Master class I with Bart van Wees
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00:00
Homogene TurbulenzDiwan <Möbel>DimmerThermalisierungDruckkraftDipolSchwebenLeistenMagnetisierungElektrizitätWindroseElektronenkonfigurationKalenderjahrCocktailparty-EffektElektrische StromdichteKlangeffektETCSPlattierenSatz <Drucktechnik>Kette <Zugmittel>MagnetGruppenlaufzeitHobelTeleportMagic <Funkaufklärung>DiamagnetismusWocheWarmumformenWasserdampfKombinationskraftwerkAmperemeterNyquist-KriteriumLeistungssteuerungScheinbare HelligkeitEdelgasatomPatrone <Munition>ZylinderkopfInternationaler Astronautischer BundGleitlagerWasserkühlungWeltraumSteckverbinderFernordnungMerinowolleFlorettElektronentheorieMechanikerinFuß <Maßeinheit>BombenflugzeugAtmosphäreColourStücklisteErdefunkstelleComputeranimation
08:11
KanoneAirbus 300ChirpSonnenuhrOrdner <Schreibware>Hydrodynamische SchmierungWechselstromgeneratorErsatzteilPatrone <Munition>Elektrisches SignalBlatt <Papier>Optisches FilterTeleportDünne SchichtSupraleiterOptometrieBombeTagGleitlagerSchusswaffeMagnetisierungElektronGleichstromKalenderjahrUhrwerkTiefgangKugelschreiberElektrizitätKlangeffektDrahtbondenMessungFACTS-AnlageSchwache LokalisationUnterwasserfahrzeugUrkilogrammFaraday-EffektSatz <Drucktechnik>Widerstand <Elektrotechnik>KernreaktorWirbelstromverlustBleistiftTransistorAngeregter ZustandImpaktMultiplizitätFraktionierter Quanten-Hall-EffektLinealAtmosphäreBauxitbergbauKristallgitterGruppenlaufzeitWeltraumStrangenessDiamagnetismusStrahldivergenzQuanten-Hall-EffektElektrischer LeiterKernenergiePotentiometerVideokassetteChirale AnomalieBandstahlWarmumformenRutschungProbedruck
16:22
SchneckengetriebeSchwimmer <Technik>Comte AC-4 GentlemanHandbuchbindereiFahrgeschwindigkeitFestkörperEnergielückeHomogene TurbulenzMunitionQuadrophonieGruppenlaufzeitPuma <Panzer>UmlaufzeitAmpulle <Technik>MinuteAngeregter ZustandElementarteilchenUmlaufzeitDruckkraftAngeregter ZustandTeleportSchiffsmastMetallschichtHalbleiterNanotechnologieKernstrahlungKalenderjahrMittwochSpinSpannungsabhängigkeitMaske <Halbleitertechnologie>GleichstromFahrgeschwindigkeitKlangeffektSpiel <Technik>SatzspiegelMaßstab <Messtechnik>Negativ <Photographie>KugelschreiberLocherBildqualitätHobelBriefumschlagSpezifisches GewichtMark <Maßeinheit>BandstrukturScheibenbremseGrauStundeDrehmasseOptisches InstrumentEnergielückeKristallwachstumErsatzteilBiegenGasturbineElektrisches SignalColourValenzbandSatz <Drucktechnik>Fuß <Maßeinheit>LeitungsbandFestkörperLuftstromAprilSpintronikMessungLinearmotorFaraday-EffektMagic <Funkaufklärung>Effektive MasseFlugzeugträgerMasse <Physik>WarmumformenPatrone <Munition>
24:32
Reziprokes GitterModellbauerHandbuchbindereiChirpGrauElektronAngeregter ZustandSchwarzes LochKristallgitterMasse <Physik>KopfstützeAbformungRotierender RadiotransientTeilchenMetallschichtModellbauerGasturbineChirpElementarteilchenEnergielückeSchalterFahrgeschwindigkeitMinuteDrehmasseAngeregter ZustandSchlauchkupplungGrauDruckkraftGleichstromKalenderjahrColourSteckverbinderKlangeffektTrenntechnikHimmelHalbleiterKristallgitterSpiel <Technik>Anzeige <Technik>FeldstärkeLichtgeschwindigkeitReziprokes GitterBeschleunigungLocherPatrone <Munition>LiegeradFeldeffekttransistorHobelBiegenMasse <Physik>GasbohrlochBandstrukturMikroskopSteckdoseFACTS-AnlageFestkörperSchwächungBahnelementMatrize <Umformen>Diagramm
32:43
Satz <Drucktechnik>NasspulvergießenVideokassetteGleichstromElementarteilchenAbwrackwerftZylinderblockLuftdruckRelative DatierungAngeregter ZustandAnstellwinkelBahnelementKampfflugzeugPatrone <Munition>KristallwachstumLeistungssteuerungFeldeffekttransistorWaferElektronisches BauelementMotorVideokassetteDVD-PlayerPolysiliciumAbendDschunkeFACTS-AnlageErsatzteilLeistungsanpassungDrehenXerographiePhotonikKristallgestaltPhototechnikKalenderjahrHandyElementarteilchenphysikWocheMatrize <Umformen>WeißKernstrahlungVideotechnikBauxitbergbauKlangeffektEdelsteinschliffChirpFahrgeschwindigkeitBasis <Elektrotechnik>SchwingungsphaseLeistenKombinationskraftwerkMikroskopFestkörper
40:54
MikroskopOptisches InstrumentKalenderjahrPolysiliciumMikroskopLichtInterferenzerscheinungBuntheitElementarteilchenBildkontrastWellenlängeDruckkraftBesprechung/Interview
41:54
ProbedruckVideokassetteBildkontrastTransistorKraft-Wärme-KopplungETCSPolysiliciumFeldeffekttransistorTeleportMikroskopDruckkraftPhototechnikMinutePatrone <Munition>FernordnungKristallgitter
43:51
MikroskopErdefunkstelleBlechdoseTransistorKlangeffektAngeregter ZustandGate <Elektronik>LadungstrennungSternsystemStarkwindMessschraubeSchnittmusterFilzschreiberAngeregter ZustandSpannungsänderungSpannungsmessung <Elektrizität>PolysiliciumKlangeffektErsatzteilElektronenstrahllithographieGate <Elektronik>Patrone <Munition>ProzessleittechnikMaschineMagic <Funkaufklärung>Faraday-EffektSchaltschützFeldeffekttransistorBildkontrastStundeSource <Elektronik>ElektrodeOberflächeTemperaturabhängiger WiderstandFlugzeugträgerNegativ <Photographie>Kraft-Wärme-KopplungMetallschichtElektronenmikroskopElektronenstrahlSchlichte <Textiltechnik>UnterwasserfahrzeugColourLadungstrennungVersorgungsspannungPhotolithographieBrennpunkt <Optik>RegenSchwellenspannungDruckkraftMinuteMikroskopKalenderjahrMotorsteuerungHalbleiterVorlesung/Konferenz
49:03
ElektronLadungstrennungMarsflugFeuerwaffeLadungsträgerbeweglichkeitMinuteElektrolytische LeitfähigkeitGasdichteAngeregter ZustandPuma <Panzer>Tau <Seil>MonatWiderstand <Elektrotechnik>MetallschichtReaktionsprinzipGasdichteHintergrundstrahlungSchwellenspannungGrauPatrone <Munition>ProfilwalzenKristallgitterTeleportQuantenfluktuationElektrolytische LeitfähigkeitKlangeffektColourFlugzeugträgerChirpHerbstWocheNegativ <Photographie>EnergieniveauAngeregter ZustandUrkilogrammMessungPostkutscheKombinationskraftwerkPassfederTemperaturabhängiger WiderstandSpannungsänderungHandyBildqualitätSpannungsabhängigkeitErsatzteilSteckverbinderLiegeradLocherErdefunkstelleBandstrukturTemperaturFraktionierter Quanten-Hall-EffektOhmsches Gesetz <Elektrizitätslehre>KristallgestaltUnterwasserfahrzeugA6M Zero-Sen
54:13
NiederspannungOhmsches Gesetz <Elektrizitätslehre>Widerstand <Elektrotechnik>MetallschichtAngeregter ZustandElektronenmikroskopEnergieniveauTeleportElektrolytische LeitfähigkeitPassfederAbstandsmessungGasdichteAngeregter ZustandGruppenlaufzeitMechanikerinSensorDiffusionStandardzelleFahrgeschwindigkeitAtmosphäreVideokassetteGrauAbtriebswelleKalenderjahrFaraday-EffektMessungElektrischer LeiterElementarteilchenProbedruckWerkzeugUmlaufzeitFeuerwaffeKlangeffektZylinderkopfRasterkraftmikroskopieTau <Seil>Schlichte <Textiltechnik>
59:23
MetallschichtMitsubishi LancerWiderstand <Elektrotechnik>DrehenBackenbremseProbedruckLunkerComte AC-4 GentlemanKotflügelZyklotronUmlaufbahnEisMinuteÜberschallstaustrahltriebwerkPelzwareTeilchenKaliber <Walzwerk>Hulk <Schiff>FeldeffekttransistorKraftstoffverbrauchFraktionierter Quanten-Hall-EffektNatürliche RadioaktivitätSpinPatrone <Munition>MessungProfilwalzenElektrolytische LeitfähigkeitStrangenessGasdichteKlangeffektProbedruckQuanten-Hall-EffektChirale AnomalieAngeregter ZustandQuantisierung <Nachrichtentechnik>Hall-EffektUmlaufbahnFaraday-EffektZyklotronElektron-Phonon-WechselwirkungDrehmasseSchmiedenGruppenlaufzeitFeldeffekttransistorMasse <Physik>ElementarteilchenLichtstreuungSatz <Drucktechnik>QuantenfluktuationPolysiliciumNiederfrequenzParallelschaltungBiegenUmlaufzeitKraft-Wärme-KopplungKalenderjahrEnergieniveauWirtsgitterKopfstützeElektrischer LeiterSchaft <Werkzeug>BehälterNachtWocheSpannungsabhängigkeitElastische SpannungErsatzteilStoff <Textilien>WindroseBand <Textilien>EisenbahnbetriebHerbstHomogene TurbulenzAugustPresspassungDschunkeTeleportBettLocherFrequenzsprungverfahrenGigant <Bagger>Temperaturabhängiger WiderstandPfadfinder <Flugzeug>SpannungsänderungRutschungWeißSpeckle-InterferometriePagerComputeranimationDiagramm
