 Okay, so this, I guess, is the first talk of the sub-theme, which is the graphene colloquium that we're having. And what I'll be talking about, though, is somewhat more general. In fact, some of you have heard some parts of it before, and there's a lot of material on this on the nano hub also. But the reason we are going over this is that it's particularly useful in terms of understanding current flow in graphene. So although the material itself is much more general than graphene, really, but it gives you this philosophy of this electronics from the bottom-up approach. And as I said, it's particularly useful in understanding conduction in graphene. So the basic problem then that we want to talk about is that we have some conductor. That's what you'd usually call the channel. Across which you have made two contacts. You call them source and drain. And you put a voltage across it. And this voltage, what it does is it separates the electrochemical potential in the two contacts. At equilibrium, all contacts would have the same electrochemical potential. You put a voltage, one side's gone down, one's up. So you've got two different electrochemical potentials. And you want to find out how much current will flow. And one of the first things, as you know, you should do is try to draw what you might call an energy level diagram. Something that tells you where the allowed energy levels are inside this channel. And what I've sketched here is some general density of states. This d of e. So this axis is energy. And it tells you the density of states. And I've sketched some generic density of states. So typically you have some non-zero value up to here. Then you might have a band gap of sorts. And then you would have some density of states down here again. And current flows if there is density of states available around the chemical potential. And the other thing I've sketched here are the two Fermi functions. That is, when you separate these electrochemical potentials, each electrochemical potential actually describes a Fermi function. And this Fermi function you have seen before, looks something like this. And for mu 1, it changes from 1 to 0 right around mu 1. For the other contact, it changes from 1 to 0 right around mu 2. Everything below this is filled. Above it is empty. Now why does current flow? The way I think about it usually is that this contact wants to fill up these states because it's below the chemical potential, wants to fill them up. This contact wants to empty them. And so what happens is it keeps filling it up and it keeps pulling it out. And that's why current flows. And that's why if you don't have any states here, no current will flow. Why don't these states contribute to current flow? Because that is one question that often comes up and causes a lot of confusion, is that should these states contribute? And what I'd like to argue is, in general, not really. Because when you consider a state down here, this contact wants to fill it up. That contact also wants to fill it up. So it just stays filled. Nothing else happens. That's all. The reason current flows is that, as far as these states is concerned, the two contacts have two different agendas. One would like to fill it up, one would like to empty it. That's the basic. Now the usual way of thinking about current flow, though, the formula that you are probably familiar with, that's this one. That's the Druter formula. And this is what is usually what you carry in your head. Like, if you're trying to understand conductivity, you'd say, well, it's proportional to the electron density and it's proportional to the mobility. And where does this formula come from? Well, the way you usually think about it is that if you think of electrons inside this material, the Newton's law would say that dp dt is the rate at which the momentum changes is equal to the force exerted on the electrons. And I'm not worrying about this minus sign. So I mean, you could put that in there, minus q. But you could think of q as being a negative number for this discussion. So it's dp dt is equal to the force. Now if you believe that, then of course the momentum would keep increasing with time. What you say is inside a solid. That's not what happens. There is always some scattering term that does this. And so there is always this friction. And so at steady state, you can drop this. And so the momentum is equal to q over tau, I'm sorry, q tau times q times the electric field. And then how do you calculate the current? Well, the usual approach is you write current is equal to q n v. This is the general equation for calculating currents. It is the charge on an electron times the electron density per unit length. It's a linear density of electrons times the speed with it moving. So if you want to calculate current there, you see you need the v. And so you put from the p, you try to get a v by putting a q tau over m. So typically, this is how you work. When you put this v in there, that's how you'd end up with the Bruder formula. Now I've written here ns because this is the electron density per unit area. Whereas what I've written here is like the electron density per unit length. So what you could do is write it as width w. I'll call this the width. This is the length. Call this the width times q ns v. That is, electron density per unit area times the width would be like electron density per unit length. But this is the one that you'd normally use to get the current, OK? So the thing is that when you think of it this way, though, there's a few issues that you have to note. One is, firstly, it gives you the impression that all the electrons are moving. It looks like it's not just these electrons, but you're thinking that, well, I should be using all the electrons, which is actually not, you know, causes some confusion. Because as I argued earlier, current really flows due to these. Now more importantly in the context of this colloquium though is that if you try to understand current flow in graphene, the first thing you run into is like what mass to use. Because if you're at all familiar with this literature, you'll know that this is a material in which the EK relation is such that there's no well-defined mass as such. That is, when you look at the EK relationship, as you know, usual semiconductor have a parabolic relationship which you write as e equals h bar square k square over 2m. But that's not the way it looks like in graphene. It looks linear, looks something like that. And then of course, it's confusing because you don't quite know how to extract a mass out of that. So that would be the first problem you'd run into in terms of trying to apply this equation to graphene. Now instead, so if you look in the literature, no one really tries to use this at all. Although this is the one that we all carry in our head. This is how you interpret experiments. This is how you think about things. Yet this is not what anyone applies though in this field. What they apply is something else and that formula looks something like this. And this comes from, they say, the Boltzmann equation. So the equation you'll see most often in the literature, they will say that, well, what you get from the Boltzmann equation is the following. U square times, so this is the same Fermi function. And here, by the way, we are talking about relatively low bias. I won't, I mean, for everything we'll discuss, its assumption is this voltage is small, so you're really talking of linear transport, nothing