 Poznači. Tako drža. Se počiš? Poželiš sva? Jih je vzout, da se Prof. Eveland Re, z Rutgerskjom universitetti, odreči, atakaj na elektronijske zvršenje z in 2D materijali. Ne znamen, da tako počišli, na 2-3... Ne bo. Na 1h. Tako, to je najbolj. Dobro, sem eksperimentalist in vznikam na 2-dimensične elektronice primarilj, z vrštih technikov, o kaj sem tukaj pravite. Vznikam tudi, dve lekture za tudi, vznikam vznikov, vznikam vznikov in teori, na Bluk diedve, that experimentalists like to think about. I think that since the discovery of graphene in 2005, it I think it is fair to say that it has revolutionized the way we think about materials. Since the discovery in 2005, there have been dozens new materials isolated or synthesized that are really two dimensional. Kaj je spesivalo na te dve materijale? Prvno, smo vse vse elektronici, vse vse kvazi partike, kaj je topologi in vse vse transporti, ki ne vse vse materijale, in vse partike vse fizika. Tako, zelo nekaj vse partike. Prvno je, ker vsi vse atomi nekaj vse vse, nekaj nekaj nekaj vse, but we can very easily manipulate their properties. So we can change, and this is, there's more and more of this coming everyday, that we can change the electronic properties at will without using any chemistry. So for example, you can stretch a piece of two dimensional material, like graphene or anything else, you can bend it, you can fold it, you can cut it, every time you get new electronic properties. zelo se pristim što je čest, da se je vsega zelo. Kaj je tega, da je však čest na pohledku, je, da se pohlede, da tvoje grafini ljudi nekaj začnje, in da se se kaj ne poznite, se počuče nekaj, kaj je se tukaj vsega zelo. In, da je ljudi ljudi ljudi ljudi ljudi, nekaj pa nekaj nekaj, nekaj, pristim. in najbolj, da sem pravda insulatora. In se jaz neskup, da sem vse superkondakter, ta je zelo vzostavljena. Daj je tvoje, da možete inžitati superkondaktivite. Sve in mnogo, je, da se neskup, inzulatora in neskup se je resenjala na nekaj vzobih, nekaj teznjah, za hodnje tk materijali. imaš tudi insulatične piste, in na dve strane imaš superkonductivne dome, je to izglednje, superkonductivne trazije, je to nekaj, da všeč imaš zelo, in imam tudi tudi tudi tukaj, kako je... Tukaj, moj pojnter je nekaj, za nekaj različ, je nekaj, baterije. Zdaj, sem spet baterije. Zdaj, bomo tudi tudi tudi. Zdaj, ko sem izgleda, je to, da se očetimo, o svoj tudi vrednjih materijali. Tudi o vsej elektroničnih prvniča, o vsej grafinji. Zdaj, v prviči vsej lektu zvom tudi elektroničnih prvniča, biasah kot naseljstva, zelo uštava i vse njič. No, to smo zdravili pozanjaj z magnetikom, zelo smo izgledaj na lanašneiate, začalje s��čno lanašneâte in vsebrand od njimlji, in vsebrand od njimlji, izgledaj več nam možno je dvečne emblantovice v disizu? Boj, nega sistersila je zelo v denominatoru. Tih je odprljene. OK, zelo, da se je začal. To je izvršen iz nekaj začeli vseh vseh, da je inštavno, da je vseh vseh inštavno, da je vseh vseh inštavno. Zato je to, da je vseh, različno vseh, da je vseh, različno vseh, da je vseh, da je vseh, da je vseh, today more graphite, the reason is people were very, after the atomic bomb people were trying to use atomic energy for peaceful purposes for atomic reactors, and what they needed is a moderator, so that Wallace Basses told him calculate the bands structure of graphite because that was a good candidate and he worked and then worked and failed, he was notay able to do this, so he said, ok as we as we usually do, I can't solve graphite, let me solve a simpler problem, let me take one layer of graphite, which we now call graphene, and let me solve the bandstructure of that. And there he succeeded. Actually it took another decade, twelve years, for a team from Rutgers to calculate a bandstructure of graphite. So this graphene, this layer was pretty much forgotten over the decades, nobody thought about it. da vsi všeč nekaj sej teoriči dobrali, ostali in vseč jazovil in vseč loli. V 1943 godine Seminov skupilšnje za vsečnjezno vsečnjezna za tredimentinje lahko in v 87-ji dan, Danken, Haldane skupilšnje za modem of krandom, in izgleda vsečnj, in zelo včasno je svoj, boješnje vsečnjezna za taj svoj, in nič nekaj stajili, da se to je na vse vsi načo, nekaj nekaj jaz nekaj skupite v labu. A jo da? Pse je, da... ... ... ... ... ... ... ... ... ... ... z nekaj izgledajših način, način je časno nečasno nekaj. Zato neka je dela, da če je način način izgledajši, je več brav, da se je izgledajši na 3. način in prišlojiti način način način. Način način nekaj nekaj nekaj magnet, nekaj nekaj superflujt, nekaj nekaj superkonduktor. 