nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 4.2: Seebeck Coefficient
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[Slide 1] Welcome back to Unit 4 of our course. This is the second lecture. 

[Slide 2] We start with the Seebeck effect. And, because you know, so far we have been talking about current flow, and in our model, current flow is proportionate to f1 minus f2. The Fermi functions in the two contacts. And, because any material, when you connect the battery, across two probes, what it does, is it creates an electro chemical potential difference between the two contacts. This mu1 and mu2, which makes f1 and f2 different from each other, and so you have a current. Now, now there's another way of making the Fermi functions different in the two contacts. Let's say they are the same potential, but supposing there's a temperature difference. And, as you know, the Fermi function, it depends on this E minus mu over kT and usually the difference comes from a difference in mu, but it could come from a difference in T. So, if you have a cold side, then the Fermi function would look, kind of a block, it would change from zero to one, relatively sharply. Whereas, if it's a hot, the contact is hot, then the change would be more gradual, looking something like this. And so, f1 is different from f2, and so that should, could give rise to our current flow, and that's very true. But, there's a difference though. With the current flow that you get with the voltage. Now with the voltage, when you apply the voltage, the electrons always flow from the negative end to the positive end. On the other hand, when you apply a temperature difference, the electrons could flow from hot to cold, or from cold to hot, depending on the material. So, broadly speaking, there is two types of materials. One is what, you could call it, n-type conductor, where when you put two probes down, one hot and one cold, electrons inside the device will flow from the hot end to the cold end, and they'll come, and externally they will flow in to complete the circuit, like so. On the other hand, there's p-type conductors, where the electrons will actually flow in the opposite direction. And this is actually a standard experiment you do in semiconductor labs, if you want to find out if the material is n-type or p-type. You put down these hot point probes, and look at which way the current flows. So, how do you understand that? Well, if you just look at this equation, what I claim is, this difference again, you can understand in a relatively, simple way. That there's-- you see the current depends on f1 minus f2. Now, if we look at f1 minus f2, this is f1, this is f2, what you'll notice is, that at certain energy's f1 is bigger than f2, that is, if you look at this positive energy's. Positive, meaning the energies greater than mu, then you see that on the cold side, the formula function is essentially zero, but on the hot side, it's still not zero, because it takes a little longer to get back to zero. So, right here, it's positive, f1 is bigger than f2. Whereas, if you look below the chemical potential, on the negative side, that's where you see this one, is one, the cold side. But the hot side is a little less than one. So now, f1 minus f2, is negative. So, when you plot it, it will look something like this. This is zero, above mu, it's positive, below mu, it's negative. Eventually, of course, they all go to zero, in the sense that, at high enough energy is both are zero, at low enough energy is both are one. But, right around there, around the formula energy, you see this change in sign. Now, would that give rise to a current flow? Well, ordinarily you would say, probably not. Because, you see, the current at any energy depends on f1 minus f2. At certain energy's you got positive f1 minus f2, so electrons flow in one direction. At certain energy's it flows in the opposite direction, and so when you add it all up, add up all the energy's, you'll get zero, nothing much. And, that's kind of true, except that, you may not have the same number of states available at all energy's. So, you can imagine two types of situations. The n-type materials of those, where, if you look around the formula energy, you got more states available above, than below. But, as the p-type ones are those where you have more below than above. So, what that means is when you actually look at the current flow, in an n-type material, the positive side will dominate. You see this positive here, and negative here, but, there's lot more states here. There's hundred states here, and only fifty down there. So, the positive dominates. And, so that will be the net direction of flow. You look at the p-type material, now you see, you have fifty here and hundred down there. And so, it's the negative side that dominates, and hence the flow is in the opposite direction. Now, that sounds pretty straight forward, you see. But, what-- the usual description is not quite that straight forward. What you normally see in many textbooks is people say that, well, you know in n-type materials, its electrons that carry current. Whereas in p-type materials, when you're near the top of a band, it's the holes that carry current, and holes have positive charge. And, I never quite liked that argument, because, you see, holes are usually meant to be a conceptual tool, it's conceptually convenient in a filled band, to think of holes as carrying the current, but what actually carries current in any material we're talking about, are the electrons, and so, it doesn't seem believable that the measurable effect, something you can measure in the lab, depends on your conceptual convenience, that is some people might prefer to think of p-type in terms of electrons, rather than holes. And, holes are just conceptually convenient, in some cases. But, that cannot change the measurable effect. So, that is why I feel that this description, to say that p-type is holes, and hence, things were reversed, that really isn't quite the full story. What is much more straight forward is to say that, current is basically f1 minus f2, above mu it's positive, below mu it's negative, if you have lots of states above mu, you have n-type material, if you have more states below mu, it's a p-type material, So, really that's straight forward. 