01:07:41
Comte AC-4 GentlemanSchiffsrumpfAngeregter ZustandBecherwerkJuniZyklotronKit-CarVoyager <Raumsonde>AnstellwinkelSpinNivelliergerätSpinZyklotronElektrische StromdichteUmlaufbahnGleichstromSchmiedenFahrgeschwindigkeitChirpAnstellwinkelSchwingungsphaseParallelschaltungWindroseEnergieniveauAbwrackwerftAngeregter ZustandFeldeffekttransistorElementarteilchenFraktionierter Quanten-Hall-EffektFeuerwaffeGeokoronaErsatzteilLocherHalbleiterStandardzelleNiederfrequenzFACTS-AnlageUmlaufzeitFront <Meteorologie>DrehenMessungFlascheMinuteHobelHadronenjetBildqualitätKlangeffektKalenderjahrTastverhältnisTrajektorie <Meteorologie>Proof <Graphische Technik>WarmumformenDipolElektronisches BauelementDiagramm
01:15:59
Kette <Zugmittel>UmlaufbahnKlangeffektChirale AnomalieHall-EffektSonnenuhrDimmerMutter <Technik>ZyklotronEnergielückeTagesanbruchNiederfrequenzEnergieniveauLichtstreuungFraktionierter Quanten-Hall-EffektMaßstab <Messtechnik>BandstrukturGitterkonstanteFaraday-EffektMagnetisierungProbedruckAbwrackwerftPatrone <Munition>ZielfernrohrSatzspiegelSteckverbinderLocherEisenbahnbetriebNachtFlussdiagramm
Transkript: English(automatisch erzeugt)
00:00
Okay, I think it's time to start. Can everybody hear me? Yes? Okay. So let me introduce myself for those who don't know me. I'm Bart van Wijs. I'm from the group Physics of Nanodevices from the Zurnig Institute of Advanced Materials in Groningen up in the north. It's
00:26
been asked to give this masterclass on graphene. Obviously, you would have preferred André Chaim giving it, yes, but I guess he's very busy. But certainly tomorrow, you will
00:41
be able to hear him. So what I'd like to do is to have a little bit of informal presentation, so please interrupt me. I have also there a whiteboard which I will try to use, so please interrupt me if there are any questions. So I'm actually aiming on,
01:00
let's say, a person who has heard about graphene, has heard a few things about graphene, and likes to know a little bit more, especially also likes to know a bit about the background, what is all the fuss about, is this now a magic material or not, etc., etc., etc. So clearly, graphene has received some new attention by the Nobel Prize of last year given
01:25
to André Chaim and Kostya Novoselov, and read carefully for groundbreaking experiments regarding the two-dimensional material graphene. So I will show you some groundbreaking experiments. I can already tell you that those experiments are not necessarily very complicated, so I will
01:44
show to you some device of graphene. Well, I believe it's graphene made from some kitchen foil, cotton wool, some graphite, and I will show you, well, at least I will try and convince you
02:02
that this is related to graphene. Okay, let's start. André Chaim, that's him, you see his age very, very young. So the nice thing about this Nobel Prize, of course, is that it shows that you do not have to be retired or, well, already in your 70s or whatever, not productive
02:23
anymore before you get the Nobel Prize. You can already get it when you're in the prime of your life and you're in the prime of your scientific career. And of course, that holds for André Chaim, that holds even more for Kostya Novoselov because, as you see, he's only 36.
02:44
So I have to say a little bit more. Clearly, the achievements which they have obtained were done at the University 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
03:03
André Chaim and Kostya Novoselov. So what you see here, this is a big magnet. It's from the high field magnet lab in Nijmegen. So what is it? It's a bitter type of magnet. So, well, I'm not going to bore you with all the details, but basically it's built up
03:23
by basically a circular plate through which a huge current is sent. And this current, of course, is very huge because you see already if it runs, it consumes about 70 megawatts of electricity. So you see that the electricity bill, that is, of course, of great concern in that lab.
03:43
So what you also need, obviously, is to get rid of this dissipated power. So these are supplies for water cooling, and this is the return, and so clearly that consumes basically a few cubic meters of cooling water per second. So what is now the goal of such a magnet? First
04:03
of all, to produce a large field. That's pretty clear, but there is also some interesting feature in that it has a bore, as it's called, which is at room temperature. So you can basically, you have a space of about, in this case, 32 millimeters, which is relatively large, in which you can put whatever you like. And I should, of course, here refer to some
04:29
interesting link with Andre Chaim, because about 10 years ago he received what is called the Ig Nobel Prize. The ceremony for the Ig Nobel Prize, I believe, is a few weeks ahead of the
04:42
of the Nobel Prize, and this is supposed to be some kind of parody on the Nobel Prize, but with a little bit of serious undertone, because this prize is given for research achievements that first make people laugh, and then make them think, or, well, sometimes
05:02
is given for achievements that cannot or should not be reproduced. So he received the 2000, the year 2000 physics prize, together with Michael Berry, Sir Michael Berry, I should say here, and because he will appear further on in the story as well, for an interesting experiment,
05:23
and that is making, levitating a frog. So let me show you how it works, and now I have a nice movie. So here is this levitating frog in this huge magnetic field. Well, I believe that the frog
05:40
survived the experiment, but of course you could say, well, what is now the, what is now the first about? Well, I can show you what is the first about, because if you try to levitate something, you will not succeed. So I have two magnets, I try to levitate one by pulling, by putting the North Pole against the North Pole, and you see I cannot get any stable, I cannot get
06:06
any stable situation, because it will basically flip. And that is actually given, there is a theorem about it, and it is called Earnshaw's theorem, which says that you cannot make any stable levitation, or a stable suspension of any magnets, or any, any, any
06:25
electrostatic arrangement, you cannot make a stable, how would I say, well, let me just show you what I mean. Well, let me show you, let me continue here. So this is Earnshaw's theorem. What is important is that you have, if you have any magnetic forces, permanent magnets,
06:42
or permanent charges, you cannot make a stable arrangement of, which makes things levitate. Clearly this is an exception, because otherwise this frog would not levitate. So there's something special going on, and that special effect is diamagnetism, because the magnetic field
07:00
actually induces a circulating current in the frog, which actually produces a magnetic field, and you can actually show that the combination of this, this big magnetic field plus the magnetic field generated by this diamagnetism can indeed produce a stable situation. So, well, what else can you make levitate? Let me just show you, I think it's,
07:27
this is a levitating strawberry, you can actually do some more, let me go. You can also levitate something, and I did not know that this was possible, because remember, so this 33 Tesla magnet, that's a really big thing,
07:44
this is just a tabletop arrangement. So you have just a configuration of magnets, well here this is the North Pole up, this is the South Pole up, and there is the third one is again North Pole up, South Pole up, and what is floating here is graphite. So let me show you this one, it's just, so here we see basically a piece of graphite,
08:19
which is again levitating via the same principle, the same principle of diamagnetism.
