 So, well, he just read my title, so that's what I'm going to talk about. And so to start, I'll just show you this, my favorite graphing slide, so simple. A question is, why do we want to manipulate graphing at the atomic scale? Why do we care about things such as add-atoms, strains, and edges? And so this slide is meant to sort of motivate that a little bit. This is kind of just a cartoon. And the idea is that what we would like to eventually do, I think what a lot of people in this room would like to do, and some of us are already starting to do it, is to carve graphing up, to cut it up into little quantum mechanical, little structures that are designed at the quantum mechanical level to perform certain functions. And we would like to be able to functionalize the graphing with atoms and molecules and control the edges, and then wire it up with metals so that we could create an integrated device on a single layer of graphings, that we could have a better video game for our children. And so this is something that a lot of people are working towards. And Fadenivores, I think, showed some nice progress in this direction. But there's still a lot of work that we need to do in order to achieve this kind of a dream. And part of that work involves better understanding how graphing behaves at microscopic length scales, at the atomic length scale. In other words, what happens around individual atoms? What happens at edges? And what happens when you strain graphene at very small length scales? And so those are the kinds of questions that I'm going to talk about today in this talk. So here they are, impurity, strain, and cutting graphene into little nano-ribbon shapes. That's what I'm going to focus on. But before I get to that, I'll say some introductory words, talking about the techniques that we've been using to do this work, which is mainly scanning tunneling microscopy. So I'll give you, I'll say a few introductory words about that technique. And then I'll say a little bit of also introductory words about the technique of STM spectroscopy, which is a technique that we have found to be very useful for exploring these kinds of issues in graphene. So first let's just start with discussing scanning tunneling microscopy as a technique for exploring small structures. A scanning tunneling microscope is a sharp metal needle, which we bring very close to a surface. We put a voltage between the needle and the surface, and we measure the tunnel current. And the rate at which these electrons tunnel from the tip to the surface is proportional to the electronic local density of states of the structure beneath the tip, right, and remember giving these talks. And this is a very useful quantity to measure, because the electronic local density of states gives you a measure of the probability of finding electron at a certain point in space, at a certain point in energy. So if you can map that quantity out, you can learn a lot about what the electrons are doing in small structures. Also by varying this voltage and measuring the differential conductivity of this STM tunnel junction, you can measure what we refer to as DIDV. And by measuring that as a function of the voltage between the tip and the sample, we can measure the energy dependence of the local density of states beneath the STM tip. And that's what this little cartoon is meant to show. And here V equals 0 is the Fermi level, and these are filled states, and these are empty states. Also, the STM allows us to measure inelastic excitations of small structures. And the way we do that is if we put a bias on the STM tip that's big enough so that electrons tunneling from the tip to the surface have enough energy to create an inelastic excitation, then that gives those electrons an additional channel to go from the tip to the surface, an inelastic channel, say by making a phonon, for example. And that then gives us a jump. See this little step right here? That gives us a jump in the DIDV at the threshold energy for creating those excitations. So this technique has been used a lot over the years to study graphene in different forms. In fact, people were using STM to study graphene back in 1992. Although they didn't call it graphene back then. And it was not isolated the way Gaim and Novaslav I have since isolated it. People used to study graphene on metal surfaces a long time ago. More recently, people have been studying graphene with STM on other substrates such as silicon carbide, silicon oxide, and more recently boron nitride. And I'll say a little bit about that. This is data from the Georgia Tech Group, the Columbia Group. This is some data from my own group. And there's a lot of groups around the world that have been using STM to look at graphene. And here I've just listed some of the people who are doing nice work with STM. But there are others too. I'm sorry if your name is not on there. So in my own group, we're very interested in using STM to perform local spectroscopy on graphene. And so a few years ago, we set out to perform STM spectroscopy on gated graphene devices, such as the one that you see here. This is a photograph of a little device we made with Alex Zettle's help. This is a graphene flake. And we can gait it from the back and perform STM spectroscopy. And what we expected to see when we did this experiment a few years ago was we expected to see a V in the local density of states in the DIDV. Because the local density of states of a material often mimics the total density of states, which is linear for graphene. And so there's the Dirac point. And so that's what we expected to see. But when we did the experiment, that's not what we saw. In fact, we saw something that looked kind of different. Here I show you the data. What we actually saw, this is a plot of DIDV versus bias voltage between the tip and the surface. This is the Fermi level. And here we have filled states and empty states. And each of these curves is measured at a different gait voltage. And we're becoming more and more P-doped as we go up, each additional curve. And so the density of states is sort of shifting to the right as you go up those curves due to the gating. And here we saw two features. One was this gap-like feature that's pinned to the Fermi level. And the other is this asymmetric dip off to one side. Now this gap-like feature is pinned to the Fermi level for different gait voltages. So that tells us that it's not a band structure feature of graphene. It's some kind of excitation. And what we believe is happening is that this is a not a true gap, but it's a sign of inelastic tunneling. Phonon assisted inelastic tunneling. And the idea is that when we come to a bias here at about 65 volts, then the electrons have enough, the tunneling electrons have enough energy to create an out-of-plane phonon in the graphene. And so that creation of that phonon helps to get the electrons in. So we get a big jump in the current due to this inelastic tunneling channel. Here, this little dip, that feature moves with gait voltage. So that tells us that it's a band structure feature. And in fact, that is actually due to the Dirac point. So you can see the Dirac point moving, as you would expect, as you p-dope the material. So I just wanted to show you this and drag you through this, because this is sort of the baseline STM spectroscopy of graphene. And this is what we see after doing some elaborate tip calibration procedures. But what we're really interested in is what happens when you modify graphene and what are the local properties of the