nanoHUB-U Fundamentals of Nanotransistors/Lecture 1.5: Energy Band View of MOSFETs
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[Slide 1] So, welcome back to Unit 1. This is Lecture 5 in Unit 1, and what I wanted to do now, and basically for the rest of the course is to dive into the insides of these transistors and understand what's going on, to produce the IV characteristics that we've been talking about. I'm going to discuss a very simple, very physical view that really explains the essence of how these devices work. And it's something that you should keep in mind as we go through the rest of the course 

[Slide 2] and start developing some more complicated equations. So, when we're talking about semiconductor devices, a really very important principle that we try to teach all of our students is the first thing you should do is draw an energy band diagram. If you can draw an energy band diagram of a device, you can usually figure out how that device works. So, here we have our cartoon of a MOSFET. And I'm going to look across the top surface, from the source, across the channel, and out the drain. And I'm going to be drawing plots of electron energy versus position along that line. And then we'll see if we can figure out how it changes as we apply voltages to the terminals and how that leads 

[Slide 3] to the IV characteristics of the MOSFET. OK. So, let's remember how we do this is basic semiconductor physics. So, first of all, let me think about having three separate regions, a source, a drain, and a channel. And they're not connected. They're just three separate semiconductors, because that's easier. So, the source is N-type. The source is usually very heavily doped N-type, that's what this N-plus means. It means that it's doped heavily there. There are lots of electrons there. So the fermi level may well be above the bottom of the conduction band, up inside the conduction band. The drain is doped just the same. So, if I look at the energy band diagram of the drain, here's my valence band, here's my band gap, here's the bottom of the conduction band. Fermi level is up there inside the conduction band. Now, the channel is a P-type layer, sometimes it's even an undoped layer. So, if I look at the channel, if I want to draw a P-type layer, I need to draw the fermi level down near the valence band. So, that's what those three separate pieces of semiconductor look like. We've got to put them all together to figure 

[Slide 4] out what the energy band diagram looks like. So, here's what we have. This is the N-type source, the P-type channel, the N-type drain. Now, very important principle in semiconductor physics is that in equilibrium, the fermi level is constant. It's sort of like the level of a lake, you know, underneath the depths of the water might vary with position, but the lake is flat on the top. Fermi level is somewhat like that. It has to be constant in equilibrium. Now, the other important point is that the electron energy is lowered when I apply a positive electrostatic potential with voltage. Positive voltage lowers all of the energies. Negative voltage raises them. Now, when I put these three pieces of semiconductors together, you can kind of conceptually see electrons are going to move from high concentrations to lower concentrations, holes are going to move from high concentration to low concentration. They have charges. When those charged particles move electric fields will get setup. When electric fields will get setup, potentials will change. The potentials will change in such a way that the fermi level ends up being constant. 

[Slide 5] So, this is how it would work. So, here we have the constant fermi level. We know things have to end up that way. If I assume that the-- I grounded the P-type layer, so its potential hasn't changed. You know, all potentials are relative. Then it means that I must have developed a positive potential on the source and the drain because positive potential lowers electron energy. That's one of the things we learn in freshmen physics usually. So we have developed the positive potential, and we would call that a built in potential in semiconductor theory, but has lowered everything. OK. So, remember, energy bands are-- their values before there was an electrostatic potential, and then we have to add whatever electrostatic potential is developed just through the movement of the charged particles, or through the voltages that we applied to the electrodes. OK. So, we put it all together 

[Slide 6] and we just connect everything with a smooth line. And this is our energy band diagram in equilibrium for the MOSFET. Here's our N-type source. Here's our P-type channel. Here's our N-type drain. This is the valence band. This is the conduction band. Here's the fermi level. That's the equilibrium energy band diagram along that top surface of the MOSFET. Now, this is an N-channel MOSFET, so it's really electrons that are giving us the IV characteristics. So, I really don't have to worry about holes and I don't have to worry about the valence band. So, I'm just going to draw the conduction band from now on, so we'll just be looking up here at the conduction band versus position. And the first question that we'll have is what happens if we apply a voltage on the gate? How does this energy band diagram change? 

[Slide 7] OK. Well, what we know is that if we apply a positive voltage on the gate, we have an insulator underneath it. That positive voltage is going to affect the electrostatic potential underneath it in the semiconductor channel. So, for now, let's apply zero volts to the gate and to the source and the drain, but just think about what happens if we apply a positive voltage to the gate. 

[Slide 8] So, here's our energy band diagram. Just showing the conduction band now. But here's the N-type source, here's the N-type drain, here's the P-type channel, so there's a big barrier here. So, this is what it looks like if I've applied zero or a low voltage to the gate. And I'm not applying any voltage between the drain and the source, so everything is the same there. Now, if I apply-- so one of the things you see there is that the electrons in the source can't get out and flow to the drain, because there's this big energy barrier. Same thing for the electrons in the drain. They can't get out and flow to the source. They're prevented from doing that by this energy barrier that got setup when the charge sloshed back-and-forth and the potential-- the built in potentials developed and the fermi level became constant. 

