nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 4.10: Summary
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[Slide 1] Welcome back to Unit 4 of our course. And I guess it's now time to sum up. 

[Slide 2] Now as we know in much of this course, we have been using this new approach inspired by nano devices which is to say that think of the device as having this contacts where all the entropy-driven processes are restricted. And that the channel has no entropy driven processes in it. And that allows you to describe current flow in a relatively simple way. That's what we had been doing for the first three units. And this unit is designed to give you some insight into what the entropy driven processes mean and what it involves. And we started with our discussion of thermoelectricity that was the kind of the first three lectures, two, three and four. And there, we introduced these thermoelectric coefficients, that is so far we had usually been talking about a current driven by a voltage, but then, we introduce also currents driven by temperature and we also introduced this concept of a heat current that is usually-- of course when current flows you have joule heating. But depending on the structure you can also have a heat current flow, namely, if you had a device looking like this, you know, where the available conduction channels let's say above mu. Then when current flows, you actually cool one contact that is you take heat from this contact and dump it out there, which means you will actually have a heat current that flows, OK? And so, that was the equation for that heat current. And so overall, you have these four thermoelectric coefficients. And what we did was we obtained the expression for these thermoelectric coefficients using this point of view. And these are exactly the standard results that we normally get from Boltzmann equation. And of course, in general as I mentioned before, whatever we are talking about in this course, I mean, it's just the way of looking at these results but the basic benchmark equation is the Boltzmann equation with all results generally would follow from Boltzmann equation. 

[Slide 3] Now, we then went on to more conceptual issues that is we say that well, you know, this heat is a form of energy but it's really very different from the usual forms of energy. In the sense that you take energy from a battery and generate heat, but you cannot go the other way. And the first law doesn't stop you from doing that but there's a second law that actually does. That is you can take energy from a battery, heat up the surroundings, but you cannot take heat from your surroundings and charge up a battery or light up a light bulb. So, you cannot do things like this. But then, you can convert heat into electricity if you have a temperature difference and these are the thermoelectric effects that we are talking about. So, there is the second law we generally governs all such processes, you know, which if you involve where heat and electricity are involve. And the basic principle is that this total amount of heat you take from the surroundings, this heat divided by the temperature, heat you take from another contact divided by the temperature, all of that must be negative when added up. That is on the whole, it then restricts it tells you that for something like this is not possible. You cannot take-- If everything is at the same temperature, you cannot take heat from the surroundings and light up a light bulb. But if things are at two different temperatures, you can maybe able to do that. So what kind of processes are possible and what kind of processes are fundamentally impossible? They're all governed by this very basic law, the second law. And one point I try to make is that by in large in this course, our focus has been on the channel. And usually, as far as the contacts are concern they are described by some Fermi function but we never talk much about the content. But the second law is really about the contacts. This channel doesn't matter. It could be very simple. It could be very complex. It could be small. It could be big. It doesn't matter. Basically what this expresses is a property of these contacts. So, there are three terms here because there are three contacts here. And each of those contacts has a very simple property namely 

[Slide 4] namely that if you take a contact and you considered a process in which you've extract energy E or number of particles N from it, then, the probability of that process divided by the probability of the reverse process is given by this factor. So if this factor is very small then the process is relatively unlikely, because the reverse process is dominant. And when you have set three contacts, then, you see in order for a process to happen, the product of things like this for the three contacts must be greater than or equal to one, and that kind of leads to this second law overall. So, bottom line is that the key property of contacts that is important that gives rises to the second law is that it is much easier to emit energy into the contact than to absorb energy from it. And this basic property then, you could use it to derive the law of equilibrium, this general law of equilibrium, that if you consider any system connected to your contact and once it is an equilibrium, the states of that system will be occupied according to this basic law. You see? And which gives rise to the formula function and all those are like special cases of this fundamental law of equilibrium statistical mechanics. See? And I kind of want to stress this point that this law really came from a property of the contact, because often people say, "Well, you know, we have this-- how can we apply this law to a small system because this are statistical law." Well, the point is it doesn't matter if the system is small, big, complicated, simple, doesn't matter. This expresses a property of the contact. It's the contacts that to be large and simple, OK? 

