 Okay, we start the recording, you can go. The floor is yours, Feliciano, thanks a lot. Okay, well, you know, good afternoon, everybody. Apologies for not being there and thank you for having me. So this is Feliciano Justino and I'm currently at UT Austin and today I'm gonna tell you a little bit about, you know, electron foreign interactions. So this is gonna be a very general, you know, light touch introduction and then trying to motivate why one functions are so important in this field. So I think I'm planning to talk like maybe 45 minutes. So if you want to interrupt at any point, just feel free to do that. But I don't see any kind of cameras. So if you want to interrupt just yell or do something like that, okay? First of all, let me tell you where I'm sitting now. So I'm actually located in Austin, Texas. So this is basically approximately here. It's in the center of the state of Texas. It's the capital city. So we are in the south of the US. So Austin is known for a couple of things. So the first one is that it's kind of a hippie city. So the motto is keep Austin weird. So as a result, we have a lot of graffiti art. So this is one of my favorites is Michael Sieben, one of the local graffiti artists in Austin. So the second important thing is that Austin is the, is a, you know, an important tech hub in Texas. So it basically hosts a very large number of, you know, tech companies. Maybe you know Dell computers. So this was born here in Austin. So Tesla, you probably heard that as relocated, it set quarters from California to here in Austin. Samsung is building the largest manufacturing plant, cheap manufacturing plant in the US. So there is a lot of activity in this area. And this also creating pressure for a university because it's very difficult to keep PhD students until the end because companies nudge them before they finish. At this moment, I'm actually sitting in what is called the Odin Institute. It's basically a unit in the University of Texas. It is entirely devoted to computational science. So the way it works is that we are approximately 40 faculty and all of us have a dual appointment, one with a standard departments like, I don't know, in my case, physics. Others are engineering, maybe chemistry. And then half of the appointment is with this Institute where essentially everybody has an interest in something ranging from applied mathematics to high performance computing. And I will tell you a little bit more about that later. But now let me get started with the electron phonon physics. So what I'm gonna do now is to give you three examples of why we care about this phenomena. And then I will try to, you know, give you a little bit of background about the theory before, you know, showing you some applications. So this first example is about transistors. So this is a pretty figure from the group of Anders Tisch at EPFL. Approximately 10 years ago, they built the first, you know, transistor made out to the material. So this would be an illustration, kind of an artistic rendering of a transistor based on molybdenum sulfide. And when you do something like that, one of the key quantities is the response of the electrons to applied fields between the source and the drain. And this is quantified in terms of the electron mobility. So the mobility is basically the, let's say the coefficient of proportionality between the average velocity of the electrons and the applied electric field. So the higher the mobility, the faster you can switch a transistor. So you can see that this quantity, this is experiments actually depends strongly on temperature. Okay, it goes down by towards the magnitude of temperature. And this is kind of a manifestation of electron for interactions. In practice, what is happening is that the atoms in the lattice vibrate with larger amplitudes as you increase the temperature and therefore there is more scattering of the electrons and the mobility is decreased. So that's the kind of thing we would like to calculate for example using these techniques. A second example that maybe is not known to many is that solar cells actually, you know, are kind of functioning mostly because of electron for interactions, if you want. Maybe it's an exaggeration, but let me explain what I mean. If you look at the solar cells you see on the rooftops, these are essentially assemblies of cells like this one. So on a solar panel, but this is essentially silicon. And you know that silicon is a indirect gap semiconductor meaning that there is a first fundamental gap at about one electrons, so 1.1. And that's an indirect gap between the gamma point and the x point approximately. And then there is a direct transition between gamma and gamma at about 3.3 electrons. So you know that when you absorb light, you know, in the present, so if you want to absorb a photon you need to conserve the momentum. So in practice, this means that the only processes that are allowed, if you have only photons in the system are direct processes. So silicon should start absorbing approximately at three electrons. And that means that the entire solar spectrum that consists of the visible light and infrared light would be completely missed, okay? So that would be useful, sorry. So the reason why there is absorption in this range is that there exists a thing called phonon-assisted optical processes where the system can absorb both a photon and a phonon or maybe absorb a photon and emit a phonon. And the emollement of the phonon