from here and go into the channel, then ten electrons must come out. So what that means is if N1 is plus ten, N2 must be minus ten. Similarly, energy, our channel is elastic. Doesn't, I guess, dissipate any energy. So, again, E1 plus E2 must be equal to zero. Now, for the second law. To state the second law, you have to consider the energy that is taken in from the surroundings. And here the idea is that, if you look at contact 1, the electrons that go into the channel take energy, E1, from the contact and take it into the channel. On the other hand, the new electrons that come in, come in right around the electrochemical potential. And so they bring in an amount of energy which is mu1 N1. And those two are not necessarily equal. And the difference must come in in the form of energy from the surroundings in order to maintain your contact at a fixed temperature. So, whatever, if the temperature is fixed, then there cannot be any net flow of energy from the contact. Similarly, if you look at the other contact, there must be a net flow from, a flow of energy from outside. Which is E2 minus mu2 N2. So that overall there is no flow. And what the second law says is that the amount of heat that contact one takes in divided by its temperature plus the heat that the contact two takes in divided by the temperature should sum to, should be negative. Negative or equal to zero. Now, you might say, well, this is, like, the fourth unit of this course, and so far we haven't even talked about the second law. Are we sure that everything we discussed is in compliance with the second law? And answer is, yes. And let me explain why. 

[Slide 3] In our discussion we have considered this elastic channel where energy, no energy is dissipated. And so all these different energy levels can be viewed as independent things. So you could just find the current flow through each one and add them up. So for this discussion, you might just as well consider a single energy level. You see rather than this distribution of levels, let's just consider a single level whose energy is epsilon. So in that case, you see the amount of energy carried by electrons going into this channel should be epsilon times N1. Similarly, E2 should be epsilon times N2. So if you use that, you could write, instead of E1, you can write epsilon N1. So that this term would become like epsilon minus mu1 over T1 times N1. Similarly, the other term will become epsilon minus mu2 over T2 times N2. And the second law says that the sum should be less than or equal to zero. So now we make use of this relation, that N1 plus N2 is zero. So we replace N2 with minus N1. So you can pull the N1 out. You have epsilon minus mu1 over T1. And since N2 is minus N1, you pick up a minus sign. So that quantity must be less than or equal to zero. Now, the next point is that you could replace this and this with f1 and f2. These are the Fermi functions in the two contacts. But then you have to reverse this inequality. This was less than, it will become greater than. Why is that? Well, we have to look at the expression for the Fermi function. Remember, the Fermi function is 1 divided by 1 plus e to the power x, where x is epsilon minus mu over kT. Now, supposing epsilon minus mu over kT, say for contact one, is bigger than that for contact two. So let's say this is bigger than that. So that's a positive thing. So what will happen to f1 minus f2? Well, if this is bigger, it means that the x is bigger. So the e to the power x is bigger. And so the f is smaller. So if this is positive, then f1 minus f2 will be negative. And vice versa. Is this is negative, f1 minus f2 will be positive. So if I replace that quantity with that quantity, then I have to reverse the sign of the inequality. Now, we have this expression. And what I claim is that this is automatically satisfied given the way we were calculating currents. Remember what I argued in, back in Unit 1 is that current is proportional to f1 minus f2. Why? Because we say that the basic driving force for currents is that contact 1 wants to bring the channel into equilibrium with itself. That is, in equilibrium with f1. Contact two wants to bring it into equilibrium with f2. And there's this difference in agenda that causes current to flow. And the direction of current is f1 minus f2. So what that means is, if f1 minus f2 is positive, electrons will flow from one into two. So N1 with be positive. N2 will be negative. So what that means is, if this is positive, N1 is positive. So the product is greater than zero. If f1 minus f2 is negative, current will flow the other way. N1 will be negative. So, again, two negatives, you get something that's positive. So in other words, using an expression like this automatically ensures that the second law is satisfied. Note that this is actually a pretty subtle point. 

[Slide 4] That is, consider, for example, a device with two contacts which are not identical. And we look at a certain energy range. And let's say in this energy range we have a very high density of states here, but not so many states out here. So let's say here we got hundred states that are available, which are, say, half full. So you got fifty electrons in there. and on the right-hand side, let's say you got ten states that are available, which are all full, so you got ten electrons in there. So which way will electrons flow? And you might say, well, you got fifty here and ten there, so it should flow from fifty to ten, from left to right. But that's wrong. The flow is determined by f1 minus f2. Here the states are fifty percent filled, so f is like 0.5. Here is the states are completely full, so f is like 1. And so current will flow in this direction. And that's this basic law of thermodynamics. Just like heat flow. Heat flows from high temperature to low temperature. It doesn't matter how much heat a particular body has, you know, how much the total heat is. What matters is the amount of heat per degree of freedom, as we'll talk about a little later in these lectures. And that's temperature. So temperature is a measure of heat per degree of freedom. And that's what determines the flow of heat overall. Similarly, the flow of electrons is determined, not by how many electrons you have, but rather this fractional occupation. And as long as you use current that's proportional to f1 minus f2, you are in compliance with the basic laws of thermodynamics. 

