nanoHUB-U Fundamentals of Nanotransistors/Lecture 4.2: Transmission
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[Slide 1] Well, we're ready to dive into unit 4 and the topic now is transmission. 

[Slide 2] So, we're comfortable now with using this Landauer approach but in using it in a ballistic case where transmission is 1. We want to focus in this lecture on the transmission itself and relate it to the microscopic scattering physics and then we'll be prepared to include that in our analysis of MOSFETs. 

[Slide 3] So, let's look in the xy plane, maybe this is the plane of a MOSFET as carriers are moving from the source to the drain and let's think of a carrier being ejected from the source. It moves for some time, travels some distance, and then it encounters something that scatters it and knocks it in a different direction. It might be a forward scattering event, so it's still traveling in the same general direction. It might then encounter some other scattering mechanism that knocks it and reverses its direction of motion. That we would call backscattering and it might backscatter again and then turn around and be heading in the same direction and, in general, there'll be some type of random walk, different types of scattering events will occur; they may be elastic, they may be inelastic. An elastic event does not change the energy of the carrier, simply changes the direction that it's moving an inelastic event could change the carrier's energy as well. So there'll be a number of these and eventually the electron might exit the drain or it might have exited the source, the same place it was injected from. So these will be random trajectories. Each one will be different and we'll be looking at the average performance as a group of electrons undergoes these types of scattering events. Now, an important quantity in our thinking will be something called the Mean-Free-Path. So if I think about the average time between scattering events and then if I relate that to the average distance between scattering events, the average distance would be the velocity times the average time, we'll call that average distance the Mean-Free-Path and we'll label it with a capital lambda. That's going to be important in our thinking about how we think about transmission. 

[Slide 4] Okay, now let's look in a little more detail about, at the scattering events that occur. Let's say we have an electron that is entering perhaps with a momentum p in some direction, maybe plus X direction in this case. It encounters and undergoes some type of scattering event. It may scatter from charged impurities, for example, from lattice vibrations, from surface roughness at the oxide silicon interface. It might scatter from other electrons or it might scatter from holes. There are a whole variety of ways that it might scatter. After the scattering event occurs the electron will, in general, have a different energy and a different direction and it will be in a different state. Now, the, just to remind you, the scattering events could be elastic, in which case the length of this arrow would be the same as the incident arrow or it might be inelastic. We could gain energy by absorbing energy from the lattice. We could emit energy, for example, by emitting energy to the lattice and in that case the length of the arrow could be longer or shorter, depending on whether we had gained or lost energy. The scattering might be isotropic, meaning an equal probability to scatter in any direction, or it might be anisotropic, there might be a strong preference to deflect it in, just to deflect it a little bit, for example. So all of these properties are determined by the detailed microscopic scattering events that occur and it's rather well understood how to treat these common scattering mechanisms in semiconductor problems. Now, one of the quantities that comes out of this treatment is tau, the average time between scattering events. Now, on the other hand, we frequently talk about scattering rate, 1 over tau. This is the probability per second that a scattering event will occur, okay? So we're not going to get into the detailed calculations of these scattering probabilities but let's assume that they could be done. You know, they're generally pretty well understood for semiconductors. If we know the average scattering time then electrons at some particular energy have a velocity that's given by the band structure so then we can multiply by velocity times average scattering time and we can find the average distance the electrons move before they scatter. Now, one other point that I want to mention before we proceed is that in general the scattering rate is proportional to the density of states. That occurs because the density of states generally increases with energy. At higher energies there are more places for an electron to scatter to so there's more probability that a scattering event will occur. The scattering time will decrease with energy generally and the scattering rate increases with energy. Okay, so we won't dive too deeply into the microscopic physics, we'll simply, we need to know that these quantities can be computed, they can be related to this Mean-Free-Path and our goal now is to relate the Mean-Free-Path to the transmission. 

[Slide 5] Okay, so the model problem that we are going to try to solve here is a slab of some length, L, that has no electric field in it but it's, say a semiconductor slab but it has impurities and lattice vibrations and things that can scatter electrons. We will write the Mean-Free-Path as lambda at a particular energy E. The problem consists of injecting a known flux, I1 of E at the left, and finding out what fraction of that flux transmits and leaves from the right. The fraction is the transmission that we are after. The problem is set up so that we are going to assume that there is an ideal contact called an absorbing contact at the right. The absorbing contact simply absorbs any carrier that comes in and makes sure that no carriers are reinjected from the right. Now, some of the carriers that are injected from the left might backscatter and turn around and emerge from the left. That would be a fraction R, and R is 1 minus T if particles are conserved and no particles are generated or lost inside the slab and we'll assume that that's the case. So the question that we like to ask is how is the transmission related to the Mean-Free-Path for backscattering and note that I've done something a little different here. I've written my Mean-Free-Path as a lower case lambda instead of an uppercase lambda and it's because the particular Mean-Free-Path that we need is related to the standard Mean-Free-Path that people use but it's a little different by a numerical factor and we'll discuss that in a few minutes. 

