 You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. So we'll go ahead and get started now on lecture six. So we've done all of the heavy lifting on transport theory, so we won't have to do much more of that, but we're going to talk about several other topics. And the first one is scattering. So I've assumed in all of these lectures that there is some mean free path and that it is probably energy dependent. So let's talk about it a little bit and see if we can understand a little bit about this mean free path. Now when I teach my course, I usually spend about a third of the course on how you calculate scattering rates in mean free paths. So we're not going to go into that kind of detail, but hopefully it will give you some kind of general sense and understanding about how these things work. Oops, get my equipment working here. Okay, so these are the things that we'll be talking about. Right, so basically we've argued that we have this expression for transmission, mean free path divided by mean free path over length. And where does this come from? And what exactly is that mean free path that's in that transmission expression? We expect that it's going to be something like the velocity times time between scattering events, right? The average distance between the scattering events that a carrier experiences. But what exactly is that relation? So we want to talk about that. So first of all, we need to talk a little bit about how carriers, how electrons scatter in a semiconductor. So we can think about doing a Gadankan experiment. And actually you can do something like this almost experimentally. Think about injecting a bunch of electrons into a semiconductor that all have their momentum pointed in the same direction. Now if we have a hetero junction, we might be able to launch them off of a hetero junction and they'll all be going and are mostly in one direction. So at t equals zero, they're all launched into the semiconductor. Now, if you wait a time, and you would call sometimes, this is called a single particle lifetime. If you wait a time, all of those will experience some scattering event. So things will be somewhat random, a scattering time tau later. But it might not be completely random. So you can see here that there's some memory of the initial momentum. There's still a net momentum to the right. If you wait a little bit longer, eventually you'll randomize all of that momentum and there'll be no average momentum in any direction. So there'll be no average current flow now. That time is called the momentum relaxation time. And it might be longer. Depends on the particular scattering events. If the scattering event happens just to deflect carriers by a small amount, each collision, then it will take many scattering events to randomize all of the momentum. If scattering events happen to take you in any possible direction, any one, then the two will be equal. But the length of the arrow here corresponds to the energy. Let's say that they were injected with excess energy, maybe 10 kT or something. So the length might, these scattering events might have been elastic. We might have been scattering off of charged impurities. The charged impurity is just some rigid thing in the lattice. It can't absorb any energy. So the electrons can change sign, but they can't change energy. So their momentum may have been relaxed, but they may still have that excess energy that they were injected with. Well, if you wait a little bit longer, they'll interact with phonons and they'll emit that excess energy and scatter from phonons. And they'll come back into equilibrium so all of the arrows are shorter now. Corresponds to 3 halves kT or something. That's the energy relaxation time. And Professor Fischer mentioned these a little bit. In general, you would expect that this would be longer than the momentum relaxation time because if you get injected with 10 kT of energy, each phonon only has 30 million electron volts or something. You have to undergo several, you have to emit several phonons in order to get rid of that excess energy. Okay, so that's what we mean by these characteristic times. So they all have a physical interpretation. Tau is the average time between collisions. Tau sub m is the average time it takes to relax or randomize momentum. Tau sub e is the average time it takes to dissipate all of that excess energy. Now the fundamental quantity, though, that we can compute is this transition rate. And so S of P, P prime. So think about an electron coming in with some momentum P. This is its crystal momentum, h bar k. And it makes a transition to some other state. So it gets deflected at some angle. I mean, this is a polar angle and 3D is coming out of the page. And it gets deflected into some other state. So the probability per second that that happens, we call the transition rate S of P, P prime. So it's the probability that you make a transition from P to P prime. And be careful about the order of the indices. Sometimes people will write this in the other direction. You always have to remember, this is the way I've written it. It's from P to P prime. Sometimes people interchange the order of those two. So it would be clearer if I put an arrow probably from P to P prime. Now how does that happen? It happens because the electron, if an electron goes through a perfectly perfect crystal, then there is no scattering. The crystal potential has all been factored into its infected mass and it just travels through without scattering. But if there's some defect, or if we're at finite temperatures and the lattice ions are vibrating out of their equilibrium positions, there will be a random potential that will introduce that scattering event. And that's the scattering potential. So to do the scattering calculation, we have to identify what that scattering potential is. And then we have to figure out how to calculate the scattering rate. Okay, but in the end what we want are these times which make physical sense to us. What we can calculate is this transition rate from one state to another state. But what we want are these characteristic times. But those are easy to calculate if we know that transition rate. So if I've got a finite volume of sample, or if it's a 2D sheet, if I have a finite area, or if it's a 1D nanowire, and I have a finite length. There's a finite number of states we can count them. Sometimes they're very, very large and we end up converting that sum to an integral. Usually that's the case. So if I want the average time between collisions, I would take that transition rate. So I'm injecting an electron with some momentum p. The transition rate tells me its probability per second that it is making a transition to some other state p prime. And then I do a sum over all of the other states that it can possibly make a transition to, and then that's the probability that it has scattered somewhere. So that would give me the scattering rate, one over tau. Now I have a little arrow on the up there and that's just to remind me that we have to be careful about these factors of two for spin that we often use. That normally when we're doing these integrals, we say, well, there's always two states there, one that can hold spin up and one that could hold spin down, so we multiply by two. But for most scattering mechanisms, if an electron comes in with spin up, the scattering potentials are only