 Philosophy of what you could do beyond. And now, the idea of having there, Sivan is to go even deeper in that. I think she will show many, Yeah, many, yeah, why their, and to many applications, and how to start from volt wave functions to get something more. And let me also comment that Sivan did the postdoc in the group of Steven Lu Love, V glupu Jefinii. Ok, več nekaj. Prostaj na berkli GW kod, ki je kompetitor del Jambu kod. Čekaj, da je tukaj in alternativne perspektiv, da je tukaj? Na GW in BSC. Zdaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj, da sem tukaj. Pistim signo. Can you hear me? Ute, bo borrow a sledadvancer z nekaj. And this one? Is there other one? And this one? Ok, zanim just bring my... Ok, vse dobro. ...my kompetitor. Vsih da. Vsne je da je here. I'm Sivan, I'm from Israel, from the Weizmann Institute of Science. And, like David said, I was in Berkeley for a while, as a postoc and before that I did my PhD also at the Weizmann. Working with Leo Konik on DFT developments. And these days, with my own group at the Weizmann, we're mainly interested in excitants and extended systems and excitant dynamics. From various perspectives. in so spetni o komplekške strukturje, tako, kaj jaz včasno boljete, pravno delelje vsev tege, in jaz pa sem tudi vse arbejte in spetne spetit. Se bo sem stajne počuk, ker sem se vseče nekaj zrednjo, bo zelo počkimo, ki smo se zelo način. Mi se način. Vse nekaj zelo... neko se priče. Tako, najbolj konečnje, Všlič sem všetko ga, da imam občanje v kompleksiti v skupnih. V nekaj razdelujem občanje, da sem položite, neko ga z vami izstavila, način se očičnjevane procesov, z tem nekaj meni, da ne bo. Češno, da ima s nekaj meni sezat, da je bilo informacija, da je nekaj meni prejzal, da je nekaj meni zelo sezat. of particles in the billion zone, so understanding k-point dependencies and so on, we just discussed that in the previous talks. And this idea is what I will try to follow, both from the electronic band structure perspective and then also from the excitonic and phononic band structure perspective. So this will be kind of the language I will try to put in front here and convince you that it may be useful. So this is how this talk will go. I will first discuss the basic concept of how we calculate optical excitations, riff overview, because you heard quite a lot about it already today. And then I will go into exciton dispersion. And specifically how can we use the concept of exciton band structure to understand how excitons depend on the crystal. The crystal shape, the crystal momentum is a translation of crystal properties, symmetry, dimensionality. So this is already a translation between the system, the structure of the system we have and the excitonic properties, which will lead eventually to the dynamics. Then in the third part, we will talk a little bit about excitons scattering, specifically about exciton phonon interactions. We heard also a little bit about that today, so I will dive in a little bit into one of these perspectives that we are also using and show you some of what we are doing as an example and also some of what others are doing these days. And collecting all this information will then try to understand together maybe if we can already describe something interesting about exciton dynamics. And when I say something interesting, I mean for me, it can relate structural effects with what is the time result phenomena, excitonic phenomena that is observed in experiment, because that will be in the end our validation or our connection, hopefully to reality. And when we try to connect to reality, then we would also like to introduce some complexity, just because actual materials can be complicated, or maybe have to be described in more than just a simple unit. So it's not just a necessity. It's also because when there are complexities, there may be interesting dynamics involved. So in the last part of the talk, I will try to convince you with some very specific cases that this is sometimes interesting to look at. So sometimes, try again, giving up. Let's try starting with the optical excitations. OK, so I'm showing at the right the outcome. And I'm going to discuss a lot to the materials today. So this is a transition mental decokogenite, a very well familiar one, one in the selenite, also similar to others. And I'm showing here a bent structure in which the gray part of the bent structure is the spin density. So we can use DFT to calculate bent structure, to calculate the projections, to calculate the spinors, how they contribute to the bends, and these are things that you know already. And we can also look at the wave functions. So what is shown here in the middle is in one of the wave functions associated with a specific k-point. In this case, if I'm not mistaken, that will be the conduction band. In the orbitals around it, OK, great. We can look at them. And this is a lot of information that we can have. So just to briefly connect that to things that we are talking about here, this is a density function, a theory calculation. So we use the Kansham equations in order to come up with these properties. And then there's a whole field of what we can do with that. Because we use the exchange correlation functional that's an approximation. And we can use local functionals and impact for these systems for other semiconductors who are interested in that. These will give pretty good wave functions. That's already a lot. We have pretty good wave functions of electrons and holes. Think about it. That means we have already the probably most important ingredient in the excitonic picture. We just need to know now what to do with it. OK, that's a very good first step. The bend gap and in general the energy levels are not well described and they should not be well described for these systems. There's interesting screening in this system. LDA, GGA screening is exponentially decaying. It just doesn't capture it. OK, so this we know. We have a problem. We also kind of know how to solve it by now. Importantly, David has showed it in the last talk as well. Exitons are not captured because of that. Non-local screening is crucial to capture excitons and exciton binding and there is no non-local screening in this kind of local DFT functionals. So we need to do something else to capture the excitonic picture. So we can use hybrid functionals. Vod of cautious. I'm going to say that very generally and not discuss it. But many people use hybrid functionals in a DFT framework to calculate excitons. This may work, but not for the right reasons. In fact excitons should be calculated, of course, within a TDDFT, time-dependent DFT framework. In fact is that many functionals have the wrong screening or approximated screening and it is not good enough for these kind of systems. So just by doing DFT, the energies are similar to the excitonic energies by accident. That's again manifestation of the fact that non-local exchange is important. When it's not there, we get energies that can be mimicking the excitonic effects without really calculating the excitonic effects. In order to really understand the exciton picture, the exciton wave function and its time propagation as we are after, this will not be enough. We really need to understand the excitonic complexity and how excitons look like. So hybrid functionals can be a very good step forward, but you have to know what you're doing. Calculate excitons always in a linear response time-dependent DFT. Then the