nanoHUB-U Fundamentals of Nanotransistors/\Lecture 3.7: Unit 3 Summary
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[Slide 1] Welcome back. We are ready to wrap up Unit 3. So what I like to do is just a quick summary of some of the key things that we covered in Unit 3 and make sure that we're all ready to proceed onto Unit 4 next. 

[Slide 2] So our goal in Unit 3 was to build on what we did in Unit 2, which was to understand charge MOS electrostatics and begin to understand transport or the velocity that electrons travel across the channel net. And in Unit 3 we -- Unit 3 was all about a ballistic treatment of transport. 

[Slide 3] Unit 4 will generalize that. There were a number of topics that we discussed and I would just like to go quickly through 

[Slide 4] and highlight some of the important things that you were introduced to. In the first lecture it was just an attempt to show you how easy it is to understand the ballistic MOSFET in terms of thermionic emission. So we showed that there is a probability of hopping over the barrier from the source and there's a probability of hopping over the barrier from electrons in the drain. And by using those two processes and understanding that they're exponentially related to the height of the barrier we could quite easily develop the IV characteristics for the ballistic MOSFET. 

[Slide 5] Now, then we switched gears and we introduced you to a much more general and more powerful approach and we derive the same equation. This approach is going to be particularly useful for us in Unit 4. So this is what we call the Landauer approach and it describes the nano devices that's having two contacts that are thermal dynamic equilibrium. We have a small device in between. We developed expressions for the current and for the electron number or electron density inside that small device. And those two expressions we can use to understand the IV characteristics of ballistic devices. 

[Slide 6] So in this Landauer approach there are some fundamental constants out front. There is a quantity tau or script T that is the transmission. It's simply a number between zero or one, which represents the probability that an electron injected from the contact goes all the way across and comes out the other contact. When T is equal to one, that's the ballistic case. That's what we considered in Unit 3. When T is much less than one that's the diffusive case. That's the regime of traditional semi conductor theory. In Unit 4 we're going to be considering the case where T is less than one, but not a lot less than one. That's the quasi ballistic case where modern-day transistors lives these days. We also have this quantity M, the number of channels in energy E, which plays an important role in our thinking. And what's critically important is the current flows only when the Fermi functions of the two contacts are different. And our two contacts have these ideal characteristics of we call Landauer contacts. 

[Slide 7] They maintain thermodynamic equilibrium. So we're assuming these special contacts. We're also assuming that electrons travel across the device in independent energy channels and then we simply add the contributions from each energy channel. 

[Slide 8] This concept of modes is an important one and we wanted to get familiar with. So in order for a current to flow there has to be a state and in order for it flow that state has to have a velocity to carry the current. So if we -- so we would expect, you know, the number of channels to be proportional to velocity times the density of states, and this h over four just introduces some factor that makes this quantity dimensionless. We also talked about what the physical interpretation of a channel is. So if this is a Ek diagram in one dimension, then if I ask at a particular energy is there a channel for current to flow. And if I find a state at that energy, I can say, yes, there's a channel to flow. So in this case I would have one state that this energy. So it's easy to visualize in 1-D, this particular expression works in 2-D or 3-D and this kind of argument could be generalized in 2-D or 3-D, it will be a little bit more challenging to draw the pictures for that. 

[Slide 9] Now we also discussed then in lecture three this concept of Fermi window. Current only flows in an energy channel if there is a difference between the Fermi functions of contact one and contacts two. That's what we call the Fermi window and we look at that Fermi window in a little more detail under small bias where we just lower the Fermi energy in contact two a little bit. We use the Taylor series expansion and we were able to show that the Fermi window is related to this derivative of the Fermi function and we got this classic expression for the conductance in the low drain bias regime. And you'll see this in many places. 