[Slide 5] Now, in terms of understanding this basic property, you see this basic property that governs all these deep things, second law, law of equilibrium, et cetera. What really governs the basic property is that it is much harder to, you see, absorb energy from the contact than to give energy to it. And where does that come from? Well, that what's going to express in terms of this idea of entropy, the idea that the contact has many states. So when you have-- when you emit energy into a contact, it gets distributed amongst many states. And that is kind of what is different here, that the energy is going into many degrees of freedom. So, if you have energy in one degree of freedom, you can relatively easily dispense it among many degrees of freedom. That is you are riding along in a car, so all the energies in that one degree of freedom, you turn off the engine, car comes to stop, all the energy goes into the road. The molecules in the road, they start jiggling and they get heated up. So, that's a process where all the energy in one degree of freedom got spread out into many degrees of freedom. What you cannot do is take energy out of the road, you know, all the road gets cold and start driving your car. The reverse process is kind of much-- highly unlikely. And so, this property then of course depends on entropy. Entropy is a measure of how many states, these degrees of freedom into which the energy is dispersed. And it's measured by this log W, this is equal to k log W. That's this major contribution of Boltzmann that is entropy was a thermodynamic concept which people had arrived at based on microscopic experiments. What Boltzmann did is he related it to a microscopic quantity W which is this number of states that are involved which take up your energy. And this k is Boltzmann's constant. And what we then did is also show that that expression could be-- there's an alternative expression we could derive in terms of this p log p that when you have a system with many states which are occupied with probabilities pi, then the entropy can be written as negative of P log Pr, OK? 

[Slide 6] Now, the advantage of this expression is you can use for all kinds of systems and the system doesn't necessarily have to be an equilibrium. That is if you apply to an equilibrium system, you get standard thermodynamic relations. I think I went through an example of that I believe in lecture eight. But you could apply it to other systems which are not necessarily an equilibrium. And one example we went through is this random collection of spins. And it showed that if you evaluate the entropy of this random collection you'd get this nk log 2. And one of the points, one of the again very interesting deep issues is the connection of this thermodynamic entropy to what is call this information entropy. That is an information theory where you consider a channel where you are sending information by sending bits ones and zeroes, let's say with 50-50 probability. Then how much information you have transmitted is often measured with a quantity like this. Note that this is dimension unless whereas a thermodynamic entropy has this Boltzmann constant in front which is joules per Kelvin. So very different things, but in the deep connection of course is that if we think of this channel where you can send ones and zeroes upon this, before you send anything it's like there are many possibilities which is this nk log 2 or n log 2 that's the number. K of course comes from this extra thing here. But that tells you is all this different possibilities and as soon as you send a particular signal, a particular six sequence of bits. I've shown it as 1111 but need not be that, whatever it is. As soon as you send that you have kind of narrowed down the possibilities. That is before you send it you had all these various possibilities, all kinds of signals you could have sent. But when you send the particular one, you have narrowed it down. So how much information have you sent? Well, it's kind of the range-- the entire range of possibilities that's what is counts as the amount of information you send by pinning it down. So, in that sense these are fundamentally in a very similar things. One involves then a number of ways your energy is distributed, number of degrees of freedom, the other is about how information is sent. And if for example you had something in a state like this where you have complete information about what it is. And it interacts with the surroundings and gradually becomes random. That's kind of like something starting out here and then gradually spreading out. If that happen then you see you should be able to use that process this process of increasing entropy to do useful work. And second law allows that. And what we did in the I guess the ninth lecture was I showed you could you actually construct the device to turn this loss of information into useful work. 