electronic wave functions are often kept. There are many interesting cases where not. The hybrid wave functions are different. Then the local wave functions, good. These are nice cases to use hybrid functionals. It's not what I'm going to discuss today. There is additional exact exchange when we include hybrid functionals. So there are ways to tune this exact exchange to describe the systems well. That's an interesting direction. In fact, that's what I've been doing in my PhD and still a very active field of tuned functionals. So there's a lot of things we don't discuss here about how to do things properly with hybrid functionals. But what these tuned functionals are trying to do is to actually mimic the screening, the non-local screening. Once the non-local screening is well described, the excitonic picture can start to be assessed. We can do that also with GW. And these days you can do that pretty easily with GW. And that's a gift that I think you should use, because why not? So if we do a GW calculation I will tell you a little bit about how we do that because we heard of different methods. In the end, when we look at the bend structure, remember our language today is a bend structure, so the spin split can be modified through including spin in the screening function. Typically in these materials not a big effect. The bigger effect is the opening of the gaps and the relative alignments between different bends and different k-points. Remember, we are going to discuss dynamics. So these alignments between how electrons and holes are in different crystal momentum is actually crucial. So we would like to have a good description of that. And there was a discussion about the frequency dependence and how to calculate this. This will be important. So we need to strive to use the best methods we have, but also sometimes we compromise a little bit when we know we can to actually be able to calculate complicated systems. So this is the GW method. I'm not going to teach you this. You already studied, but it is within the Green's Function representation of a self-energy. We calculate a self-energy that takes into account a heart-feelike term and the screen exchange and within a Dyson picture. But what I would like to mostly stress is how do we go from DFT to GW? Because I don't know if that's so easy to understand. What exactly is good enough from DFT? What is exactly that we're using when we do GW? We have a specific DFT starting point. What does it mean? The GW is already an approximation in the way we truncate the interaction in the self-energy form. Now it's also approximation in the way we include the DFT. I just told you the DFT is going to be very different with different functionals. So what does it mean about the GW we use when we use different functionals as a starting point? This can be a complete mess. It's not well-defined always. It can be well-defined, but once you start to use that as a user, you will find that you can get different answers if you use different approximations. That's because we use approximations. But we want to use approximations. We, in a way, have to use approximations and it's a good thing because we learn things from it. But we need to do it wisely. To do it wisely, we need to understand what is the effect of the way we describe the wave functions in what we eventually get. For example, here, we calculate the polarizability. We calculate how electrons interact with other electrons and how holes interact with other holes before we put them into the excitonic picture. We calculate this kind of M. This is in the Berkeley GW language, but I pretty sure you look at very similar matrix elements also in the Yambo language. These matrix elements here will be block states interacting, but these block states are going to be calculated from DFT. These are DFT wave functions. These energies in the spectral representation of this polarizability of this polarizational function will be the energy differences between DFT energies. We know already that they are not always exact. Well, they're not exact. Sometimes they're good enough. What does it mean? It means that we compromise a little bit when we go this direction and there are ways to improve it, but here I'm describing the RPA, screening calculation. We're going to use DFT wave functions and DFT energies as long as we know what we're doing. If we end up finding that we have a problem, maybe this is where the problem started. We need to know where to go back. That is probably good enough, also empirically. It's probably good enough for the materials that we're working on. Now we can go ahead and calculate the screening function, almost trivially, from having this polarizational function with the Coulomb interaction. Everything in G and Q and you're familiar with the block representations. Everything is in crystal language. Then we correct the GW energies like we saw this week. GW self-energy minus the DFT self-energy, which is the exchange correlation potential. An exchange correlation potential is a certain approximation for our self-energy. So, this we are all saying, that's what we're doing. The reason I'm standing here and giving all this to you again is because I feel that for me it's very important to break the language of GW into step-by-step kind of process so that we know later on where to come back if we think we need to improve something. It is possible to improve the theory and there's a lot going on here this week about this. It is also possible to improve the approximation to be more appropriate for what you're trying to get. And it's not always similar. It really depends on what you're trying to get. For example, in these materials, this is one paper by the Louis group describing the behavior of the dielectric function. So, this will be the two-dimensional dielectric function in real space. It is, look at it. I mean, it's hard to describe and it's hard to predict. And it was hard to predict before looking at actual calculations of the key dependence in the dielectric function to understand that will be such behavior. And we looked at it already. You remember all these field... I don't have it in the slide because I had enough, but all these kind of field lines in the particles. This is the reason because the short-range interaction will be so different than the intermediate range, will be so different than the long range that everything is complicated. And once we look at this picture, we can say, ah, yeah. Well, it's one in the short range because there's no screening. And it's one in the long range because there's no screening because we go into the vacuum, right? So, the limits we understand. Because there's a range where we feel the material very strongly, the screening of the 2D material. So, of course, it has to behave like that. But good luck with predicting it without seeing it from an actual calculation that is trust. OK, and that's the power of actually calculating screening. And even within a non-perfect approximation, and back then, believe me, it was not perfect. It is still not perfect, but it's getting better and better. But even in 2016 without subsampling, without really looking at all the different aspects in the algorithms, you can have to have a perfect sampling of the grid. Still, we can learn something about the function. And in this paper, actually, also connect that to previous models and to understand why they are almost good enough at some limit. And other models will be better at another