nanoHUB-U Fundamentals of Nanotransistors/Lecture 3.6: Revisiting the VS Model
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[Slide 1] Hello, everyone, and welcome back. We had been talking about ballistic MOSFETs in this unit. Now it's time to revisit our virtual source model. You'll recall that we developed this model in Unit 1. We're sort of using it as a framework, as a traditional device that we can go back to and relate our new models to. So let's see if we can relate our ballistic model to the virtual source model. 

[Slide 2] Just by way of reviewing, we derived our ballistic model from the Landauer approach, this is the general expression. If we apply a small voltage between the drain and the source, we can expand f source minus f drain by Taylor series expansion, and we get this classic expression for the conductance, and their -- in the linear regime. And if we go to high drain voltages, then fs is much bigger than f drain, and we get this expression. And all we have to do is to evaluate these expressions and we have our IV characteristic for the MOSFET. 

[Slide 3] This was relatively easy to do in the non-degenerate case. And we found this expression for the linear current, and we found this expression for the saturation current. And we found this expression for the current in general that goes between those two limits. And that was easy to do with non-degenerate statistics. Now, it's also a little curious, I'll just mention, the drain saturation voltage, VDSAT in the ballistic model is actually quite low. It's just a few kT, 2 or 3 kT. It's the voltage needed to make these exponentials negligible. So a ballistic MOSFET displays a rather low drain saturation voltage. 

[Slide 4] Now, we have also developed this -- using traditional MOSFET modeling concepts, this model we call "the virtual source model." And when we did that, we have this classic textbook expression for the linear region current, we have this classic textbook expression for the saturation region current, and we indicated that there are some concerns about the physics of these expressions, especially when we push channel links down to the nanometer regime. In order to get a smooth curve that goes from low drain voltages to high drain voltages, we stitch these two solutions together with an empirical drain saturation function. And by doing so, we developed really a quite good virtual source model. 

[Slide 5] And I'll just remind you from Lecture 2.8 how that model works. We begin by writing the drain current as width, times product of charge, times velocity. We developed a semi-empirical expression for the charge from sub-threshold to above threshold in terms of a few physical perimeters like the inversion gate capacitance, threshold voltage, and some empirical perimeters as well. We then expressed the average velocity as a function of drain voltage using this empirical drain saturation function, which introduced another empirical perimeter, beta, that we tweaked to adjust data -- to match data. And we have a drain saturation voltage, which separates the linear from the saturation regimes. That was our basic model. And it has only 10 parameters in it; most of them are very physical device parameters. There are a couple of empirical perimeters. But with that small number of parameters, we can fit the characteristics of most transistors, if we just make a few adjustments in things like mobility and saturation velocity. 

[Slide 6] Okay, so what we would like to do now is to connect this new ballistic model that we've developed in this unit to the traditional model that people have been using for many years, and, you know, with physics we're uncomfortable with the nano scale. This is our expression for the ballistic linear region current. This is our expression for the diffusive current, when there's lots of scattering in the linear regime. These two expressions look much different. But if I simply divide by L, multiply by L, then I can bring a W over L out front. I'll end up with another L inside. I'll make this definition of the quantity we've been calling the "ballistic mobility," and when I do that, we end up with an expression for the linear regime current that is more familiar than the one that we started with, and which looks more like the traditional expression. We've only done one simple thing, we've replaced the actual mobility of electrons by the ballistic mobility. So that was a very simple change, and we can relate our new model to our old model very simply. 