essentially allows the electrons to change momentum in the proven zone, okay? So this entire kind of section of the absorption of silicon comes from electron-phonon interactions. And actually, even more interestingly, it comes from zero point fluctuations, okay? So if you go to a very low temperature, this remains pretty much the same. The third example I just want to, you know, show you is about superconductivity. So that's probably the most striking manifestation of electron-phonon interactions. And the phenomenon is the following, you know, in two words, that you have a metal. For example, you can measure the resistivity, then you go down in temperature, resistivity decreases. At some point, the resistivity drops completely to zero. So you lose entirely the electrical resistance and you have a superconducting phase. So the first mechanism to be proposed and, you know, the one that we really understand today is a BCS mechanism whereby, you know, two electrons form pairs as a result of an attractive interaction due to phonons. And the plots that you see here are from this work where essentially they found that by compressing, you know, hydrogen-based materials at extremely high pressure. So we're talking about 200 GTA here. You can form a metal that actually has very high superconducting critical temperature. So in this case, it's maybe 200 Kelvin and more reason has been worth showing that one can even reach a room temperature. So the smoking gun, as I said, the kind of experimental proof that you have phonons involved is usually the isotopic experiment, isotopic substitution experiment, which consists of taking this compound and replacing some elements with something lighter or heavier in terms of isotopic content. So this example is essentially going from sulfur hydride to sulfur deuterite. So deuterium is heavier than the hydrogen. Therefore, you expect the vibrational frequencies to decrease. And if the BCS theory is correct, what you should observe is a decrease in the critical temperature, as you can see here, okay? So that's another manifestation of, you know, electron-borne interactions. So how do we study this phenomenon? So what I want to, what I usually do in this kind of introductions is to start from a very simple, like elemental introduction to the problem using some heuristic arguments, because usually that is easier to digest. And today I'm gonna change a little bit the angle. So I want to start from the difficult side, just to give you an idea of, you know, the complexity of these problems and the kind of tools that have been developed over the years to address them, okay? So we start from the most general, you know, Hamiltonian for a couple system of electrons and phonons. So in this case, the wave function would be, a wave function is coming with electrons. And in this case in nuclei, so the electron position would be R sub i, and the nucleus position would be tau sub k. And here you have all the interactions between these particles. For example, you have, you know, the red bits here and here would be the electronic Hamiltonian where you have kinetic energy and then the Coulomb repulsion between electrons. You would have blue bits here and here, again kinetic energy, the nuclei and then the Coulomb repulsion between nuclei. And then there is this magenta term that is a Coulomb interaction between electrons and nuclei. So that's an attractive component, okay? So V here in all the cases is defined below and it represents very simply the Coulomb interaction between two charges. So in principle, we would like to solve this equation, but actually this is very difficult. Now you, I'm sure you know that one could use a quantum Monte Carlo techniques to solve for the electronic problem, you know, to attack a Hamiltonian looking at that. And that is fine. But if you try to move on to incorporate also the nuclei, things become much more challenging. And the reason being that the nuclear wave functions are much, much narrower than electronic wave functions. So not only you have the problem of correlations, but you also have a problem of massive scale nature because, you know, there are really different length scales for nuclei and electrons. So one approach that has, you know, essentially being popular and it's probably the most rigorous way we have today to describe these problems is to move from a representation in real space to a representation in Fox space. Now we try to explain what I mean in a second. So the idea essentially is to try not to work with the complete wave function that depends on the coordinates of all the particles in the system, but to express that as a linear combination of the later determinants. So this is very typical in quantum chemistry. For example, what we could do is to take a slated terminal. So this is not here made of, for example, the villains, ComSham states, okay? So that's not exact, but it's, you know, one possible determinant. And then we could construct a linear combinations of modifications of those. For example, I could apply this destruction operator to remove an electron in the state N and then the construction operator to add it to another state M. And this would be a modified determinant where the N state is empty and the M state is occupied. And this is what we call a single excitation. Now this is not gonna be the description of the ground state or the excited state we want to look at, but I can take linear combinations to be more accurate, for