Master class I with Bart van Wees

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Master class I with Bart van Wees
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I will give basic introduction into the physics and technology of graphene, a one atom thick hexagonal lattice of carbon atoms. Starting from the elementary electronic sp2 bonding states between adjacent carbon atoms, I will show how the two-dimensional electronic bandstructure of graphene is be obtained. The role of the Schrodinger equation is replaced by the so-called Dirac equation, which decribes a two-component wave function. This leads to very rich physics and a interesting analogy with high energy physics. From an experimental point of view I will give a demonstration of the Scotch tape technique which made it possible to obtain single graphene layers for the first time. This made it possible to observed new effects, such as the anomalous quantum Hall effect, in field effect transistors based on single graphene layers. Various techniques to improve the quality and/or the quantity of the graphene layers will be discussed, including suspended graphene and techniques to grow graphene on various substrates. Finally a future outlook will be given.
Keywords graphene physics
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OK I think it's time to work starts everybody knew me yes sir so let me introduce myself for those who don't know me I'm mad but from a I'm from the group physics of 9 devices for all that only cases advanced materials in cloning and hair in open North also it's a pleasure for me there to have been asked to give this I have to give this masterclass on graffiti obviously you would have preferred company high giving it just so but uh I guess is busy but certainly motto you where you will be able to hear him so what I like to do this is to have a little bit of informal presentation so please interrupt the IAF also there a white board which I will try to use air so please interrupt me if any questions so I'm a actually aiming at all let's say a person was heard about caffeine has heard a few things about 2 feet and likes to know a little bit more especially also likes to know a bit about the big round what is all the fuss about this is now a major material or not etc. etc. so clearly curfew has has said he sees some new attention by the Noble Prize off last year given to Anaheim and Garcia Nova sell-off and we've carefully for groundbreaking experiments regarding the two-dimensional MeteoGroup goofiness I'll show you some groundbreaking experiments I can already tell you that those experiments are not necessarily very complicated so I will show to you some device of coffee while I believe it's coffee made from some kitchen foil cotton wool some care fight and I'll show you well at least I will I will try and convince you that this is related to goofy OK let's start on attendance you see his age very very young so the nice thing about this noble cries of course is that it shows that you do not have to be retired or a lot already in your seventies or whatever not productive anymore before you get the Nobel Prize you can already get it when you're in the prime of life and prime view of your scientific career and of course the holds for other high that holds even more From for Garcia but also because you see is only I have to say a little bit more there clearly the achievements which they have have obtained for dinner at University of much of Manchester but there is a very interesting Dutch connection and I will show that here because it's also related to several aspects of the work of of all the time and cost journalism so what you see here this is a big magnets it's from the Highfields make no method let in namely so what is it what is it is a bit of type of making it so why not go to bore you with all the details but basically it's bill that might basically a circular plate from which you couldn't is essential and discovered the cause of the huge because you see on any evidence it consumes about 70 megawatts adversity so you see that in this city built that is of course have great concern in that laps so what you also need obviously is to get rid of this dissipated power so these are supplies for water cooling and assisted return and anticipated it consumes basically a few the cubic meters wall of cooling water 2nd so what is not the goal of such a magnitude 1st of all to produce a large field that's critically but that is some also also some interesting features it added that it has a or as it's called which is at room temperature and so you can basically have a space of about in this case study to be made to switch rather than lodged in which you can put whatever you like N yeah I should of course he
referred to to N link is on the side because about 10 years ago he received what is called the ITC Noble Prize at the ceremony for ignoble price I believe is a few weeks ahead of you there the normal price and this is supposed to be some kind of parody on the noble price but with a little bit of serious undertone and because there this this prize is given for research achievements that 1st make people laugh and then make them think or well sometimes of this given for achievements that cannot or should not be reproduced so he received the 2000 a year 2000 physics prize together with Michael Berry Sir Michael variations say here and there because he would be a further role in the story as well a foreign investing experiments entities making levitating a frock so let me show you work and now I have a nice movie so it is is levitating stroke in this huge magnetic fields while I believe that the Frocks survive a the experiment a bit of course you could say Well what astounded what is now the firm's about well I can show you what is 4 is about because if you tried to levitate something he will accept succeed so as to make I tried to levitate 1 head by pulling by putting the North Pole against the North Pole and you see I cannot get any stable I cannot get any stable situation with basically flat and that is actually given there is a feeling about that and has called Burdeshaw theorem which says that you cannot make any stable levitation orders stable suspension of any magnets or any any air any electrostatic arrangement you cannot make a stable How would I say what leverage let me just show you what I mean well let me show you the continuing so this is conscious start shows theorem what is important is that you have if you have any magnetic forces permanent magnets or permanent charges you cannot make a stable arrangement of which makes things Leavitt TD this is an exceptional because otherwise
this would not let them so there's something special going all and a special effect is dire magnetism yet because the magnitude mechanism magnetic field actually induces circulating currents In the frog but actually produces a magnetic fields and you get show that the combination of these this McNair big magnetic field plus a demand the magnetic field generated by these dire magnetism can indeed produce a stable situation