2-dimensional crystal because there is no long-range order. And here is an illustration of why that is. What I have here is 400 carbon atoms, and they're thrown into a cubical box, and they're numerically heated up to 2,000 degrees Kelvin, and let's watch what happens. You see the atoms jiggling around. Every once in a while, they will grab a neighbor and form a chain, and if you watch long enough, they even form a ball, a bucky ball, which is kind of stable, and this can go on forever. It can go on and on and on, and no matter how you even may be able to do a nano tube, but no matter how long you wait, you will never get a 2-dimensional surface because fluctuations out of the plane are too cheap, and the fluctuations are going to destroy the long-range order. In 2004, Gaiman Novoselov from Manchester announced that they were able to isolate 2-dimensional atomic crystals. People did not believe them. In fact, they were not able to publish their work because people said, you can't do that, you must be lying, you must be cheating. So they got very nervous, they invited everybody in the field to come to their lab to teach them how to do this. So they were able to isolate graphene, niobium-dyselonite, boronitride, et cetera, about five different 2-dimensional materials. So what was the trick? So they thought, okay, you cannot have long-range order in two dimensions, can we cheat nature? So you have layered materials, like graphite or transitional metal-diacalcogenite that have very strong in-plane bonds but very weak van der Waals bonds in the third direction. So you get layers, and it's very easy to exfoliate them. So their idea was, okay, let's play like a conjurer trick. We pull one card out of the deck, and let's see what happens. So this is what they did. In fact, if you write with a pencil, we've been writing with pencils, I think since the 16th century when the graphite was first discovered, the pencil was first made. And every time you write, you bring pencil to paper, in the debris that is left behind, there will be graphene. And we've been making graphene for centuries, except nobody knew. And what's the reason that nobody knew? The reason is that it's impossible to see a layer that is only one atom thick. There's just no contrast. You don't have the tools to see them. So this was the well-kept secret for all this century until a gymen of Novaselos completely accidentally tripped over the following. We said, OK, we're gonna take some scotch tape and just put it on a substrate. And they had sitting in the lab a very old wafer, a wafer covered with 300 nanometer of silicon dioxide. This used to be used in the 80s in the semiconductor industry. And the reason this has 300 nanometers of silicon dioxide, nowadays this is totally obsolete. So they just had this lying on the shelf and they put their graphite. They just wrote with a pencil, if you like, on top of it. Now, the reason 300 nanometers is when you shine light of it on this wafer, you have reflection from the top layer and reflection from the interface with the silicon. And there is a destructive interference between the two rays at green light. That makes it so that any piece of dust will appear, will be visible because you change the phase of the interference. And the reason they had to do it this way because in those days you inspected the wafer manually just by eye. OK, so there are these extension bands. This is wavelength versus thickness of this layer and you see everywhere where you have dark you're going to get, you're going to be able to visualize dust or anything else. So if you work at 500 nanometers wavelength, which is green light, where our eye is most sensitive, 300 nanometers is a sweet spot. So they had this and they put the graphene on top and this is what they saw. So with the 300 nanometer oxide they immediately saw this is single monolayer, bilayer, trilayer. I see the step in atomic force microscopy. If they had 200 nanometers of oxide and white light they would have seen nothing and 300 nanometers of oxide and white light it would have been much less of a contrast. So you see there was really serendipity at play here. So how do we make graphene in the lab today? Let's see if this works. Take a Scotch tape and gently lay it down on a flat surface. Next, take clean metal tweezers and pick a thin graphite flake and then place this gently onto the Scotch tape. Next, fold the Scotch tape at the edge of the graphite flake. Peel it off gently and do this step several times until you obtain a nearly transparent region on the Scotch tape. After this, take a clean silicon wafer to transfer the Scotch tape graphene onto the wafer. Use plastic tweezers and gently rub the area of the Scotch tape where graphene may potentially be. Slowly peel off the Scotch tape so as not to break any potential graphene sheets. Use an optical microscope to view and find graphene. Graphene appears as a purple spot on the screen. The center of the screen is multi-layer graphene and at the lower right corner of the screen is single-layer graphene. This is how we do it. Nowadays, there are techniques where chemical vapor deposition where you can grow huge areas like a whole wall full of graphene but those are not perfect. They have dislocations, they have grain boundaries, but this technique by exfoliation is the best sample. When you want to learn about fundamental physics, this is the technique that you use. Still today. Just a few properties of graphene so you must have heard that it's the strongest material known. How was this measured? This was the group, the Manchester