nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.10: Summary
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[Slide 1] Welcome back to Unit 1 of our course, The New Perspective. Though this is the last lecture it's time to sum up what we did. 

[Slide 2] Remember we started with defining this concept of density of states. That's very important as I said. And if you want to describe current flow through any device the first step is always to draw this density of states diagram and locate the Fermi energy. Fermi energy or electrochemical potential, which we just denote with this mu 1 and there should be a mu 2 on this side. And they're separated by the amount of the applied voltage. A positive side is lowered by the amount of this voltage, V. Q times V. So that's the first picture that you need to draw. And then what we showed is that if you used this idea of elastic transport, that is electrons go from one contact to another without changing energy, then you see you can write the current in this form. That you can write the current at any energy as some conductance function times the difference between f1 and f2. Because, as we discussed, the essential point why current flows is because f1 is different from f2. So one contact tries to bring it into equilibrium with f1 another contact tries to bring it into equilibrium with f2. And in the process electrons keep coming in and going out. So that's the current through a given energy. And since every energy channel is independent because the way you're thinking is electrons, whatever energy they have coming in, they have exactly the same energy going out. So you can take all these different energies and just integrate it over energy to get the total current. So that's the basic current formula that we obtained in, I believe, lecture three. We then went ahead and said let's assume the applied voltage is small and obtain an expression for this current versus voltage. Rather current divided by voltage. And that's the measured conductance and this is given by an average of what we call the conductance function. Something that depends on energy. And if you have to average it over energy in order to compare with measured values, and if you're at low temperatures, than this function is very sharp. And so the value of this only at the Fermi energy is what matters. Fermi energy of this chemical potential. So what matters is only the values right there. Whereas if you're at relatively high temperature then you have to average over energy window of the order of kT. Now what's this expression for conductance? It depends on density of states and the time it takes for an electron to get from left to right. We then talked about the ballistic conductance where the time is given by length divided by velocity. And so you have an expression for the ballistic conductance, the Sharvin conductance that we talked about. And then we say that in general when it's not ballistic, see, then you can write it as the ballistic conductance times this mean free path over L plus lambda. And what's in the numerator here? GB lambda is what you can identify as conductivity times the area. So this point of view then, kind of, connects this ballistic conductance, which is what you normally learn in a course on mesoscopoic physics with the diffusive limit, which is what you learn in a normal course on devices or on solid state physics. And this, kind of, connects the two. You see it smoothly interpolates from one limit to the other. So we don't have a separate story for ballistic and another for diffusive. It's just one continuous story. If L is small you get the ballistic limit. If the L is large compared to lambda you get the diffusive limit. And this lambda here we defined as this diffusion coefficient two times the diffusion coefficient divided by the average velocity. And then we went ahead and-probably it was 

[Slide 3] in lecture eight that we obtained an expression for this lambda in terms of this velocity and time. That is we showed that the average velocity would be this v, which corresponds to a particular energy that is electrons and a certain energy, let's have a certain magnitude of the velocity, that's v. And in different dimensions the averaging gives you is different geometrical factors. And similar diffusion coefficient is v square tau. And then with these, different geometrical factors, again. And if you look at lambda from those you can deduce that's equal to v times tau velocity times the mean free time, which is normally what you'd have called a mean free path. But then we have this extra factor in here that depends on whether you are in 1D, 2D, or 3D. Now why do you have these factors here? Why not just define this as v tau if this has mean free path? You know, v tau is a mean free path. Well if you did that then what would happen is these equations would look a little more complicated. It would be like it's L plus - you'd have L plus some geometrical factor times lambda, which would depend on the dimensions. But the way we have done it, it's like this is a general expression and then you redefine your mean free path depending on the geometry. 