so well what else can you make levitate let me just show you think this is a levitating strawberry you can actually do some more legal you can also levitate something and I did not know that it is was possible because the members of the certificate that slamming into the city big saying and this is just the tabletop arrangements so you have just a configuration of magnets for you this is the North Pole work this is the South Pole and that is the 3rd 1 is again both polyps Stealth bomber and what is floating here is gonna fight let me show you this 1 so here see basically piece of graphite which is again levitating fired the same principle and the same principle of of dire magnetism so it's not this dire magnetism there obviously is not a very set a very rare phenomenon it happens in most in most materials what is of course special is that you can do a tabletop experiment here and of course most of you probably have seen a very similar experiment but then with superconductors and than the Diamond that indeed very strong and because in superconductor is superconductors basically all magnetic flux is expelled the only part of it is expelled but nevertheless that it still works even with non superconducting materials here in the United States of America amendments the draft that is because the front moved in and out of and yet don't appeal it's a very good question because yeah I'm not sure if you have seen the experiments with the high DC superconductor because that was basically the same arrangement of course you have to call it a little bit with liquid nitrogen and then you also have some typical movement like this so I don't want any beyond what the answer is basically the that the motion is probably given by by some damping because if it starts to move them in a magnetic field basically you generate some so-called eddy currents which stands the motion so but I'm not understood this perceptual which is that if this is the philosophy but would only thing I can say is that this typical motion very much the same whatever you everything so local good fight so this sounds like a very boring material because as a matter of fact until a few years ago people were basically interested in graphite obviously for using it pencils and so let's pop Logan it is a very no material as a moderator in nuclear reactors because it slows down the nutrients which need a basic rule have to have 2 years of had the multiplication effect in nuclear reactors it is obviously has a layered structures and course that is the reason why you write you could easily take off layers of but but still call it graphite it is used in steelmaking industry part of the graphite which makes use for for making this casino devices actually comes from a from steelmaking industry while it's a good lubricants for the same reasons and of course it's also very important is very good electrical so this is a slight which I borrowed from my from behind because I think is a very nice 1 because it shows the history while here we have the graphite and clearly it's made of layers and these layers are called could and so what is what is isn't typical is that you have a carbon atom which is connected to only 3 days so that difference from diamonds because then we have a carbon atom then we have 2 carbon atoms sitting there to sitting there and that basically has 4 neighbors so in a very primitive way anything it is justified you can say well prominence for electors to make bombs In diamond they're all used and he and I have only free elections to make bonds with the adjacent atoms and then 1 1 electricity and can actually move through the From the lake obviously it can also move in this direction but it's a bit more complicated so if you measure for assessing graphite you measured the electrical conductance in this direction I think it's about a factor of 10 to 100 times better than in this is no material the same thing is that was what was basically known well basically the 1st let's ignore trivial material which was known as buckyballs and so they're called off the air buckminsterfullerene which he designed buildings like this you see again you have this this this terrible coordination here in this case you have alternation of of of effects of gold Pentagon and it's actually it's very much like a soccer ball that's actually the same exactly the same see 60 years exactly the same so this was basically the 1st type of a single trivial carbon material to be discovered and then delivered later people found carbon nanotubes they not very rare because if you have a candle and you have you make it a bit Psuty a new collector sold and you start to look carefully you find that that beasties buckyballs and discover nanotubes actually produced so it's not very what is a bit surprising is that this material basically was isolated only so late on the ball 5 years ago so I think the story is a bit of L given better story why this was so it also said it was personal not to exist in the Free state which is always a bit of a mystery uh use would be supposed that the material people say Well effect if I basically look at all the possibilities for letters vibrations and add up everything basically the whole thing diverges this material would not be stable but of course that's a bit of a taking argument because if you peel it off from some other material he peeled off from fight it might take a long long long long time before it's basically disintegrated could be a lifetime of the universe has so these argument is is not a very available this is a nice 1 was made by year by year Post Oak of mine where it was made somewhere in 2008 and actually describes the number of publication on goofiness function of dying pay you see high the high impact journal Nature Science and you see a bit more officers physically but also on a nice during office the obviously also so what you see this is the evolution of the number of people publication Griffin as a function of time and while he has not yet exponential but here you see some kind of exponential growth and of course I should remember remind you that this was actually made somewhere in the middle of 2008 so that the total number of publications into 2008 was already here and obviously we are well far far far far but this so and of course is the important papers this was the paper by the end of a of time actually who 3 guns this entire thing and this is why they got basically the Nobel price I will you but to look at it says electrical for elective field optometry thin carbon filters than Sesay signal this is a single-A and because the funny thing is that this experiment very likely was not the almost single layer of graffiti was still not in multilayer of and only 1 year later in 2005 data there well before the and together with another group would the show that they could measure Single-A and I'll show you how this works here we entered its 3rd micrograms of two-year graffiti early somebody you could say that was a bit too late but fortunately the that's as if the curve would really go like this and we would already have a lot of approach the end of the escarpment fortunately of course we still valid OK this is what I like to
work to discuss with you there obviously it's only a tiny fraction of what Kenwood what can be discussed about graffiti that I have to make a choice a also a bit of a consistent Joyce that's so let me I will give introduction into electronic properties of graffiti how do I make atrophying device of course how how do we use this this famous Scotch state methods what is this enormous quantum Hall effect which is actually the proved that you could make seawall at make em measure a device which is only based on a single layer of comments in a layer of graffiti by letter graffiti and you can go turn the story back if you can make a single layer could also to make to what is the difference how they should behave what book this is a game you can play with graffiti you can suspend if you can make a device which can to consist of a suspended signal laid off all and I will show you how it works and what we can do there are some nice games which you can play with graffiti electron optics optics I would basically call it I'll show you an example a few applications of defeat and then probably the 2nd hour I will discuss what we are doing with graffiti and not well of course of course we also looking in electronic properties and will use the charge of the election but of course we also going to use some other property which is spin so we don't to spintronics L and was do-it-yourself good OK there is a lot of mystery in a while there are a lot of mysterious words about conceding that almost looks like it's it's not the real material it's something magic uh however we should realize that is basically a very nice an example of a solid state materials so I would like to summarize a little bit what he's basically should know what should be remembered for from therefrom solid-state physics so 1st of all that we have a nice periodic lattice of atoms so that means that the electrons basically experienced periodic potential by write down its editions 1 coordinator of course this could be basically also are be explains he doesn't matter basically the potential is periodic on Wednesday