going to scatter it to another state with spin up. Unless we have a magnetic impurity and then that could flip the spin. So if an electron comes in, it can only scatter to half of the states for the common scattering mechanisms that I'm going to be talking about. Professor Dada might talk a little bit more. He's going to talk about spin transport. Maybe he'll talk about spin-dependent scattering. So we have to be careful about that factor of two. We can only, the number of final states you can go to, you can only go to a final state that has the same spin as the one you were coming in at. Okay, what about the momentum relaxation time? Well, S of P P prime gives me the probability that you make a transition from one state to another state. But if I need to weight that by the fractional change in momentum between those two states, if I just made a small deflection, I wouldn't have changed the momentum by very much. So that scattering event wouldn't count very much for the momentum relaxation time. So we simply weight each state by the fractional change in momentum. And then we weight by the probability that that particular transition occurred, and we sum over all the possibilities. That's the mathematical prescription that would give us the momentum relaxation time. The energy relaxation time, we would do the same thing. So we come in, we undergo a scattering event. We look at the energy of the final state. If the energy of the final state is different from the energy of the initial state, then we take the fractional change. If we came in with an energy E sub zero, if the scattering event, if we absorbed the phonon or emitted the phonon, we have a different energy. We just weight by that fractional time. That gives us the energy relaxation rate. So if it's an elastic scattering event, there is no change in energy. And even though a scattering event occurred, it has no effect on the energy relaxation time. So how do we compute those transition rates? So if you take a course in quantum mechanics, people spend a lot of time talking about scattering calculations for various types of physics problems. There's one particular technique for calculating scattering rates that's first order perturbation theory. That's the simplest possible technique and which usually works pretty well for semiconductors. Most scattering rate calculations in semiconductors are done by Fermi's Golden Rule. Sometimes that's not quite good enough and people go to second order perturbation theory or more sophisticated techniques. But most calculations that you'll see done for semiconductors simply use Fermi's Golden Rule. And you can find it in any introductory quantum mechanics book. And we're not going to get into it here, but I just want to acquaint you with how people do these calculations. So the basic idea is that an electron comes in with some crystal momentum P. It's got some wave function. I'll call that size of I. That's the wave function of the incident electron. It encounters this scattering potential, which is a short range potential that we describe quantum mechanically. And it gets scattered out into another final state. And when it's far away from that scattering potential, it's another plane wave and that has a wave function phi sub f for the final state. So the prescription that you use for scattering rate calculations is that the transition rate from P to P prime is 2 pi over h bar times the magnitude squared of this matrix element. Now notice the order of the P and the P prime there. So this is where the potential for confusion. You always, when you write a matrix element, you always sandwich in the initial state is on the right. The final state is on the left and the scattering potential is sent sandwiched in between the two. That's always the order. So sometimes people will write S of P prime P to mean what I'm meaning S of P P prime. You always have to be careful about that. So this is a transition from P to P prime and that involves a matrix element and it goes backwards from P to P prime. So and it involves a delta function which tells us that we have to conserve energy. So the way we calculate these scattering rates then is that you identify what the scattering potential is. You compute this matrix element. You ask yourself whether this was an elastic scattering event or an inelastic scattering event. The final energy is the initial energy plus the change in energy that occurred for the scattering event. If you have a static scattering potential, like an ionized impurity, it can't change the energy of the carrier, it can just deflect it. So delta E is zero. Then that delta function says you can only scatter to final states that have the same energy as the initial state. If you have a time dependent potential like a phonon oscillating at some frequency H bar omega, then you can absorb a quanta of energy H bar omega or you can emit the quanta of energy H bar omega. Delta E is plus or minus depending on whether you absorbed or emitted a phonon. So the delta function there tells me I can only scatter into states that are H bar omega above the initial energy or H bar omega below the initial energy. It just expresses energy conservation. Okay, all right, so the calculation procedure, you identify what the scattering potential is. You'll have a different potential if it's an ionized impurity. If it's a lattice vibration, there'll be a scattering potential. It might be roughness at the surface. It might be some kind of crystal defect. It might be scattering from another electron or from another hole. You identify what it is, what the scattering potential is. You compute this transition rate using Fermi's golden rule. Typically works most of the time for semiconductors except in some special cases. And then once you have that transition rate, you can perform these various sums, weight by the fractional change in momentum, weight by the fractional change in energy. And you can compute these characteristic times. And those are the times that we are most interested in when we do transport theory. And one of the things you should remember is if the scattering mechanism is such that it likes to just deflect electrons a little bit, then the momentum relaxation rate is much longer than the scattering rate. Or the momentum relaxation time, the momentum relaxation rate is much smaller. The momentum relaxation time is much bigger. But if it's isotropic, the thing can come in and it can scatter anywhere with equal probability, then those two times are equal. Some scattering potentials are anisotropic. They like to deflect electrons by just a small angle. That typically happens for charged impurity scattering, ionized impurities, or if you have phonons in polar materials. The scattering potential is an electrostatic one and it likes to deflect electrons by a small amount. Things like acoustic phonons in non-polar materials tend to be isotropic, so the two scattering times are the same. All right, then we can compute the mean free path. This is what we're trying to do. Because if we want conductance, we want see that coefficients, we have to compute this energy dependent mean free path. That's what we're trying to do. And we think it should be something like the velocity of the incident of the electron times how long before it scatters. But we're not there yet, so we have the time. And let me just mention, I'm