nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 4.1: Introduction
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[Slide 1] Welcome back to our course on the Fundamentals of Nanoelectronics, and this is the fourth unit. And this is our introductory lecture, so I'll try to give you a feeling for what this unit is about. 

[Slide 2] Now, as you know, so far we have been-- whatever we have been discussing has been on this basis of this concept of an elastic resistance, the idea that electrons get through this channel without losing, exchanging any energy and all the energy exchange happens at the ends. And the thinking is that, of course, when current flows through a device, there is heat that is generated but that generation of heat is all in the contacts. And this assumption allows us to get a relatively simple equation for the current in terms of the Fermi functions in the two contacts. And as I mentioned earlier that what makes this relatively simple conceptually is that these two very distinct types of processes are spatially separated in this model. That is where this generation of heat which is this entropy driven process, this thermodynamics as opposed to just the pure force driven, what we call the mechanics and the entropy driven processes are in the contacts and you are not getting into the details of it, you are handling it just by arguing that the contacts are all just maintained in equilibrium, and so they have these Fermi functions, f1 and f2. Now, what we want to do in this unit is go a little deeper into this. And first let me point out that these entropy driven things or heat generation is kind of different from usual mechanics, in the sense that-- you see, although, you know we understand that heat is a form of energy and energy is conserved, that we all understand. So, the way you would think is, here is a battery which gives energy, and that energy is getting converted into heat and that goes out into the contacts. But could we reverse the process? Could we imagine a situation where the electrons, instead of going this way, actually flow the other way? Of course, in that case, electron is getting from here to here, is picking up energy. So, it only dissipating energy, and what it will be taking is taking in energy from the surroundings. In other words, the normal process we all understand is electron, the energy comes from the battery, gets converted into heat. But could you turn this around? Could the electron just take heat from the surroundings and charge up the battery, or maybe even light up a little light bulb? Now, you don't need me to tell you that that really never happens. So, you can see there's a-- Although this wouldn't be against energy conservation, you know, we all understand that energy is conserved, but this would also conserve energy. You'll just be taking heat from your surroundings and lighting up your light bulb or charging up your battery. So, no problem with energy conservation. Something else is involved. And that's largely what we'll be talking about in this unit. And in the simplest terms I guess the difference is this, that heat is a form of energy but it's this random form of energy and it goes into this random motion of atoms and into these many degrees of freedom. So the basic point is you can take energy from a battery and dispense it among many degrees of freedom, that's easy. What's-- what you cannot do is take energy from many degrees of freedom and concentrate it all into one. So that's the distinction. That's why one way is easier than the other. 

[Slide 3] But then you-- doesn't mean though that you cannot convert heat into electricity. It is just that as long as your surroundings are all at the same temperature, you cannot just take heat and light up a light bulb. But, if you had for example the source and drain at two different temperatures, then you could extract a certain amount of energy and charge up your battery, light the bulb, and that's one of the first things we'll talk about in this unit, the Seebeck effect, which is a very interesting effect even from a practical point of view as well because it allows you to convert waste heat into electricity. Another related thing we'll talk about after that is the Peltier effect, you see where just by running a current, you can cool one side and heat up another side. And that's an effect that is used for refrigeration, for cooling tanks for example. So, in other words, it's not like you cannot go between heat and electricity, you can convert heat into electricity and use electricity to create a temperature difference. You can do that but there are limits on this. And to express these limits, what we then consider is a more general model. 