limit. And then you can combine analytical models and just get this picture. But how do you understand that and how do you understand it? OK, so that's an example. But now we are here for excitons. So once we have these bent structures and the understanding of the screening, we can use them to start a couple bends like we just learned and to look at how these bent coupling will form. So let's just have one slide on how we're doing that within the beta-serpita equation. Again, without the full theory behind it, you know it already just to claim that we are looking at the exciton wave function and we are looking to solve the problem in the exciton basis set for now. So we define an exciton wave function by a basis set spanning of pairs of electron and whole... See, they're here. Electron and whole wave functions. And these a's will be the basis set coefficients, the spanning coefficients of them. Good. Now within a beta-serpita equation we can calculate first the energy difference on the diagonal, the energy difference between electrons and holes. From GW, I didn't write here GW, do it however you find that is good for you. If you can compromise or choose, forget about the compromise, if you can choose an energy difference that will not be a GW energy difference, so be it, it is your choice. But have the way that works for you while you understand what's going on in the equation. So that will be the starting point, that will be what you're going to bind. If you underestimate it and you can do that on purpose, just remember you did it. OK, because it means that your exciton energy will not be perfect, but perhaps the binding energy will be OK. OK, just know what you're doing. Then the second part is the coupling. In the second part we take three electron-hole pairs, these are these, VC, V prime, C prime, three electron-hole pairs walking around and we bind them together. OK, so what is called here electron-hole kernel K to get within this eigenvalues equation, the excitation energy associated with this system and all along this problem we have the eigenvectors. You can see already that the eigenvectors are the components that will build eventually the exciton wave function in this basis set. We just transfer to a different basis set. OK, of course, it's almost trivial, but we do it like that. I vote GW energies, but you know now that you don't have to. So this electron-hole kernel that again you already saw, but just very briefly, the way we usually truncate, it's a two-particle Green's function here and the way we usually truncate here we choose two terms, two leading terms in this expansion. One of them is direct, so electron goes to hole, hole goes to electron, but it basically tells you that almost like a Hartree picture you just have particles remain in their same description but just change their occupation, bend occupation. But on the other hand we have the exchange term, quantum mechanics particle can also switch and we need to include both of them. When you look at it like that it's almost Hartree-Fock for excitons. You almost have the Hartree-like term, the direct interaction of the folk-like it's not folk-like term, but where you need to include the exchange interaction. And when we do both of them in this excitonic basis then we get the coupling that we typically get within BSE. And there was a reference earlier today to the many body book of Robert von Juven. There's an interesting discussion about that in this book. But also in these papers is an example. And good. Voila, we have absorption spectra. So, you know, it takes some time to understand that here something is really important so don't do that without doing the grid something properly. But once you know what you're doing you can get an absorption spectra. That's the excitonic picture we were looking for. We have the A and B peaks. How do I know these are the A and B peaks? I know because I'm not an experimenter that actually did that on the computer and the things came from. I know, for example, to put the bands, this will be the conduction bands participating and this will be the valence bands and to look at the contributions at each one of the excitations. Where did it come from? Can also look at which k-point it came from. You can do whatever you want with these excitons. You can break them to pieces and understand exactly their composition. So this exciton is coming from A, that's my definition. The second exciton is mainly by the second valence and the lower conduction. That's a B peak. If you know TMDs and you know the band structures that we're looking at, then you will also know what are the excitons. Let me just tell you that life is not always perfect. It's not just a band to band transition. There are more bands participating. Sometimes more spins participating. Sometimes momentum can participate. I haven't showed you yet, then it starts to get messy. If you're doing exciton calculations, my own opinion is that it's your responsibility and it's your job to understand the mess. Because this is where things really start to be interesting. If you have an exciton that has both spin channels in it, oh man, this is not what they see in experiments. This is not what they see when they apply magnetic field. Wonderful. Why did it happen? What was the selection rule that was supposed to be kept and was broken once you introduced the ab initio wave functions? Because something happened. You go into the reducible representation of these excitations, this should not happen. It did. Good. Because the wave functions in these systems, okay, let's go back again. Because the wave functions in these systems are not really atomic wave functions. And sometimes if you represent something with a d orbital that would be a very nice approximation. But you don't really need to because you have a representation of the ab initio calculated wave function and sometimes it breaks the symmetry a little bit. And sometimes things will happen that you cannot expect. So I think it's nice to remember that while we all want to do a calculation and get an answer that we understand. Yeah, that follows the theory and that follows maybe also what we see in experiment so we can understand it. But sometimes we see something more. The computer tells us we put in the code, we put in the approximation and we put in the ingredients but we get some gift back. We get information that sometimes is unexpected. This is where ab initio is important. You got this information, you may get scared, don't be. This is exactly why you went into all this effort of learning all this. Because you learned something that cannot be understood without going into all this effort. And now it starts to become interesting. Ok. So let's talk about exciton dispersion. We talked about electronic band structure. Now we can talk about exciton band structure. I'm going to bomb you already with an actual one. We typically expect excitons to behave nicely. There will be parabolas dispersed in this. Why do we think about parabolas when we think about excitons? I don't know if you do, but why do I think about parabolas when I think about excitons? Because it would be nice to think about an exciton as a hydrogen atom. Some kind of a hydrogenic model. We would like to think about some valley where the excitons are the electrons and holes are close together. And that would be great, because then we can think about relatively simple approximation for the exciton and move on with that. Sometimes this