[Slide 5] Now, one thing that made it relatively easy to comply with the basic laws is that we considered an elastic channel. More generally, you might have a channel that exchanges energy, E0 with another reservoir. I guess that's the technical name, reservoir. We could call it a contact. I mean, these are, of course, physical contacts, source and drain. But you could have other conceptual contacts, with which you exchange energy, if not electrons. And as I say, the technical name is reservoirs. In our context, we could call them contacts. So if we had a third contact like this, then for the first law we should add E0 to this energy balance. That's just a statement of the conservation of energy. Now, for the second law we have to add a similar term here. E0 divided by T0. And the meaning of the second law, if all the temperatures are-- were equal, then it would be quite straightforward to see. What it would mean is, if all the temperatures are equal, then you see the numerators must add up to less than zero. So what's this numerator? It just tells you how much heat you are taking in from your surroundings. So what this basically tells you is, if everything's at the same temperature, you cannot take heat out of your surroundings. That is, the amount of heat you take out of your surroundings must be negative. Which means less than or equal to zero. Which means, basically, on the balance you should be giving out heat. And you'll notice also that the first law, this energy conservation makes E0 plus E1 plus E2 equal to zero. So that mu1 N1 plus mu2 N2. Notice the minus signs. So what it means is that sum must be greater than or equal to zero. Less than or equal to becomes greater than or equal to because the minus signs. So what are these two terms? Well, that basically tells you how much energy comes in from the battery. Because these are the electrons that come in from the battery. And so what it says is that, if everything's at the same temperature, you must be taking in energy from your battery. You cannot, there's no way you can charge up your battery. Doesn't matter how cleverly you design the device. Now, in connecting up to the next lecture, I want to make one important point now. And that is that so far you see in this course we are focused on the channel. The contacts were kind of incidental. We didn't even have the third contact. These two contacts have these Fermi functions, f1 and f2. But other than that, we never talked about the contacts. But the point to note is that this second law is really a property of the contacts. It really doesn't matter what's in here. If we look at these terms that appear here, it all has to do with each of the contacts. How much heat you are taking from a contact at what temperature. So in other words, it really doesn't matter this channel structure in here. We could forget that. It's really about the contacts. And the point I want to make is 

[Slide 6] that in a way the second law can be related to a basic fundamental property of all contacts. That is, if you consider a contact at temperature T electrochemical potential mu and think of a process in which you're trying to take an energy, E, or number of electrons, N, from it. So the interaction that let's you take this energy also causes the reverse process that kind of wants to reverse the whole thing. And the basic property of contacts is that, if you look at the ratio of the probabilities of the process you want to happen and the reverse of it, that's this r here. Then that ratio is exponential of that quantity. And if no electrons were exchanged, then you could drop the N, and you'd have P as exponential minus E over kT. So what that means is that, if E's positive, then this would be a very small number. Which means the process you want to happen, which is to take energy, E, that would be a relatively low probability thing. More dominant would be the process of giving it up, if E is positive you can see that. So if we are to put it in one line, it's like all contacts have this property that it's kind of exponentially harder to take from it compared to giving to it. And this property automatically leads to the second law. Because the way you can think about it is consider a process where you're taking these amounts of energy from the surroundings. So then the chance, the probability of that happening is this P E1N1, P E2 N2 and P E0. And consider that relative to the reverse process, which wants to reverse all of this. And this we'll talk about more in the next lecture. The overall, that ratio must be greater than or equal to one in order for this process to actually happen. See, equal to one means it's kind of happening very slowly. That's what you call a reversible process. But it needs to be greater than 1 to make it happen. Now, for this ratio, if we put in this property that I mentioned, then you'll get exponential of that quantity, that quantity and that quantity. And, of course, when you take the product, the exponents all add up. So exponential of that quantity must be greater than one. Which means this quantity here, the exponent must be greater than or equal to zero. And there's these negative signs in there. So if I take out the negative signs, the greater than becomes less than, and we get the second law. So the point I'm trying to make is the second law that we talked about kind of comes from this elemental property of all contacts. That is, all contacts, it is always 

[Slide 7] harder to take energy from it than it is to give energy to it. And why that is so, that brings us to this very important concept of entropy, which we'll talk about in the next lecture.