[Slide 4] Think of a situation where you have a channel with certain levels, contacts, and let's say N1 electrons come in from this contact and N2 electrons come in from that contact. Now, of course, I'm just-- these are just reference directions. In practice of course, if 10 electrons come in here, 10 must be leaving over there. Right? So, and do-- if N1 is ten, N2 must be minus ten. And that's kind of reflected in this equation I've written up here. You see, N1 plus N2 equals zero. So in other words, these numbers are kind of reference things used to do the bookkeeping. So N1 plus N2 is zero. Similarly, as electrons enter the channel, let's say they bring in a total amount of energy E1, and on this side total amount of energy that comes in let's say E2. Again, what energy conservation requires is that E2 should be the negative of E1. You know, whatever came in should leave, steady state. So that's this E1 plus E2 equals zero. Now, what I want to point out is that there is another rule that isn't as apparent, and that is think of what happens at this contact. You have electrons leaving which are taking out an energy E1. The new electrons that come in, however, come in right around the chemical potential. These are called conductive regions. We have a chemical potential there and the new electron that comes in has an energy right around it. And so, if N1 electrons come in, they actually bring in an energy of mu1, N1. So think of this contact. What came in is mu1, N1, what left is E1. So if this was everything, then of course the contact would be getting colder, because more is leaving than coming in. So it's steady state, what must be happening is there must be a certain amount of energy just coming in from the surroundings in the form of heat, which is E1 minus mu1 N1. And the point I want to make is this is just a reference direction. In practice, energy could either be coming in or going out. And similarly when you look at the other contact, what's coming in is E2 minus mu2 N2. And the basic principle, this is what's called the second law of thermodynamics, is that if we look at the amount of energy that came in from here, divided by the temperature of that contact, look at what came in from there divided by the temperature of that contact, add it up, then overall it must be negative, negative or equal to zero. Now, if the two temperatures are equal, then what it basically is telling you that the total amount of energy coming in from the surroundings, from total amount of heat that you are taking in must be negative, which means basically you'll have to be generating heat. That's the kind of the first thing I said that you can't just take heat from your surroundings and charge up your battery. So, when the two temperatures are equal, you can kind of see what this is saying. And if the temperatures are unequal, then you see this is a more general principle, in the sense using this principle you see you could understand why this is allowed, what the limitations are, why something like this is allowed, what the limitations are, and why something like this never happens, for example. This is a basic principle that-- And it's an inequality so in that sense anything you observe would be in accordance with this second law. And the second law, this is called the second law. Of course, the first law is just the conservation of energy. So, what we wrote down there, you could call that the first law of thermodynamics. Whereas, this is the second law which is I say again, much less appreciated and much more subtle and generates a lot of discussions. Now, you might say, well, how come you're telling me about all this at the end of the course? I mean, whatever we have been discussing, are you sure? Is it in accordance with the second law? As you know, much of what they have been discussing has been based on that equation. And one of the things I'll show is that that equation automatically ensures that the second law is satisfied. So, in that case, you don't need to have--you see, you don't need to have any worries on that score. Not that whatever we have done is in accordance with that. And what makes it relatively straightforward again is because we assumed that the channel part is elastic, no energy is exchanged. 

[Slide 5] The more general situation would be if in the channel, there is an exchange of energy E0 with the surroundings. So, this would be a more general case. In that case, it wouldn't be an elastic resistor then. Yes, or this Landauer resistor that we talked about. Now, here, of course we see easily how to modify the first law. You just add the E0 there. Total energy is conserved again. So, whatever comes in, that's all equal. That's equal to zero and that sounds fine. Similarly, for the second law, you would add in another term which should be is E0 over the temperature of that source. Now, these three terms you see are kind of what's usually called entropy. I guess it's the negative of the entropy that is anytime you have a reservoir and you add energy to it, the entropy increases by the amount of energy you add divided by its temperature. And--So here, of course you are not-- you're taking energy N, so that's like reducing the entropy. So, these are kind of the negative of the N. So, this is the negative of the entropy change of reservoir-1, reservoir-2 and this reservoir-0. And that all these negative entropies added up must be less than zero so what that basically says is that the overall entropy must be increasing. That this delta S must be positive because minus delta S is negative, so delta S must be positive. So, this is what we'll talk about in the later lectures in this unit. This entropy and how this concept of entropy leads to the law of equilibrium. And the point you notice that basically the second law then requires that this entropy is increasing. So, what that means is, you know, earlier I pointed out that if you had a source and drain at the same temperature, you couldn't just take energy from the surroundings and charge up your battery or light up a light bulb. But if this channel communicates with another source like you dump some heat into it, then you could do that. So, you could be taking heat from these contacts, dump a part of it in here and use the difference to light up a light bulb if this is at a lower temperature than those for example. So-- But all that follow from the second law. So, the second law kind of tells you what is possible. Of course, what it doesn't tell you is how to design a device that will do that or even do that efficiently. That is up to you to figure out. But what it tells you is what you fundamentally cannot do no matter how well you design it. So, it's important to be aware of what the requirements are, what the limitations are based on this second law. And these are the things we'll be talking about. And one of the interesting aspects of this is when you see, here we talked about the exchange of energy with the contact or reservoir whose temperature is T0. 

