nanoHUB-U Fundamentals of Nanotransistors/Lecture 3.1: Introduction
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[Slide 1] So welcome back to Fundamentals of Nanotransistors. We're ready to start unit three now. Now in units I-V and two we covered some introductory material. Unit one was some very basic material about the transistor I-V characteristics and definitions of things like device metrics. Unit two, we discussed in detail how the voltages we apply to the terminals affect the electric fields inside the MOSFETs. That's a very important component of MOSFET operation. It's the underlying physical ideas have not changed much in the 50 plus years of transistor evolution. It's gotten much more challenging to design electrostatically well-tempered MOSFETs. But the basic concepts have not changed. And if you've had a course in MOSFET device physics before you've probably seen that material in Unit 2. Now in Unit 3 we switch our attention to the velocity. How do carriers flow across from the source to the drain? And things have changed here. Over the course of the evolution of transistor channel lengths from quite long to incredibly short these days, our descriptions of carrier transport need to change. Textbooks haven't kept up with this, but modern-day transistors operate in a way that is fundamentally different from the way that transistors operated 20, 30, 40 years ago when MOSFET theory was first developed. So we're really beginning the really unique part of this course. 

[Slide 2] There are a number of topics that we're going to discuss. In Unit 3 we will focus on the ballistic MOSFET and what I want to do in this introduction is to derive the I-V characteristics of the ballistic MOSFET in a very simple way. Sort of based on an approach that has been used in semiconductor physics for a long time. But then we'll change the emphasis beginning in Lecture 2 to a more general approach called the Landauer approach, which will allow us to derive the same ballistic I-V characteristics, but is more generalizable in order to treat real effects in real devices. 

[Slide 3] Okay so let me remind you what we mean by ballistic versus diffusive transport. We talked about this picture of a MOSFET. We control the current flow from the source to the drain by controlling the height of this energy barrier between the source and the drain. When we push the barrier down electrons flow across. If they flow across unimpeded just in a straight line and don't encounter anything that they can scatter or bounce off of we call that ballistic transport-- now if the channel length is longer, or there are impurities or surface roughness, or lattice vibrations the electrons might scatter, they might turn around and go back out the source, they might bounce around a few times and eventually go out the drain. If they scatter many times we call that diffusive transport and that's the domain of the original MOSFET, the traditional MOSFET theory was really worked out assuming diffusive transport. If they scatter a few times then we call that quasi-ballistic transport. It's somewhere between this ballistic and diffusive limits. That's where modern-day transistors operate. And that's what we need to understand. So we're going to be focused in Units 3 and Units 4 on this quasi-ballistic transport beginning in Unit 3 on the ballistic limit, which is considerably simpler. 

[Slide 4] And we can treat it just as an introduction in this lecture we can treat the ballistic MOSFET using this concept of thermionic emission. So the idea of thermionic emission is that if I have a barrier and if I have particles like the electrons in the source that are in random thermal motion, then the probability that one of those particles can hop over this energy barrier is just e to the minus barrier height over kt and that applies to any kind of particle. Whether it's charged or not. Now if I have particles, you know, electrons in the drain bouncing around in random thermal motion, the probability that they can hop over their energy barrier is e to the minus their energy barrier over kt. We're going to use this simple concept of thermionic emission and see if we can derive the I-V characteristics of this ballistic MOSFET. I'll point out that the energy barrier from the electrons in the drain to the top of the barrier is larger than the energy barrier from the source to the top of the barrier because we apply a voltage to the drain and pull the energies down. Okay. All right. 

[Slide 5] Now, we're going to be interested in current flow so the electrons that hop over the barrier from the source they give me a current from the left to the right. The electrons that hop over the barrier from the drain they give me a current from the right to the left. And the net current that we're interested in is just the difference between the two. 

