nanoHUB-U Fundamentals of Nanotransistors/Lecture 4.1: Introduction
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[Slide 1] Welcome back everyone. We are ready now to begin Unit 4 where we pull everything together in this course. 

[Slide 2] What I would like to do in this lecture is just tell you a little bit about what is in store for unit four. So, there are a number of lectures. We are going to be discussing a number of topics, beginning with a little discussion of transport theory and what determines the transmission when it is not ballistic, when it isn't equal to one. Then we will apply that understanding to MOSFETs and we will develop our general theory of the nanoscale MOSFET. That also works for larger MOSFETs. We will connect, once again, to the virtual source model for the final time and see how we can interpret the parameters in the virtual source model in a very physical way, from the ballistic limit to the diffusive limit and anywhere in between. I want to discuss, just briefly, how we can use this model that we have developed to analyze experimental data and extract physically meaningful parameters from measured data from nanoscale transistors. And then we will end the course just with a very brief discussion about what the ultimate limits of MOSFETs might be. 

[Slide 3] So, we had been discussing in unit three the ballistic MOSFET, focusing on the virtual source or the top of the barrier, where we assume that we know the charge, given by equilibrium MOS electrostatics, as we discussed in Unit 2. And we focused on trying to understand how fast a charge moves in a ballistic MOSFET in Unit 3. We derived this -- we began with this expression for the drain current, product of charge times velocity. We assumed ballistic transport transmission equals 1. And we derived an expression for the average velocity at the virtual source as a function of drain to source voltage. It was the ballistic injection velocity times some function of the drain to source voltage. 

[Slide 4] Now let's look at that a little more carefully. So, here is our current. Here is our assumption of ballistic transport. Here is our expression for the average velocity as a function of drain to source voltage. For Maxwell Boltzmann statistics, things simplified enormously. Our drain saturation function is given by this simple expression involving exponentials of the drain to source voltage. And we can understand this easily in terms of thermionic emission over the barrier from the source over the energy barrier in the channel, from the drain over the energy barrier in the channel. And, in this case, the ballistic injection velocity is just our old friend, the unidirectional thermal velocity, which depends on the effective mass of carriers in the semiconductor. More generally, especially above thresholds and especially for III-V semiconductors, where the effective mass is light, we should use full Fermi-Dirac statistics. So if I assume two-dimensional electrons, that's electrons moving in the plane of a MOSFET, then our drain saturation function is a little more complicated. It involves ratios of Fermi-Dirac integrals instead of exponentials. The ballistic injection velocity now is the unidirectional thermal velocity, but now it depends on a ratio of Fermi-Dirac integrals. And the argument of those Fermi-Dirac integrals is this term eta F, which tells us where the Fermi level in the source is located with respect to the conduction band at the top of the barrier. That is gate voltage dependent, so now our ballistic injection velocity is gate voltage dependent. So the expression up here assumes nondegenerate carrier statistics. More generally, this saturation function depends both on the gate to source and drain to source voltage, and this ballistic injection velocity depends on the gate voltage, because that determines where the Fermi level is with respect to the top of the barrier. 

[Slide 5] Okay, now, the on-state or the on-current is an important device metric that we discussed in unit one of this course. If we look at the on-state, then we have applied a large drain to source voltage. We can simplify our saturation function. Our drain saturation function reduces to 1 in the on-state. In that case, the average velocity is just the ballistic injection velocity. If we assume nondegenerate conditions, that is just the unidirectional thermal velocity. So the average velocity at the virtual source in the on-state is this unidirectional -- is this ballistic injection velocity. So this is a very important metric for devices, and device designers focus a lot on maximizing the injection velocity in nanoscale devices. 

[Slide 6] So, the question that we are going to be addressing in Unit 4 is: if we begin with this ballistic treatment, how does it change if we -- how does it change in terms of what changes do -- if we relax the assumption that the transmission is 1 and allow the transmission to be anywhere between 0 and 1, now how does that drain saturation function change? How does the ballistic injection velocity change? We no longer have ballistic cases, but we will have an injection velocity. 

[Slide 7] So we might try to guess what the answer would look like. If we go back to the ballistic IV characteristics that we derived in Unit 3, we had this general expression assuming transmission was one. You know, we understand what it reduces to for low drain to source voltage, what it reduces to for high drain to source voltage. Now we have to relax the assumption that the transmission is one. We'll have some finite transmission. 

[Slide 8] So we might guess that what is going to happen is that we will simply multiply the linear current by some average transmission. Average meaning average over the energy channels that are populated at that particular gate voltage. And we would do the same -- at least we would guess that we would do the same in the on-state. Simply multiply the ballistic current by some average transmission averaged over the energy channels, which will be less than one, and therefore the current will be reduced. Well, we also might guess that our general expression we would do the same thing. So it looks like it could be quite easy. Now, it turns out that it is really a little more interesting than that. This works for the linear current; it does not work for the saturation or on-current, and it does not work in general. And the reasons that it doesn't work are really quite interesting and we'll spend a little bit of time discussing that and we will derive the appropriate IV characteristic expressions in the case where the transmission is not 1. 

[Slide 9] Okay, now, the first thing that we will dive into in this lecture -- beginning in the next lecture is trying to understand transmission in a little more detail, relating it to what goes on inside the device. So we'll think about injecting a flux of electrons into a nanodevice. This will be the small region near the top of the barrier. There will be some mean-free path, some average distance, between backscattering events due to impurities, lattice vibrations, whatever is going on inside this region that might scatter carriers. And some fraction of the carriers will emerge from the other side. The fraction is our transmission. And we will set up a model problem where we have something called an absorbing contact that does not inject anything from the right, and we will see if we can calculate this transmission in terms of the mean-free path and relate that to the scattering physics. Okay, so we expect that the transmission is going to be some function of the average distance between scattering events. That average distance between scattering events is known as the mean-free path. And our first task will be to derive an expression for the transmission in terms of the mean-free path, and that relates the transmission to the scattering physics and then we are ready to use it. 

[Slide 10] Now, in the limit that the transmission is very small, then we are in the region of long channel lengths, classical MOS theory as it was first developed in the 1960's or so. That's what we call diffusive transport. If the transmission is one, then we are in the ballistic limit. That was the subject of Unit 3. What we seek to do in Unit 4 is to develop a way to understand MOSFETs that are anywhere between the ballistic and diffusive limits, and that is the transmission model that we will be developing in these lectures. 

[Slide 11] So, our objectives for this -- for this particular lecture are very simple. We, first of all, want to establish the relation between the mean-free path for backscattering and this transmission and then we want to evaluate the Landauer expressions for the current and derive the IV characteristics in the case where the transmission is anywhere between 0 and 1. That is what we set out to do in this unit, and then we will try to understand those IV characteristics and apply them to real devices a little bit as well. So, with that, we are ready to begin Unit 4.