[Slide 7] And this is what we call this info-battery consisting of this antiparallel spin valve with two magnets that are pointing in opposite directions and a whole bunch of localized spins which are all pointing up. And because they are all pointing upward to do is the kind of interact with itinerate electrons in the channel taking negatives and flipping them-- taking down spins and flipping them to up spins and those up spins then come out of this contact and cause this to become negative. And the missing down spins are filled in from here and that makes it positive. So this is what we discussed in the ninth lecture. But the main point here was you could start from this collection of localized spins with the zero entropy and get a battery out of it with an open circuit voltage. Of course as you draw energy from it, it will gradually run down. And running down basically means this perfect alignment of spins will go away and it will have this random collection. And once you have a random collection there's no voltage anymore. That's this highest entropy state. And this device I should mention kind of has an interesting parallel with something that has been discussed extensively for over a century and that's this Maxwell's Demon. And so the Maxwell's Demon this was something that's due to Maxwell, you know, back around 1850, where what it said is consider a box like this where you have say some black molecules and some white molecules. So let's say the black ones are fast moving ones and the white ones are the slow moving ones. And let's say we have a demon who knows everything, he knows exactly what those velocities are. And so whenever he sees one of these white things come along, the slow moving ones he opens the door and let's them go through. And whenever he sees a black one coming from here he opens the door and let's them go through. So after some time you'll have lots of black or fast moving atoms on this side and you'd have slow moving atoms on that side which means this side will become hot and that side will become cold. And of course once one side is hot and one side is cold you could do useful work from it, you know, using a conventional effect like the Seebeck effect for example. So the idea was that this demon because of this information that is available to him which is he knows the velocity of each atom. He is able to create I guess a situation where you have a temperature difference. You started from no temperate difference and went to the situation and then it can use that to do useful works. So it's kind of converting information into energy. I'd argue that this device is kind of similar. The demon is this collection of localized spins. And what this collection of localized spins does is kind of lets these up electrons come up here, pile up, and the down electrons come in from here that's why this side becomes positive. You see. Now question is can a demon like this function forever? This is something that's discussed at length. I guess one of the very interesting readable discussions as you might imagine is Feynman lectures that you could look at. And I think what Feynman says is well, the demon after he has been doing this for a while it kind of after interacting with all these atoms he becomes hot and at some point he's not able to tell this velocity which is moving which way anymore. You can see that discussion. I guess the equivalent thing here is you have this up spins that's like the fresh demon hot at work but as he does his work he gradually becomes hot if you will and loses his ability to tell the difference because as long as it was all up he would just take a down spin and converted into up but it wouldn't let convert the other way. But as long-- but once he is random he converts it both ways. And now you don't really have this preferential. You don't have a battery anymore. So that is-- this is a good concrete example of this very well known thing that has been discussed at length in the literature. 

[Slide 8] So that-- On that note then I guess we could like to end this unit, the forth unit. As I said earlier our basic point here was to explain the nature of this entropy driven force or entropy driven processes something that in earlier units we kind of glance over. We said that will-- let's say assume this entropy things happen and-- the entropy-driven things happen in the contacts and which maintains the contacts in equilibrium. So, this unit was designed to give you a better feeling for that kind of thing. So, what comes next? Well, this kind of is the end of this first part. And the difference between the first part and the second part is that the first part was essentially all based on the semi classical picture which means the way you think is you have this density of states which of course require some amount of quantum mechanics which we did in Unit 2, but very little of it and done in a very really elementary way. So you say that you have some density of state but then electrons move along the states just like particles. The wave naturally is not very-- does not need to be considered. So that's what you'd call the semi classical models. And everything we discussed is based on the semi classical model which means the proper mathematical based is for it would be the Boltzmann equation, the semi classical Boltzmann equation. The second part of the course is about the quantum model. You see, that's where you bring in the quantum mechanics. And to connect from one to the other I guess we have a little epilogue after this which kind of sums up all four units and introduces the next one. Thank you.