nanoHUB-U Fundamentals of Nanotransistors/Lecture 3.3: More on Landauer
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[Slide 1] So welcome back to our discussion on the Landauer approach to carrier transport. So we are trying, in these two lectures, just to get comfortable with this way of treating electron transport in small devices. In the last lecture, I derived the expression. We looked at it a little bit, what the various terms, transmission, number of channels, and in this unit, we want to get a little more familiar and actually do some calculations. And then we will be ready to apply this approach to the ballistic MOSFET in the next lecture. 

[Slide 2] Okay, so I'll just remind you, this is our nano device, a generic nano device. A couple of things to remember is we have these very special contacts that I've called Landauer contacts. They're large, they're maintained to near thermodynamic equilibrium by frequent inelastic scattering that mixes all the energies and the momentums up, and consequently they are each described by a Fermi function. The Fermi levels may be different because we may apply a voltage to the second contact and lower its Fermi level with respect to the first contact. Current will flow and you know when current flows, you are out of equilibrium, but we're assuming that these contacts are so large, that the perturbation away from equilibrium is small enough to be neglected, and we can describe our contacts with Fermi functions. We're also making the very important assumption that we have independent energy channels. Electrons flow from contact one to contact two without changing their energy. Once they get into the contact, the inelastic scattering mixes everything up and changes energies and restores the thermodynamic equilibrium and the Fermi function. But the device itself has no inelastic scattering. Okay now these are the results that we derived. We derived this expression for the current, and we want to work more with this in this lecture and get more familiar with this. We also derived an expression for the electron density within the device, and again, we are not trying to resolve anything spatially within the device. To do that properly, we would need to do something like a Boltzmann equation. We are just thinking of this device as a small, uniform device. The electron density is uniform within that device, and we have a way to calculate it in terms of the density of states, and in terms of the Fermi functions of the two contacts. 

[Slide 3] Okay so let's look at the current expression in a little more detail. And in particular, let's look at this term f1 minus f2. So you can see that when the Fermi functions of the two contacts are the same, no current will flow. Current only flows when f1 is different than f2 over some energy range and the range over which that is true is what we call the Fermi energy. That is what we focus on when we want to understand current flow. Those are the range of energies that are important. 

[Slide 4] So we can get a picture of that from these diagrams. This is that zero kelvin. At zero kelvin there is a probability one that all the states below the Fermi level are filled, and then a probability zero that the states above the Fermi level are filled. If we have some finite temperature, then the transition from one to zero occurs over a range of a few KT, just not as sharp. Now, if these are the Fermi functions of contact 1 at 0 K and at room temperature, say, then perhaps at contact 2, I have applied a positive voltage and lowered the Fermi energy. If I'm at zero K, I will just pull the Fermi energy down, at contact 2, and then I'll have a range of energies over which f1 is bigger than f2, and that is the range of energies over which current will flow. I would call that the Fermi window. Now, at room temperature, if I pull the Fermi level of contact 2 down with respect to contact 1, I'll just pull everything down. The transition is not an abrupt one, but there is still this region of a width depending upon the bias that I've applied, and the temperature which smears out the transition. I'll call that the Fermi window as well. So that is what the Fermi window looks like when there is a fairly large bias on contact 2. 

[Slide 5] If there is a small bias on contact 2, I just pull the Fermi energy in contact 2 down a little bit, and the Fermi window begins to look like a delta function at the Fermi level. If I do the same at room temperature, I'll just pull the Fermi level down a little bit. There will just be a small energy range now, and now it will depend on the temperature, the width of that energy range. That also begins to look a little bit like a delta function, and we call that the Fermi window as well. So this range over which f1 does not equal f2 is called the Fermi window. 

[Slide 6] Now, let's look at this a little more carefully in the case which we apply a small bias. This is like the linear regime of a MOSFET. If it is a 2D, just a two-contact conductor, we would call this the regime of linear response. So here is our general expression. We are going to focus on f1 minus f2, and see what it looks like when both contacts are the same temperature. That is an assumption we're making here throughout all of this unit, all of this course in fact. But we are going to assume that we apply a small voltage to contact 2. So here is the Fermi function of contact 1. I can write the Fermi function contact 2 by saying it is going to be close to the Fermi function of contact 1 because we've just applied a small voltage to contact 2. So I can write f2 as a Taylor Series Expansion of f1. It is f1 plus DF1, with respect to the Fermi level, plus delta Fermi level. Because all I've done is to change the Fermi level in contact 2 a little bit. So I'll write that Taylor Series Expansion. Now, if you look at this Fermi function, you can see that if I take a derivative with respect to the Fermi energy, I get the same answer except for the sign as if I take a Fermi of a derivative with respect to the energy. So the derivative with respect to EF is the same as a derivative with respect to E, except for the minus sign. So this is the way we usually like to write it. So I'll write it that way. Okay now I can say that, well, the change in Fermi energy, this occurred-- I changed the Fermi energy in contact 2 because I applied a small voltage. A small positive voltage lowers the Fermi energy so the change is minus q times the voltage that I applied. Okay, now all I have to do is subtract f1 minus f2, and I'll find what this quantity looks like when I have a small bias applied, and what I get is it is just related to this quantity Df1DE, just a derivative of an equilibrium Fermi function, something that I can take and it is proportional to the voltage. q times the voltage. Okay, now if I take this and put it in to this, and evaluate the expression, what I'm going to find is I get a current that is proportional to voltage, that is why we call this the regime of linear response. The constants out front would be the conductance. And the conductance would be given by this expression. I simply use this Taylor Series Expansion for f1 minus f2, the expression I get is for the conductance is given here. So that is a very important expression that is widely used 

