nanoHUB-U Fundamentals of Nanotransistors/Lecture 4.7: VS Analysis of Experiments
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[Slide 1] So hello, and welcome back. So now that we've developed our theory for the nanotransistor, it's time to take a look at experimental data and see what we can learn by applying that model to experimental data. 

[Slide 2] So just to remind you very briefly, we're going to apply this virtual source transmission model that we've developed to real data. The transmission model that we've developed expresses the linear current. We can write it in terms of an apparent mobility. The apparent mobility is related to the transmission in the linear regime. The transmission in the linear regime is related to the mean free path and the length of the channel. The saturated current is related to the injection velocity. The injection velocity is related to the transmission in saturation into the thermal velocity. The transmission is related to the mean free path, the same mean free path we're assuming as in the linear region, but the length is different. The length, script l, is a small fraction of the channel length. 

[Slide 3] So that's our basic model. Now it's these basic ideas have been implemented in a much more comprehensive model by the MIT group. We refer to that as the MVS model, MIT's virtual source model, that model is summarized here. We begin by saying that the drain current is width times charge times average velocity at the virtual source. We have an empirical expression, or semi-empirical expression that describes the inversion layer charge continuously from below threshold to above threshold. So some key parameters are the inversion capacitance, C inversion, m the subthreshold slope parameter, k T over q. There's a parameter alpha here which is an empirical parameter which accounts for the fact that the threshold voltage is really different in weak inversion and strong inversion, and that occurs because we can bend the bands a little more than 2 psi b in strong inversion. Okay, we account for DIBL, the fact that the drain -- that the drain voltage can lower the threshold voltage with a DIBL parameter. And then we write the average velocity. So charge, we just discussed the charge. Average velocity is written as a drain saturation function times the injection velocity. The drain saturation function is an empirical function that smoothly connects the linear current to the saturated current and there's an empirical parameter in there, beta. And there's a drain saturation voltage which is related to the injection velocity and the apparent mobility. All right, so the nice thing about this model is that there are only 10 really device-specific parameters here, and most of them have clear physical significance. There are a couple of empirical parameters, which once you're working in a particular technology they get adjusted to that technology and they don't change much, but the capacitance, channel length, and the other parameters are parameters that have a clear physical meaning that we either input, like the channel length must be known, the gate capacitance must know, or we extract the key parameters that we're going to extract are the apparent mobility and the injection velocity, and the series resistance too. So the fixed series resistance of this device is also an important parameter that can be extracted. 

[Slide 4] So my objective in this lecture is to show -- to illustrate how we can use this virtual source transmission model to analyze experimental data. And we will be looking briefly at silicon  N-MOSFET's and at III-V n channel HEMTs. 

[Slide 5] So the silicon devices that we'll be looking at are extremely thin SOI devices fabricated at IBM Research. You can find more details about the device itself and about the analysis that I'm going to briefly describe here in this reference. Just some key parameters, the silicon layer here is about 6 nanometers thick. The inversion capacitance is about 2 microfarads per centimeter squared. The effective mass -- the appropriate effective mass, that's a little bit bigger due to quantum confinement, bigger than the bulk transverse effective mass of silicon, which is .19. The appropriate effective mass is about .22. If we plug in numbers for the unidirectional thermal velocity we get 1.14 times 10 to the 7th centimeters per second. Now if we measure the mobility on a long channel device where the apparent mobility is the actual mobility, then we find a mobility that is about 350 centimeters squared per volt second. And that implies a mean free path that is about 16 nanometers long, about half this channel length. So we expect to be operating in the quasi ballistic regime. 

[Slide 6] Now we will also be looking at III-V HEMTs, so the device structure is a little more complicated. It consists of a number of layers that are grown expitaxially on the substrate, the semi-insulating substrate. The key part is this -- indium arsenide channel, and that indium arsenide  channel has a very high electron mobility. It has a very light effective mass and a very high mobility. That's why this transistor is called a high electron mobility transistor, that's what HEMT stands for. 

