nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.2: Two Key Concepts
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[Slide 1] Welcome to the second lecture of this unit. And, what we'll do now is introduce these two important concepts that we'll be using throughout this course when trying to describe current flow in nano devices. 

[Slide 2] And, these two concepts then, let me try to explain. So, we are interested in the current that flows in a device when you apply a voltage across it. But, let's first start with the equilibrium case where there is no voltage applied. Okay. Now, the first question is what are the energy levels available in the channel? Now, as you know, if you had a hydrogen atom, you'd have discrete levels which are separated by quite a bit like 5 electron volts or 10 electron volts. As you get to bigger things, the energy levels come together. So, they look say more like this more dense like this. And, when you get to a solid, usually these levels are awfully close together. And, rather than draw lots of lines like this, it's more convenient to define something called the density of states which tells you how many states you have per unit energy. So, a typical density of states plot might look something like this. This axis is energy and this takes a little getting used to because, you see, usually, you plot the independent variable horizontally, I mean that's the x axis usually. But here, actually, it's convenient to draw the energy axis vertically because that's the way we draw the energy levels, see. And, what the density of states, like this value here, what it tells me is how many of these energy levels I have per unit energy in there. So, when you go to this region, for example, the density of states is 0. So, what that means is there are no states here at all. And then, down here we have, again, states available at this energy. I have shown them here. I should've drawn something like this down here as well. You see. So then, that would correspond to this picture that I've drawn. So, this is a density of states plot. That is kind of the first thing you need when you want to talk about current flow in a device. So, how do you know what's the density of states? Well, experimentally, usually, you measure it using different types of experiments. The most common one is this photo emission for example. Photo emission means, electrons here, if you hit it with light, you can excite an electron and knock it out into the vacuum. The idea is an electron inside the solid, of course, stays there because the energy here is lower than if it is outside. So, this being outside, that's what you call the vacuum level. So, if an electron went outside, that's what its energy would be. Inside the solid, it's something less than that. And so, if you take that as 0, it would mean all energy levels in a solid would be negative quantities. I mean, that's why they stay there otherwise they'd jump out. And, you can hit it with light photons, which have sufficient energy, and then they have a photoelectric effect whereby these electrons will get knocked out of the solid. Okay. And, typically, that energy that is needed to knock these highest electrons out of the solid is on the order of say 5 to 10 electron volts. So, these are like what you call the valence electrons. The ones that are really, have the highest energies here. Now, as I mentioned before, there's lots of other stage down here, the core electrons which will take a lot more energy to knock out, this would be like kilovolts down there. But, as far as current flow is concerned, what really matters is, is top part, the part near the, I guess, that's the next concept I'll explain is this Fermi energy. 

[Slide 3] Now, what's that? Well, if you look in the context, we have the density of states in the channel, and in the context I've drawn this continuous density of states up here. And, over all that equilibrium, you see, there is a level up to which all the states are filled. And, as I explained in the introductory lecture, electrons naturally want to go to the lowest energy state. But, because of the exclusion principle, they can't all go into the lowest one. So, instead, they fill up a lot of states. If you had a million electrons they'd fill up a million states. And, one point that sometimes causes a little confusion is people say well isn't it like every state can hold 2 electrons up spin and down spin? Well, I think the right way to say it is, every state only holds 1 electron. But, usually you have 2 separate states, 1 for up spins and one for down spins. So, it is as if the states come in pairs. But, the point is each states holds only 1 electron. That's the exclusion principle. Okay. So, there is this level then which you call the electrochemical potential or the Fermi level which separates all the filled states from the empty ones. So, for 0 temperature, everything below it would be filled, everything above it would be empty. Well, what if I raise the temperature? Well, then you see some of these electrons from down here will have enough thermal energy to actually jump up and occupy these higher states. And, the distribution at equilibrium will be given by something called this Fermi function. So, what I've plotted here is this Fermi function, again, the axis on the, this vertical axis is e minus mu that means this is the energy reference to mu which means at mu there, this axis is 0. And then e minus mu here e's greater than mu. Here e's greater than mu. Here e's less than mu. And, it's divided by this quantity kT. That's something that also appears all the time in our discussion. So, this is called a thermal energy. The k is called a Boltzmann constant and T is the absolute temperature. And the actual mathematical function describing this, you see, this function, you'll notice at energies far below mu is 1. And that's what you'd expect because far below mu all states are filled. And, what this function tells you is what fraction of the states are filled. So, down here, we expect 100% to be filled. So, the function is 1. Up here, we expect them to be completely empty. And so, the function is 0. You see? So, the function goes from 0 to 1. And the mathematical form of this function looks something like this. It is 1 divided by 1 plus exponential e minus mu over kT. So, when that quantity is negative then it is exponential of a large negative number which is 0 and so the function becomes 1. When that number is big like exponential of a positive number, that's a very large number. And so, 1 divided by a large number, that's what makes it a very small number. It goes to 0. So, you could take that function, plot it out, and see how, you know, these days it's very easy to plot it out on MATLAB or Mathematica, whatever you're comfortable with, and you can see how that function will look. And, one thing, as I said, is very important that comes up often in your thinking and your discussions is this thermal energy kT. And, that kT is roughly 25 mili-electron volt. You see, it's a dimensions of energy and the MKS unit for energy is joules. Joule is like a coulomb times a volt. But, what is commonly used though, in terms of describing electronic energy is rather than joules, we usually use electron volts. And this is a mili-electron volt. So, if you write it out, 25, the mili is the 10 to the -3 and electron volt is like 1 electron times 1 volt. So, how do you convert it to MKS units like if you want joules? Well, 1 electron is this 1.6 times 10 to the -19th coulombs. So, you can see that if you wanted to write energy in joules, you'd usually be always carrying around this 10 to the -19th type of thing with you. And, that's inconvenient. And so, what's commonly done is, when you talk of energy, you talk in terms of electron volts. But, you should be clear on how to handle the units and how to convert them. Well, one last thing, you see, so, 

[Slide 4] I've introduced these two concepts, the density of states and the Fermi function, but we talked about what the picture looks like at equilibrium. If you apply a voltage go away from equilibrium, then what happens is, as I mentioned in my introduction, the two contacts instead of both have the same electrochemical potential, they shift with respect to each other because when you apply the voltage, the positive side, all the energies including the electrochemical potential which tells you how far they're filled, everything sinks, everything goes down. So, that's why, on this side everything is lowered by qV. So, this is the picture then that we'll use 

[Slide 5] in our next lecture to discuss a formula for current. So, we'll talk about how electrons flow. Thank you.