 Okay. Good afternoon. I had assumed that you're starting at 4.30, so I thought I had a lot of time, but you got an extra 15 minutes of rest. So I decided to talk about the things that I got lots of questions yesterday. So I really changed my talk this morning to kind of address the questions that were asked. So people asked me for the notes, lecture notes, so I posted them on my website here. You can copy this. And the first two are there. So I'm going to focus mostly on research topics in our lab, but I realized that yesterday we finished too early and I didn't get to the point how we get the graphene Landau levels from using Anzager's relations. So we'll do that. We'll finish that, do density of state. And then what we're going to do is see how we can change the bandstructuring graphene with defects. What we do, we pluck out a single carbon atom and look around and see that there's huge, hugely different properties that occur near the vacancies. And one of them is what I'm going to start with. We can create an artificial atom that resembles atomic collapse in real atoms. I'm going to tell you exactly what that means. And the vacancy develops a magnetic moment which we can tune with either curvature of the substrate or with an electric field. We can tune an electrically tuned vacancy that is condo screen depending on the conditions. And the last part will be I'm going to be talking about flat bands and twisted bilayers and how they develop correlated phenomena. In particular I'm going to show you the latest results on superconductivity and insulating phase. So let's get started. So this is what I'm going to talk about. Starting with finishing up yesterday's session on density of state and Landau levels in graphene from Onsager. And then we're going to be moving on to how we do experiments, scanning, tunneling microscopy and spectroscopy. And then I'm going to bring examples of modifying the density of states with vacancy and with another substrate with a moiré pattern. Okay, so let's get started. So remember from yesterday graphene is sp2 hybridized carbon and there's three ingredients that go into its unusual band structure, two-dimensionality, honeycomb structure and the fact that we have identical atoms on the two sublattices. This gives rise to this band structure which consists of two Dirac cones that are related by time reversal symmetry to each other at the K and K prime points. At the Dirac point here, which is where the Fermi energy sits at charge neutrality, the conduction and valence band touch at one point. The dispersion at low energy is linear in momentum and the pre-factor is the Fermi velocity which is about 300 times slower than the speed of light. And the quasi-particle, as I've shown you yesterday, are chiral or helical and they have a, and there the pseudo-spin is either aligned or counter aligned with the local momentum. And that gives rise to very long mean free path and to the absence of back-scattering. In addition, it has a berry phase of pi which is very important for many things, but for one of the things which is important, as I'm going to show in a moment, is for the Landau level sequence. Now if you calculate the density of states, now this is a little exercise which I have on my next slide. But unless you ask me, I'm not going to show it. If you really want to see the derivation of this, come to me in the break. So it's a pretty straightforward derivation, but the bottom line is the density of state, the number of available states at a given energy in a unit area, per unit area is linear in energy and it vanishes at the Dirac point. And this is very important because when you sit at the Dirac point, the density of state is zero. There's no room for an electron to sit there, essentially. So we can, and because the density of state is so small and linear in energy, we can tune it with a gate. It's very easy to do it. You cannot do it in regular metals where the bands are flat. But here we can very easily tune it and we make a device like this. This is a typical device that one looks at. So it's silicon substrates, it's doped silicon and we have 300 nanometer of insulating silicon dioxide. Graphene sits on top, we have some electrodes on top. You apply a gate voltage and that allows you that basically it's like a capacitor, it's capacitive charging. But because the density of state is so small, a little bit of voltage already moves your Fermi energy along the band. Let me repeat the animation here. So I'm gating it, positive gate, I'm moving in the electron sector, negative gate, I'm moving in the whole sector. Completely smooth. I can really change the Fermi energy, you cannot do this in regular metals. Because the densities of states are much too high. And this is how much it takes. One volt buys you 7 times 10 to the 10th carrier per centimeter squared. That's a lot. So you can really move the Fermi energy. And we use that in all our experiments. Okay, here's the derivation, I'm going to skip it. Okay. Okay, so now let's go back to Anzager relation. Yesterday we derived the Anzager relation and that tells us that the area of the n-cycleton orbit in reciprocal space multiplied by the magnetic length squared is 2 pi times an integer n. And these are the corrections that come from quantum mechanics. One half comes from the zero point energy. And this gamma is the buried phase. Now for graphene, the buried phase is pi. So look what's happening. A gamma over 2 pi is a half. So the buried phase completely swallows the zero point energy. No more zero point energy. Okay, so the bottom line is that now it's 2 pi n. The Anzager relation on the right-hand side is 2 pi n. N is the number of the cycloton orbit, the order of the cycloton orbit. So if you take a circular orbit, you get that pi k n squared. Lb squared is 2 pi n. And you can immediately derive what k n is. It's just proportional to squared of 2n. And then plug it in, in the dispersion, and you immediately get the Landau level sequence. And notice what it looks like. So this is h bar vf over the magnetic length. You can rewrite vf over the magnetic length as a frequency. So it's an effective cycloton frequency. Remember, Lb is h bar over eb. It's interesting proportion of the squared of the field. So the bottom line is that the sequence of Landau levels in graphene goes like square root of the Landau level index and like square root of the magnetic field. It's huge contrast with the Landau levels in non-rativistic electrons. And notice, most importantly, it has a level at zero energy. You don't see that very much. Because of this Berry phase, it swallowed the zero point energy. So we do have a level at zero energy. Okay, so I derive the energy sequence, the Landau energy sequence, using the Onsager relation. Now you can do the same thing using quantum mechanics. You can use the Dirac-Weil equation here. So this is, we have a 2 by 2 Hamiltonian. You go to the, you write down the, instead of the, in the momentum, you write the canonical momentum here or mechanical momentum. And then you chug along, solving it. It can be solved pretty straightforwardly. You bring it to a harmonic oscillator form. It's a little bit more complicated. But at the end of the day, what you get is the sequence of Landau levels, which is exactly, which is identical to the Landau levels we got from the Onsager relation. Okay? So the Landau level sequence looks like this. I'm drawing it here. So the band structure has broken up into these Landau levels. There is one sitting right at zero here. And that one is very special because it's a consequence of this Berry phase. It's a consequence of the chirality of these quasi-particles. And it's very special in many ways because, remember, the degeneracy of each level, of each level is the same. It's just equal to the number of flux lines that thread your sample, multiplied by the internal degree of freedom, which in this case is 4, 2 for spin, 2 for valley. Okay? So this is a degeneracy. This is how many electrons you can put in into each Landau level. So but look, but notice here that on the negative energy side, it's holds, excitation, where on the positive energy side we have electron excitation. But what's going on at n equals zero? Look what's going on. We have half electron, half holds. And they all are sitting at the same energy. So that is very promising. There's lots of things that can happen there. If you put your Fermi energy there, you can have, for example, exit on condensation. There are lots of interesting things that can happen that there is predicted. Maybe we've seen it. Maybe we haven't. But it's a promising place to go and look. Okay. So now let me compare this to the Landau levels in a non-relativistic system because that's pretty instructive. So here is, we have a standard two-dimensional electron system in a semiconductor Gali-Marsen. Sorry. Yeah? What is the width of the Lorentzian? So this is a very good question. What is the width of the Lorentzian? It is determined, I mean, if you go to zero temperature, if you have no random potential, it is still finite width. And that is electron-electron interactions. We'll determine that, yes. And we have measured that. I'm not going to talk about that. But it's definitely to do with electron-electron interaction. If there were no interaction, it would be a delta function, right? And I don't remember offhand what it is. And the bandwidth depends on which Landau level you're in. And it's of the order of one or two millivolts. But I vaguely remember this. Okay. So here we have the non-rativistic electrons. So the band structure is parabolic because we have a mass. Energy goes like p squared. But