[Slide 6] Okay, so let's see if we can calculate the transmission. Here's our problem. We inject the known flux at, from the left at x equals 0. Inside the device there is some scattering so the forward flux gets backscattered and becomes a reverse flux. The reverse flux gets backscattered and adds to the forward flux and there's a lot of complicated scattering physics that goes on inside the slab. Okay, what we're interested in is what comes out on the right. The fraction T that emerges is the transmission that we're after and we will try to compute that. Okay, particles are conserved so it means that we will also know the fraction that returns to the left and say returns to the source if this is a MOSFET. Now, we can write a simple equation describing how the forward flux grows and shrinks inside the slab due to the scattering processes. So, under steady state conditions we can write the derivative of the forward flux with respect to position is minus the forward flux divided by lambda. So now we're interpreting this lambda in a very special way. It's the probability per unit length that a forward flux is converted to a reverse flux. That's what we mean by Mean-Free-Path or backscatter. So there's a negative sign here because backscattering decreases the forward flux but once we have a reverse flux inside the slab, if that flux backscatters it will become a forward flux so there's a plus sign there. All right, so that's a physically sensible equation we can write down, we can derive it directly from the Boltzmann equation for those of you that are familiar with that. Now, I can also ask what is the net flux. The net flux is just the difference between the forward flux and the reverse flux. That's constant, independent of position, as long as no carriers are generated or recombined within the slab. So the net flux is constant everywhere. Now, I can use that, in fact, if the net flux is constant I can recognize that I have an I plus minus an I minus up here. I can group those two together and I can rewrite my equation for the forward flux as the derivative of the forward flux with respect to position is minus the constant net flux divided by the Mean-Free-Path for backscattering. So we have a very simple equation that tell us that if we inject some forward flux it's just the magnitude decays linearly with distance according to this equation. So I could easily integrate that equation and find the forward flux at any location within the slab, it's simply the flux that was injected minus the net current times the ratio of distance divided by Mean-Free-Path. 

[Slide 7] Okay, so let's see what we can do with that relation. We now know how this forward flux varies with position inside the slab. Okay, so we have an equation for the forward flux. Let me expand the net flux now in terms of if the difference between the forward flux and the reverse flux and now let's see if we can evaluate the flux that emerges at X equals L. So we'll put X equals L in this equation. This is our injected flux. This is I plus at X equals L. This is the negative flux that is injected at X equals L but we've set the problem up with an absorbing boundary at X equals L such that no flux is injected at X equals L. So we know that that's zero, that's a boundary condition that we set the problem up in. Okay, so now we can use that in this equation. We now have an equation for the emerging flux in terms of the infinite flux. This is what we're after, now we can compute the transmission. 

[Slide 8] So let's use this relation between the emerging flux and the incident flux. Let's solve it for the emerging flux in terms of the incident flux. Let's take the ratio of the emerging flux to the incident flux. That's what we mean by transmission and we have a very simple expression for transmission. So we've done what we set out to do. We've derived an expression for the transmission. We have a very simple expression for the transmission in terms of the Mean-Free-Path for backscattering and length of the sample. This Mean-Free-Path divided by Mean-Free-Path plus length. In the effusive limit, when the length has many Mean-Free-Paths the transmission gets very small and approaches 0. In the ballistic limit, where the length of the sample is very short compared to a Mean-Free-Path then the transmission approaches 1 and it works anywhere in between and it actually works rather well in practice. So we have a simple expression for the transmission 

[Slide 9] that we can use to understand how scattering affects MOSFETs. Now, let me just go back for a minute and talk about this new Mean-Free-Path that we have introduced. The Mean-Free-Path that most people talk about is this capital lambda, simply the average distance that carriers move before they scatter. And we set out to try to show, you know, what is the relation between main-free-path and transmission and what we found is that there's a natural relation between this related Mean-Free-Path, the Mean-Free-Path for backscattering and the transmission and we derived this simple formula. So the question is, what are these two different Mean-Free-Paths, what is the difference between them? 