[Slide 6] But another interesting situation is when you are interacting with a non-equilibrium system which may not be describable by a temperature. Of course, you could still use the second law but this first term, first term here, you cannot write as E0 over T0 because T0 isn't quite well defined. So-- But you could write it as a negative of the entropy change. So, let me give you an example. Suppose we consider a system of spins. What I mean by that is say things like magnetic ions in your solid. And these magnetic ions, let's say, could be pointing up or down like magnet in these ions. And so, let's say your solid has a lot of these ions sitting there and if it's in equilibrium and they're not interacting with each other or anything. So, if it's sitting there, they'll all be random. Some would be looking up, some would be looking down. So, that's sort of the equilibrium state would be like. Now, supposing I take all of them, make them all look upwards and connect them to the channel, connect them in the sense that as the electrons flow through the channel, they interact with those ions, exchange spin with them in the sense they can flip those spins as they go along. So, when electron comes in with up spin, seize an ion with down spin, can flip it and move on, so they interact with it. So, let's say we have a situation like this and then the point is that you started it out this way and as the electrons interact with it, it will gradually become this random equilibrium thing. And the-- And in the process though, no energy is exchanged. Why? Because whether a magnet is spin-- looks up or whether it looks down, it has the same energy. So, we are assuming these are what we call paramagnetic ions. Up and down have exactly the same energy here but no energy is exchanged, but still the entropy changes. 

[Slide 7] Entropy changes because when they're looking all up, you see, if you have a collection of say one hundred spins and if they are looking up, there's only one way that can happen. It's because they are all looking up. And entropy, as we'll discuss, is like related to how many ways this can happen. So, it's like-- it happen only in one way and so the entropy is zero. It's like log of one. On the other hand, the equilibrium state is one hundred with fifty up and fifty down and that can happen in many ways. So, that actually has an entropy that is nonzero. And we'll discuss it, it is over this Nk log 2 where N is the number of spins here and k is this Boltzmann constant. So, in going from here to here, you see the entropy has changed. Entropy has increased. So, because the entropy has increased, it allows you to extract some energy and light up a light bulb. But the point is the energy is not coming from the spins though because this state and this state have exactly the same energy. But since the-- you're paying a cost in terms of entropy, you are now allowed to take energy from the surroundings and light up the light bulb. You know, something that's basic-- I started by saying that never happens but here you can actually do that because, again, it's-- this-- here, you're paying the entropy cost of it. So, if you did that, it wouldn't violate second law. Now, the point I want to make again is, it's not like if you hook it up, you'll be able to extract that energy. You have to figure out how to design your device appropriately to do that. But what second law assures you is that you should be able to do that and it won't-- it shouldn't be impossible. So, if you are clever and you think about it, it will come up with a design. And in the last lecture, I guess I'll try to show you a concrete design which will do just that. So, this is what we call like fuel value of information in the sense that it is almost like here everything was up so you had a lot of information about you knew exactly what state it is in. And by letting it become random, you're kind of creating information for energy. So-- Anyway, so these are interesting, these are not meant to be necessarily of any practical importance really. It's either designed to illustrate the concept and it's also related to another very interesting thing you'll see a lot in the literature which is called this Landauer's principle. And Landauer's principle, you'll see this phrase about information erasure, erasing information. That is if you are in this state and you have turned it into all up, that's kind of like erasing information in the sense that there could be many states which are half up and half down but irrespective of where you start, you do something to it so that the end result is independent of where you started. So, that's kind of like erasing information. Whatever was there is gone. And the Landauer's principle says that if you erase this and turn it into that, you have to spend an amount of energy of NkT log 2. And the way it kind of follows from what I say it is that once you are here, you can design a device to extract the NkT log 2 from it. And so, if you didn't have to spend any energy getting from here to here, then you see you could be continually generating energy, that is take this, turn it into that, use your device to extract NkT log 2, turn it back, use it-- use your device again to extract so you could be continually extracting energy. And that is one that-- that you cannot do. You cannot be extracting forever. And so, the process of erasing must fundamentally require that amount of energy. So, this is something, again, you'll see a lot in the literature, discussed in the literature. And this also has Landauer's name to it. So, Landauer was a very thoughtful person, you know, had-- is a deep thinker and kind of ahead of his time. And the two important contributions of his that had a lot of impact on the field, in a sense inspired a lot of people to work on them and take it much further. One of them is what we started with, this idea of an elastic resistor or-- and you'll see the Landauer formula, that we have been talking about in our course. And then there is this other thing that you'll see his name with-- associated with the Landauer's principle. And usually these are two very different things and this means people who are aware of one are not necessarily as aware of the other one. But the way I feel is both of these actually served to clarify this very deep issue of mechanics and thermodynamics and their relationship. And so I feel it's indeed fitting that we kind of start this course with one of Landauer's ideas and end it with the other-- the second of Landauer's ideas. Anyway, so we have a lot of concepts to cover. So, it's time to get started. 

[Slide 8] So, we'll get started with the next topic or the first topic here and that's this Seebeck effect, how you convert temperature differences, use temperature differences to extract energy.