nanoHUB-U Fundamentals of Nanotransistors/Lecture 4.8: Limits of MOSFETs
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[Slide 1] Hello, I'd like to talk about something a little different today. So we have been talking about a model that was developed to describe very short channel transistors. It's interesting to ask ourselves, how small can a transistor be made, what are the fundamental limits, how close are we, you know, how well might our model work in the future. So what I would like to discuss in this lecture is just some very simple estimates of what the limits of transistors are and try to gauge how close we are to some of those limits. 

[Slide 2] So this is our model that we're familiar with now, our transmission model. We have a model that describes the device from the linear to the saturated regime, we can express it in this virtual source form, then in terms of two key physical parameters, the apparent mobility and the injection velocity. Now the question that we have in this lecture is, what are the fundamental limits of these transistors and how close are we to those limits. 

[Slide 3] Now we have been using, throughout these lectures, a very useful picture of the MOSFET in terms of energy bend diagrams. This is an energy bend diagram along the source channel and out the drain of a MOSFET under high drain bias and low gate bias so it's in the off-state. So in the off-state we can see where we have a large barrier between the drain and the top of the barrier and we have a large but smaller barrier between the source and the top of the barrier. And the barrier is too large for electrons to get over, so only leakage currents flow the device is off. In the on-state we push the barrier down. Now electrons can flow across the channel. When they enter the drain they undergo a rapid inelastic scattering, they lose their energy and they get thermalized very quickly. Under the on-state the energy barrier between the drain and the source is much smaller and we've essentially removed it between the source and the top of the barrier. 

[Slide 4] Okay, I'm going to use some very simple arguments to roughly estimate ultimate limits of MOSFETs and they're going to be similar to arguments that were used in this paper by Victor Zhirnov, our approach will be just a little bit different but we'll arrive at similar conclusions. 

[Slide 5] So let's first of all ask ourselves, what is the minimum energy to switch a 1 to a 0 or vice versa, the minimum switching energy. Okay, so let's take a look at our device and assume that we've pushed the barrier down and it is on now. Electrons travel from the source across the channel, dissipate their extra energy in the drain, and come into thermal equilibrium in the drain. Now, the electrons are in random thermal motion in the drain. We have to make sure that they are not thermionically emitted over the barrier from the drain, back into the channel and go back out to source. If that happens, we have not undergone a switching event. So the probability that the will be emitted over there has to be low. Well, we're talking about ultimate limits, so I'll just say the probability has to be less than 1/2 to say that I've had a switching event. 

[Slide 6] Okay, so probabilities, we have seen that the probability that an electron can hop over an energy barrier is e to the minus height of the energy barrier divided by kT. And to have a switching event, we're going to take a very loose definition of a switching event, we just want that probability to be less than 50%. So we can say that the minimum barrier height has to be such that we make that probability 1/2, solve for the minimum energy, and the minimum switching energy we conclude is kT times log 2. Now that's a very simple argument to get a very general result that has -- that can be derived by much more fundamental considerations, so that is really quite a solid argument about the fundamental lower limit of the switching energy. 

[Slide 7] Now I want to switch and talk about what determines the lower limit of the channel length. Now practically we worry about two-dimensional electrostatics and maintaining gate control of the channel, so those are practical factors that are getting more and more challenging. But fundamentally we have to worry about quantum mechanical tunneling. These are some calculations, these are some simulations of the electron flow through a nanoscale transistor. So what you're seeing here is the conduction band under high drain bias, low gate voltage, so the device should be off. The color here indicates the magnitude of the current. If we look here at a device with a channel length of 13 nanometers, you can see that the current is flowing over the top of the barrier. This device is operating the way a transistor is supposed to operate, the barrier height controls the current that flows. If I look at a shorter channel length, this is a 10 nanometer silicon nanowire MOSFET, you can see that, again, most of the current flows over the top of the barrier. You can begin to see a little bit of penetration under the barrier, that is quantum mechanical tunneling. If I look at 7 nanometers, then you can begin to see a significant tunneling of the current underneath the barrier. And if you look at 4 nanometers, you can see that there is a lot of tunneling. The electrons sort of don't even know that there is an energy barrier there. It is very hard in this case to make a good transistor because the operating principle of a transistor is that we control the current flow by pushing this barrier up and down. If quantum mechanical tunneling makes that barrier transparent to the electrons, we lose our transistor action. So quantum mechanical tunneling is going to set a fundamental lower limit to the channel length. 

[Slide 8] So let's see if we can roughly estimate what that is. When the device is in the off-state, to say that it is in the off-state there has to be a low probability that electrons will tunnel through that barrier. Well, we're going to be generous here, we'll say we will consider the device to be off if the probability is less than a half that the electron can tunnel through that barrier from the source and go to the drain. 

[Slide 9] Okay, now, we'll need to do a quantum mechanical calculation of tunneling through a barrier and there's a classic approximation called the WKB approximation that people use. And you can estimate the probability of that tunneling occurring and it's going to be related both to the height of the barrier and to the thickness of the barrier. The thickness of the barrier is the channel length. We want that probability to be less than half to consider our device to be off. That's going to set a minimum length for the channel, it has to be longer than this value in order for the probability to be less than 1/2. If I assume that the height of my barrier between the source and the drain -- between the source and the barrier is the same as my minimum switching energy, and I would probably design my device to do that so that everything is symmetrical, then we can plug numbers in, and we get a very simple expression for the minimum channel length of a MOSFET so that I can ensure that it's off when it's supposed to be off. It's given by this expression, related to the effective mass and to this minimum switching energy. 

