nanoHUB-U Fundamentals of Nanotransistors/Lecture 4.9: Unit 4 and Course Summary
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[Slide 1] Hello everyone. So we are at the end of this course. And before I say goodbye I just want to spend a few minutes, summarizing some of the key concepts and understanding that I hope you'll take away from this course. So this is both the summary of what we did in Unit 4, but more broadly a summary of what we did throughout the entire course. 

[Slide 2] So what we set out to do is very simple. Really two things, the first was to develop a solid physical but simple understanding of the operation of a modern day nanotransistor. The second objective was to relate this new understanding of very small transistors to the traditional theory of MOSFETS. If you've had a course in MOSFET theory before, chances are you've learned the traditional theory, it's still widely used. But one of the objectives in this course was to relate a physically sound way of understanding carrier transport to this conventional way that was developed for longer channel length MOSFETS. 

[Slide 3] Now we found in these discussions that are very useful way to understand MOSFETS was to draw an energy band diagram. This is an energy band diagram of a MOSFET under high drain bias. We focused on this top of the barrier, which we also called a virtual source. That's what we took as the beginning of the channel. Now in a well designed transistor, one that is electrostactically well designed, the height of this energy barrier between the source and the channel. Is controlled strongly by the gate voltage. And is only weakly dependent on the voltage that we apply to the drain. So that's becoming more and more of a challenge as MOSFET channel length scale down. But that's the essential goal of transistor design to maximize the control of the gate over that potential energy barrier and then we modulate the current that flows out of the source just by pulling the barrier in the channel up and down with the gate voltage. As we also discussed, there is a very short region near the beginning of the channel, where the electric fields in the channel direction is low. This is a bottleneck for transport. Carriers have to get across this bottleneck without scattering or without scattering too many times. Once they merge into the high field part of the channel, then they're simply collected. So we could think of this low field region as analogous to the base of a bipolar transistor. And we could think of the high field portion of the channel, as analogous to the collector of a bipolar transistor. As something that is operating as an ideal absorbing contact. And the electron that makes it to that region is simply absorbed and exits through the drain. So this is a very simple conceptual picture that illustrates the operating principles of transistors, whether they're long channel or short channel. And you know when you take a course in MOSFET theory sometimes you can get all tied up in some rather complicated mathematical analysis. Sometimes it's all based on diffusive transport models that are even questionable at the nanoscale. And you can lose track at how simple the operating physics of this device actually is. This is something that's been known for a long time. Was first pointed out in a paper that is still worth reading, a paper that was published in 1973. 

[Slide 4] All right now the description of transport that we used in our studies here was a so called Landauer approach. Which provides a simple conceptually sound way to treat transport at very small length scales. We learned that it also works at very large length scales, and that makes it very attractive to go seamlessly from short channels to long channels. We described the current in terms of some fundamental constants. A transmission that's just a number between 0 and 1, which is a probability that an electron injected from the source comes out the drain and vice versa. The number of channels at that particular energy which is related to the band structure of the semiconductor. And the Fermi functions of these two contacts which had some nice properties we call them ideal landauer contacts that are maintained in thermodynamic equilibrium at all times. So this was our starting point for the transmission theory that we developed. 

[Slide 5] And we developed that theory by learning how to apply it to a MOSFET and our goal was then to evaluate the current voltage characteristics and we did that throughout the entire voltage regime. What I'd like to just remind you of, is what the solution looks like under low drain to source voltage the linear regime and what it looks like under high drained source voltage, the saturation regime. 

[Slide 6] So in a linear regime, we can write the final result in a form that looks very much like the traditional result, W over l apparent mobility charge times drain voltage. Now this apparent mobility is related to the transmission in the linear region of operation. And that transmission in a linear region is related to the mean free path for back scattering and determined by the length of the channel. Now in the saturation region, we had an expression for the current was width times charge times something we called injection velocity. And the injection velocity is related to the transmission under the saturation conditions and to this ballistic injection velocity, or thermal velocity. We had an expression for that transmission in saturation again it's related to the same mean free path for back scattering and to the length of this little bottleneck region that we just talked about. And that little bottleneck region is a small fraction of the entire channel length. 