[Slide 6] Okay so let's look at the net current. So we can factor out a current from the left to the right and get this expression. And then we can realize that the ratio of these two currents is just the ratio of their probabilities to hop over the barrier. So the ratio of those two currents is e to the minus the difference in barrier height over kt. That difference in barrier height comes because we've applied a voltage to the drain and increased the barrier for electrons from the drain. So we have this expression. Now we're not quite done because we really don't have a convenient way experimentally to control this current I left to right. We know from Unit 2 that we can control the charge on the top of the barrier through MOS electrostatics that's simply related to the gate voltage. So we would rather express the current in terms of the gate voltage, something we can control. So we saw that in a well-tempered MOSFET above threshold the charge at the top of the barrier is negative and its approximately negative inversion capacitance times vgs minus threshold voltage. Let's see if we can get that into this expression we have for the ballistic MOSFET. 

[Slide 8] Okay, so we can always write current as a product of charge times velocity. So the current from the left to the right is carried by electrons with positive velocities, electrons moving in the plus x direction. So I'll write that as n sub s, because it is sheet electron density. Electron density per square centimeter in the inversion layer. I'll write that as n sub s plus. Because they're moving with a forward velocity in the plus x direction. So the charge is q times n sub s plus the velocity that they move at is this thermal velocity, this random thermal motion that they're in. And the width is just the width of the MOSFET. Okay so that's the current from the left to the right. There is a similar expression for the current from the right to the left. >From the drain toward the source, but that's carried by electrons that have a negative velocity. Magnitude of the velocity is the same, that's this v sub t here. But it's carried by the electrons that happen to have a negative velocity. Now we can control through the gate the total charge at the top of the barrier. And the total charge includes electrons with a positive velocity at the top of the barrier. And electrons with a negative velocity. Okay, so we can solve these two current equations for the amount of charge due to the positive velocity electrons, for the amount charge due to the negative velocity of electrons. We can add the two and now we've expressed the charge which we can control by our gate voltage in terms of these two left to right and right to left currents that are inside the device itself. Okay, I can factor out a current from the left to the right and write the charge in this way. And then I can recognize that the ratio of these two currents is the ratio of their probabilities of hopping over the barrier. And the difference in the barrier is just due to the drain voltage. So now I can relate the charge to-- that internal current and then I'm ready to get my final expression for the ballistic MOSFET. 

[Slide 9] So here's what we had. This is the net current, the difference between the left to the right and the right to left currents. This is our charge in terms of the left to the right current. We solved the second equation for I left to the right. We insert it in the first equation and we have our final result. This is the I-V characteristic of a ballistic MOSFET. So that was relatively easy to do. Just as easy as the traditional textbook MOSFET derivations that are based on diffusive transport. So the ballistic limit is really quite easy to handle. 

[Slide 10] All right let's take a look at that expression in a little more detail. So let's look first when we apply low voltage between the drain and the source, and we're in the linear region. You'll remember that our traditional diffusive model we had an expression like this. This is the textbook expression that you'll see everywhere. Now the question is what does our ballistic MOSFET look like in this linear regime. 

[Slide 11] Here's our general expression that we just derived. Let's assume that we have a small drain to source voltage. Small compared to kt over q. And we can remember our Taylor series expansions. e to the x is approximately 1 plus x, when x is small. And if we apply a small voltage then the argument of the exponential is small. We can expand both the numerator and the denominator for small exponential arguments of the exponentials and we get this expression. This is our expression for the linear region currents of the MOSFET. So there it is. 

[Slide 12] Let's take a look at it again. Here is the traditional expression in terms of charge above threshold. This is c inversion vg minus vt. Here's the ballistic expression that we just derived. They look quite different. One thing to note that the ballistic expression is independent of the channel length. Because it doesn't matter how long the device is if electrons are going through unimpeded, the length of the channel will not limit the current. In the diffusive approximation the longer the current, the more scattering that occurs and the harder it is for electrons to flow from the source to the drain. Okay so I could also express those if I'm thinking just above threshold I could express the charge in simple terms of c inversion times vg minus vt. So these are the two expressions that we have. They look quite different, but let's look at them a little more carefully. Let me take the ballistic expression and let me divide by length so that I can make it look like my diffusive equation with a w over l out front. And then if I divide it by length, I have to multiply by length and let me group all of those quantities together. Now if you look closely at these quantities, let's look at the units. Velocity is centimeters per second. Length is centimeters. kt over q has the units of volts. This quantity in parenthesis has the units of centimeters squared per volt second. That rings a bell. Something else we know has the units of centimeter square per volt second, that's mobility. Mobility is related to scattering, but we have a ballistic MOSFET. But we can lump these terms together and we can call this a ballistic mobility. Whatever that means. Now, we'll return to this later on in the Unit and we will discuss how we can actually attach some physical significance to this ballistic mobility. But right now we just might think of it as some algebra to make the ballistic expression look more like the diffusive expression. 