let this period and now there is a very important theorem which is called Lost them that is you now want to calculate their the wave function 1 a mechanically you couldn't see item like this when you consider where French is a product of away from Xinhua here this is the way function which actually describes the way function on the scale of the of the unit-cell but basically repeats itself so this is through periodic way function and its modulated by some envelope wave function which looks like a plane so this is a this is basically the expression for wave function of the elected has this is a war already very interesting and of course he described at the uh that he this basically tells you that this at least this part of the wave function periodic with period but the entire function of course is not here because it's modulated by sector what is this thing this is the way factor so this is a very important parameter is also very important because it's related to property of the but of the of the electron which is called the crystal momentum would of course the momentum it's just given by by taking this wave number and multiplied by age by why is this so important because if I now apply a force on this election which is moving in its periodic potential it basically sits in this this momentum toward dying derivative of the momentum basically obeys Newton's 2nd law foot of the particles is strange because I mean this very complicated problematic elected feeling this periodic potential moving out nevertheless for this "quotation mark could well this quality momentum I just have to equation at that time derivative of the momentum is equal to the applied force which can be electric field it can also have this is still a loss for this very special features and also very nice feature what you can also get is a band structure what you will get so you will get us for each value of k value you off the way effect you have a set of states that a different energy and which are labeled by and which are called events and this is also an important 1 we have a group velocity if you were not really want to know that the velocity of the particle you have to take the derivative of the energy With respect to and I will show you how this works again this is good old solid-state physics From from left so
now why did I emphasize this because now you can look at the difference between a metal semiconductor and and defeat so this is the energy this is the momentum so this is a typical pop artist impression of a metal you see the energy the depends on a momentum while in a parabolic way it goes with the momentum square we immediately recognize this here because this is the expression for the kinetic energy and there's only if cost 1 important difference there because if you would have a free election of course the mast would be 9 times 10 minus 31 times the kg which is the loss of a the free-electron here it's modified his now we get an effective Mossad's modified that again because the selected is moving in letter but there is no problem he added this is simply a particle moving with positive effective this is a semiconductor what you see immediately that is something interesting happening what is is failing me you see that the energy gap so there is an energy arranged out states of course it's very crucial for semiconductor and above that you have elected states which you can immediately recognize again with this as particles which which have a positive effect of mask and these other holes and you see they actually behave more much much the same but not energy goes down with increasing momentum and it actually means that they have negative effect of Moss ball sounds a bit alter negative effective March and that it is purely a consequence all 4 officers the specific band structure Of course we call the selected states that because if you are if you if you now we have to introduce a thing which is called the Fermi energy and so it basically means that in in in semiconductors you can have the election transport would you make particles in the conduction band when you create electrons but on the other hand you can also take away the elections here from the valence band then you have and empty states they can also move on as well obviously they're called holes this and we the several different 1st of all it is not the metal and because it it has a very very special bends the chip because now the energy of the state's depends linearly although all the moment so it is very strange because then you cannot say that it has an effective master because if it's nicely quadratic we could say it says it has a disk positive affected Mosier the negative on what you read don't that is this is 1 April 11 featured the other 1 feature is of course that if I feel graffiti if go blessing of confused neutral then the Fermi energy of course I have to fill all the states until I basically used up all the elections but then it turns out that you sit exactly at this point and this is called the neutrality .period it's also called the Iraq neutrality .period 46 which I will just look at the dynamics lessons let's use Newton's law again suppose now I have let's say I haven't and that a particle sitting here and I apply a force in this direction of force means that the momentum will change linearly with so look a year ago no moment yearly and I go along this good for the energy starts increasing linearly but us solve this entails so deep decay remains the same as the velocity of the velocity is not changing its own applying a force to this there to this election and the velocity is not changing its costs so this is very typical aspect of the scene get a good feeling always will over the same velocity what is this velocity it's about into the 6 meter thousand kilometers per if this will be the only thing of caffeine it would be even why because the situation sometimes also occurred in the band search of metals and semiconductors maybe not at the Fermi energy but a bit tired a bit lower and sometimes you experience this kind of call 6 but then if it's block it arises of course from from a lot of trouble at St. Hambletonian insist so let's see let's see how it works OK now we
have to to do some physics again this is caffeine what you recognize is that 2 sublet the these guys and and and the sky so that's already 1 important ingredient so if you want to describe the if you were described their the way functions clearly you have to describe them on this side and on the site and what you're going to do we go to describe them together I like it this by the way is now suing the saying only 1 troops statement we
were saying OK this separation by the way is about is 1 . 4 2 banks and adjust to do due to be complete so to make their connection with the solid-state picture I go now again to support the reciprocal space so now I have to go into different directions an Arab ACT III whatever lower double he is said to have and not also is innocent separable space you get the unit-cell which looks very much like the 1 in real space and the 2 important points out these guys they had a K indicate Prime 10 How do you
arrive at events such as this was done already a long time ago what so what you do is use a tight binding model dying by the model is a little bit like a mystery I'm not trying to explain the microscopic a justification for this but you basically what you say it's OK I haven't elections which could settle on a site and beside has history a side is coupled to 3 B b sites and B side of course couple to free a sites so what actually did this term good does it actually moved an electron from the B side and it puts it on and adjacent a so this is sometimes called the hopping hoping hopping matrix element and the strength of this 1 is called the that's NT is measured in In energy so this is the 1 which could connects the nearest neighbors if you like you can also includes coupling between a sites and so this a company discovered side coupled with this 1 is couples with this 1 at center and of course also the company between the besides visited by what would you get from this was already calculated again in 47 in 1947 this is the mathematical expression it doesn't say too much it's