there's absolutely no reason in these materials to have electron-holes symmetry. So the mass is the inverse of this curvature here. So you have no reason for them to be the same for the electrons and holes. The density of states is flat. And again, because they're not symmetric, you have different density of state. Most importantly, you have a gap. These are electrons and semiconductors. And when you do Landau levels, the sequence is just a harmonic oscillator. It's linear in the Landau level index N and linear in magnetic field. And there is a one-half here. So you have no level at zero. It's one-half offset. They're all equally spaced. Another important difference is for a given material, you either have electron doping or hole doping. You can never have both. Another important difference is the Fermi energy sits wherever it wants to sit. You cannot change it. It's determined by the chemistry because the density of state is huge. You cannot do anything to it. And let's contrast that with graphene. The electron holds symmetry. That is more or less true. This is almost true. The dispersion is linear in momentum, which means these are ultralativistic quasi-particles. Density of state is linear. Vanishes at the Dirac point here, and that's why we can gate it. The Landau level sequence goes like square root of the index and the magnetic field. The higher you go in energy, the tighter they go. The higher you go in field, the sparser they are. And we can walk on here with the gate. We can actually probe every possible state here by gating the system. Very, very versatile. Okay, so let me summarize the... So there is a Landau level at zero energy in graphene, and that is a reflection of the baryphase of pi. It's electron holds symmetric, more or less. Energy goes like squared of n squared of b. And the internal degeneracy is four, four spin and valley. To contrast that with non-relativistic electrons, there is a gap at zero energy, and that is because the baryphase is zero. It's a trivial system. Equally spaced levels. The levels are linear in magnetic field, and the degeneracy is two. Internal degeneracy is two, because there's just spin there. Okay, and these are some numbers. So actually these are interesting numbers. The spacing between the first zero and first Landau level is about 33 millivolt times square root of b. So if you're at one tesla, the spacing is 33 millivolt, and that is roughly room temperature, which means that in principle, not only in principle, also in practice, you can see the quantum Hall effect at room temperature in graphene. Quantum Hall effect, which I didn't talk about, is when you put your Landau, we offer me energy in the gap between two Landau levels. That's when you get these edge states that are perfect conductors. And there's a lot of room to put, because of this huge number here, it's very easy to see integer quantum Hall effect at room temperature. People have done that. You can see fractional quantum Hall effect at very reasonable temperatures. We've seen it at 12 tesla. We've already seen it at 4 Kelvin. And in normal standard two-decks, you have to go to the magnet lab to 30 tesla to see the fractional quantum Hall effect, yes? Is there a way to tilt it? Yeah, to make it asymmetric, sure, yes. What? You can do these things by stretching. You know, you change the hopping parameters. But you will not lose the Dirac point, but you will lose... I mean, there's several ways of doing this. For example, if you put it on an externally... You put it on stripes. So you make a periodic potential that's striped. You're going to get the velocity that is along the stripe is dramatically reduced perpendicular to the stripe. It remains the same. This is equivalent to having a tilt, right? Because the firm velocity is the slope of the cone. Yeah, you can make it... If you make it an isotropic... Now, what do you mean by tilting? When you tilt it, you have different slopes in different directions. That's exactly what I was talking about, right? No, no, but this is... The story is very changes. This was working for the special condition that I gave you, the three conditions. If you add stuff like asymmetry and all sorts of things, things become murky. No, you cannot just think in terms of the Dirac equation anymore. You can do a perturbation theory. It depends how strong your perturbation is, yes. Otherwise, you have to start from scratch. Okay, so how do we popular lambda level? Okay, so let's say that I have a two-dimensional sample, a perpendicular magnetic field. If I want to throw in electrons, I have to remember that the wave function has to be single-valued, and then it's a very nice way to interpret that you break up the magnetic field into flux lines. So instead of this having one magnetic field, I'm breaking it up, the same flux, into this array of flux lines. Each one takes up a circle whose radius is the magnetic length, 25 nanometers times squared of B. Each one carries a flux, a fundamental unit of flux. Each flux line corresponds to an electronic state, so you can put one electron there, multiplied by the degeneracy, so you can put two electrons for two spins, or four electrons for two spin and two valleys. So each of these flux lines accommodates four electrons in graphene. Okay, so the orbital degeneracy gives you the number of flux lines, the internal degeneracy gives you spin and valley here, so it's four. So this is how many we have. So now I'm going, I'm turning on my gate and I'm going to throw in electrons. Where do they go? So each one can take up, or each four take up a spot, and that's it. I finished filling up the slanted level. There's no more room for the next electron. Next electron has nowhere to go. So what does it do? It goes to the next Landau level. Now we have a node in the wave function. And again I can fill it up, and the distance here is h bar omega c, and now I can fill it again. And then my next electron is going to have to go to the next Landau level, which has three nodes in it, and so on and so forth. So this is how we fill, how we gate, how we fill the Landau levels. So the number of filled Landau levels is equal to the total number of electrons divided by the total degeneracy. And the number of electrons per flux sign is the total number of electrons divided by the orbital degeneracy. Okay, so here is a summary of this stuff. Okay, so the quantum unit of flux is a very small number here. It's expressed in fundamental units, a Planck's constant and electron charge. The flux enclosed by a cyclotron orbit has to be an integer multiple of the fundamental unit of flux. Onsager relation is the k-space area of the cyclotron orbit times magnetic length squared is 2 pi times the index of the orbit plus 1 half for zero point energy minus the buried phase in units of 2 pi. And that allows us to find the Landau level sequence for all sorts of systems if we know the dispersion. So we've shown that for non-relativistic electrons we get the sequence which looks like a harmonic oscillator. And I mixed it two up. This should be two here and this should be four. And for graphene we got the sequence of Landau level at zero energy. The sequence goes like square root of field and Landau level index. And the degeneracy is internal degeneracy times orbital degeneracy. The two should go here and the four should go here. Okay, no, no, no. This is area in k-space. Right, this is area in k-space. So it's 1 over length squared. So multiply by length squared it's a number, right? Okay, yes? I have done it yesterday. Look at the notes if not you can come down and ask me in the break. Okay? I think I've done it yesterday. If not, we'll talk later. I can't do it on this spot here. Okay, any other questions? Yes? Can you speak up? No, and I remember I didn't do it yesterday. It's in the notes that when you solve for Landau levels when you solve the harmonic oscillator Hamiltonian, that's where it comes out because you write your lengths you want to write something in dimensions unit and the natural unit that comes out is the magnetic length. I think I've done it. If not, I can put in my notes. But if not come down and I'll show you. Yes? So disorder doesn't play any role to broaden the Landau levels for graphene. You said only it is electron-electron repulsion That's a very good question. What's the role of disorder? And I have a whole lecture about that. It's huge. So, okay, the bottom line is the following. As long as the magnetic length is small compared to the length scale of the disorder they don't care. So as long as you can fit this line in a region that is roughly flat you're going to see at least one or two Landau levels. If the disorder is on the order if the disorder so there's two energy scales there's the energy scale of the disorder, the fluctuation of the potential if that is large comparable or large to the Landau level spacing you're going to mess up your Landau levels. Then there is the length scale that's determined by magnetic length. If the orbit encompasses some of those things you're going to mess up your Landau levels. But otherwise so you know roughly the conditions for this to work. These are also the conditions for observing the quantum whole effect. Any other questions? Okay, so we're done now with yesterday's lecture. So let's move to today's. So now I'm going to tell you just a little bit about our technique. I know most of you are theorists so this is going to be fascinating for you. Okay, scanning tunneling microscope. It's a very very simple device. Actually this is a high school project you can actually build your own STM for about $50. Okay, so the idea is the following. Take a sharp metallic