[Slide 7] So just some key parameters from this particular device. The thickness of the channel is 5 nanometers, the inversion capacitance is about 1 microfarad per centimeter squared. The effective mass is very light. It's about 10 times lighter than that silicon example we just saw, about .022 times the electron rest mass. When we evaluate the unidirectional thermal velocity in this case we get a value that is more than three times higher than what we computed for silicon, 3.62 times 10 to the 7th centimeters per second. And if we measure the mobility on a very long channel device where the apparent mobility is the actual mobility, we find an extremely high electron mobility, over 12,000 centimeters squared per volt second, which suggests a mean free path that is very long, about 153 nanometers. That is significantly longer than the 30 nanometer channel lengths that we're going to be looking at, so we expect this transistor to be operating very close to the ballistic limit. 

[Slide 8] All right, so let's look at the silicon data first. Here's the actual measured output characteristics and the solid line, the red lines here are the MVS fit. So you can see that the MVS model does a good job of fitting these characteristics. The key parameters that we extract in doing this fitting are the apparent mobility and the injection velocity, and it's also useful to extract a series resistance as well. 

[Slide 9] Okay, so let's see if we can make sense of some of this data. Let's look in the linear regime. Remember in the linear regime the key parameter in our transmission model was the transmission in the linear regime, T LIN. And by -- we argued that theoretically it should be mean free path divided by mean free path plus channel length. What we extract is this is apparent mobility, and the apparent mobility is a combination of the ballistic mobility and the real mobility. Okay. So if we divide the apparent mobility by the ballistic mobility and work out the result, this is our expression for the real mobility. It's related to the electron mean free path and this is our expression for the ballistic mobility, it's related to channel length. And if you work this out almost all of the constants drop out and we find that the ratio of the apparent mobility that we extract from the MVS fit to the ballistic mobility, which we know how to compute, is simply mean free path divided by mean free path plus channel length. That's the transmission. So we conclude that the transmission in the linear regime is just the ratio of the apparent mobility to the ballistic mobility. So it should be easy for us to estimate the transmission in the linear regime. We extracted an apparent mobility of 220. We can compute the ballistic mobility because we know the thermal velocity, the channel length is 30 nanometers, we get about 660 centimeters squared per volt second. We compute the ratio and we conclude that the transmission in the linear regime is about 1/3, so this device is operating in the quasi ballistic regime, it's not transmission -- isn't close to one but it's pretty sizable, about 1/3. So 1/3 of the electrons injected from the source make it across and exit through the channel, through the drain. Okay, so linear region transmission is something that is easy to extract from the experimental data. 

[Slide 10] Let's look in the saturation region and see if we can extract the saturation -- the tranmission in saturation. In this case it's the mean free path divided by the mean free path plus this critical -- the length of this critical bottleneck region. The injection velocity we've seen is some fraction of the ballistic injection velocity, or the thermal velocity, and the fraction is transmission divided by 2 minus transmission. So we can solve this equation for the transmission, and it's simply related to the thermal velocity, which we can compute, divided by the injection velocity, which we can extract from the MVS fit. So we have a simple way to estimate the transmission and saturation. We computed the thermal velocity from the effective mass of electrons in silicone. We extracted the injection velocity when we did the MVS fit. We put the numbers in and we find a saturation transmission of 84%. Okay. So that analysis is easy to do. It's easy then to analyze experimental data and extract the transmission in the linear regime and extract the transmission in the saturated regime. 

[Slide 11] And what we find when we do this analysis is as expected, as we argued earlier, the transmission in the saturated regime where the carriers are more energetic and are expected to do more scatter -- undergo more scattering, the transmission in the saturated regime is distinctly higher than the transmission in the linear regime. Our analysis of the experimental data bears that out. 