graphene when you change it, when you manipulate its properties. And one of the ways of doing that is to drop atoms and molecules on the graphene or to have impurities in the graphene. And I just want to point out that we already know something about how graphene behaves when we add adsorbates to it from previous measurements. For example, this is Amir Yakobi's measurement using scanning SCT. And you can see the electron hole puddles that arise due to impurities derived from the silicon oxide layer. Also, from transport, you can also get some idea of what happens when you drop charge impurities onto graphene. And this is some work from Michael Führer's laboratory. But these measurements actually don't, even though they give us a lot of good information, they don't tell us what's happening locally in the vicinity of these individual atomic-sized impurities. And so that's something that we're really interested in studying. And one of the reasons we're interested in it is because if you have a charge impurity in graphene, then it's been predicted that the electrons in graphene will behave around that impurity differently than they behave in other materials. For example, if you consider impurities in silicon and gallium arsenide, you expect a different kind of behavior for the electrons around an impurity in graphene because the electrons in graphene behave as massless Dirac fermions. And so it's been predicted by people such as Levitov and others, as well, that if you look at how these relativistic electrons behave around a Coulomb impurity, then there are different physical regimes, depending on the strength of the Coulomb impurity. And you can get what is referred to as subcritical behavior, where there are no bound states, no resonances, or a critical regime where there are bound states that have a particular narrow energy width and spatial distribution. So this is the kind of thing that we would like to measure experimentally and to better understand. And so motivated by this, we dropped atoms down onto graphene. We dropped different atoms. And one of the first atoms that we looked at is cobalt. And so we evaporated cobalt atoms down onto a gated graphene device at low temperature so that the atoms would stick and would not move all around. And then we took STM images of it. And so here you see an STM image of a gated graphene device on silicon oxide where you can see two cobalt atoms. There they are. And what we did was we put our STM tip on top of those cobalt atoms so that we could measure their electronic structure. And so here I show you that electronic structure. So this is the electronic structure of a single cobalt atom sitting on top of a gated graphene device. This is DIDV as a function of sample bias. Here's zero. That's the Fermi energy. And each of these curves is we're p-doping the sample. So we're increasingly p-doping it as we go up. So you can see there's a lot of features there. There's a lot of resonances there. This is all due to the electronic structure of that atom just sitting on cobalt. And so it looks kind of complicated, but it's not so bad. There's really three main features. This first feature here at the center, at zero, there's this DIP at the Fermi level, which is gate-independent. And we believe that that DIP is simply an inelastic feature due to the vibrations of the cobalt atom. So I don't want to focus on that in this talk, unless you force me to. But what I'd rather focus on is these other features, these features which change with the gate voltage. And so in order to understand those better, I've plotted the energy of all these resonances here. This is the energy versus gate voltage. And I've also plotted the Dirac point, how it moves with gate voltage. So you can see that some of these features move with the Dirac point, and other features don't. They move opposite in energy of the Dirac point. Now these features that move in red that I've outlined in red, these ones that move with the Dirac point, because they move with the Dirac point, that implies that they are density of states features. They're features embedded in the band structure due to that impurity. And so the idea, which you can see in this little cartoon, is that the cobalt atom, if you take the cobalt atom, an isolated atom, it has atomic energy levels. And so if you take that atom and you drop it onto a piece of graphene, then those atomic energy levels are going to hybridize in some way with the graphene. And that's going to lead to defect states, which I have drawn schematically here. So this is not the result of a calculation. This is just a cartoon. And so here are defect states in the density of states of graphene. Now we can see those defect states with the STM in our spectroscopy. And something that's kind of nice, something that's kind of new I think, is that we can change the energy level of those defect states with the back gate. So we can gate our atom, we can gate the surface, and we can move those defect states up and down in energy. And the reason that's significant is because that allows us to fill or deplete those defect states with charge. So here, for example, I'm showing you a scenario where we've gated the sample at a gate such that these defect states are below the Fermi energy. And I have a negative sign there to indicate that they're filled with charge, negative charge, because they're below the Fermi energy. And here you can see the actual data. This is the DIDV versus voltage, and these are the impurity states below the Fermi energy. Now here I'm showing you the same atom, but with a different gate voltage. So now those states have been pushed above the Fermi energy, and I put a little plus sign there because now the charge has flowed out of those states. So now we've depleted, we've removed charge from that cobalt defect. And here you can see that in the data. Here are those same states, but now pushed above the Fermi energy here. Now you might also have noticed these other little states right here that I've marked with the S. Those states are due to tip-induced charging of the defect. And a nice way to think about it is we have an atom sitting on the surface. We gate the sample, and that determines a charge state for the atom. But then we can bring the tip in, and the tip is also a gate. It's a movable gate. It's another gate. And we can bring that tip in, and then we can use the tip to change the charge on the atom. For example, here in this case, I have electrons in the defect state, but at this bias voltage, the STM tip is sucking the electrons out of the atom. And that's why we get this little bump. Whereas in this case, where I've already removed the charge from the atom with the gate, at this bias voltage, the STM tip is pushing electrons in to the atom. And so that's why we get this little bump. So this is kind of a nice new capability that we have, which is to control the charge state of this defect using the back gate. And one of the things that we would like to do with this new capability is to use it to investigate what happens in the vicinity of Coulomb potentials on graphene. Because now we can change the charge state. And what we would like to do is then measure around these charge impurities, what the electrons are doing in the graphene. So that's something that we want to do. But we have problems. Nothing is ever perfect. And the problems that we've been having, because we've been trying to do this, one of the problems that we have is that the graphene is so inhomogeneous. This is an STM image of