nanoHUB-U Fundamentals of Nanotransistors/Lecture 3.2: Landauer Approach
========================================
[Slide 1] Hello, everyone. So our topic in this lecture is the Landauer approach. So this is an approach that is not specific to MOSFETs. It is applicable to any small device. Actually it's not widely known it's also applicable to large devices. So it provides us with a very good way to treat transport in devices from large to small. We're going to apply it specifically to MOSFETs but it's an approach that's worth learning if you're interested in nanodevices of any kind.
[Slide 2] So let's look at this.
[Slide 2] We'll, think of this as a generic nanodevice. We have two contacts. I'll call them 1 and 2 now. Later on, they'll become source and drain. Our device is in between. It has some length L. It's connected to these two contacts. The two contacts are going to be special contacts. We're going to assume that they're large. There is a lot of scattering that maintains thermodynamic equilibrium. So the population of the energy states in these contacts are given by Fermi functions. But the Fermi functions or the Fermi levels might be different because we might apply a voltage to one contact versus the other. And what we want to get familiar with in this lecture is how we describe the current flow through this nanodevice. And the expression we're going to use here is what we call the Landauer approach. There's some fundamental constants. There's a quantity we call transmission, which is just the probability, a number between 0 and 1 that an electron that is injected from the contact 1 goes all the way across and makes it out contact 2. M is the number of channels in which current can flow. And f1 and f2 are the Fermi functions of the two different contacts. So what we would like to do in this lecture is to see if we can derive this expression and get comfortable with the various terms inside the expression. And in the next lecture, we'll spend some time using it and then we'll be ready to consider MOSFETs.
[Slide 3] So let's do the derivation and I'll do it in 2D, because I'm thinking of a planar MOSFETs, where the electrons are flowing in a two-dimensional sheet, like an inversion layer. So think of this as looking down on the top of a two-dimensional conductor. This is the x direction in which current is flowing. This is the y direction, which will become the width of our MOSFET later on. Inside the device, you know, there is some mean free path. Electrons might scatter and bounce around. We're going to assume there's no real electric field inside the device. We have a contact over here specified by a Fermi function. We have a contact over here specified by a Fermi function and electrons can come in from both of those contacts. Those are like the electrons hopping over the barrier in our thermionic emission approach, the ones coming in from the drain and hopping over the barrier or the ones coming in from the source and hopping over the barrier. If we inject electrons from contact 1, some of them might come out the other end. Some of them might backscatter and come out the end that they were injected in that. Same thing for electrons that are injected from contact 2. Some might come out on contact 1 side. Some might backscatter and reemerge in the contact 2 side. So that's the general picture we have of our nanodevice. We're going to assume-- at least two very important things. We have these ideal contacts, which I may from time to time call Landauer contacts, the contacts maintain thermodynamic equilibrium. Now, of course, you know that when current flows, we don't have thermodynamic equilibrium but we're going to assume that these contacts are so large that they only need to be perturbed an infinitesimal amount away from equilibrium in order to carry whatever current flows through this nanodevice. We're going to assume that there is lots of scattering, inelastic scattering that mixes all of the energies and the momentum and maintains thermodynamic equilibrium in these contacts. So our Fermi function, which describes thermal equilibrium situations, is valid in those two contacts. We're also going to assume that as electrons travel through this nanodevice that they travel in independent energy channels. If it enters in one energy, it doesn't undergo an inelastic scattering event which would change its energy to another channel. So we're simply going to treat each energy as an independent channel for current to flow and then we'll add them all up at the end to get the total current. Now these may seem like restrictive assumptions and they have to be examined carefully in each case that we apply them to but it turns out that-- they can provide a very good starting point and often a very accurate solution in many problems of interest. And we'll see how that all plays out during these two lectures.
[Slide 4] So let's see if we can derive this expression for the current. So to begin with, here's our situation. If I look at what comes out the right side due to the injection from the left and due to the injection from the right, some fraction transmit across, some fraction T of the flux injected from the left transmits across and comes out, but some fraction, 1 minus T, gets backscattered from what was injected from I2 and comes out there as well. So we have to add both of those up. Same thing on the left. Some fraction of what was injected from contact 2 emerges, that's this term. And some fraction of what was injected from contact 1 backscatters, that's this term. Okay so we can specify the incident fluxes are given. That's what comes in from the contacts. The emerging fluxes are what's determined by the scattering and the things that go on inside the nanodevice. Now I can find the net current, just the difference between the positive and negative here, or I can do it here. Assuming that there is no generation of electrons anywhere, the net current is the same everywhere. I'll get the same answer and this is my answer. So in any energy channel, the current is the transmission at that energy times the difference between the two currents that were injected from the two sides. If I want the total current, I would just add up the contributions from each energy. Now there's a very important assumption I've made here. I've assumed that the probability that an electron transmits across from left to right is the same as the probability that it transmits across from right to left. That's an important assumption. We can justify that when we have elastic scattering, these independent energy channels. And so it's an assumption to keep in mind. It's going to work well for us in MOSFETs and we'll discuss a little bit later in Unit 4 why it works so well in MOSFETs.