[Slide 12] And the physical argument, you'll recall, was that under low drain bias the scattering matters in the entire length of the channel, so the critical length is the entire length of the channel, and the transmission is mean free path divided by mean free path plus the channel length. Under high bias; however, scattering only matters in the low field region near the beginning of the channel and the transmission is mean free path divided by mean free path plus the length of that very short bottleneck region, which is why the transmission is higher in the saturated case. So the point is that the transmission in general is a function of the critical length, which is a function of the drain bias. 

[Slide 13] All right, let's turn to the  III-V data and take a look at it. Again, we can take the measured data, these are the measured output characteristics, the dots, and the MVS fit, or the red lines, we see again that the MVS model produces a nice fit to the experimental data. We extract from this fit an apparent mobility of 1800. We extract an injection velocity of 3.5 times 10 to the 7th centimeters per second, and we also extract the series resistances at the source and the drain. 

[Slide 14] Okay, so let's use our data, our extracted apparent mobility and injection velocity, the thermal velocity that we computed earlier and see if we can do the same analysis that we did for the silicon MOSFET. So to extract the linear transmission, that's just the ratio of the apparent mobility to the ballistic mobility. The ballistic mobility we can compute when we have all of the parameters. Channel length is 30 nanometers, the thermal velocity is 3.62 times 10 to the 7th. We find a ballistic mobility that is a little over 2000. So we can put numbers in our formula and we find a transmission in the linear regime that is 86%. Much closer to the ballistic limit than the silicon device was. In the saturated regime we also have the numbers. We can easily compute a transmission in the saturated region and we find in this case it's 98%. We are essentially operating at the ballistic limit in this III-V HEMT. 

[Slide 15] So just quickly to compare our analysis of these two devices, what we find is that in the linear regime we are in the quasi ballistic regime in both cases, but were much closer to the ballistic limit for the III-V HEMT than we are for the silicon device. In the saturated regime, even the silicon device is operating relatively close to the ballistic limit, but the III-V HEMT is essentially there. The injection velocity is a little less than 1 times 10 to the 7th in the silicon device. It is significantly higher than 10 to the 7th in the III-V device because of the light effective mass and the high thermal velocity. And all of this could have been expected because in the long channel devices if we look at the mobilities, those mobilities imply a mean free path that is on the order of the channel length in the silicon case, but which is much higher than the -- much longer than channel length in the III-V case, so we would have expected the III-V transistor to operate very close to the ballistic limit, and indeed it does. 

[Slide 16] Okay, now what I want to do in the remainder of this lecture is just to look a little more carefully at an analysis in the linear and the saturated regime. So remember, in the linear regime we extract this parameter, the apparent mobility. We know how the apparent mobility is related to the mean free path, we know how the ballistic mobility is related to the channel length, we can put these expressions together and we can derive an equation that looks like this, 1 over the apparent mobility is 1 over the scattering limited mobility plus the ratio of the mean free path over the scattering limited mobility times 1 over channel length. What this says is that if the mean free path is independent of channel length, then if I plot 1 over the apparent mobility that I extract as a function of 1 over the channel length of the transistor, the y inter-- that will be a straight line, and the y-intercept will give me the actual scattering limited mobility and the slope will give me the mean free path. 

[Slide 17] Okay. So it's another way to get those two parameters. If we do that, here is an example of how that data looks. Indeed it does fall on a straight line. >From the intercept we extract a mobility that is a little over 12,000, very close to the value that's measured on a long channel transistor, and if we extract from the slope, the mean free path, we find it's a little over 170 nanometers, again, close to what we estimated from the long channel results earlier on. So this is another way to estimate scattering limited mobility and mean free path in a short device. 

[Slide 18] Okay, now let's look a little carefully, what if we plot the data and we find it's not on a straight line. That is likely due to the fact that the mean free path depends on channel length. So in that case we can manipulate these equations and we end up with an expression that allows us to deduce the mean free path at any particular channel length if we know the apparent mobility at that channel length. So by doing that, you know, I point out that in this case we must know the thermal velocity. In the previous case I didn't have to know the thermal velocity, in this case I have to reliably know that thermal velocity, but if I do I can extract the mean free path. 