much better to look at the at the picture and so now we really have to be Picture This is the energy of the states that is the the it's This is the KY and the interesting thing is these are display indicate Prime points and you see that basically you would you you would have 6 if you go around and that this 1 is equivalent to this 1 this 1 is equivalent to that 1 and that was so basically we have 3 2 in equivalent something and you see a recognize immediately and what already bolted before this is your neutrality .period if I go you basically you get a cold so the energy as a function of the while deviation from this game .period increases linearly and of course you decreases
this is a OK this is
asymmetric you see you see this 1 instead of much bigger change in energy there for elections than 4 holes and the reason for that is because of this presence of this
deep pride a if you forget
about the but the PT pride for the moment you only connect you take the nearest neighbor hoping that you get a very nice symmetric bent structure like this where again you can recognize the 2 different so now that the nice thing about caffeine as I will show you a step in the neutral case all the state's occupied to defer the energy however you can very easily changed they're the charge in the graffiti can make a field effect system so you can clearly more states firmly playing up and down and then you really see what's going to happen if you will down to and said that the whole issue if you move it up you enter electors and that he did you can very nicely measure why do
call people call these elections In paragraph in Mosul you forget about that but it isn't but is it now Marsalis well this is basically what I already explained that if you'd of classical particles which can be characterized by some loss I have a kinetic energy like this which can reunite In French as a function of momentum like this and MCA obviously Is there is the mosque is only but there is only 1 loss in classical physics at you have of course Newton's 2nd loss to assess the forces should be given to is equal to the mosque and acceleration and it's a diving to derivative of the moment but now Einstein comes in and says Hey wait a minute this is not fully correct this is only correct for ferocity switch smaller than the falsity of light I should actually use this equation so you already see that I want to create a particle which doesn't move upward momentum 0 I still already have to putting an energy and COC square of which for no 1 expression and good if if this particle starts to move if the momentum starts to increase and of course the energy starts to increase according to these relations the interesting thing of course is that you don't was Newton's law still holds so if I apply a force on the relativistic particle still means that the force equal to the change of momentum as a function of the time it also then this was also still holds the velocity is illegal To the derivative of the energy respectable mentor so if I put here I get something like a broker so this will be abated energy this is the moment and I created electron here it will take some years degradation of energy non going to increase the momentum what's going to happen lowly citizens in this region is still parabolic has so this is the classical achieved but then you clearly see when this particle is going to approach the speed of light and then the energy starts to increase more more linearly with the with the momentum and of course you also would know what it means because the velocity is not increasing anymore you've achieved more or less the speed of light and the only thing which is increasing as the mosque so you recognize that for graffiti we had this many of its provisions so what is special alone about this because this was the energy we needed to to create a particle it was M 0 square it looks like you for graffiti all that these particles more-or-less imitate relativistic behavior except for the fact that an 0 0 will not need any energy had to create these particles so many lesson is that he's electors in graffiti be they behave like but of course we know and other particles which had this dispersion than that of their photos nevertheless elections even if he did behave like shots photons so there is not a real fault which are sold in free space but with a 300 times more like velocity and so everything is like you model a maker bait mate basically make a copy of a particles moving in free space approaching the speed of light and defeat a the particles would approach this problem would basically a hazardous a speech which is which is about into the 6 meter 2nd I cannot say too much about it that many many Nice analogies between the behavior of electors in defeat and and and high energy physics I'll just leave you with this match admitted to that it's something like a solid-state laboratory for high physics this part I like because he added because any non-trivial and I'll try to explain it to you so now there's this call like this dispersion come from and so what is now can we say something about it turns out that yes we can because we can describe the Hambletonian on a pro approximated close to either did you not while did the cave the Oct .period or decay prime recently said relatively simple the expression you see that this is the Hambletonian it's at the White 2 matrix it acts on the way functional on effect or which describes the way functional they sites and on the basis of the disease and special 1 and because it is only off diagonal elements it has some common combination of the momentum in the exam election and the wider direction and you see therefore DKK Prime well if the present mind 9 aside changed so the singer emphasizes that if you now calculates the way functions then they look like this I've basically I should mention I've forgotten here the exponent I take spotted which which which will according to the block fearing should be there I only look at this prefecture and now and I wanted say that he had something funny happens and then this so let's assume that we haven't particle which is moving with a certain momentum and assist the pigs direction this is the white direction so I define an angle the time like these so you're less like this 1 so it's the Arctic SE ties the Oct engine Of fixed component divided by the white components sold then you see that's it corresponding to situate to the direction that is a the face vector for the way function on the a side and there's also face facts of waterway functions on the eve of the side well you could say so what let's assume that the guy is 0 yes so that we get all lot said we had something like 1 1 so that would correspond with a particle going this way now we increase c . 2 so now we get this direction I think we get and the other 1 is so -minus I think to be minus but now the interesting thing is now we make the cut to buy which actually means that we have this particle which is moving in this as indirection 1st move is urging the status of moving this election for some reason go around and then we're back to you in so many would say Well we should get the same way function and because it's describes the same state lies in and so if we put to by in this 1 we get to buy over 2 exponent minus and he also the so like the wave function is multiplied with effective of mine 1 and you could say well who cares about that factory lines 1 week because we know that we always have to mobilize way Finchem said we have to normalize its reports some normalization factor in involves so well and indeed the wafer devolved into this is the starting with functional he built applied by minus 1 who cares that turns however to be very very important and as establish late to keep that in mind and that theory experiments