[Slide 5] Okay, let's continue our derivation. This is what we have for the current at any energy. Now if I try to look a little more carefully at what these currents are involved, you know, what do they consist of, let me first of all will look at how many electrons I have with a positive velocity. So the electrons that are injected from contact 1 come into the device with a positive velocity. The states are populated according to the Fermi function of contact 1. And the number of states is given by the density of states times dE, but only half of the states. Everything is symmetric. Have of a positive velocity, half of a negative velocity. So I can write down this expression for the number of electrons with a positive velocity at this particular energy range, E plus dE. And then if I want the current I1, I simply take the number multiply it by q because I have charge, charge times velocity, and then the width because, you know, the width is like resistors in parallel, so I have to have a width there. So this quantity v sub x plus, this is the average velocity of the electrons that are coming in at that energy in the x direction. So remember, this is a two-dimensional structure. There's an x and y direction. They might come in at an angle. The brackets here denote an average over angle. It's the average x-directed velocity. We'll talk about that in a minute. So we have this expression for I1 in terms of fundamental quantities like densities of states and velocities of carriers and Fermi functions. We can write down exactly the same type of expression for the current injected from contact 2. It just has the Fermi function of contact 2.
[Slide 6] All right, let's see where that leads us. This is our net current. This is how we write these injected currents at the two ends, I sub 1 and I sub 2. If I put them together, I get an expression like this. And now I've clumped a bunch of things together, average velocities and densities of states together in these curly brackets. Let's look at that a little more. I'm going to define a quantity we call M, which is going to be very important in our discussions later. And I'm going to define as Planck's constant divided by 4 times the average velocity in the direction of current flow at the energy E times the density of states and we're talking about 2D now. We could do this for a 1D nanowire or 3D bulk semiconductor but we're focusing on 2D because we're thinking of an inversion layer in a MOSFET, and then I've divided by 4. Now if you look at that, let's look at the units of this quantity. You know, what are the units of this thing M that I've just defined? Well, width has units of meters. Planck's constant has units of joule-seconds. Velocity has units of meter per second. Density of states in 2D is the number of states per unit energy range joule per square meter, because we're talking about density per square meter. So if you look at all of that, everything cancels out. This quantity has no units. It's just a number and, you know, what is the interpretation of that number? Well, the interpretation is quite simple. Here's the formula. But let me think -- Let's go back to 1D now. I'm going to think of a nanowire because it's easier to visualize it in 1D. Here's a band structure, energy versus k, you know, typical type of a band structure we might sketch for a semiconductor. And we are at some energy E and I might ask the question, how many channels are there for a current to flow in at that energy. Well, if I look at that energy, I need a state and the state has to have some velocity in the plus x direction, then that can provide a channel for current to flow at that energy. And if I look at that, there is one place where there is a state at that energy. So I would have one state, two if I would count spin but we're going to account for spin later on. So what we have here is we would say M at this energy is 1. We have one channel for current to flow or sometimes people call this a mode. It's like a combined mode in a waveguide. So that's what the interpretation of M is and we can figure out M in any case, if we know the velocity and the density of states.
[Slide 8] So let's do M in 2D because we're going to need it 2D when evaluate the Landauer approach for MOSFETs. So here's our expression for M. And in order to do this in 2D, I'm going to define some quantity M. It's going to scale with width for large devices. If I make it twice as wide, I'll have twice as many channels for current to flow. So I'll take M and divide it by W and I'll call M sub 2D the number of channels per unit width. All right, so then I just lose the W in my expression. This is the quantity that I have to evaluate and then I'll know M sub 2D. Well, I need to remember the 2D density of states. So here it is. It depends primarily on the valley degeneracy and on the effective mass. I'll need to know this average velocity in the direction of current flow.
[Slide 9] So let's talk about that for a minute. This is the two-dimensional sheet electrons that we're talking about. And electrons are moving in the x-y plane and we are interested in the average velocity in the direction of current flow in the x direction. Okay, so let me assume that I have simple energy bands. So energy is 1/2 mv squared. So at any given energy, I can compute the magnitude of the velocity if I know the effective mass. Okay, but the velocity is pointing at some arbitrary angle theta. I'm interested in the x-directed velocity. So that's magnitude times cosine theta. I'm interested in the angle average x-directed velocity. So I have the average over all of the forward velocity states. So I'll integrate from minus pi over 2 to plus pi over 2, v of E cosine theta d theta and I'm doing an average over angle. So when I sweep from minus pi over 2 to plus pi over 2, I'm going over pi radians. You do that integral and you can see that the average x-directed velocity is 2 over pi times the magnitude of the velocity at that energy. Okay, so that's how we do that calculation. That's what the brackets mean. That means we average over angle at that particular energy. Okay, now we know what the average x-directed velocity is.