the electron hole puddles in graphene that you've already heard about. And this electronic inhomogeneity in graphene makes these kinds of measurements hard for us because it obscures what's happening. Because if I put one atom down, say, put an atom right there, then if there's some charge distribution around that atom due to the Coulomb impurity, it gets obscured by all of this charge inhomogeneity. Another issue that we have is that this is a problem we have with cobalt atoms is that the impurity itself has what we call an inhomogeneous charge state. And what I mean by that is that we can set the charge state of the atom with the back gate, and then we want to map what the electrons are doing around it in the graphene. But the problem we have is that sometimes the STM tip will cause the charge state of the atom to switch. I mentioned that to you earlier. And so we have trouble then mapping what's happening around the impurity because sometimes the impurities charge will switch. And you can see that here by this ring. This is a phenomena that people have seen in other STM experiments as well. This is what we sometimes refer to as an ionization radius. And what it means is that when the STM tip is outside of this radius, the atom is in one charge state. But when we go inside of the radius, it switches to another charge state. And that's due to the fact that the tip is a movable gate. So these are problems for us. Maybe sometimes we try to pretend that it's all really great, but actually it's not. But we have made progress at solving these problems. So I want to tell you about some of the progress that we've made. And we've made a lot of progress because mainly recently through the use of boron nitride as a substrate. So this is a really great thing. And the first people to do it was the Columbia group. And they showed that if you put graphene on boron nitride, then you can greatly enhance the mobility. And so we were inspired by this great result to try this ourselves. And so we built some devices in this fashion. We put boron nitride flakes on silicon oxide. And we took CBD graphene and put it over. And then we stuck it into our scanning tunneling microscope. And these devices, Alex Zettle's group also helped us to fabricate these devices. And so we looked at those devices with STM. And it turned out that they looked really great. And so this helped solve one of our problems, which is this inhomogenity problem. Because here you can see that graphene on boron nitride. Here we compare graphene on boron nitride to graphene on silicon oxide. This is the topography, the comparison. And you can see that graphene on boron nitride is orders of magnitude more flat. And also down here, we compare the charge inhomogenity for graphene on boron nitride compared to graphene on silicon oxide. And you can just see from the color scale that graphene on boron nitride is much less, is much more homogeneous. So that's really great for us. That solves one of our problems. I also want to point out that about the same time we were doing this work, Brian Leroy's group also did similar work in collaboration with the Columbia group. And they got results that are very similar to ours. And so that's a great substrate to use. And so now we're using that substrate, graphene on boron nitride. And it allows us to solve another one of our problems. Because now that we have such a nice flat surface for the graphene to sit on, we can perform atomic manipulation on the atoms that we put on the graphene. And we can actually engineer our own defects by moving the atoms together. And so we can create new defects that have different charging properties than the ones that we have before. For example, before we were limited to say monomers, the cobalt monomers. But now we can take the cobalt monomers and move them together. You can see these three have been moved together like that. Then these two, we move together with the STM tip to create a dimer. And then we can move this one over here to create a trimer. So now we can create these small clusters which have different charging properties. And that's actually been good for us. Because when we look at this cobalt trimer, for example, we find that these clusters have much nicer charging properties than the monomers. And we can put them into stable charge states. Here, for example, the cobalt trimer, we can gate the cobalt trimer such that when we bring our STM tip and image the electrons around the trimer in a particular charge state, the trimer stays in that charge state. So we can put the trimer into different charge states. And then we can map out the electrons around it. And the charge state will not change. And so that's for us, that's progress. And so this is a work in progress. We're not done. But we're very excited about this because now we're starting to map out what the electrons are doing around these coulomb impurities. And we can control the charge on these coulomb impurities. So now we have a chance of actually testing some of these ideas. For example, here you can see an uncharged cobalt trimer. Here you can see a charged cobalt trimer. And this yellow halo, that's the electrons in graphene rearranging themselves around that charged trimer due to that coulomb impurity. So the physics that we want to get at is living in that halo. So that's what we're in the process of analyzing now. So putting atoms down onto graphene is one way in which you can change the properties of graphene. But another way, well, but you might want to change it in different ways. For example, you might want to induce an energy gap in the graphene. You might want to quantize the energy levels in graphene. And so a way in which you can do that, if you want to change it in that way, one of the many things you can do is to turn on a magnetic field. And a magnetic field is nice because when you turn on a magnetic field, it causes the electrons to go around in circles. And when you quantize that motion, that circular motion, then you get Landau levels. And the degeneracy in energy spacing between the Landau levels is going to depend on the strength of that magnetic field. So this kind of behavior has been seen a lot for graphene. And since I'm giving an STM talk, I'll tell you about some of the nice STM data that shows these Landau levels. And some of the nicest data has been taken by Eva Andres' group, as well as Joe Strossio's group in collaboration with Phil first. And they've seen beautiful Landau levels with their STM up at magnetic fields around five teslas. But a question that I want to ask is, what happens if you take a sheet of graphene and carve out a little tiny device, a little tiny sub-micron device, like here, like I've shown in blue, and what if you wanted to create Landau levels right there in that region, but did not want Landau levels here in these contact regions? How might you do that? Well, the answer, I think, for normal materials is that there's no way that you could do that, because you can't shrink your superconducting magnet down to sub-micron dimensions. Superconducting magnets are about this big. You can't shrink them down that big. And so this would normally be impossible. But with graphene, this kind of experiment actually is sort of possible, because graphene has a very peculiar property, which is that if you strain it in a particular way, then the electrons in graphene will behave as though there's a magnetic field turned on. They're going to want to go around in circles. And that's a very kind of peculiar effect. I don't