nanoHUB-U Fundamentals of Nanotransistors/Lecture 1.6: Traditional IV Theory
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[Slide 1] Welcome back. This is lecture six in unit one. And what I want to do in this lecture is to review, very quickly, the traditional way that MOSFET IV Theory is presented in textbooks. This is the way it was originally developed when people first started making MOS transistors. And then it was extended in the 70s and it's basically the viewpoint that you see in almost every textbook that is out there today. The reason that we're doing this is we want to relate this new view of MOSFET Theory that we're going to be developing to the traditional view. One of things we'll find is that the traditional theory works better than one would've expected. 

[Slide 2] And we're going to explain why that's the case. So we're trying to understand the IV characteristics of MOSFETs. We know qualitatively we can understand them in terms of energy bands and it's all about manipulating energy bands, pushing barriers up, pushing them down, pulling them up with gate voltages. That gives us a qualitative view. We want to derive the quantitative IV characteristics now. So here's our device. We are interested in the drain current. Notice, when I apply a positive voltage on the drain and when I turn the transistor on, current flows in the drain. We'll call that ID. Now there's also some possibility that some leakage current may flow from the drain into the semiconductor itself and not out the channel. What we're most interested in is the part of the current that flows between the source and the drain. That's most of the current, except for these nuisance leakage currents that are an effect that you learn about in more advanced MOS courses. So we're going to primarily be concerned in this course, in fact, totally concerned, with the drain-to-source current, the part of the current that flows in the channel, not the other leakage currents that might flow. And our goal is to understand how this drain-to-source current depends on the various voltages. For the most part, we're going to focus on the voltage between the gate and the source and the voltage between the drain and the source. We'll assume that the source is grounded and the body is grounded. You know, some transistors these days, like SOI transistors, don't even have a body. So the main terminals are the source, drain and gate terminals 

[Slide 3] and the voltage is between them. All right, now it's important to differentiate between two classic types of transistors. Early transistors had very long channels. So this is a plot of a transistor, actually fabricated recently, but it has a 980-nanometer-long channel length. That's almost 1 micrometer. That's pretty long these days. And if you look at this IV characteristic, each one of these lines, remember, is a different gate voltage. And if we look out here in the saturated regime, as we increase the gate voltage, what you'll see is that the saturation current increases as the square of the gate voltage minus the threshold voltage. This is called a square-law device. Now you don't see many of these devices these days because channel lengths are much shorter. You see transistor characteristics that are more like this. So this is a 30-nanometer channel length and the same SOI technology. But now you can see that if you look in the saturation regime, if we look at how the saturation current increases with gate voltage, it goes linearly with VG minus VT. Now that's caused by saturation, by velocity saturation in the channel. And we'll discuss that several times throughout this course. So usually, when you observe an IV characteristic like this, you say, ah, that's a signature of velocity saturation. I know what's going on inside. 

[Slide 4] So we need to develop an IV theory for both of these cases. All right, so here's our transistor characteristics. And we're going to be content. It actually gets a little bit complicated and if you've had a course in MOSFET Theory, you spent some time doing this. But we're going to focus on just understanding the linear regime and the saturation regime and then we'll connect them with a smooth curve later. So we have this low drain-to-source voltage regime we call the linear regime. Things act like a resistor. We have this high drain-to-source voltage range we call the saturation region or the current. It almost saturates. And we'll try to develop a theory for each of those two and not a full theory that goes between the two. 

[Slide 5] All right, so remember our energy-band diagram under a high gate voltage and low drain voltage? This is what the conduction band looks like along the surface of the device. Now if I've applied a large voltage to the gate and turned the channel on, then I'll have a charge in coulombs per square centimeter in the channel. And that's the charge that's going to flow and give me the current. Now you might ask where did that charge come from. Well, it came from the source and maybe the drain. When we pushed the barrier down, the charge hopped in from the source and the drain and fills up the channel. That's where it comes from. Now if we apply a small voltage, because we're thinking about the linear regime, we pull things in the drain and down a little bit. We have a constant electric field in the channel. And that's the situation we want to try to analyze what's going on, how much current do we get in this case. 