not going to go through any of these calculations. That would take more time than we have for the summer school. But one of the things I want to mention is that frequently, a first order assumption to understand what the scattering rate is like. Is to say, well, if I have some probability that I'll go from one state to another state, then if it's elastic scattering, what matters is how many states are there at the final state. The density of states tells me that. So one over tau is the probability per second that I'll scatter somewhere. And that should be proportional to the number of states that are at that energy. So frequently people start if they want to understand how the scattering time varies with energy, they just say it varies as the density of states. If I have two states, I have twice as many ways to scatter as if I have one state. If I have a thousand states, I have a thousand ways I can scatter. Now if I have some scattering mechanisms that tend to select out certain preferred states, then you get slightly different answers. Like ionized impurity scattering just likes to deflect you by a small angle. It doesn't have equal probability for all of them. If you're absorbed a phonon, then you need to find a state at an energy h bar omega above the initial energy. So your proportional, the scattering rate will be proportional to how many states there are, how many final states there are at that energy. If you emit a phonon, you have to find the state with a lower energy. So scattering rates for the simplest mechanisms just go proportional to the density of states. Okay, so just a little bit about scattering. As I said, we calculate these by Fermi's golden rule, but you can see what some of them look like. Acoustic phonon scattering is going to look like acoustic phonons typically don't carry much energy and it's not a polar interaction. So they're isotropic, they scatter you anywhere. So acoustic phonons will give us a scattering rate that's proportional to the density of states at the incident energy. Optical phonons in non-polar materials are, they're not elastic, but they're isotropic. So the scattering rates will be proportional to the density of final states. Now if you go to a 3-5 semiconductor, then the optical phonons have an electrostatic scattering potential and they select out certain preferred states and it gets a little more complicated. What if you have charged in purity scattering? So if you think about a semiconductor that's doped. Usually we just think, well there's N sub D dopants per cubic centimeter and we just think of it as a uniform background charge. But if you look at it microscopically, on the scale of this electron wavelength, there are discrete dopants. Every time there's a dopant, it can, depending on its sign, it'll pull the conduction band down a little bit because of the electrostatic potential or it'll push it up, depending on whether it's a donor or acceptor. So if you look, that conduction band is rough on an atomistic scale. If there's a dopant there, it pushes the conduction band up or down. That rough conduction band profile will scatter electrons. That's the scattering potential for ionized impurities. So when I do the calculation with Fermi's golden rule, we're asking ourselves, how will the electrons reflect off of a rough potential like that? Now you can, one of the things you want to remember when you do that calculation is that it's anisotropic. It likes to scatter, it likes to deflect electrons by a small amount. When you do the Fourier transform of the Coulomb potential, you just find that the conserved momentum, there aren't any many components that are out there. The other thing that you want to remember is that if your electron is very high, it doesn't really see these little fluctuations down there. Meaning the higher the energy, the less the electrons are scattered by ionized impurities. And that can sometimes be important for thermoelectric problems. Because remember this little delta above the bottom of the conduction band. If you can get the current to flow a little higher above the bottom of the conduction band, you get a little higher CBAT coefficient. If you're dominated by ionized impurity scattering, it sort of helps the current to flow at higher energies. So thermoelectric people, that's the reason that people worry a lot about what is the energy dependence scattering time or scattering length. Because it affects the parameters that you measure. Okay, so an important point for ionized impurity scattering is that the scattering rate decreases as the energy increases, or the scattering time increases as the energy increases. That's kind of unusual because in most scattering mechanisms, the higher the energy, the more states there are to scatter to. Because the density of states generally increases with energy. So generally when carriers get more energetic, they scatter more frequently because there are more states to scatter to. Ionized impurity scattering is an exception. Now you'll frequently see when you read papers and people are trying to do analytical calculations. You have to simplify the scattering, have some simple functional form. And for many of the common scattering forms, you can write the energy dependent scattering time in this power law form. It's some constant times the kinetic energy, E minus bottom of the conduction. And we usually normalize that to KT to some characteristic power, S. Now you can see if the scattering rate is proportional to the density of states, and in a parabolic band the density of states is proportional to the square root of energy, then the scattering time is proportional to one over the square root of energy, so S would be minus one-half. So for acoustic phonon scattering, people will say S is minus one-half. Now the calculation for ionized impurity scattering is more complicated. The final expression looks approximately like this, but S is plus three-halves. So a lot of times when people are doing calculations of seabed coefficients and conductivities, they'll assume that they can treat scattering by a functional form like this. Okay, so that's a little bit about scattering. That's about all I'm going to say, but now we want to switch and see how we can take that scattering time and calculate a mean free path. Because once we have the mean free path, then we're home free. Then we know what to do with it, how to calculate all these transport parameters. So we have to calculate the transmission. And the key point is it's related to the mean free path. But I always try to be careful when I talk about this and I call it a mean free path for back scattering. It's a specially defined mean free path to make things work. So we'll talk about that a little bit. And it's related to these microscopic transport processes. So we want to talk about two things. So first of all, let me put on hold the actual calculation of how we relate mean free path to velocity times time. And ask the first question, why is the transmission lambda over lambda plus one? I kind of argued early on that in the diffusive limit, this gives you the right answer. In the ballistic limit, it gives you the right answer. It's actually better than that. So we'll see if we can derive this. And then we'll get back to how we find out how lambda is related to V and tau. Okay, so we have to do a little bit of algebra here, but