Electronic Excitations in Two-dimensional Materials and van der Waals Heterostructures

[Music] like to welcome everybody to this afternoon's event the structural thesis defense and most especially I'd like to welcome the doctoral candidates questioned some of Susan and the two official opponents professor Liu Qi are earning from Ecole Polytechnique in Paris in Paris Oh in France and professor de Coulomb at Sahni they called police Nick fédérale de Lausanne in Switzerland also welcome to Professor hitting freeze polls in the Department of Physics at DTU is the chairman of the assessment committee my name is Rasmus Larson I'm the provost and a chief academic officer at this university and I'll be the chairman of the proceedings this afternoon so these proceedings will be as follows so after my small introduction here the doctoral candidates Christian summit recent will take the podium and present his thesis that's entitled electronic excitations in two-dimensional materials and Fanta bars hit zero structures in a lecture of approximately 30 minutes length thereafter we'll have a small break and when we come back from the break the two official opponents will enter into discussion ask questions expect answers from the candidate after the two official opponents have exhausted their lists of questions or exported the knowledge of the candidate whichever comes first there'll be the opportunity for questions its auditorium now if there are unofficial opponents in the auditor auditorium please contact me in the break after the lecture now before we begin I should also add there's one more important item so when all discussions have ended in here and the defense the formal part of the defense is over then the Technical University of Denmark will host a small reception outside in the lobby immediately outside the auditorium and I hope all of you will join us out there now before we begin I'd like to say that in these years there is a very dominating agenda in that we see how digitalization is changing every industry product service and almost every human endeavor this is really dramatic our information and communications technology big data and Internet of Things bring new opportunities to all engineering areas but I think it's very important to remember that this is not the only thing that is happening there are other basic disciplines that bring disruption to biology is one biology and bio engineering will bring many new opportunities into the engineering disciplines will see bio-inspired solutions confluence of electronics and biology and so on now a third trend is in material science so material science has and we'll continue to bring dramatic changes in all engineering fields new materials with amazing and sometimes unexpected properties emerge and gives us new possibilities for engineering products in all the engineering areas and this I think is something that we will learn more about today about materials on the smallest scale about molecular electronics representing new and exciting low-cost nanoscale alternatives to the conventional semiconductor devices that we have in all our IT equipment potentially making it possible to reduce all these things in size I think there are great perspectives there now with this I'd like to end my small introduction and invite questions so much using to the podium to present his work that's cool thank you very much I suppose I just need the pointer here so welcome to all of you I'm very happy to see so many people that showed up here today as last ones already set the title of my thesis is electronic excitations in two-dimensional materials and Fant about hetero structures and I'm sure there are a number of you who don't really know what these words mean so I will start by explaining in simple terms what a two-dimensional material is so a two-dimensional material is atoms that are bonded together by chemical bonds in a two-dimensional fashion like shown up here this is the case of graphene where carbon atoms are bonded together the thickness of such a material is then just a single atom or maybe a few atoms so this is less than one nanometer this corresponds roughly to 100,000 times thinner than a human hair and these materials can be on the order of square centimeters large when they are produced so this means we are talking about extremely thin and because they are so thin these materials they can have a number of very special properties that are very different from the properties we know of materials that are thicker and more conventional so to illustrate that I'm using graphene which is by far the most well known two dimension material also the first that was discovered graphene is a fantastic material it's the best electrical conductor that we know it has a room-temperature mobility which is 10 100 times larger than that of silicon which is the basic material that is used in electronics today it's the strongest material known it's roughly 100 times stronger than steel it's an exceptional heat conductor one of the best-known heat conductors it's almost transparent so it has good electronic conductivity and is transparent at the same time which seems to be in conflict with our normal perception of a metal it's lightweight and it's highly flexible it can be stretched by 20% before it breaks and then in addition to all these fantastic practical properties it also hosts electrons so electrons in this material they behave very different from electrons in all other known materials and that has led to a number of very exciting fundamental discoveries 4/6 discoveries over the past decade and some of them are listed here ok so the field of two-dimensional materials has developed extremely rapidly over the past decade since graphene was discovered back in 2004 and let me try to give a very brief review of this history focusing on some of the