[Slide 4] OK? So one point I stressed is that this point of view not only gets you results for nanoelectronics for very small devices, but also connects smoothly to large devices. And that's kind of important, you see, because lots of devices that are of interest are really not quite in the ballistic limit. It's usually somewhere between ballistic and diffusive. It may not be completely in the diffusive limit but it's somewhere in there. And so what you really want is a theory that goes smoothly. That allows you to think in a continuous way from one limit to the other. Now you might say that, well, for large devices how is it that we are able to get this expression for conductivity, this general expression for conductivity that we talked about that normally you need advanced formalisms to get? You know, the one that is obtained from Boltzmann equation or from what you call the Kubo formula. We obtain that expression for the conductivity, actually. Just from this elementary discussion. And what made this possible is because we were using this idea of a elastic resistor. The idea that electrons go through the device elastically and all the dissipation occurs in the contacts. As I've mentioned before in the introduction that these are two very different types of phenomena. These are two branches of physics that developed separately, you see? And what usually makes transport such a difficult subject is that this is all mixed up. You see? So usually what you need is this Boltzmann equation for semi-classical transport. For quantum transport you'd need the quantum version of it. Where it would be Schrodinger's equation plus entropic forces. Whereas when semi-classical transport what you use is the Newton's Laws plus entropic forces, which is what the Boltzmann equation is. So in general your benchmark, your gold standard, is this Boltzmann equation. So all results that we are talking about here you can get from the Boltzmann equation. See? But the Boltzmann equation is takes some time to get used to. There are many things that you have to get familiar with and you have to spend a semester or two, you see, playing around with it to get there. Whereas what we were able to do is get results very simply because we made this assumption. Because when you make this assumption then you see the mechanics are separated from the thermodynamics. And so you can take care of the thermodynamics in a way that's so simple that you don't even realize what you're doing. You see the way we took care of it is by saying, that you know, we've got these two contacts, these big contacts, and there's all kind of thermodynamic processes going on there, and what they do is they maintain the contacts in thermal equilibrium. There's very complicated processes that maintain a thermal equilibrium and that's what is reflected in this f1 and f2. So just by legislating that the electrons and the contacts are distributed according to the Fermi function, we took care of the thermodynamics. But when an electron comes out in the contact, you see, it is a hot electron. It's got much more energy than it should have. And then you see there's all kinds of complicated processes that restore equilibrium. But we never got into those details. We just said that's OK. We know that what all those thermodynamic processes will do is they will maintain this reservoir at thermal equilibrium with a Fermi function f2 or f1, depending on which contact we are talking about. So that's how, you see, we were able to get our results in a relatively simple way without explicitly involving the Boltzmann equation. But whatever we are getting though is exactly what you'd have got from the Boltzmann equation and we'll talk a little bit about that in the third unit of this course. But you might say, well, but that assumption that an electron does not lose energy in the middle that, well, I can buy that for small devices but how can it justify that for long devices? 

[Slide 5] That's where, again, as I mentioned in the introduction, the way we are treating long resistors is we are thinking of it as if it's a series of little nano-resistors. Little elastic resistors. Electron goes elastically for a while and then loses some energy. Again, goes elastically for a while, loses some energy. Of course, in the real thing its continuous but here we, kind of, approximated as if it's going discontinuously. And then what we do is focus our attention on one of these little units and deduce an expression for conductivity and so on. And the thing is the answers we are getting is exactly what you get from a rigorous treatment of the Boltzmann equation. So not that you're losing anything by way of, you know, rigor here. The actual answers come out exactly the same. See? For low bias. Now for if the bias is higher then, of course, then this only gives you an approximately physical picture and that's something we'll talk about a little in the next unit. Now one subtle question that I wanted to stress then is that when I go from this expression for current to this expression for conductance we use this Taylor series expansion. And that expansion requires you to assume that the applied voltage is much less than kT. We say well, Taylor series expansion is something that only holds for a small value of X. And the X in this context is applied voltage over kT. So you might say, well, but then are you saying that you have this linear dependence of current on voltage only for small voltages? Because our flow kT's like 25 milli-electron volts. So it means if you apply a voltage bigger than 25 millivolts you'll think this expression won't work. But we all know that when it comes to big resistors they're often linear even if you apply 10 volts across it. Like if you got a little resistor from Radio Shack it will probably be linear. You could apply volts across it and it'd still be linear. But this theory seems to say that anything more than 25 millivolts and it won't be linear because I won't be able to use Taylor series expansion. And that is where, of course, what if you remember is when you have a long resistor like the one you buy from Radio Shack - see in order to apply this point of view you have to break it up into little resistors. And the length of each section should be, kind of, much less than an inelastic scattering length, in a length that an electron traverses before losing energy. So you should use a relatively short section here. See? And the point is when you're applying this theory we're applying it to one of this short sections. And so in order to be linear the voltage drop across one of these short sections should be less than kT. So the idea is you shouldn't have more than 25 millivolts applied across a small section who's length is roughly how far an electron travels before it relaxes energy. 

[Slide 6] OK? So with that then I guess we are done with this first unit of the course. Note that in this discussion we kept things kind of general in the sense that we expressed everything in terms of density of state without talking too much about what exactly the density of states is. How do you model it? Where do you get it from? What it is? Right? So the results are very general and could apply to any solid. Any solid meaning, you know, amorphous solids. We could talk about molecules. Could use this basic results that we talked about. In the next unit what we'll do is we'll adopt something called this energy band model. Something that is widely used for discussing crystalline semi-conductor. Crystalline solids, in general. And see how you can obtain expressions for the density and apply it to the formulas, the expressions that we developed in this unit. Thank you.