know any other material that has this effect, and so I just want to spend a minute or two trying to give you a hand-waving explanation of why graphene has this really interesting behavior that when you strain it in a particular way it behaves as though there's a magnetic field. And we call that a pseudo-magnetic field. Now, the reason that this occurs is because of a very intimate relationship that exists between strain in graphene and the electromagnetic vector potential. And one way of looking at this is to consider graphene, and now this is unstrained graphene. Now consider graphene that has undergone some amount of strain. You stretch it, and what that does is it causes the atoms to change their relative distance between each other, that then causes the quantum mechanical hopping amplitude to change for an electron to hop from one carbon atom to the other. And what that does to the quantum mechanical eigenstates, the electronic eigenstates, shift then in reciprocal space like this, that little delta k that's meant to represent the shift of the graphene eigenstates in reciprocal space due to that strain. Now, if you remember some basic quantum mechanics, then you might remember that if a charged particle is moving through space and you turn on a magnetic field, then what that does is it changes the momentum of the particle by an amount that's proportional to the vector potential. So take a look at these two equations, and you can see that they're very similar. And so what we can do is we can create a mapping between this change in momentum and this vector potential. And if you're smart and you work through the details as smart people have done, then you find that for a particular strain tensor in graphene, that will lead to a pseudo vector potential that you can write down as I've done here in this formula. And you can see that the pseudo vector potential depends on the uniaxial strain or the normal strain components as well as the shear strain in this way. You can then take the curl of this pseudo vector potential, and that will create a pseudo magnetic field. And so the electrons will then feel that pseudo magnetic field, which arises purely due to strain, even in the absence of an actual magnetic field. So a couple of years ago, some smart people wrote a really clever paper on here they are. It was Paco Guinea, Katznelsen, and Andre Geim, honorary theorist. And so this is a really nice paper where they came up with this clever idea of taking this mapping. And they figured out a particular strain geometry, in which case the resulting pseudo magnetic field would lead to a constant magnetic field, which would cause the electrons to go around in a circle leading to Landau levels, pseudo Landau levels. And what they figured out is that if you create a trigonal strain pattern, one that has triangular symmetry, then if you work through the details of this formula, then that should lead to a constant magnetic field. The question, though, and subsequently Landau levels. The question, of course, though, is how to test this idea. And in that paper, they actually suggested an idea of using differential thermal contraction. And so that's what we did. And so we tested this idea by using differential thermal contraction. And the idea here is that if you put graphene on a substrate and then cool it down, graphene does not shrink that much when you cool it down, whereas other materials will shrink more. And that can lead to a large buildup of strain in the graphene, which could then provide possibly highly strained regions, which might show some of these effects. So that was actually an idea that they mentioned in their paper. And so we tried out this idea by epitaxially growing graphene on platinum at high temperature. And then we cooled down the graphene and looked at it with our STM. And I just want to mention that other people have grown graphene on platinum long before us. And so we copied their recipes. For example, Combsheld's group and Salmiron's group. So we just copied their recipes and grew it. And then we looked at it at low temperature and we saw something kind of interesting. This is what we saw. This is an STM image of graphene grown on platinum 111. This region right here is a patch of graphene, if you can see the outline of it. And these bubbles, which appear at nanometer length scale, we refer to as nanobubbles. And we saw these nanobubbles all over the place on the graphene that we grew on platinum. And these nanobubbles have a triangular shape. They're like little pyramids with three sides, sort of tetrahedral. And what we saw, we then did spectroscopy on this surface. And when we did STM spectroscopy away from those nanobubbles, the spectroscopy was pretty featureless and pretty boring. But when we did spectroscopy right on top of the nanobubbles, we saw something very interesting. We saw these peaks in the local density of states appearing at different energies. So we saw these very pronounced peaks. And what we think these peaks are, we believe that these peaks are due to the Landau levels occurring in these nanobubbles as a result of the pseudo-magnetic field that is induced by the strain in these little strain nanobubbles. And in order to further test that idea, we looked at a lot of these nanobubbles. We performed spectroscopy on a lot of them. And we took all of the different peaks. And we plotted the distribution of peak energies as you see here. And we see that this peak distribution follows the behavior that we would expect for Landau levels in graphene, a square root of n behavior. And so from this behavior, we can then extract a pseudo-magnetic field from the spacing of these peaks. And when we extract that pseudo-magnetic field, we get a field that's very high on the order of several hundred tesla. Here, for example, I show results from one particular nanobubble. Here's a picture of it. And here you can see the pseudo-magnetic field that we extracted from this nanobubble up here at around 400 tesla. Now, in order to further test this interpretation, we did some theoretical calculations in collaboration with Paco Guinea and Castronero. That means that they did the calculations. And they used continuum elasticity theory. And they simulated nanobubbles having roughly the same size and strain as the ones that we saw experimentally. And this is the pseudo-magnetic field that they calculated, which is very comparable to the one that we saw experimentally. So this is further evidence to support this interpretation. And this is a result that I'm very excited about. And one of the reasons is because these pseudo-magnetic fields are so high, several hundred tesla, it gives us access to a new physical regime. For example, if you consider DC magnetic fields, the largest DC magnetic field that anyone has ever created in the laboratory is less than 100 tesla. But now we're looking at pseudo-magnetic fields up at around 400 tesla. So this allows us to potentially explore a physical regime that could never be accessed by any other means. OK, so now what I've done is I've told you two ways in which we can modify the graphene, one by dropping atoms down onto it, another by straining it locally. And now I want to talk about the third and final part of this talk, which is to explore what happens when you modify graphene by cutting it into little narrow strips that we refer to as nanoribbons. Now it turns out, and I think it's kind of obvious that if you cut graphene into a little ribbon, you're going to get some size quantization. So you