milestone discoveries that have been made so the starting point is the isolation of graphene in 2004 then in the years between 2004 and 2010 all these remarkable properties that I just mentioned of graphene were discovered and that led also method to the Nobel Prize award to entry gaim and Konstantin Novoselov in 2010 and around the same time research in other types of two-dimensional materials took off and one of the most interesting classes that were discovered at the time where the transition metal dye Calcott units of which molybdenum disulphide is the most well known example and in contrast to graphene which is the so called semi metal it doesn't have a gap between the occupied and unoccupied electronic states these materials were semiconductors and that means we can begin to use them for electronics and optics and already one year after their discovery the first field effect transistor based on a single layer of MOS 2 was produced the year after in 2012 it was discovered then one can excite shot carriers in these semiconductors selectively in different valleys of the semiconductor using circularly polarized light and that adds another degree of freedom to the charge carriers in the material so in addition to charts and spin now the valley index can also be controlled and that led to a new field called Meli tronics that took off at that time in 2013 and reg I'm and co-workers produced the first so-called Fender bass hetero structures which are hetero structures produced by combining different types of two-dimensional materials forming a vertical stack as shown here and that opened up completely new perspectives because now we can begin to engineer materials also taking advantage of the third dimension in 2014 and other by now very famous two-dimensional material called phosphor Wien was discovered and in 2015 different types of light-emitting diodes based on both lateral and hetero and vertically layered hetero structure devices were developed and fabricated last year it was discovered that some of these two-dimensional materials can host quantum light emitters that is they can emit single photons and that holds great perspectives for different disciplines and in quantum technologies like quantum communications this year the first ferromagnet two-dimensional fur magnet was discovered in the form of chromium iodide 3 and then with all these materials discovered one okay how many are there and I think a first good answer to that came also this year by different groups in particular nikola Masari who is one of the opponents here today who looked at all the known crystals all the crystals that have been synthesized today and looked at how many of those are actually layered and could be exfoliated to single layers and the answer is more than 1000 so there is a huge number of two-dimensional materials out there that we can potentially take off and study and recombine again by using the concept of of hetero structure layering so I should say at this point perhaps mainly as a curiosity that the idea of designing materials by stacking two-dimensional materials is not new the American physicist Richard Feynman who was a very visionary person already back in 1959 gave a very famous talk of plenty of room at the bottom where he said imagine what we could do with layered structures if we just had the right layers what would the properties of such materials be if we could arrange the atoms all the layers in exactly the way we wanted so I think these days or these years we are experiencing this this this vision of Feynman is is about to come true we can really arrange these two-dimensional layers in the way we want them here's an example just a couple of weeks old published in nature where wafer scale semiconductor films so these are our films that are square centimeters in area were produced and you can see here they consist of single layers of different two-dimensional materials such semiconductor heterostructures really form the basis of all digital electronics today and typically these films are tens of nanometers thick and there are various problems there they are very hard to produce you can only stack materials in this way if the lattice constant of the materials match but these types of materials lattice mismatch is not a problem and the reason is that the materials themselves are chemically inert and therefore they don't form covalent bonds in the third dimension you can see these this is a at sea I am picture ask transmission electron microscope picture you can see black lines in between and that's because the bonding between the layers this week this is called fender valves bonding and does the name van der Waals hetero structures this means we can put any types of two-dimensional layers together here's a tri layer of three different semiconductors so this is an illustrative but very simple calculation one can try to do so imagine we have 10 different two-dimensional materials as at our disposal in reality I think we know about 30 today that have been isolated how many possible combinations can we make how many three-dimensional materials can we make if we stack these layers to form a hetero structure with 10 layers that number is 10 to the 10th so that's an astronomic number and as I said today we know that they are probably not just 10 but more like 1,000 different two-dimensional materials out there so that means this number is huge and this also shows that we really need to understand and we need methods that can predict in a relatively quick way the properties of these types of materials if they're experimental they should have any chance of of designing and producing materials with tailored properties so after this general introduction I come to my own contributions which is described in the thesis so what I've considered in the thesis are electronic excitations