is almost the case. So what we're looking at here is the exciton dispersion. We'll talk about it how we get it in a second. Of the pentazine molecular crystal. I know it can be scary to listen to this combination, but the pentazine molecular crystal is just a crystal composed out of molecules interacting with one of our interactions between them. Very well ordered sometimes. They have nice symmetries and they have very molecular properties of the wave functions, but they also have interesting crystal effects. Let me convince you. In these systems we would expect the excitons to be almost molecular. I'm going to take a risk that we think will be the bend structure of these excitons. They are very localized in the molecules. How would the bend structure look like? What is the bend structure? The exciton energy is a function of momentum, of exciton momentum. So suppose the exciton is very localized if it was atomic. What would be the bend structure of it? Would the bend be very dispersed or very flat? Flat. This is the bend structure of an atom, it will be flat. Why? Because it doesn't care about the crystal environment. Whatever momentum you give it will be the same. That's a flat bend. The bends I show you here are the exciton bends for singlet states. I will soon talk also about triplet states. They are really almost flat. It tells us that it's almost a molecular picture. But they are not completely flat. They are not completely flat, so the singlet states have some probabolic shape. Let's see how we got it and then discuss it some more. BSc equation, same as before, but now we have this large q. This large q is exciton momentum. How do we get exciton momentum? We go from an electron at one k-point this friend to a whole at a different k-point and the momentum difference between these k-points is this big q that's the definition of the exciton momentum. This would be for a specific exciton s with a specific momentum q and this is really just trivially expand the BSc equation to also have this electron momentum here. The electron in this representation has a finite momentum, the whole doesn't and the difference between them is what we care about. Now we do that for each q, for each exciton. There was a question I think on Zoom before if this is a more difficult calculation than regular BSc. This is regular BSc. It's the same calculation, different parameter. We can do that in Belkaca GW back then that was the original paper where this implementation was introduced. Now things also start to be interesting. Again we have two terms in the kernel, we discussed it before, direct term and exchange term. The exchange term actually behaves like it goes like the Coulomb interaction. Let's stop for a second. Here is something else about the structure we are looking at here. Should this be interesting in this system? We may find, you see that I calculate here these single states and these single states have slightly different energies and different q's. We went through all this effort of learning and using and some people like Diana Chu here of developing this code in Belkaca GW just to find that these are almost flat ok, we could kind of guess that from the structure. But when we calculate the triplets we see that the triplets in this structure have opposite dispersion. I'm not going to go into multi exciton generation now. There was a question about bi excitons before. We actually calculated bi excitons for these systems to evaluate what is called singlet fission mechanism. Singlet fission is a transition between a singlet state that is optically excited and decay directly into two triplet states. You may see that their energies is pretty much half of the singlet energy so that can be energetically allowed. This property of the opposite dispersion tells us a lot about the intermolecular interaction that leads to these triplet states and this actually results in direct coulomb coupling between singlet states and bi triplets. Already from this bench structure I can tell you that we have learned a lot. We developed it and we formulated it and we calculated it. But already from this picture I can tell you that you can see these interactions between single exciton and bi excitons. Only from looking at the bench structure. You shouldn't be able to just see it from looking at it and just telling you that paper here below. But we can skip another step now and in the future let's look closer. Let's focus on these singlet states but now we look at very small momentum. And Julio presented this already in this morning and I'm very thankful for that. There was a good preparation. When we go to very small queue we see the exchange behavior depends on the details of the crystal and I would mainly like to convince you about the functionality of the crystal. Pretty simple considerations. This is the behavior of the Coulomb interaction and in turn of the exchange term in the kernel at different dimensions when queue goes to zero. That will be the analytical limit of this behavior. And already from these considerations we see that there will be interesting things going on. So let's start with concentrating on what we see here. This is a 3D material. It's a bulk material. We should expect some theta dependencies. What is theta? Theta is the angle between the polarization direction the light polarization where the light is shine to the excitation direction to where the preferred excitation will be in the crystal. And excitons in these systems are not the same in different directions. These are pi systems stacked in one of our stacking and the stacking direction is going to determine a lot Let's hold for a second. This is a crystal effect. You will never see that from a single molecule perspective from taking two molecules. This is really something to think of. There will be interactions here between the molecular crystal and the molecules in the crystal. It will be anisotropic so there will be dimensionality effect and maybe we can learn from the way they interact how the system is built. I'm plotting it here. How exactly the phase we are at. How exactly the molecules are oriented. Ok, good, now we have all this information. Let's go back. What do we see here? We see a split and this split is just because the excitation at the x-direction is much stronger than the excitation at the y-direction in this system. Are you standing? Ok. I got scared for a second because we didn't touch dynamics yet. What does it mean? Even if you don't touch dynamics. For me today this is the essence. That's the most important ingredient. Because now we have a connection. We see an actual effect of the crystal structure. It comes from the way the molecules interact in the crystal but we can assign that to exchange and we can assign that to the behavior of the exciton energy at very small momentum. This is what we call longitudinal transfer split of the excitons and indeed it's very similar in spirits to the longitudinal transfer split in phonons. Ok, so that's an example. Why does it happen only to one exciton? It happens to the bright exciton and not to the dark. If you're familiar a little bit with excitons you may already know the answer. Bright excitons come together with large exchange interactions. Ok, it comes together with large electron-hole coupling and there is an interaction between them. This loud exchange interaction is also manifested in this behavior. Dark excitons are very small exchange so we don't see the split. It's very, very