to do with any nonlinear issues at all. Now, so this F is basically what you'd have at equilibrium. That's the F that you use here. And this D is the density of states per unit area. So I'll usually define the density of states as the density of states just per unit length, rather, sorry, as per unit energy, rather than as per unit energy per unit area. So what we'll do is divide this by WL. And by the way, there's a W over L out here. So this is what you'd normally call the conductivity. And you say that this is what comes out of Boltzmann equation and there's usually no simple explanation where it came from. But this is the starting point for a lot of the work in the literature on graphene. And this quantity, by the way, is what you usually might recognize as the diffusion coefficient. That is, I wrote here Vx square, that is, this direction is x. And of course, electrons are going at all various angles and when you do the angular average, then Vx square would be like half of V square. So that's what people usually do. They write that as V square over two. They take that and put it as V square. So this is the formula that you'd see widely used. And here you see you're not having to ask this question about mass. It's just density of states diffusion coefficient. That's it. Now the other point I'll note here is this philosophical difference. And that is that when it comes to this formula, the feeling is all electrons are moving. All electrons have this electric field, they're all moving. When you use this formula, that's not true. What contributes to this conductance is only electrons around the chemical potential. How do you see that? Well, you see, look at this derivative of the Fermi function. You see, that Fermi function looks like this. If you take its derivative, it's non-zero only right around here. That's it. When you look at that important function in this integrand, regardless of everything else, the point is that function only peaks up right around here. And so when you do this integral, what really contributes is electrons with energy right around there, like I had argued before. That's what really should be contributing to the current flow. Now another formula that you'll see in the literature. In fact, let me write a slightly simplified version of this here for our comparison. That's Boltzmann. So Boltzmann would be, I kind of simplified it a little bit. You see, this looked, this has an integral and all that in it. Now, for this discussion, I'm kind of simplifying it by saying, let's assume this temperature is low enough that derivative is almost a delta function right here. What you can show is that the area under this curve is always one. So in that sense, no matter what the temperature is, that area is one. And if it's very sharp, you can think of it as a delta function. So what I'm saying is, let's assume it's kind of at very low temperature. So this is a delta function. And so this integral can be dropped. And all that contributes is this. And that's what I've written here. Now you might say, well what happens if you go to higher temperatures? Well, what it means is if you understand the conductance at some energy, then you can always average over energies to get the effect of temperature. And when you do that, what will happen is you'll pick up all kinds of these Fermi Dirac integrals. You know, there'll be F3 halves, depending on what you're calculating. And I'm kind of trying to keep out of that so that we can focus on the main thing. Okay, so that's then the Boltzmann result. Now another expression that you'll see in the literature often is the following, and that's this Landauer formula. And that goes like this. It says q square over h. That's the quantum of conductance. Times integral de minus del f del e. And then there is a quantity that people call the number of modes for unit energy. And this is supposed to apply only to ballistic conductors. And so usually people say that well, if you assume that your conductance is ballistic, is if your conductor is ballistic, that is electrons get from left to right without any scattering, then we'll use a formula like this. But then it's not clear how to go between this ballistic limit and this diffusive limit. This is the one that's supposed to apply in the diffusive limit. And what I'd like to describe to you is this bottom up point of view where you have a nice continuous way of going from the ballistic to the diffusive. And you can see clearly what's happening here. So again, if I focus on low temperature, so I won't worry about that integral and all that. And I can write g is equal to, now I think I missed a q squared here. q square over h times the number of modes. That's the formula. Okay, now let me explain a little bit as to where that modes come from. So the way you can think about this is the following. We're starting from this case of a ballistic conductor where you have electrons that are going through like this and they're electrons, you have states, half the states are going from left to right, the other half are going from right to left. And the idea is that you could write the current that is carried by the right moving states. See, those, we could write it as q times the electron density times the velocity, these. So you could write the velocity v of x, that's in the x direction. And this electron density, that we could write by saying that we have the density of states is d over two because I only want to consider half the states, the ones that are going from left to right. And I want the electron density per unit length, so I'll put a length here and I'll multiply it by f plus. What's f plus? Well, f plus is the function that tells me how the occupation of the right moving states. How well they're occupied. By the way, this f one and f two, those are Fermi function. Those are the equilibrium things in the context. Inside the device, it's a non-equilibrium situation. The positive going ones are a little more filled than the negative going ones. And so f plus tells me how well the positive going ones are filled. And then there's the negative going ones. Those are also filled a little differently and that's why the current is the difference between the two. And you could write it this way and I could write I is equal to integral dE. So this is at a given energy. If you just looked at q and v, you'd get q, that's the v. This times this, that's the n. And you have things that are going from left to right and you have things that are going from right to left and the difference when you sum it over you'll get there through the current. That's the idea. Now in the case of the ballistic conductor though, the simplicity that comes is the following and that's this observation that everything going to the right is really came from this contact and so is basically has the same occupation as the Fermi function in the left contact. So this of course is a very important non-trivial thing and I'll try to explain it a little better and then the ones that are going the other way they come from the other contact. So what you do is you'd say this is F1 and that is F2 and this of course is the important observation and you can kind of see it's intuitive. It's like this, this is electrons are going from left to right. So let's say you know this is still Lafayette and this is Chicago and it's like going from South to North and let's say there's something important happening in Chicago so there's lots of people