in these two-dimensional materials and in particular I have looked at when I say I you should always understand me and all my fantastic co-workers that have contributed I'll come back to that in the end of the talk but we have considered three different types of excitations in these materials called quasi particles exit zones and plasmids a quasi particle that's an electron or a hole in a solid together with its screening cloud okay and the energy of the quasi particle forms what is known as the band structure of the material and exits on that's a photo excited or that's an electron hole pair photo excited electron hole pair which is due to the attractive interaction between the negativity electron and positively charged toll is bound together and forms a bound state with an energy that lies inside the band gap of a semiconductor a plasma is a collective quantized oscillation of the electron system in a metal in combination these three types of excitations determine the electronic and optical properties of the material so therefore understanding them and developing computational methods that allow us to predict and rationalize the properties is extremely important they can be measured by various types of experimental probes STM optical absorption electron energy loss etc before we can begin to understand and calculate electronic excitations in any material we have to understand the concept of screening we have to understand how two electrons or an electron and a hole interact with each other inside the material if they're in free space this is Coulomb's law this is just something that's games with one over the distance but here is a two-dimensional material and there is a positive and negatively charged particle in here and what you can see is that the field lines most of them actually grow through vacuum and that means that in in in contrast to three-dimensional materials where all the field lines are inside the material then the screening will be very weak so that's the first thing to remember screening is very weak in two dimension material the other thing is as you can imagine when these particles are close together a relatively large fraction of the field lines are still inside the material but when the particles are very far away almost all the field lines are in the vacuum outside and that leads to a very non-trivial dependence of the of the screened interaction which does not follow what we know from three dimensions and mathematically this means if we look at what is called the dielectric function as a function of wave vector Q that for a two-dimensional material we get the green line this is hexagonal boron nitride a single layer that's the green line that's the dot static dielectric constant and the dashed line is bulk hexagonal boron nitride we can see four large Q vectors here where Q is larger than the inverse thickness of the material the screening properties are similar to in three-dimensions okay this corresponds roughly to the situation in real space where the particles are close to each other but when the particles are far away and that means we have wave vectors in reciprocal space that have wavelength that is longer than the thickness of the material then there is a clear deviation and the dielectric function becomes linear and q rather than constant as we know from three dimensional system so this is the mathematical consequence of these peculiar screening effects in two dimensions so there is a number of consequences of this particular form of the screening in two dimensions and I've listed some of them here and I say unique because all these effects here are very different in two dimensional materials compared to the well-known phenomena from three dimensions and they can all be traced back to the peculiar type of screening in two dimensional materials now two dimensional materials are usually not alone in the universe they are typically put on a substrate or they are embedded in a Fender valles hetero structure and since the cool long way interaction is long range it means that the effective interaction between the two particles now will be very sensitive to the environment of the two dimension material so that's of course a complication if you want to do calculations or if we want to interpret experiments we have to take the environment into account but it also has a very attractive opportunity because it means we can begin to tune the effective interaction between particles in this material and thus the electronic structure of the material by controlling the external dielectric environment so we have a way of now engineering properties and electronic excitations in the two dimensional materials so how can we calculate dielectric functions or screening properties of fanta bars hetero structures we've developed what is called the QE of what we call the quantum electrostatic hetero structure model or QED eh in short which calculates the dielectric properties of a hetero structure shown here by taking each of the two dimensional entities here and then computing what we call the dielectric building block for each of them so this is for an isolated 2d material we calculate a low dimensional representation of its full dielectric function using full-fledged quantum mechanical calculations and then we coupled the different layers using the dielectric building blocks we couple them together only by the Coulomb interaction between the layers so that's a classical coupling mechanism and that provides us that that and enables us to calculate the full dielectric function of this material so with that we can now calculate the effects of interaction between an electron located up here and another election located down here so we have full control over the dielectric properties of the hetero structure using this method so does it work yes here's an example