nanoHUB U Fundamentals of Nanoelectronics: Quantum Transport/L3.6. Magnetic Field
========================================
 >> 

[Slide 1] Welcome back to Unit 3 of our course on quantum transport. This is the sixth lecture. 

[Slide 2] Now, as you know, in lectures three and four of this unit we developed this general approach for calculating the self-energy, and as one example of how you might want to use that approach, we went through this example involving graphene and that was the last lecture. Now, in this lecture, we'll look at another example which superficially looks a little like what we did in lecture two. Here we have a square lattice, two dimensional conductor. Of course, in lecture two we didn't really need this advanced method, this general method. We could do it in a simpler way. But, what we'll be looking at, the problem we'll be looking at in this lecture, does require this general method and the reason is that in this problem we actually have a magnetic field present as well. And, so your current is flowing from left to right along this conductor and your magnetic field is perpendicular to the plane of the conductor either coming out or going in to the plane of this slide. Now, as a result, what happens is there's a Hall voltage that is developed. This is this famous Hall effect and the idea is that when you put a magnetic field the electrons want to curve to the left or to the right and giving rise to a potential in the transverse direction so that's what you usually call the Hall voltage, or transverse voltage. And you look at this transverse resistance which is the ratio of this voltage to the current that is flowing. Now one way to treat something like that would be to actually introduce two additional contacts. You know, usually we have two contacts, sigma 1 and sigma 2 but, in experiments the way you'd measure this voltage is you'd have additional probes here, additional contacts and in that case you'd need two additional self-energies, sigma 3 and sigma 4 and you'd have this multi-terminal conductor to deal with. So that would be one way of doing it. What we'll be doing is something simpler that is, we won't really explicitly put in these contacts. So, we'll take those out. Instead, what we'll do is, just look at the potential here by looking at the occupation of the states in this region. So, we are doing this again at low temperatures so at the given energy you look at the occupation of a level and that occupation can be related to the local electrochemical potential because, the electrochemical potential is a measure of how well the states are occupied. So, what you'll do is simulate this structure with the current flowing and use the NEGF method to calculate this electron density, as you know this, function Gn, that's a matrix and if you look at its diagonal elements, it gives you the electron density, per unit energy, actually, 2pi times that, whereas this A which is a spectral function it's diagonal elements gives you the density of states per unit energy. So, when you look at the ratio it tells you the fraction of occupation and this fractional occupation is assumed to 1 in this contact, 0 in this contact, and you look at what it looks like inside, and that directly translates into what you might call the electrochemical potential at different points. And, so for a given current then, what you'd look at is the electrochemical potential difference between the two edges. 

[Slide 3] Now, when you do that we'll use this NEGF method and you can calculate the alpha's and beta's and then you use this general method to find the sigma's ect. And, when you do all that, you can then calculate this transverse resistance which is the voltage in this direction divided by the current in this direction and plotted as a function of the magnetic field. Now this Hall Effect is a very old phenomena, goes back to the 19th century but what was known in those days that's usually described by what's called a Semiclassical theory. And what it says is, that the resistance is linearly proportional to the magnetic field. A major development in Mesoscopic physics was the discovery of what's called a Quantum Hall Effect and what was found in 1980 is that when you look at the Hall resistance at high magnetic fields you see these amazingly accurate plateaus, that is the R is given by this quantum of resistance that we have discussed times 1 over M where this M is an integer. Now this is something we discussed in the context of the quantum point contact but in the context of the point contact usually the resistance is described by this formula but not very precisely, in the sense that M may be 2.03 or 2.1 rather than exactly 2. But, in this quantum Hall effect what was amazing about this phenomena when it was discovered is that this is like 2, M is equal to 2 or equal to 3 or equal to 4 out to many places in decimal. Indeed it's so accurate that it's even used as the standard for resistance and this was a of course a seminal paper that appeared in 1980. Now, using the NEGF method that we discussed if you set up these H and sigma's correctly and calculate R xy as a function of B, the point is this is the black curve, this solid curve is what you'll get. So, in other words, this amazing result will come out automatically. And, as I've often mentioned, I view this NEGF as a method that you can use to improve your understanding of phenomena because you can calculate things without necessarily understanding it very well to start with. But then, you can look at it carefully and try to understand things. So, I won't go too much in to the physics of this quantum Hall effect, what leads to these plateaus. I'll just briefly mention it at the end. What I really want to explain in this lecture is how you can use the NEGF method to perform this calculation. 

