ECE Purdue Essentials of MOSFETs/Lecture 4.8: Unit 4 Recap
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[Slide 1 L4.8] Okay, welcome back. So we've been working up in this whole course towards the material that we have covered in Unit 4. So if you understand Unit 4 you really have a very good, solid, deep understanding of how nanoscale MOSFETs work. 

[Slide 2] What I would like to do in this lecture is just quickly go through some of the topics that we've discussed in Unit 4. And make sure that we highlight the important takeaways from each of these lectures. 

[Slide 3] We began with the Landauer approach. This is a very important and general way to treat electron transport in small devices, and large as well. And we discussed we didn't derive it, but we discussed what it means and where it comes from. And what each of the terms in this expression, transmission, number of channels, and Fermi window. What each of those terms is all about. 

[Slide 4] We then specialized that general Landauer expression to the small voltage difference between the two contacts that will be the linear regime of our MOSFET, and large voltage difference, that would be the saturation regime. And we showed how the more general Landauer expression simplifies in those two cases. 

[Slide 5] We discussed the key parameters in the Landauer Approach. The transmission is a key parameter. It's related to mean-free-path for back-scattering and the length of the conductor by this very simple expression. We also pointed out that there is a relation between the diffusion coefficient and the mean-free-path for back-scattering. That relationship is important because it's the mean-free-path that is for back-scattering that is important and we would like to be able to deduce that from routine, commonly available information. The commonly available information is the mobility of a semiconductor. We can use the Einstein relation to determine the diffusion coefficient from that mobility. And then we know that that mobility, that diffusion coefficient, is simply related to the mean-free-path for back-scattering. And that way we can estimate the mean-free-path for back-scattering. The velocity in this expression is the unidirectional thermal velocity. This is a very important velocity in our understanding of nanoscale MOSFETs. And we discussed the physical interpretation. This is the average velocity of the electrons moving in the plus x direction. The direction of the source to the channel. For a non-degenerate semiconductor, there are some complicated Fermi-Dirac integrals that appear when we have to include degenerate semiconductors. 

[Slide 6] Now, in the Landauer approach there's a term called F1 minus F2. We call this the Fermi window. This is an important range of energies over which F1 does not equal F2, because this is the range of energies that matter for current flow. And we illustrated that by thinking of a small piece of semiconductor with some parabolic E versus k. A Fermi level in contact one, a Fermi level in contact two, that is a little bit lower because we've applied a small voltage and pulled the Fermi level down. And then we've thought about how things work inside this device. And we see that if we're thinking about low temperature, where it's just easiest to explain everything, that if I'm below the Fermi level in contact two I'm also below the Fermi level in contact one. So both contact one and contact two want the states below that energy to be filled. Since there always filled, nothing is really going to happen. It will not contribute to the current flow. It's the states between those two Fermi levels that are important. Those are the states where F1 does not equal F2. Those are the states that matter for current flow. If I look at a state in that energy range that state is below the Fermi level of contact one. So contact one would like to see it filled. It sends in an electron. That state is above the Fermi level of contact two. Contact two would like to see that state empty, so it removes the electron, it flows out of contact two and contributes to current flowing in contact two because of the negative sign of the electrons. So electrons flow in from the source, or contact one. They flow out through the drain, or contact two. And that represents a current flowing in the opposite direction. That's how we understand current flow in these nanodevices. 

[Slide 7] Alright, we then took those expressions, those Landauer expressions for linear regime, saturated regime, and found even in the ballistic case we could do the full regime. We derived an I-V characteristic of the MOSFET. We derived a general characteristic from small drain to source voltage to large drain to source voltage. That's the expression you see here. We showed that in the small drain to source voltage regime it simplifies to our linear current expression. And we saw how we can express that in terms of a ballistic mobility. And that allowed us to make this expression look like the traditional expression. Which is really a nice thing to have. We saw how that general expression for any drain to source voltage simplifies for high drain to source voltage. And we saw that it looks much like the traditional expression. It's just that the high field saturation velocity has been replaced by this unidirectional thermal velocity. 

[Slide 8] So this ballistic mobility, that we introduced. We had a well-defined definition of it. It was some quantity that had the physical units of mobility, so it made sense to call it a ballistic mobility. It was very similar to the diffusive mobility with one change. In the diffusive mobility, the mobility is limited by scattering. This occurs in the bulk, where the mobility is well defined. We just replaced the mean-free-path due to back-scattering by the channel length itself. Because in a ballistic device, the furthest that an electron can travel is from the source to the drain. It scatters in the source, it scatters in the drain. So that's the distance between scattering events. 

[Slide 9] Then we examined how the velocity at the virtual source varies as a function of drain to source voltage. We found that in the linear regime, small drain to source voltages, the average velocity is mobility times electric field. But it is ballistic mobility times electric field. So very similar to what occurs in a bulk semiconductor, but in a bulk semiconductor it's the mobility due to scattering that appears here. Under high field, the velocity saturates. Now initially, years ago, when people were first thinking about ballistic MOSFETs, that surprised a lot of people. Because we generally think of velocity saturation in semiconductors as occurring because of strong scattering. Which makes it impossible to push the electrons any faster because it just increases the scattering. We see the velocity saturate even when there is no scattering. So that's a quite interesting feature. 