group. They took a sheet of graphene which was about half a micron, about actually five microns long. They founded it with copper nanoparticles and they measured how much it deflected at the end. They found that it deflected by what nanometer you immediately can calculate that the Young's modulus is about two terapascals. To give you an idea what this means if I had a piece of paper here and it was as large as this room, if I put all of you on this piece of paper, if it were graphene, it will sag by about one centimeter. This is what two terapascals means. By comparison to other materials, here are all the materials that we know. Up here we have ceramics. Steel is somewhere here too and if you put graphene on the same sheet, it's stronger than anything we know. That is one of its unusual properties, remarkable properties. Another is optical properties. This is an optical microscope where you're looking at the reflections through graphene, so this is no graphene, one monolayer graphene absorbs 2.3 percent of light, two monolayers double that. In fact, because of the electronic properties that we're going to hear about, the amount, the transmission is one minus alpha, alpha here is the QED, fine structure constant, this is the only thing that enters into the physics of graphene and pi, so it's 97 percent for one layer, it's 2 alpha pi for two layers and so on and so forth. This is when it's a charge neutrality, when you gate graphene and you put in electrons or holes, so you can actually change, for example, this is a gate voltage here, so this is charge neutrality, so the transmission, it's almost completely transparent, you put 50 or 30 volts on it, it becomes dark, so you can use that, for example, for screens and so on and so forth. One last property that I'm going to mention is it has, it's sensitive to single molecules, so for example, if you measure the change in the hole resistance when you add sodium or ammonia or carbon monoxide, every molecule that comes in gives you a step in the resistance, so you can actually measure single molecule, very sensitive nose. One other thing that is very important today is that it's completely impermeable, you can put a pressure gradient of one atmosphere of graphene, of helium, for example, not a single helium atom will go through, perfectly permeable, even though it's one monolayer thick. So since the discovery there were many new two-dimensional material that discovered about 40 of them, transitional metal dicalcogenate for phosphorine family, monocalcogenites here and so on and so forth, and they each have different properties, superconducting, insulating, semiconductor, and so on, and what's even more, now it has become an industry, you can just stack them one on top of each other and you can put various properties so you can design your material basically without any chemistry. So these kinds of stacks, if you try to actually synthesize them chemically, many or most of them are not stable, but doing it this way these are, of course, they will probably metastable, but they're stable enough to make a device out of them and to examine their properties. So here is, for example, how we do them in our lab, you pick up a graphene piece with boronitride, you have a polymer here, which we call PDMS, and you just press it onto a substrate and then you can put another boronitride layer on top and this is what it looks like in an optical microscope. So this is graphene boronitride, boronitride graphene and on top of that boronitride is the same kind of material that was used in order to discover the superconductivity and the twisted layers and this is a TEM image and you see how beautiful, perfect they are. The boronitride layers are here, graphene is here OK, so let's go, let's talk about carbon now. So carbon has six valence electrons, has four valence electrons, six total electrons and when it makes a chemical bond there's two ways it can make a chemical bond. One is the SP3 where the four valence electrons hybridize together and you form a symmetric structure which is a tetrahedron and this is gives rise to diamond basically. It's a very strong insulator, insulating material. Now the other kind of bond that graphene can make that carbon can make is SP2 where we have we have three of the valence electrons hybridized into and they use into these, to form these sigma bonds these are flat bonds separated 120 degrees from each other and there's one pi orbital that sticks out. This is the SP2 hybridization and this gives rise to graphite which looks like this and of course a single layer would be graphene. What is actually very interesting, I mean when you ask your girlfriend to marry you some people give her a diamond because there's this myth that diamonds are forever but in fact diamond is a metastable phase of carbon, the stable phase of carbon is graphite. If you look here graphite so this is pressure versus temperature so this is where room ambient conditions are the stable form is graphite. So don't tell it to your girlfriend eventually the ring will turn into soot but she can wait that it's