that's not too bad. So we're going to think about doing a Gdunkin experiment. And this is something that's easy for us to do by Monte Carlo simulation. And you can think of yourself, let's say we have a slab of a semiconductor. No electric field, no electric field inside it to accelerate them or anything. It's a homogeneous slab of semiconductor of some finite length L. And it has some mean free path lambda. And let me inject some electrons at some energy into that slab at x equals zero. So I'm going to assume that the scattering is elastic. So I can just take one energy channel. And we're trying to find what the mean free path is. Or we're trying to find what is the transmission at that energy. So if I were to do this, and we do these calculations on a computer, you shoot 10,000 electrons at this slab. And you count the number that come out the other end. If it's 6,000, then your transmission is 0.6. So what comes out the end is some fraction of what you put in. So the current that comes out is transmission times the current that went in. And the rest of them end up scattering and going back out the side they were injected at. And that gives us the reflection coefficient. So if the transmission coefficient is 0.6, the reflection coefficient is 0.4. Because I'm not allowing carriers to recombine or disappear or anything inside the semiconductor. That's just conservation of particles. Okay, so there must be some relation between that mean free path in that slab and the transmission. And that's what we're trying to find. Now notice in general, I could be injecting electrons from both sides. I could inject some electrons from the right side and ask how many transmits over to the left side. Now if I were to do this, you might ask, well, is the transmission coefficient the same? Now one thing you can show is that if, and you can show this from Fermi's Golden Rule or other ways, is that if the scattering is elastic, then the scattering is always, the scattering from contact one to contact two is always exactly equal to the scattering from contact two to contact one. That's why I've never, in all the other expressions, and we didn't get a question on this, I always had a T. I had two contacts that could fill up the device. The electrons could come in from two sides, but I only had one transmission. I was really assuming that they transmit across in both directions with the same T. That we can justify rigorously for elastic scattering. Now, since we're near equilibrium, we have no electric field in there. Everything is symmetric. So even if there's a little bit of anise of inelastic scattering, we don't expect any asymmetry in this problem. So I'm only going to talk about one T. What is the T from the left to the right? So here's the Gdunkin experiment that we're going to do. We'll inject electrons from the left side. We'll try to figure out how many come out the right side. We'll take the ratio of those two and that will give us the transmission. We're going to ignore any changes of energy inside there. We're going to assume that we can do all of this at one energy channel. And then we'll just add up all the results when we're done. Now, I am going to assume that the mean free path is constant. Otherwise, things just get messy in there. Okay, so here's the picture. We inject electrons from the left. We're just injecting electrons with positive velocities. Once they get inside the slab though, they can back scatter. And now I have a mixture of positive and negative velocities. At the end of the slab, if they have a positive velocity, they can emerge. And the amount of flux that emerges is just transmission T times the flux that came in, that's the definition of transmission. And some of them with negative velocities, if they're at x equals 0, they can leave the slab and that gives me the reflection coefficient. I'm going to assume that I have a perfectly absorbing boundary over here. So that I'm not injecting anything from the right side. Just because there's no reason to, I can calculate T without doing that. Okay, now I want to write a little differential equation. I'd like to know how these positive and negative fluxes vary with position inside the slab. So I can write a little differential equation. I expect that they're going to change with position. So there'll be some i plus dx. So if that plus flux that comes in back scatters, it's going to decrease the plus flux. And here's where I'm defining my main free path. I'm defining my main free path. One over my main free path is the probability per unit length that a positive flux back scatters and becomes a negative flux. That's why I label it a mean free path for back scattering, okay? That's my definition of mean free path, right? So it might scatter, but it might scatter in the forward direction, right? Then it's not a back scattering event, and that's why it's different from the average distance between scattering events. Only the ones that turn the direction around are the ones that are going to matter. That's how we're defining the mean free path. So the first term just says that my positive flux will decrease because the probability per unit length that it will back scatter is one over lambda. But then I build up a population of negative fluxes. If they back scatter, they become a positive flux. So there's a plus sign there. And if they back scatter, they increase the plus flux, okay? Now, I can also say that the net flux has to be constant. This is current continuity. It's because particles are conserved, so that's my continuity equation. So if I take the difference between the plus flux and the minus flux, that difference is constant. It's the net steady state flux, all right? So I have to do a little bit of algebra here now. So I can solve that equation for the negative flux and put it in the first equation. And that's going to give me, when I do that, you can see those two terms are going to cancel out. And I get a nice simple equation for how the positive flux varies with position inside this slab. It's just going to, the slope is constant, negative. It's going to decrease because the back scattering is going to drop it down. The current I, I don't really know yet. But it turns out I'm not going to need to know that. But I is just the constant. It's the net, net current. Okay, so this is what we have. That tells us how the positive flux varies with position inside the, inside the slab. Okay, now let me integrate that. So I'll integrate that from some place where I know the positive flux. That's at x equals 0 to some arbitrary location inside this slab. That's i plus of x. So I'm integrating from the right side to from x equals 0 to some location x. And the result is that we get this little equation. This just tells us if I inject the positive flux at x equals 0, then it's just going to decay linearly inside this slab. Okay, so we just have to be careful about going through the algebra here step by step. So this is where we are. Now, let me go back and let me remember that the current i is the difference between i plus and i minus. It's the net current. So I'll just put that back in now. All right, now what I'm interested in is finding the current that comes out at the end of the slab. Because if I can divide that by the current that comes in, I've got the transmission. So the current that comes out at the end, I just have