example. And I could also do the same by looking at more complex constructions. For example, I can remove two electrons and then I add them somewhere else. So this would be double excitations and then I can do triple excitations and so on. So this is just to say that by working in the space of slated determinants, it is possible to transfer the complexity from a real space dependence on these variables to the coefficients that we don't know here, okay? In practice, in solid state physics, we don't usually work with these coefficients because things get complicated quite quickly. And there is one extra step which makes our life much, much simpler. And that is the following. So what we do is to take linear combinations of these field operators or these creation and annihilation operators. For example, this is a linear combination of these structure operators where the coefficient is the value of let's say a conscious state at a given point, okay? When you do this, you are defining an operator that depends parametrically on the coordinate in real space. So why this is useful? Well, this is extremely useful because if you take the Hamiltonian I showed in the previous slide, you can discover by just looking at the standard textbook on second quantization that that can be rewritten in a very simple way. For example, if you have a sum of single particle potentials, for example, imagine the interaction between electrons and nuclei and this will be a summation of all the electrons, this can be rewritten as a single integral of the potential over this expectation value of this potential over this field operator. So the advantage here is that essentially we get rid of all the coordinates, all the particles and we are left with a single coordinate, okay? Clearly the disadvantage is that we are not talking about the wave function by the operator. So that complicates our life a little bit but that's not too much of an issue. So what happens to the Hamiltonian I showed you earlier? Well, in practice, we will have five terms as before so kinetic energies and then the electrostatic interactions and these electrostatic interactions in this new representation take a very simple form. For example, the electron-nucleus interaction can be written as a essentially a Coulomb integral of two densities, the electron density, the nuclear densities. And so that's essentially just a pure electrostatic and notice that I don't have any more variables of all the electrons in all the nuclei, I just have the A1 variable for the electron density and one position variable for the nuclear density. Similarly, if you look at the electron-electro interaction that also becomes extremely simple in this kind of at this formula that is essentially just a Hartree interaction. So you take the electron density, multiply by itself and then you divide by the distance within two electrons by electron interaction. And in this case, you also take care of removing the spurious self-interaction of electrons. So it's like if by using this formalism you really reduce the problem to something looking like a kind of Hartree problem. The only difference between these and Hartree is that these are not kind of C numbers or no real numbers, these are actually operators, okay? And just to be clear, the electron density operator is essentially the product of two field operators, our creation and the destruction operators. So what do we do with these things? So once you have these amethystones, what the people would like to calculate is the Green's function. In this case, it's for an electron but you can also be calculated for phonons, for example. So this is a particular version of the Green's function called time ordered and it's just a version that is useful for certain applications in the ground state. So time order means essentially the times here increase towards the left, but that's only a detail. So that's a definition of Green's function. What it is, as you can see, it's say the expectation of these field operators over this state that I call N. So this is essentially the ground state of the system, let's say neutral crystal, that we imagine an insulator with field valence bands. So let's try to understand what people mean by this expression and why it was invented in this way. So if I focus on this side, so what I just highlighted in blue, so this corresponds to taking the ground state and creating an electron at the point r prime and at the time t prime, okay? So this represents the ground state plus an electron at the spacetime point r prime t prime. If you now look at the left hand side and just ignore for a second the VIX operator, so what you discover is basically that we have essentially the Hermitian conjugate of what I just described. So here we have the ground state with an extra electron at the spacetime point r and t. So really this Green's function is a scalar product or inner product between two states where an electron has been added to the system and is now in two different places. So this is what people call the propagator of the electron essentially is the probability amplitude for an electron to go from an initial spacetime point to a final one, all right? And this contains a lot of information about the electronic system. For example, one can prove that if you take the Fourier transform in time, so in t minus t prime and you go in the frequency domain, the poles of this function give you the excitation energies of the system. It means the energy