[Slide 19] And if we do that for the silicon ETSOI device, what we'll find is that for very short channel lengths the mean free path drops a bit. Now this is something that is sometimes seen in transistors, not in all transistors, and people are still trying to understand in detail what exactly is going on here. One possibility is that this has to do with so-called long-range Coulomb oscillations setting up oscillations of the carriers in the sort and heavily doped source and drain regions that become very important to that short channel length. Another possibility is that it is related to various details of the processing that introduced more scattering near the edges of the channel that affect the short channel devices more than the longer channel devices, but it's something to be aware of. There is a possibility that the scattering limited mobility is actually different in a very short device than it is in a longer channel device. That wasn't the case in the III-V HEMT but it seems to be the case in the silicon device. 

[Slide 20] Now let's look carefully at a saturation region analysis. So in a saturation region we know how the transmission is related to the mean free path and the critical length, we also know that the transmission is related to the injection velocity and the thermal velocity, we derived that a little earlier. That's how we estimated the transmission in the saturated regime. So we can solve these two equations and we can find -- we can relate the injection velocity to the mean free path and to the length of the critical region. Now, if we assume that the length of the critical region is proportional to the length of the channel, now that's a little hard to justify rigorously but it seems to work in practice rather well and there is even some support from some simulations for that assumption as well, but if we make that assumption we can test it later, then we can rewrite this equation in this form, 1 over the injection velocity is 1 over the thermal velocity plus some parameters times the length of the channel. So if we plot 1 over injection velocity versus channel length, then we ought to be able to extract from the intercept of that plot the thermal velocity. And that's useful because sometimes it's not so clear what effective mass to use in these very thin channels, it may be strained, it may be undergoing quantum confinement. >From the slope of the line we can extract the length of this critical region. So that is also something that is very useful. 

[Slide 21] Here is such a plot on the III-V HEMT data. And you can see that we get close to a straight line from the intercept of that straight line we deduce a thermal velocity that is very close to the value that we estimated from the -- from the -- what we thought was the effective mass, the electrons in the indium arsenide, and from the slope of this line we extract this parameter, a squiggle which tells us that about -- the critical length of the bottleneck region under high drain bias is about 10% of the channel length. The channel length is 30 nanometers, so that means that that bottleneck region is about 3 nanometers long. So it is indeed very short as we've been arguing. 

[Slide 22] Okay, so just to summarize the analysis that we can do. We can easily do an analysis by fitting MVS -- the MVS model to experimental data and then we can quickly extract the transmission in the linear regime and the transmission in the saturated regime. We can do a little more careful analysis and if the mean free path is independent of channel length, then we can plot 1 over the apparent mobility versus 1 over the channel length and we can extract the mobility due to scattering and the mean free path from such a plot. If the mean free path is channel length dependent then we can plot the data in a different way and we can extract for any given channel length. We can extract the mean free path -- an apparent mobility at that channel length and then we can extract the mean free path at that particular channel length. And finally, if the mean free path is independent of channel length, then a plot of 1 over injection velocity versus channel length will be a straight line, and from that we can extract the value of the thermal injection velocity and we can extract the length of this critical bottleneck regime. So those are some very useful ways that we can analyze experimental data and extract some very important physical parameters. 

[Slide 23] So just to summarize, this MVS transmission model provides us with a good way to extract important physical parameters from the measured characteristics of good transistors. Now I'll just emphasize, we're discussing here good transistors. The MVS model in our transmission model is really a model of well-behaved transistors. In early-stage development many things can go wrong, there can be many non-idealities, and you wouldn't learn very much from trying to fit an MVS or transmission model to data like that. But when we have a well-behaved transistor then we can learn a lot. And for those of you that are interested in more discussion about analyzing experimental data, I would encourage you to have a look at these two references. So, thank you.