[Slide 4] Now-- One word about this semi classical theory here, the semi classical expression for the Hall resistance, that's basically this Rxy, is that its magnetic field divided by charge on an electron times the electron density per unit area. Now, in our calculations we'll be performing our calculations at a given energy using this tight binding lattice. And, as you know, in this tight binding lattice you usually have this cosine dispersion relation which if you choose your parameters correctly can reflect this parabolic dispersion accurately within a certain range of energies. And, our calculation here is done at the given energy. And where do we choose this energy? Well at the electrochemical - wherever the Fermi level, or the electrochemical potential is. Because, at low temperatures the conductance or the transmission coefficient, these are all determined by the properties at that energy. So, how is this energy related to this electron density that appears in the semiclassical expression? Well, that's where you could use this result that we discussed back in Part A that is when in the context of the semiclassical theory which says that the electron density is related to the momentum, maximum momentum up to which the states are filled. And, of course, in semi-- in Part A we always talked in terms of momentum, in Part B now that we're doing wave mechanics, now that we're treating the electron as a wave, we talk in terms of k, and the two are related by p equals h-bar k. That's a particle property. That's a wave property. And then that k you can relate to the electron density using what we discussed in this lecture in Part A.  00:09:41,196 --> 00:09:45,586 

[Slide 5] Now, the question then is, how do we choose the parameters of our lattice so as to include a magnetic field? Now, electric fields are usually described in terms of their potential which we know how to include. Magnetic fields require what's called a vector potential and we haven't quite discussed how that is included and that's what I want to explain a little bit. Now first, let me just remind you how we normally choose the parameters if no vector potentials are involved. The idea is that given any tight binding parameters like this, the E k relationship is given by that expression and so if you stand at some point n, and sum over all the surrounding m's you come up with these five terms. This is what we did in Unit 1 of this course. And then, if you put it together you get this cosine dispersion relation. Now, if you want to mimic this parabolic dispersion relation then how does this relate to that? Well, the idea is, if the k is small, that is for small values of k, if you do a polynomial expansion of this cosine functions then you get something that looks like a parabola. So, this is the general approach and if we want to choose our tight binding parameters, epsilon and t to reflect the known dispersion relation for the material, you should choose your epsilons and t's so that this quantity is Ec and that quantity is h-bar squared over 2m. So, this is what we discussed in Unit 1. 

[Slide 6] Now, how do you include a scalar potential, for example? Well, scalar potential simply means if the potential is V you have to add minus qV to the energy. Well, that's easy enough if we put a minus qV we just adjust the epsilons accordingly. What about a vector potential? Well, if you want to include a vector potential then the p's get modified. So, px becomes px plus qAx. Py becomes py plus qAy. Now, how do we include that? And the other point I want to make is, that I guess instead of p, if you put kx, I guess h bar kx then you can see what's happening is where we used to have a kx we know have k plus qA over h-bar. And the way you could accomplish that, the way you could modify your original parameters to get that is by including these phase factors. You see, previously we had te to power ikxa. Now what we can do is we put t in the phase factor like this, exponential i phi x. Similarly this one we put exponential minus i phi x, i phi y, and e to the power minus i phi y, and then if use your standard trig identities and pull it together you'd get something looking like this. So which is just like what we had before but kxa has gotten replaced by kxa plus phi x. And, kya, by kya plus phi y. And now, if you do your polynomial expansion this would look just like that. So, the trick then is for a given vector potential you choose your phi x to be equal to qAxa over h pi. They can just see by looking at what is needed to make this look like that, and you get phi y. 

[Slide 7] Now, in terms of this lattice then, what it looks like is diagonal element is still epsilon. The connection to the right is te to the power i phi x, connection to the left is te to the power minus i phi x. Similar to i phi y and e to the power minus i phi y. So, what would your alphas and betas look like? Well, the alpha is supposed to be one of these columns so it still looks like before. Epsilons on the diagonal but the upper diagonal and the lower diagonal have picked up this phase factor. Is it still Hermitian? It is because you'll notice that this is the complex conjugate of that. So, this is still a Hermitian matrix and you have picked up this phi x and phi y. You have just picked up the phi y, not the phi x. The phi x goes in to the beta. If you look at the beta which is the connection between one column and another, then you have te to the power phi x on the diagonal. So could we use, could we treat this using the methods we discussed in lecture two of this unit? Answer is yes, if this was the case we could. Why? Because beta is still essentially a identity matrix. What that means is, if we transform basis it would still remain an identity so we could do a basis transformation and diagonalize this. And once everything's diagonal, you can use the methods of lecture two. But, the point is we still actually haven't included a magnetic field which we need to. Because, what you assumed here is a vector potential that is constant so just as a scalar potential which is constant of course doesn't give you electric field. Electric field means its varying spatially. Similarly, magnetic field requires that the vector potentials vary spatially. 