expect subbands and energy gaps to occur. And indeed, that happens. But there's also some other physics that occurs, which is I think not so obvious, which depends on the details of this edge. You can get some very interesting physics depending on the symmetry of the edge. And so I want to talk about that a little bit. So let's talk about the edge. If you take graphene and you cut a nanoribbon out of it in this particular direction, then you'll get edges that we call armchair edges. So here's the two sides of the nanoribbon. The nanoribbon extends in this direction. So if you cut an armchair nanoribbon out of graphene, then it's been predicted, and here are some theory papers for various people who have made these predictions, it's been predicted that you will get an energy gap. And that energy gap should behave as one over the width. And I think that makes intuitive sense, because as you make the width narrower, it's like particle in a box. You make the box smaller, you expect the energy level spacing to get bigger. And so for the armchair nanoribbons, you can really think of this energy gap as a size quantization effect. If you turn on electron-electron interactions, it doesn't change things all that much. So you just get that gap. And there's no edge state for the armchair nanoribbons, or at least that's the prediction. On the other hand, if you take graphene and if you cut it at this other angle, 30 degrees from the armchair angle, right here at that angle, then you get what is known as a zigzag nanoribbon, where these edges are zigzag edges. And the zigzag nanoribbons have been predicted to have very different behavior from the armchair nanoribbons. For example, in the absence of electron-electron interactions, it was predicted a while ago that you should get one-dimensional metallic bands that are localized at the edge, edge states. And since these edge states are metallic, then it was initially predicted that you should not even get an energy gap for a zigzag nanoribbon. But then people thought about it some more, and they realized that if you include electron-electron interactions, then what happens, because this edge state has a big density of states at the Fermi energy, it has been predicted that you would get a magnetic transition that the edge would become ferromagnetic. And it was predicted that each edge would become ferromagnetic and that there would be an anti-ferromagnetic correlation between the two edges. And this anti-ferromagnetic correlation is predicted to open up an energy gap. And that energy gap was predicted to vary as one over width. So here with the zigzag nanoribbon, you also are expected to get a one over width energy gap dependence, but for very different reasons, very different physics. And I just want to point out that some of these ideas have been thought about since the mid-'90s and Millie Dressel House is on some of these early papers. So some of these ideas are due, in part, to her. So the question then is how can we actually experimentally measure this kind of effect? It's kind of tricky to measure this kind of effect, because to really do it right, you need to do two things simultaneously. You need to measure the electronic structure of the nanoribbon while simultaneously measuring the geometric structure at the atomic length scale. You have to do both of those things. And so that's a big challenge to do both of those things. But people have been trying and have been investigating this system for a number of years. Here, I show some work a few years ago done by Philip Kim's group, where they use transport measurements to look at the electronic properties of nanoribbons that they define lithographically. And they were able to see some energy gap behavior. And other groups did measurements as well on nanoribbons, such as Fadenivores's group and others. So transport is a very nice way to measure electronic properties of nanostructures. But one of the problems is that it doesn't tell us what the local geometrical structure is. So it's hard to know what the edges are doing in these nanoribbons. For example, it's hard to know the level of disorder at the edges, whether it's armchair or zigzag. And so because of that, that then motivates people like me and people like me who like to use microscopy. And so luckily transport doesn't measure everything, so there's still some stuff for me to measure. And other people have been doing microscopy on these graphing nanostructures for many years. Here's some early STM work. And one of the earliest things that people did was to look at the surface of graphite. Because if you can look at the step edge on the surface of graphite, that can give you some insight into the behavior of graphing edges. And so that work was done early on. More recently, people have been looking at graphing nanoplatelets, put on different materials. People have been doing TEM of graphing edges. And there's been some very beautiful recent work by Klaus Mühlen's group making nanoribbons using molecular precursors that I think is really pretty cool. And in my own group, one of our strategy, what we decided to do was to look at nanoribbons that were made by unzipping nanotubes. And the idea, the hope, was that by looking at nanoribbons made by unzipping nanotubes, the hope was that this would leave the edges pristine. And so that's what we wanted to do, was to look at nanoribbons with nice edges. And my group, we don't do the unzipping. We don't unzip nanotubes. But there are other people who know how to do it. For example, one of the first groups to do it was Jim Torr's group. They developed a way back in 2009 to unzip nanotubes to make nanoribbons. And more recently, Hange Dai's group has developed a new method for unzipping nanotubes that involves sonocating the nanotubes in a special kind of way. And so what I want to show you now is some results that we got looking at these nanoribbons. And here's the recipe for how to make them in this paper. So Hange Dai gave us some of these nanoribbons. And we spun-coat them onto clean gold crystals. And we looked at them with STM. And this is what we saw. This is a room temperature STM image of a single nanoribbon on gold. And it's a beautiful nanoribbon. Believe me, we've looked at a lot of unbeautiful ones. But using this technique, the nanoribbons are almost always really nice. So you can see that it has very nice straight edges. Here's a cross-sectional slice at this little black slice right there, is this cross-sectional slice. And you can see that near the edges, we see this curvature, which we did not expect. It was kind of an unexpected feature. And that kind of threw us off at first, because we thought that maybe we were seeing crushed nanotubes or graphing that was folded under. But we put a lot of effort into looking at different samples, and we actually investigated folded nanoribbons to really see what's happening at this edge. And we determined that this edge we see here is actually not folded under. It's not a crushed nanotubes. It's not folded under. We believe very strongly now that what we're seeing is it actually is a terminal edge, as I show in this little cartoon. But we just happened to have some curvature right here at the edge. And so now that we know that we have these nice terminal edges of our graphing nanoribbons, we can look at them with higher energy, higher resolution at low temperature, and investigate their atomic structure and their electronic structure simultaneously. And that's what we did. So here I show a high resolution