[Slide 10] Okay. Now, that expression actually allowed us to talk about this on some of the experimental work that was done in the last couple of decades of the 20th century, where people made very small resistors, which they could electrically vary the width of and measure the conductance as a function of width. If we take that expression and realize that at low energies the Fermi window it becomes a delta function, then we get a simple expression for the conductance. It's 2q squared over h, which we call the quantum of conductance, times the transmission at the Fermi energy, times a number of channels at the Fermi energy. Now, in our thinking in this course, the number of channels vary smoothly with the width. If the width is twice as wide, we have twice as many channels. That works as long as things aren't too narrow. When things become -- when conductors become very narrow, then we have to worry about quantum mechanical confinement. The fact that the wave function has to go to zero at both ends of this resistor at both of these two ends and the width direction. This is why channels are sometimes called modes because they look like confined modes over an electron wave guide. And when you do that, we find that the number of channels takes on a discrete nature in a nano structure. So when you vary the width the conductance just steps up discreetly as the number of modes, which is countable increases one by one. We are primarily concerned, in fact, exclusively concerned in this course with wider MOSFETs for which we can assume that the number of channels varies continuously with width, but you should keep in mind if we're dealing with very narrow MOSFETs, you might see this quantization in terms of the number of channels. 

[Slide 11] Okay. Then we switched in lecture four and we tried to apply this Landauer approach to the ballistic MOSFET. We talked a little bit about how we think about the ballistic MOSFET as a nano device. We focus on the top of the barrier. We think about that small region near the top of the barrier. This is our nano device. The region to the left and the region to the right, these are our two contacts to the nano device and we put our focus on the top of the barrier. And we treat it as a nano device using the Landauer approach. 

[Slide 12] So then to treat a ballistic MOSFET, we have to evaluate this expression f sub S is the Fermi function of the source. f sub D is the Fermi function of the drain. For load drain to source biases we can expand f sub S minus f sub D in a Taylor series expansion and we get this expression. For high drain biases f sub s is much larger than f sub D and we get this expression. And all we have to do is to work out those expressions and we have our IV characteristic of the ballistic MOSFET. 

[Slide 13] In the non-degenerate case that's easy to do. We went through that calculation and this is our IV characteristic of a ballistic MOSFET assuming non-degenerate carrier statistics. For small drain voltages it simplifies to this expression and for the linear current, for large drain voltages it simplifies to this expression for the saturation current. 

[Slide 14] Now, we also indicated, but didn't spend much time deriving what those calculations look like if we don't assume non-degenerate conditions. You know, in that case we get a linear current that is more involved, involves Fermi Dirac integrals. We get a saturation current that looks the same, but this ballistic injection velocity actually involves Fermi Dirac integrals. We can a general expression that involves Fermi Dirac integrals. The same basic ideas apply, which is -- it gets a little more difficult to see them with these Fermi Dirac integrals. Things look more complicated, but really it's just a simple as the non-degenerate case. If you're interested in seeing more discussion about that derivation, you should consult the lecture notes. 

[Slide 15] We also gave a prescription for how one would calculate the IV characteristics using Fermi Dirac statistics. So one would start by saying I know how to calculate the charge based on MOS electrostatics. So that's known at a given gate and drain voltage. We know how it's related to the location of the Fermi levels, so we can solve this equation for the location of the Fermi level. Okay. Once we know the location of the Fermi level we can solve for this ballistic injection velocity. All right. It depends on this ratio of Fermi Dirac integrals. Once we have that we can evaluate our current at this particular set of voltages. We know the charge. We deduce the Fermi levels so we can calculate everything that we need. That's the prescription that we would use to calculate the IV characteristics using Fermi direct statistics. 

[Slide 16] Now, we also looked in a little more detail about what is going on inside this device as we vary the drain voltage. So we looked at the top of the barrier. We examine the velocity distributions inside the device. We saw how they changed from a symmetrical distribution at zero drain voltage to an asymmetrical distribution, which gets increasingly asymmetrical. And in the element of a high drain voltage becomes just a hemi Maxwellian or hemi Fermi Dirac distribution. The average velocity of which is what we call the ballistic injection velocity. Now, there's one factor here that I haven't discussed very much, but I should mention at least. It's a very interesting effect due to the self consistency with the Poisson equation. So you can see here we have lost half of the distribution. We have no negative velocity electrons. But we know that MOS electrostatics really doesn't care about these microscopic details that are going on inside the transistors. The charge on the gate must be balanced by the charge at the top of the barrier. So the charge never changes, even though the distribution of velocities is changing drastically. Now, how can that be? Well, if you will look carefully at this, you'll see that the positive half under high drain bias is larger than the positive half under low drain bias because now all of the electrons with positive velocities have to match C inversion VG minus vT. That's what MOS electrostatics tells us. How did that happen? How did the positive half get bigger? If you look at these calculations, you can see that the top of the barrier has been pulled down. That's the self consistent solution with of the Poisson equation, it's pulling the barrier down, which lets more electrons in. It gives me more positive velocity electrons to give me the required number that I need to balance the charge in the gate. So there's some very interesting physics that's going on inside this ballistic device. 