where we computed the so-called plasmas in a hetero structure composed of two key to graphene layers here with insulating hexagonal boron nitride in between and what we find is two different types of plasmas one is a symmetric and an anti-symmetric combination of the sheet plasmas in the graphene layers because they hybridize and and interact with each other and what I'm showing down here is the plasma dispersion of the two modes calculated using the q8 model that's the full lines and the symbols is what we get from a full quantum calculation of the entire hetero structure and you can see the agreement here is I would say almost too good we can do these full quantum calculations up to three layers of hbn but with the qe8 model we can calculate that for any number of HB and layers in between so we can go to much larger structures so the method works ok so now I'd like to turn to band structures and how screening effects span structures in two dimensional materials here is a band structure and the band structure gives the binding energy of electrons and holes in the material they can be measured experimentally by shooting photons into the material and it's ejecting electrons and then by measuring the kinetic energy of this electron we can infer what is the energy of the hole that it has left behind we can do the same thing for the unoccupied states here by inverse photo emission theoretically we can calculate these energies by solving what is known as the quasi particle equation the quasi particle equation has a simple part which is the kinetic energy and the external potential set up by the crystal and then it has a complicated part which is called the exchange relation cell energy and that self energy physically describes the interaction of electron with its screening cloud so the electron polarize is the material and then the electron interacts with that polarization cloud and that's described by the self energy that self energy in the widely used GW approximation which is the approximation that has been used throughout the thesis takes this form mathematically what I want to show you here is that the screened attraction between electrons ends is explicitly in this expression so this shows that by manipulating the screened interaction we should be able to manipulate the self energy and thus the band structure of the material okay I'm just showing you what we can do with computational material science today here we calculated the band edges so the top of the valence band and the bottom of the conduction band for 50 different two-dimensional semiconductors and there's a lot of information in this plot that I don't have time to go into detail with today but one thing we were interested in here is finding good candidates for doing water splitting and hydrogen evolution which is in a very important chemical reaction for the production of chemical fuels but what I want to show now is how could the agreement is with experiments when we do these calculations so I should say that there are two types of calculations up here the red path is this GW method that we are using and the blue is a more standard single particle approximation called Lda now if we take the GW result and we compare the conduction band minimum and valence band maximum two experiments the gap between the two is calculated to 2.5 and that's in very good agreement with experiments so on the bandgap we're in very good shape we can really predict those very accurately if we look at the absolute position of the conduction band here and compared to measured electron affinities the agreement is not so good and the reason for that is is a technical one but we are missing what is called the vertex Corrections in the self energy this is probably mainly for the opponents and the vertex Corrections are really important in in in predicting the absolute energy of these levels and if any one else should be interested this I should say that we have a PhD defense on Tuesday but here Schmidt who's working in the group who's looked at this problem and developed vertex Corrections that can actually fix these problems to a large extent and this is also work together with Thomas Olsen okay so coming back to the effective screening the calculation I showed you before we're all for isolated two-dimensional layers what happens if these two-dimensional layers are in contact with other two-dimensional layers or a substrate so we look at this system here which is example boron nitride which is an insulator with a large band gap and graphene here's the band structure calculated again both with GW and LD a this is where the we have the band gap of of hexagonal boron nitride or HB n between the red bars what is the energy of an electron in this material well it has the energy that it has in hexagonal boron nitride but now there is a graphene layer on top that negatively charged electron will induce a positively charged image charge in the graphene layer and there will be a positive interaction between the two that's a correlation effects that we have to take into account and in the simplest classical description we could use an image charge model to model this and then we would expect a 1 over distance dependence of the band gap in hexagonal boron nitride depending on the distance to the graphene and that's exactly what we find so this is a 1 over distance fit to the GW calculations of the band gap in hexagonal or nitride you can see this is the gap in freestanding hbn so we can actually reduce the gap quite significantly by putting it onto a layer of graphene the problem with this from a practical point of view is that it's very difficult to perform calculations of these hetero structures GW calculations of these hetero structures this system here is roughly the largest you can do today so to solve that problem we've developed another