gonna take many years. Now carbon is the only element on the periodic table that where SP2 is the ground state all the other elements all the other elements in the fourth column the stable form is SP3. So people are trying very very hard to make silicon that's called silicon germanine but it's very hard because they don't want to do that and the reason is carbon is because of the ratio of the size of the orbits because it's only in the first level that your valence electrons are and their coulomb energy is strongest whereas in the lower elements silicon and germanium etc. atom is larger the p orbitals are further out and it pays if you do the calculations it really pays to be SP3 only carbon has a ground state of SP2 so only carbon can create something like graphene except if we try hard as you will see later Okay so SP2 carbon allotrop there are several forms so it's a very flexible kind of structure because it's two dimensional and it's bound in the third direction with another layer by very weak van der Waals forces so this is a zero dimensional form this is a bucking ball discovered in 1985 actually it was discovered by astronomers 1996 Nobel Prize the second form is the carbon nanotube this is the one dimensional form discovered in 1991 no Nobel and it's the reason there is no Nobel for this is because there are too many people fighting each other and whenever that happens there is no Nobel because they can't make up their minds and the third one was two dimensional Nobel in 2010 and this is graphene and of course three dimensional is graphite okay now we're going to the next topic which is the electronic properties the band structure now I'm going to just do a quick reminder of tight binding because the tight binding gives a very very good it's a very very good approximation to the actual band structure of graphite of graphene okay so if you have a periodic potential in red here we have the atomic orbitals which sit on the atoms and then if you have phi's are the eigenstates of the isolated atoms and now if you have a little bit of overlap between the orbitals and if it's small enough your Hamiltonian is the original Hamiltonian plus the perturbation you can do perturbation theory so this is the new Hamiltonian new wave function and we can just expand in this correction delta u so tight binding works very well when you have small overlaps so the way this works is okay if you have a periodic structure and in order to have a periodic structure you have to have a Bravais lattice and this is a real kicker here and as you will see in a moment if you have a Bravais lattice which means that you have a periodic structure you can write a block function block function is a good solution for your Hamiltonian so you write it down like this it's a plane wave multiplied by something that is periodic and this is the atomic wave function on each one of the atoms and then okay we have the block wave function we plug it into the Hamiltonian so these are the conditions the wave function has to satisfy normalization number one it has to be at eigenstate of the Hamiltonian number two and then you just do some algebra and you find a solution it's as simple as that although the algebra can be pretty involved but there's only these three steps and then you write you find a solution in terms of the transfer or hopping integral which is these are nearest neighbor and we call it tij and also in terms of the overlaps between adjacent wave functions and this is another way of writing down the Hamiltonian with raising and lowering operator but it's the same thing but the reason I'm writing it like this is because it has explicitly here the transfer integrals so so so so when you do tight binding for very simple periodic metal like gold or silver you usually get a very simple band structure that can be approximated but it's basically quadratic in a momentum and if whenever your energy is quadratic in a momentum you completely say that must be momentum square whatever is the numerator here must be a mass so we define an effective mass which is absolutely nothing to do with the actual mass of the electron it's just a way of saying that the energy is proportional to the square of the momentum so we get something that typically looks like this this is a parabola wave function these are equi energy surfaces now how is graphene different now the graphene is different because honey comb lattice it's not a bravel lattice the games that I've done here don't work for when you don't have a bravel lattice so when you don't have a bravel lattice you get all sorts lots of lots of interference and your band structure looks like a mess looks like this and these are the equi energy planes so you get a very unconventional dispersion by the way this is exactly the dispersion that Wallace got except that it was not in three dimensions but his solution was absolutely correct even in today's terms so now what I'm going to do I'm going to walk you a little bit through how we get the solution and I'm going to walk you through the quasi particles that corresponds to low energy