STM image taking at low temperature of the edge of a nanoribbon that has a width of 20 nanometers. And I'm just zooming in on the edge. So this is the edge. This is gold. And this is the nanoribbon, this greenish stuff. And this orange means that it's taller here. And this is the edge. So first, I want you to see that we have a nice, straight, well-ordered edge. And because we can get atomic resolution for this region near the edge, that allows us to determine the chirality of the edge. So now we can actually directly determine the chirality of the edge by simply looking at this angle with respect to that angle. And that allows us to determine that this nanoribbon, for example, has an 8-1 edge, which means that it goes zig, zig, zig, zig, zig eight times and then zag. Zig, zig, zig, zig, zig, zag. And on and on. And that's about a 20 angstrom periodicity. And so now that we know the chirality, we can measure the local electronic structure. And we do that using STM spectroscopy, as I described before, the technique. And we perform STM spectroscopy at the edge and then at points as we move inward toward the center of the nanoribbon as we move in from the edge. And here I show the data. This is a plot of DIDV versus voltage. The first point, I can't barely see it, but there's a black dot there on the gold. That refers to this top curve. So on the gold, you see the spectrum is featureless. But here, as we go in to the edge, I want you to focus here on this low energy regime, which I refer to as the elastic regime. And what we see are these peaks suddenly sprouting up here in low energy. And those peaks fall exponentially in amplitude as we move away from the edge. And you can see that here I've plotted the amplitude of the peaks. And because those peaks fall exponentially, that indicates that they are due to an edge state. Because that's one of the characteristics of an edge state is that its amplitude falls exponentially away from the edge. That's what an edge state is. And so here we're seeing an edge state on these nanoribbons. But also, another significant fact is that we see two little peaks. Sometimes they're asymmetrical, but we see these two little peaks. Here you can see a close-up. So you can see these two peaks. And we believe that this indicates that there's an energy gap in the edge state, where the energy gap is the energy difference between those two peaks. And so we measured this behavior on many different nanoribbons. And we saw that for different widths of nanoribbons, we got a different energy gap. And so here I show that data. We see that the energy gap gets bigger as we decrease the width of the nanoribbons. So we see a 1 over width dependence on the energy gap of this edge state. So now, oh, also, we measured how the edge state varies parallel to the edge here in this data. And we see that the amplitude of the edge state oscillates. It goes up and down and up and down and up. So there's a periodicity to the amplitude of the edge state, which is approximately the same periodicity of this 8-1 edge. So now we have the atomic scale structure and the electronic structure of the edge. So now we can start to compare our results to theoretical calculations. And so these calculations were done by Steven, Louis, and Oleg Yassyev and their coworkers. And they used a tight binding model to simulate the electronic structure of nanoribbons that have precisely the same symmetry of the same edge structure and the same width as the nanoribbons that we measured. And here I show the results of their calculations. This is the electronic structure and the density of states that they calculated for this nanoribbon, an 8-1 nanoribbon with a 20 nanometer width. And this is a calculation done in the absence of electron-electron interactions. And what you see here is a metallic edge state at zero, which leads to a big peak in the density of states. Now that's not what we saw. And so in order to simulate what we saw, they had to turn on the electron-electron interactions. And they did that using a Hubbard model, using a mean field solution of a Hubbard model. And when they turned on the electron-electron interactions, that caused a gap to open up in that edge state. And that gap that opened up was of an order of magnitude very close to what we saw experimentally. So there's some agreement here between their theory and our data. And I just want to mention to you what the physics is, what's happening as they turn on the electron-electron interaction. That's causing that edge state to become magnetic. Each of the edges becomes magnetic. They become anti-ferromagnetically correlated. And that anti-ferromagnetic correlation causes this gap right there to open up. So that's what happens in the model. So we actually saw that same gap in our experiment. In the model, it arises due to magnetism. But we have not measured magnetism in our experiment. We just see the gap. So now that we have this nice theory, we can actually compare our other results, the spatial dependence of the edge state to their theory. So here's the spatial dependence as we move perpendicular from the edge, parallel to the edge, and the width dependence of the edge state energy gap that we measured experimentally. And here's what they calculated using their tight binding model in red is the theory. And so you can see there's pretty good agreement between experiment and theory here. So I feel that this is very strong evidence that we are seeing an edge state for these graphene nanoribbons. And this edge state has an energy gap that is width dependent. So that's where I want to stop. And I'll stop just by concluding that I think carbon still has a few surprises left. I told you a little bit about what happens when we sprinkle atoms down on graphene, how we're able to measure pseudo-field effects on graphene, and how we have discovered the presence of an edge state on chiral graphene nanoribbons. I think there's still a lot of things to do in the future. And we'll talk about them in the future. But for now, I just want to tell you who my collaborators were. A lot of people collaborated on this work. Here are the different PIs who collaborated. I won't read their names, but you can read them yourself right there. But here below, more importantly, are the students and postdocs who collaborated on this work. And for the Atatom spectroscopy, this project was done by Victor Brar, Regis Decker, Yang Wang, Hans Michael, Solawan, Chowler, Girret, Kevin Chan, Hongkyung Lee, Will Regan, Will Gannett. The pseudo-magnetic field work was done by Niv Levy, Sarah Burke, Casey Meeker, and Melissa Pan-Lisigi, in addition to help from these guys. The graphene nanoribbon work was done by Chenggang Tao, Yanxia Chen, Liying Zhao, Zhuanzhuang Feng, Xiaowei Zheng, Oleg Yazhev, and Rodrigo Capaz. And the funding for this work was funded by the Department of Energy, Office of Naval Research, and the US National Science Foundation. All right, that's it. We still have some work to do. I'm gearing up to do it. I mean, at least we have the sample processing down. That's good. At some level, the hardest part, we know how to prepare these samples, but now we have to start preparing the tips. And part of the problem is that the system where I'm doing the add-atom, moving the atomic manipulation of the atoms, that's the same system that I would use for the spin-polar as STM. So we want to finish up that project before we do the spin-polar. Because it depends on the anisotropy energy. I do not expect a strong high anisotropy