[Slide 7] Now we spent some time trying to understand the physics of scattering in these very small devices. We derived this expression where region that has a very small or no electric field. So we had a simple expression that actually works remarkably well for the transmission. That expression works in a linear regime. And in the linear regime the entire channel is susceptible to back scattering, an electron that back scatters anywhere in the channel can turn around and exit through the source and not leave through the drain. So the appropriate length over which the scattering is important, is the entire length of the channel. Now in the saturation regime we pulled the energy barrier down, we have a very small region where there's a low electric field. We learned that in a structure like this, a low electric field region followed by a high electric field region that overall transmission is controlled by the transmission across the low electric field region, that's our little bottleneck region. It has a length l, which is a small fraction of a channel length and that's why the transmission under saturation conditions is distinctly higher than it is under the linear region of operation. So in general this critical length, that varies from the entire channel length to a small fraction of the channel length. Depends on the drain to source voltage in a way that's actually rather difficult to calculate and we didn't go into in much detail. 

[Slide 8] Now we also related our transmission model to something we call the virtual source model in particular to MIT's implementation of this virtual source model. The MIT virtual source model goes like this. We expressed the drain current as width times charge times velocity. The charge we determined from MOS electrostatics using a semiempirical expression that describes the charge from below threshold to above threshold accounts for the finite subthreshold swing greater than 60 millivolts per decade if it is, the finite DIBL that's there. We then described the velocity of electrons at the virtual source and by multiplying the injection velocity by this empirical drain saturation function. This empirical function with an empirical parameter beta is a function that smoothly takes us from the linear regime to the saturation regime. And VDSAT is a separation in drain voltage between the linear and the saturated regions of operation. The nice thing about this model, is that it has only ten parameters and most of those ten parameters are very clear physical parameters that we can extract or input to the model. A couple of empirical parameters that we need to fit the IV characteristics to. 

[Slide 9] Okay, so what we learned is that if you look at this virtual source model this was a model that was developed to describe nanotransistors. I derived it in a different way, I derived it from a traditional prospective and then we saw that a model of nanotransistors is very similar to the traditional model for long channel transistors with two changes. The two changes are that the actual scattering limited mobility is replaced by something we called an apparent mobility. And the high field saturation velocity due to scattering in a high electric field is replaced by something we call the injection velocity. 

[Slide 10] And we learned how to clearly interpret those two parameters from a physical point of view using out transmission model. So the replacement of the real mobility by the apparent mobility was a first step. The apparent mobility is related to the transmission in a linear regime. We can also express it as a combination of the scattering limited mobility and the ballistic mobility. The scattering limited mobility is proportional to the mean free path. Now you may be use to seeing mobility written as as q tau over m where tau is the scattering time. This is just another way to write the same thing. In this case, instead of using the average time between scattering events we're using the mean free path for back scattering. And in the ballistic mobility we just replace the actual mean free path by the length of the channel itself. The second change to the traditional model is to replace the high field saturation velocity which doesn't have much physical meeting in short channels. By this quantity we call the injection velocity. The injection velocity is related to the transmission and saturation. And we saw that we can express this as this velocity as, the lower of either the ballistic injection velocity or the velocity at which electrons diffuse across this thin bottleneck regime. Okay, so we have a physically sound transmission model and we understand how it connects into the traditional model. And now we understand a little better why the traditional model works so well at the nanoscale. 

[Slide 11] Now just to go back to the Landauer formulation, this was the basis, this was our starting point for developing the transmission model. This is a very simple expression, it captures the essential physics of transport at the nanoscale. It is a model that is not a replacement for detailed numerical simulations, indeed much of the insight and intuition that I put into the development of this model came from doing detailed physical simulations and trying to interpret them. One can actually derive the Landauer approach from the semiclassical Boltzmann Transport Equation. One can also derive it from a more rigorous description of dissipative quantum transport, so for example the non-equilibrium Green's function approach is widely used these days. And this provides that clear connection of this simpler model to these more detailed rigorous microscopic models. And so this shouldn't, this Landauer approach should be viewed as something that compliments some more detail models. Help us develop some understanding and insight into what those detailed models are telling us. Maybe even helps us pose questions that we go to the detailed models to seek answers for. So the detailed models and the simpler model that we've been discussing in this course actually go together rather well. 