tip and bring it close to a surface. Put in a battery and an ammeter to measure current. And you bring it closer and closer until you start seeing a current here. And that's when you stop. Now you have to make sure that you don't actually touch the surface. So that's how you do it. Now the current that flows here is given by this expression here. It depends on two parameters on the density of the state in your substrate. There's a density of tip which I'm taking out as a constant. We are assuming a flat band for the tip. And this is multiplied. This comes from a matrix element but the bottom line is that it has an exponential dependent on the height here because this is after all it's tunneling through a barrier. So it has an exponential dependence on the height here. Okay, so this is the current is given by these two quantities and then you have to integrate from zero from your Fermi energy to the bias voltage to get the current. So for example, if I have a bias voltage so this is the tip band structure and this is my sample band structure you're going to tunnel from the tip into available states in your sample. So you can actually move this whole thing by applying a bias. So if the bias is zero, this whole structure is going to move up. If it's positive it's moved down, negative it moves up. So you have electrons coming from the Fermi energy. This is not true here. It's mainly from the Fermi energy and then you can actually see each one of these peaks showing up in your spectrum. Okay, so there's two measurements that we can do with STM. Just basically what we do and that's about it. So the first measurement is called topography. What we do, we keep that we keep the height of the, we make sure that we keep the current constant, sorry. We keep the current constant by we have a feedback. So if the, because of this exponential dependent I can put a motor to the surface is close to the tip. I can lift the tip if it's, because the current goes down and vice versa. So I have a motor, a feedback that actually follows the topography of the sample because of this exponential dependent is extremely sensitive. So what you do, you raster the sample and you make sure the current is constant and you then measure or you map the amount of how much you need to change the height of the tip. And that gives you a picture like this and this is actual graphene. Every corner here is a is a graphene atom. Now the other measurement that you can do is you can go into this integral here and you can measure the density of state. The way to do this you take the derivative of the current with respect to the bias voltage that gives you the density of state and you make sure that the height doesn't change. You sit in a fixed point, fixed height and that gives you a spectroscopy and then you can get the density of state as a function of energy by changing the bias. So this is for example a graphene with a lot of disorder. This is graphene on silicon dioxide and the density of state this is supposed to be the density of state. It doesn't look anything like what I showed you, like the linear nicely linear dependence that vanishes of the Dirac point. So now as you have actually brought me into this topic what you see in the density of state when you look at graphene is hugely influenced by the environment, in particular the substrate. So if you have a crappy substrate with a lot of random potential with charging, you're not going to see anything that looks like what you expect from theory. Now the best kind of substrate to use for graphene can anybody guess what it is? The smoothest, flattest most beautiful substrate? What? Suspended. Excellent, very good point. So we tried that. The problem is that when you suspend graphene it's a membrane. It's a very stiff membrane and we come with your tip, it has a normal mode it has drum modes and it shakes like crazy. So there's a lot of noise in your signal and there are ways to get around it but absolutely best would be suspended but it's tough. The next best something that is kind of similar, the most similar is graphite. They put graphene on graphite and you get the best possible results. Okay, this is our STM, it's all home built because we are cheap, we can't spend so everything was home built here so this it goes down to 2 Kelvin, we have a magnet this was not home built, 15 Tesla and we have a course motor that allows us to scan a millimeter but then we can zoom into subatomic resolution. Okay, so we can measure topography, we can do spectroscopy density of state at zero field, we can apply magnetic field and look at Landau levels. And we use graphite as a substrate and this is a topography which I showed you before, absolutely perfect you can go like microns and not find a single defect. Now this is density of state it's beautiful linear density of state in the function of energy here is the Dirac point the Dirac point is the minimum here, the Fermi energy in STM is always zero so in this particular sample we have a little bit of hole doping, Fermi energy is below the Dirac point and it's very nice that you can separate the physics of the two and finally we apply magnetic field and look at these beautiful Landau levels. N equals zero and equal one, two, three and we're going up to I think seven on both sides. Now what information can we get out so here in a zero field we can find the local doping and the changes, if you have a rough substrate this fluctuates a lot and we can find the electron phonon coupling, these are these two wings here that we find I'm not going to discuss, there's a con anomaly there that you can see in the Landau level sequence you fit this and it fits beautifully you can extract the Fermi velocity and you can measure the local Fermi velocity from point to point from that we extract the quasi particle lifetime with a few millivolts and you can extract the coupling strength to the substrate so it's a very useful and useful tool yes excellent question excellent question I was wondering if someone is going to wake up to this okay I mean this is like an oxymoron right graphene on graphite so normally you saw the video yesterday what we do we peel we take Scotch tape and we peel off a layer of graphene and we separate that's beautiful it turns out that if you look at what's left behind after you peel the graphite quite often you will find a layer of graphene sitting on top that is decoupled electronically decoupled so the distance in equilibrium in graphite the distance between layers it's about 0.3 nanometers okay now you find layer that is 0.32 nanometers it's like 10% off and because the coupling it depends on tunneling it's completely electronically decoupled so it kind of floats on it and there's no tunneling down into the graphite so that's what we are looking at it's a decoupled layer so in a sense it kind of by accident we found it but these are the best results that we got okay so now I'm going to the next topic so far we've discussed perfect graphene and now the question is okay let's mess up the graphene a little bit let's put in defects so what we are going to do is we are going to pluck out a carbon atom and see what happens okay so let's talk about magnetism in a moment you are going to see why so magnetic materials typically are characterized by having a localized electrons in partially filled inner D or F shells so most of the magnetic materials have D or F shells now carbon doesn't have D or F it just has pi shells has p-shell has a partially filled p-shell now partially filled p-shell horns rule the carbon atom is a magnetic however when you make a compound out of carbon, graphite or any kind of ST2 compound nanotubes graphene graphite it's absolutely non-magnetic in fact it is quite diamagnetic it's one of the most diamagnetic materials we know you can actually levitate a magnet with graphite so it's one of the most diamagnetic materials which we know and the question is why and the answer was given a long time ago by Elliot Leib who is who told us that if you have a repulsive Hubbard model with the bipartite lattice I'm going to translate this in the half filled band the spin of the ground state with na and nb populated site is one half times the difference in population now what does bipartite mean bipartite means exactly this lattice where you can take two colors red and green and you color your lattice so that every red atom has only green atoms around it and every green atom has only red ones about it and this is true for all the square lattices for example three dimensions or two dimensions it's not true of course for triangular lattices so if you have a bipartite lattice with a Hubbard model and it's and it's half filled as graphene would be at charge neutrality so if you have na atoms on sub lattice a and nb sub lattice b the spin of the whole sample is very simple given by one half times the difference in population so if you take if you take perfect graphene pristine graphene where na is usually equal to nb you must have spin equals to zero it's going to be non-magnetic so in the at the beginning of this millennium in the year 2000 people started reporting having found magnetism in graphite in bucky bowls and all sort of samples that came from meteors and there was huge controversies so these people were told that they're either lying or they don't know what they're doing one of the papers that was published by Tatiana Makarova and collaborators in 2000 in nature she was forced to retract the paper because people said you know it's not really graphite that's magnetic it's you have some impurities you're not careful of how you handle your tweezers and it was retracted so this is how and people