nanoHUB-U Fundamentals of Nanotransistors/Lecture 4.6: Connection to the VS Model
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[Slide 1] So, hello again. So, in this lecture we're going to be talking about the virtual source model again. So, the virtual source model has been providing a framework for our discussion. as we go through this course. We formulated it using very traditional MOSFET theory concepts. Now that we have the full transmission theory of the MOSFET we can relate our transmission theory which is physically sound at the nano scale to the traditional model and we'll find that with some very simple reinterpretation the traditional model serves quite well. 

[Slide 2] So, just by way of review, we've developed now and discussed some of this transmission model for the nanotransistor. In the linear regime we have the current related to charge and velocity, and the key parameter here is the transmission, channel transmission, in the linear regime which we have seen is related to the mean free path for backscattering and to the length of the channel. In the saturation region, current is width times charge, times velocity. The appropriate velocity now is what we call the injection velocity. The velocity at the beginning of the channel. It's related to the thermal injection velocity by the transmission in the saturation region. The transmission in the saturation region is also directly related to the mean free path near equilibrium. But, the length here is the length of this bottleneck regime which is a tiny fraction of the channel length. So, that's our transmission model. 

[Slide 3] Now, the traditional MOSFET model that we first developed in Unit 1 expresses the linear current as width divided by length, times electron mobility charge, times drain voltage in the linear regime, and in the saturation regime we use the velocity saturation model, width times charge, times the high field scattering limited saturated velocity of electrons. So, these are the two models that we want to connect. 

[Slide 4] So, let's discuss the connection in the linear region first. Our traditional model is here. I'm assuming above threshold, so this is the charge in the inversion layer. Our transmission model is here. So, how do we connect these two? Well, the transmission is related to the mean free path so, we can expand this expression, rearrange it, and it turns out that there are two different ways that I could do the algebra, and both of which offer some insight, and so I'll show you both of them. You can do this algebra and you can write the current as width divided by length, plus mean free path. So, this is one way to account for ballistic and quasi ballistic transport in the traditional model. We simply replace the length in the denominator with length plus one mean free path. As the channel length gets very short, much shorter than the mean free path for backscattering, we simply have W over mean free path. Remember that there is also a mean-free path in the mobility. So, those two mean free paths cancel out and we're at the ballistic limit. Now, another way to do this algebra here, you know, in expressing the transmission in terms of mean free path is to rearrange things so that we end up with a mobility. And, when we do that what we end up with is this apparent mobility. And, the apparent mobility is given by this Mathiessen's Rule, it is basically the lower of the actual mobility, or this ballistic mobility. 

[Slide 5] So, let's talk about the apparent mobility because this plays -- this is the way we formulate the virtual source model and it plays an important role in our thinking. So, we're going to extend our traditional model by introducing this apparent mobility. The apparent mobility is the combination of the actual scattering limited mobility and this ballistic mobility, and it's combined in a way that will look familiar to some of you. This is called Mathiessen's Rule for combining mobilities through the different scattering processes. The mobility itself, you know the real mobility which you would measure in a long channel device or a short channel device if you do it carefully, the mobility due to scattering in the channel is given by this expression. It's proportional to the mean free path. Now, the ballistic mobility, we simply replace the mean free path by the channel length, and what this describes is the ballistic case where there is no scattering in the channel, all of the scattering takes place in the source in the drain contacts. So, these two different physical processes combine together to give us the apparent mobility in our linear region currents expression. 

[Slide 6] So, just a quick example to get calibrated on how this all works. Let's take some rough numbers for current day technology and estimate the electron mobility as say 200 centimeters squared per volt seconds. We need to then calculate the ballistic mobility. If I calculate the thermal velocity for electrons in silicon under non-degenerate conditions, we end up with 1.2 times 10 to the 7 centimeters per second. Plug the numbers in and we'll get a ballistic mobility of about 500. If I put the two together to find the apparent mobility, I'm going to get an apparent mobility that's a little bit less than the lower of the two. The apparent mobility turns out to be 140 centimeters squared per volt second. So, the apparent mobility of this MOSFET is a little bit less than the scattering limited mobility because, this device is operating in what we would call a quasi-ballistic regime. Not diffusive, not ballistic, somewhere in between which is often the case that we're in these days with silicon MOSFET's. Now, I'll just point out, and there's some more discussion about this in the lecture notes. You might ask, why are we using the transverse effected mass for electrons in silicon when computing the thermal velocity? Well, that has to do with quantum confinement and which valleys are the lowest. There is some discussion of this in the notes. The appropriate mass to use when we have a channel in which carriers are confined in the 100 direction, is the transverse or light effective mass of electrons in silicon. 