tiny. It should be there from these considerations but the exchange is small so the split will be small. Ok. So how can we connect that to something that can actually be observed? We will look at it soon. In this case it's more in disulfite and I was discussed before in this case there are two degenerate at zero-q bends but not in other queues parabolic and linear bends another manifestation of the dimensionality effect. Ok. Now I should stop for a minute and try not to confuse you too much to tell you that this property is in this basic set that we calculate. In the actual exciton it will be much relaxed the interaction with the light field for example will relax this property but still when we want to look into dynamics we don't only look at interactions with light we also want to understand the fundamental excitonic properties that will dominate dynamics so that is a fundamental property but to observe it may be complicated. Ok. So from here we keep in mind exciton dispersion, exciton bends structures things can be less trivial than what we would expect but this is where they become interesting. I did not show you yet that they become interesting but hopefully soon I will be able to convince you. Let's go to scattering now. Why? Because I want to touch how these exciton bends structures and how these exciton properties are related to time result phenomena so we need to look into time what happens at long times but we know that in all these systems once we let time in we have to take into account exciton scattering and one of the main mechanisms will be exciton scattering with phonons. Right? So how do we do that? There are different ways I don't think any of them is perfect yet but I'm going to take you through one of them briefly. Ok, so for example if we think about these diagrams of these electrons and holes you can think about phonon interaction between the electrons in the excitonic basis so we have coupled electron hole pairs but the electrons can interact through phonons. That's a very specific diagram. Ok, and there isn't paper by Antonius and Louis and also this follows work from 2008 by Marini here there are others I'm just showing you a specific example because I like the diagrammatic form that they came up with and that's a paper about a self-energy understanding of the exciton phonon interactions and for me it's helpful maybe it will also be for you so now we look at the kernel of the exciton phonon interactions and we ask ourselves what is this self-energy for the exciton phonon and there are some possibilities with their names so these will be these electrons and holes interacting separately but within an excitonic picture there will be other terms like exciton phonon exchange terms and interesting terms if you want to look at localizations that we're not going to discuss today this is just to promote a little bit that now there's literature to go in and learn about this and I think you will learn more about it tomorrow about the separate variables we just say that this we can calculate you can calculate and it will be easy for you you will learn how you now have all the ingredients ok, you should be so happy really it's wonderful really, I was not that lucky and I'm really happy for you that you have this possibility but now how do you do that what will be the right way to put things together not this, this is just the first example to show how to go the direction of the right way it will be right for some things but we want to be in the end as exact as we can so let's start let's start with these two terms what do we do if we want to account for these two terms in the BSE equation simple ok, this is just let's couple the two excitons here s and s prime with one term that will take into account the electron electron phonon coupling and one term that will take into account the whole phonon coupling it looks like many terms it is really not that much of an issue to put these terms together if you can python you can do it with the ingredients you get from a BSE and some DFP calculation but it's slightly more complicated than that because you need to know what are your parameters here what are you accounting for and note that we have both large q and small q here and that's because the exciton dispersion and the phonon dispersion are both important that will be the phonon dispersion of pentazine but you can calculate for any material that you want basically so that will be the phonon dispersion for the system the electron phonon coupling can be calculated on top of that good you have this and you have this it needs to be coupled to the excitons and it needs to be coupled to the excitons with different momentum because why not there is no reason it should be coupled only to zero momentum excitons the only reason is that we used to think that's it, that's not the physics of scattering if you want to include scattering mechanisms you have to be careful about the way you include the momentum even if it's flat it needs to be there otherwise you just exclude many important parts of the interaction and then behold I'm showing a rate we'll talk about rates but we talked also before about the fact that we can gather this information into a firmist golden rule what is a firmist golden rule is a way to calculate lifetimes or inverse rates from summing all these coupling terms that we just discussed we came up with coupling terms, good now we have matrix elements to tell us what are the coupling between the particles and the occupation don't be scared about these ends just tell you if the phonons are going to be occupied in the temperature that you choose and the energy conservation transition energy between the two excitons and the phonon so if the phonon can create this transition good V it is written as a delta function we always expand it it will never be a delta function first because numerically it's a nightmare and second because it's never really a delta function we know that the energy conservation should be we should give it some broadening so we usually put gaussians or Laurentians to take into account this broadening it is really simple okay sometimes you will hear that people tell you that these kind of things are complicated because it was complicated for us sometimes for you it is really simple this is where you start this is what we get when we do something like that for the pentas in crystal and let me just take you a little bit into what's going on here remember I showed you this split well I'm actually treating a little bit with the title my bad comparing tetrasine to pentasine here so pentasine is in the right and there's another crystal very similar the molecules of four rings instead of five and that will be tetrasine I'm showing also tetrasine because in the tetrasine case the split of the bright band is below the dark band so we have a well ordered set of dark states and bright states well in pentasine the energies mix because of that so what we end up having in pentasine is intersection we have intersection intersection of the excited states of the dark and the brices you can already see it here but if you look at it as a function of the angle you see it also in different angles and these matrix elements is a little bit of a show off just to show that we can calculate them this is just what I just showed you for the lifetime this is one over the rates that we just looked at between the bright excitons and look how the picture gets more complicated in pentasine