needed to basically add or remove electrons to the system or electronic excitations and things like that, okay? So in practice, from this Green's function, you can obtain things like many body bulk structures. The way people do it is to Fourier transform also the position operator here, the position variables in case space and therefore you will have a function that is absorbed both in momentum and it ended in the frequency. So from this function in practice, what one does is to build the spectral density function that is simply the imaginary part of the Green's function and that is if you want a density of states, not the consumed density of states but the many body density of states. So typically what happens when you construct these things is the following. I will give you just a cartoon example and then I'll show you with a couple of prelations. So you start from something like DFT calculation, you will have a density of states. In this case, I consider only one state just for simplicity. This state in DFT is basically a stationary state, a solution of the constraint equations. So that means it is infinitely linked and the density of states is a sharp dirac delta functions, right? So if you switch on interactions between electrons or between electrons and nuclei like phonons, what happens typically is the following. This density of states gets deformed and the deformation can be described qualitatively in the following way. First the peak has moved, okay? It's no longer in the original position that means the excitation may have just changed a little bit and this is what we will call the passive particle shift. The other change is that the peak is no longer sharp so now it's slightly broad and this is a signature of the fact that the electron has acquired a finite lifetime. It means it's not gonna be there forever. And then you also have additional structure here. Usually this is in cohenous structure that is much, actually in this case it's exaggerated but for me in general cases it may be 10 or 20% of the weight of the main peak and it has to do with the interactions with for example bosons that could be phonons, could be plasmons, could be magnons and maybe electron-hopal excitations and things like that, okay? So once you understand that this is what one can calculate is that the next question is how do we obtain this green function and this spectrum function? So the way it works is quite straightforward. In practice, we define field operators that depend on time by simply using the Heisenberg representation. And then from that you can derive the equation of motion of these field operators essentially is like a Schrodinger's equation for these operators. In this commutator you can evaluate it using the Hamiltonian I gave you a couple of slides back and one can prove that that's the result that you obtain. So you have a kinetic energy and then some kind of potential acting on the electron field operators. So this expression actually is very beautiful because if you think about it, if you remove the hat in these operators that will be exactly the equation of motion of an electron in the Hartree approximation, okay? So this is a Coulomb potential and this will be the density or for example the electrons. The challenge here is that we have the hats so these are operators so things are not so simple. So what one does at this point is to consider if you have an evolution equation of motion for the field operators and since the green function is defined in terms of field operators you can use this expression to derive the equation of motion for the green function and then to obtain it's let's say calls and for excitation energies. So the equation of motion for the green function derived from the first slide here looks like this. So the left hand side is a standard let's say Schrodinger type kind of part of the equation. This will be the energy of the manual and this will be the kinetic energy. So what you see on the right is essentially the potential part and here the complexity starts to show. So remember that the electron density here for example is a product of two field operators. So here we really have four field operators and that's really complicated to describe. So this is what contains things like the Hartree energy, the Fock exchange energy. There is a two particle green functions these electron fall interactions. So this term here contains a little bit of everything. So at this point this will be pretty hopeless but some kind of the pioneers in our field maybe in the 50s they derive techniques to deal with this in a kind of systematic fashion. And the basic idea is to replace this difficult term here by the product of a self energy and that means function itself. So if you want the self energy here is actually defined in such a way as to have these integral being equal to the difficult expression I showed you in the previous slide, okay? And what one can do is to do some maths to find out precisely what the self energy is. So what I want to say here is that in practice in this expression we can see already something looking like a Schrodinger's equation and the electron fall interactions enter through this potential here. This potential depends on the exact let's say ground state density and these are also the contains a nuclear quantum fluctuations. For example, it contains the fact that nuclei are not strictly localized in a point because they're quantum particles. And this term here contains