to put, for x, I just have to put the length of the slab, l in. Now I remember that my experimental condition is that at x equals l, I'm not shooting any flux in. So by definition, i minus of l is 0. That's the experiment, the Gadunkin experiment that I'm doing. So that means I have this expression, which only involves the plus flux at x equals 0 and the plus flux at x equals l. So I can solve this little equation for the flux emerging at x equals l divided by the known flux that I put in. That's my transmission. All right, now we're almost home free. So we just solved that equation for i plus of l. And we see that it's related to what came in. And in fact, we see that the ratio of the emerging flux to the incident flux is just lambda over lambda plus l. So that's the derivation of the transmission. And notice nowhere in there did I make an assumption that the slab was many mean free paths long or that the slab was ballistic. It applies both in the ballistic limit and the diffusive limit and anywhere in between. I assume that there wasn't much in elastic scattering. And I also mentioned earlier in response to a question that we frequently use this technique as a way to, we have a more sophisticated Monte Carlo simulation that's got a complex band structure and has all kinds of complex scattering physics. And we ask, what's the average mean free paths that an electron experiences? We'll just go in and shoot 10,000 electrons in one end and count the number that come out the other end, track them with a computer, take the ratio, and that tells us what the average mean free path is. And it works very, very well because if you plot it versus l, it obeys this equation. So that's the equation and the point is that it is more than just the ad hoc expression that gives you the right answer in both limits. It works in the quasi ballistic regime too. All right, good. So now we, okay, so now we know what the transmission is. Now we have to get back to this question about how is lambda related to the scattering time. And so here I'm just going to try to give you a sense as to how it works. So let's think of a wire, think of a nanowire. An electron comes in, it encounters a scattering potential. And let's say it's isotropic scattering. It can do one of two things. It can scatter with equal probability in any direction. There are only two directions. It can scatter forward or it can scatter backwards. If it scatters forward, it hasn't decreased the positive flux. If it scatters backwards, it's a backscattering event. So in this case, you would say only half of the scattering events really count for backscattering. So in this case, I would say that the average time between backscattering events is two times the average time between collisions. And if I were calculating the mean free path for backscattering, it would be two times V tau. Most people will define mean free path as being V tau. But if we want things to work, if we want a device that is one mean free path long to have a 50% probability that an electron injected in one end will come out in the other, we have to define mean free paths this way. Okay, now if you're in 2D, it gets a little more complicated. And I'm going to refer you to a paper so you can see how it's done. But you can kind of see what happens. This is a forward scattering event, but you lose some forward flux because you're deflected at an angle. This is a backscattering event. This is a backscattering event and you've lost a lot of flux in that case. So you have to do this average over energy properly to figure out what the factor was 2 in 1D, it's going to be something else in 3D or in 2D. The factor was 2 in 1D, it's going to be pi over 2 in 2D. If you have isotropic scattering and if you have an isotropic energy surface where you have the same density of states in any direction, you'll get a factor of pi over 2. Now, if you want to know where this comes from, I'm going to refer you to a paper. But you can actually give a proper mathematical definition to this. Its definition is 2 times the average Vx squared tau divided by the average of the magnitude of V. So what is this by the bracket? This means at an energy E, Vx is the average x directed velocity at that energy, but we need that quantity times tau. And then we need to divide by the average magnitude. Remember we're near equilibrium, so since we're near equilibrium, the velocity is equally distributed in all directions. So if I just did the average of Vx, it would be 0 because I'm near equilibrium. So if I take the average magnitude, I'm only getting the positive half. So you can work that out for any band structure and you can work that out for any dimension. And when you do that, this is the answer that you'll get. You get the 2 and 1D, that's intuitive and easy to see. It's sort of intuitive that the back scattering mean free path should be longer than the actual mean free path because it takes longer to reverse things and the amount that it's longer is pi over 2 and 2D. And the amount that it's longer is 4 thirds in 3D. And I'll refer you to a lecture on the nano hub if you want to see this worked out in a little more detail or I'll refer you to a paper by Changwook and JAP where you can see how you can do this for a more complicated band structures. Okay, so let's see, we just have a few things to discuss then. So I'll talk a little bit now. I'll get back to these questions about how we estimate mean free paths from measurements. I've done a little bit of that, but I sort of had to pull things in from the air. I had to say diffusion coefficient is B times lambda over 2 and where does that come from? So we'll talk about some of those things and mobility. And let's see how this goes. Okay, so let's consider that I have a 2D resistor, maybe a channel of a MOSFET. And I would like to determine what the mean free path is in this MOSFET. Well, I can easily measure the conductance and I have a theoretical formula for the conductance. So the question is, if I've measured the sheet conductance of this 2D film, how do I deduce what its mean free path is? Okay, so a little bit of algebra here. The top line is the expression we've been using for the conductance, the sheet conductance, 1 over ohms per square. Now if I do a little bit of algebra, I can just rearrange these things. Let me divide by the integral of m times df de and multiply by the same quantity. That quantity is what I've been calling the effective number of modes. The other quantity here, it looks like I'm averaging the mean free path and the weighting function that I'm using is m of e times minus df de. So that is my mathematical definition of what I mean by the average mean free path. So if I know the scattering physics, if I know how lambda varies with energy, I can perform that integral and I can find the overall average mean free path for all of the electrons. But if I'm doing an experiment, what I've done is I've measured the sheet conductance and I'm just trying to deduce what that quantity is, what that average mean free path is. This is its mathematical definition. I'm just trying to deduce it from an experiment. Okay, so in order to deduce it from an experiment, all I have to do is to take the measured sheet conductance divided by 2q squared over h and divide it by the effective number of channels that are participating in conduction. So I need