08:26
So it's not, this diamagnetism obviously is not a very, very rare phenomenon, it happens 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
08:43
superconductors, but of course then the diamagnetism is indeed very strong, because in superconductors, basically all the magnetic flux is expelled, only part of it is expelled, but nevertheless it still works even with non-superconducting materials. It's a very good question, because I'm not sure if you have seen the experiment
09:19
with the high TC superconductor right, because then it's basically the same arrangement,
09:24
of course you have to cool it with liquid nitrogen, and then you also have some typical movement like this. So I do not know exactly what the answer is, basically the motion is probably given by some damping, because if it starts to move, then in a magnetic field
09:42
basically you generate some so-called eddy currents, which damps the motion. So, but I'm not 100 percent sure which is the, if this is the full answer, but the only thing I can say is that this typical motion is very much the same whatever you levitate. So, okay, graphite. So this sounds like a very boring material, because as a matter of fact,
10:04
until a few years ago, people were basically interested in graphite, obviously for using it in pencils, pencil lead, pot loading, it is a very known material as a moderator in nuclear reactors, because it slows down the neutrons, which you need a basic to have the,
10:23
let's say the multiplication effect in nuclear reactors. It obviously has a layered structure, and of course that is the reason why you can write, you can easily take off layers of, well let's still call it graphite, it is used in the steel making industry, part of the graphite which we actually use for making this graphene device actually comes from
10:46
the steel making industry, well it's a good lubricant for the same reasons, and of course it's also very important, it's a very good electrical conductor. So this is a slide which I borrowed from, from Andre Chaim, because I think it's a very nice one,
11:02
because it shows the history. Well here we have the graphite, and clearly it's made of layers, and these layers are called graphene. So what is, what is very typical is that you have a carbon atom which is connected to only three neighbors. So it's very different from diamonds, because then we have a carbon atom, then we have two carbon atoms sitting there, and two sitting there,
11:23
that basically has four neighbors, so in a very primitive way, and I think you can even justify it, you can say well, carbon has four electrons to make bonds, in diamond they are all used, and here I have only, I need three electrons to make bonds with the adjacent atoms,
11:40
and then one, one electron is free and can actually move through the, through the layer. Obviously it can also move in this direction, but it's a bit more complicated. So if you measure for instance in graphite, you measure the electrical conductance in this direction, I think it's about a factor of 10, 200 times better than in, than in this direction. So this is known material, but the strange thing is, that was what was basically
12:05
known, well basically the first, let's say non-trivial material which was known, was buckyballs. So they're called after Buckminster Fullerine, who actually designed buildings like this, and you see again, you have this, this, this triple coordination here, in this case you have
12:24
an alternation of, of, of hexagons and pentagons, and actually it's very much like a soccer ball, it's actually the same, exactly the same, C60 is exactly the same. So this was basically the first type of, let's say non-trivial carbon material to be discovered,
12:41
and then a little bit later, people found carbon nanotubes. They are not very rare, because if you have a candle, and you have, you make it a bit sooty, and you collect the soot, and you start to look carefully, you find that, that these, these buckyballs and these carbon nanotubes are actually produced. So it's not a very rare material.
13:02
What is a bit surprising is that this material basically was isolated only so late, only, well, five years ago. So I think the story is a bit, I will give a bit the story why this was so. It also said it was presumed not to exist in the free state, which is always a bit of a mystery why this, why this would be. Suppose you have a 2D
13:23
material, people say well, if I basically look at all the possibilities for lattice vibrations, and I add up everything, basically the whole thing diverges. So this material would not be stable. But of course, that's a bit of a tricky argument, because if you peel it off from some
13:40
other material, if you peel it off from graphite, it might take a long, long, long time before it basically disintegrates. It could be a lifetime of the universe. So this argument is not a very valid one. Okay, this is a nice one. This was made by a postdoc of mine.
14:00
It was made somewhere in 2008, and actually describes the number of publication on graphene as a function of time. Hey, you see, well, the high impact journal Nature Science, and you see a bit more, well, physics letter is also a nice journal, this would be obviously also. So what you see, this is the evolution of the number of publications in graphene as a
14:22
function of time. And well, here it's 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 the total number of publications in 2008 was already here. And obviously, we are, well, far, far, far, far above this. So, and
14:44
of course, this is the important paper. This was the paper by an author, Salaf Chaim, actually who basically triggered this entire thing. This is why they got basically the Nobel Prize. I will, but you look at it, it says, electric field effect in atomically thin carbon films. It doesn't say single, it doesn't say single layer. And because
15:05
the funny thing is that this experiment very likely was not done on a single layer of graphene, it was still multi-layer of graphene. And only one year later, in 2005, they, well, Wozlov and Chaim, together with another group, could actually show that they
15:21
could measure a single layer. And I will show you how this works. Here we entered, my group entered the graphene arena somewhere here. You could say that was a bit too late, but fortunately the, if the curve would really go like this, we would already have, well, approached the end of the S-curve, but fortunately, of course, we are still
15:41
going up. Okay, this is what I like to discuss with you. Obviously, it's only a tiny fraction of what can, what can be discussed about graphene. I have to make a choice, also a bit of a consistent choice. So let me, I will give some introduction into electronic
16:01
properties of graphene. How do I make a graphene device? Of course, how, how do we use this, this famous scotch tape method? What is this anomalous quantum Hall effect, which was actually the proof that you could make a, well, make and measure a device, which is only based
16:20
on a single layer of carbon, a single layer of graphene, bilayer graphene. You can go turn the story back. If you can make a single layer, you can also make two. What is the difference? How does it behave? What, this is a game you can play with graphene. You can suspend it. Yeah, you can make a device which can consist of a suspended single layer of, of atoms. And I will
16:43
show you how it works and what we can do. There are some nice games which you can play with graphene, electron optics, optics I would basically call it. I will show you an example here. A few applications of graphene, and then probably in the second hour, I will discuss what
17:01
we are doing with graphene. We are not, well, of course, of course we are also looking in electronic properties, where you use the charge of the electron, but of course we are also going to use some other property, which is the spin. So we are going to do spin-tronics in graphene. And I will end with do-it-yourself graphene. Okay, there is a lot of mystery,
17:23
well, there are a lot of mysterious words about graphene. It almost looks like it is not a real material. It is something magic. However, we should realize that it is basically a very nice example of a solid-state material. So I would like to summarize a little bit what we
17:42
basically should know or should remember from solid-state physics. So first of all, we have a nice periodic lattice of atoms. So that means that the electrons basically experience periodic potential. Well, I write down X, this is one coordinate, of course this could be
18:01
basically also R, it could be X, Y, and Z, it does not matter. Basically the potential is periodic and with a lattice period A. And now there is a very important theorem, which is called Bloch's theorem, that if you now want to calculate the wave functions quantum mechanically, you can actually write them like this. You can say the wave function
18:22
is a product of a wave function here. This is a wave function which actually describes the wave function on the scale of the unit cell, but basically repeats itself. So this is a true periodic wave function and it is modulated by some envelope wave function, which looks like
18:41
a plane wave. So this is basically the expression for a wave function of a free electron. So this is already very interesting, and of course here I describe that this basically tells you that at least this part of the wave function is periodic with period A. But the entire wave function of course is not, because it is modulated by
19:06
this factor. What is this thing? This is the wave factor. So this is a very important parameter. It is also very important because it is related to a property of the electron which is called the crystal momentum, or the quasi-momentum. It is just given by taking
19:24
this wave number and multiplying it by h-bar. Why is this so important? Because if I now apply a force on this electron which is moving in this periodic potential, it basically, this momentum or the time derivative of the momentum, basically obeys Newton's second law for a free
19:44
particle. It is very strange because I mean this is a very complicated problem here, the electron feeling this periodic potential moving around. Nevertheless, for this quasi-momentum, I just have the equation that the time derivative of the momentum is equal to the applied force,
20:01
which can be electric field. It can also have this as the Lorentz force. So this is a very special feature and also a very nice feature. What you can also get is a band structure. For each value of k, each value of the wave factor, you have a set of states at different energy which are labeled by n, which are called the bands. This is also an important one. We
20:25
have a group velocity. If you now really want to know the velocity of the particle, you have to take the derivative of the energy with respect to k. I will show you how this works. Again, this is good old solid state physics from...