things like dynamic normalization and other correlation effects that one wants to incorporate, okay? So one can do a little bit of maths to figure out what is the self energy. So I'm gonna just show you the main terms without deriving it. And I will use a graphical representation just because it's easier to remember. So the self energy contains essentially three terms. There are two other terms that are very small so we're not discussing. So the first one is something that you might have seen in other contexts. So this is essentially the GW self energy meaning that you have a product of an electron line. So this is a function and the screen interaction between electrons. And this is a vertex that in typical calculations one neglects because it's very difficult to evaluate. Then there is another term that is similar in shape but in this case, instead of having the screen through interaction, we have the phonon propagator here. So that would be the phonon green function. So this is the form of a electron green function times phonon green function times a electron phonon matrix element here and here times the same vertex we have on the top. So in practice also in this case, the vertex is usually neglected because it's very difficult to evaluate. And this is what we call the farming of self energy. And then there is one last term that is a little bit weird. It's essentially a self energy where there is no electron line and the phonon line folds upon itself. So this is what is called the Debye-Waller self energy however I should be a little bit careful here. So that's not exactly a self energy. Actually, if I can go back a couple of slides this comes from this potential term here and in particular from the fuzziness of the atomic nuclei, the positions around the equilibrium. So it's not a dynamic, it's a finite general way. So what can we do with this self energy? So once we can calculate them, so what I'm gonna do now is to show you some applications of this concept and then I will go back a little bit to more details on how we perform calculations. So I wanna show you three applications. One is gonna be about mass renormalization. The second one is gonna be about photo emission satellites. And the third one is gonna be about inelastic excess carbon on phonons, okay? So the first one is about, this is just an example that we've been working on in the past on peroskites. So the peroskite structure is basically a material with atoms arranged in octahedral that are connected by the corners. In this case, you have lead in the middle, iodine in the corners. And then in the middle of these cavities, there is a molecule called metal ammonium. So this is CH3 and X3. And that's, this material is as attracted attention because it, some modifications of this compound have led to solar cell materials with efficiencies that are competitive with kind of the cell of the art in silicon. So a lot of people are working on that. So one of the questions in these compounds in the physics of this compound is, what is the effect of phonons on their properties? So what you can see in the middle is a calculation of the spectral density function where I was showing earlier. And basically you have the wave vector on the horizontal axis, the energy on the vertical axis. And what I'm doing here is to show is showing the conduction band and the valence band. And I'm basically setting the energy axis so that the bottom of the conductor is zero and the top of the valence is zero. And you need to imagine that in the middle there is approximately a band of 1.6 selectables. So if you do DFT calculations, you will find this dashed blue line. I'm not sure you can see it, but I'm trying to point at it with my pointer. So this will be standard DFT bar structure. If you use this familial self energy that I mentioned, you obtain the plot shown in yellow and purple. So you basically see the spectral density function that is a yellow line that is a relatively sharp band at the bottom here. But you can see that it's slightly flatter than the previous one. So what we say is that the mass has increased in this case. So the mass has increased as a result of electron-phonal interactions. And this can be seen in this diagram here at the bottom. So basically we typically quantify the mass increase by a parameter called lambda, that it defines so that one plus lambda is the increase in the mass. So if lambda is equal to 0.3, it means that the mass is increased by 30%. So in this system, phonons increase the mass by something between 20 or 40% of low temperature depending on the specific approach used to perform the calculation. The second property you can obtain from this calculation is that you see that the bands are very sharp. So this yellow line up to a certain energy. At some point, everything becomes very, very broad. You almost lose the notion of a kind of a sharp band or a band structure. So this can be quantified by looking at the width of the spectrum. So this is the broadening of the quasi-particle and it's plotted here as a function of energy. So at low energy up to, let's say here, you have almost no broadening. So it will be this range here. And then you have a sharp jump and then it remains something of the order of only 15 million electrons. So what is happening here is that the electron of this energy is able to emit a longitudinal optical phonon. And as a result, it's lifetime gets decreased. And in particular in this case, the lifetime