to know the effective number of channels that are participating in conduction. Well, we get that by just weighting the number of channels by this function that picks out the ones that are near the Fermi energy, right? That's the integral that we do. And we know what the number of channels is in 2D for a parabolic band. We can perform that integral and unfortunately we're going to get a Fermi Dirac integral, but that's the way it is. We just do the integral, that's what you get. So, if I'm doing an experiment, how do I compute that? I have to know what the Fermi level is in order to compute that. A to F is the normalized Fermi energy. How do I get the Fermi energy? Well, the Fermi energy determines the carrier density. So if I can measure the carrier density too, and I know that the carrier density is related to the Fermi energy. And I'll find the precise relation by integrating the 2D density of states times the Fermi function. In this case, I get a Fermi Dirac integral of order 0. If I can measure the carrier density, I can deduce where the Fermi level is to give me that carrier density. I can put it in my expression for the effective number of channels. And then I'm home free. So the procedure works like this. We measure the sheet conductance, we know it's equal to that expression on the right, then we deduce the average mean free path from the expression number two there. But in order to perform that calculation, we need to know where the Fermi energy is. So we do that by measuring the sheet carrier density also, and then deducing the Fermi energy. So, if I do that for a non-degenerate semiconductor, things simplify, those Fermi Dirac integrals all become exponentials. So if I just do a little bit of algebra and simplify that expression that I had, this expression that I have here, I can simplify this on slide 32, this expression number two. I can just simplify that for Fermi Dirac statistics, rearrange some terms. And the result is fairly simple, 2KT over Q divided by Q times the thermal velocity times sheet conductance divided by carrier density. The thermal velocity here is square root of 2KT over pi M. Remember, there are lots of different ways to define thermal velocity. There's the RMS thermal velocity. This is the thermal velocity directed in a direction. This is the average thermal velocity of electrons in the positive x direction. Okay, so for example, let's see what would happen if I did this. This is my expression for the mean free path. If I substitute in for sigma s NQ mu, then I can solve that expression for mobility. And I get mobility is thermal velocity times average mean free path divided by 2 divided by KT over Q. That looks like an Einstein relation. It looks like something divided by KT over Q. The something must be the diffusion coefficient. So the diffusion coefficient must be the average thermal velocity, 1.2 times 10 to the 7th centimeters per second under non-degenerate conditions. Times the average mean free path divided by 2. Okay, I used that earlier. I just pulled it out of the air. But you can see it. It just comes directly from the expressions that we've been using. Let me look a little bit more generally about that. What that diffusion coefficient is. And a way to do that is to go back to our slab. And let me look at x equals 0. If I want to know what are the number of electrons that have a positive velocity at x equals 0. Well flux is always number times velocity. So the number is just a positive flux divided by the average x directed velocity of those electrons that I'm squirting in. The number with negative velocities there is just a negative flux. And here I'm going to make assumption that we're near equilibrium. The positive flux is moving at the same velocity as a negative flux. That means I know, let's see. N plus of 0, what am I doing in the next line here? N plus, oh, the total, okay, I'm getting confused. So N plus of 0 is the total electron density. I have to add the electrons that have a positive velocity to the electrons that have a negative velocity. So the third line is the sum of the first two. That's just the total number of electrons, whatever velocity they have. And then I remember that t plus r is equal to 1. So I could also write that as 2 minus t times injected flux divided by the average velocity. Okay, I can do the same thing over here. And I can get an expression for the total number of electrons at the end of the slab, same kind of algebra. And if I look a little more carefully, I can convince myself that the line is linear in between. So what I'm getting here is, we've deduced what the carrier density is at x equals 0. I've deduced what the carrier density is at the end of the slab. This looks like a diffusion problem. It looks like carriers are just diffusing across the slab. And I've got the carrier density at x equals 0. I have the carrier density at x equals l. The difference in the carrier density, I just subtract those two. And I get this expression. And if I want to find out what current that comes out, the current that comes out is just t times i plus. So I can substitute the second equation on the right into the top equation on the right. I can solve it for the current. And the result is this expression. Notice that it's proportional to the gradient of the carrier density in the slab. This looks like fixed law. Now, so what we've discovered here is that just by doing this algebra is that I can write the current as minus something times the gradient of the particle density in this slab. All of those factors out front must be the diffusion coefficient. This looks like fixed law. So we just said, what we've deduced is that current flows down a concentration gradient. And that we have an expression for the diffusion coefficient. But I know what t is. It's lambda over lambda plus l. So I can put t back in here. And finally, we get a very general description for the diffusion coefficient. It's average value, average x directed velocity times mean free path divided by 2. Similar to what we got before. So we're seeing this over and over again. Now there's something very interesting about this exercise. Maybe there was just too much algebra. You appreciate it. This is fixed law. It says that particles flow down a concentration gradient. One of the things that people, you've seen oftentimes, people worry about. Well, if your structure is small compared to a mean free path, the fixed law shouldn't work. Because fixed law assumes diffusive transport. This says fixed law works all the way to the ballistic limit. That's very interesting. There was no assumption that this slab was many mean free paths long. Current always flows down a concentration gradient. And that's kind of interesting. It's not widely known, but I think it's true. William Shockley wrote a paper in 1961 who pointed this out. And people haven't paid much attention to it. But you can treat diffusion across a region that's one mean free path long using fixed law. You just have to be very careful about the boundary conditions. That's where things usually go wrong. Okay, so we have this expression for the diffusion coefficient. This is the diffusion coefficient of electrons at an energy E. Because we're doing this calculation at a specific energy. That bracket means we've averaged the x-directed velocity. It's the average x-directed velocity over all of the