nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.3: Why Electrons Flow
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[Slide 1] Welcome back to our, the first unit of our course. This is The New Perspective, so this is the third lecture where we'll talk about this current flow and obtain an expression for the current through a nanodevice. 

[Slide 2] So as we discussed in the last lecture, the first thing you need is this density of states and the electrochemical potential which tells you how far these states are filled. At equilibrium, both contacts have the same mu, same electrochemical potential. Out of equilibrium, you have to 2 different electrochemical potentials, the positive side is lowered with respect to the negative side by the amount qV. Now why does current flow when you apply a bias? Well, the idea is that if you look at these states this contact wants to fill them up because it wants to bring it into equilibrium with itself, but this contact wants to empty it because again, it wants to bring it into equilibrium with itself see, so every time the state is empty, the source fills it up and the drain takes it out and they electron then goes out of the drain into the battery and a new one comes in and so this process goes on forever. Now why do states down here don't contribute to current flow? Well, because as far as these states are concerned both contacts want to keep them filled, so it just stays filled, end of story, nothing else, no current flows, okay? And this is the part that is hard to understand. If you think of electric fields as driving electrons, because in that case electrons down here also you might say should start moving, okay? But, what is well-known, widely accepted is that current flows only in the small energy window around the Fermi energy or the electrochemical potential. States deep down here play no role in current flow. 

[Slide 3] Now, let's try to make this quantitative, let's try to get an expression for the current and let's first do this approximately and the next slide I'll do it a little better. So first let me assume that over this entire energy window of where we have applied the volt, you know, which is opened up by this applied voltage. Let's assume the density of states is almost constant, so then how many states do we have in this window? Well, it will be D times qV. So, how many electrons will we have in here? Well, it's like half of that. That's because if you think about it you have these states, the left contact wants to fill them up, the right contact wants to empty it, so what you'd expect is that these states would be kind of filled half the time, so we'd get filled up, taken out, okay, that's filled up, taken out and as long as it's equally connected to both contacts, you would think these electrons would be, would fill up the states like half the time. And so the number of electrons in the channel would be this. Now the flow of electrons, that's this electrons per second, that's this current divided by q. That is current is the amount of charge that flows per second and if you divide it by the charge on 1 electron then what you get is how many electrons flow per second, and this is the amount of electrons in the channel, this is the flow of electrons, and they are related by, so number of electrons is equal to the flow of electrons per second times the time that an electron spends in the channel. And I'll try to explain this point a little better shortly, but if you accept this then you see you immediately get an expression for the conductance; current divided by voltage just simple arithmetic and you get algebra and you get q squared D over 2t, so that's the density of states and that's this time it takes for an electron to get from left to right. So, where did this equation come from? Well, one analogy I've often used is supposing we have a graduate program, let's say where say a 100 students graduate with a PhD every year and let's say every student take the 4 years to get a PhD. Then what would be the steady state number of graduate students in your program? So, say a 100 per year times 4 years per student, so you would have 400 students in your program. If you think about that, but this idea is, you know, is something that you would use to estimate the steady state number of PhD students in a graduate program. You see new students continually come in and leave and graduate with a degree, so same here you see, new electrons continually come in and leave, but steady state if you want to know how many electrons are in channel that's like your graduate program, then that number of electrons would be the flow times the time that each electron spends in the channel. Okay? So, this is an equation, this express, relation is often quite useful. You see sometimes it may be easier to estimate I you know I and then you can estimate t, or if you know t then you can estimate I etcetera, so you have occasion to come back to this later. Okay. So, but if you accept this as you see, you can get the conductance right away, but I assume here is that the density of states is constant over this energy window, so let's now do it a little better, that is let us assume that it is not constant, 

[Slide 4] so we consider a small energy range dE. Okay? So, let's us consider the small energy in d so the number of states in that energy window is density of states times dE. And so the number of electrons is that divided by 2. So, previously we done D times qV over 2, now we are D(E)dE over 2 and then there's the current and then there's the time spent and in principle the times spent can also depend on the energy because at certain energies the electrons might have a bigger high of velocity and zip right through, other energies it might take longer, so we could write it this way and then you can rearrange it and write the current as dE qD over 2t. Now this is then the current that you'd have if the states at this were completely full one and this end were 0. On the other hand, in general, these states may be partially full, this states may be partially empty, so you might have like, so the occupation on this side is described by the Fermi function here, occupation on this side is described by the Fermi function over here and so in general you would have this factor here f1 minus f2. So for example, if both sides have the same f then as we discussed there would be no current flow. Current flow is because, you know, one wants to fill it up, the other one wants to empty it, but energy is down here for example, f1 is equal to f2 both are equal to 1 and so there is no current flow, so this factor here automatically takes care of this part; the current is driven by the difference in this Fermi functions, that is, and the two Fermi functions in the two contacts are different and that's what causes current to flow, so this is the current at a single energy and the thing is that in this viewpoint we are assuming that an electron goes from left to right at the same energy without changing energies and that makes it very simple, you see, you can think of different energy levels as all being independent, so this is elastic resistor idea; that current is carried by different energy channels all independently. And so if you want the total current, you just add it up, and add it up means you integrate it. So once you have done that, you have got this expression for current which you could rewrite in this form that current is equal to this f1 minus f2 times the quantity here that we could call the conductance function and that conductance would be written as q squared D over 2t. See, so this would be the general form of the current equation and what we had done earlier was done a simpler version where we assume that this conductance is independent of energy in that energy window and so we were kind of able to pull it out, that's kind of what we did in last slide, but this is the more general expression. 

[Slide 5] So with that then we're now ready to move on and obtain an expression for the conductance in general and that's what we'll do in the next lecture. Thank you.