really felt very strongly about it because they didn't read the small print in leap theorem because okay perfect graphite cannot be magnetic however all you need to do is to unbalance it so you can do that by making holes by plucking out a carbon atom or you can do that by cutting the edges you have zigzag edges it'll be one side is going to have spin up the other side is going to have spin down actually it's kind of neat go google diamagnetic materials and check which one of these satisfy leap theorem it's kind of fun many of them do actually okay so people woke up to the fact to leap theorem and then there's lots of theoretical papers that started to show okay if you have defect if you have vacancies you can make graphite magnetic okay so let's make let's try to make graphite magnetic we're going to pluck out one carbon atom, make a vacancy and see what that does to the band structure or into the density of state so here we have perfect graphite this is the density of state of perfect graphite linear density of state vanishes at the Dirac point and now I'm going to pluck out one carbon atom what happens so you pluck out one carbon atom you've taken away four electrons right you've taken three away three sigma electrons that bind to the neighbors and one pi electron in the pi orbital so here we removed it and what we have is going to have degenerate bands here the sigma bands the sigma bands from the bonds that were left behind the dangling bonds way down here and the pi bond you have a localized state that sits at the Dirac point that's from the missing electron in the pi band that would be your lib electron because that electron is missing its brother so it upset the balance these electrons lib didn't talk about but then there is a crystal distortion yantela distortion and things get a little bit more complicated so two of these bonds hybridize so this is from the yantela distortion and the third one is you have a third orbital from this electron that stays hanging alone so this one has a spin of a half so you have a dangling bond that carries a spin of a half okay? now what about the missing electron in the pi band so the missing electron in the pi there is a kind of almost delocalized state here it's sitting around the fermi energy and that also has a spin of a half okay? so that gives you the missing pi electron gives you a quasi-localized state on the other sublattice so if I plucked out a carbon from the A sublattice and this carries not a full bore magneton somewhere between a half and 0.7 because of the spreading out and the interaction with the other electrons so now the density of state what happens this is calculated density of state and I don't know why all these wiggles they probably didn't do enough actually I don't know why these wiggles but what you have here this pi electron state gives you a peak at the density of state and that peak is sitting at the Dirac point and it is a manifestation of the broken ab symmetry and the more you break it the more it's going to move actually so this is a manifestation of broken symmetry when you charge it this peak is going to move okay so vacancy properties it has a magnetic moment and looking at the vacancy gives us the opportunity a unique opportunity to study the interaction of ultra relativistic electrons with magnetic moments can't do it in any other system now there is another part to this story the vacancy carries a charge and I'm not going to show you the calculation but I'm going to show you a little bit later it carries a charge plus one electron that is spread out on this vacancy here so this was this was done by DFT calculations so that gives us the opportunity to study the interaction of ultra relativistic electrons with a point charge lots of good physics that one can do here okay so I'm going to talk now about interaction with charge okay so remember a Klein tunneling I told you that it's impossible to confine the electrons in graphene with electrostatic potential and this was the cartoon that I showed so the electron turns into a hole under the barrier and turns back into an electron and because there is no backscattering okay so basically we cannot confine the electrons not in a simple way anyway so the question is can we use a very strong charge a local point charge to actually confine the electrons so that's the question can you use a point charge to control the carriers a point charge which now we know we can create by plucking out a carbon atom okay so here I need to talk to remind you a little bit about what happens when you have an electron interacting with a proton or with a charge z so we start with classical physics so I have an electron charge z the non relativistic classical case you have kinetic energy p squared over 2m and potential energy z squared over r and we know that in classical physics the electron is just going to fall into the into the nucleus there's absolutely nothing to stop it from just going in there so there's no bound state solutions now let's go to not quite quantum mechanics I'm going to go to semi classical approximation and when you do a semi classical approximation all you have to do is the trick is replace p by h bar over r and there you are so I'm replacing the kinetic energy by h bar over r and now this doesn't look anything like what I showed you because I'm using dimensionless unit I'm going to show you in a moment what they are basically now the energy is mc squared I like that because that's easy to remember times rho is a length and let me show you what I've done here so I'm expressing the distance in units of the Compton wavelength so this is a characteristic length scale in the problem so that I'm calling that rho I'm expressing the charge here in terms of the Coulomb coupling constant which is the charge of the fine structure constant now the fine structure constant is the measure of the strength of the Coulomb interaction compared to the zero point energy so the fine structure constant is a tiny, tiny number in our world it's only 1 over 137 and we are lucky for that so I'm everything here is expressed in dimensionless units and fundamental constant now you see the kinetic energy here rho squared so it grows as you come to the origin here so the potential energy here is negative it goes like 1 over rho so basically you have a minimum and the minimum happens exactly at the Bohr radius which is the Compton length simply the Compton length divided by the charge by the Coulomb coupling constant so that's our Bohr radius and then you have the Rittberg scale which is this minimum here it is one half mc squared beta squared I really like this way of writing it because I can never remember where e and h bar and all that goes here just mc squared times the charge squared so that's the Rittberg so now you have a nice bound state and you have a Rittberg gear so you need the quantum mechanics in order to have stable atoms as we all know so now this all this works beautifully when I'm starting from classical physics from non relativistic physics and when beta is very small and my Rittberg is very small I'm sitting out here everything works nice because the energy scale here is much smaller than the rest energy of the electron so I don't need to take that into account it's completely outside a different scale I don't need to take into account relativistic effect now let's go towards heavier and heavier atoms you go down the periodic table z becomes larger and larger er becomes larger and larger and what will happen is the Bohr radius becomes smaller and smaller you get closer and closer to the nucleus and eventually eventually your energy your Rittberg becomes comparable to the zero the rest energy of the electron so our starting point is completely wrong we have to start from relativistic physics because here soon enough this is going to fall into the Dirac continuum can't happen so we need to use relativity to describe this problem so to use the relativity all you do is basically add the rest energy of the electron mc squared so let's see how that changes the story so now I'm going to instead of writing p squared over 2m I'm going to replace that by the relativistic expression here so this is what it was without relativity when I add relativity all that changed here I have a 1 here which comes from the rest mass okay and how does the Bohr radius change there's a minus so there's a 1 squared of 1 minus beta squared here which comes into the Rittberg as well so you might think okay so why should I care about this well in fact it's immediately obvious that you should care about this because what happens when beta becomes 1 or larger than 1 there's no solutions so the Dirac note is that a long time ago said this when you have a charge of 137 so you have an atomic number of 137 all hell breaks loose the solutions are no solutions basically they are imaginary so there's a bunch of from a run-through and Sordowski and somebody else whose name I didn't write down Zeldovich they fix this they fix this by regularizing the potential they say okay the electron comes very close to the nucleus but it can't get in so let's put a cut-off which is the size of the nucleus and that doesn't doesn't give you imaginary solutions it gives you so then the solutions survive until 170 but this is what happens with the Rittberg series as a function of Z so here this is where the normal periodic table lives this is our 1S state this is a 2S state and you see when you start having heavier and heavier nuclei the 1S all the level starts dropping down and that's okay until they drop into the rock continuum and that's when you have the so-called atomic collapse so at about 170 in