because of this mixing and we also we can also look at the transitions between the bright and dark that will have a certain structure and it's not going to it now but if they will have a certain structure we could never predict a priori that we know from these calculations in a different structure from pentasine the important thing is that the lifetimes are much longer than the intra band transitions so we can look at intra band scattering and intra band scattering and how it's related to the specific phonons that dominate this scattering and so on, there's a lot of information again, this is a complicated picture now your job, break it to pieces understand where things are coming from this is what we're here for these kind of things were also done for example this is from Davide and Marco Bernardi in this group we looked at it also before for HBN similar, these are lifetimes same method, lifetimes calculated with the exiton band structure and also the terms for the different momenta for different states that will show us where are the places where the exiton momentum, the exitons are strong where are the brilliance on places where the exiton momentum will dominate the interaction again, take momentum into account and as a follow up I would like to stress that if you want to look at exiton photo luminescence exiton emission this kind of scattering is something that you really need to think of very carefully so this is from a paper we looked at also earlier today in this kind of analysis is done now by more and more people which I think is very nice can we look at the emission spectrum or the photo luminescence but it is a little bit more comparison between absorption and emission then just the selection modes of the exitation and the decay to different momentum look it will include already some decay to different momentum but how time plays a role now in emission absorption can be thought of as a linear response but emission cannot be emission is a scattering process so you have to take into account when time becomes an issue phonons become an issue temperature becomes an issue emission is a very interesting set of information you can look at the broadening of the emission peaks also give you an estimation of this scattering as also calculated here and so on so there are many things to take into account when you start to think about time result processes and the emission for me at least is one of them before we even touch dynamics this is already dynamics emission and of course many other properties that will include scattering we reached exit on dynamics Davidi, can you tell me something about the time? ok so let's let's go through exit on diffusion we talked about emission, now we talk a little bit about diffusion this is from a nice review paper by Naomi Ginzberg which I like a lot because I would like to understand the experimental perspective Naomi is an experimentalist looking into exit on diffusion doing this kind of experiments as well as others these days and we looked at it also before there is a kind of an exit on excitation let's call it a wave packet and now it propagates in time and they can look at the broadening of it really looking at it this is very interesting time result for the luminescence and briefly they look at the change in the broadening and assign that to diffusion coefficients and from that they can look at the diffusion mechanism is it normal diffusion or is it special diffusion is it ballistic multi-heterogeneration what not how do we describe this from theory ideally from real-time propagation methods I'm not going to go into that of course you heard that this morning you are walking on the code I think masters this and this is wonderful keep doing that because one day we would like to describe these things with time result properties this is so to touch TDGW TDGW is a way to build excitonic picture is it a way to look at exit on dynamics complicated discussed before a little bit exit and foreign interactions within this picture it is going this direction for the scattering in my mind I would think that excitonic dynamic should be captured in TDBSE but again it depends how exactly how are we going to do that in terms of scattering it's more complicated than just what will be the right method and let's discuss this complexity a little bit so that's one way which we're not going to touch today how do they think about it from experiment in experiment they usually talk about these excitons within a kinetic equation kind of picture so we saw that we can calculate rates so for example we can think about some diffusion coefficient within this kinetic equation what is n and is the exciton density and the exciton density will change the function of this coefficient and also a decay that is related to the scattering lifetime wish how nice it will be to think about the exciton diffusion this way but that means that the excitons should be very very stable things are in the exciton basis and remain in the exciton basis for a very long time and other things don't happen maybe sometimes it's the case maybe for sometimes this is the case from the theory perspective I would really recommend following the work of Mikhail Glasov from the Jaffa Institute where he looks into these kind of terms in a more ordered fashion this is a wonderful paper I think about exciton phonon interaction from a Green's function perspective but semi-classically Newtonian maybe excitons are boson they propagate if that will be the case what will happen this will be the theory there is rate here for exciton phonon it will be taken into account as well as a bunch of lifetimes associated with that if you want to see how this looks like analytically go to Misha's papers but it's not just about how do excitons diffuse it's also about do excitons diffuse I mean is this always the case that we can look at it as Newtonian or boson particles that just go along the material when should we be a bit more careful about that ok, so I may take a little bit of the question time but if we go back to the exciton dispersion that we talked about earlier for transition metadecocogenides let's think about this problem now we have bends, what is diffusion Boltzmann, diffusion is the relation between bend velocity and scattering lifetime or the collision lifetime of the particle and we tend to meet some other particle phonons and we know it's bend velocity maybe we can say something about the way it propagates good, let's think about it in the semi-classical picture and build wave packet theoretical wave packet on top of it that would be this wave packet and let's propagate it good, we propagate it on the parabolic bend at 150 femtoseconds nothing happens, why because it's very flat nothing really happens no matter when we are great that would be exactly the picture we want we can go much further from that because we know here phonons will come into play so they start to be scattering but in the ballistic regime ok, on the linear bend oh no, what happens that makes sense there is another analytical term at the queue the wave packet just goes away from it this looks very surprising at first but there are papers showing that this ring forming is the property of exciton propagation in 2D materials but it's hard to say that this predicts this property because that's the very short lifetime in these papers the theory for this ring forming will be coming from exciton phonon scattering that actually induces where the light excitation