becomes something of the order of 20, 20 seconds. So this is something that people are interested in for things like transport or study for the citations and things like that. So the second example is a much simpler compound. So this is a rock salt compound. It's called a European oxide. Essentially it is a kind of a cubic arrangement of European and oxygen. So just to give you some background, so this compound actually is important conceptually because it is the only material. It is the only, let's say semiconductor that is as a fully spring polarized bands. So for example, if you perform a calculation using the FD plus U, you will find that both the valence band top and the conduction on bottom have a spin oriented in the same direction. So this product has a perfect spin filter. And this actually was found experimentally maybe 10 years ago. The only complication is that this phase is not stable at room temperature. So this become the European cess pu oxide. So it's not really useful for applications, but it's a proof of concept that one can achieve fully spin polarized semiconductors. So here our experimental collaborator 15 at the St. Andrews in the UK was interested in looking at the electron phone effects in the conduction band. So you perform ARPES experiment. So then that means the angle is off of the electron spectroscopic experiment. So in ARPES, what happens is that you shine light typically in a synchronized source that extracts electrons. And from that you can reconstruct the bath fracture or the compound prior to extraction. So in this case, for example, you see that you have the P bands that you can see here in the ARPES spectrum, the European F bands here. And then in this experiment, what they also did is to dope the system with gadolinium. So your opium has a two plus, gadolinium is three plus. So that means that the linear injects electrons in the system. So we should see a little bit of signal coming also from the conduction on bottom. So here it's not very visible because there is a lot of signal coming from the valence. But if you remove the valence and you enhance the signal here, you actually discover that there is indeed a signal coming from the conduction. Now, in this plot, we were expecting a conduction band looking like a small parabola, okay? But in the experiment, what they observed is something looking like a blob, let's say. Maybe if you use a bit of imagination, this could be a parabola, a small parabola. But then there are also other features, a satellite here, a satellite here, maybe a little bit of a satellite here. And if you look at, it used to go measuring the separation between the satellites, that's always 60 million electrons. And 60 electrons by coincidence is the energy of the highest longitudinal optical form is compound. So the proposal was that these have to do with the, you know, some electron form interactions in the systems. So what we did is to try to do calculations of this effect using this self-energist. And that, you know, leads us to reproduce this spectra. So this will be again the same experiment these are calculations as obviously, we don't have all the broadening including the experiment, but you can probably see this first band, the satellite and another satellite. And just to make it more clear, I'm also showing a cut in the middle of this band. So if you go down this line, you follow essentially the energy here. So you have a first quasi particle peak, that's the blob here. Then a first satellite, a second satellite, a third satellite, okay, that are equal in space. In the calculation also you draw a line in the middle here that we find a quasi particle peak, satellite, satellite, and so on, okay? So this is just a way to say that, yes, one can confirm that the satellites are an effect of electron phonon interactions, okay? And in particular of an interaction with a specific polar optical form that is longitudinal optical mode, okay? So that's the idea of the level of accuracy one can achieve now with these kind of calculations. And then the last example I want to give you before moving to technical details, is about calculations of phonons. So in the previous two examples, I told you how phonons affect electrons. So for example, by changing their back structures, but also the reverse is true, electrons affect the property of phonons. Usually we don't discuss that because when we perform DFT calculations, you calculate the dynamical matrix and you diagonalize, those again, so the phonon dispersions already taking into account a lot of electron phonon interactions in the static sense, but there is additional interactions that are not captured by the standard density function perturbation theory approach and can be captured using a Green's function formants. So to do this, one needs to build a self energy for phonons that is similar to what I showed you for the electrons. So the example I want to show is about diamond. So diamond is a wide gap insulator. You're looking at here at the valence bands and in principle, this should be completely filled. In this experiment carried out by Maurice Hirsch at the ESRF in Grenoble, what they need is to dope diamond with boron. So since borons are as three electrons, this corresponds to doping poles into the valence bands and that creates a very high doping a tiny Fermi surface near the top of the valence band. Okay, imagine a spherical Fermi surface. So if you perform a calculation with density functional