[Slide 7] in many different cases, not just for small MOSFETs. So we have now determined that this Fermi window here that we sketched a few slides ago is actually minus the derivative of the equilibrium Fermi function with respect to energy. So we know what it is. So this is what we call the window function. So the window function is just this. Minus this derivative of the Fermi function, if you integrate that to cross all energies, you will see that it integrates to one, so when it's narrow, and you know, the width is a few KT, so when it's narrow and it has an area of 1, it's going to look like a delta function, and in some cases it will be convenient to treat it as a delta function. 

[Slide 8] Now, this brings us to a topic we can discuss something that you may have heard before. The quantum of conductance. So let me look at zero kelvin. At zero kelvin, this Fermi window is very sharp. KT is very small. So minus the derivative of f naught with respect to energy can be considered a delta function at the Fermi energy. That makes this integral easy to do. So we find that the conductance at zero kelvin is just 2, q squared over h, times transmission at the Fermi energy, times the number of channels at the Fermi energy. We get a very simple expression for the conductance at low temperatures. Now, there is something important here that I want to point out. In deriving the expression for the number of channels at the energy E, which we did in the last lecture, we assumed a 2D density of states. Remember, the number of channels, we need a velocity, so that was the X directed velocity at that energy. But we need a state, and that brought in the two-dimensional density of states. But the expression that I use for a two-dimensional density of states is one that is derived for a large, two-dimensional area. In small structures, the density of states can be modified by quantum confinement, and we can start to get discrete and countable numbers of states. So for the large structures, and we're going to primarily be thinking about large structures, in a large structure the width of the resistor, say the width of the MOSFET, we are going to find our expression tells us that a number of channels just scales with the width of the resistor. But in small structures, for which we get quantum confinement, we can't use this large, this density of states that comes from a large material. We have to compute the density of states properly, assuming boundary conditions on the wave functions. This is where the term modes comes in for channels. The density, the number of channels, comes in discrete, countable units in a nano structure. So that is an area that is fascinating. There is a lot of work in nano science and nano technology in that area, and it is something that is, by now, 

[Slide 9] a well-established experimental fact. For example, these are some experiments that were done in the late 1980s with very small, two-dimensional resistors, with a very short channel, such that the transmission across those channels was near ballistic when the measurements were done at low temperatures. The width of the channel could be electrically varied by reverse biasing Schottky barriers, and this is the measured conductance versus that width. So in a large structure, you know, our expression for the number of channels, we would just say the wider the structure is, the higher the conductance is. The conductance should be proportional to W. What you find, however, is that the conductance goes up in discrete steps and we can, you know, so this is an example of quantized conductance. We are seeing the number of channels increase discretely as the number of states that can fit into that width and make the wave function go to zero at the two edges of the resistor, as that increases we get discrete numbers of modes or channels. 

[Slide 10] So that is an example of quantized conductance. As I said, we are going to be working with MOSFETs in which the width is wide enough that we don't see this quantization, but the general expression that we work with handles both cases. Alright, now let's look a little more carefully at the conductance for T is greater than zero. Then the mathematics can get a little more involved. I don't want to work this out too much here now, but if we take our expression, if I divide by this integral, and multiply by this integral, and I define a quantity, you know, M in brackets as that integral that I divided and multiplied by, I am going to interpret that integral as the number of channels in the Fermi window. So that makes sense. Here's my Fermi window. Here is my number of channels. If I do that integral, it is the number of channels in the Fermi window. If I define an appropriately average transmission across this range of energies, then you can see 

[Slide 11] that I can write the conductance in a general way. There is this quantum of conductance, 2q squared over h, times an average transmission, you know, average in the energies of interest that are within the Fermi window, times the number of channels in the Fermi window. So it's just a convenient way to write that expression in order to get numerical answers. I've got to do all of these integrals. At zero Kelvin, things simplified. If the quantum of conductance times the transmission at the Fermi energy, times the number of channels at the Fermi energy, and this is the expression that is used to interpret those experimental results, we saw a few slides earlier. 

[Slide 12] Now, let's look a little more carefully now at the ballistic case, and look at the conductance. You know, when the temperature is greater than zero because we are primarily interested in MOSFETs at room temperature. So this is our general expression for the conductance. This is our expression for the number of channels as a function of energy. And we are assuming wide structures where the discrete nature of the channels won't be an important factor for us. So it is just proportional to W. We're assuming ballistic transport, so the transmission is 1, and we know the Fermi function, so we can work out this derivative. So if you do that, then we'll get an expression like this. I'll rewrite the minus Df naught / dE as a plus d/dEF, which now I can bring outside the integral, then I can do the integral, then I can take the derivative. 