"quotation mark statements what is it 2 1 0 there will be a hands-on demonstration of so you can you can try the Scotch tape as it yourself to see if you're good at it let just explain it power works so this is a nice piece of graphite it's nice crystal this case it's a joke which highly ordered oriented political fighters of the United now comes the take so we need so much sticky date doesn't have to be scotched the idea sticky tape and reduced the pressure told the on the site nothing special that would be laid off so remember cooking if it is this weird of what he got was going to happen well you only need not be surprised that you will work pull off a flake which is developing the thick of thousand of thousands of layoffs all sometimes it'll break already have its business in general you will get get something which still contains thousands thousands of layoffs now
comes the thick playmaker substrates which consists of silicon silicon oxide so let me let me just inside here this is silicon and then we have a layer of silicon oxide and and here we have the flake for many many many many many caffeine layers and he was a sticky tape and we're going to pull of like so what would you expect if you look at this of the many many flakes start to pull OK well maybe pull of completely ordered about you will a lot of players will stick and then maybe thousand layoffs and a little bit further all the other players and then I know but if that will be the case this entire thing would not work Hey because you will have a collection of junk on a substrate and you you you you might just by chance have a small piece which is just 1 layer and that he will not be able to find it the trick is of course that if you put pressed status this flake it's still relatively flexible if you press it against a substrate then I suppose that this last defeat layered likes to stick better to the substrate then to the next layer have because we already know that the sticking between the layers is not so great then there is another big chance that you will pull off just year steeple of all the flakes except the last and of course is not going to happen all the time a lot campus about 10 per cent of the cases out of habit that you have a relatively big chance off of producing single and this is 1 of the tricks of the Scotch tape that is much more magic to it because everybody has his own technique and that this is 1 of the ingredients if this will not happen he would not with Hyman Novoselov would not be able to to get that nice itself so you can OK this
is the microscope at you see on microscope you see a lot of interference nice color and this is very bright 1 why because it's relatively thick and this is still a thousand thousand late so he be really get interference of the light and which is scattered back-and-forth is also scattered back forth in this layer so we have the silicon oxide layer and also denied that the success of this also matters because you can imagine that that if you choose your players in such a way that you you match it with a number of years of wavelengths and that you become more sensitive to what you put on top and so this has also been up there this is also a bit of of the takes you have to use it now look at these guys he had this looks like the bright that the to the lightest you can find the particle that is not so good so you might say Hey if this is the lightest which I can find it cannot find anything else this might be single-A right so what you do used now atomic
force microscope so this is now and where you have basic you tape which wants to keep the same force between the substrate or whatever you put on it and it did so of course if there is a goofy layer it has to go up to this to go out there Morris said the etc. and you can make a picture of the heights of This Is this contrast basically indicates heights you see that is already quite soon could engage in cases such as this because of silicon oxide silicon oxide is not flat has some small variations of Europe and and here there is the graffiti on top so you see you read the graffiti actually follows the the congregation's here we see the next layer the next Leyland we go back to Israel so we scan if its candidate here that a step here there is another step pianist deputy collectors here and this step is in this case 0 . 2 9 nanometers which corresponds more or less With the thickness of just 1 1 good now you will immediately say Hey wait a minute you know fooling with because if I go from here to here I goal from the theme of carbon character photography but if I go from here to here I go from silicon oxide to graffiti to differently so the interaction between the tape and the material might be different so this height might not necessarily be the realize that it could be bigger or smaller so this experiment does not really proved headed if you if you look at this 1 that this is single-A Is it basically proves that you hear you go from any layoffs to and was 1 and 2 at the school that you can also use it as a proof that you have a single-A so have to do something else so let me see
luxury what we do now would now it gets a bit more expensive then there the the problem with this technique
a lot of small flakes and so this is about should mention this this is about the not that Michael me so that that that that it's not going to be very commercially relevant technique and because you have 1st as to act locate where they are so this is of course a great of markers show you say Well it's between Marcus 6 on the of between 6 and 7 on the x-axis and between what is at 4 and 5 and here we want to make it better so now we have to be
more expensive part see so we need UBL so we have just electron-beam this basically the electron microscope In but instead of scanning the election you like in TV a Yukon actually program the electron beam only to scare in those areas where you want to make Her contacts and it is done here and so here we have a good few Flake which happens to be very very elevated and we have told the machine anyway minutes scanning electron beam he had here and here because we want to make contacts the and all these position this is a device we
made we stood at about 4 years ago it's a fellow magnetic devices should mention that the effort along with medical a close and I will tell you in the next hour like we did this you can't really recognize the coffee inflate this is again a picture was elected microscope so you the reason why you can't see it so nicely it's only 1 layer thick that cost the elections which are scattered from the graffiti had discussed scattered differently than the silicon auctions so that is why you have such a nice contrast between this single-layer graffiti and silicon oxide substrate here we have contacts and this is a typical the scientists
yeah all this is made with electron beam lithography and that is you can't see because in our experiment we have to get the spacing between the context relatively short like a Michael smaller you see that these contacts on all of with this is much smaller than micrometres of this kind dimensions it's difficult to to obtain the optical with life optical lithography so this is done with elected me because the election be weakening the focus is relatively small I want to use this process what is this that what again the good question and in most cases today it and so you can imagine that of course this this this this and this solvents can create underneath the silicon oxide and the graphing and all all of that sort that is what sometimes happens right away because we sometimes he is evident in most cases it will it simply stick the remainder don't ask me a citywide that that's that's a lot of chemistry in this is a caffeine field-effect consists of so this is history this is why actually works and it's a bit of a mystery I have to say it's a bit of magic here because what we do either of already bolted there normally what you have is a field effect insists its silicon silicon oxide and we have metal gate of have because you interested in st in inducing carriers at the interface between the silicon silicon oxide so this will be the gate this will be the interface and we apply gate voltage between here and so we wanted to control the charge here this is the In the 1st because we now I've made this this casino pop of the silicon oxide and now you want to change the charge in the in the defeat for what we do now is we use silicon again but it's very highly doubt so does no behave like a