[Slide 6] So remember, current is the flow of charge. So it's charge per unit time. So we can easily write down an expression for current. It's charge times velocity. And then we have to multiply by the width of the transistor coming out of the page. The wider it is, the more current that we get. And you just check your dimensions. This is width maybe in micrometers or, you know, in MKS units, it'd be width in meters. This is charge in coulombs per square meter. This is velocity in meters per second. We get coulombs per second, we get amps. So we have the right units. The negative sign is just because the charge is negative because it's due to electrons and we to have to find a positive current when current flows into the drain terminal. So remember, capacitance is charge divided by voltage. Now we're going to spend some time talking about how we relate this electron charge in the channel to the gate voltage and capacitance, but for right now, I'm just going to write down this equation. So remember, the basic law of a capacitor says charge is voltage times capacitance, but in order to get a significant amount of mobile charge that carries a significant amount of current so that we call the device on, we have to be above threshold. So really what matters is how much above threshold is the gate voltage. And the capacitance there is just the capacitance related to this parallel plate capacitor, given by the gate oxide. So that capacitance is dielectric constant of the oxide divided by thickness of the oxide here. So we're going to assume the charge is related to gate voltage through this simple relation. And as I said, we're going to talk about that quite a bit more later on in unit two, 

[Slide 7] when we talk about MOS electrostatics. All right, so with that, we can see if we can derive the IV characteristics in the linear regime. So in the linear regime, we have a small voltage between the drain and the source. We really have no potential drop along the channel, no significant potential drop, so everything is uniform. We just have a negative electron charge that's oxide capacitance times gate-to-source voltage minus the threshold voltage. So we know the charge. Current is charge times velocity times width. Charge is capacitance times voltage above threshold. The average velocity, so remember from basic semiconductor physics, when we have an electric field, it exerts a force and pushes carriers along at a constant velocity, the average velocity is proportional to the electric field. The constant of proportionality is the mobility, the electron mobility. And the negative sign is because a negative charge gets pushed in the opposite direction of the electric field. So we have charge. We have velocity. We just multiply the two together. We need the electric field. We have a constant electric field in the channel. It's just the voltage across the channel, which is the voltage between the drain and the source divided by the length of a channel. Put that all together and we've got this IV characteristics for the linear region of the MOSFET. So this is the region that it's operating as voltage-controlled resistor. All right, so that was pretty easy. And it actually does very well. Even when we do more sophisticated theories, we're going to end up with this IV relation. 

[Slide 8] Okay, now let's look at the high-Vds regime. So this is what it looked like in equilibrium. Under high drain voltage, we pull everything down. We have a high gate voltage here. So we're on a small barrier for electrons to hop over and flow to the drain. Now things vary with position along the channel. Now let's look at the beginning of the channel. So the beginning of the channel, we're going define as being the top of this potential barrier that separates the source from the channel. Now at the top of that barrier, I'm just going to assert inside this channel because things are varying so rapidly in space, it's actually rather difficult to figure out what the charge is at any arbitrary spot inside the channel. But at the top of this barrier, there is no lateral electric field. It's zero by definition at the top of that barrier. And the charge at that point is just what it was given under low-bias case, basically minus Cox VG minus VT. So actually, this particular point is very important and we're going to mention it many times. We call this point the virtual source. It's really the actual place that we say is the beginning of the channel of the MOSFET. And one can envision, you know, one might ask, well, why does this simple relation continue to hold under these very high biased conditions. That's really the challenge of the MOS designer, to control MOS electrostatics in a way that this simple relation between charge and voltage continues to work, even under high bias. And we'll talk much more about that when we talk about 2D MOS electrostatics in unit two. 

[Slide 9] Okay, so with that, now we can see if we can derive the IV characteristics of a long-channel device, the square-law device. So the situation looks something like this. We have a very large voltage on the drain now. So there's a very large potential drop along the channel, meaning that the charge is not going to be uniform along the channel. It's going to vary with position. We still have it at Cox, but what matters is the difference between the voltage in the gate and the voltage in the underlying silicon. And the voltage in the underlying silicon is zero at the source end and it's the drain voltage at the drain end. So that varies quite a lot. This V of x along the channel varies quite a lot. And then we have a threshold voltage in there, too. Now if you look at that, you can see there's a potential problem here, because if the drain voltage is VG minus VT, then this expression tells me I have no charge. If I have no charge, I'm going to get no current. Well, really, we've pushed this approximation a little too far. Now what happens is, when that happens, we have a very high electric field and a very small charge and this relation is no longer valid, but the current doesn't stop flowing. Now this is actually one of the most difficult things to explain to students the first time they see an MOS transistor, you know, what this pinch-off means and why current doesn't stop the flow. But you can kind of see that this relation is going to say that the charge is going to go to zero. It can't be exactly zero. It's going to be close to zero. But it's going to be moving very fast, so it'll carry the same current. Okay, so that's what we call the pinch-off point, where the mobile charge goes to approximately zero. And when that happens, we're going to get a very large electric field there. 