[Slide 17] We also talked about the ballistic injection velocity. How it's simply a constant under non-degenerate conditions. Unidirectional thermal velocity and how it increases under degenerate conditions. We quoted the limits for complete degeneracy and we gave this expression in terms of Fermi Dirac integrals that allows us to compute the entire curve. 10 to the 13th electrons per square centimeter is about where you're at in a modern-day high-performance transistor under on current conditions. So you can see the Fermi Dirac statistics really should be quite important. 

[Slide 18] Now, I just want to make a couple of comments. We've assumed two-dimensional carriers flowing in the XY plane. So we used two-dimensional density of states. We've assumed parabolic energy bands to compute that two-dimensional density of states. We've assumed that a single sub band is occupied. Due to quantum confinement in this extremely thin layer that we're dealing with. And when I say "EC," what I'm really meaning is it's the bottom of the first sub band. That's what I mean by "EC". So for example, if you were to do this calculation for a 1D nano wire MOSFET and nano wires are more and more interesting and that may be where the technology ends up going. It will be a different calculation. There will be a 1-D density of states. You may still use parabolic energy bands, but you'll find Fermi Dirac integrals of different orders, but the same general principles that we discussed here will apply. In fact, that's a useful exercise for you to do. You know, work out the ID characteristics of a ballistic nano wire MOSFET. So if you like to see more discussion about this Fermi direct integrals and one 1-D nano wire transistors, again have a look at the lecture notes, lecture 13 in the lecture notes. 

[Slide 19] Okay. Then sort of to wrap up this unit, we returned to our virtual source model and tried to connect what we did, the ballistic model that we developed to that virtual source model. So in the virtual source model we developed it from traditional arguments. This is our traditional linear region current. This is our traditional saturation region current. We use an empirical function to fit those two together to get a continuous curve. And then we asked how does this ballistic model relate to that. 

[Slide 20] And what we showed is that we can clearly relate this we simply interpret the mobility in that traditional model as the apparent mobility and in the ballistic case that apparent mobility we showed is the ballistic mobility. We simply interpreted the saturation velocity in that traditional model as the injection velocity and we showed that for the ballistic MOSFET that injection velocity is the ballistic injection velocity, which in the non-degenerate case is simply the unidirectional thermal velocity. So we were able to give a very clear physical interpretation for the terms in that virtual source model and to explain why it continues to work so well, even for these very small devices. 

[Slide 21] So just to summarize, our goal in Unit 2 was to understand MOS electrostatics. Our goal in Unit 3 was to treat-- understand the velocity from a ballistic treatment perspective. Bottom line is that if we assume that the transmission is one, we have a ballistic model then we could calculate this average velocity. In the case of non-degenerate or Maxwell Boltzmann statistics that average velocity at the virtual source only depends on the drain to source voltage. If we do the more general case of Fermi Dirac statistics, then that velocity depends on where the Fermi level is located and that depends on the gate voltage. So in that case, the velocity at the top of the barrier depends both on the gate voltage and on the drain voltage. We get this complicated looking expression involving Fermi Dirac integrals. We also get an expression for the ballistic injection velocity under high bias, and there will be times when analyzing data that we'll need to use expressions like this. So we're ready to proceed to Unit 4. In Unit 4, what we will do is we'll say that well, we have to relax the assumption that the transmission is one. Real devices have some scattering. How do we include scattering in this model? How do we describe realistic devices where the transmission is less than one, but not a lot less than one where the traditional models apply. That will be the subject of Unit 4 and I'll look forward