method where we compute the energy of an electronic state of a two-dimensional layer embedded in a hetero structure so this would be layer I a state NK we've write that as the energy of the electron in an isolated two-dimensional layer which we can easily calculate and then we add Corrections due to hybridization between the layers that's easy we can do that with G of T or and there is a correction due to this screening effect which is much more complicated but we can calculate that using the qe8 model because we can calculate the change in the electronic screening between the electron and and the other electrons in the system so we can do that very cheaply and that leads to the G G Delta W method that we've developed with browsers now to calculate quasiparticle band structures of hetero structures with hundreds of layers so I think that really opens up new perspectives in the field here you can see the calculated band gap of different semiconductors when they are embedded in two layers of graphene and the point here is just that we can really tune the band gap quite significantly here by by modulating the dielectric environment by embedding the semiconductor in different types of two-dimensional materials okay so now let me switch gears and talk about exit tones as I've already said an exit Sun is an electron hole pair in a semiconductor which are forming a bound state in the band gap due to the Coulomb interaction between the electron and the hole okay so the binding energy of the excitation is the distance between the exits on energy and in the band gap that's an important parameter cause the exits on binding energy another important parameter is the size of this excitation so the average distance between the electron and the hole in the semiconductor that's called the exit on radius in a many-body a quantum mechanical language this excitonic wavefunction is a superposition is a highly correlated state a superposition of electron hole pairs created over the entire breeding zone we can take the expansion coefficients here and then one can show that the Fourier transform of these expansion coefficients which are defined in reciprocal space satisfy this simplified reading our equation which is called the mod linear model so this is a drastic simplification and if the screening is simple so if it's just 1 over R times epsilon as it is in three dimensional crystals where we have a dielectric constant then this is just a screened hydrogen atom in three dimensions ok the point is this doesn't work in two dimensions and let me show you why here are some measurements these are measurements of the five lowest lying exit zones the five lowest energies of the route Berk series in tungsten dye selenite that have been measured and here's a fit to a hydrogen atom and you see that fit doesn't work very well so what they did in this paper to explain this was they used the standard expression for the energies of a hydrogen atom but then they used epsilon C R dielectric constants that were state dependent and by doing that they could actually fit the experiments very well of course it's not a very satisfactory to fit these values to experiments so we asked ourselves can we calculate those a be neutral so without any fitting parameters so the starting point is this a hydrogen model now in two dimensions rather than three dimensions and we have this screening constant down here so what to use for that the problem is that depends strongly on Q right so it goes to one here so for Q going to 0 there is no screening at all in a two-dimensional material so we can't just use a dielectric constant that's the whole problem so what we did this was work together with Thomas Olsen what we did there was instead to average the dielectric constant here over the inverse radius of the exit on okay that's a technicality but that's roughly the screening wavelengths that are important for the exit zone if we do that and we combine that with the analytical expressions we have for a two-dimensional hydrogen atom then we can actually end up with analytical expressions for the dial effective dielectric constant now state dependent but calculated analytically the energy and the radius of the exit zones does it work yes it works very well here's a comparison again what do we get with the analytical results compared to more accurate calculations if these dots had fallen on this diagonal here that would be perfect agreement so there is a bit of scale of scatter but it's actually quite good overall a very interesting thing here is if we look at our analytical expression then the mass actually drops out to a good approximation so the mass of the electron and the hole the effective mass doesn't enter the expression for the binding energy this is again very different in three dimensions the energy is proportion to the mass so that's a novel two-dimensional effect another thing is that the model is explains is the observed linear scaling between the exit unbinding energy and the band gap of the material because the polarizability alpha here or the dielectric constant of the material goes roughly as 1 over the band gap of the material okay so that's also well explained by this expression so coming back to the initial example here these were the fitted values for the dielectric constant and what we get with our model calculated up in each show is a very good agreement and therefore the exits on energies we calculate also in very good agreement with the experiments I should add here that it's very essential to takes substrate screening into account so if we do these calculations for an isolated layer of of tungsten dye selenite we get the green energies over here so the experiments are the Reds if we use the qe8 model to include the the screening from the from the substrate what we get is the blue here again in very good