[Slide 10] Okay, let's also ask, how fast can this device be? What is the minimum switching time? Well, when I push the barrier down, electrons flow across the channel. The minimum time is going to be the time that it takes for electrons to flow from the source to the drain and cross that channel. In practice, we saw earlier in Unit 1, that the speed is usually limited not by this transit time but by the time it takes to charge and discharge various capacitors in the circuit. But fundamentally the ultimate limit should be this transit time. Transit time is just the velocity, they're going across at the thermal velocity, divided by the length of the channel. 

[Slide 11] All right. Okay, so if we look at the switching time, then, the minimum switching time is the transit time, the transit time is the length of the channel divided by the velocity. If the length of the channel is the minimum length that we have just established, and if the velocity is the thermal velocity, and if I assume that the minimum energy is related to kT, I can insert the minimum energy here and then I can throw away some constants on the order of unity because I'm just trying to get an estimate for what the fundamental limit is, and I end up with a very simple expression. The minimum switching time is the Planck's constant by 2 pi divided by the minimum switching energy. 

[Slide 12] Okay, so these are some very simple arguments. We've established the minimum switching energy, the minimum channel length and the minimum switching time. We've been very generous with our definition of what it means for a device to be off, what it means for a switching event to have occurred. So these are really probably some very lower limits to what might be achievable. The minimum switching energy is .017 electron volts, the minimum channel length from this argument is 1.5 nanometers. I'm assuming that the effective mass is just the electron mass in the vacuum and the minimum switching time is 40 femtoseconds. 

[Slide 13] All right, how close are we to those limits? Are they relevant in any way to us in when we think about practical technology? So if we look at current day technology and estimate some of the numbers, let's assume we have a 7/10 of a volt hour supply. Can estimate the capacitance for this particular device in farads per centimeter squared and then I need the capacitance in farads so I'll multiply by the width and the length. I'll assume a 1 micrometer width and the length is 22 nanometers. The on current is roughly 1 milliamp per micrometer. So I've got some numbers. I can compute the energy that is stored when I'm storing a one, it's 1/2 gate capacitance times power supply voltage squared. Plug in numbers and that turns out to be 57,000 times its minimum switching energy So orders of magnitude larger. In fact there's a lot of parasitic capacitance in modern transistors that we're not including, so the situation is, you know, the actual switching energy in real technology is even larger than this. Now if I look at the channel length, channel length is about 20 nanometers in this technology, that's about 13 times the minimum channel length. So that's, you know, we're within almost roughly an order of magnitude of a really very fundamental lower limit. And the switching time, again, I can deduce the switching time by saying C times V, that's the charge, the on current is charge divided by time, so my switching time is CV divided by I, that turns out also to be roughly an order of magnitude above the fundamental limit. So we conclude that both the channel length and the switching time are within an order of magnitude of some very fundamental lower limits, and we're still pushing these dimensions down. The energy and the power, they're orders of magnitude bigger and that's just a reflection that in order to do real circuits and have low error rates, we need much higher voltages, higher barriers. We need on/off ratios that are more like 10,000 not like 1/2. So as a result of that, energy and power are well above fundamental limits and will probably remain so. 

[Slide 14] All right, so just to summarize. We've just taken a quick look at some fundamental limits, we established them with some very simple arguments. And one of the things that we learned from that is that modern day transistor technology is actually getting remarkably close to some of the fundamental limits. Now, in practice, real technology considerations are going to determine, you know, how small we ultimately make transistors. Things like series resistance, which doesn't scale down as we all like it to, parasitic capacitance which just gets bigger and bigger in comparison to the intrinsic gate capacitance. Various leakage mechanisms, things like band-to-band tunneling that give us leakage currents through the drain, things like that. These practical technology considerations, the necessity of maintaining good electrostatic control of the channel by the gate, these practical technology considerations are probably what's going to set the ultimate limit, not the fundamental consideration that we've discussed in this lecture. Good transistors operate semiclassically by this energy band model. We push the barrier up and down, we control the amount of current that thermionically flows over the barrier. So that's the way a good transistor operates, that's a very semiclassical description. So we conclude that this semiclassical, you know, simple physics description of transistor that we've developed in this course is probably going to apply to any good transistor in the future no matter how small we make it. We won't need some fundamentally new quantum mechanical theory of the MOSFET. Quantum mechanics is important, it affects effective masses, it effects tunneling currents, and leakage mechanisms. But in order for a transistor to operate well, a barrier controlled transistor has to operate semiclassically and our semiclassical model should capture the essential physics. So that's a summary of where the technology stands today, and we are basically done now with Unit 4. What I would like to do in the next lecture is to summarize, not only what we've learned in Unit 4, but to give a broader summary of some of the takeaway messages and what we've learned throughout this entire course. So I'll see you at the next lecture and we'll wrap up the unit and the course. See you there.