nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 4.9: Fuel Value of Information
========================================
 >>

[Slide 1] Welcome back to Unit 4 of our course. This is Lecture 9. 

[Slide 2] Now, as you might recall in the last lecture, we talked about this expression for entropy and how that kind of corresponds to information. That is if you have a collection of spins and physically what we have in mind is the whole bunch of magnetic impurities which can either be up or down and they don't move or anything, they're just all localized spins inside the channel of a device, and if the-- all the spins are up, there's only one way that can happen because they're all up. That corresponds to this entropy equal to zero. Whereas-- And that corresponds to having information, you know what it is. On the other hand, if it just interacts with its surroundings, comes to equilibrium and goes into this random collection, that corresponds to a higher entropy state. And here, we are assuming there's no interaction between the spins. So that whether they are up, all up or whether it is like up, down, doesn't matter. The energies are all the same. So, there is no change in energy going from this to this, just the change in entropy. And the point I'd like to make, I think I mentioned this at the end of the last lecture, is that if these localized spins were coupled to an appropriately designed device, so we have this device where you have two contacts, electrons can flow in and out of this channel. And let us say, these spins were coupled to it in the sense these are localized impurities which are probably sitting inside the channel. And if I designed it properly, then I could make use of this loss of information to actually do useful work like charging up a battery or lighting up a light bulb. And that's the reason for the title of this lecture. The fuel value of information that is when using information, you're converting information into useful work. What the second law says is that it should be possible to do so. Because ordinarily what the second law would have said is, that there is no way you can take heat from your surroundings and do something useful. Why? Because whenever you take heat, the entropy goes down. And overall, second law requires that entropy has to go up. But in this case, because there's a third terminal where the entropy is going up, it means that without violating the second law, you could take some amount of heat from the surroundings and do useful work. Note that none of the energy is coming from this terminal at all because whether these spins are up-- all up or random doesn't matter, energy stays the same. So, you're not getting any energy out of it. But the increase in entropy is kind of letting you take heat or energy from elsewhere and do useful work. So the second law says that you can take a certain amount of heat out of your surrounding and convert it into useful work as long as that amount is less than T times the entropy cost that you have paid here, this delta S, which means you should be able to convert an amount of energy up to nkT log 2. What the second law doesn't tell you though is how to do it. It just says in principle, it's possible. But this is an inequality, it's not equality. So, if you don't design your device right, you won't be able to. But if you design it right, you should be able to, in principle. So in this lecture, what I'll describe to you is a concrete design that will achieve just that. And the purpose is again, not that it's a very important device, the purpose is just to illustrate an important conceptual point about entropy. 