excitation for graphene but they have weird and interesting properties those graphene quasi particles ok so we start so there's only three ingredients that go into the unusual band structure of graphene really only three the fact that it's two dimensional the fact that it's a honey comb lattice which means that it's non brave and the fact that we have identical units on the two sub lattices now what do I mean by two sub lattices notice that a brave lattice is one is when you move no matter where you move how many lattice spacing you move your environment is exactly the same but here you see immediately we have two group it's bipartite we have two groups of atoms I can color it red and green the green atoms have a red neighbor down south the red atoms have a green neighbor up north so this is not a brave lattice it's two interpenetrating triangular lattices which are brave so what we have to do we have to write a wave function two wave functions one for the green atoms and one for the red atoms and the total wave function will be a linear combination of the two so here are just some concepts to give you some idea don't get too hung up on this I'm defining here the lattice constant A so that's the lattice constant for graphen is 2.46 angstroms this is the first Brillouin zone that looks like it's a hexagon these are the reciprocal lattice vector and there are special points in the Brillouin zone these are the k-points that we're going to talk about a lot and one k-point is at 4 pi over 3a this one the m-points in between the 2 here and the gamma point is the center of the Brillouin zone so we're going to mention these a few times so because we have two wave functions the wave function for the electrons in graphene is a linear combination of the two so it can be written as a linear combination of the wave function on the A sub lattice and one on the B sub lattice so green sub lattice red sub lattice and you can write down the prefactor here this looks exactly like a spinner it has two components because we have two difference of lattice each component corresponds to the part of the wave function that it's on the green or on the red sub lattice so immediately you see we have a new degree of freedom this is like and it resembles spin up and spin down for a spinner but instead of acting on the spin we're going to be acting on the sub lattice degree of freedom which we call now pseudo spin it's not a real spin, it's like a spin we call it pseudo spin ok so I'm going to do a little bit of the derivations here because there's a few concepts that I want to bring home so the Hamiltonian of course is a 2 by 2 Hamiltonian and on the diagonal here we have the self energies here on the a sub lattice on the b sub lattice here and the off diagonal here we have the phase which is basically the sums of the phases of the nearest neighbors e to the ik delta j sum of all these ok so these are the things that go in now for graphene epsilon b and sorry this should be epsilon a this shouldn't be a b ok epsilon b and epsilon a are equal and we call them epsilon zero and this really simplifies the algebra enormously and by the end of the day we get a constitutive equation like this where the energy has two where we have two solutions which I'm going to draw in a moment but notice that it depends on two things one it depends on the hopping integral t and two it depends on the phase here so it depends crucially on two things the two most important things are a the geometry which comes in through f here because it is triangular geometry and b it's t here which is the hopping integral ok now here is a note of caution if we have different species of the two sub lattices the solutions are extremely complicated and most importantly you will get a gap at the fermi energy you will get a gap at charge neutrality it's going to be an insulator so one example is for example boron nitride where it's identical to graphene instead of having carbon everywhere one sub lattice has boron the other sub lattice has nitrogen and that has a huge gap has a 6 electron volts gap so the best insulator we know because because you avoid this crossing here which you're going to see in a moment so this is the solution I'm writing down the solution and it's really very simple algebra I strongly recommend that you do this as a homework and these are the two solutions we have an antibonding and a bonding solution these are for the plus and for the minus and you see they cross here this is what we call the Dirac points they cross at these points now if you look on the brilloenzo this is the K and K prime so these are the two points but of course this has to be a two dimensional structure so we have six of these K points so you see we have six of these paper this is what the three dimensional image of the mass structure looks like but what we're going to be interested in is the low energy properties so we zoom into this point here the crossing what we call the Dirac point and this is what it looks like now we have six points here I only draw two why do I only draw two why should I not draw all of them