[Slide 10] We can go back and see if we can derive the number of channels. So the density of states times the average x-directed velocity, which we now know. We know the magnitude of the velocity, because we're assuming parabolic bands with this calculation. We put it all together. That's a simple expression for the number of channels as a function of energy and that's an expression that we're going to use frequently now in this unit.
[Slide 11] Okay, so let's go back just a little bit so we can return to our derivation. This was the net current in terms of transmission and the two injected currents. We wrote the injected currents in terms of density of states and velocities. We defined this new quantity we called the number of channels at that energy or the number of modes. And then we can use that definition and we can write our current at any energy with this expression, some fundamental constants, transmission, number of channels times the difference in the Fermi functions at that energy. If we want the total current, we have to add the contributions from all energies. So we have to integrate over energy and we have derived our Landauer expression. So there it is. This is the expression that we're going to use extensively in Units 3 and Units 4. And we've seen it's not too difficult to derive.
[Slide 12] But let's discuss it a little bit because we want to get familiar with this expression. There are some fundamental constants. We will discuss those a little bit later. There's this quantity we call transmission. In Unit 3, we're going to assume that the transmission is 1. That means ballistic transport. That means if I inject an electron from contact 1, it's going to go across the device and out contact 2. It will not backscatter. In the diffusive case, the transmission is much smaller than 1. There's lots of backscattering and it's very hard for an electron that's injected from one contact to make it through to the other. Unit 4, we will discuss this entire range from ballistic to diffusive and see how we describe that, but for Unit 3, transmission is just 1, the ballistic case. M, as we've seen, is the number of channels. People also call that the number of modes at that particular energy. And the difference in Fermi functions comes in because we're assuming these ideal Landauer contacts that can be characterized as equilibrium contacts with two different Fermi levels and two different Fermi functions.
[Slide 13] Now we're going to need one more thing and then we'll be able to do some calculations and dive in and get some results. We might also ask how many electrons are there in this nanodevice. We're going to need to know that as well. Now we're thinking of this device as very small. We're not spatially resolving anything within the device. We're just assuming it's a small device and we want to know the number of carriers in that device.
[Slide 14] Well, do you remember the way we would do that in a large semiconductor is we would integrate the density of states times the probability that the states are occupied and we would integrate from the bottom of the conduction band, usually to infinity, because the Fermi function will make sure that the probability of states being occupied decays to 0 rapidly and we don't have to worry about what goes on at the top of the conduction band. Okay, we know the Fermi function. If we assume parabolic energy bands, then we know the density of states in two dimensions. There's an integral that we have to do. It's a little bit complicated, but it's not that bad. If you write that, it's a bunch of constants, which we call the effective densities of states. It depends on effective mass and fundamental constants and valley degeneracy and temperature and things like that. And then it's logarithm of 1 plus e to the eta. Eta F is simply a dimensionless number that tells us where the Fermi level is located with respect to the bottom of the conduction band in units of kT. The bigger eta F is, the higher the Fermi level is. So it's effective density of states times logarithm of 1 plus e to the eta F. Logarithm 1 plus e to the eta F also has another name. it is a Fermi-Dirac integral of order 0. And various types of Fermi-Dirac integrals pop up in these calculations that we do. Now in the non-degenerate limit, eta F is much, much less than 0, because the Fermi level is several kT below the bottom of the conduction band. This logarithm simplifies to an exponent and we just get effective density of states times exponent. We will use that frequently just because it makes the math simple, the derivations easy to do. In practice, it's often important to consider the full Fermi-Dirac case and to use Fermi-Dirac integrals.
[Slide 15] So we have an expression now for this, the number of electrons in equilibrium but we're not in equilibrium. We're interested in what happens when we apply a voltage to the two contacts. So it turns out that it is easy. I'm just going to state the answer here, but you can see that intuitively it makes sense. We can generalize this. We have a group of states in this device that are occupied by electrons coming in from contact 1. So the probability that those states are occupied is given by the Fermi function of contact 1 but only half of the states are filled by contact 1, because only half of the states have the positive velocity that contact 1 can fill them from. Now we have another contact. It may have a different voltage applied to it. It has a different Fermi level. It has a different probability of occupying states. Half the states can also be occupied by contact 2. So we can generalize our equilibrium expression. Now we have some of the states being occupied by contact 1, some of the states being occupied by contact 2. According, the probability of being occupied by those two contacts is given by the Fermi functions of the two contacts. So that's a way to generalize it. We could do that integral and we could express the results in terms of effective densities of states and Fermi-Dirac integral of order 0.
[Slide 16] Okay, so we've succeeded. This is our final result. We have an expression for current that we're going to work with frequently. We have an expression for carrier density that we will also need frequently. With those two expressions, we can derive the IV characteristics of nanodevices quite generally. And in particular we're going to be applying these two formulas, not in the next lecture, but in the lecture after that to the ballistic MOSFET and see how that plays out. So in the next lecture, we will use these formulas and get a little more comfortable with using them before we apply them to the MOSFET. I'll see you there.