trying to get from Lafayette to Chicago. So F1 here is almost one. You got lots of people trying to get on the highway and it's all ballistic in the sense once you get on the northbound lane you can turn around you keep going. On the other hand there's hardly anyone in Chicago wanting to come back at this time. So this is all empty. Now if you looked on the highway what would you see really? You'd just see all the northbound lanes filled up bumper to bumper, all the right bound lanes completely empty. That's what would happen. So that's exactly what you expect. You see you could draw a picture something like this. If you looked at the occupation of these levels from left to right this axis is the x-axis. At this end you have F1 which I'm assuming let's say is one. That is these states are all filled at this energy on the left. And then as you go through this thing it's all filled and then somewhere inside the contact of course it goes down to zero. Somewhere inside the contact. And when you look at so these are all the right bound or the northbound lanes. These are plus K states. And these are all the minus K states which are filled from here. They do this and inside the contact it matches again. Now you might say well but if there are so many states that how is it that once you get in here they have all the occupation goes down so much. The idea is that's what happens inside a contact. You see because once you it's like you got only two lanes here but once you get to Chicago it's like there's lots of lanes to get out into. So once you go out there of course it becomes just a normal thing. So that's the idea of a contact. It's this delusion. Anyway so this is what you'd expect inside a ballistic conductor. And because of that you see you can write the current very simply. You know it's Q and V. N is the density of states, half the density of states per unit length times F that's like N times V. This is it. And so this leads us then to the ballistic current which would be integral DE times M of E should be put here and then put F1. So what I did here is I've called this quantity M divided by H. So that's this number of modes that I talked about. That's the M that appears here and I'll try to connect it up better. But basically you see it's the density of states times the velocity. So just as density of states determines electron density when you want to calculate current you need density of states times velocity. It's not enough to have a state there. It has to be moving. So that's why this is DV. So the expression for M then would be something like DVX over 2L. This is M over H. It's just dimensionally this has the same dimensions as H. M is a number. So instead of that quantity I've written here M. Now how do you get to this conductance formula? Well that's where basically what you do is this F1 minus F2. For low bias we say that we are taking the difference between two Fermi functions and you could write it as this Taylor series type of thing del F del E times mu1 minus mu2. And this integral DE minus del F del E and this M of E and mu1 minus mu2 is q times the voltage. And so current becomes proportional to voltage and whatever's inside that's then like the conductance. This would be the standard derivation of the ballistic formula. So finally what you'd have then is conductance is equal to q squared over H integral DE M of E times minus del F del E. And this is where as I said earlier is I'm simplifying things by saying let's do it at low temperature. So that thing is just a delta function and so you get the Landauer formula. Now the reason I went through this in a little more detail because some of you I'm sure most of you have seen this somewhere in some form is this, that the question arises is how do you apply this Landauer formula to something that's not ballistic. Now not ballistic basically means that you see once you get on the northbound highways not like you have to stay in the northbound you could continually turn around. In fact there's a mean free path like every, so if the mean free path is one mile it's like every mile a certain fraction will be turning around, see. So what you'd expect would happen and this you can show is that instead of going like this the distribution would look something like this. This would continually go down and if you look at the negative case states it will continually go up and inside the contact it will be in contact. So the point is that you are coming in here this is a bumper to bumper right here but then most people change their mind and come back. So by the time you get a little further down it's not quite filled as much anymore. The highways are a lot more empty and what you can show is that that of course when I want to calculate the current I still want to use F1 minus F. I still want to use F plus minus F minus. That is what is the difference in the occupation of positive going states and negative going states that's what I still want. It's just that now this difference is only a fraction of what it is between the two contacts. So you had one here and zero here but now in the middle the difference is a whole lot less. So some fraction thereof. And what you can show is that that fraction this is like F1 minus F2 times a mean free path divided by L plus mean free path where L is this length. So if the mean free path is very long then that factor is one and you have ballistic transport. Then the difference is exactly equal to whatever you have at the ends. That's what you've been talking about so far. But once the length becomes comparable to a mean free path that's when what you get is a whole lot less. Now the result of course is that the current here then instead of being what we what I've written before is now reduced by lambda over L plus. Now this version of lambda formula is of course very well known. What I haven't seen as much as this version which is which says that if you had a conductor that was a few mean free paths long then you should reduce the conductance by that factor. And that's what I have not seen as much. And this is usually what I'd say is most useful when it comes to analyzing real experiments in graphene because most of the graphene experiment is not like full in the diffusive limit necessarily. On the other hand it's not ballistic either. It's usually a few mean free paths. And what I'd say is well this would be a good way to analyze such things. And this mean free path then if I had to write an expression for it in terms of tau because the mean free time as you know is this tau that entered the Drude formula. So mean free path would be velocity times the time. Only thing I'd add is that actually there's a factor of two in there. And that's because you see the mean free time is how far an electron goes before it gets scattered. And if you assume that the scattering is isotropic. So supposing you have an electron going in some direction. So mean free time tells you the time after which it's scattered but then if it's isotropic then you see the chances of getting back scattered. What I mean is all of these are like northbound lanes. The current is reduced of course only if you get scattered into a southbound lane. As long as you're scattered somewhere here it really doesn't matter. So it's like only as if half the scattering matter. Which is why you usually put in this too. So I'd say then from the Landau view point if you proceed in this is the expression you would come up with. And then if you apply it to