angles. And if we have a complex band structure, we have to do that average over all of those angles. We do this in, if I have an isotropic band structure, then we can look at this in 1D, the average velocity. There's just one velocity in the x-direction. So that average velocity is just V of E. In 2D, we did this early on, maybe the second lecture. We did that angle average in 2D, and we found the average x-directed velocity is 2 over pi times V of E. If we do this average in 3D, the average velocity is 1 half the magnitude of the velocity. So the average velocity in the direction of transport is related to the magnitude of the velocity by these statistical factors. Now, if I put those factors back in to my expression for the diffusion coefficient on the top, then we find that in 1D, the diffusion coefficient is V squared tau. In 2D, it's V squared tau over 2. In 3D, it's V squared tau over 3. You'll frequently see this expression, and this is where they came from. Now let's see, here we're going to talk about mobility. So frequently, if you read a paper, people will, usually they'll do a measurement where they'll measure the conductivity. They'll measure the carrier density. They'll divide the two, and they'll report the mobility. So if you're reading a paper, maybe the only thing you have is the mobility that they told you. So if you want to deduce the mean free path from the mobility that you have, how do you do that? Well, we've measured it by using this expression. And we take the measured carrier density and the measured conductance we divide the two and we get a mobility. And from the measured mobility, we wanted to deduce the mean free path. So the idea here is that we have these two different ways that we could write the conductance. The fundamental way that I've been encouraging you to think about it, and the way in which we write it in terms of the mobility. If we equate these two expressions, it tells us what the mobility is. And this is what people sometimes call the Kubo Greenwood formula, that this is the definition of mobility just by equating those two expressions. Okay, now, somebody has measured the mobility and they've given it to us. And we want to determine what the average mean free path is. So we have this fundamental expression for mobility. This is what it is. So again, I can do a little bit of algebra here. I can divide by the integral of m minus df de and multiply by the same quantity. And that means that I can write my mobility as 1 over the carrier density times 2q over h times average mean free path. And this is a mathematical definition times the effective number of channels. And experimentally, we're given the measured mobility, the measured carrier density. We can deduce the average mean free path if we know the effective number of channels. And the effective number of channels we can compute, same expression we had earlier. And we find that then that there's a simple expression for the average mean free path. If I've got the measured mobility and I need to know where the Fermi energy is, but I get that from the measured carrier density, then I can take these two ratios of Fermi Dirac integrals and I can deduce what the mean free path is. If we're non-degenerate again, all Fermi Dirac integrals reduce to exponentials. So I have e to the eta divided by e to the eta, that's one. Things simplify and I get that the average mean free path is just 2k t over q times mobility divided by thermal velocity. I used that a long time ago. I think in one of the early lectures to try to estimate the mean free path in the MOSFET. But I told you that I was assuming Boltzmann statistics and I probably shouldn't do that in a MOSFET above threshold. So that's the way we could do it properly. Now I'll just point out that a lot of times if you want to take into account that the fact that this mean free path has some energy dependence, the simplest way to express it is in terms of a power law energy dependence. I'll use r there. I'll use r for mean free path energy dependence, s for time energy dependence. Then you can work out these mean free paths in terms of these characteristics exponents and sometimes you see people doing that. So let's go back to this silicon MOSFET now. We estimated the mean free path earlier. Let's do it again but more carefully. So the measured mobility is 260 centimeter squared per volt second. The measured carrier density at the highest gate voltage is I guess 7.9 times 10 to the 12th centimeters per square centimeter. So we can take this expression we developed for the mean free path. We can take the carrier density. We can relate it to the Fermi energy through the conventional expressions. We can deduce a to f, plug it in, and we'll get a mean free path of about 7 nanometers. I think in the earlier estimate when I did Boltzmann statistics I estimated 15 nanometers. So if you really want to get the right numbers when you're doing this, we're usually working in this regime where we're not completely degenerate and you can't assume t equals zero type of things and we're not nondegenerate. We're somewhere in between and if you want to get the right number, you've got to deal with Fermi Dirac integrals. Remember you can get a little app for your iPhone, first Fermi Dirac. All right, so that's it and then we'll see if you have any questions. So we talked about how transmission is related to the mean free path and we showed how you can derive this simple expression lambda over lambda plus l. It's important to realize when you're reading papers and people are coding mean free paths, unless they're using this land hour approach, they're probably quoting a different mean free path. So you'll be off by factors of pi over two or four thirds depending on what dimension. So if you want to compare numbers, you have to be careful about that. But we have a prescription, we can relate it to backscattering processes. And we have ways that we can deduce this from the measured conductivity or measured mobility, we can extract what the average mean free path is. Okay, so I'll stop there and see if we have any questions. Again, we have this fellow coming around with the microphone, so wait for him. The question is, is there any theoretical way that we can determine what kind of scattering we have in our device? Yeah, so that's a very good question. So, and part of the way people phrase this is they think about these power loss scatterings, and you have an s equals three halves for ionized impurity and s equals minus one half for acoustic phonons. So in general, it's very difficult. But one of the reasons that people like to characterize all of these thermoelectric parameters is that when you compute these averages, like of the C-beck coefficient or the Peltier coefficient and the conductivity, the energy dependence of the scattering processes plays out differently for the different coefficients. So, people try to see if they can explain a set of parameters, if they can deduce what the right scattering mechanisms are to explain the C-beck coefficient at the same time that they can explain the conductivity. The other thing that you can do, and we'll talk about this in the measurement section, is you can do temperature dependent measurements. And sometimes you can see signatures of phonon scattering and signatures of ionized impurity scattering from the temperature