[Slide 3] So, the device I have in mind is what you might call an antiparallel spin valve. You see spin valves have, in the last couple of decades, have got a lot of attention as useful devices. What it is, is it's a normal device but with magnetic contacts. And here, what we have is an antiparallel spin valve in the sense that if one magnet is up, the other magnet is antiparallel, pointing in the opposite direction. And what we assume is that these magnetic contacts are perfect, 100% efficiency. What that means is that if you have an up contact magnet, it only talks to the up-spin electrons in the channel. So, up-spin electrons can flow in and out of this contact. Similarly, the down-spin electrons only talk to the-- this down contact. Now, in practice, real magnetic contacts are-- don't have 100% efficiency. They might talk, say 80% to that one and 20% to that one. But what we are assuming, because this is entirely conceptual, it's designed to illustrate a conceptual point. So we are assuming these magnets are perfect. So, this only talks to up-spin, that only talks to down-spin. So, what that means is that in this device, ordinarily no current can flow. Why? Because up-spin electrons are connected to the left contact, down-spins are connected to the right contact. But there is no path going from the right contact to the left contact. You see, up-spins can come in but then they can't go anywhere. Down-spins can come in from the drain but then they can't go anywhere. Now, if we take this channel now and couple it to these localized spins, you know, these spins that I talked about, these are these just localized impurities. Let's say we couple them. Now, actually, current flow becomes possible. Current flow becomes possible because we have this interaction between them. There's exchange interaction which says that if you have an up-spin electron in the channel, these are the moving electrons, and you have a down-spin impurity, they can interact and flip. So, the up-spin becomes down-spin and the down-spin becomes up-spin. So, it would be something like this. An up-spin electron in the channel interacts with the localized down-spin. The localized down-spin becomes an up-spin, and up-spin electron becomes a down-spin. So, this is what-- we would call this exchange interaction. If both had up, nothing happens. If both had down, nothing happens. But up down becomes down up because of this interaction. Now, as a result, what would happen? As you see, you have all these up-spins here. They would let down-spins converge to up-spins because they would interact. You see the down-spins will interact with all the up-spin localized ones and turn them into up-spins. And these up-spins would want to come out into the contact. The key to the design of this device is that up-spins has to come out from the left contact, down-spins only talk to the right contact. So, anytime you create an up-spin, it goes to the source. And of course if it cannot flow out because it's like open circuited, then what will happen is the source will become negative as I have shown you. On the other hand, you lost the down-spin so a new down-spin has to come in from here making this end positive. So, the point is, if you take a structure like this with all-- interacting with this collection of up-spins, up-spins in the channel, these up-spin localized things, then it will be like a battery. There would be an open-circuit voltage, one side will be negative, one side will be positive. So, how do we estimate this open-circuit voltage? Let us try to write the current. You could see that there is a process which converts up-spins into down-spins, up-spins into down-spins. And anytime an up flips into a down, it is able to flow out. So, it gives you a current from left to right, electron current from left to right. So, that's this f up times 1 minus FD. This is the occupation of an up-spin. This is the probability of a down-spin state being empty so that the electron can go into that. Similarly, there's the reverse process. Down turns into up, that will depend on down-- availability of a down-spin electron and an empty up-spin state. And there's another factor we have to include. In order for an up-spin to convert to a down-spin, you need a localized down-spin impurity. That's this n sub D. Similarly, for this process, you need a localized up-spin impurity. Now, what we can assume is that this up-spin channel is essentially in equilibrium with the left contact. What that means is instead of f up, I can write f1. But f1 is the Fermi function in this contact. Similarly, instead of f down, I can write f2, which is the Fermi function in that contact. Now, you might say, well, didn't we always say that current is proportional to f1 minus f2? So, how come you have such a complicated expression though? And the answer is, yeah, this is a good example why current need not always be proportional to f1 minus f2. That only applies to these elastic resistors and where no entropy driven processes are occurring in that contact inside the channel. They're all in the contact. Now, here of course we have actually have an entropy-driven process in the channel. Note however that if nD is equal to nU, which is the usual equilibrium state, that if all these spins where like 50, 50 because it's an equilibrium, then you see that nD would be equal to the nU and so the current would be proportional to f1 times 1 minus f2 minus f2 times 1 minus f1. And now you'll notice that these cross terms f1, f2 cancel out, and so we get our old result that current is proportional to f1 minus f2. But in general, the point is that because we now have entropy driven processes being considered in the channel, current is not necessarily proportional to f1 minus f2. What we have is this expression. So what we want to do though is use that to find the open-circuit voltage. 

[Slide 4] To find the open-circuit voltage we say, well, current has to be zero. So, if current is zero, it means the first term has to be equal to the second term. And so with a little rearrangement, you could write it this way. Now, here there's a little bit of algebra that's often comes in handy and that is if this f is a Fermi function then, as you know, it has this form of one divided by one plus E to the power x. And so, 1 minus f over f, you can show it a little algebra, is E to the power x. So, what that means is I could write-- instead of 1 minus f1 over f1 and I could write exponential E minus mu1 over kT because that's the x basically. E minus mu1 over kT. And here, you have exponential E minus mu2 over kT. Again, just rearranging, dividing this by this, you'll get this exponential mu1 minus mu2 over kT is equal to n up divided by n down, which you could write in terms of probabilities, that if you have a total number of localized spins n, then n up is n times the probability of an up-spin and down is n times the probability of a down-spin. So, the ratio of n up and n down is the same as this ratio of probabilities. And that immediately gives you now an expression for this open-circuit voltage that left to itself, this battery develops this voltage of kT log P up over P down. And of course if it's an equilibrium, 50, 50 P up is equal P down, there is no open-circuit voltage but if it is in the state with all of them up then you do have an open-circuit voltage. 