[Slide 10] Well, it's easiest to see this in one dimension so let's think about a nanowire MOSFET, for example. Electrons can only move in the plus X or minus X direction. If an electron moves in a plus X direction and scatters then there are two possibilities, it can forward scatter and now it is still contributing to a positive flux so it's as though nothing has happened, or it can backscatter and when it backscatters it contributes to the negative flux. So only backscattering matters. If the scattering is isotropic, which means that there's an equal probability to scatter forward and reverse, then we can see that the average time between backscattering events is 2 times the average time between scattering events. So in this case it would be very simple, the Mean-Free-Path for backscattering is simply twice the standard Mean-Free-Path so there's a numerical factor there. 

[Slide 11] It's a little harder to see what that numerical factor is in 2D but I'll just give you the answer. In 2D the picture is this, electrons move in the xy plane. When they scatter they're deflected by some angle. They may end up still contributing to the forward flux. They may back scatter and start contributing to the reverse flux. They may backscatter even more and then contribute even more strongly to the reverse flux. So there's an averaging over angles that has to take place. And I'll simply quote the answer here, some of the references give the details about how this calculation is done, but in this case the numerical factor is pi over 2. So the Mean-Free-Path for backscattering is always longer than the standard Mean-Free-Path because only the backscattering events reduce the forward flux and those are the ones that reduce our transmission, okay? So this distinction between Mean-Free-Path for backscattering and the standard Mean-Free-Path is something you need to keep in mind because when you're reading papers you'll have to be sure that you're clear about which Mean-Free-Path are the authors talking about. We always use the Mean-Free-Path for backscattering because that's what goes in the transmission expression. 

[Slide 12] Okay, now, let's look at a different problem. Let's look at a problem in which there is a very strong electric field in this lab and see if we can compute the transmission in this case because part of our channel near the drain under high drain bias will have a very large electric field so we'll need to treat that part of the channel as well. Well, it turns out that this is not an easy problem. It's actually one of the more difficult problems in semi-classical transport theory and people generally do this with detailed numerical simulations. But the numerical simulations show something very simple. So let me, let's take a look at some of those simulations and see if we can understand simply what's going on. 

[Slide 13] So here are some calculations. These were done by Peter Price some years ago, in which a computer program followed electrons as they went through a high field region, you know, this is a conduction band diagram with a very steep slope, so a strong electric field. Electrons are injected, they're accelerated by the electric field, they scatter and that scattering and acceleration process is tracked until the electrons exit either from the right or turn around and come back from the left where they were injected from. Now, Peter found something very interesting when he was trying to find the fraction that came out the right he found that if the electrons just got a little ways down this potential drop, a very short distance, then they were bound to come out the right and contribute to the transmission. So there was a very short, critical layer, if the electrons didn't scatter across of that then it didn't matter whether they scattered and the simple answer is that the electric field is so strong that it turns electrons around and just make sure they get swept out the right and they don't go out the left. Now, there are other reasons, if there are inelastic scattering events the energy is lowered and it's harder for the electrons to get out. But the bottom line is quite simple, there is a very short critical length. If the electrons get across that critical length, and it's easy for them to do that because it's so short, then they come out the right so the transmission in a high field region is very close to 1. In fact, we could say that these high field regions are very good carrier collectors. They're very much like this ideal absorbing collector we sketched in our derivation of the transmission. So, the actual calculation is fairly involved but the final result is fairly simple. High field regions are excellent collectors. 

[Slide 14] Now, in the MOSFET we have a low field region near the source and a high field region near the drain when the drain bias is high, so we have a structure that looks something like this. We have a low field region in which we know how to compute the transmission, simply given by Mean-Free-Path over Mean-Free-Path plus the length of that low field region, and then we have a high field region where the transmission can be expected to be close to 1. So if I ask what is the transmission across the entire structure, well, it's just limited by the transmission across the low field part of that structure. So it's simply the transmission across region 1. So that's a very simple result that we will use in our analysis of MOSFETs. The transmission is dominated by the low field part of the structure. 

[Slide 15] Okay, so that's what I wanted to cover. We have enough understanding now of transmission and what controls it and how it works out in regions without an electric field and regions with a strong electric field. So the key points are if a transmission is related to the Mean-Free-Path for backscattering, when there is no electric field or a very small electric field we have a very simple relation between the Mean-Free-Path for backscattering and the transmission. In the ballistic limit, the transmission approach is 1, in the diffusive limit the transmission gets small, it's just the ratio of the Mean-Free-Path to the channel length, which is much longer. We also discussed how high field regions act as very efficient carrier collectors, the transmission is close to 1 and the bottom line is that that means that the transmission in a structure like this, a low field region followed by a high field region is controlled by the low field part of that region. So, we now understand transmission and we can dive into using this knowledge to develop our transmission theory of the nanoscale MOSFET. We'll start that in the next lecture and I'll see you there.