semiconductor it's basically a metal and now you can see this is this is a nice a few hundred nanometres insulated from ECU basically make a capacitor between this metal metallic silicon and and and the graffiti so you apply all you connect this but this is what you connect finally to work to some voltage source you applied voltage to use silicon relative to the typically attend vaulted the fault or something like that and the ideas of course if you apply positive voltage she will attract negative charges defeat at least if we have the firm energy sitting here if you want to get negative charges and we move up the Fermi energy if you and negative gate voltage we expect positive charges we toll for managing the graffiti goes down so that is a very nice thick what is also very nice if that's all you can already understand that that the caffeine all this the all the states we have talked about the stated the graffiti so old the charge which you induced in the graffiti is going to be used as for electron transport is not that you have surfaced states where where electors can sit and do nothing it's not bearing so that's a very also important to be sold in the interview that he had invited to to measure that exists between the source and that the rain got to past credits from here to here and we're going to change the voltage which apply to the gate so this is a
typical could be called as the darker and the reason is of course that this is the good feelings or the existence and he retains the gate voltage so you immediately see a bit of nicely submitted occurred when we reach some kind of maximum level is another reasonable that we do that then the answer is yes because I suppose we applied bake negative gate voltage a week ago to attract a lot of calls and says we have a lot of whole carriers that means that the resistance of the 70 million relatively on the other side also we apply positive gave voltage we attract a lot of electrons s so that justice will again be a relatively low you see that's more or less a Medicaid because showed you that if the but they only the hopping is only between the a and B sites and then we get a nice bent structure which is symmetric for elections and hopes that looks like in most cases indeed this is the case here because you really see pop it's not fully symmetric it's almost what is surprising you may and I should emphasize this form is disappointed at this point we should achieve neutrality and so we're moving at the Fermi energy all this is the maximum we can achieve so we set exactly at this the upward . while you see that we have we need to find out of gate voltage for this and why is that because there is always some background charge present finances in the oxide you have to cope inside said for that charge and you do that basically there is no charge anymore in the graffiti and then you might ask had this reaction to this the . is only a point fall .period how can still be there the financing 270 shouldn't have resisted can be infants so this is an important question was going these are measurements
of this uh property and I should mention that I should mention
that the value we have this is typically a few key now you might say Well cares about the long slow I do because they said it was the worst was 1 . 4 4 combination of natural call center stage in the square which has the units this is given the bank's calls of his election charts squared if you calculated is 25 . 8 kilo so this is very important quantity it's the unit of conductance fought off resistance effort into very irrelevant Cornwall effect and you see now that the peace again Union bollards not exactly it it this is some fraction of that value by the way you can check for yourself that this the I made myself also has a few so there might be some express a very special reason for that so let me let me see this is the Dokic these are measurements where we have changed the property of the graffiti of what we have changes the quality and this is measured in mobility and we see that this this maximum "quotation mark maximum resistance minimal connectivity is always around uh this 40 squared if you have time I cannot say too much about what I'm Oblast I think
give you that I will skip this part in explaining why the care just mentioned a few
things Seoul this is this is
not a parameter which renewed his call the density of states how many states do we have available for the transport as a function of the energy give said exactly that that up .period Yosef created 0 about 2 . 6 0 to go away from the deal I want you know that we have these nice called going this way and go the other way it goes like this so it is you can actually show but the density of states of the number of states available in energy range increases linearly on either side and now the nice thing is of course well can we know in the city here and there are several reasons why we cannot 1st of all is that the potential is never flat as fluctuates a little bit so this entire diagram fluctuates a little bit as a function of position so if you then calculates the average density of states it will always be fine that's getting going on materials not perfect so that basically means that all the states have a finite lifetime and now we can use some kind of answer the mutilation we say Well Delta II is something like age over the lifetime and sold the shorter the lifetime the bigger the broadening and again if you will broaden all the states that kid also here that you did density of states will increase confine finally there is the effect of the I am not going through this calculation but you can actually show but if you take into account this finite lifestyle of the states in the region 1 of important reason is that the elections they they don't move for a long distance they scattered already offer something like 30 nanometers wide because of all kinds of perfection if you take that into account you can actually and that was the conductance which is a this 2 dynasties spread rates multiplied by some affected by and the reason is you can that some kind of compensation mechanism so if the materials key and there is no broadening the density of states very long now you start to induce scattering soldiers there so so you will get broadening of the energy levels of the for the density of states will start to increase that because you could say that the density of states on either side is also going to produce a final density here but of course in the end in the same time the dynasty scattering time becomes more and that compensates for so there is some fundamental argument why for a single layer of graffiti at neutrality .period you would get the conduct something like that and again that's dirty piece of material also business so the question is is that now also single-A of graffiti already niceties but it's not yet enough proof that we have a single layer of graffiti and because we haven't certainties of effective tool have shown you this ayurveda measurements were when the 1st step you're not certain about 3 measures the height so hard now and we do not have a resolution he would say Well you put a nice electron microscope on it so that you can really all the uh the individual atoms you cannot do that you can do that in transmission but not in the air and let's see if you have a standard electron microscope but does not have the resolution has sold so I'll OK we now get approved that we have a single and this is a very important experiment which was turned into a thousand 5 sold 1 year after that the 1st experiment it was done by again Hyman an overzealous but also by the group of 1 them of the announced which 1 this 1 here and