principle the periodic table should end and what happens there the electron folds in there and what happens the electron folds in the nucleus it merges with the proton and becomes a neutron and you emit a positron so and that goes to infinity now our friends in nuclear physics have been doing colliding heavy ions for decades trying to look for the spontaneous generation of positrons when you have atomic collapse and that never worked because you don't have such heavy nuclei okay so it's very interesting to see if you add relativity to quantum mechanics the atom again collapses at a critical charge okay and I don't know if you know that but this is the last element in our periodic table it's called ununachtium 118 I think it now has a name actual name I don't know what it is okay okay so now what happens in in graphene so here we have a mass we have a finite mass so we have a mass gap in the spectrum and I want to take this to graphene which means I have to take the mass to zero I have to go to two dimensions and I have to switch from the speed of light to the Fermi velocity and when you do all that and you do all that through these relationship which I used before you get the dispersion that looks like this so the energy is h bar vf over r and there's one half minus beta here one half comes from the angular momentum so this is very interesting notice that there is no scale here no characters this is a scale free problem whereas before we had the Bohr radius we had the minimum this thing doesn't have a minimum it either goes to plus infinity or it goes to minus infinity no minimum this is a typical scale free problem there are a few of those in nature this is called an FMF series it okay so now let's put it here let's put graphene here so e is h bar vf over r times one half minus beta beta here is the Coulomb coupling constant for graphene so z is the charge but the fine structure constant now has to be replaced the effective fine structure constant for graphene has to be renormalized by the Fermi velocity so you have to replace the speed of light by the Fermi velocity and something amazing happens when you do that you also need the dielectric constant here the effective fine structure constant for graphene is 2 it's huge so if we were made out of graphene we would not exist because there wouldn't be any periodic table okay so it's a scale free problem the critical charge now is 1 so you should already see collapse at 1 so now you see where I'm going if I can put a charge of 1 on graphene maybe I can see the effect of atomic collapse okay so now let's take small charge beta is less than the critical what we have the total energy function goes like this remember this is radius so so the electron just goes through when it comes in from infinity it just goes through there it's not it's basically not doing anything it's not seeing it oops we have clearly we have no bound state this is basically client tunneling okay just go through it doesn't see it it doesn't get bound now what happens on the other side of this transition this is a quantum phase transition because it's not driven by temperature it's driven by a parameter in the problem so what happens on the other side of the transition now the total energy is negative and we get quasi bound states and and we get these are kind of almost chaotic orbits and this is a series that resembles the Rydberg series but not quite because this is a scale free problem so the sequence of energy is linear in log of the energy of the level index it's linear in log of the level index because it's a scale free problem so this is the energy quasi bound energy sequence let me show you one more thing here what's going on here so this is our one over R potential I'm writing I'm showing both sides and this is the so when you are when you're these the rock points should be actually riding on top of the potential I took this from this paper here so this is wrong so this should be here and this one should be somewhere down here but the bottom line is that the type of carriers is reversed when the Fermi energy crosses the rock point as I shown you in the Klein tunneling example so here we have holes electrons holes so sign of carriers is reversed and so you have two radii one radius of the orbit outside here and another one radius of the orbit inside one for the orbit outside so you have a centrifugal barrier in between them at R star okay so so basically and things depend on angular momentum and so on so one thing that I want you to notice is that the Klein tunneling couples electron like state to hole like state so these are not really bound state they will be quasi bound states so they extend a little bit outside and this is a memory of the positron creation okay so that's the manifestation of the positron creation in atomic collapse is the fact that you have coupling you have a broader this is a quasi bound state and it kind of leaks out to infinity so what is the experimental signature of atomic collapse in graphene this is a simulation by my collaborators here so when so this is the energy as a function of beta when you have when beta is less than a half you have unbound state just Klein tunneling and as beta is above a half you're going to start having these collapse states and this looks like an artificial atom but this is not quite because the sequence is not a Rydberg series but they the level start dropping down into the into the negative energy sector just to compare it to the atoms this mass gap here which you have in atoms is absent in graphene okay and the collapse here but still the collapse happens on this side here when you go this is the positron production here we have the when you go above the supercritical charge you have um you have quasi bound state so in the 2013 my chromies group at Berkeley what they did is okay they try to try to see this collapse it's very difficult to charge graphene it just everything becomes screened so they piled up five calcium dimers one on top of each other it's nobody ever reproduced this this is a total feat because it's like a bomb you know they're charged so they managed to bring them all together five dimers and they saw in the density of states this is the Dirac point energy here so as they uh as they come closer and closer to this complex they saw a bound state just below the Dirac energy and that they attribute to atomic collapsing graphene and where where they they were here they were almost you know almost at the boundary here okay so then we ask can we do better let's try to use a vacancy where we don't have to pile up charge the vacancy is naturally charged uh so can one host a stable charging graphene the answer is yes now the vacancy has two has two has two ground state and the meta stable state so the ground state has a charge of one where the graphene is flat the meta stable is off by about 50 millivolts up here you have the apical atom kind of sticking out now when we prepare our vacancies as you will see we always prepare them this state so the vacancy is not charged it has a spin but it has no charge and this is how we how we prepare them um so this is our STM this is pristine graphene we bombard this sample with uh with low energy helium ion about 100 electron volts and we create these spots here and when you zoom in you see this triangular structure which is the signature of a single atom vacancy am I running out of time here okay let me go for for at most five more minutes okay uh okay so what's going on so remember I talked about the zero mode which has to do with the unpaired electron in the in the pi band and it gives you a state here however when you charge the vacancy the zero mode very rapidly moves down and the amount by which it moves down is a measure of the charge of the vacancy so that's a very useful tool to have so here is um and then if you continue charging it then you should start seeing these the quasi bound state this thing drop to out of sight uh so this is a spectrum of a vacancy that was just created here is the zero mode and you see it sits right at zero it hasn't moved at all that means the vacancy has zero charge almost zero charge okay so what do we do next so we were really we didn't know about this metastable state so we were very puzzled theorists tell us that it should be charged we see nothing so uh yuhan my postdoc decided to do something very brave and she started banging on that vacancy so what she did she applied voltage pulses to the vacancy site and every pulse this zero mode moved down which means that the thing started getting charged and uh so if we plot these these points for every pulse they just ride on top of the and then we can we can calculate what the charge is for with every pulse from the position of the speak and then she continued to pulse it and then there's a new sequence that appeared here that you can map on to this quasi bound states uh so uh so you charge it this is atomic collapse this is the atomic collapse in graphene uh and now uh if you look it's very interesting now so you say oh these are just peaks that are moving down but they're qualitatively different if you do topography the zero mode is very tightly localized within two nanometers of the vacancies whereas the collapse mode are really spread out as you should expect for a quasi bound state uh and last thing I'm going to show you is here we're going to look at the highest charge which is about beta 1.09 and we're going to do maps energy maps so we are moving uh as a function what I'm plotting here is the density of state as a function of position as you move out from the center and this is uncharged now for an energy of uh at the 2s energy here