20:45
So now, why did I emphasize this? Because now you can look at the difference between a metal, a semiconductor and graphene. So this is the energy, this is the momentum. So this is a typical, well, artist impression of a metal. You see that the energy depends
21:01
on the momentum, well, in a parabolic way. It goes with the momentum squared. And we immediately recognize this because this is the expression for the kinetic energy. There is only of course one important difference because if you would have a free electron, of course the mass would be the 9 times 10 minus 31 times the kilogram, which is the
21:21
mass of a free electron. Here it is modified. Now we get an effective mass. It is modified again because this electron is moving in this lattice. But there is no problem here. This is simply a particle moving with some positive effective mass. This is a semiconductor. What you see immediately, there is something interesting happening.
21:43
I hope this is not failing on me. You see that there is an energy gap. So there is an energy range where there are no states. And of course that's very crucial for a semiconductor. And above that you have electron states which you can immediately recognize again with this as particles which have a positive effective mass. And these are the holes
22:01
and you see they actually behave much the same but now the energy goes down with increasing momentum. And that actually means that they have negative effective mass. Well, sounds a bit odd, negative effective mass, but it is purely a consequence of this specific band structure. Of course we call these electron states because now we have to introduce a thing
22:27
which is called the Fermi energy. So it basically means that in semiconductors you can have electron transport where you make particles in the conduction band when you create electrons. But on the other end you can also take away electrons here from the valence band. Then you
22:45
have empty states. They can also move and well obviously they are called holes. This is and you immediately see several differences. First of all it is not a metal because it has a very,
23:00
very special band structure because now the energy of the states depends linearly on the momentum. So that's very strange because then you cannot say that it has an effective mass because if it's nicely quadratic we could say it has in this case a positive effective mass, here it is a negative one. Well, here we don't know. This is one relevant feature,
23:22
the other one feature is of course that if I fill graphene, let's say if graphene is neutral, then the Fermi energy, well of course I have to fill all these states until I have basically used up all the electrons, but then it turns out that you sit exactly at this point. And this is called the neutrality point. It is also called Dirac neutrality point for reasons which
23:44
I will show. Just look at the dynamics. Let's use Newton's law again. Suppose now I have, let's say I have a particle sitting here and I apply a force in this direction. Well a force
24:00
means that the momentum will change linearly with time. So, okay here I go, the momentum changes linearly and I go along this curve. So the energy starts increasing linearly. But the slope doesn't change. So dE decay remains the same, but that is the velocity. So the velocity is not changing. So I'm applying a force to this
24:25
electron and the velocity is not changing, it's constant. So this is a very typical aspect of graphene. Carriers in graphene always move with the same velocity. What is this velocity? It's about 10 to the 6 meter per second, 1000 kilometers per second.
24:46
If this would be the only thing of graphene, it would be a bit boring. Why? Because these situations sometimes also occur in the bend structures of metals and semiconductors. Maybe not at the Fermi energy, but a bit higher, a bit lower. Sometimes you
25:01
experience this kind of crossings. But in graphene, it's, well it arises of course from, well from let's say the Hamiltonian in the system. So let's see how it works. Okay, now we have to do some graphene physics. Again, this is graphene. What you recognize is
25:23
that there are two sub-platases, these guys and these guys. So that's already one important ingredient. So if you want to describe the wave functions, clearly you have to describe them on this side and on this side. And what we're going to do, we are going to describe them together like this. This by the way is, no it's really failing on me.
25:49
I have another one here, if it works, just a second. What was I saying? Okay,
26:13
this separation by the way is about, is 1.42 angstroms, just to be complete. So to make the connection with this solid state picture, I go now again to
26:25
receptacle space. So now I have to go into different directions. And basically what I will see is that, well also in receptacle space you get a unit cell which looks very much like the
26:40
one in real space. And the two important points are these guys. They are the K and the K prime valleys. How do you arrive at the bend structure? This was done already long time ago, Wallace. So what you do is you use a tight binding model. And a tight binding
27:01
model is a little bit like a mystery. I'm not trying to explain the microscopic justification for this. But basically what you say is okay, I can have an electrons which can sit on an A site and on a B site. So the A site is coupled to three B sites. And a B site of course coupled to three A sites. So what actually this term does,
27:25
it actually removes an electron from the B site and it puts it on an adjacent A site. So this is sometimes called the hopping matrix element. And the strength of this one is called T. And T is measured in energy. So this is the one which connects the nearest neighbors.
27:46
If you like, you can also include coupling between A sites. So this side couples with this one, this couples with this one, et cetera. And of course also the coupling between the B sites with a T prime. Well what you get from this was already calculated again in 1947.
28:05
This is the mathematical expression. It doesn't say too much. It's much better to look at the picture. And so now we really have the 2D picture. This is the energy of the states. This is the kx. This is the ky. And the interesting thing is these are these k and the k prime points.
28:24
You see that basically you would have six if you go around. But this one is equivalent to this one. This one is equivalent to that one and that one. So basically we have three, two inequivalent valleys. And you see, recognize immediately what I already plotted
28:42
before. This is your neutrality point. If I go, you basically you get a cone. So the energy as a function of the, well, deviation from this k point increases linearly. And of course here it decreases linearly. This is, okay, this is asymmetric as you see. You see this one is,
29:05
there is a much bigger change in energy for electrons than for holes. And the reason for that is because of this presence of this t prime. If you forget about the t prime for the moment, you only connect, you only take the nearest neighbor hopping, then you
29:22
get a very nice symmetric band structure like this where again you can recognize the two different valleys. So now the nice thing about graphene, as I will show you, is that in the neutral case all these states are occupied up to the Fermi energy. However, you can very easily change the charge in the graphene. You can make a field effect
29:42
transistor. So you can clearly move this Fermi plane up and down. And then you really see what's going to happen. If you move it down, you enter the hole regime. If you move it up, you enter an electron seam. And that you can very nicely measure. Why do people call
30:04
these electrons in graphene massless? You've heard about that. What is this now massless? Well, this is basically what I already explained to you. You just have classical particles which can be characterized by some mass. I have a kinetic energy like this which you can rewrite
30:23
as a function of momentum like this. And m zero obviously is the mass. There is only, well, there is only one mass in classical physics. You have, of course, Newton's second law which says the force is equal to the mass times the acceleration. And it's the
30:41
time derivative of the momentum. But now Einstein comes in and he says, hey, wait a minute, this is not fully correct. This is only correct for velocities much smaller than the velocity of light. I should actually use this equation. So you already see that if I want to create a particle which doesn't move, well, where the momentum is zero, I still already have
31:03
to put in an energy m zero c squared, which is the well-known expression. And if this particle starts to move, if the momentum starts to increase, then of course the energy starts to increase according to this relation. The interesting thing, of course, is that
31:20
Newton's law still holds. So if I apply a force on a relativistic particle, this still means that the force is equal to the change of momentum as a function of the time. It also, this one also still holds. The velocity is still equal to the derivative of the energy with respect to the momentum. So if I plot it here, I get something like
31:44
a blue curve. So this would basically mean this is the energy, this is the momentum. I create an electron here. This will take me this amount of energy. Now I'm going to increase the momentum. What's going to happen? Well, you see this region is still parabolic. So this is the classical regime. But then you clearly see when this particle
32:04
is going to approach the speed of light, and then the energy starts to increase more and more linearly with the momentum. And of course you also know what it means, because the velocity is not increasing anymore, and you have achieved more or less the speed of light. The only thing which is increasing is the mass.