so the from here 1 of the main reasons that goes becomes larger than that those this is the expression of this density of states as a fraction of the Epsilon if you if you go away if you increase the energy so eventual this increases linearly so if you say that this neutrality .period the density of states should be easier right now my argument this is material has a finite we lost so that hazardous states have a finite lifetime but always won over this is usually passed over the firm's velocity because lecterns moved off a certain length that means ,comma how long they take to travel the length which basically this expression this gives you some energy broadening its Delta something like age bar over told the Tribune and so basically we will be basically put in this energy broadening in this 1 it's a bit of a hand-waving waving argument but that and we again OK I should mention this situation I forgot to mention this equation because this is a very important what is in the eyes of relation at all the collectivity is the density of states multiplied by by the square times of parameter which will be used this year's review because the person really goes through this means the past goes to infinity then obviously it becomes more and more perfect union density of states and to the point becomes smaller and smaller and but nevertheless if you evaluate this formula and its CEO energy this 1 goes to 0 but this 1 goes to infinity because you read boss electors can travel for whatever longer distances and the and then still and there was something like that and waving arms have to say that this is a very different if you in the middle than your density of status almost constant then of course you see that if you if you reduced to me the parties to material becomes dirtier than immediately upon conductors goes up and that he had because if he said that this year .period because of the special situation this is what you always get can you can change a factor of 2 were effective 3 of where the site that's the diffusion constants they tells demotion since the means the boss is the motion of the selectors for videos diffuse 39 which is much smaller than a typical site collectibles like this and those living like this so this is diffusion and this motion is described by diffusion constant and is given in 2 dimensions by half time the firmly velocity of title because again this is the typical velocity of particles and this has to be year of because it is that it is the 1 more never that in the event of president union and the composition of the absolutely it's not quantized an innocent and Serbian exactly 2 weeks grant overreach but there is a sense that there is an argument why this should be called off by itself this order of magnitude there was here yeah this is also a member of the company also the on going I didn't have a good question In this type of preparation of graffiti could most nights but it's always prepared on the substance and the substrate is silicon oxide is a said it's not flat so there is always getting along and this type of scattering actually produces disbelieving pop for instance be determined by the static potential fluctuations and of course you can say there is other contribution which is due to the fall onstage to the let's operations but that is usually not strong enough to add to what an issue he did very much it's not determining this music so casino on a substrate is basically uh that he did the me the bosses East this is determined by the scattering from the imperfections and it can also be some junk on top if you make a good fit suspended which I will also show that he can make this wheezing much much longer I can move on this is not enough evidence
for Single-A what was done in
2005 by kind of a sell-off but also from a group of given what admitted is 1 thing and now I'm trying to do well forget about and just tried to focus on this 1 so what we know is if we have let her have to schedule a bit emotional we promised the which so so I bought a certain couldn't from here to here I measured the for the whole voltage and I can find a halls resistance which is you know that I know I can also define a whole conducted switching of over and that is given should be careful sized energy giant the story so this is so that it could make sense so if I pass this current animated his Hall voltage fight inquiries the story of a yes if I increased the
density of obviously my whole conductors go to increase and because the market if I increased my magnetic fields and my whole conductors is also is going to decrease so this is the standard behavior is nothing special is simply don't the Hall effect In this case into dimensions soul if you properties as a function of mainly look at this 1 this is the collectivity as a function of and this 1 is a function of and I should get more or less straight lines this is the classical they're all effect what that in this case is the quantum Hall effect and I will explain what it is that you can just show you why it is a bit strange because this is a 0 there and this is this is 0 hole conductor now I gets a plateau and this plateau forms and one-half times this 40 squared away why is this voice but overreach the symbolism it's 4 times the elementary units so I bet apparently have 4 possibilities for this doing well I already had 2 valid selectors considered 1 rally and they considered sit in the other guests effective too but I also have a spin can has been principal you always have to take into account that each state can be occupied by 2 elections for the president of the gives another factor so the elementary unit is 40 squared away which would recognize you from year to year which is see the 1st step that is only to be squared away so that already deviates from the norm long-haul because then all the steps would be the same year the 1st store with greater range than the next on you would you act spread rates than you do at 4 you go to 60 squared arrangements for go to bed so it's an enormous sequence of also let us switches actually prove that you have seen a layer layup of and I will try and show you what it is some more about the
the the the simplest way to understand the quantum Hall effect is to take a sample which is fully so we have a two-dimensional sample this is extensive wine is perfect and we applied perpendicular magnetic fields the rest of the system I was going to have a lot inside there then there the material now elections are going to 2 touring around it so cited in August so they are pretty well but useless didn't because they're going nowhere just make circles that said if there are they are at the idiots it's a different story because imagine now there what he wants the election was let's let's forget about impunity for the moment unless you look at this election goes like this is reflected in our hits the boundary it president flies again so you see that at the edges you get Sokol skipping or its electors are not actually producing channels where elections on this page can go from there to there and all this image can go from there to there it's exactly the opposite so this is I think 1 already 1 1 important ingredient what is also ingredients imported please just remember this and this is related to wider upon white wine something like a quarter can be selected suppose there is just 1 impurity which what which wants to interfere with this nice BA well elect incomes "quotation mark around and that's kept back makes 1 circle hits the impurity again and continue to slide impurity doesn't do anything here so as long as the electors which move along this section are not able to to go by a series of skating events to this edge you and sell the security of doing nothing that is the reason why qualification be so accurate and can be let's say although of course samples real samples on what I want to emphasize here is that you if you do do the calculation you find that they make cited an albeit with a frequency some policy which is given
by the be all and stock and so again these are particles regular particles which have mosque they would behave like this so the cyclotron frequency does not depend on the energy so how do we get this is the classical picture how do we get the quantum picture of well I started of made Linux transparency but
I'd like to to show it I am periodic motion with all modesty that's the frequency so what I might want levels well let's assume that this will be there at the bottom of the Barents Sea particle would not bold ideas get a nice set of states of the pieces a half the the half part of With the half so the