[Slide 9] OK. Now, let's see what happens if we apply a high gate voltage, and the principle is that when the voltage increases, the electron energy decreases, because the electrons have a negative charge. That's-- well, first of all, let's leave the gate voltage where it was and let's apply a high drain voltage and see what happens there first. So, if I do that, the high drain voltage is going to pull everything on the drain side down. All right? So, we just pull all of that down. If we have a well-designed transistor, the potential in the channel is controlled by the potential on the gate electrode, so nothing changes there. We just pull everything on the drain side down. So, we have a large voltage between the drain and the source. That's what this energy band diagram is showing. But we have a low gate voltage. So, we still have that energy barrier. We still can't get electrons. A small fraction of the electrons can hop over that barrier and give us the leakage current. That's I-off. But the device is basically off. 

[Slide 10] OK. Now, let's ask what happened if we apply a large gate voltage. So, if we apply a high gate voltage, we're going to pull the energy band down. When we pull the energy band down, we lower the height of that potential energy barrier. And now if we take a look, we can see that what's going to happen is that we have a significant probability that electrons can hop over that energy barrier and flow from the source across the channel and come out the drain. So, we push the barrier down, electrons can get out of the source and flow to the drain. 

[Slide 11] OK. Now we could look at it another way. We could say "Well, what if we keep the source and the drain voltages the same and look at what happens if we just apply a voltage to the gate?" So, this is our equilibrium energy band diagram. If we apply a voltage to the gate, we're going to lower the potential in a channel and push the barrier down. So-- it-- we're going to look like that. Now, still, no current will flow, because there's a probability that electrons can hop over that barrier from the source into the channel, but there's an equal probability that electrons in the drain can hop in and flow in the opposite direction, so everything is cancelled out. No current is flowing. When there's no voltage between the gate, between the drain and the source, even though we have a high voltage on the gate and we've turned the transistor on. 

[Slide 12] OK. Now, let's add a small voltage to the drain and see what happens. So, if we have a large gate voltage, so we have this line down here. Now, if we apply a small voltage to the drain, we'll pull this down just a little bit. So, if we pull that down a little bit, we pull down the fermi level, we pull down the conduction band, and now we have a little bit of slope in the conduction band inside the channel. So, if we look there, remember the conduction band is changing because the electrostatic potential is changing. So, the slope of the conduction band is related to the slope of the electrostatic potential. The slope of the electrostatic potential is basically the electric field. So, what we're seeing here is that there is a small, constant electric field in the channel. 

[Slide 13] OK. So, this basically explains how a transistor operates. So, here are the output characteristics of a transistor. If we look in the linear regime, in the linear regime we just have a small voltage difference between the drain and the source. If we apply a large gate voltage, we push the barrier down. If we turn the transistor on, you'll see there's a linear slope between-- along the channel that means a constant electric field. And the device behaves as a voltage controlled resistor. If I look up here at high drain voltage, our energy band diagrams will look like this. The high voltage pulls everything in the drain way down. But as long as the gate voltage is low, we have a big barrier, the device is off. If we apply a high voltage, we push the barrier down, current can flow, can hop over that barrier. And you can kind of see now why the current saturates, because what happens is once the electrons hop over the barrier, it doesn't really matter how far down the drop is. They're going to go out to drain anyway and contribute to drain current. What matters is the rate at which they can hop over that barrier from the source. So, the current is more or less independent of drain voltage. So, that simple picture basically explains how transistors operate. It's all about manipulating potential energy barriers with the voltages that we apply in the channel. If we can design the transistor such that we manipulate those correctly, 

[Slide 14] we have a good transistor. So, if that's all there is to it, then what's the rest of the course about? Well, this picture-- part of the course is about electrostatics. Is exactly how do we control these barriers and do it in a way that the transistor has good characteristics. That's MOS electrostatics. These sketches are-- of the energy band diagram are a qualitative description of the MOS electrostatics. And we'll spend a whole unit on MOS electrostatics. Now, we've said that the electrons can hop over and they can go across. If they just zip across unimpeded, don't encounter anything that they can bounce or scatter off of, we call that ballistic transport. And we'll have a whole unit on ballistic transport. That will be Unit 3 of this course. But, most transistors have channel links they are a little bit longer. There's a possibility that you'll encounter some impurity or some lattice vibration or some roughness at the interface that will cause the electron to backscatter. It may go back to the source and not contribute to drain current. Or, it may scatter again and undergo some kind of random walk and finally end up going out either the drain or the source. If there's a little bit of scattering we call this transport quasi-ballistic. If there's a lot of scattering we call it diffusive. And this transport is really what makes nanoscale transistors interesting, and it's this move from diffusive transport to quasi-ballistic and even ballistic transport, that has occurred as device dimensions have gotten smaller and smaller. And that's really our major task in this course is to understand how we treat these effects in a simple but physically sound way. 

[Slide 15] OK. So, this energy band view, this is a really simple, qualitative way to get at the essential physics of what makes a transistor operate. MOSFETs and actually most transistors, bipolar transistors, HEMTs, other types of transistors, JFET, are barrier controlled devices. We manipulate the current by modulating this energy barrier up and down. And if we can do that properly, we have a good transistor. Now, the next topic that I'm going to talk about is a review of the traditional way that we treat MOSFETs. This is a way that would-- that one would not think should be suitable for modern day nanoscale MOSFETs, because it assumes very long channel lengths. So, we'll discuss that traditional viewpoint, and a lot of the course will be about how do we replace that traditional viewpoint with a much more physical viewpoint that is more suitable for the small dimensions that we're currently operating at. So, we'll continue with that discussion in the next lecture.