32:22
So you recognize that for graphene we had this linear dispersion. So what is now special about this, because this was the energy we needed to create a particle. It was m0 c squared. Here it looks like you, for graphene, well that these particles more or
32:41
less imitate relativistic behavior, except for the fact that m0 is zero. You do not need any energy to create this particle. So the main lesson is that these electrons in graphene, they behave like, well, of course we know that other particles which have this dispersion, they are photons. Nevertheless, electrons in graphene, they
33:05
behave like charged photons. So they're not real photons, but charged photons in free space, but with a 300 times smaller light velocity. So everything is like you model, basically make a copy of particles moving in free space, approaching the speed
33:22
of light, and in graphene, the particles would approach this, well, would basically have this speed which is about 10 to the 6 meter per second. I cannot say too much about it. There are many, many nice analogies between the behavior of electrons in graphene and high energy physics. I'll just leave you with this message that it's something
33:45
like a solid state laboratory for high energy physics. Okay, this part I like, because here it becomes really non-trivial, and I'll try to explain that to you. So where
34:00
now does this cone-like dispersion come from? So what is now, can we say something about that? It turns out that yes, we can, because we can describe the Hamiltonian or approximate it close to either the Dirac, well, the K Dirac point or the K prime with a relatively simple expression. You see that this is a Hamiltonian, it's a two
34:25
by two matrix, it acts on the wave function on a vector which describes the wave function on the A sides and on the B sides. But you see it's a very special one, because it has only off-diagonal elements, it has some combination of the momentum in the X
34:40
direction and the Y direction, and you see for the K prime it's, well, just a plus and a minus sign are changed. So the thing I want to emphasize is that if you now calculate the wave functions, then they look like this. Basically I should mention I've forgotten here the exponent ikx part, which according to the Bloch theorem should be there, I only
35:07
look at this pre-factor. And now I now want to illustrate that here, something funny happens. So let's assume that we have a particle which is moving with a certain
35:28
momentum. This is the X direction, this is the Y direction. So I define an angle theta like this. So more or less like this one. So basically theta is the arc tangent
35:44
of the X component divided by the Y component. So then you see that corresponding to such a direction, there is a phase factor for the wave function on the A side, and there
36:02
is also a phase factor for the wave function on the B side. Well, you could say so what? Let's assume that theta is zero, so then we get, well let's say we had something like one one, so that would correspond with a particle going this way. Now we increase
36:24
theta to pi, so now we get this direction, I think we get I and the other one, and one is, so I, or minus I I think, that should be minus I. But now the interesting
36:46
thing is now we make theta two pi, which actually means that we have this particle which is moving in a certain direction, first it moves in this direction, then it starts to move in this direction for some reason, go around, and then we are back to here. So then you would say, well we should get the same wave function, because it describes
37:04
the same state. Well is it not so? So if we put two pi in this one, we get two pi over two, exponent minus I pi, and here also we get same thing. So it's like the wave
37:20
function is multiplied with a factor of minus one. And then you could say, well who cares about that? Factor of minus one, because we know that we always have to normalize wave functions, right? We have to normalize that, we put some normalization factor in front, so whether the wave, well this is the starting wave function or it's multiplied
37:43
by minus one, who cares? That turns however to be very very important, as I will show you later. So keep that in mind. Enough theory, experiments. Scotch tape method. What do you need? Tomorrow there will be
38:04
a hands-on demonstration. You can try the Scotch tape method yourself, see if you are good at it. Let me just explain it, how it works. So this is a nice piece of graphite, it's a nice crystal. In this case it's HOPG, highly oriented pyrolytic graphite, doesn't really matter. Now comes the trick. So we need some sticky tape. It does not
38:26
have to be Scotch tape, just sticky tape. And we just press it on the side here. Nothing special. Then we peel it off. So remember, okay, if you just peel it off, what is going
38:41
to happen? Well, you will not be surprised that you will pull off a flake which is relatively thick, I mean this is still thousands and thousands of layers. Well sometimes it will break already, doesn't matter. In general you will get something which still contains thousands and thousands of layers. Now comes the trick. Now you make a substrate
39:01
which consists of silicon and silicon oxide. So let me just illustrate it here. This is silicon and then we have a layer of silicon oxide. And then here we have the flake, many, many, many, many, many graphene layers. And then here was the sticky tape
39:23
and we are going to pull it off like this, right? So what would you expect? If you look at this, well there are many, many flakes. You start to pull, okay, well maybe you pull it off completely or, well, a lot of layers will stick and then maybe
39:40
thousand layers and then a little bit further on, 300 layers and then, I don't know. But if that would be the case, this entire thing would not work because you will have a collection of junk on your substrate and you might just by chance have a small piece which is just one layer. But you will not be able to find it. The trick is of course
40:01
that if you press this flake, it is still relatively flexible. If you press it against your substrate, then suppose that this last graphene layer likes to stick better to the substrate than to the next layer, because we already know that the sticking between the layers is not so great, then there is a relatively big chance that you will
40:20
pull off, just you pull off all the flakes except the last layer. And of course it is not going to happen all the time. Well it happens about 10 percent of the cases or not, but you have a relatively big chance of producing single layers. And this is one of the tricks of the scotch tape. There is much more magic to it because everybody has his own technique, but this is one of the ingredients. If this would not happen,
40:45
you would not, Hyman or Vazilov would not be able to get their nice results. So here we peel it off. Okay, this is the microscope which you see on the microscope. You see a lot of interference, nice colors. And this is a very bright one. Why? Because it is
41:04
relatively thick. This is still thousands and thousands of layers. So here you really get interference of the light which is scattered back and forth. It is also scattered back and forth in this layer. So we have the silicon oxide layer here. And also the thickness of this layer also matters because you can imagine that if you tune your layers in such a way that you
41:25
match it with a number of wavelengths and that you become more sensitive to what you put on top. So this is also a bit of the tricks you have to use. But now look at this guy here. This is the lightest you can find. The contrast is not so
41:45
good. So you might say, hey, if this is the lightest which I can find and I cannot find anything else, this might be a single layer. So what you do is you use now atomic force microscopy. So this is now where you have basically your tip which wants to keep the same
42:04
force between the substrate or whatever you put on it and the tip. So of course, if there is a graphene layer, it has to go up two layers. It has to go up even more, etcetera, etcetera. And you can make a picture of the height. So this contrast basically indicates heights. You see there is already quite some corrugation. This is the structure.