a long device then you'd say okay let me drop the lambda here. You can drop the mean free path from the denominator and you have a conductance that goes down as length just as it should from Ohm's law and all that. And you'd have something looking just like that. Now how do you connect these two things? Well you can kind of see how it works. You could write this as we cancel the W. You could write it as Q square dvx over 2l. So I'm taking this standard Boltzmann expression and just writing it out separately like this. And if you compare the two then I'd say if I put this thing equal to m over h and this is the lambda over l that's basically the Landau expression. That's it. The advantage of course is that this one is relatively harder to derive. You have to go into the Boltzmann equation and it takes a while to understand the Boltzmann equation and then there's various approximations that go in before you get here. So it's a lot harder to derive which is why the expression you usually carry in your head is more this one, not this one. And the other thing is this only applies when you have a very long diffusive conductor. Whereas this way you see you have something that interpolates nicely between the ballistic and the diffusive regime. This goes continuously. Now what you might wonder is how could it be this simple then? You see because as I said to derive this would be a lot more work usually. And I'd say that the assumption we made here though here is that and electrons as they go through from one end to another, they sort of retain the same energy. There is no inelastic scattering in the process. So, and that is why you can write the current. I mean the key to our derivation was that we wrote the current as integral dE times something times F1 minus F2. This was of course a key point. And once you have F1 minus F2 then we went to the linear response and got all the other stuff. But the thing is this expression doesn't really apply to a very long device with lots of inelastic scattering. You can convince yourself of that. That if this was a real long device with lots of scattering you could put 10 volts across it and it would still be in linear response. Of course, 10 volts across say a 10 kilo ohm resistor from Radio Shack, I mean it's still linear. But if you take two Fermi functions separated by 10 volts and try to apply it you'll get an answer. So this kind of a formula, this kind of formulation you can really apply rigorously only to conductors that are relatively small where you can argue that there's no inelastic scattering from left to right. Then how are we applying it to a big conductor? You know after this Boltzmann expression is supposed to work for big conductors. So how are you applying that? I'd say the philosophically the way you can justify it is by saying that when you have a very big conductor you can still think of it as lots of little things in CDs. And it is as if we are taking one of those little things applying this kind of a viewpoint to it extracting the conductivity from that. And then saying well if you put a lot of them in CDs that's still the same conductivity. And it seems you're doing it right. I mean the part you can convince yourself is I mean this and this they're essentially the same expression. But then of course the Boltzmann equation is more rigorous in the sense as I said the reason it's harder also is that you have to go through certain arguments and you can show that this expression only holds when you have, when scattering is either elastic or isotropic but if it is inelastic and isotropic then it doesn't quite hold. And I believe that in that case some of the simple parts simple assumptions were made wouldn't quite work. But the other point I'd like to mention here is that the nice thing about this viewpoint and this is kind of the basis of what we have been doing a lot in terms of our bottom up viewpoint of all this is that bottom up viewpoint means instead of trying to take big conductors like and where Boltzmann equation is applied and all that and then trying to figure out what happens in small things we do it the other way start with small things and go. And what we believe is when you take your experience from big conductors and project it down it unnecessarily complicates the story a whole lot. And so if you're really interested in small conductors this is of course a whole lot simpler and the point I was trying to make is it even might give you a lot of insight into big conductors as well. And this is particularly true of things like thermoelectricity that's something I won't have time to get into today but as you know in conductors one way to drive current is of course put a voltage across it but another way to drive it is to put a temperature difference one end is hotter than the other and one of the very interesting results that I'm sure you all know is that if you put a hot probe and a cold probe then which way the current flows depends on whether it's n-type semiconductor or p-type semiconductor. That's an experiment actually even I have done this. And it's a very nice simple result and yet trying to understand why the current flows in the opposite direction is not very easy. See if you think about it, why is it that in p-type semiconductors the current actually flows in a direction that's opposite. On the other hand if you take this expression and just look at f1 minus f2 the answers will just follow. It's just that the difference between the two Fermi functions now comes not from a difference in chemical potentials but just from the difference in temperatures. That's all, nothing more to learn. So it's sort of like once I've understood conduction thermal electricity and all that just falls right out of it. There's nothing else to it. No new things to learn really, just fun. So there's lots of things where this bottom up viewpoint where you basically kind of start from here essentially and which applies to a lot of small conductors and even to some big conductors. And you can get all sensible answers that you can use. Okay, now to take this a little further then I need to bring in EK relationships. That is so far you see I haven't even argued about EK relations. I just said well you have some density of states. How do you get density of states? Well usually for crystalline conductors the way you do it is get it from an EK relation, okay? Now the first case you might want to consider then would be one dimensional conductor. So let's say I have a one dimensional conductor, this direction and we'd like to know what is the density of states. And the way you do it usually is that you assume any EK relation, let's say need not be parabolic, could be anything. And the idea is that if you put it in a box then the allowed values of K are separated by two pi over L, right? And that comes from imposing this periodic boundary conditions and as I always say that it's not that any real solid actually has periodic boundary conditions on it. But what you believe is that in big conductors it doesn't matter too much what actual boundary conditions you use. And so you can use whatever it is that is mathematically convenient. That's really the argument. Because physically of course you don't really, it's not in the form of a ring at all in general. Anyway, so this is just a way of counting the density of states and the feeling is that even if you did the actual boundary conditions things wouldn't change all that much at least as far as big