[Slide 12] Now I haven't talked much about the limitations of this model. Our goal was to capture the essential physical ideas. Now when you look at the physically details simulation you can see that things get quite complicated. We've tried to extract the important first order principles and we found that when fit the experimental data they actually do rather well in describing the experimental data. But there are some limitations that you should be aware of. Let me just mention a few of them. Now we write current as product of charge times velocity, that's fine. But we have taken the charge from equilibrium MOS electrostatics, now when current flows we're out of equilibrium. Can we use the same charge? You might expect that there would be less charge when it's now flowing and it's harder, it's harder for that charge to maintain itself. So this is an assumption that people worry about, it's a little bit of work going on these days in trying to improve that assumption. And treat the non-equilibrium effects that occur. When current flows and the charge has to be modified. 

[Slide 13] Now this very simple treatment of scattering that we have we can also you know think about exactly how good is this? How good is our assumption that even under high bias, where were very far from equilibrium that we can still use the near-equilibrium mean free path to describe the transmission? We argued that at the beginning of the channel we're still near equilibrium but exactly how close are we and how good is this? This is something that we need to go the physically detailed simulations to answer. There have been a number of detail studies that have addressed these concerns. This critical length for back scattering. You know exactly what is it? How do you compute it? There have been a number of papers, it's a non-trivial problem we can get some simple estimates if we want to do it properly again we need to look at very detailed physical simulations. 

[Slide 14] Another assumption are these special contacts. Our device was assumed to have these ideal Landauer contacts that always maintain thermal equilibrium, that can always be described by an equilibrium Fermi function. How well do those contacts apply to real devices? In fact it turns out especially in some III-V transistors that in effect called source starvation can occur and that's really a breakdown of this ideal Landauer contact picture. 

[Slide 15] So there are a number of limitations. The fact that when we apply this model experimentally to very small silicon and very small III-V transistors, tells me at least that we are describing the essential physics. But we have think very careful about what the limitations are as we refine the models and look more carefully at experimental data as transistors continue to evolve. It's very important to be aware of these limitations and to see if we can, what we can do to address them and to improve the model and for those of you that are interested in learning more about this. There's a discussion about some of these issues in the lecture notes. 

[Slide 16] So let me just summarize what I hope you've have taken away from this course. So we have been talking about an incredibly important device, it's had a really profound impact on the world that we live in. It's really hard to envision the modern world without billions and billions of nanotransistors. Now not only are these devices important from an economic and technological perspective but they're also really interesting scientifically. And they provided us with an example in which we can study current flow at the very smallest dimensions that we're able to manufacture these days. We've also learned I hope that the Landauer approach provides a very convenient way to understand electron transport in these small transistors. And I hope you've taken away from this course, the understanding that the operating principles are really quite easy to understand. There really as easy as the conventional MOSFET theory that was developed in the early 1960's. Sometimes it takes people a little while to get comfortable with them because the concepts are quite different from the traditional concepts but they're no more difficult, it's no more, it's no harder to treat the nanoscale transistor properly than it is to derive the I-V characteristics of a long channel transistor. 

[Slide 17] And just a couple of final thoughts before we sign off here. So I hope you understand now, you leave this course with a good firm understanding of the essential operating physics of these nanoscale transistors. I hope you also understand how this transmission model that we've developed is related to the traditional model that you may have been familiar with from previous courses on MOSFETS and I hope you've gained an appreciation for the usefulness of the Landauer approach in analyzing very small devices but also very large devices that provides us a way to go seamlessly from large to small devices. And it's applicable to small devices quite generally not just to the transistors that we've been discussing in this course. So I hope you've found the course interesting and useful. We've enjoyed putting it together for you. If you have comments or suggestions for things that we might do for the next time we offer the course. I hope you'll pass those along to me. So for those of you who are taking the final exam, good luck on it and I hope you've enjoyed the course. Thank you.