theory and you set a Fermi level like that in the calculation, you'll do this cover that the original phonon dispersions in blue become softer, so you get the dashed lines. And the reason is that you have more screening coming from this Fermi surface, okay? However, the one issues at this point was that this softening was too large as compared to experiments. So for example, I can show you the experiment. So these are essentially inelastic excess scattering experiments carried out at the synchrotron in Grenoble. And what you can see here, it basically is the phonon dispersions corresponding to this upper branch here of the tantalina, okay? So the first plot is for the undoped diamond. As you increase doping, you see that the brolin here is increasing and there is a little bit of more pronounced bending. And then as you keep increasing doping to very high levels, the bending continues and the brolin here is such that you almost lose the quasi-particle in this region, okay? The problem is that if you do then see functional theory calculations, you find a softening of about 20 million electrons, but in experiments of the order of five million electrons. So there is a significant discrepancy. So to fix that, one has to go to many body techniques like the ones I showed you earlier. And this is a many body calculation where you see the dispersions of undoped diamond. What happens if you start doping? So you have a little bit of bending and increasing brolin. And then the fact that if you keep increasing the doping, so here you lose completely the quasi-particle signal and the bending is now more pronounced, but this bending is more in line with the experimental value than a brute kind of a DFT aesthetic, let's say a DFT calculation. So that's to say that there are also techniques to look at how electrons affect the properties of phones. So now let me tell you a little bit about the technical details of these calculations. So what I showed you is that, you know, for example, for the electrons, we want to calculate something like a family that self-energy. It was looking something like that in the previous slides. There was a vertex. And as I say, we neglect that because it is difficult to calculate. So what I want to make a discuss here in the next two slides is that the most important point and also the most difficult to calculate is this red bit, which is the electron form of its element. So first let me tell you what it is. And then I will explain the challenge in calculating it. So this is essentially a sandwich between the two electronic states and the change of the potential associated with the phone. So nothing really complicated. So these are the two periodic parts of block-wave functions and it's the change of the potential. So you can see this as the probability amplitude to scatter for an electron to go from a state k to a state k plus q, by a phonon or wave-electron 2. So this, the calculation of the states like the Konshaw-Megen states is very simple and also, you know, computationally cheap today. So what gets complicated is the calculation of this change of the self-consistent potential. So I want to give you the full expression. So you can see what this is made of. So this is the complete expression. So the way it works, so at least conceptual is the following. I take the self-consistent, let's say the Konshaw potential of a supercell, for example, I move an atom, so atom, kappa, direction alpha and even unit cell. And then this creates essentially a dipole. And then I do that for all the atoms in my system. Then I perform a linear combination of those objects, these potentials using the phonon edging mode. So this is the phonon polarization vector. This tells me in which direction each atom is moving, not even phonon mode. Then I multiply this by the amplitude of each displacement. So this is in the direction, but I don't know by how much they move. So this quantity tells me how much they are moving in those directions. And then I add the pre-factor which takes into account essentially the, that essentially brings a prerequisite to the system and that's the entire potential. So the complexity of calculation is potential is that, so this can certainly be done with codes like phonon or inabinit, and essentially can be done by density function perturbation theory. The challenge is that each calculation for each wave vector is almost, or let's say approximately as expensive as a DFT total energy calculations. And one could say, okay, that should not be too complicated because I can just calculate maybe a dozen matrix element and I'm done. So the problem is that that's not really the case and one needs a lot of matrix element to perform calculations. So we're thinking how to explain this concept in a way that sticks and then you can remember it without going back to these slides. And this is probably the best way I have to explain it in just a few seconds. So let's take an example of the inverse of the electron lifetime coming from electron from interactions. So if you want, this is proportional to the imaginary part of the family that self energy mentioned earlier, okay? And the reason I'm gonna make now applies identically to the real part of self energy. So this calculation can be essentially written as an integral where you have some electron from elements, some prefactors here that we're not gonna bother with. And then there is a direct data function. So this data function essentially enforce energy