[Slide 3] Now, let's do this mathematically. So, if you remember, when we were talking about conductors, we took this f1 minus f2, and did the Taylor series expansion to get the conductance for low bias. We can do a similar thing now, where we allow both the chemical potential to vary, as well as, the temperature. So, when you write f1 minus f2, it is like f for mu1 T1 and minus f for mu2 T2, and you could do the Taylor series expansion, but now you will have two terms. One, what we had before, that there is, you take the derivative of f, with respect to mu, multiply by the delta mu. But then, you'll have another term which is the derivative of f, with respect to temperature, multiplied by the delta temperature. And, what you can show, and I'll show this in a minute. You see, again, back in unit one, what we had shown as that the derivative with respect to mu, is the negative of the derivative which is respect to energy. And so, that first time, had become minus del f times delta mu. What we'll now show is, in a minute, is that del f del T can also be written in terms of the derivative with respect to energy, but you pick up an extra factor of E minus mu over T. So, finally, what you have is these two terms, and written as del f/del E, times this delta mu, plus this factor, times delta E. So, let me explain how you show that del f/del T. The derivative with respect to temperature is related to the derivative, with respect to energy. So, for that, what you note as, that you see f is a function of this x, and x is E minus mu over kT. So, when I'm writing derivative with respect to mu, what you do is this chain rule. You first take the derivative with respect to x, and then multiply by the derivative of x, with respect to mu. Similarly, if you want derivative with respect to energy, you first do df/dx and then take the derivative of x with respect to energy. And, if you want derivative with respect to temperature, again, the first step is the same, df/dx. But then, take derivative of x, with respect to temperature. So, all these three things then, the first term is the same. They differ, only because of the second term. So, what's that second term? Well, if we look at x, its derivative with respect to mu is, minus one over kT. What is the derivative of respect to energy? That's plus one over kT. And that's why, of course, this is exactly the negative of that. This one though, when you take derivative of x with respect to temperature, that's a little more complicated, it's one over T, so the derivative is like 1 minus over T squared, and hence. But what you can now show is, if you compare these two, that this one, is like E minus mu over T, times that one. And, that's the factor we have here. So, if you use this, what you can do is. 

[Slide 4] So, this f1 minus f2 then, you're writing as dfde times this, and you can now get an expression for this, linear response that is current related to the delta V and delta T. Remember the convention we're using in all of this, is when we say current, we mean electron current. Actual current is, in the opposite direction, similarly when you say electron-- when you say volt you kind of mean, you mean electron voltage. Now, how do we get G0? Well, that's what we've done before. When you substitute this quantity in there, you get this conductance, which is this integral over energy, it's a df/dE, times the conductance function. Now, for GS, which is related to the delta T, you get a similar expression. So, this is what we're adopting in Unit 1. Not the new result, but what the new part, is the second term. And, that has this extra E minus mu over T in it. So, the expression for the Seebeck coefficient involves this derivative of the Fermi function times that factor. And, these expressions you see mathematically capture what we have discussed physically, in the first slide, that is, when you look at the conductance, you realize the conductance is like an average of GE of this conductance function over around the Fermi energy, because, df/dE looks like that. But, when you look at this second coefficient, the one related to temperature, you notice there's an additional term factor here, which is anti-symmetric. So, when you plot that function, it doesn't look like that, it looks more like something that changes its sign. So, when you do this integral, it kind of takes the conductance function above the mu, and subtracts from it, the conductance function, below mu. Which is what we are physically argue, that it's this difference that determines how much current you get under the temperature difference. 

[Slide 5] Now, the next thing you could ask is, so what's this Seebeck coefficient then? So, what we have written down is an expression for the current for a given voltage difference, and the temperature difference. And, in general, what you could do is, you could take a system like that, and use it to drive some electrical load, it could be the lightbulb, it could be charging up a battery, it could basically extracting energy from it. That's this load resistor. Now, the Seebeck coefficient is, tells you, what is the open circuit voltage you would get if you connected it to a very high resistance, that is, if this was open circuited, I mean, that's like a very high resistance, it means no current is allowed to flow. Then what is the voltage difference that you'd get? And, if you put current equal to zero, you can see, that in order for the current to be zero, delta V over delta T, must be this minus GS over G0. So, that's what you usually call this Seebeck coefficient. And, the way it is defined, is negative for, n-type materials. This is open circuit voltage, for a unit temperature difference. So, what you have now seen then is, this very important physical phenomenon, the Seebeck effect, which allows you to convert a temperature difference in to an electrical voltage. And, in the next lecture, what I want to talk about is, kind of the reverse effect, which is called a Peltier effect. Where you can take an electrical voltage, and use it to create a temperature difference, make something colder, and something else hotter. But, use this is an effect that can be used to, for refrigeration to make something cold, for example. 

[Slide 6] So, that's what we'll talk about