it has to do with the fact that that all the other guys are connected to either K or K prime by reciprocal lattice vector so the physics is identical so only two matter all the other ones are going to just repeat the same physics because you can move from one to the other by adding a reciprocal lattice vector now one very important thing you probably learned in your first course in condense math in solid state that if you have bands that intersect you have level repulsion you open a gap and it happens so why isn't anybody asking me how come this survives now the reason it survives is that it's protected by three symmetries you have to the symmetries by which it is protectors is time reversal inversion and C3 rotation by 120 degrees so you have to break at least two of them to open a gap very very difficult to gap out graphene which is both a curse and a blessing ok so those the rock points are very very stable I mean you can stretch the thing you can bend the thing they move around in case space but they don't open a gap until you do something dramatic no no ok very good question this is only nearest neighbor now when you include next nearest neighbor now you see we have electron hole symmetry what happens if you include next nearest neighbors you're going to break the electron hole symmetry not open a gap next nearest neighbor doesn't do anything to the symmetries so you introduce next nearest neighbor you're going to get instead of it being beautiful and symmetric this is going to be a little shallower than this but really qualitatively nothing new happens ok so some nomenclature so it charge neutrality the Fermi energy sits exactly at the the rock points in the upper band is called you have conduction electrons the lower bands we have valence is the valence band so conduction and valence band as usual with semiconductor it's just that there's no gap ok a little bit more of formalism and don't get too hung up on it there's a few points that I would like to make so first of all we go near the k points and then we do a linear expansion so the k, the 2k points here are psi, psi is an index these are called valleys so the psi is an index for valleys so plus 1 for k minus 1 for minus k and then the position of 4pi over 3a and 0 on the 2 sides here right? if you expand in small momentum term here so so we define k as the momentum relative to this the rock point to this k points here we define little k and this is by how much we deviate from this point so we deviate a little bit around here so we get what we call what we will now call the momentum relative to the rock points so these are the momentum operator here now I'm doing the linear expansion for this phase here and the Hamiltonian, so if you put in these approximations here you get an approximate Hamiltonian that looks like this and the prefactor here vf, which is we call the Fermi energy is proportional to the overlap integral times the lattice spacing with some factors here and this number here is about 10 to the 6 meters per second it's a velocity it's a velocity of the quasi particle as you see in a moment but it is about 300 times smaller for example in the speed of light you will see in a moment why I'm mentioning that ok, so this is the Hamiltonian when you have close to the rock point it has this form here so it will have a plus here in the k point in the k prime point and so this is exactly a Dirac-like equation with eigenstates are the projection of the wave function on sublates A sublates B so this is again our Hamiltonian as I wrote it before in shorthand notation I can replace I can write px times sigma x and py times sigma y so I can write it in a shorthand notation where the sigmas are the matrices so I can write it in terms of poly matrices so I have a very nice compact notation so the bottom line the Hamiltonian is equal to the Fermi velocity times the projection of the momentum on this pseudo spin sigma is a poly matrix but remember it operates on the sublates degree of freedom not on spin we haven't talked about spin at all here so this is a Hamiltonian extremely simple this is exactly what we call the Dirac-bile Hamiltonian which is for ultra relativistic particle that are massless same Hamiltonian as for neutrinos except that this is in two dimensions and except vf here replaces the speed of light for neutrinos you will not have speed of light instead of vf ok so poly matrices they operate on the sublates degree of freedom so if we take into account the k-points and the two k-points then instead of having a two component wave function we have a four component wave function and we can just the Hamiltonian breaks down into two blocks one for k and the other for k prime what's important to notice and this is very very important is that Hamiltonian for the k point is vf sigma dot p whereas for k prime it's vf sigma star dot p this is the time reversal brother of the k so the physics in the k prime valley is time reversal brother of the physics in k so whatever if you go like this in the k valley the electron in the k prime valley is going the other way so these are