[Slide 10] So if we look at pinch off in a channel, if you look at it in an energy-band diagram, and this is another reason to get comfortable with energy-band diagrams, then it doesn't look mysterious at all. In this pinch-off region here, you know, you can think of this as a region of very high resistance because we have very few charge carriers. If it's very high resistance, we're going to get a very large voltage drop across that part of the channel, the pinch-off part. If I look on my energy-band diagram, I look down here where the slope is the steepest, that's the high electric field region. That's the pinch-off region. But now when you think about it, electrons will hop over the barrier. They will flow down the channel. They'll hit that region, that region we call the pinch-off region, where the electric field is very high. There's nothing to stop them. They'll just flow down the energy barrier, drop down, go out the drain and continue. Current will not stop flowing. So the pinch-off region doesn't stop current from flowing, okay. 

[Slide 11] But the pitch-off region is going to be very important when we analyze the IV characteristics of this square-law device. So let's see how that works. So in general, current is charge times velocity, okay. Now the difficulty is it's very difficult to compute the charge at any point within the channel, but there's one point in the channel that we know what the charge is and that's at the virtual source or at the beginning of the channel. So if I look at the beginning of the channel, x equals zero. Since I know the charge there, if I can figure out the velocity there, I've got the current. And the charge there is given by basic MOS electrostatics. We know what that is. The velocity is mobility times electric field. And now what's the electric field? So here, I have to be a little bit careful. Remember, the pinch-off region is special and we're not really considering that as part of the channel, because once the electron encounters the pinch-off region, it just gets swept out, goes out the drain immediately. So the channel itself is just a little bit shorter than the physical channel. And the voltage at the pinch-off region is Vg minus Vt. So the voltage across the channel, at this end, it's Vg minus Vt. At this end, it's zero. The distance is about the length of the channel, just a little bit shorter. So the electric field is about the voltage at the pinch-off point divided by the length of the channel. That's minus VG minus Vt divided by L. All right, I put these all together in my expression. And you can see, I'm going to get a Vg minus Vt for the charge. I'm now going to get a Vg minus Vt for the electric field, which is proportional to the velocity. So I put it all together and we have the square-law characteristic of the long-channel device. Now there's a factor of two down there. Where did that come from? Well, the reason where that came from is the fact that the electric field is actually strongly non-uniform inside this channel. And this little estimate here, this was just sort of a rough estimate of the average electric field. If we do it little more carefully, we find what the electric field at the virtual source of the beginning of the channel is actually half of this average value. If you look in the lecture notes that are provided with this unit, you'll see a derivation of this result and it's done in all standard textbooks. 

[Slide 12] Okay, so we have a description of this square-law device, but most devices that we encounter, especially nanotransistors, aren't going to be square-law devices. They're going to be linear devices. How do we describe the saturation current 

[Slide 13] for a linear device? Now that turns out to be even easier. So one of the things we have to remember from basic semiconductor physics is that if we measure the velocity versus electric field in a large slab of silicon, initially, as I apply an electric field, the electrons move faster and faster, but if I apply a large enough electric field, I'll get to a point where I can't move the electrons any faster and the velocity saturates. And that happens physically because they gain so much energy. They find lots of ways to scatter and randomize their motion and you just can't push them any faster. So that particular value is called the saturation velocity or high-field saturation velocity. It typically occurs somewhere around 10 kilovolts per centimeter electric field. Okay, so what's the electric field inside one of these small nanotransistors that we're interested in? The power supply voltage is about 1 volt. The channel length these days is about 20 nanometers. That's 500,000 volts per centimeter. That is, that means we're way out in this velocity-saturated regime. The velocity is not proportional to the electric field. The velocity has saturated at VSAT. 