[Slide 5] So, how much energy can you extract from it? So, the idea is you have this open-circuit voltage and now if you put something here like a little light bulb then you'd let-- you'll kind of draw electrons out from here and they would flow to your load. Now, the thing is that every time you draw an electron out, of course a new one has to be created, and in the process you need-- this has to be flipped into this. And in the process, one of these localized spins will flip the other way. So, the energy you extract is mu1 minus mu2 times dn, n being the number of electrons. But dn is actually equal to minus dn up, because every time you have to create an electron, the number of localized up-spins goes down by 1. So, you could write the energy then as dn up times mu1 minus mu2 and if you saw, put in the mu1 minus mu2 in terms of this P up and P down, what you're doing is-- then you have this kT and log P up minus log P down. And instead of n up, I'm writing n times P up. Now, with a little algebra, what you can show is that what we have here could be written in this form, it's T times integral of dS, S being the entropy. And that is of course the basic result, that way I'm trying to prove that the maximum energy that you could extract from this device is actually equal to what second law allows you to do which is T delta S. Now, in order to see that this is T dS requires a bit of algebra, that is you first write S using this second formula for the entropy that we discussed in the last lecture, that you have these two possibilities P up and P down so the entropy is P up log P up plus P down log P down. And so when you find dS, when you take the differential of this, you get these two terms. And when you take the differential of that, you get two terms again. But the point to note is because the system is either up or down, it means the dP up plus dP down, that must be zero so you can take that out. And here, instead of dP down, you could put minus dP up. And this DdS, this expression will basically become what you had here. So this was just in-- designed to show that the expression for energy becomes T times integral dS, initial to final. And so, what it shows is that you can extract this entropy. That what happens is, you started with all ups, that was kind of like information which gradually got randomized in the process, the entropy increased, information got lost. And you've extracted an amount of energy, T delta S, to do useful work. 

[Slide 7] So this is what you could call an info-battery, sort of, one that converts information into useful work. And this battery consists of this antiparallel spin valve, ideal. Two magnets pointing in opposite direction with electrons that can-- flowing in from one and here you have down-spins that are connected to this contact. Here, you have up-spins connected to that contact. And they are interacting with this background of localized spins which have all been put in an up state. And that will given you an open-circuit voltage. You could use that to light up a light bulb. But of course, as you draw energy from it, these localized spins will get randomized. And eventually, it will all be random and there won't be any voltage left. And it will run down like a battery, of course, to the run down. And how much energy could you extract from it? Well, the maximum is T times the increase in entropy. Now, one point you might say is that you should be-- that's interesting to note is that S equals 0 doesn't mean that all have to be up. It only means that you have to know exactly what state it is. It's this information. So, for example if you knew they are all down, that will be fine too. You could build the battery easily, that's trivial. But let's say that you knew they are half up and half down. So let's say you had something like this. Half of them you know is pointing up and other half is pointing down. Now, you might say, well, now if I put up-- try to make a battery the way I described it, you don't get anything, don't give you any voltage. Yes, that's true, that old design won't work. You have to now figure out a better design. For example, what you could do is separate out the two parts, don't keep them together. So, in that case, if you now put two terminals here, this is negative and that's positive just like before because you have all up-spin localized electrons. Here, you have down-spin. And so, here the polarity would be reversed. So, you'd have plus here and minus there. And what you could do is, you know, you kind of have now two batteries you see and the way you do, take two batteries and put them end to end to get the series combination so you could connect up this end and then take your power out of this one, so you could build a battery. Now, remember again, of course, if you had a very complicated, I guess, collection of spins then designing a proper device could be pretty difficult and building it could be impossible. But the real point here is just the conceptual point, that if you knew exactly what the distribution is, you could in principle build an info-battery to extract this information, I guess, in the form of useful energy. You could convert information into energy in principle. In practice, building the device, as I said, could be very hard. 

[Slide 7] So, basically as you know in this unit, our whole purpose was to give you a little insight into this very deep topic of what exactly this entropy driven processes are about that, as we started in this course, we say that, you know, if what makes it possible to describe nanodevices in a relatively simple way and what gives you a lot of insight into big devices as well, is this separation between mechanics and thermodynamics. And by and large so far, we never talked much about the thermodynamics. So, we just said, well, you have these contacts where they are held at with certain Fermi functions. And the purpose of this unit was to give you a better feeling for what these entropy driven forces are-- processes are about. And this last example in particular shows you a very interesting example involving entropy driven processes in the channel which actually do not involve any energy exchange so it is kind of strictly speaking still elastic but it's entropy driven and how that could invalidate this equation that we had been using for example. So, I guess with that, now it is time to sum up. And that's what we'll be doing in the next lecture.