[Slide 13] And when we do that, we see that we can fix our diffusive expression to work for ballistic MOSFETs by doing one simple thing. We simply replace the real mobility of electrons in bulk silicon with this quantity that we defined called the ballistic mobility. And then we have an expression that looks the same. So the bottom line here is that we can make our two current expressions look the same. Remember here that even though the expression has a w over l for the ballistic current, the ballistic mobility is proportional to l so the ballistic current is independent of length as it should be. We just rearrange terms to make them look the same. 

[Slide 14] Okay, now let's do the same thing in the saturation region. Let's look for large drain to source voltage. This was our traditional expression. The velocity here was the high-field saturated velocity that's due to electrons that are scattering frequently due to the energy that they've gained in the strong electric field under high drain voltage. Let's see what happens in the ballistic case. 

[Slide 15] So again we began with our basic expression. Now we assume that the voltage is very large. The arguments of the exponential then are e to the minus x where x is large so that's about 0. So this expression simply reduces to width times charge, times thermal velocity. So here is our expression for the saturated current for the ballistic MOSFET. 

[Slide 16] So if we compare the diffusive expression, the velocity saturation expression with the ballistic expression, we see that they're almost identical. We simply replace the high-field saturated velocity of electrons in silicon with the thermal equilibrium velocity of electrons in silicon. And I can write it equivalently by above threshold by writing the charge as c inversion vg minus vt. 

[Slide 17] So here's where we stand. This was our traditional textbook model which we then evolved into what we call the virtual source model. We were able to connect the linear currents and the saturation current with a drain saturation function and then we had an expression that went smoothly from the linear regime to the saturation regime, which is empirical drain saturation function. 

[Slide 18] In our ballistic model we have manipulated our expression so the linear current looks very similar. We just have a quantity called the ballistic mobility. Our saturation current looks very similar. We just replace the saturation velocity with the thermal velocity. And we have an expression to go smoothly over the entire IV characteristic. In this case it's not an empirical expression that just fits the two together, it's an actual expression that we derived so this drain saturation function has this expression here. 

[Slide 19] All right so this is basically the ending point here. We have derived an expression for the ballistic MOSFET. It was relatively easy to do. We made some assumptions. One of the assumptions is that we've assumed non-degenerative or Maxwell Boltzmann statistics for carriers. That's why we have exponentials and not Fermi-Dirac integrals. We'll talk a little bit about how to fix that later. We could fix that within a thermionic emission treatment. But we'll see how to do that within this Landauer approach that I'm going to introduce next. The other assumption here is that there is no scattering. Now, in a realistic MOSFET electrons are going to encounter defects, and surface roughness, and lattice vibrations, and things that they're going to scatter off of. It is hard to include that scattering in a thermionic emission model. And that's really the power of the Landauer approach. It's going to allow us in Unit 4 to include carrier scattering. 

[Slide 20] Okay so we're going to be focusing in Units 3 and 4 on applying a different way of treating transport to these very small devices. The traditional way with which we can really justify in the diffusive case, where there's lots of scattering between the source and the drain, begins the traditional approach begins with a drift-diffusion equation. We're going to begin with a different equation, we're going to begin with something called the Landauer approach, which works in the diffusive limit, works in the ballistic limit which we just discussed, but which works in between, which is very important for modern transistors. So the Landauer approach is the subject of the next lecture, and I'll see you there. Thank you.