[Slide 7] Okay, let's talk about the saturation region and see if we can connect the traditional and the transmission models. Here are two expressions. Again, the transmission in saturation is related to the mean free path and also to this critical length of the bottleneck region where the scattering really matters. Well, if I rearrange this expression using the definition for the transmission in saturation, I can re-express it in a different way. So, if we do the algebra, it's width times charge, times the velocity. And, this velocity looks a little different. It's 1 over the thermal velocity, plus 1 over D divided by l where D is just our normal scattering limited diffusion coefficient. This is an expression that we developed earlier in Unit 4. Thermal velocity times the mean free path, divided by 2. So, the question is, you know, what does this mean? How do we make sense of this expression? 

[Slide 8] So, let's go back to our energy band picture which has been a very useful way for thinking about how transistors operate, and you'll recall that we focus on the top of the barrier, and there is this short bottle neck region of length script l, and if the carrier's scatter there, there's a chance that they can go back to the source and not contribute to drain current. So, that's really what we focus on. That's where the scattering matters. Now, you could think of that short region. This energy band diagram looks similar to an energy band diagram of a bipolar transistor. So, for those of you that are familiar with bipolar transistor's we could think of this bottle neck region as the base of a bipolar transistor. The high field part of the channel, we could think of as acting like the collector of a bipolar transistor. And, in order to get current then, electrons have to get across this bottle neck region which is like getting across the base of a bipolar transistor. The average velocity at which they do that is given by this expression which just says it's the slower of two velocities. So, traditionally we think about carriers going across the base of a bipolar transistor by diffusing across the base. D over l is the effective diffusion velocity at which carriers diffuse across that thin bottle neck region. But, we have to be a little bit careful when this script l gets very short because, D over l might end up being faster than the thermal velocity, and that is unphysical because diffusion is random thermal motion. Particles can't diffuse faster than the thermal velocity. But, this expression will automatically take account of that because if D over l gets bigger than the thermal velocity, then it's the thermal velocity that will limit the velocity across that region, then we would be in the ballistic limit. Turns out people could build very narrow based bipolar transistors a long time ago, in the 1970's actually. And, expressions like this were used to describe those bipolar transistors in the 1970's. Now, they're relevant for MOSFET's. 

[Slide 9] So, just a little more discussion about this injection velocity and what it all means. If we look again at our energy band diagram and we focus on this bottleneck region, then we can think of what's happening as three processes. Electrons are thermionically emitted from the source to the top of the barrier at some velocity, v1. They diffuse and transmit across that region at some velocity, v2, and then they enter the high field region, and they get swept out some velocity, v3. Now, intuitively we would think that the appropriate velocity limiting the current for the MOSFET is the slower of these three velocities. So, we can say that one over the net velocity is 1 over v1, plus 1 over v2, plus 1 over v3, and that will ensure that whichever one of these three velocity's is the slowest, that will be the bottleneck, and that will be the limiting factor, and that will limit the current. Now, our assumption is that in the high field region, transport is very efficient. We saw, early on in some Monte Carlo simulations that there is strong velocity over shoot for n-channel MOSFET's, electrons move very rapidly across there, so our assumption is going to be that v3 does not limit the current. We only have to worry about the first two processes, thermionic emission from the source to the barrier and then the process of getting across the barrier. So, that's what this expression means. This is the first velocity, thermionically emitted at the thermal velocity from the source to the top of the barrier. This is the second velocity, electrons diffusing across that thin bottleneck region. And, the third velocity is not there because we assume that it is so fast that it doesn't limit this. So, that's how we think about the injection velocity in these devices. 

[Slide 10] Now, our virtual source model, we developed a virtual source model in a different way than it was originally developed. We developed it based on very traditional MOSFET theory concepts. The kinds that were worked out for MOSFET's in the 60's, 70's and early 80's. We developed it. We said, current was width, times charge, times velocity. We have an expression for the velocity that is some drain saturation function, times the high field saturation velocity. VDSAT was given by this expression. The two key physical parameters were this high field saturation velocity, and the electron mobility. And, as we saw early on, we're very concerned in a nano scale MOSFET that these two parameters are losing their physical significance so this doesn't seem quite right. 

[Slide 11] Now, what we understand now, as the result of developing this transmission theory and relating it to the traditional theory, is that all we need to do is to replace the real mobility in the traditional theory by the apparent mobility, and the high field saturation velocity by this thing we call the injection velocity. And, each of these velocities has a clear physical reason, we don't have to change the formulation of the traditional model, we simply need to reinterpret the physical parameters in that model in a way that makes sense at the nano scale. 

[Slide 12] So, the particular model that we have been developing is a version of MIT's virtual source model. When those authors, when these authors developed that model they really based it on concepts of transport at the nano scale. I've been doing it a little bit differently. I've developed this model based on traditional concepts of MOSFET theory. We end up with similar expressions, and then I've shown how, if we simply reinterpret those traditional parameters in terms of these new parameters, we get a very physically sound model. So, with that, this model provides a very nice description of a wide variety of transistors at the nano scale in current day technology, both in manufacturing and in research. 

[Slide 13] Now, in the next lecture, I would like to discuss how we can use this virtual source/transmission model to analyze experimental data and to extract some physically meaningful parameters from nano transistors. So, that will be the topic of discussion next, and I'll look forward to seeing you there. Thank you.