nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 3.7: Electrostatic Potential
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[Slide 1] Welcome back to Unit 3. This is Lecture 7. 

[Slide 2] Now so far we are focused on this electrochemical potential and how it varies inside a device. And in this lecture what I'd like to tell you is about this electrostatic potential because you see, a lot of people would tell you that well you know electrochemical potential is kind of difficult concept, hard to define out of equilibrium. It's much better to stick to electrostatic potential. I don't quite share that view and I want to explain why. Now so what do you mean by the electrostatic potential or how do you look at it? Well one way to think is see we draw this electrochemical potential here, that's mu1. And I said earlier what it denotes is the level up to which the states are filled, at low temperatures at least and at high temperatures it's spread out a little but it's still a good measure of the level of filling. Now what I've drawn here below is the bottom of the bank, that is you have a band of energy levels that are available and this is like the place where the density of states ends. And for this discussion we're assuming you have just one band. Now the point is that when if you apply a voltage I've always drawn the electrochemical potentials and say that they are separated by the applied voltage. But the point is that if mu is separated by qV, that amount, the bottom of the band must also be separated the same way. In other words, when you apply a voltage you see this side, the positive side, goes down with respect to the negative side, but all the energy levels go down as well. So if this was the bottom of the band on this side, on this side it will be down by this qV. So that when you look at the electron density you would see much the same in both contacts because this contact, these are the states that are full, it comes towards the electron density in here. Same electron density in here, not that it has extra electrons or anything so it should also go down. And the question is, what does this bottom of the band kind of look like in the middle? Because this bottom of the band generally follows the electrostatic potential, except for a minus sign of course. That is because we are drawing electron energies, the energy is minus q times V, which is why when you have a negative voltage the energy levels are all higher. The question is what does it look in the middle and should this more or less follow what I drew here? Because remember what the mu denounces roughly the level of which things are filled. And you might think that well the bottom should also follow that. But if it did that then the electron density would remain unchanged from what it was before. And that of course cannot be quite right because you see if the electrostatic potential is changing like this, it means there isn't electric field in this region. Why? Because as you know the electrostatic potential, its gradient gives the electric field. So anytime the electrostatic potential is changing, there is an electric field and electric field goes from positive potential to negative potential. So this side is the positive potential, which is why the energy is lower. This side is the negative potential so that's the electric field. Now in order to give rise to an electric field you need charges. And so what you must have in order to be consistent with this variation of the electrostatic potential is positive charges on this side and negative charges on that side. Now this is what often people call the residual dipole. That any time there's an obstacle there'll be a dipole around it, which also sounds intuitive. There's this obstacle, electrons are piling up kind dearth on the other side. But the point is you need that in order to create this electric field which needs to be consistent with this variation and many electrostatic potential. But if U and mu are to follow each other exactly, then you see there will not be any change in electron density because after all this distance is a measure of the electron density. So in order to accommodate these extra electrons or this deficit of electrons, this curve cannot quite follow that. It has to be a little different. On the other hand, if this medium is very conductive, what that means if the density of states is very high, then what I can argue is that this electrostatic potential will approximately follow the electrochemical potential. Now let me explain why. Now if you want to write down the number of electrons, the way you are supposed to do it is you take the density of states and multiply by this filling factor, the fraction of occupied states. And the fraction of occupied states that's given by this f of E minus mu. And the density of states itself of course is shifted by the electrostatic potential energy. So if U is high at some point it means the density of states has moved up. If the U is low the density of states has moved down. So that's what is reflected there. Now what we could do is rewrite it a little bit by redefining e minus u as the new energy. So you could, since the integral runs from minus infinity to infinity, we could do this change in variables and write it as d of E. But then this argument now picks up U. So this is a simple transformation of variable from here to here. What it allows you to do is put kind of all the change into the Fermi function rather than into the density of states. Now you could then do a little Taylor series expansion that we always do, namely look at the change in N, that's delta N, due to a change in mu minus U. And these changes are then related by this integral. This is exactly the same Taylor series expansion we have been doing ever since unit one. But the point is then that it relates the change in N with the change in mu minus U but I mentioned instinctively of course electron density depends on this mu minus U. And it's related by this average density of states. That's the point I'm trying to make, that you could generally write the change in electron density and relate it to the change in this distance from the electrochemical, the difference from the electrochemical and electrostatic potentials through the density of states. Now what that means is if you had a very conductive medium with a high density of states so it's large D0. Then you see for a given electron density change you need a very small change here. So that's the argument that if this happens to be a very conductive medium, then all this extra electrons that we need to create the electric field, they can be accommodated with only a slight change in mu minus U, slight enough that on this scale you won't even notice it. So in a very conductive medium you could argue that the change in U would more or less mirror the change in mu. And if so, of course there are advantage people would say conceptually, but okay that's good. I don't really have to go into all these conceptual arguments about how you define electrochemical potential inside a device. We can just talk about the electrostatic potential. But the point is that usually we're not quite in this limit. I mean this is true only if you had a very conductive medium. More generally, especially when you're talking of small structures and semiconductors, it's really the density of states is really not high enough to make that argument. So in general what do you see? 