energy for the cobalt. And this is at 4 degrees Kelvin. Yeah, but I don't have a magnetic field. I know I'm the only guy, I'm like one of the few guys who doesn't have a magnetic field. I have no magnetic field. I have a pseudo-magnetic field. I have a pseudo-magnetic field. So at least I have that. So I should be happy, but I don't have a real. First, I want to point out that we do not know what is the actual termination. You know everything that I know, because everything I know is right there, right here on this slide. You know STM too. So this is our edge. So we definitely know the chirality of the edge, because we know the structure. But the actual termination, like is it terminated by a hydrogen atom or a carboxylic acid or an oxygen atom? That we don't know. So we don't know the absolute edge termination. However, even if it's terminated with different atoms, at some level, you still expect these edge states to exist. Because if the termination is just bonded to that sigma bond sticking out, as long as the pi network is not all messed up, then we still expect this edge state to occur. So that will lead to band bending on the graphene. And in that case, I mean, you notice that there is a bump at the edges. This curvature. This curvature. That's structural. That could be a natural consequence of the electronegativity. It will decay in an oscillatory manner inside the ribbon. A lot of the observations will be the same with a classical model. And in this respect, I'm just wondering, when you saw the Trimer, the Cobalt Trimer, were there any Friedel oscillations seen? I didn't notice any. Yes. I didn't show all of the data there, but there are Friedel. We have seen some Friedel oscillations in certain parameter regimes. So they should be also present in the nano ribbons. Oh, so you're saying that you could have a Friedel, like a Friedel oscillation decay? Yes. Yeah. But one thing is that I don't see exactly how that would lead to an energy gap, especially a width dependent energy gap. If you have dipoles at the end, if there are, say, OH groups or something, and the shorter ones, there will be interaction. In the what? There will be interaction between dipoles. And that will open up a gap. Oh, you mean across the width? Yes. It will be a width dependent gap. Well, OK. I mean, we'd have to think about that. That's an interesting observation. You said the two opposite edges, the ferromagnetic, were anti-ferromagnetically aligned, right? That's the prediction. We have not seen magnetism. Because I mean, in the prediction, that must be due to an RKK by interaction, which is a bit like the Friedel interaction he's talking about. But of course, that depends on having a Fermi surface. I mean, let's see, you've hardly got a Fermi surface where there's almost zero gap, have you? I'm not sure what you're asking. Well, I mean, you know the RKK-Y interaction between magnetic impurities in a metal, that you get an oscillatory interaction. Can I answer this question? Yes, please. In graphene, at zero density, there is RKK-Y interaction, which decays as 1 over R cube, unexpectedly for two-dimensional system. So the way how you can figure this out is you take the distance between impurities, then their RKK-Y interaction will be carried by electrons with typical momentum of the order of inverse of this distance. You take the density of states corresponding to this momentum, which is non-zero, and then plug it into the usual RKK-Y formula, which will give you 1 over R cube. What determines the wave vector for the oscillation? Is it normally given by the range? It depends where exactly you put the impurity. If the spin is on the side. It's normally given by the dimensions of a Fermi surface, right? Yes. In this case, I don't know. Yes, in this case, there is no firm surface. But in this case, there is a typical range of momentum which contributes to the polarization, which is of the order of the inverse distance between magnetic atoms. And the details depend on where exactly on the lattice you put the magnetic atom. If you put it on the side, then it will oscillate between A and B sublattices. Just don't see where the oscillation comes from without the Fermi surface. The oscillation comes from that you have two sublattices and that you have the corners of the brillian zone, which is finite momentum. Yeah, I just wonder about the pseudo fields. And if it can really generate so high magnetic fields, like several hundred tesla, do you think it will influence your measurements because of the electron facilities? The thing is, though, that it's not an actual magnetic field. It's an orbital effect. So it's just how the electrons are moving. They're moving as though they feel that field. But there's no real magnetic field there. For example, there's no Zeeman effect. The spins are not feeling it. It's purely an orbital effect. So I don't think that there's a magnetic field, then that's really going to affect us. It's just affecting the electrons moving in the graphene. So also related to that pseudo field, do you expect a quantum hole effect as a result of it? If you say it's an orbital effect, then the quantum hole effect is an orbital effect. But yeah, we're just looking at these very small regions. Could this be a way of measuring the quantum hole effect in zero field? Yeah. You're saying if you could perform transport measurement on it, on just those strained regions, yeah. I think, in principle, you should be able to do that. You should expect an effect on transport from this magnetic field. But I measure no effect on transport for reasonable wrinkles of a few nanometers. It doesn't show up. Is there any critical angle of the wrinkle of the curvature? Typical scale to get a field of one Tesla would require a strain of, say, 1% over one micron distance. So the fields are so large because Michael's bubbles are so small and strained within those bubbles, I believe, 10% or something like that. And when you have one Tesla field, it's probably in typical samples on silicon oxide. They're just covered by impurity fabric. It can also depend on the symmetry of the strain field. Because I think we kind of got lucky here. It really has this trigonal symmetry, because it's the 1, 1, 1 FCC. Yeah, exactly. If you just have a homogeneous strain, it would be plus minus field until it's canceled each other. That's it. Additional scattering rather than anything else. So we've got kind of lucky here. Your nano-rebolts are lying on gold. And gold, presumably, should dope it. So you think that they just can it in there? OK, so you're asking to what degree, what is the effect of the doping? I'm asking to comments, why don't you what do you expect from doping by gold and so on? OK. So yeah, I mean, take a look. We do see doping. I mean, like right here, look, see how this thing is offset from zero. Let me make two comments on that. First, I want to point out that people have grown graphene on gold 1, 1, 1. And they see that the gold 1, 1, 1 does not interfere too much with the electronic structure of the graphene. It still has the Dirac cone. And there's not a lot of charge transfer for that perfect system. However, here we have an imperfect system, because we have these nano-ribbons. And it is not a perfectly clean system. There is also adsorbates down there. It was transported through air. And so what we see are fluctuations on the order of plus or minus 20 millivolts from ribbon to ribbon variations. And these variations are almost precisely what people have seen in the past for