[Slide 13] There is a little bit of work to do that. I don't want to go through that. It's done in the lecture notes. I'll simply quote the result. This is the result of that integration. So this brings up another Fermi direct integral. This time an integral of order minus one-half. Now, we can simplify that for non-degenerate statistics, and that makes life easier. For non-degenerate statistics, Fermi direct integrals always reduce to exponentials. So we simply replace the Fermi direct integral with an exponential. Okay, now if I look at my expression for the electron density, we also worked out in the last lecture an expression for the electron density. It involved Fermi Dirac integrals of order zero. They reduced to exponentials. So I can write the electron density as the 2D density of states, effective density of states, times an e to the eta F, and remember, eta  F measures the location of the Fermi energy with respect to the bottom of the conduction band in units of KT. So you can see that I can relate the electron density to e to the eta F, and I can use that in the first equation, then along with the definitions of the 2D effective densities of states, I can put that all together, and I can rewrite this ballistic conductance in this form. So I simply eliminated the e to the eta F, we've used the definition of the effective densities of states. This quantity v sub T was a collection of constants that came out of this algebra, and it is a collection that has units of velocity. This is the unit directional thermal velocity and we've seen it before when we were doing the ballistic treatment in lecture one of this unit. And we will discuss it a little more in this unit. It will come back again and again. It is an important velocity in our thinking about ballistic MOSFETs. So if we would like to, now, we could divide by L, and we could multiply by L, and when I do it that way, you can see that we are achieving this quantity that has the units of mobility. This is the ballistic mobility we saw earlier in lecture one. So the bottom line of all of this is 

[Slide 14] that if we take our general formula with a conductance in a linear regime, if we evaluate it in the ballistic limit when T is equal to 1, we can write the result in the traditional textbook way that the diffusive conductance is written, nq mobility times width, divided by length. We can write it in the same form as long as we define this thing we call the ballistic mobility. So that ballistic mobility is something that is going to come up in our discussions as we talk about ballistic MOSFETs 

[Slide 15] and quasi-ballistic MOSFETs. Okay there is one more thing I'd like to do in this lecture, and that is to consider the case of a large bias. This would be like the on current of a MOSFET. So we have these two Fermi functions. If we apply a large voltage, we lower the Fermi energy in contact 2, and that means f1 is much bigger than f2. We can ignore f2, and our current simply is given by this integral. We just ignore f2. 

[Slide 16] Well, again, that is an Well again that's an  integral that we can do. We are considering the ballistic case of the transmission as 1. We know the number of channels. We know the Fermi function of contact 1. We put it all together. We do the integral. 

[Slide 17] We can do the integral. If you work that integral out again you're going to get Fermi direct integrals. This time an integral of order f plus one-half. If we work on this a little bit, use our definition of the thermal velocity, the unidirectional thermal velocity, our definition of the effective density of states, we can do a little bit of algebra and we can rewrite this on-current as thermal velocity times effective density of states divided by two times this Fermi direct integral of order one-half. Alright, so a little bit messy algebra, but no rocket science. 

[Slide 18] We can go through and work that out. Let's look at it a little more carefully and see what we can make of this. This is the result we get from evaluating those integrals in the non-degenerate limit. Fermi direct integrals always reduce to exponentials. Again, we can look at our carrier density. It was related to Fermi direct integrals. Eta F2 is very small, because we've applied a large voltage to the drain, and lowered the Fermi level in the drain, so we can ignore that. You know, we can simplify this for nondegenerate cases and then we can ignore the contribution from eta F2, so we have a simple expression for the electron density as a function of eta F. We can insert it in the first expression and what we will find is that we can write the ballistic on-current in a very simple way. And we're going to see this again when we do the MOSFET. It is charge, q times the sheet carrier density times the velocity. And the velocity is the thermal velocity. So take some time to go through this algebra. Look at the lecture notes if you'd like to see how it's done. The final result is simple and easy to remember, and it is something that we'll see from time to time. 

[Slide 19] Alright so that's it. Hopefully by sitting down, going through some of this algebra, reviewing this lecture, you'll get comfortable with using these equations and applying them to different situations, and you'll be ready to consider the ballistic MOSFET in the next lecture. These are the takeaway messages. We've been working with the current expression in the ballistic limits, where the transmission is 1. We considered the low bias case, where f1 is about equal to f2. We saw that current is proportional to voltage. G sub B is the ballistic conductance. The ballistic conductance, if we choose to do so, we can write it. We can make it look like the traditional diffusive expression, which varies as W over L, but to do that we have to define this thing called a ballistic mobility, whose physical significance we have not yet discussed. But we will. And we will come back to that in a later lecture. Under high bias, f1 is greater than f2. And we can write something we would call the on-current simply as charge times velocity. The velocity is the thermal velocity. So with that, and the previous lecture, we are ready to apply these approaches to the ballistic MOSFET.