nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 1.7: Connecting Balistic (B) to Diffusive (D) ======================================== [Slide 1] Welcome back to Unit 1 of our course, The New Perspective. We are now on Lecture 7 and our objective is to connect up this ballistic and diffusive conductance that we talked about in the last two lectures. [Slide 2] So what we're shown was that starting from our general expression for conductance you can obtain and expression for the ballistic conductance or the diffusive conductance by using this expression for the time, that the basic differences in the ballistic limit the time is proportional to the length, in the diffusive limit the time is proportional to length squared. Now the way you can get the general answer that would work for both limits and in between is to write the time as L over v bar, the average velocity plus L squared over 2 D. Now it might look a little surprising how you're adding the 2 see, so one way to justify it would be to say something like this, that you are writing the time kind of like a polynomial expansion, L, L squared in principle there could have been an L cubed, et cetera. And of course what you expect is then when the length is very small then the first lowest power should dominate things. When the length is large that's when the higher power still dominates and you could say well for very long things I know the answer from the diffusive limit, that it's L squared over 2 D. So that tells me that first I shouldn't have L cubed, L fourth, et cetera because we know that for long things it goes with L squared, that's one diffusion theory and for short things we know that it should go as proportional to length so that tells us that the lowest power should be L. So that could be one way to justify this. If you accept that then you could have written it and read it in this form because this first time is the ballistic time so I can pull that out. So the second what we have here is like the ratio of the diffusive time to the ballistic time, that's what we have written here, which you could rewrite in the form L over lambda where this lambda is like the mean-free path as we'll show but for the moment we have just defined the lambda as the ratio of this diffusion coefficient to the average velocity two D over v. So if we accept this then this is just a little algebraic rearrangement of things. So let me say a little more to justify where why we might expect this, why time why the transfer time should be given by the ballistic time times this 1 plus L over lambda. [Slide 3] Now what we did in the last lecture if you remember when I was trying to justify this diffusive time the way we did is we said well let's just look at the electrons in the channel and relate it to the flux. A fundamental relation that I've often brought up is that the average time an electron spends in the channel is equal to the number of electrons in the channel at steady state divided by the flux of electrons, the flow...the rate of flow of electrons is current. Now based on that what we saw was that for ballistic transport the amount of electrons that are stored in here was the D over 2 and interestingly even in the diffusive limit the answer is still the same, D over 2, the only difference is that in the first case all the electrons are in there like the northbound channels as I said but you can think of this channel as having half the states that are northbound, the other half that are southbound. So it's like the ballistic case all northbound states are filled, all southbound states are empty. Whereas in the diffusive case northbound and southbound are about equally filled but the occupation goes down as you get from this end to this end. Right because electrons come in from here and so there's lots of electrons at this end but then as you go through the channel you have less and less that are remaining because many of them have turned around and gone back. So what happens in a situation that's got an intermediate between the two? This is something we'll talk more about as I said in the third unit of this course when we talk about this what and where is the voltage but for the moment let me just give you a simple description and that is let's say T and we'll use this capital T to define this fraction of electrons that make it from source to drain. That is in the ballistic case any electron that came in from the source goes all the way to the drain and that's why all the northbound lanes are completely filled, all the way steady state. Now let's assume that we have kind of semi ballistic transport so that the electrons that come in from here let's say 50% get through, so what that means is by the time you come out of this end the northbound lanes have become kind of half filled 50% or 30% filled or 70% filled, so that's this T. So in that case when I try to plot the electron density you'll see something like this and this end is D over 2 L but then the occupation of electrons as you go...occupation of this number of electrons in the northbound lanes, that's these electrons per length for unit of energy will have gone from D over 2 L down to T times D over 2 L. As far as the electrons going the other way is concerned, that will start out at zero at this end but will gradually build up and the reason it builds up is again because electron are turned around. In unit 3 of this course we'll actually do this more quantitative, we will write down the proper differential equations. But this is what it will look like and the first one I want to make is that if you look at the number of electrons that are in the channel not surprisingly the answer is still the same. You might have guessed that you know after all when it was ballistic it was D over 2, when it is completely diffusive it's D over 2, so when it is in between it's probably still D over 2 and that's what you can