conductors go. So it's separated by two pi over L. If you look up to a certain energy E and ask how many states do I have then the argument would be, then the number would be something like, so this is plus K minus K corresponding to a given energy that's the maximum. So what you first write down is the total number of states which is two K divided by two pi over L. So the idea is the total range is two K, states are separated by two pi over L. So the total number of states in here is two K divided by that which is K L over pi. Now once you have this total number of states the way you get density of states is take its derivative with respect to energy. The idea is that if I increase the energy a little bit how many extra states do I pick up? So you put D of E and what you'd get is L over pi D K D E and as you know this D E D K that's what is usually called this group velocity. That's the velocity you should associate with an electron in that state. So usually you should write it here because we'll be needing this more. H bar V is equal to D E D K. So that's a general relationship that does not depend on E K relations. Now I'm sure most of you have seen some versions of these things. The point I'm trying to do though is I'm trying to get these relations in a way where I do not use any special E K relation because remember we are kind of always trying to do this because we want to understand graphene. And graphene has this E K relation which is not the standard parabolic one. So we don't quite want to use a parabolic relation in anything we are saying because then it would be special and you wouldn't be able to use it for graphene. So this is general. Doesn't matter what your E K relation looks like, that's true. Similarly what I wrote here, that's perfectly general. No matter what, this is still true. So when you do density of states then this D K D E that would be let's suppose this is one over H bar V. So that was the K L over pi. So now if you want to calculate this number of modes you know because in this way of thinking of course modes plays a very important role in this thing. So let's say we want to calculate the number of modes. So it's equal to D V over two L. So that's equal to D is L over pi H bar V. That's D, taken all that there. And then I need a V over two L. So you'll notice the answer you get is I guess this is M over H, D V over two L and so M is equal to one. So for a one D conductor you see originally I introduced this concept of modes as by saying well let's take a ballistic conductor see what current flows through it and what you get is something like density of states times velocity. And in all of this of course there's no H in the thinking, D V. It's all classical as far as we are concerned. We're just talking of some states with some velocity. It could be as I said cars on the highway. I mean no H involved, nothing else. We're just taking the states times the velocity nothing else. But then now of course we have brought in the E K relationship. Which of course builds into it the quantum view point, wave view point, et cetera. And once you bring that in then you can see this M D V over two L for a one D conductor basically becomes one. It's like as if this is a single mode. And so that gives this concept, this mode of rather a lot more importance and significance. In the usual solid state physics you often you see a lot about density of states. That's what you'd usually see in the literature. Modes is something no one talked about till mesoscopic physics started. Till people started looking at small conductors. Where people started using this Landauer method and mode started playing a very important role in that thinking. And what this shows is this M is yeah when you put in these numbers comes out as one and of course one of the most important experiments that happened like in the late 80s which kind of helped people start using the Landauer formula was this conductance quantization. Where people took a piece of semiconductor continually made it thinner, narrower and narrower and showed that the conductance instead of going down linearly as the width actually went down in steps. It is more like this where the M went as one, two, three, four things like that rather than as a continuous variable like that. And the general expression for M then which is what I'll try to obtain next. What happens is let's do this in 2D now. So again I want to calculate M for that I need density of states and so for 2D we first write down the number of electrons that I would have if I were to occupy all the states up to a certain energy E and in that case the way you think about it is so two dimensional things kx, ky and I'm assuming the ek relationship is isotropic so the question we are asking is if you go up to a certain maximum value of k how many states do I have in here? Some maximum value of k corresponding to some energy and then the idea is if I increase the energy a little bit how many extra states do I pick up? That's the density of states. So N of E would be suppose this pi k square that's the area of that circle and then you divide it by the density of states and the states of course in this direction are spaced by L over 2 pi in this direction by W over 2 pi whatever the width is. So you basically have W over 2 pi I'm sorry 2 pi over W and 2 pi over L and that then comes to LW k squared over 4 pi. So that's the total number of states. Now if you want the density of states you'll get I'm supposed to take derivative with respect to energy LW 4 pi 2k and then dk de. Now at this stage usually you then invoke the ek relationship and do that which I'm trying to avoid invoking of course. So what I'll do is instead for dk de I'll write h bar v. Now that way that's general that doesn't depend on ek relationships. So the expression you then get let me write it on top is that the density of states is LW over 2 pi h bar v and then a k I've done this right. So you'll notice the density of states is proportional to k divided by v and you see k is like momentum. So usually if instead of h bar k you write MV then you'd get the standard parabolic band density of states and all that that you could do if you want it but as I said that's what I'm trying not to do. Now if you want m number of modes then I'd use this relation and this is where of course what appears here is the velocity in the x direction and I want to replace it with the net velocity. That means you got modes in various directions which have different components in the x direction and what you can do is if you average it again assuming there's a big conductor so I don't have to worry about graininess just average it. So if you average over that angle then what you'll find is instead of vx you could write 2v over pi. This 2 over pi is what you get by this angular averaging of cosine theta. So I think what you should get then is try out h over 2L and then the D is whatever I wrote there L times kw over hv you know 2 pi h bar is just h and then 2v over pi. And I think this should be equal to if I've done this right kw over pi. So this would be another way of thinking about the number of modes. Indeed this viewpoint is usually more well known. This is the way you think about this is you've got a certain width how many wavelengths fit in because you say k usually think of as 2 pi over the wavelength. So this kind of tells you that in a given width how many half wavelengths actually can fit into it and that's the viewpoint that