[Slide 14] That makes it very easy to compute the IV characteristics. So the electric field is very high in these short-channel devices. Current is still charge times velocity. I can evaluate my charge from MOS electrostatics. Now the velocity is just the saturation velocity because I'm assuming the electric field is large everywhere. Plug it all together and we get this very simple expression that describes the IV characteristics for a velocity-saturated MOSFET. And now it goes as Vg minus Vt to the first power. This is a linear device. All right, so it's even easier to describe these velocity-saturated devices than it is the longer channel devices, where we have to worry about pinch off and what all that means. 

[Slide 15] So we can recap. We've looked at this IV characteristic. We've just focused on the linear regime and we've got a simple relation. That was relatively easy to do. We've look at the saturation regime. And as long as we assume it's velocity saturated, that's really easy to use. And later on, we'll draw a smooth curve between these two, but what we have now is a two-piece approximation to this IV characteristic. The drain voltage that separates the linear regime from the saturation regime, this is our VDSAT, drain saturation voltage, okay. And we can quantitatively deduce what that VDSAT is. It's just where these two curves give the same current. So now we have an expression for the first time for the drain saturation voltage. 

[Slide 16] So this is where we stand. We've developed this IV characteristic of the MOSFET. Relatively simple to do, using very traditional ideas that people first developed in the 50s and 60s and then velocity saturation in the 70s. The difficulty is we've pushed technology far beyond the 

[Slide 17] validity of the simple approach and here's an example. So these are some simulations that were done by some researchers at IBM Research a number of years ago. Here's the conduction-band diagram, you know, by simulation of a small MOSFET with a 30-nanometer channel length. Modern-day MOSFETs are even shorter than that. And each dot here represents an electron that the computer is tracking through the MOSFET as it encounters forces due to the electric field and random scattering events that knock it around in random directions. And we're just stimulating this at a very detailed physical level to see what goes on. Well, we can go to any position here and we know the velocity of every one of those electrons. We can just add them all up and we can compute the average velocity versus position from the source, across the channel, and out the drain. So if you do that, what we'll find is a velocity versus field curve that looks like this. So the average velocity is low in the source, though there are lots of electrons so it doesn't take much electric field to carry the current. It pops up and becomes large when we get into the channel. As you get deeper and deeper into the channel, the velocity gets higher and higher. And then, when the electrons enter the drain, they lose all of their velocity and there are lots of electrons to carry the current in the drain, so the electric field is very, very small. Now there's something that's interesting here. The high-field saturation velocity of electrons and silicon is almost exactly 1x10^7 cm/s.  And that's what you see right here. So our best understanding of the physics of electron transport in these short-channel devices tells us that there is no velocity saturation in them. And the reason is easy to understand. The electrons just don't spend enough time in the channel to scatter enough to get slowed down. In bulk silicon, we have a long region. The electrons can gain a lot of energy. They can scatter a lot. Everything, all of the transients can decay. They slow down. They end up at VSAT 10 to the seventh centimeters per second. But in short-channel devices, they just zip across and scatter once or twice and they're out the drain. They never saturate. So the assumption we made about velocity saturation to derive 

[Slide 18] that IV characteristic is fundamentally wrong, all right. Now it's really quite surprising, even though it's fundamentally wrong, it doesn't work badly. It works remarkably well. Even though it's based on physics we know to be wrong. One of our goals in this course is to explain why this can be and it's very interesting and very easy to explain. We'll see that later on in the course. So what I want to do in the next lecture, which will basically end up unit one, is to extend this two-piece model into a full model for the IV characteristics of a MOSFET. We'll call that full model the virtual source model. It's actually a version of a model that is widely used these days to describe nanotransistors. And we will be using that model as sort of a framework. As we go through the course and understand the physics in more and more detail, we'll go back to the model and incorporate that physics into this simple model and we'll end up in the course with a very simple but very physically detailed and sound model that will explain some of the reasons why these traditional models, that are based on the physics, that we know to be so erroneous, tend to work reasonably well and what it is that they miss and what it is that they capture. So in the next lecture, we'll extend this basic model here to the virtual source model, and I'll see in the next lecture. Thank you.