energy is any stress a half the universal there so it's it's periodic mentioned efforts like Moore was like a harmonic oscillator if you .period you get the levels for less than half of the year monikers like that where quality this is the energy this is my long low-level secrets so this is to see you could say this is the 0 . of this is the half this is the the hospitals said where else I should say something if I now start to approach the he gets so this is a function of the guiding center of this of show you the guiding
center of this decided that or so in the center obviously there's another way to put it reported here already the energy is not going to change so if I do the quantized version of it should for says here this is my series of along the levels as they're called for security if I start to approach the edge situation is changing a bit because then you see that this periodic motion not decide this circle emotionally disturbed I also get the premier compulsion but now they are skipping the collection show the the energy goes up so if I if go toward the edge the the energy of these low levels go up like this so why is this not so important because if I now change my firm Unity supposed to here then I know that I should only worry about the state which at the Fermi energy all states below the Fermi and fully occupied I had a little worried about it so the only way to get elected transport used to play a little bit with the state's here or play a little bit the state's you but these are of course exactly this categories so if I put in some more elections here I get in that could this way if I put them some more elections here I get in that condition so so this is a way to understand the qualifications of the jets but the remember the sequence you get is this is the way that you need a little bit energy to reach the 1st low-level then you have a nice a fixed sequence of of state what happens in good faith the different and because in this field then the velocity of the particle is only makes the same it's always the same velocity no matter what energy usage has sole so let's let's not this Ivanov tried to plot different the trajectories both sides at an orbit and different energy but the velocities are always the same it means that if you have if you go if you reduce the energy that and then you go closer to the center .period clearly the site frequency is going to increase In a particularly it's going to diverge become become infamous because if you sit very close to the airport you can the election only has to make very visible on this is a very complete this is already important difference between the policy that the and standard-setting and if there's not a
lot and I would like to do this before the break and that's the very very I've already told you a bit about the fact that if an elected in graffiti moths in a certain direction is more sensitive momentum then in the way action itself than to change his face which looks very much by the way I like it a expire so if you have spin pointing in space you can also be presented by 2 components to speed up and spin down component and a superposition not basically superposition of that and you can make any that any angle in space so this this and this quantity is called so that sort of thing so why is it important about this is a very they face being I am close to the North Pole yeah and basically I start to work to moved 4 this is and this could be my my left then I start to move to the site of the election of my long love left on the year then I moved backwards again and so whatever have for work like this and I moved to the site now and then moved like this so I basically said well and this is my diamond like this like this and then there so got my Celestyna still pointing in the same direction but if I do it on the stadium is obviously not true because this was my left arm you go like this and not really I had to have an angle of degrees if I if I use let's say this part of the that is an example of the benefit sold them the relevant 1 here is the very face for spin half bottles this is now electoral units and medics the lack speed I applied magnetic fields so now it's a let's say aligned with few then I moved field spins goes to move like this and it's slowly then I field like this then I'm off the field like and I'm back to the original situation so I would say Wait a minute by the way function has not changed that is because I mean I'm backing a go like this and I go like this :colon and go back to the original situated in the same direction but of course remember what can change is the face factor the community like his mind is 1 which shown some face that involved in that X is exactly what happened you can show this was being a half particles Barry faces a half times the solid angle and so this means that if I wanted to have a spin which actually most in the plane which which I am have magnetic fields which have made little fully and the spin follows this magnetic fields then the wave function requires an extra Fayville 5 Hey wait minute the Member now and that was exactly the same situation which we sold it when we had selected in which might have momentum basically went like this and then goes around comes back but that is exactly that this kind of
motion and because he is also the momentum of the particle changes animation goes back to the same situation but it has made the city as long as it to buy rotation and because of the analogy between this side of Spain and Israel spin this bad faith actually tell you that an extra faced by this is the last thing which I will show before the break but is it every now brought along the levels in graffiti we find now that is isn't the level at 0 energy because he was not allowed he said If you particles of the mosque mutant this this year on motion but basically now we have an extra phase of by which which which which we take into account so that the total face into energy already this year so that means we have it is there .period leaders effect we have to to lie low levels 1 an electron like lower-level which is actually a lover level which has moved down due to his duties very face you could say 4 holes normally 4 holes was fired Moss you have a sees same series but then you go down all this now has moved up so we have this special policy at all not low-level sitting there would you also see is that the energy spacing is not called as I've already told you that is because it is for for for small entities at the elected only has to make a very very small cited in all rich and then frequency because much larger and then if you go higher let's stop and have a break until thank unless the questions congress from which it has to be formed because of interventions that goes on and on things that structural review might deviate from the U.S. operations and still service number levels of no it's not correct so where I did not say that the Bensinger deviates because the band structure is determined to reveal a lot of holes if you ever wondered about that and let's say this bothersome inanimate so that is that's already have learned uh let space has sold so invincible these imperfections on escaping only only modifies while it does not modify the band structure with the basically gives rise to excess skeptics so if you look at this picture of what is the scope go to do lots except the picture which showed here and this is
the way in which House Cantonese of most of the cases the elections are indeed making these nights the motion as if there were a perfect profane and that then had some here at at some locations that when can and it's pretty clear what you need to get this not disqualification want this motion to be completed the Soviet space the elections should be with the scattering time they should be able to complete this motion so this is a good criteria like all my transport should be larger than what so that is why you general you need also to other behind fields because of how magnetic fields are high at the site of frequencies very very high so the electors connections completes their books before what was it please you can find you're
going to tell the eukaryotic which has 2 layers of it and also that of the finale of the system