comes naturally if you're thinking of it from the point of view of microwave waveguides and that's how usually people think of modes and which is kind of nice and what it shows you is that number of modes really depends on k again in kw over pi irrespective of the ek relationship. Again remember I avoided using that you see. So this is true no matter what doesn't have to be parabolic this is what you'll always have see. And this viewpoint of course the reason I kind of started here because this is the one that comes naturally doesn't even require you to know quantum mechanics. This is the one that comes by saying well you're just counting states in a finding the current in a ballistic conduct is density of states times velocity et cetera. That's what comes naturally you see. And what I tried to show you is that that is what will also give you this other viewpoint of what modes is kw over pi. Now that then leads you to a natural way of thinking about the conductance. You see so let me write this up here. So I'll write it here m of e is equal to kw over pi. And let me also write the electron density. So as you know the total number of electrons that we wrote up to a given energy if all the states here were filled up to a given energy would have been this pi k square. And this thing what we had was WL times k squared over 4 pi. That's what we had by this pi k square and then dividing by 2 pi over L and 2 pi over W. This would be that. So you could write the electron density per unit area. So this is n divided by WL. You could write as k squared over 4 pi. So if you look back then in this Gruder formula conductance is proportional to ns. And that thing is what you interpret as mobility. And so anytime you see any data on conductance first thing you do is divided by the electron density that you have measured and plotted and then ask the question of what is this mobility? On the other hand if you believe this formula then the way you might want to interpret your data would be like this. You see m is proportional to k whereas ns is proportional to k square. So what you can easily show is that m is proportional to square root of ns. That's easy enough to see. And so from this point of view when you write the conductance instead of being like ns times something that you call mobility you tend to get square root of ns times this mean free path from here. And the square root of ns is simply because it's proportional to the number of modes and ns is proportional to k square and m is proportional to k, you see this out. So if you believe this then when you measure conductivity you'd want to divide it by square root of ns and then whatever you get you try to interpret as mean free path and make sense of it. As I said what picture you carry in your head kind of determines what you do with your experimental data really. And what everyone does is of course this one. As I said when they actually calculate things they often use this one for graphene but when they think they usually think with this one. So I think in the handout one of the picture I think that's the last one in your handout that I have there but I show that with the same data if you plot the conductivity divided by ns you'll get something like this as a function of ns and of course as you change the electron density what you're really doing is changing the Fermi energy changing the energy up to which things are filled and you tend to see something like this and you look at it and you interpret it as saying well you know the mobility is actually going down with energy so more energetic electrons less the mobility that's how you interpret it. And the other hand when you take that same data and divide by square root of ns you tend to see something that's more flat and you think that okay this is more like the mean free path and it's constant with ns. Because when I divide by square root of ns with the right appropriate factors which you have in the notes I gave you I think the right factors are all there. But the point is you look kind of flat. And you might say well why is it that this one's going down and this one's flat? After all this is supposed to be the mean free path and this is supposed to be q tau over m that's the mobility. Now usually you'd say well mean free path is like velocity times time and when you increase the energy what happens is the velocity is also changing. And so you could always have something where the tau is changing but lambda isn't just because of that. Except that in graphene when you look at the e k relation it's linear d, dk is constant. It's the same velocity at all energies. And which is why often people say well the mass almost looks like it's infinite it may put an electric field of velocity doesn't change. You know looks like something very heavy can't change its velocity. So the thing is from this point of view you'd say well velocity is the same so if lambda is the same shouldn't tau also remain flat? And my answer is well what happens is in this case it is as if the mass is increasing with energy. And I'll try to explain why. If you really want to interpret it in terms of mobility then you ought to remember that the mass is not a constant with energy but it keeps increasing with energy. That's all. So how would you show, why do I say that the mass is increasing with energy? So that's this last point I want to make and then we can stop. And that is if you equate these two viewpoints that we have here. If you say that I want to and this of course is equivalent to that that I've argued. If you say that I want this to give me exactly the same results as that in two dimension. Actually in three dimensions as well. Then what you'd find is that the right way to define mass is equal to velocity divided by momentum. Which is kind of different because normally often we think of mass as the expression that you learn that we usually teach. I mean which you usually would need to know for your qualifiers. I mean you'd be in trouble if you use this one. What you should use to get through qualifiers is this one. And if you use this one then usually you'd get a mass of infinite in this case because in graphene the velocity does not change. You see momentum is proportional to k. I mean p is equal to h bar k of course in this discussion. So in graphene the velocity doesn't change with k. It's the same at all momentum. So based on that this is zero so m is infinite. If you use that formula it doesn't look right at all. But if you use this one then what you get is something's interesting and that is you see v I put h bar k and for graphene e is equal to h bar v times k. And so what I could do is for mass I could try to eliminate this in terms of e. So what I would do is let's say instead of this h bar k I could write e over v. And so that then will give me mass is equal to e over v square. So what that means is as the energy increases the mass also keeps increasing. For a linear relation that's exact. Now if you had applied this to a parabolic dispersion law you'd have got just the standard mass. It would be energy independent there would be no problem. But when you apply it to a linear thing if you use this relation where mass is velocity over momentum. I mean one over m is v over momentum. This is the expression you get. Which is kind of reminds you of e equals mc square. Kind of like that. But and this is also the mass that people measured in cyclotron resonance experiments. Though why that so that requires some