is very rapid multi exciton generation so the assumption is that the excitons just go away maybe but we also see that this can be seen as just a starting point of the propagation just from the bend shape is this real, is this true please join the party we want to understand but we can, ok, assign that to some bend velocity some diffusion coefficient we can put that into a rate equation do we want to do that probably not in our case because this representation of this rate equation is nice to understand things but we think that very quickly we go out of this out of this exciton basis and we cannot really describe it this way anymore so how should we describe it good question and in the very last part I will show you some cases for why it may be interesting to describe it properly and I will not answer the question how if we meet again in few years I hope we can discuss more about that this is an open question and my personal opinion it is a very interesting one I think you people can be the ones who actually give answers where can it be interesting, three examples promise first of them is defects people can now go very precisely into creating defects, for example in TMDs and shine light on them and see what happens and this is the emission picture I can just tell you defect states and the defect states will live for longer time than expected they should recombine very quickly they don't and there will be interesting exciton dynamics and scattering associated with that there was a question here before about magnetic properties of excitons this is not a magnetic system but let me tell you something about that when we calculate excitons in more than a self-made width and without and with defects so we can for example make an exciton picture and the transitions dominating it we will not go into that and just tell you that there are transitions between very flat defects remember, we discussed it before that the dispersion of very localized bands is very flat, that what happens with defects but it is mixed with non-defect bands so we have this wonderful mixing of flat and dispersed bands this creates very special excitons and when we calculate the g factors of these excitons from BSE if you know what it is good we can talk about it later but it is basically a measure of the magnetic response of these excitons the response they have under magnetic field that can be observed in experiment and is observed in experiment and we see that while for the non-defect state everything is simple there are A and B peaks with magnetic response that is pretty much what is observed in experiment and people did that introduce one vacancy one calcogen vacancy and this is what you get magnetic response is everywhere and that happens because all the properties that dominate the spin selection walls here break and they break wonderfully in a very complex way depending on all the possible exciton transitions the average of that will become zero and this corresponds nicely to our collaborators in Munich who measure zero g factors for this kind of defects but you know this is just one group and we can discuss if this is really true or not there is an outcome of the exciton complexity so there was a question before how can I look at a single exciton from BSE do not why should you I mean do if you want but you can use BSE to capture all this wonderful complexity so then you can also look at one exciton this one is pretty simple it's this one, one bent to another but look what also happened so why should you exclude these effects they are very interesting so dynamics? no the spin scattering and the valley scattering is going to be crazy and the decoherence associated with the valleys will break so we better look at this very carefully this will be strong interactions in the exciton scattering second example I will not well allow it into I will just tell you that TMD heterostructures are another interesting case we have a recent paper on that with the group of Tony Heinz the momentum effect this was discussed before by Davida there may be transitions of excitons with direct momentum to excitons with indirect momentum in emission they can measure now interlayer excitons in this kind of structure and look at their energies and their temperature dependencies at absorption versus emission with I think pretty good proof that there will be these momentum transitions involved in emission experiments can go very deeply and exactly now into what is observed energetically and that's wonderful by the way they also see in absorption the two spin channels in this case not only one spin channel which means that only having an heterostructure heterostructure breaks the spin selection what we talked about before and there are no dark and bright excitons excitons become bright just because of the symmetry effect ok and the last thing I wanted to to show is the case of strong exciton lattice coupling that's another type of coherences that may occur and this is for the case of perovskite and perovskite are very interesting systems with very interesting structures and they have been shown by many people that they can hold self-trapped excitons what are self-trapped excitons excitons that really don't want to diffuse why? because lattice there are strong interactions with the atoms this is very interesting with the assumption that exciton polarons are actually the main particles there and not just independent excitons and this is a very nice paper from the Chernikov group in Dresden where they showed that this at some limits of the interaction looking at these terms of photodominescence you actually get what they call negative diffusion or no diffusion which really cannot be explained by the theories that we know and this I think is kind of a nice way to think of how can this be explained it definitely has to go beyond this picture but think about even the phonon description here is problematic it's collective, it's not localized okay so how are we going to do that how are we going to take that into account this is for us and this is for you to think of but trying to convince you that the observations are interesting and the theories are interesting and the combination between everything is really interesting so that's the summary you have the ability to compute many things now you have the law scale computing really accessible, you have exciton phonon scattering pretty much ready at certain level exciton properties from BSE exciton dispersion you have a lot of information about DFT once you learn that you can even skip some steps use DFT if you think it's more appropriate it will be system dependent it's for you to understand and to choose that eventually you can use all this to have some starting point of the excitonic picture to understand what should be the dynamics and if this should be the dynamics how should we describe it from a theory perspective so this is yours you can choose it if you want I would like to acknowledge the people who are involved in some of the results that I showed you here happy to take questions and thank you so thank you very much Sivan so let me stress what Sivan said so there are very many useful applications which you can try to tackle with the BSE and beyond BSE of course here you are in a school where first you want to learn a BSE but then there is also plenty of room for developing and improving the theory so I mean if you will learn DFT then you could be the one who will go beyond the present knowledge it would