it looks like this so this is where the electric where the function sits so you can measure how big it is about 15 nanometers then I'm moving to the energy of the 1s prime 1s prime is a satellite peak which has to do with the fact that you have a broken symmetry uh with you wouldn't have it was a charge that is not a vacancy without breaking the symmetry and then there is the 1s and 1s has a halo and this is the you know this is the positron if you like this is the leaking out of the wave function on the whole side so there we have a captured electron state with its halo so basically this gives us a mechanism to confine electrons in graphene so you completely change the Hamiltonian basically and you can trap them and this is really the last uh and you can untrap them so if you apply a gate voltage so this is I'm changing the carrier density so this is beta as a function of carrier density so as I approach the Dirac point there is less and less screening so the charge increases but as soon as I'm on the other side boom the whole thing goes away no more bound state because the charge is negative so basically by moving with a little bit of voltage across here we can trap them and let them go as as we please at the vacancies and uh so this is a summary of part of this part so let me read it to you so in quantum electrodynamics the fine structure constant measure the strength of the electromagnetic interaction and it is very very weak and that's why we have such a long periodic table uh in graphene the fine structure constant is huge it's about one and therefore we can see atomic collapse and probably other things as well and we can ask many more questions okay break okay what collapse like uh yeah okay okay welcome back so now we're going to the last part uh and we're going to talk about the magnetic moment in in graphene uh and in particular about the condo effect in graphene okay so if you have a impurity a magnetic impurity in a metal in a standard metal what happens you get um the and you at a high temperature it's nothing happens to it there's no screening the magnetic moment is bare however in normal metals when you reduce the temperature there is a temperature below which the conduction electron forms a nice screening cloud here that completely hide the magnetic moment and what are the and this is called condo screening what are the conditions for this screening to happen you have to have anti ferromagnetic coupling between the conduction band and the impurity this is usually the case um and you have to have a finite density of state of the fermi energy and if that if these conditions are satisfied you this critical temperature below which the magnetic moment is just going to disappear uh is exponential in one over the density of state and the coupling constant in the um in the sample between the impurity and the conduction electron so all you need is to have a finite density of state at the fermi energy and you need a positive coupling constant to have screening of magnetic moments so if you look at an insulator nor in normal metals this is practically always the case there's always some temperature below which your magnetic moments disappear if you look at insulators uh the density of state of the fermi energy is zero you never the magnetic moment survive to all temperatures so the question to ask is graphene is kind of in between a metal and an insulator the density of state at the Dirac point if you are charge neutrality is zero uh however if you just move epsilon away it's finite so what's going on there so that's the question that people started asking in the 90s with the advent of high temperature superconductors that have Dirac points in their band structure and this is what they came up with there were lots and lots of theoretical works uh that if you have a pseudo gap system where the density of state is not flat but it's a power law of the energy things become kind of interesting uh so if the power is one as in graphene and in high tc superconductor it turns out that if you sit at charge neutrality here uh condo screening can happen but only above a critical coupling in normal metal a condo screening could happen at any coupling in if you have pseudo gap system there's a critical coupling above which you have screening below which the system is on screen so if you move across here this is a quantum phase transition if you change your coupling constant now what happens if you have a finite doping so you move up on the chemical potential axis here uh so you have a region so below the critical coupling you have a region where it's on screen and if you have sufficient electrons there it's going to get charged even though you're below the critical this is this is a critical regime here fly so uh if you're able to if you're sitting somewhere here in a sample uh you can actually tune your magnetic moment with an electric field with doping so if I have a sample that sits here uh I can see the magnetic moment I apply and dope it the magnetic moment is going to disappear and this is kind of appealing because in principle you can have electrical control of magnetism no no no there's a finite region here and we know exactly exactly how it behaves yeah it's not any finite mu there's a yeah yeah yeah but this is still this is a this regime this yellow regime it's a kind of a critical regime and the physics is totally different uh look at all these guys this is all discussed there uh and I can tell you tk well okay tk goes is uh exponential in in one side it's it's uh it's uh it goes like square of the so you're right it's very small but it goes like square of the dope of mu and on the other side it goes like some exponent of mu so it's not strictly speaking not zero but the value yeah theoretically you're absolutely right but for any realistic experimental situation you have this so so it's true at zero temperature you just have a line here true now experiments don't work at zero temperature and these scales are exponential so when we're at four Kelvin I have a finite regime here yes um okay so what how what are the experiment experimental signature of condo screening this was found a long time ago so you look at the resistivity in a metal that has magnetic moments so the resistivity goes down your phonons are freezing out if it were a metal without magnetic impurities just just goes to zero if there are magnetic impurities you hit the condo temperature scale and the resistivity goes up and the reason it goes up you have this condo cloud that the electrons that provides extra scattering and that scattering increases as you decrease the temperature so the the canonical signature of condo screening is a minimum as in the resistivity of the function of temperature the second signature of condo screening is in magnetization you so curie the curie behavior is the susceptibility goes like one over t all the way down to the lowest temperatures if you have um if you have condo screening it saturate at the scale that is comparable to the condo temperature and that saturation against has to do with the fact that you've screened out the magnetic moment of your impurity and the third signature is a peak a peak at the at the Fermi level that peaks come from the fact that these electrons that screen the screen your impurity all come from the Fermi energy so they you have an enhancement of the of the density of state of the Fermi energy that sticks right that it gives you a peak right at the Fermi energy so what's the situation experimentally in 2011 the Michael Führer's group measure resistivity they saw a minimum so they made vacancies in graphene just as we do and they may measure the resistivity a function of temperature they saw a minimum and then they said okay there is condo screening so there is condo screening graphene and we can measure it uh one year later the Manchester group did magnetization measurement and they saw curie behavior down to the lowest temperatures and they said no condo screening graphene impossible and it's very interesting the theorists sided with the Manchester group and they tried to find reasons why Führer's group is wrong they were saying oh he's not seeing condo screening all he's seeing is a localization and this kind of work was considered to be wrong so we decided to try to measure signature to see if there is a peak to if we can find a peak at the Fermi energy and here's the experiment so we have a vacancy that we created as I told you before and we look with our STM we go far from the vacancy and we do look at a density of state it kind of looks okay this sample is graphene on silicon dioxide the Dirac point is here the Fermi energy is here so we on purpose doped it so to separate the Dirac point physics from the Fermi energy physics so we whole doped it here with two 10 to 11 holes per centimeter square and then we move to the center of the vacancy here and this is what we see two peaks appear one peak at the Dirac point which we identify as the zero mode that we discussed before and another peak right at the Fermi energy this is our candidate for the condo peak we need to check but this is our candidate okay so what kind of checks do we do first of all we look at a function of position it's very tightly the peak appears only very close to the vacancy you don't see it anywhere else it's localized within two nanometers of the vacancy we look at the line width and the line shape this is a typical funnel line shape funnel line shape if you have a resonance within a band the line shape has to be like this and then you can fit it and extract the condo from the width you can extract the condo temperature and we get about 70 degrees