[Slide 7] And I discussed in the last lecture briefly, and let's do it just one more time, what this ballistic mobility means. So the physical picture in the linear regime is that we have a very small voltage applied to the drain, electrons enter, they cross the channel ballistically and do not scatter. But they scatter frequently in the source. They scatter immediately when they enter the drain. So the average distance between scattering events is the length of the channel. And that's the argument we use for replacing the mean free path in the real mobility by the length of the channel in the ballistic MOSFET. And that's how we're interpreting and giving physical significance to this ballistic mobility. We simply replace the-- appropriate distance between scattering events and a ballistic MOSFET is the channel length itself, not the mean free path in the bulk. Now, it's important to understand that the ballistic mobility is proportional to the channel length. The current is proportional to width divided by length. And since I have a length here, the net result is independent of channel length; which is what it should be for a ballistic MOSFET. Doesn't matter if the channel is twice as long, nothing is slowing the electrons down and preventing them from getting out of the drain, so the current should be independent of channel length in a ballistic MOSFET. 

[Slide 8] Here's a quick example. Here are some IV characteristics of a 16-nanometer channel length MOSFET. This particular device has a real mobility of about 250 centimeters squared per volt second. Let's see if we can calculate the ballistic mobility. We know all of the terms. The unidirectional thermal velocity for silicon is about 1.2 times 10 to the 7th centimeters per second. kT over q is .026 volts. L is 60 nanometers. Put numbers in and we get almost -- a ballistic mobility of almost 1400. Now, we'll discuss this in a lot more depth in Unit 4. But intuitively you can see that it's the lower of the two mobilities that's going to matter. That's the one that's going to limit the current in this device. And since the lower of the two mobilities is the real mobility, this device is strongly influenced by scattering, and is definitely not a ballistic transistor. That's what we conclude from this calculation. 

[Slide 9] Okay, let's look at the saturation region now, and connect the ballistic model to the traditional -- our traditional virtual source model. So on the ballistic theory, here is our expression. In the traditional theory, here is our expression. No work needs to be done. We see all that we have done is to replace the saturation velocity the high-field saturation velocity of electrons in bulk silicon with this unidirectional thermal velocity in our ballistic model; so a simple replacement. 

[Slide 10] The physical picture in the traditional model is the high electric field in this regime causes lots of carrier scattering, and causes the velocity to saturate. Okay, but we know that that's not physically sound in a very short channel length device, because the carriers just are out -- across the channel and out before they have a chance to scatter often enough to limit their velocity. So that's not a physically sensible thing to do for a nano scale MOSFET. What we've seen is that in a ballistic MOSFET the velocity saturates at the velocity the electrons come in at, at the velocity that they possess at the beginning of the channel. That's where the velocity saturates in our drain current expression. And that velocity we call the "ballistic injection velocity." For non-degenerate carrier statistics, it's this unidirectional thermal velocity, v sub T. So we talked 

[Slide 11] about how if we take this virtual source model that we've developed using traditional concepts, and if we simply replace the physical perimeters in that theory that we developed, which we're uncomfortable in using at the nano scale, if we view them as adjustable parameters, the mobility becomes the apparent mobility, the saturation velocity becomes the injection velocity, then we can tweak those parameters and we can fit measured data extremely well. And we suggested that the fact that we can do that with simple adjustments for this theory suggests that there is something deeper going on, and there is something that makes this theory work at the nano scale. 

[Slide 12] Well, we can examine that now in the case of a ballistic MOSFET, since we understand ballistic MOSFETs. So the question is if we say calculate the IV characteristics of a ballistic MOSFET, and then take our semi-empirical virtual source model and try to fit those calculations, can we do that? If we can do that, what is the physical significance of those two perimeters, or those -- that we have adjusted to fit our computed ballistic IV characteristic? 

[Slide 13] Well, here's an example. So we took some typical dimensions, oxide capacitance and channel length is irrelevant for a ballistic device, series resistance, and we computed the lines here are then computed IV characteristics of a ballistic MOSFET modeled on this real device. So those are the computed ballistic IV characteristics in the non-degenerate limit. Then we fit the virtual source model to that to see if we could do that. The circles are the virtual source fit to that model. And you can see that we can fit this ballistic MOSFET very well. In order to do that, we had to adjust our mobility to this fudge factor -- not really a fudge factor, and we needed a value of about 650 centimeters squared per volt second. And we needed to adjust the saturation velocity to this thing we call "the injection velocity," and the number we needed was 1.24 times 10 to the 7th. So the question is, you know, do those empirical adjustments in order to fit this computed IV characteristic, do they make sense for a ballistic MOSFET? 