get actually if you analyze this properly, now how many electrons do you have in the southbound lanes, how many do you have in the northbound lanes, add them up you'd get exactly the same answer, D over 2. What's different now, what's different in all three cases, this, this and this is the current that is associated with it, the reason the times are all different is because the q's are all the same but the currents are different so the current has the highest value in this case corresponding to the shortest time, here the current is relatively low giving a longer time and this is in between, you see. So if we try to write the current what you might expect then is that whatever current you had in the ballistic case now the current will be T times that because only 50% of the electrons get through so previously if you had a ballistic current of let's say 2 mA now you will have only 1 mA. So this current would be like T times that. Now what that means is when I try to calculate the average time an electron spends in the channel I'll get Q divided by I and I is T times the ballistic current and Q divided by the ballistic current is the ballistic time. Remember the Q stays the same in all three cases, whether they are ballistic, whether they are diffusive, whether they are half way there, all three cases, same Q it's the I that's different and so it's tB divided by T. Now this transmission coefficient are the this fraction of electrons that get from left to right, that's actually given by mean-free path divided by length plus mean-free path. In fact one could use that almost as a definition for mean-free path because you could say that well when the length is very short compared to a mean-free path that's when I can drop this so the T is 1, which means 100% of all the cars, all the electrons get through for the short thing. When the length is approximately 1 mean-free path then you see if you put L equal to lambda I'll get a T of half, so you could say well that's the way I'll define my mean-free path, it's the length such that only 50% get it. Of course you could have defined it as length for which 30% go through or a length for which 60% get through. The conventional definition is not exactly this, we have chosen this definition because that way this expression comes out nicely. You know it's like lambda over L plus lambda whereas if you have chosen a different definition you might have got say 1.5 lambda over L plus 1.5 lambda so exactly how you define the mean-free path that's somewhat can be different from one definition to the other but...and we'll in the next lecture see what the exact expression for this mean-free path is. But the bottom line is that this fraction of electrons that make from source to drain can be written in this form, lambda over L plus lambda and if you combine that with this then what you get is that the time is equal to the ballistic time divided by T and the divided by T means like L plus lambda over lambda which can be written as 1 plus L over lambda. So this is the expression that I mentioned earlier that the time in the general case can be written in this form, when L is small it is the ballistic time, when L is large this is what will become the diffusive time. [Slide 4] So if you now combine that so for ballistic case you see we have that expression for ballistic conductance that we obtained before by putting the ballistic time for T but now let's say we use this one so we have the time is ballistic time times that factor so let's use that here so what you get then is the conductance would be the ballistic conductance divided by that factor, why because the T is in the denominator so when I put a ballistic time there I got this, now I'll put something a little bigger, I'll put the ballistic time times that factor. So when I put that in the conductance I'll get will be less by that same factor so this is the expression you'd get which you could rearrange to write as GB lambda over L plus lambda. This is something you can then identify with, this expression, conductivity times area over L plus lambda so you have this expression for the conductance. Note that if I have a very long channel then I can drop the lambda from here and so the conductance will be sigma A over L just like you learn in freshman physics, what it should be Ohm's law for large conductors that's what you'd get but then when length gets really short the point is the resistance doesn't become zero, conductance doesn't become infinity, it tends to some constant value, see. The thing to note there is the sigma A you can identify with is GB times lambda. So usual understanding of conductivity as I say starts from large conductors where as in this new perspective that we are using here, the fundamental thing you first look at is the ballistic conductance that any material the first thing you ask is if I made it really short what conductance would I measure? That's this ballistic conductance, that's a material property and now you ask the question and now if I make it long what is the conductivity, the answer is take your ballistic conductance multiplied by the mean-free path and that will be it's conductivity times area. So that now brings together all these concepts that we talked about so it puts it all on the same page now. [Slide 5] We had this expression for conductance, we looked at the ballistic limit, looked at the diffusive limit and if we write it altogether in this form where this conductivity is given by this GB lambda I notice I missed an area here, this should have been sigma A okay, so now what we'll do is we're ready to move on to the next step which is we talk a little bit about these two quantities that appear and one of these appears in the expression for the ballistic conductance and the other appears in the expression for the conductivity, that's what we'll talk about in the next lecture. Thank you.