agreement with the experiments ok so the last example if I have five minutes concerned plasmas and in particular the different nature of plasmons and exit ins and again there are some things when we compare exit zones and plasmids which are very different in two dimensions compared to three dimensions so this was a collaboration with super stem at the darris buri which is a scanning transmission electron microscopy facility which are specialized in making very high resolution both spatial and energy wise so it's it's a TMS that are well suited for studying electronic excitations so what we looked at were different flakes of MOS 2 so here is a sample where you can see from this TEM image different terraces here corresponding to different thicknesses of the MOS 2 so the thinnest around 3 layers and the thicker one here are are close to bulk MOS 2 so this is a Tim image and over here is shown eel spectra so this means you take the electron beam you place it at a particular point over the sample and then you measure the energy loss at that particular point and you get a spectrum out as a function of energy you can now take that energy spectrum and you can integrate it and here it's integrated over the area where the exits on are sitting in the spectrum and then you can produce a spatial map of that energy loss spectrum and you can see it agrees very well with the TEM image so the exit zones are are sensitive to the thickness of the material and that's something we started here but that's not the point I'm going to make today what I want to say today is the following so if we calculate the île spectrum of this material here in the Q going to 0 limit so in the long wavelength limit which is what is relevant in optics where you have photons photons don't carry any momentum really so you can only probe the small Q regime then we get the black curve up here so it's completely flat in this energy regime where there's a lot of spectrum in the experimental spectrum okay so all these features here are positioned where we would expect the plasmas to be and we can infer that from from plasma and resort from from plasma calculations of ballgame os2 but they don't show in our spectrum so at this point we have to remember that a very localized electron beam is not a plane wave in the lateral dimension right it's localized so it has all kinds of momenta in plane so really to reproduce the île spectrum of a localized electron beam we have to sum over all the Q values and then we get the blue curve and you can see now we get a very nice agreement well we get a better agreement for sure with the experimental spectra there is still a lot of room for improvement but we get these features and they are indeed due to the plasmids the plasmids developed at finite Q so they are completely absent for Q equal to zero and that's very different from the three dimensional case where plasmas typically dominate the île spectrum for Q equal to zero there is another interesting thing here because what I've just said is that for small Q values that is slowly varying fields they can only compos with the exit zones where as rapidly varying fields they coupled to plasmas and that you can actually see if you look at the yo spectra here this is the eel spec the spatial eel spectrum integrated over the range of the exit zones and you can see this is where the border of the and this regime the electron beam is actually parked outside the sample and still there is a lot of signal that's because the electron beam has a slowly decaying tail the small Q values of the electric exciting the exit ins if we look at the plasmids instead which we can look at by integrating the L spectrum over this peak up here then we see a much sharper transition from the sample into the vacuum region so that's because we can only excite plasmids by the low Q values by the rapid variations of the electron beam okay this is again very different from the three-dimensional case and with that I would like to summarize the talk so I hope you you have a feeling now share my feeling that two-dimensional materials are really a fascinating playground for exploring and controlling electronic excitations and electronic and optical properties of the materials we are close to realizing I think Feynman's missions from 59 namely stacking two-dimensional materials in order to produce decide materials with tailored properties targeting different types of applications and finally I hope I convinced you that these types of calculations provide basic fundamental understanding of the physics of materials and also that they can provide and are really essential for providing guidelines that could lead to the rational design of novel materials in the future and that brings me to the most important slide of this talk which is the thank you slide there is a very large number of people who I owe a lot Thomas Olsen has been involved in a lot of the work I've talked about Karsten Jacobson and and Jakob shirts my colleagues and they're always very inspiring and have very many good ideas and then all these young and talented people here have contributed a lot to the work and I wouldn't stand here without work from all these these people here everyone in in C and G have just listed the people that I have common publication with not least computing and Administration Ola and ginger on are always making a perfect infrastructure handling that in the most professional way my n is helping a lot in the section in general and with the work of this thesis in particular and then a lot of exceptionally good external collaborators that I've worked together with in in the past years here and thank you to all of you for for coming and listening [Applause] [Applause]

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Length: 38min 14sec (2294 seconds)

Published: Tue Dec 05 2017