[Slide 14] Well, let's take a look at that. Here are the parameters that it took for us to fit our virtual source model to the ballistic IV characteristic. We know now that the injection velocity in a ballistic MOSFET should be the unidirectional thermal velocity under non-degenerate conditions, which is what we assume to calculate those IV characteristics. I can plug in numbers for silicon, and I can compute that. Turns out to be about 1.2 times 10 to the 7th centimeters per second. In order to fit that curve, we needed a 1.24. They're about the same, so this empirical adjustment of the saturation velocity to 1.24 times 10 to the 7th, what we were doing was just adjusting it to the ballistic injection velocity. So we can interpret that injection velocity as the unidirectional thermal velocity. If I look at this mobility, we can calculate the ballistic mobility; we assumed a 30-nanometer channel length. We know what the unidirectional thermal velocity is. We put numbers in, we calculate for this ballistic device a ballistic mobility of 690 or so centimeters squared per volt second. That's close to what it took to fit the IV characteristics, so we conclude that the apparent mobility that we used to fit the IV characteristic is actually the ballistic mobility. So we can attach physical significance to these parameters that we're adjusting in the virtual source model to fit our ballistic MOSFET. 

[Slide 15] Okay, now let's look at the measured IV characteristics of this device. So what you're seeing here and the points, these are the measured IV characteristics of this device. The one volt on the drain and-- a half a volt gate voltage, as I recall. The line that you see here is the virtual source fit to those measured results. And you can see the fit is quite good. Now, the black line here is the corresponding ballistic IV characteristic of this device. That's what it would look like if there were no scattering, using the same series resistance, the same oxide capacitance and parameters like that. So what you can see is that this silicon transistor, 30 nanometers channel length, operates at a reasonable fraction of the ballistic limit, but is distinctly below the ballistic limit. Okay, we're going to talk about that much more in Unit 4. 

[Slide 16] Now, if we look at a III-V HEMT, so this is a field effect transistor, high electron mobility transistor, made on a semiconductor that has a very high mobility, so little scattering, again, what you're seeing here are measured results, and you're seeing the virtual source fit to those measured results so we can fit the measured data very well. What you're seeing in the black line is the corresponding calculation from the ballistic model that we have developed in this unit. And now you can see that the story is a little different. A III-V HEMT, a high electron mobility transistor, actually operates rather close to the ballistic limit. And we will discuss this again in Unit 4, and try to understand why the III-V MOSFET operates very close to the ballistic limit, and why the silicon MOSFET operates less close. 

[Slide 17] So just to summarize what we've covered in this lecture; what we've learned is that the apparent mobility and the injection velocity in the semi-empirical virtual source model that we've developed and have been using, these are not just adjustable parameters to fit data. We've shown that they have a clear physical interpretation, and we've discussed that in the ballistic limit. We'll discuss it in a more general case in Unit 4. We've seen that silicon MOSFETs operate at a significant fraction of the ballistic limit, but well below the ballistic limit. And we'll discuss that more in Unit 4. And we've seen that III-V transistors operate actually quite close to the ballistic limit. And we'll talk about why in Unit 4. 

[Slide 18] So in Unit 4, we will -- what we will add to our model is carrier backscattering. The reason that devices deliver less on current than the ballistic MOSFET is because the transmission is not exactly 1. It is less than 1. And we will discuss how we treat scattering, and how we generalize our model to these more realistic cases in Unit 4. Now, before we do that, in the next lecture I'll just briefly summarize the key things that we've covered in Unit 3, just to be sure we're all ready to prepare to head onto Unit 4. Thank you.