related by time inversion symmetry now the third point that I want to make is no we can solve for the Hamiltonian very easy it's a 2 by 2 Hamiltonian you can solve for the energy spectrum and you can solve for the wave function and the energy spectrum is extremely simple it's vf times p or you can write it as h bar k and s here is the band index so it's going to be positive in the electron in the top band and s is negative in the bottom band here and the wave function looks like it again s it changes side for electrons and for holes so you have this is a plane wave multiplied by a spinner now phi here is the polar angle it's just the ratio the arc tanges of p y over p x it's the polar angle so what we see here immediately this dispersion is no longer p squared over 2 m it's actually vf times p this is exactly the dispersion that we have for photons exactly the dispersion that we think we have for neutrinos these are massless particles mass is zero mass is linear in momentum not quadratic very important so it is like photons except these are very slow ones they move 300 times slower than photons or neutrinos now this two component vag spinner it's a pseudo spin vector and notice phi appears here when s is plus one the pseudo spin is parallel to the momentum we have e to the i phi one here when s is negative it's antiparallel to the momentum and you're going to see in a moment what that means and number four absence of backscattering within a dirac cone so this is our wave vector if you take if you take the projection of the wave vector in direction zero so this is basically the angular scattering probability angular scattering probability around a circle you take the projection of psi at angle zero onto projection of psi at angle phi the square of that is cosine square phi over two what you see immediately that it's zero there's absolutely no backscattering at when phi is equal zero so under pseudo spin conservation that's what this means backscattering with one valley is suppressed there's no backscattering with one valley so an electron that moves in one part of the cone cannot scatter back to the other part of the cone no backscattering within a valley so this is very important for transport properties and number five is the helicity so we've seen that Hamiltonian can be written like this sigma dot p now you can pull out actually the magnitude of the momentum this is sigma dot n this is known as the helicity operator is the projection of the spin onto the direction of the momentum it's called helicity finite mass so you see that this is a good quantum number so the projection of the pseudo spin on the direction of on the momentum or on k is a good quantum number it cannot change so helicity is a conserved quantity and we can have in conduction band the helicity is plus one so it goes let's say clockwise in the valence band it's minus one they go the other way and here is an illustration of that so in conduction band p and pseudo spin are parallel to each other in the valence band they're anti-parallel to each other now in the other cone everything is reversed because of the time reversal symmetry so if you have because helicity is conserved in the other cone everything is opposite we have no backscattering between cones so what does this all mean so here are the two cones this is the Hamiltonian in this k and this is the time reverse Hamiltonian in this k conserved helicity and you see one we have plus helicity in the k cone minus helicity in the k prime cone what this mean that there is no backscattering between the two but we also cannot backscatter because because you cannot backscatter within one cone we just did that so the bottom line it's no backscattering which mean you send an electron in your sample it can't go back it just that means that you can have very large mobility why is there what oh it's a good quantum number this is helicity so if an operator commutes with your Hamiltonian that means that you have a common basis vector and that means that corresponding observable is conserved unless you disturb it somehow ok so this brings us to climb tunneling now here I have a barrier that you can do an electrostatic barrier I have an electron this is graphene and I am applying a voltage here ok so it causes a barrier now one thing that maybe I didn't emphasize is that the Dirac point rides on top of the external potential so the external potential here is this is flat here so Dirac point is here here the Dirac I apply a voltage so the Dirac point is here now notice what's going on so I bring this is the Fermi energy I'm bringing an electron it's moving to the right that means it's moving on this branch here it's moving to the right now it's coming under the barrier now it's no longer an electron it's a hole there's lots of places for it so it's not it doesn't have to go back but it also cannot go back because there's no back scattering it does that by turning into a hole and then it comes out again so it is as if this thing completely didn't exist it's completely transparent so this is I tried to do this little animation