discussion we won't go into it now. But this is also this is the mass they measure. And what they find is while in ordinary semiconductors as you change the energy as you know the cyclotron resonance frequency is supposed to be qb over m. That as you change the magnetic field the cyclotron frequency is something like this. And usually in a normal semiconductor that's independent of the carrier density. To change it same mass is fixed resonance. Whereas in graphene as you increase it the cyclotron frequency actually goes down. That's an experimentally measured fact. And to understand cyclotron resonance also the same mass actually enters. And the main point here I'm making though because we haven't really discussed cyclotron resonance. The main point I'm making here is that if you want those two things to be the same if you look back at how one deduced the root formula it was like from Newton's law you got a momentum. And then divided by mass to get the velocity so you could calculate the current. Remember that's the first thing I did. If you think of that you can see that the right mass to use is the ratio of velocity to momentum. And so if you use that then of course you'll be able to interpret you know you would be able to use this formula and interpret things. But the important thing to remember then is that this mass is not a constant it's energy dependent. Whereas if you go at it from here then of course mass never even enters your thinking. You can just go straight with density of states or something or the number of modes and that never ends. So that's the point I wanted to make here though. I guess as I said the purpose of this talk was kind of two fold. One is just the general introduction to this electronics from the bottom up viewpoint. And where I'd say when you try to understand current flow it's like you start from ballistic things and then ask what happens as you put in more complications. Instead of the other way around where you start from big resistors and start and then wonder what happened when you make it smaller. And what I feel is when you do it that way lots of things including thermal electricity and all kinds of other things get a lot clearer actually. So that's the one general philosophical thing. In this context though it's particularly relevant because to understand graphene as I said it's like you really get into some of these important issues where it's important to understand the difference between these three types of conductance formulas. This kind of summarizes this. So what I'll be doing in the next lecture after the break is we'll talk about the EK relationship of graphene and where that comes from. You know where this comes from and the theory behind that. Thank you. Yes, please. Okay, I guess the question is about the origin of the term chemical potential, right? Where does it originate? So usually I would think of it as like in general if you have any density of states that you have, yeah. So density of states at zero temperature we are generally accustomed to thinking that all the lowest states are filled, all the highest ones are empty. So there will be some line here that will then separate the lowest states from the higher ones. I think you call that the Fermi energy. Now when you raise the temperature that's when I think you still have that demarcation somewhere is just that it gets a little diffuse. So that instead of having this very sharp thing like this where everything here is occupied and everything here is unoccupied it is kind of smeared out over kT, over a few kT which seems reasonable except that that of course needs discussion why it is that way where the Fermi function came from et cetera, those are things that need discussion but roughly you'd say but it doesn't seem unreasonable. I mean at low temperatures it is all filled up to here. Now it's separated. And I think that standard terminology is that at non-zero temperatures people call it a chemical potential. And again to that I should add this thing that I think strictly speaking what we are talking about should be called a electrochemical potential. Electrochemical in the sense that there's the electrostatic potential also. So for example, if I put a voltage here like a positive voltage that would lower everything. The entire density of states. If this was like under a gate where you put a positive voltage makes it much easier for electrons to get in all the electron energy levels go down. And if the electrochemical potential stayed right there there'd be a lot more electrons which won't happen. Usually what would happen is it would want to stay neutral and so just as this goes down the entire level will go down somewhere. So I think people say that the electric so this part of it they might call the location of the this Fermi energy relative to the bottom they might call the chemical potential and the whole thing the electrochemical potential. But as far as I'm concerned in our discussion all we need is this electrochemical potential. And one point I do keep making often and that is that you often tend to say that in a j equals sigma e, that is current is driven by an electric field and the point I make is that's really not true because in a PN junction there's an enormous electric field no current flows. Current is really driven by sigma grad mu. And what really matters is the gradient in the electrochemical potential. So it will flow. What drives everything is because up here the electrochemical potential is high and electrochemical potential is low somewhere. That's really what is driving current. It's not electric fields at all really. Yeah, that's the part. Yes, so the question was that now this idea that electrons want to go from higher chemical potential to lower where does that come from? Where, how did it? And I'd say that in thermodynamics to me it is kind of analogous to temperature and just as deep, namely just as temperature you could ask why does heat flow from hot things to cold things? And a very important realization is that what really drives it is not the content of heat but the temperature. So you could have, so for example, supposing you have these two materials, one has a density of states like that and the other has a very low density of states, hardly any. And this has a chemical potential that's somewhere up here, let's say. And if you actually counted electrons, you'd say there's hardly any electrons here. There's lots of electrons there. But the point is the current will really flow this way and not the other way. So it's like when you write a diffusion equation, you often think that well current flow is gradient of the electron density but that kind of the implied thing is that the density of states is the same. That's kind of implied there. So it's a more special thing because as I said, if you had a very low density of states here and you fill it up to here, there's very little few electrons here and here there's a high density of states but you fill it up to here. So lots of electrons there. But the point is electrons will still go this way. It's not driven by how many electrons you have. It's rather the degree to which it is filled and in that sense, it is kind of like temperature. So I guess the question is that in the case of gravity when something goes down under gravity, you can see where that force comes from. I mean, we understand that. See, in this case, what is it that is driving it? That is the question, right? And I'd say this is purely