Kelvin you can do an independent test by actually changing the temperature and looking at the behavior of the line width and fitting to theory we get something that is consistent with the line width about 68 Kelvin within the arrow bar so this is all very nice so the next thing that we do okay but the peak condo peak has to be stuck to the Fermi energy so since we can dope the system we can move the Fermi energy so we can see if it's stuck there or not maybe this is an artifact so we dope it so now we increase the dynamical potential minus 92 up to 73 this peak is completely stuck to the Fermi energy the other peak is the zero mode it kind of moves in it moves in so this is our zero mode so this is a good sign this looks like a condo peak let's continue to dope it now sorry now this is inverted now I'm increasing the doping downwards this graph is inverted so I'm getting closer to the Dirac point going this way this is what I showed you before it's just upside down condo screening is on now I'm moving closer to the Dirac point no condo peak this is the Fermi energy condo peak completely disappeared and I'm coming out on the other side boom it reappeared again so what's going on here so I can measure I can extract the condo temperature so on the hold-up side there's condo screening as I get closer to the Dirac point the whole thing disappears and then it reappears again so that reminds us of the thing I showed you that you move the chemical potential close to the critical coupling so it looks like this has this vacancy is below the critical coupling and J is less than JC otherwise you'd have condo screening everywhere but this is as much as we can do as experimentalists in order to figure out what actual value of J is we need help from theorists so but the bottom line is that you have an electrically tuned magnetic moment here okay I'm not going to go through this this is all published this is with our theory collaborator so you have two orbitals you have the sigma orbital and you have the pi orbitals and when the Fermi energy changes you get the two interactions the problem is really messy with two orbitals and so our collaborators use normalization groups the calculation to calculate the band structure to calculate the phase diagram and one can approximate it with a single band just with a sigma band by doing this kind of approximation I'm not going to go into it I'm just going to show you the bottom line is the phase diagram that they got so if you want to read about it it's all published so this is the phase diagram that we got chemical potential versus reduced coupling so when the critical coupling is sitting right here so here you have the local magnetic moment here in the pink regions our condo screen and here you have a frozen impurity basically it's where you have double occupancy of the vacancy so you have a spin up and spin down sitting in that state so the magnetic moment is disappearing and now let's put our vacancy on here and and this is where it is so the coupling was indeed less than critical and it's 0.8 of the critical coupling so then we looked at other vacancies and all hell broke loose we got coupling all over the map some vacancies were subcritical some were supercritical some were very close to the critical point what's going on are we doing something wrong do we have irreproducible results so that's the question what's going on here why are they all different so in order to understand that let's go back and to figure out where this comes from so we have one magnetic moment that comes from the dangling bond here but that dangling bond is a sigma bond and that's orthogonal to the pi electron there should be no coupling j should be 0 we cannot have screening this magnetic moment cannot be screened by the conduction electron j should be 0 for this guy can't have condo screening of this moment now let's look at the other moment that comes from the missing electron in the pi band but that one has ferromagnetic coupling to the band so again j should be 0 so the theorists having sided with the Manchester group based their arguments on this the coupling has to be 0 and therefore the transport measurements were wrong so what's going on we do see condo screen we do see a condo peak there's no question about it so what's going on there's really a mystery going on here so the rescue came from this group here they said oh but you know graphene is a membrane it can bend if you're putting it on a substrate that is not flat and your orbital your sigma orbital is sticking out of the plane then you should there's no the orthogonality is gone and you should be able to couple you should have a finite coupling so yes we do have a corrugated substrate so the coupling is actually very simple function of the the local curvature okay so then we did the experiment on all sorts of substrate so this is on a graphene directly supported by silicon dioxide which is very rough and we can measure the height fluctuation with AFM you get about 2 nanometers amplitude here along this line and in this sample here we get the tk the maximum tk is 180 Kelvin it's huge coupling and most of the vacancies we look at have a conda peak most of the vacancies are screened then we go to a slightly flatter substrate so we have two layers one two layers of graphene one on top of each other on silicon dioxide and now the corrugation is about half as high tk is about half it's about 70 and only 30% of the vacancies show conda screening and finally we go to boronitride which is the flatter it's about 10 times flatter no conda screening not a single vacancy shows conda screening so everybody was right in a sense so the J depends on local corrugation and we have so in a sense we have a mechanically controlled magnetism not by electric field by mechanically controlled now let's go back to the experiments here are versus Manchester the thing is that when you do a global measurement you're doing an average over all these magnetic moments and all their screening you don't see individual ones so transport and magnetization measure complementary properties when you do transport what you're looking at is screening off the condo is scattering off the condo cloud if there is a condo cloud there you're going to see it you're not going to see those moments that are on screen you're only going to see those that are screened when you do magnetization you only see those the moments that are screened with the saturation so if you have vacancies that are screened you're just not going to see them so that's why you see this curie behavior because you're only looking at vacancies that are on screen and these guys are only looking at vacancies that are screened so if you have both kinds you're going to get two totally different results so therefore it's absolutely essential to have a global to have a local probe okay so let me oh I thought I have a summary of the whole thing at the end so I'm done with this part now we can stop here or we can go on to twisted graphene I mean you're the boss are you guys too tired to continue or shall we go what? okay let's try to do it in 15 minutes so we're getting to twisted graphene so now we're talking about the substrate changing the band structure with the substrate not with defect but with the substrate now let's say that I have graphene here a honeycomb structure and I'm going to bring another graphene layer on top okay and I'm putting it right on top and now I'm going to start twisting it now when you put two patterns on top of each other you create a super structure which we call moiré and I'm sure you've seen this take two pieces of cloth or two patterns so the period of this moiré pattern is inversely proportional to sign of the rotation angle okay I'm going back now as that angle becomes smaller and smaller the period here kind of diverges so you have a super period here and I'm stopping at three degrees okay so you have a super period so the bright parts are you have AA on top of each other okay now this is something that people have known for a very long time so here is a sample that we have this is a suspended CVD grown sample and we have twisted layers in it so this is a magnification of four this is the moiré pattern the twist here is 1.79 degree we have a period of 7.5 nanometers clearly these are not atoms you zoom in to the atomic resolution these are the atoms so we have a moiré pattern here and we can go from moiré pattern to no moiré pattern at the boundary and then we do now people have known about this moiré pattern ever since the STM was invented the 90s moiré pattern and graphene everybody saw moiré pattern and graphene we talked about looking at spectroscopy everybody showed those patterns and said okay so we always look at spectroscopy so we looked at a spectroscopy so now this we take the spectrum outside the moiré pattern somewhere here it looks kind of a regular graphene spectrum and now we go inside the moiré pattern boom we see this huge peaks that appear and those peaks are everywhere no matter where you look and you can see the same on this twisted layer and this was really puzzling what's going on the band structure has completely changed with this twist and then we look at various angle 1.16 the period is 12 nanometer 1.8, 7.8, 3.5 so you see the period get smaller and smaller as the angle becomes larger and larger and then we do spectroscopy and you see those peaks are always there so the larger the larger the angle the further apart they are here they are also out of the range here and the smallest one here you see the peaks have merged not only have they merged but there is a gap that opens a 12 millivolt gap that opens here in this