hole, it's completely insensitive to electrostatic barriers this is called client handling it used to be called the client paradox how come these massless chiral particle can just go through barriers it's a paradox it's this it has to do with having basically the conduction and valence than touching so what are the consequences we cannot have electrostatic confinement we cannot use electrostatic to do anything to the electrons in graphene they do whatever they want can't control them with electric fields so you can't do all the things that we know and love like make quantum dots which make transistors you can't guide them and so on and so forth however I told you that there's no back scattering but that you could have side scattering and the sides so now this is transmission probability as a function of angle of incidence so zero if you come ahead on the transmission is absolutely one but if you go at one angle and it'll depend you have regions where there is almost no transmission and the red and green depend on the energy on the density of your carriers I'm going to come to the last point and that is the berry phase so this is a paper by Berry in 1984 so if you have a Hamiltonian and you very slowly change something in the eigenstate and very slowly changing the adiabatic approximation that tells you that the wave function is going to follow is always going to be an eigenstate of the Hamiltonian so you take your Hamiltonian through some sort of a circle you come back to the same point the Hamiltonian is the same the wave function will have changed by a phase and people knew that forever since quantum mechanic was invented phases don't matter in quantum mechanics let's forget about it what Berry showed is that this phase carries the information about the trajectory that you went on and phases can be seen in quantum mechanic when you have interference effect so it's not completely invisible so this is what Berry said and he showed an example how to calculate the Berry phase by how much does the phase change of the wave function change when the Hamiltonians come back to the same spot so you go along a circle you have a parameter lambda equals 0 that your wave function depends on you go on a circle and you come to lambda equals 1 so you increment it continuously and when you come and the circle at lambda equals to 1 and then you calculate the Berry phase like this is the imaginary part of the integral over on this loop of your integral over lambda u lambda d by d lambda you take the derivative of your wave function with respect to lambda and I'm going to show you that if you do this exercise with the wave function for graph this is pretty neat so what does the Berry phase mean when the phase of your wave function changes that means that something in the geometry it measures something in the geometry or topology of your of your environment of space take for example a Mobius bed of course there is that the topology is not simple and if you follow g for example I am going around g is out here I am not seeing it and then it is going here and then it is going to appear again I have to go twice around the circle for g to show up on the circle tells me something about topology of space another very nice example is parallel parallel transport let's say that you have a car and with a little handle pointing sideways and you are going along a geodetic so I am coming down here my car is pointing this way I am doing parallel transport so now I am going to move parallel to myself and then I am going parallel to myself like this you see it is always pointing in the same direction the car knows it is a weird car and I am coming back to the exact same point but now the nose is pointing in different direction this is the very face and it measures the curvature of the space on flat space you get zero it comes back to exact space but this is a way to measure the curvature of your space in optical fiber this is a very neat thing you send a polarized light in optical fiber you twist the fiber around the polarization is going to change direction is going to change and the dirac belt is amazing but I don't have time to do it I can illustrate it with my hand but I am not always successful in the break maybe I will show you ok so if you do this thing for the wave functioning graphene it is pretty straight forward you just do this integral you define lambda as 2 pi theta over 2 pi and you take lambda from zero from zero to one you take this integral and you end up with pi so the very face of graphene for the quasi particle graphene is pi and that has to do with the singularity of the dirac point it's measure the topology of space we have a singularity of the dirac point this is what it measures and as a result it has all sorts of weird consequences which are seen in quantum whole effect and in landau levels ok ok if you want you can do this as a homework exercise ok so I think I am pretty much done with my time right for the first part I have 3 more minutes ok if I do 3 more minutes here I am or if you are tired we can do it after the break let's do it after the break