42:24
And this is because of silicon oxide. Silicon oxide is not flat. It has some small variations of the order of a nanometer. And here there is the graphene on top. So you see really the graphene actually follows the corrugations. Here we see the next layer and the next layer
42:40
and here we go back to zero. So if we scan it like here, there is a step here, there is another step here, a step here and you go back to zero. And this step is in this case 0.29 nanometers which corresponds more or less with the thickness of just one graphene layer. Now you will immediately say, hey, wait a minute, you are not fooling me because if I go from here to here,
43:06
I go from carbon to graphene. But if I go from here to here, I go from silicon oxide to graphene. It is a different layer. So the interaction between the tip and the material might be different. So this height might not necessarily be the real height. It could
43:24
be bigger or smaller. So this experiment does not really prove that if you look at this one that this is really a single layer. It basically proves that here you go from n layers to n plus one and to n plus two, but you cannot really use it as a proof that
43:41
you have a single layer. So you have to do something else. So let me show you what we do. Now it gets a bit more expensive. So the problem with this technique, and you already see it here, is relatively small flakes. So this is about, I should mention this, this is about 10 micrometer. So it is not going to be very commercially relevant
44:02
technique. And because you first have to locate where they are, so this is of course a grid of markers, so you say well it is between marker 6 and 7 on the x-axis and between what is it 4 and 5, and here we want to make a pattern. So now we have the bit more expensive part. Let us see. So we need EBL. So we have just the electron
44:23
beam. It is basically the electron microscope, but instead of scanning the electron beam like in a TV, you can actually program the electron beam only to scan those areas where you want to make contacts. And that is done here. So here we have a graphene
44:43
flake which happens to be very very elongated, and we have told the machine hey wait a minute, scan your electron beam here, here and here, because we want to make contacts at all these positions. So this is a device we made. We studied about
45:00
three years ago. It is a ferromagnetic device, I should mention that. We have ferromagnetic electrodes and I will tell you in the next hour why we did this. You can really recognize the graphene flake. This is again a picture with an electron microscope. So the reason why you can see it so nicely, it is only one layer thick, but of course the electrons which are scattered from the graphene are scattered differently than the silicon oxide.
45:25
So that is why you have such a nice contrast between this single layer graphene and the silicon oxide substrate. Here we have contacts and this is a typical size. Yeah?
45:40
This is made with electron beam lithography. And the reason you can see, because in our experiment we have to get the spacings between the contacts relatively short, like a micrometer or smaller. You see that these contacts are all, the width is much smaller than a micrometer. So these kind of dimensions, it is difficult to obtain with optical, with
46:00
light, with optical lithography. So this is done with the electron beam, because the electron beam we can really focus in a relatively small area. What graphene stays throughout the process of applying a resistor? That is a good, again a very good question. In most cases it does. It does. Yeah? So
46:23
you can imagine that of course this solvent can creep underneath the silicon oxide and the graphene and it will roll up. That is what sometimes happens by the way. We sometimes see it. But in most cases it will simply stick. It will remain there. Don't ask me exactly why, but that is a lot of chemistry and all kinds of things.
46:44
This is the graphene field effect resistor. So this is the, this is why it actually works and it is a bit of a mystery I have to say. It is a bit of magic here. Because what we do, I have already plotted it there. Normally what you have is a field effect resistor. It is silicon, silicon oxide and we have a metal gate on top. Yeah? Because
47:05
you are interested in inducing carriers at the interface between the silicon and the silicon oxide. So this would be the gate. This would be the interface and we apply a gate voltage between here and here. So we want to control the charge here. This
47:21
is the inverse because we now have made this, this graphene on top of the silicon oxide. And now we want to change the charge in the, in the graphene. So what we do now is we use silicon again, but it is very highly doped. So it does not behave like a semiconductor. It is basically a metal. And now you can see this is a nice, a few
47:42
hundred nanometers, an insulator. So you see you basically make a capacitor between this metal, metallic silicon and the graphene. So you apply, well you connect this, well this is what you connect finally to some voltage source and you apply a voltage to your silicon relative to the graphene. Typically a 10 volt, 100 volt or something like this.
48:04
And the idea is of course if you apply a positive voltage you will attract negative charge to the graphene. And that means if we have the Fermi energy sitting here, if you want to get negative charges in, we move up the Fermi energy. If you apply a negative gate voltage, we attract positive charges, we attract holes, then the Fermi
48:21
energy in the graphene goes down. So that is a very nice trick. What is also very nice is that, well you can already understand that the graphene, well these are all the states we have talked about. There are no more states in the graphene. So all the charge which you induce in the graphene is going to be used for electron transport.
48:41
It is not that you have surface states where electrons can sit and do nothing. It is not there in graphene. So that is a very also important ingredient. So in this field effect device we are going to measure the resistance between the source and the drain. We are going to pass current from here to here and we are going to change the voltage which we apply to the gate. Here it is. So this is a typical curve.
49:09
We call this the Dirac curve. And the reason is of course that this is the graphene resistivity or the resistance and here we change the gate voltage. So you immediately see a bit of nicely symmetric curve where we reach some kind of maximum. Well is
49:24
it now reasonable that we do that? The answer is yes because suppose we apply big negative gate voltage. So we are going to attract a lot of holes and since we have a lot of hole carriers that means that the resistance or the resistivity will be relatively low.
49:40
On the other side also we apply positive gate voltage. We attract a lot of electrons so the resistance will again be relatively low. You see that is more or less symmetric because I have showed you that if only the hopping is only between the A and the B sides then we get a nice band structure which is symmetric for electrons and holes.
50:02
And it looks like in most cases indeed this is the case because you really see well it is not fully symmetric habit. It is almost symmetric. What is surprising here maybe and I should emphasize this one is this point. At this point we should achieve neutrality. So we are moving up this Fermi energy. Well this
50:21
is the maximum resistance we can achieve. So we sit exactly at this direct point. Well you see that we need a finite gate voltage for this. And why is that? Because there is always some background charge present. For instance in the oxide. You have to compensate say for that charge. And if you do that basically there is no charge anymore in the
50:45
oxide. I have showed you this direct point. It is only a point. Well two points. How can there still be a finite resistivity? Shouldn't the resistivity be infinite? So this is an important question here. What is going on there?
51:02
So these are measurements of this. Okay I mean I should mention the value here. This is typically a few kilo ohms. Now you might say well who cares about kilo ohms? Well I do. Because there is a combination of natural constant. It is h and e squared
51:28
which has the unit. This is the Planck's constant. This is the electron charge squared. If you calculate it, it is 25.8 kilo ohm. So this is very important quantity. It is the unit of conductance or of resistance. And for instance very relevant in quantum hall
51:44
effect. And you see now that it appears again here. Well it is not exactly. This is some fraction of that value. By the way, you can check for yourself that this thing I made myself also has a few kilo ohm. So there must be a very special reason for that.
52:02
So let me see. This is okay. These are measurements where we have changed the property of the graphene. Well what we have changed is the quality. And then this is measured in mobility. And we see that this maximum or let's say the maximum resistance or the minimum conductivity is always around this four e squared over h. In view of the time,
52:26
I cannot say too much about, well okay. I think in view of the time I will skip this part and explain you why this is. Just mention a few things. So this is another
52:42
parameter which we need. It is called the density of states. How many states do we have available for the transport as a function of the energy? If we sit exactly at the Dirac point, the answer is clear. It is zero. Well two points at zero. If you go away from the Dirac point, you know that we have this nice cone going this way and if you
53:01
go the other way, it goes like this. So you can actually show that the density of states or the number of states available in a certain energy range increases linearly on either side. And now the nice thing is of course, well can we now really sit here? And there are several reasons why we cannot. First of all is that the potential is never
53:21
flat. It fluctuates a little bit. So this entire diagram fluctuates a little bit as a function of position. So if you then calculate some average density of states, it will always be finite. There is scattering going on. The material is not perfect. So that basically means that all these states have a finite lifetime. And now we can use
53:42
some kind of uncertainty relation. We say well, delta E is something like h bar over the lifetime. So the shorter the lifetime, the bigger the broadening. And again if you would broaden all these states, it is pretty clear that also here, the density of states will increase and will become finite. Finally there is the effect of the temperature.
54:04
I am not going through this calculation, but you can actually show that if you take into account this finite lifetime of the states, and the reason, one important reason is that the electrons, they do not move for a long distance. They are scattered already after
54:21
something like 30 nanometers. Why? Because of all kind of imperfection. If you take that into account, you can actually end up with a conductance which is this 2 times e squared over h multiplied by some factor pi. And the reason is you can, there is some kind of compensation mechanism. So if the material is very clean, there is no broadening.