dependence of the mobility, say. Yeah, skip. Back at the beginning, we had an equation landing equal to average many times tau. And I think the tau was an average scattering time. I was just doing a thought experiment, say we pooled the device tau, so we had my phonon scattering just left with ionized impurities gathered. Then it seems the tau would be able to see, well, because it's all elastic scattering. Tau E would, well, that's not quite, you know. So if you injected, if you injected energetic electrons into a cold semiconductor, there wouldn't be any phonons to scatter from, but you could emit phonons and generate them, right? So you could relax the energy by emitting phonons. You could scatter by that way. So there would still be energy relaxation. Yeah. So that kind of is true. That's good. Yeah. Yeah, but you know, when we, so for example, when we talk about phonon transport, very similar things will happen. You use similar transmission formulas for phonons. When you cool a sample and you look at the lattice thermal conductivity, the mean free path can get very long. And then it's sort of determined by the boundaries of the sample. They're scattering off of the edges and things. So it's the same idea that you take the shorter of the dimensions. Well, I've actually made a question still, it's still a bit confusing. So I guess we could still emit acoustic phonon if we're not going to change the energy, I'm not sure. But you can emit optical phonons. So if you inject electrons, you know, with 50 MeV of energy, they can emit a 30 MeV optical phonon, right? Yeah, let's say we're keeping it low energy. Well, they're about 69 electron volts. Yeah, so you know, if you were, so it's possible, you know, and you could have hot electron effects. If you injected an electron at 20 MeV, it couldn't relax its energy by emitting an optical phonon because that would drop it down below the band edge where there are no states. So you would have hot electron effects there. So at low temperatures, it could be, you know, it could be a challenge to maintain these near equilibrium conditions, right? That's the way I would phrase it. You might have to apply microvolts across the sample, right? You'd be doing, otherwise you'd be doing a hot electron problem, right? Yeah. Yeah. It's like if you made an assumption, P1 is equal to P21. Yes. So that included this vaccine's relevant assumption, but I didn't see it come up anywhere in that derivation. Yeah, no, you're right. I didn't attempt to derive it. And you could actually show it from Fermi's Golden Rule. I might discuss it in one of my 656 lectures. I don't know. Slide 18. I'm not sure that I have a simple one-line derivation of that. Maybe there is one. But what you find is, you know, when you do the scattering calculation for elastic scattering, you always get the same transmission probability from one side to another. But that's not the case for inelastic scattering. And you can kind of see why. Let's say, you know, let's say I have a large potential drop across a transistor, right? But we're doing this, you know, this E here is not kinetic energy. It's total energy. So, yeah, let's say I have an energy heat and I'm trying to do this channel and I'm trying to calculate the probability that I'll transmit from contact two to contact one and asking if that's equal to the probability that I'll transmit from contact one to contact two. These carriers have much higher kinetic energy. They're going to be much more prone to scattering and emitting optical phonons. They have a much higher density of states because they're much further above the band edge. These ones have much less kinetic energy and they're going to scatter less. So these two transmissions are going to be very different if I have any elastic scattering processes going on. But you can show that if there are no any elastic scattering events, if I only have ionized impurities or reflections off of these barriers, then the probabilities are the same. Why is this important in a division that follows? Oh, yeah. Oh, okay. So why is this an important? Well, it's because I'm doing all of the bookkeeping and energy channels. You know, I'm assuming that everything is coming out of this energy channel. You know, if I have inelastic scattering, it could be coming out of other energy channels. And then I have to, you know, if I inject something at one energy, it'll start coming out. My total flux that comes out will come out at a whole bunch of different energies. And I would have to add up all of those contributions. So the whole assumption that we're trying to make is that we can just deal with each channel independently when we get all done. We just add the contributions to get the total current. But that's what Professor Dada calls vertical flow. And that complicates things a lot when you're moving between energy channels. Okay, we have another question down here. So the question is about tunneling. And I'm not really... I mean, if you had a problem like this, you might be interested in what's the probability that you can tunnel through here, right? So if I want... And people do these kind of calculations. And the way they would normally do them is you'd say i is 2q over h integral m of e. Let me just say f1 of e minus f2 of e, dE. And, you know, now I have a problem in which there's a spatial variation. But I could quantum mechanically compute the tunneling probability as a function of energy here. And this T of e would be that tunneling probability. So people do these kinds of calculations this way. Sometimes when they do it in this manner, this is sometimes called a su-isake formula. So in that case, you first of all do a quantum mechanical solution of the tunneling probability. That gives you the transmission. Again, if you assume that everything else is ballistic and there's only elastic events going on, that's also the probability in the other direction. That's why I have only one T here. But that's how people do these kind of calculations. So you see people do this for superlattices and things. They'll use an expression like that. And you'll just quantum mechanically calculate T of e. So you mentioned that the scattering rate of the quantum increases with energy because the density of space increases with energy. So that's another one-dimensional thing of the structure. Oh, yeah. Yeah. So, good question. So we know that the density of states in 1D, the density of states versus energy, if I'm at the bottom of the conduction band, there's a singularity and then it drops as 1 over the square root of energy. And this can lead to some interesting effects in 1D. The more energetic they are, the less likely they are to scatter again. So it could lead to some interesting effects. But if you have many subbands, each one has a 1D density of states. So as you start getting up here, you'll now have an additional state. This is due to subband 2. Let's say that this is confined state 1. Then confined state 2 has some additional states. So if you go up high in energy, you may start populating many of these subbands and the general trend still could be that it could increase with energy. But you do sometimes see these. They appear unusual because we usually think that density of states and scattering rates should increase with energy. But you do see some effects like this in 1D. Things go in the other direction.