[Slide 8] In what way? Well, B is the curl of A and since our B's in the z direction, you know plus z, minus z what we're interested in is this expression Bz equals del Ay del Ax minus del Ax del y. So one way to represent a given magnetic field Bz is to assume that Ax is proportional to y. So that, that term is nonzero and choose the Ay to be zero. Now, if Ax is proportional to y then you'll notice that phase factor also needs to be proportional to y. So phi x is proportional to y, what that means is not every diagonal element is the same because as you go down this way, you're actually going down in y. So, you'll have one value of phi x here, another one there, another one there which means each diagonal element will be different. And, if each one is different, then when you transform basis it won't stay diagonal anymore. Now, another choice of vector potential because there are choices actually, not unique because any vector potential whose curl is equal to B is fine. So, another possible choice is you could put Ax equals zero and Ay is proportional to x. And, in that case, phi y would be changing with x what that means is it's this alpha that would be effected not the beta. So, beta would still be an identity matrix, essentially, but, alpha would be changing with x and so again, we'd have trouble diagonalizing it. Now, in general, the method we use, we use this choice and for this vector potential. And, what it means is that this beta has a phase factor that is changing as you go along the width. 

[Slide 9] Now, once you use these beta's and use these standard equations from the non-equilibrium Green function method, you'd automatically get this plot of the transverse resistance versus magnetic field that I mentioned earlier, you know with these amazing steps that come about. Of course, this still doesn't explain to you exactly why you get these amazing steps. Right, for that you have to study it more carefully and the notes describe some of the ideas and I won't go in to it in this lecture. Let me just mention one thing that allows you to study it more carefully. That is one of the very powerful things about the non-equilibrium green function method is, not only do you calculate the currents, the terminal currents, you can also look inside. You can look at the density of states inside for example. Now, if you do that, so that's this A, is the spectral function. By the way, I used A for vector potential in the last slide but this A is different. This is the usual A in the NEGF method. And, as you know, that's two pi times the density of state. So, if you look at its diagonal elements and what I'm plotting here is this axis is energy, this axis is the width of the conductor. And, I'm using a gray scale plot which means if you see a white spot that means there's a lot of density of states there, and black means there aren't any states. So, this striking thing you'll notice is how there's a density of states near one edge and some density around the other edge but the middle doesn't have any states. And, as you go higher in energy, you kind of see two states there, two states here, and then three and three, and these are what are called the edge states. And, this just comes out of the quantum transport method. Now, as I said, I won't be going too much in to the physics of this but the way people usually understand this is by saying that well, what happens in a magnetic field is the electrons want to go around in circles in a magnetic field, because of the force due to the magnetic field and they describe what are called the cyclotron orbits and this is what's called a cyclotron radius. And, as the magnetic fields get higher, the radius gets smaller so, these circles get tighter and tighter. And, if this radius is much smaller than the width then what can happen is an electron coming from here tends to be bent by the magnetic field, hit the wall, come back, again get bent, hit the wall, come back, and so the electron continues hugging this edge and never gets out in to the rest and these are how you often visualize these edge states. And, at the other edge, you have electrons going the other way but, once again, hugging this edge. So, what is remarkable about this high magnetic field region is that you have states that are going to the right which are all around this edge and states that are going to the left which are all around this edge and when you apply a potential, so usually they're all equally occupied but, when you apply a potential what happens is if mu one is the potential on this end the electrons that are coming from here they all populate this edge and so this edge tends to be in equilibrium with the left contact. Whereas here, they tend to be in equilibrium with the right contact and so it has a different electrochemical potential, and so there's this voltage difference which in this limit is equal to the applied potential difference. So, if you look back at our discussions of the quantum point contact you'll see that the physics are similar and what we had talked about in Part A of the course where we talked about the electrochemical potentials, semi classically in a ballistic conductor, it is a lot like that. But, the difference is that what you now have is a amazingly ballistic conductor. Why amazing? Because ballistic means when you're going in one direction you don't get turned around easily. Normal conductors, no matter how short you made them, there's always a chance of getting turned around. Here, there's a divided highway. So, anybody going to the right is on one side, anybody going to the left is on the other side, so it's extremely difficult to turn around. So, the result is the conductor route with amazing ballisticity, essentially. If you get launched in this lane, you just stay there forever, and that is why the corresponding resistances are so perfectly quantized. So, as I mentioned earlier, I won't really be doing justice to the physics of this instead what I really wanted to convey, more or less, how you could use our formulism, this NEGF method, to model all this and then look at things like density of states, electron density, inside the device so that you can study it and try 

[Slide 10] to understand it better on your own. Well, what we'll be doing next is something a little different and that is, again, we look at this general method that we used for calculating the self-energy and see how to connect it to a very well-known result that is widely used in many different contexts called this Fermi's Golden Rule. That's what we'll do in the seventh lecture.