54:42
Your density of states is very low. Now you start to induce scattering. So you will get broadening of these energy levels. So the density of states will start to increase because you could say the density of states on either side is also going to produce a finite density here. But of course in the same time, the scattering time becomes
55:04
smaller and that compensates. So there is some fundamental argument why for a single layer of graphene at your neutrality point you would get a conductance something like this. And again, that dirty piece of material also does that. So the question is, is that now also a single layer of graphene? All very nice this, but it is not yet enough
55:29
proof that we have a single layer of graphene. Because we have uncertainties of a factor of 2, I have shown you this AFM measurement where the first step, you are not certain
55:42
about it. It really measures the height. So how now? And we do not have resolution. You would say well you put a nice electron microscope on it so that you can really resolve the individual atoms. You cannot do that. You can do that in transmission but not in let us say if you have a standard electron microscope. It does not have a resolution.
56:01
So how now can we now get proof that we have a single layer? And this is a very important experiment which was done in 2005. So one year after the first experiment, it was done by again, Chaim and Ovezelov, but also by the group of, yeah, which one?
56:27
This one. Yeah. So can you see from here, somehow maybe if the mean free path goes, becomes larger, then the conductance does not increase, but, or?
56:40
No. This is the expression of this density of states as a function of the epsilon. If you go away, if you increase the energy. So if epsilon, this increases linearly. So if you sit at this neutrality point, the density of states should be zero, right? But now my argument is this material has a finite mean free path. So that the states have a finite lifetime.
57:01
The tau is one over this, is the mean free path over the Fermi velocity because electrons move, are scattered after a certain length. That means how long do they take to travel that length, which is basically this expression. This gives you some energy broadening. It's delta E is something like h bar over tau. There should be, yeah?
57:21
And so basically we, we basically put in this energy broadening in this one. It's a bit of a head waving argument. But then, and we again, okay, I should mention this, I forgot to mention this equation because this is a very important one. This is the Einstein relation that tells you that the conductivity is the density of states multiplied by e squared times a parameter which is called the diffusion constant.
57:44
But if here the graphene becomes, I don't know, kind of perfect, the mean free path goes to infinity. If the mean free path goes to infinity, then obviously it becomes more and more perfect. The density of states at the Dirac point becomes smaller and smaller. But nevertheless, if you evaluate this formula at zero energy, this one goes to zero.
58:07
But this one goes to infinity because your mean free path. The electrons can travel for longer and longer distances. And that compensates. And then still I end up with something like this.
58:21
Hand waving argument. I have to say that this is very different if you do it in the metal, then your density of states is more or less constant. Then of course you see that if you, if you reduce the mean free path, if the material becomes dirtier, then immediately your conductance goes down. But here if you sit at this Dirac point, because of the special situation,
58:43
this is what you almost get. You can change a factor of two or a factor of three. Sorry? That's the diffusion constant. So it basically tells you the motion since the mean free path is,
59:00
the motion of these electrons through this material is diffusive. 30 nanometers much smaller than the typical size. The electron moves like this and then goes a little bit like this. So this is diffusion and this motion is described by a diffusion constant, and that is given in two dimensions by half times the Fermi velocity times L. Because again this is the typical velocity of the particles, and this is this B3 path.
59:22
Yeah? I'm a bit confused by the word quantized, because the way I understand it is it's not really quantized, it's more like universal conductance, independent of your sample. Because I don't see anything like quantization, like flux quantization or whatever. Absolutely it's not quantized in the sense that it should be exactly 2 e squared over h.
59:42
But there is, as I said, there is an argument why this should be, well, why it's of this order of magnitude. Yeah? Yes sir? I was wondering about the mean free path. Is there also some energy dependence of this mean free path?
01:00:01
Again, a very good question. In this type of preparation of graphene, you can say, well, it's nice, but it's always prepared on a substrate. And the substrate is silicon oxide, as I said, it's not flat. So there is always scattering going on. And actually this type of scattering actually produces this mini-3 path. So it's, for instance, determined by the static potential fluctuations.
01:00:26
Of course, you can say there is another contribution, which is due to the phonons, which is due to the lattice vibrations. But that is usually not strong enough to add room, well, unless you heat it very much, it's not determining this mini-3 path. So graphene on a substrate is basically, there the mini-3 path is determined by the
01:00:46
scattering from the imperfections. And there can also be some junk on top. If you make the graphene suspended, which I will also show, then actually you can make this mini-3 path much, much longer. I like to move on.
01:01:06
This is not enough evidence for a single layer. So what was done, 2005, by Chaim and Novoselov, but also from the group of Philip Kim, what they measured is the quantum Hall effect. And now I'm trying to, well, forget about, I'm just trying to focus on this one.
01:01:24
So what we know is if we have, let's, I have to show a bit about the Hall effect. So I pass a certain current from here to here.
01:01:52
I measure the Hall voltage and I can define a Hall resistance, which is V over I. I can also define a Hall conductance, which is obviously I over V.
01:02:04
And that is given, I should be careful, by N e times B, over B, sorry. So this is, so that it makes sense. So if I pass this current and I measure this Hall voltage.
01:02:22
If I increase the, sorry, yeah, if I increase the density, obviously my Hall conductance is going to increase because I have more carriers. If I increase my magnetic field, my Hall conductance is also, is going to decrease. So this is the standard behavior.
01:02:41
It's nothing special. It's simply the Hall effect, in this case in two dimensions. So if you plot this, as a function, you should maybe look at this one. This is the conductivity as a function of N. This one, as a function of N, I should get more or less a straight line. This is the classical Hall effect.
01:03:00
What you get in this case is the quantum Hall effect. And I will explain what it is, but I will just show you why it is a bit strange because this is at zero density and this is zero Hall conductance. Now I get a plateau and this plateau forms at one half times this 40 squared over h.
01:03:22
Why is this 40 squared over h? Well, the simple reason, it's four times the elementary unit. So I apparently have four possibilities for this, doing this. Well, I already had two valleys. So electrons can sit in one valley and they can sit in the other valley. That's a factor of two. But they also have a spin. They can have spin up and spin down.
01:03:41
You always have to take into account that each state can be occupied by two. Elected spin up and spin down. Then it gives another factor. So the elementary unit is 40 squared over h, which you recognize from here to here. But you see the first step, that is only two e squared over h. So that already deviates from the normal quantum Hall effect because then all the steps
01:04:03
would be the same. Here the first step is two e squared over h, then the next one, you add two e squared over h, then you add four. You go to six e squared over h, you add four, you go to ten. So it's an anomalous sequence of plateaus, which is actually a proof that you have
01:04:21
single layer graphene. And I will try and show you why that is. Some words about the quantum Hall effect. The simplest way to understand the quantum Hall effect is to take a sample which is fully clean.
01:04:40
So we have a two dimensional sample. This is x, this is y. It is perfect and we apply a perpendicular magnetic field to the system. What's going to happen? Well, inside the material, now these electrons are going to run around in so-called cyclotron
01:05:02
orbits. So they are pretty useless, right? Because they are going nowhere. They just make circles, that's it. If they are at the edge, it's a different story. Because imagine now, the electron wants to make, let's forget about this impurity for the moment. Let's just look at this.
01:05:20
The electron goes like this, it's deflected and now it hits the boundary. It hits, tries again, tries again. So you see that at the edges, you get so-called skipping orbits. Electrons are actually producing channels where electrons on this edge can go from there to there and on this edge can go from there to there.
01:05:41
It's exactly the opposite. So this is I think already one important ingredient. What is also important, please just remember this, and this is related to why something like the quantum Hall effect can be so accurate. Suppose there is just one impurity which wants to interfere with this nice behavior.
01:06:02
Well, electron comes, moves around and then scatters back, makes one circle, hits the impurity again and continues. So it's like the impurity doesn't do anything. So as long as the electrons which move on this edge are not able to go by series of scattering events to this edge, and so these impurities are not doing nothing.