[Slide 10] We understood the physics of that through some numerical simulations that looked inside a nanoscale MOSFET. As we increase the drain to source voltage, and looked at what happens to the distribution of electron velocities at the virtual source as we increase the drain to source voltage from zero to some large value. This is what we found. At zero voltage, we found a thermal equilibrium distribution of velocities. No average velocity in any direction. When VDS is zero, we expect zero current through the MOSFET. The positive half of that equilibrium distribution was injected from the thermal equilibrium source. The negative half of that thermal equilibrium distribution was injected from the thermal equilibrium drain. Apply a small drain to source bias. Positive half doesn't change, but the negative half gets much smaller. It gets smaller because we've lowered the Fermi level in the drain. That means there's a smaller probability that negative velocity electron states are going to be occupied at the virtual source. So the negative velocity component decreases. Increase the drain bias a little more, it decreases even more. Increase the drain bias still further, and we completely remove any electrons with negative velocities. As the negative velocity component is dropping, the overall average velocity is increasing. But once we removed all negative velocity electrons the velocity cannot increase any further. The velocity saturates at the average value of this positive half of the distribution of electron velocities. 

[Slide 11] Now, it's important also that that's an understanding of how the velocity varies with drain voltage, the velocity also varies with gate voltage. The more higher the gate voltage, the more electrons there are at the virtual source, at the top of the barrier. When I do the calculation of the average velocity of the positive velocity directed electrons, I can do the calculations for a non-degenerate semiconductor. I'll find that the average velocity is this unidirectional thermal velocity. Square root of 2kT over pi m. But as I increase the electron density then the influence of Fermi-Dirac statistics begins to be felt and the average velocity increases. Under on-current conditions we will typically achieve, in a high performance, silicon MOSFET, we'll have an inversion layer charge density of about 10 to the 13th electrons per square centimeter. Under those conditions the electrons are quite degenerate, and we'll have to include the effect of these Fermi-Dirac integrals in our expressions and we'll find that the injection velocity can be significantly higher than it's non-degenerate value. Now that's a complication that we haven't been dealing with in these lectures. Life is much more simple with non-degenerate carrier statistics. But you should keep this in mind if you're analyzing data, Because it can be important to get quantitatively accurate values. 

[Slide 12] Okay, so our transmission theory of the MOSFET. We can express it in traditional language with apparent mobilities and injection velocities. But we can also express it in terms of transmission coefficients themselves, too. And one of the things that we learned and argued is that the transmission in a linear regime is smaller than the transmission in a saturated regime. This is actually counterintuitive. In the saturation regime, we apply a high drain bias, we'll give carriers a higher energy. Higher energy carriers should be able to find more ways to scatter, because they're higher up in the density of states where there are more possibilities for scattering. But, counterintuitively, we find that the transmission is actually higher at high drain voltages where there is more scattering. 

[Slide 13] We explained the physics of that, in terms of these energy band diagrams. Under low drain bias, back-scattering anywhere in the channel can be detrimental. Because an electron can back-scatter at the end of the channel and then return to the source. Under high drain biases, there is a very short bottleneck that occurs right at the beginning of the channel. Just a few nanometers long. If an electron gets across that bottleneck without back-scattering, then it doesn't matter whether it back-scatters. Because that strong electric field there, the fact that the electrons will begin emitting lattice vibrations and losing energy means that it's going to be very difficult for electrons that back-scatter in that regime ever to get back to the source. They'll all come out of the drain. So the back-scattering doesn't really matter if it occurs beyond this critical bottleneck regime. That explains why the transmission can be higher and under high drain biases even though there is more scattering, the more scattering occurs in a region that doesn't limit the current. 

[Slide 14] Then we also pointed out that we have developed in Unit 3 on MOS Electrostatics general expressions for the charge in the inversion layer has a function of gate and drain voltage. They go from subthreshold to above threshold. And if we express our currents in terms of those charges, and used the relations we developed in Unit 3, then we can describe these transistors from subthreshold to above threshold. Two key parameters in our model, then. The MOS Electrostatics is a very traditional part of MOS Theory. What's new in this way of describing nanoscale transistors, is these two parameters, mobility and diffusion coefficient need to be replaced by apparent mobility and injection velocity. And these are not fudge factors we use to fit data. They have a sound physical basis and interpretation. 

[Slide 15] So that allowed us to generalize our virtual source model. The level one model, which was based on traditional MOS Theory. And simply replaced the high field saturation velocity by the injection velocity, the actual mobility by the apparent mobility. And then we have a physically sound model for transistors at the nanoscale. 

[Slide 16] We talked about how this model fits experimental data for short channel transistors really well. And how the fitting process, we can use it to extract some key parameters from these transistors and we can interpret and analyze these parameters and learn something about the operation of these transistors. 

[Slide 17] Two of the key quantities that we can extract relatively easily are the transmission in the linear regime because the apparent mobility is extracted from the fitting process. When we fit the I-V data well, we know what the apparent mobility is. And the ballistic mobility is a well-defined quantity that we can calculate with reasonable accuracy, pretty easily. The saturation transmission depends on the unidirectional thermal velocity. Which we can compute reasonably accurately if we know the effective mass and the temperature. And the injection velocity, which we deduce from the fitting process to the experimental data. So it allows us to understand relatively easily what the transmission of this particular transistor is. 

[Slide 18] So, Unit 4 really brings together everything that we have been discussing in this course from day one. So we now have a good, comprehensive understanding of modern nanoscale MOSFETs. All the way from the traditional ballistic case, the diffusive case, to the ballistic and quasi ballistic regime that modern transistors operate in. So this is what we have been working towards and this is what we've been discussing in Unit 4. There's one more unit. There are a few extra, additional topics that I think anyone who works on transistors should know about. And we'll be discussing a collection of various topics in Unit 5.