a statistical thing here, right? And I have sometimes used this word entropic force in this context that in a way, why does energy go from hot things to cold things? It is just that if it flows that way, then I mean, energy always wants to distribute itself in as many degrees of freedom as possible. And what you can show is that by going from hot to cold, you make that possible in a way. But that requires more discussion. So in that sense, it is not like mechanical force. The example that you used is mechanical force. And I've sometimes talked about this as being something like an entropic force. And lots of things in real life are really driven by something like this. Now, in the case where the density of states is constant, then you have the diffusion equation. And then you would say, well, that's because you've got lots of electrons here and very few here. So that's why they are going just as if you had a box where it was all filled on one side and the other side was evacuated. And if you suddenly remove the partition, it would just distribute. That's also an entropic force, of course, weight distribution. And in that case, it has said, yeah, lots of electrons here, it goes over there. So what's not so easy to understand is why even when you have less electrons, what really matters is the degree to which it is filled. So just as heat flow does not depend on the total amount of energy, but rather the temperature, which is actually the amount of energy per degree of freedom. That's KT, really. So it goes from hot things to cold things because hot things have a lot more energy per degree of freedom. Isn't that like... Prefer to go to a... We have lower energy. Work can be in a dose. So the example I have sometimes used is the following. That, take the simplest case, like say a hydrogen atom, something very simple with just two levels. So the question was, what it is that it's like, why does it, the electrons always want to go down, but normally not up? So for example, that's what we normally learn. And it's one of those things where, you see, beginning students sometimes ask me this question that, why is it that it always goes down and never up? But usually graduate students, I mean, by the time you're a graduate student, you probably stop asking questions like that. Not that it gets any clearer, but... And, but if you think about it, it's really a very valid question that, and after all, this doesn't come from Schrodinger equation by any means. You know, if you take, there's nothing to do with quantum mechanics really, because if you take the Schrodinger equation, anything that takes you down will also take you up. It's a Hermitian Hamiltonian. If you have a h1 to, there'll be h21. Really doesn't help explain how you go down and never up. And the argument that I think in Feynman's statistical mechanics, it has this like on the first page actually, on the first couple of pages. And the argument he makes is that, that this is coupled to a reservoir which has a certain density of states. And the thing is, so if you look at the density of states, let's say it's something like this. Now, let's say the electron is here and the reservoir is somewhere here. Now, if the electron goes down, the reservoir has to go up. Energy, overall energy exchange, it's conserved. And the reason it's much easier to go down than up is simply that all normal reservoirs usually have an increasing density of states. So if you have 10 states here, you'll have 1,000 states there. So it's like when you're going down, it is like you're going down to one level, but your reservoir has 1,000 levels. When you're going up, you're going to one level here, but the reservoir has only 10 levels. And so it's a whole lot easier to go down than up. So this is again essence of this entropic force that it's like, yeah, this reservoirs always have this and logarithm of this density of states, that's what you call entropy basically, K log W. But these are of course all very deep issues that require lots of discussion to get it. So what happens if you have ballistic trend this reservoir is often the probability that if you get into the drain and then scatter back, how do you take that into the picture? Because I see that you have scattering mode within the channel, but not on the drain side. So you, scattering is going on the drain side. What's the probability of that getting back? So is that also incorporated into this as lambda 1,000? Now yeah, in terms of the, what happens in a transistor at the drain, probably Mark has thought about this a lot more on this. But in general, in this context, at an abstract theoretical level, what I'd say is that what really defines the contact, what makes it very special relative to the channel is the fact that here you have only a few modes, like say 10 or so, whereas here you have like millions of modes. It is as if you have this two-lane highway that suddenly opens up into lots and lots of lanes, right? Because this is the one point that Landauer always stressed very much that the picture I drew, that where I said that there is a certain separation between positive going states and negative going states. But once you get into the contact, there is no separation. And people say, well, yeah, sure. Once you get into the contact, it's all in equilibrium. But if you think about it, you see current is the same everywhere. So if there is no difference between plus and minus, then how is the current being carried? And the argument really is that you got 10,000 modes in there or 1 million modes in there, and the slightest difference is enough to carry that current. So because you have only 10 modes here and because you have a million modes here, you only need a small fraction of this separation to carry that same current. But that's really the essence of a contact. But in a real device like a transistor, where you can draw the line and say that this is a contact and that's a contact and that needs serious discussion. Because this is an idealization that tells you that in any device, of course, somewhere there must be a contact of some kind. That's essential. Somewhere. For practical reasons, of course, you would like to draw it as close as possible. I mean, in principle, you could always draw it somewhere up here, way outside. So include everything as your device and draw your line somewhere where you have put the solder. But for practical reasons, of course, you'd like to draw it close. And then the question is, what can you get away with? How close can you put it? But that, in the context of transistors, I think Mark has thought a lot more about it. So we should discuss it with him. I think you're using the dark point at the end of the reference. Yeah, though, all I did was, though, I was just, yeah, I was using this relation, E equals h bar vk. So something like this. Right, right, right. So, right, because this is zero at this point, right? Right there, you'd say, based on this, yeah, mass is zero. Carrier density is also going to zero, yeah. Now, experimentally, what happens is, as you approach the direct point, other issues get into the picture, like disorder and all that, right? Whereas if you just do it theoretically, then I think it wants to blow up around there, right? What do you see? Now, though, experimentally, what happened, hard to compare directly with experiment because around here, usually all kinds of disorder issues, kind of don't let it blow up, essentially, right? Smooth it out.