[Slide 3] Well in general the way the math goes is something like this, we still have this relation between delta N and delta mu minus U. But to calculate the actual change in the potential, this delta U, you start from the poisson equation, see, of electrostatics. And what we can do is take the delta n from here and replace it with the expression on top so that you get del squared of U is given by that quantity. Now this factor here That has the dimensions of inverse length squared and it can be defined actually as the inverse of the screening length squared. So you could rewrite this equation in terms of the screening length in this form. So you see what you have here is the first term is the del squared delta U. That's that. The next term is minus 1 over lambda S squared delta U. That's actually this quantity which is 1 over lambda S squared times delta U. And then the delta mu we have kept on the other side. So one could say that given a delta mu, this differential equation defines how the delta U will respond to it. Remember, delta U is like this electrostatic potential that we're talking about. Delta mu is like the electrochemical potential that we talked about. So how do you calculate delta U for a given delta mu? Well actually it's very easy, if the screening length is very short. Because then what you could argue is if this is very small, like a very conductive medium, because if this is very small, very high density of states, then that quantity is so big that you can drop this first term here, the del square. And if we drop it then we see you just get delta U equal to delta mu. That's kind of what I argued in the last slide that in a very conductive medium the change in mu is kind of followed or mimicked by the change in U. But in general you see if the screening length is not very big, then you have to solve this equation. Now how do you solve this equation? Well one way to think about it is, find the impulse response of this equation. That is, if the delta mu had been a-- if this change in mu had been a delta function at one point, like what you call an impulse, what would be the corresponding delta U? So you could define a screening function which is like the impulse response of that equation. And that screening function will look something like this where its width is roughly the screening length. Now this is the response to a delta function so what is the actual response to a given delta mu? Well it's like the convolution. So the change in electrostatic potential is like the change in the electrochemical potential convolved with this screening function. And when you convolve something what it basically does is takes what you had and spreads it out, spreads it out by a screening length. So finally what you'd get is not this red curve but rather this smoothed out curve. So this is the point I'm trying to make. That you see when you looked at the electrochemical potential that we had this sharp variations that helped us locate where the resistance is and so on, like you could look here and say, here is the resistance and that corresponds to this barrier. And here is the resistance and that's corresponding to the interface. Here we have a resistance corresponding to that interface. So we have a nice clean story that went with our intuition. But now if you look at the electrostatic potential, everything's got smeared out, smeared out the by screening length. And all that information is now kind of lost. Now of course this wouldn't be the case if the medium was very conductive. Then the electrostatic thing would follow that one and you could make the similar arguments with the electrostatic potential as well. But in general that's not the case. Now let me just say a few words about the pn-junction device that many of you are familiar with, those of you who are familiar with semi-conductor devices. Select the first device you talk about, pn-junction, where you have something similar, you see? 

[Slide 4] The pn-junction, as you know, there's a p-type material, there's a conduction band edge and valence band edge and there's a Fermi level that's closer to the valence band and the p-type material. And then there's an n-type material where the Fermi function is closer to the n-type, to the conduction band edge. At equilibrium the electrochemical potential or this Fermi level must be the same everywhere. What that means is this side is offset from this side by a certain built in potential. Now when you apply a forward bias that means you put a negative voltage on this side and it boosts this side up and so you have this separation of Fermi levels. And then you often draw this quasi-Fermi levels. There is a Fermi level, Fn, for the electrons, which is controlled by electrons coming in from the inside. Whereas there's a Fermi level for holes, or the valence band, which is controlled from the P side, and in this region they're actually separated around the junction and then they come together by the two bands. Now this, if you have taken a course in devices you have probably seen pictures like this and this quasi-Fermi level gives you a lot of insight into how current flows. On the other hand if you look at the electrostatic potential, the first point to note is that even at equilibrium there is this built in potential. And that's one key difference with electrochemical potentials. At equilibrium, electrochemical potential is absolutely flat. I mean that's the basic equilibrium statistical mechanics. What it says is just as at equilibrium, the temperature is the same everywhere, similarly, here at equilibrium the electrochemical potential is the same everywhere. That's an exact statement. On the other hand electrostatic potential is already got a major variation right there. And so if you believe that current flows due to electric fields, we have a problem here because there is a built in electric field even at equilibrium with nothing flowing-- I mean no net flow. And when you apply a voltage, of course, that electrostatic potential is reduced. So if you looked at the change in the electrostatic potential you'd see something like this, see what was like this has become less pronounced so the change would look something like this. But the point I'm trying to make is that that's not giving you much information again because electrostatic potential again is smeared out by the screening length and so has kind of lost all the information that you had when you looked at quasi-Fermi levels. 

[Slide 5] Now why do we still look, why do we look at this instead of electrostatic potential then why don't we just always look at this electrochemical potential? The answer is that again because it requires additional concepts. Because conceptually the electrochemical potential is a little harder to define and so requires more discussion. I mean we talked about all in the earlier lectures and it requires this introduction of quasi-Fermi levels, see, and we talked about the boundary conditions of the quasi-Fermi levels, for example. But all this requires additional discussion and justification. And what I'd like to do in the next lecture is go on to this proper justification of these which involves the Boltzmann equation. 

[Slide 6] So that's what we'll talk about in the next lecture. Thank you.