nanotubes on gold 1, 1, 1. When people have performed spectroscopy of nano-ribbons on gold 1, 1, 1, they also saw shifts in their electronic structure of this same order. So I cannot really tell you what is causing that. So we do see some charge transfer on the same order as the nanotube guys. But I cannot distinguish whether this charge transfer is from the gold or from adsorbates. We have measured the transfer length of graphene on gold. And it is of the order of three microns, which is huge. That is, the efficiency of transfer, charge transfer, is very, very weak. Jack Blakely from Cornell. My question was, are you sure there are bubbles? Or is the substrate restructured underneath? The reason I asked the question is, in the 70s in my group, we did a lot of work on forming graphene. Graphene forms as a stable monolayer phase on nickel 1, 1, 1, where the epitaxial fit is excellent. But if you go off that orientation, then you still get the same phase transformation. But it's accompanied by a formation of pyramids on the substrate. In other words, the whole substrate restructures to form to expose 1, 1, 1 facets on which the graphene then grows. And so my question was, are the bubbles just the pyramids of the substrate covered with graphene? Well, I mean, we tried tearing them apart to see what was underneath them, but it got kind of messy. So that would be the obvious thing to do. So we were not able to peel it back. I think that there is not a pyramid underneath. One reason I think that is because one of the things that we've noticed is that when the graphene is touching a metal, actually in full body contact, like on the platinum or on the gold, then the features, this electron phonon feature, the Dirac point feature, they go away. They get swallowed up by that electronic wave function. And on these nanobubbles, we're seeing, as soon as we get to the edge of the nanobubble, all of these features come back. We start seeing the features that we expect for the suspended graphene. Whereas I would expect to start seeing the metallic, the signs of the metal underneath. And we don't see that. So that's a little kind of experimental answer. But other than that, I don't really know what the energy of formation is for these pyramids, that I don't know. Sure. I suppose the test would be to actually look at a substrate that was restructured to see whether you still see the same SDM features. And to see. But you're saying that then the graphene would have to provide enough strain to cause the platinum to do that. Well, if you form the graphene at high temperatures, where there is enough mobility, atomic mobility of the substrate, they do restructure into pyramidal. Oh, yeah. But remember, people have been looking at this at room temperature for many years. And so going from high temperature to room temperature, it's not seen. So it's really something that we saw as we went from room temperature to low temperature. OK. Thank you. I mean, the effect of your pseudo magnetic field will be washed out at a certain temperature, right? Proportional to the field, just like the Hartzmann-Alphen is washed out. I mean, have you observed such temperature dependence? Well, it's too big. The energy spacing is hundreds of millivolts. So I think that we would need temperatures on the order, you know, kT on the order of hundreds of millivolts to wash it out. So I see. Because, I mean, a normal field you can put on the Hartzmann-Alphen are washed out at, you know, about 1k or something. Yeah, but these, I mean, but we have these, but we have these separations now of big, big, you know, 100 millivolts separations. How much? 100 millivolts, or a few hundreds of millivolts. So that would be, you know, gigantic, enormous temperature. So. OK. But you should maybe at least see the beginning of some, you know, whatever it is shown. As we start moving from 4k to room temperature? Yeah. Yeah, I hadn't thought of that. Have you seen anything that close to the armchair edges? We see a random distribution of chiralities. We would very much like to see the armchair, because then there should not be an edge state. And so that would be very important observation. We have, I can't remember the closest that we got to it, but we haven't gotten the armchair. The armchairs are very rare. You know, they're the most rare, because it requires a zigzag nanotube, you know. And those are the most rare ones, a perfect zigzag nanotube, because we believe they cut axially. So I think it makes sense that there's not a lot of them down on the surface. But we would like to observe that. One way, actually, that I want to try to do it is, you know, this guy, Klaus Muhlen, who's doing his molecular precursor growth, he only grows armchairs. So this molecular precursor grows only armchair nanoribbons. So we're actually trying to do that to get the armchair. Relating to that, I think if you go to the really kind of smaller width, at some point that the bulk start kind of opens up the gap. And then you should, when you get into smaller width, eventually bulk gap is big enough so that you can pick up that the gap features from your spectroscopies inside of the nanoribbon, not only the edges, right? I wonder whether you can pick up those kind of bulk gap. Something like that, if I just look at the curve, say 80 nanometers, or 80 angstroms of the width, probably bulk gap becomes appreciable so that you should be able to detect that using that STM spectroscopy. I see. Yeah, I mean, that's something for us to look for. But I'll tell you a problem that we have. One of the problems that we have is that as we go into the middle of the nanoribbon, it's in close contact with the gold. And so then we don't see the graphene features when we do spectroscopy, when the nanoribbon is directly in contact with the gold. So as we go into the bulk, then it's washed out. So we need to take the nanoribbons and put them on a different kind of surface. Luckily, this curvature takes the nanoribbon off of the gold, and so we're able to see the exponential decay before the graphene reaches the gold. So I believe that the exponential decay is an intrinsic effect and not due to the gold. But as we go further in, we cannot do the measurement on gold. And now, from extra university, I just wondered how reversible your nanobubbles were. So if you heated it up, do they disappear? Do they come back again if you recall down? In the same place? Well, the problem is that once we, for this particular system, we do not have the capability to warm up and cool down and find the same microscopic location. So I'm pretty sure that we've taken, I'm 90% sure that we've taken the same sample and warmed it up and cooled it down. Yeah, it's actually 95% sure. That we've taken the same sample and warmed it up and cooled it down in the ultra-high vacuum chamber to room temperature and cooled it down. And look at other regions of the surface. We did not do a careful study at room temperature. So I expect that they disappear based on the measurements that other people have done, but we did not do a careful study at room temperature. We did all of our measurements at low temperature. So I expect that they disappear because then the differential thermal expansion is not so strong. But I don't know. I'm pretty sure we did not do that measurement. I'll have to ask my students again if they did the room temperature, but I don't think we did. Mike, thanks very much for your fascinating talk and for being so patient with all the questions. Let's thank the speaker again.