be great and so I would have many questions but first I would like to let the students to do them there are some in the chat here maybe I can try it with a simple one so you have discussed there are from the others okay okay so Tom Sayer Tom Sayer, ja, you can try to unmute yourself and make the question Hello Sivan, me again I like the picture of the end, that's good thank you, it was a very challenging talk so this MOS2 dispersion where you have this sort of linear and parabolic band this is calculated around the gamma point and like you said you have an optical excitation that starts it off it necessarily has q equal 0 and then there can be phonon scattering to give it finite q momentum but in MOS2 you also have this band nesting effect where there are these sort of parallel linear dispersion bands near gamma but away from it where optical excitation then leads to spontaneous dissociation of electron and hole in k space 1 to the left and in my mind that almost conserves momentum so do you know about that, does it need phonons to do that and would the dispersion of the exciton band be negative in that case, would it lower the energy to gain finite q and what does that mean ja, these are great questions and what I'm showing here is really the dispersion that you naively get by using the method they showed what you're asking is about what would be the exciton dispersion upon more or less reveal electron dispersion how they will this be combined I can tell you that this is not very far from what you get when you have defects in different types of them where you also have additional channels that will also be inserted into the exciton dispersion and into the exciton phonon scattering I can tell you that we're walking on that in this case even the numerical steps are not revealed you enter a supercell picture that you have to treat very carefully with momentum that will be my tip for now but I agree with you that this is a very non-trivial and interesting case to look at I don't have an intuition I don't have a result or a slide for what will happen I can tell you some of the intuition that I will have but I don't know if I should in the defect case my intuition will be that similarly to the case of different excitons you will also have different interactions with phonons and this will be just have an ensemble picture of things interacting together so that's a many body picture of different states in this case of different bends different excitonic bends different electronic bends and different phonons have a well defined method and you will see the effect together I'm glad you're trying to solve this because for a very certain case that's another issue that I may raise that this is a difficult structure what about finding methods to calculate these difficult structures in other methods you guys and girls probably like machine learning that will be one way to think about how to come up with the effect of these kind of complicated structures and build it into this picture from the wave function properties there are many ways to go in this direction I'm not saying that these are easy calculations they're not the defect calculation that we did for the excitons to cast which is the stronger GPU computer in the US right now but it's doable and it will be easier probably next year already OK so we have another question so Burak Odzdemir if you can unmute yourself and pose the question yes actually I wrote in the chat because I was curious whether it is possible to calculate exit on lifetime in external electric field applied whether it is possible magnetic or electric electric well, DFT you probably have it in amble you can easily compute the effect of electric field on DFT bends and wave functions these days now you can think about how to put that into the excitonic picture not trivial, it's not just a wave function effect it should also affect the interaction so go back to the derivation go back to the screening and think about what would the electric field effect I will think about it as a dipole effect and I agree that it's very interesting and I think maybe people already do it it should be doable what for me is more interesting is then think about that, then you create dipoles how will they affect the excitonic picture you will have this information from the electron hole coupling how will this affect exciton dynamics you may also have this information so I think this is indeed an interesting property of excitons OK, thank you and then we have another question from Yadong Wei if you can unmute yourself OK, otherwise I will read the question is it possible to calculate phonon assisted photo luminescence via exciton phonon coupling it is very desirable and starts to be possible a certain level of approximations and that's an example it is, I would say very much wanted by many and this will be an example the Bernadi paper here also has photoluminescence in it this is one way to go the two other ways are presented here the left part of the slide so very happily the answer is yes I couldn't give that answer not too long ago but due to very interesting work by groups here and also in other places now the answer is yes my question should be can it be done in a way that will describe the interactions we want to describe for these kind of systems that are described here this is the proper way for other systems with other type of excitons there may be other ways and so on so, any question from the audience here OK, fully up so, first of all, thank you for the amazing presentation and I just have a technical question about the calculation of the exciton phonon matrix element and we have seen in the expression that you need basically both grade in electron momentum and some grades in transfer momenta and the question is do you compute these matrix element in the irreducible part of the beluen zone grades and then to get for example q plus k plus q you apply some symmetry rotations or you compute everything in the full beluen zones both in transfer momentum and momentum we kind of a brute force kind of group we compute everything because we don't really trust ourselves to know how priori what would be important but I think it definitely can be folded into the properties you are describing if you know what you are looking at and if you come up with the right symmetries for the system it should be doable in some cases I would say maybe be careful about that because extended picture actually captures some of the symmetry breaking so you don't want to go back to the irreducible symmetry but I think in this case that I showed for the pentasyn crystal it can be done in fact we have some reason to walk with Felipe Junada at Stanford where we did something like that to understand a multi-ceton generation from tight binding picture we took here in the unit cell we only have two molecules per cell and still that's kind of a super cell so you can think about how can this be described in the unit cell picture what will be the appropriate symmetries and only from this you can understand what will be the bend structure of the electrons, the excitons and then the multi exciton generation that I discussed before that will be I think going into the direction of investing to use the ab initio understanding to now reduce the level of complexity and solve the problem I will vote for that for sure if you don't lose information about the complexity Ok, so I think now it's really time we go for lunch we thank Sivan again Thank you