peak that merged let's see what's going on here now we've seen this way back like in 2010 ok so here I'm taking two graphene layers one on top of the other in reciprocal space here are the diracons so we have a pair here and so this is a pair of diracons the point where they intersect they intersect the energies in the electron and whole sector that corresponds to the difference in the k vector which is 2k sin theta 2 k is this big k here so the distance here the position of the energies the energy where they merge goes like h bar vf delta k it's proportional to sin theta 2 so the merging is important now if there is no hybridization between the two layers nothing happens you just have two independent graphene layers and that's it however if you have some hybridization if they're sufficiently close so that you allow tunneling between them then you get a reconstruction of the band structure so I'm showing you here that I'm going to a larger angle so the energy where they cross of course increases increases like sin theta over 2 but if you have hybridization and the energy scale of hybridization let's call it w you create a saddle point in the band structure and that's the energy now that separates the two sandal points is the original crossing minus the interaction energy and this is the actual density of state done with this was done with tight binding okay so if you have a saddle point in the density of states it's a flat region in the band structure that gives you a peak in the density of state it's called van Hove singularity so the density of state that corresponds to these saddle points is a peak and this position of the peak is going to follow the position of the saddle point and this is our data this is the calculation and they fit pretty well this is for 1.79 degrees and this is what's going on you get reconstruction of the band structure due to this twist so now here this is a small very small angle what happens at small twist angle so when the twist angle is very small your moire pattern is going to be huge the period becomes huge this is 1.08 degrees this is the density of state map these points are AA the A atom the top graphene is sitting on top of A atom in the bottom graphene the dark region is AB like that and this is the Brillouin zone you have a moire Brillouin zone this is the small segment here which does all the new physics so this is at one K point and this is the other K point so in the moire in the twisted graphene this is the Brillouin zone that matters this one you have to forget about it one very important thing gave these small angles at the magic angle that I'm going to get to this huge there's about 13,000 atoms per unit cell it's impossible to calculate this exactly so you have to do all sorts of tricks and you're going to see lots of papers that calculate the band structure and you get different results and you get different approximation and we have no idea who's right we only know that what's going to happen the bands are going to flatten so here so here is what I showed you before the saddle point and now I'm getting them closer and closer the saddle points get closer and closer and the band get flatter and flatter let me show you an animation here of what's going on as a function of angle angle is decreasing, decreasing and when flat band so when w is exactly equal to h bar vf delta k you get a flat band and now you go even smaller angle it opens up again and then again you're going to get a flat band so flat band happens when these things touch you get a flat band and we really like flat band and condensed matter because that's where correlations happen so ok so the magic angle is where these two touch and that happens now we don't know exactly what w is but w if you take it from the tunneling in bilayers in actual bilayers people put in about 100 millivolts and they get this magic angle of 1.09 degrees another thing that happens when you get to smaller angles because the bands become flat the velocity is renormalized it becomes smaller and we actually measured the Fermi velocity as a function of angle using Landau level spectroscopy and we found these points theoretically it should go like this so you see that really when you go to a very small angle the whole thing stops you have a completely flat band your quasi particle don't move anymore so when we did this we had no idea what's going to happen here I mean because in some sense it has to be singular because at some point it becomes a bilayer and you have a totally different band structure however when you're at more than 10 degrees you recover the single layer graphene band structure and we use that in order to screen the random potential in the substrate so we use two layers large angle and that gives you much cleaner STM so why do we like flat bands is when the energy scale of the electron-electro interaction is comparable to the bandwidth that's what you care about correlation effects become dominant and so when you put your Fermi energy at half filling of a flat band you get all sorts of good stuff superconductivity, magnetism topological intercellator and so forth the coulomb energy now has the moire period in it so it goes down as you decrease the angle but the bandwidth goes down much faster so this is our data here so we actually we're almost at the magic angle we saw a gap opening at the Fermi energy of about 12 millivolts so these are the results that were published in March by the MIT group so what they did, they actually tuned the twist to the magic angle and they were looking for interesting effects that are going on, it took them a very long time to get there but they were able to find some very cool it down, they had to cool it down to dilution refrigerant temperatures and they found the following what I have here so one thing to remind you we have a super period the moire period is now like your new crystal so a full band is when you have one electron per moire unit now if you have a degenerative 4 it would be 4 electrons per moire unit very very low density in graphene we have a degenerative 4 so the full band would be 4 electrons per moire cell so you have to really tune your density very low density to see that half full is going to be 2 electrons per moire period and the moire period remember it's many nanometers it's like, I don't know what it is here like 20 nanometers, 15 nanometers or so so they tuned it there the carrier density this is and this is half full there should be an NS over 2 half full band here on the electron side half full band on the whole side this is conductance and you see that it goes to 0 so they have an insulating phase here and this is done this is what I call poor man's STM but you see definitely there is a gap here which goes away with magnetic field for some reason they they can measure the gap size with arenas plots by looking at the temperature dependence and on the 2 sides here, on the 2 sides of the insulator on the whole side they see superconductivity so they see that and they say it's superconductivity because it's magnetic field it disappears by magnetic field and the resistivity goes to 0 so you see they have 2 domes on the electron doped side of the insulator on the whole doped side of the insulator so this resembles high TC superconductors a lot except that unlike in high TC superconductors we can walk through this phase diagram without making a new sample for every point without chemistry you just do it with a gate and okay what did I miss here okay the maximum TC is 1.7 Kelvin and an important criterion for what kind of coupling you have is the ratio of TC to the Fermi energy and the ratio here places it flat into the strong coupling regime exactly like high TC superconductors so here is the here they have a plot of the chemical potential versus Fermi temperature this is where the high TC superconductors lie this is magic angle graphene their measurement and high TC YBCO is here BISCO is here so it's on this line here so here they have TC over TF is about 0.1 yeah 0.1 so that puts it really into the strongly coupled regime so this of course now everybody started working on this theorist and experimentally the theories are much faster as of a couple of weeks ago I counted about 70 theoretical papers and there's a zoo of scenarios that people propose so we want to know what's the pairing mechanism what are these cooper pairs what's the gap symmetry what's the nature of the insulating phase all these are open questions a lot of work for both experimentalist and theorist and that brings me to the end so let me just summarize part 4 so in this part I talked about condo screening so we have seen that condo screening in graphene does it exist it occurs about the critical coupling strength the magnetic moments in graphene we have seen can be tuned with a gate or with a local curvature so you have both mechanical and electrical control over your magnetic moment pretty unusual if the coupling strengths so if you have J that varies on your sample one of the conclusions from all these contradicting results that you cannot tell what's going on from a global measurement you need a local measurement and then the second part we talked about twisted graphene so the biostructure of bilayer graphene can be tuned by a twist with an angle at small twist the density of the state develops van Hof singularities which are very interesting because that's where correlations can be observed at the magic angle a flat band forms at the charge neutrality point and if it's half full you develop an insulator we don't know what that insulator is and on the two sides a superconducting phase and that's the end thank you for your attention