01:06:27
That is the reason why quantum Hall effect can be so accurate and can be, let's say although of course the samples, real samples are not perfect. What I want to emphasize here is that if you do the calculation you find that they make
01:06:43
cyclotron orbits with a frequency omega c which is given by E b over m star. So again these are particles, regular particles which have a mass. They would behave like this. So the cyclotron frequency does not depend on the energy.
01:07:01
So how now do we get, this is the classical picture, how do we get the quantum picture? Well, I've tried to, I've made it on the next transparency, but I like to show it here. I have a periodic motion with omega c, that's the frequency. So what are my quantized levels?
01:07:22
Well let's assume that this would be the bottom of the band, see the particles would not move. I just get a nice set of states. This is a half, this is three halves, five halves, seven halves. So the energy is n plus a half times h bar omega c.
01:07:45
So it's periodic motion, so it's like more or less like a harmonic oscillator. If you quantize it you get the levels more or less of the harmonic oscillator. That is what I plotted here. This is the energy, this is my Landau level sequence.
01:08:01
So this is the zero, you could say this is the zero point motion. This is the half, this is the three halves, five halves, etc. What I also should say something, if I now start to approach the edge, so this is as a function of the guiding center of this, well let me show you, this is the guiding center of this cyclotron orbit.
01:08:22
So in the center obviously it doesn't matter where I put it, if I put it here or here the energy is not going to change. So if I do the quantized version of it, I sit for instance here, this is my series of Landau levels as they're called. If I sit here or here, if I start to approach the edge, the situation is changing a bit because then you see that this periodic motion, well this circular motion is disturbed.
01:08:46
I also get the periodic motion but now they are the skipping orbits and you can actually show that now the energy goes up. So if I go towards the edge, the energy of these Landau levels go up, like this.
01:09:01
So why is this now so important? Because if I now change my Fermi energy, suppose it sits here, then I know that I should only worry about the states which sit at the Fermi energy. All states below the Fermi energy are fully occupied, I don't have to worry about it. So the only way to get electron transport is to play a little bit with the states
01:09:22
here or play a little bit with the states here. But these are of course exactly the skipping orbits. So if I put in some more electrons here, I get the net current this way. If I put in some more electrons here, I get the net current this way. So this is a way to understand the quantum Hall effect in terms of these edge channels. But remember the sequence you get is, this is zero energy, you need a little bit energy
01:09:45
to reach the first Landau level, then you have a nice fixed sequence of states. What happens in graphene? Very different, because in graphene, the velocity of the particle remains the same.
01:10:05
It's always the same velocity, no matter at what energy you sit. So let's now see. I have now tried to plot different trajectories, well, cyclotron orbits at different energies. But the velocities are always the same.
01:10:23
And that means that if you go, if you reduce the energy, then you go closer to this direct point. Clearly, the cyclotron frequency is going to increase. And in particular, it's going to diverge, become infinite. Because if you sit very close to the direct point, the electron only has to make very, very small orbits.
01:10:44
So this is a very, this is already important difference between the quantum Hall effect in graphene and in standard semiconductors. There is another one, and I would like to do this before the break, and that's the Berry phase.
01:11:01
I've already told you a bit about the fact that if an electron in graphene moves in a certain direction, if it moves in a certain momentum, then in the wave function itself, there is a change in phase, which looks very much, by the way, like a spinor. So if you have a spin pointing in space, you can also represent it by two components,
01:11:22
a spin up and spin down component, and a superposition, well, basically a superposition of that, and you can make any arbitrary angle in space. So this quantity is called pseudo-spin. So why is it important? Well, this is the Berry phase. Berry phase means I am close to the North Pole, yeah,
01:11:43
and I basically, I start to move forward, yeah. This is me, and this could be my left arm. Then I start to move to the side, in the direction of my left arm, until I am here. Then I move backwards again, yeah, so what I've done, I've moved forward like this,
01:12:03
and I've moved to the side, yeah, and then moved like this. So I basically said, well, this is my right arm, I go like this, I go like this, and then I go back, yeah.
01:12:21
So my left arm is still pointing in the same direction, but if I do it on a sphere, it's obviously not true, because this was my left arm, you go like this, and now clearly I have an angle of 90 degrees, if I use, let's say, this part of the sphere. That is an example of the Berry phase.
01:12:41
So the relevant one here is the Berry phase for spin and half particles. So this is now an electron, here is a magnetic field, this is like a spin. I apply a magnetic field, so now it's, let's say, aligned with the field.
01:13:00
Then I move the field, so spin is going to move like this. I do it slowly, then I move the field like this, then I move the field like this, and now I'm back to the original situation, right? So I would say, wait a minute, the wave function has not changed, yeah? That turns, because, I mean, I'm back in, I go like this, then I go like this,
01:13:23
go back to the original situation in the same direction. But of course, remember, what can change is the phase factor, like this minus one which I've shown, some phase factor in front. And that is exactly what happens. So you can show that for spin and half particles, the Berry phase is half times the solid angle.
01:13:42
And so this means that if I, for instance, have a spin, which actually moves in the plane, which I have a magnetic field which I make rotate fully, and the spin follows this magnetic field, then the wave function acquires an extra phase of pi. Hey, wait a minute, remember now,
01:14:02
that was exactly the same situation which we saw when we had this electron in graphene, where the momentum basically went like this and then goes around and comes back. But that is exactly this kind of motion, because here also the momentum of the particle changes and then basically comes back to the same situation,
01:14:21
but it has made a 30, well, let's say a 2 pi rotation. And because of the analogy between this pseudo-spin and this real spin, this Berry phase actually tells you that there is an extra phase of pi. And this is the last thing which I will show before the break.
01:14:40
What does it now mean? If we now plot the Landau levels in graphene, we find now there is a Landau level at zero energy. Because here it's not allowed, we said if you have particles with a mass, we need this zero point motion. But basically now we have an extra phase of pi, which we take into account.
01:15:01
So now the total phase at zero energy is already zero. So that means we have at this Dirac point, as a matter of fact we have two Landau levels. One is an electron-like Landau level, which is actually a Landau level which has moved down. Due to this Berry phase, you could say for holes, normally for holes with a finite mass, you have the same series, but then you go down.
01:15:22
Well, this now has moved up, so we have this special zero Landau level sitting there. What you also see is that the energy spacing is not constant. As I've already told you, that is because for small energies, the electron only has to make very, very small cyclotron orbits,
01:15:44
and the cyclotron frequency becomes much larger than if you go to higher energy. Let's stop and have a break until nine. Unless there are questions now. One question. I have a short question. You said before that because of imperfections in the lattice
01:16:04
or other kind of things, the band structure of graphene might deviate from the theoretical picture, and still you can observe the Landau levels as you can see.
01:16:22
No, it's not correct. I did not say that the band structure deviates because the band structure is determined really on a relatively small scale if you have 100 atoms, 1,000 atoms. Let's say these three paths are 30 nanometers. That's already 100 lattice spacings.
01:16:41
So in principle, these imperfections or the scattering only modifies, well, it does not modify the band structure, but basically gives rise to extra scattering. So if you look at this picture, what is the scattering going to do? Well, it's exactly the picture which I showed here. This is the way how scattering is introduced.
01:17:00
So most of the cases, the electrons are indeed making this nice motion as if they were in perfect graphene. But then here at some occasions, they're basically scattered. And it's pretty clear what you need to get this quantum Hall effect because you want this motion to be completed. So basically electrons should be within the scattering time.
01:17:24
They should be able to complete this motion. So this is a criterion like omega c times tau should be larger than one. So that is why, in general, you need also relatively high magnetic fields because if magnetic fields are high, the cyclotron frequency is very, very high,
01:17:41
so the electrons can actually complete their loops before they are scattered. Yeah? Why is this a proof of single layers? Because you still have these 2k prime values in two layers, right, or not? You are going to tell me, you ask the question, what happens in two layers?
01:18:02
Yeah, basically. And I will answer that after the break. But now we have a break.