conservation. And it tells me that my initial state can decay into a final state that is basically removed in terms of energy by plus or minus the photon energy in both these products, okay? So let's see what happens in terms of requirement for this matrix element. If I consider a simple two dimensional system with a parabolic electron band. So imagine this being our, let's say a metal that you want to study. You want to look at the, I don't know the lifetimes, a certain energy. So the energy is something that we set by, deciding this pink plane. So what is happening is that my initial, my mistake I want to consider is something lying on the circle. And what I need to do here is to perform calculations where I add up all the processes where the state can scatter into states that are actually at an energy that is very close to itself because it's plus minus one photon energy. So approximately we need to say that, we can say that the state scatters on other states that are lying on the circle, okay? So what we want to do is to describe this integration carefully. Suppose you use a grid of K points in the Bruin zone similar to what you do for the electron density in total energy calculations. So let me be brutal and say that we use a four by four grid. So really what you want to describe is a circle in the Bruin zone. But if you use a four by four grid you actually obtain a square, okay? So instead of doing, for example, a scattering between a state here and other states on this white circle, you're scattering with something that is lying on a square. So you lose all the phase space kind of geometry that you want to describe. So to achieve something that looks more like a circle, well, you need a finite grid. So if you do, for example, 40 by 40 you discover that now the circle looks more or less like a circle when you pixelize it with this this is Bruin zone, okay? So the bottom line in this line is that the problem in calculating this self-energies is that usually we are trying to perform integrals on a domain which is a lower dimension than the Bruin zone itself. So for a 3D Bruin zone, we perform integrals in the time on a two-dimensional surface. And that's what adds complexity to these calculations. And that's why we need to evaluate many matrix elements usually. So how do we do that? Well, calculations of hundreds of thousands of matrix elements by direct density function perturbation theory are essentially impossible, okay? There's just no chance. So in this context, you know, 1A functions become extremely useful. So this is just a cartoon to remind you what the 1A function is. And obviously I don't need to say that in a 1A function school, but the basic idea that is described in detail in many of the papers, this team and then also this very nice review article is that we can try to perform a transformation. So a linear transformation between a standard, you know, block-periodic wave functions and localized objects, okay? And these localized objects, you know, can be described using, you know, a generalized 1A Fourier transformation where there is both a kind of phase factor here in the unitary rotation between the bands. Now, in the early work, Marzare-Marder, we thought they realized that the arbitrariness associated with this unitary matrix can be exploited to make the 1A functions as localized as possible. And it's very useful because then you can exploit concepts that come from, you know, type binding, for example. So that's basically the idea. And what we did 15 years ago is to basically just generalize this concept to the case of electromagnetic settlement. And the key to this is the fact that an electromagnetic settlement has three components. So two block states. So for those, we can do a rotation to 1A functions. And then there is the change of the potential coming from the phonon. So this change, if you remember, maybe I can go back a couple of slides. If you remember the expression, is a linear combination of essentially dipole potentials coming from the displacement of a single atom. So what one can do is to actually re-extract this type of potentials from the periodic components. And in practice, what happens is that one can write this, you know, perform a unitary transformation to re-extract the kind of atomic dipole potential associated, you know, that is contributing to the kind of phonon perturbation. So this is by definition something localized, at least in metals. And then what one can do is to exploit the fact that these objects and these objects, they are all localized. So if you make a magnetic settlement and they're sitting in different unit cells, probably the magnetic settlement will be very small. And therefore you can do concepts like, you can use concepts like Fourier padding and then interpolation. So in practice, what happens is that one can write down a relation between the magnetic settlement in the block representation, the magnetic settlement in a one-year representation. So using these three things, this is again a generalized one-year Fourier transformation. So the capital U refers to the electronic part. The E is the polarization method. So it's the phonon part. And one can go back and forth to exploit this expression. What one does in principle, in practice, is that one calculates this magnetic settlement on a coarse grid, as usual, then use this expression to exactly determine the magnetic settlement in a real space. Then pads the real space with zeros and goes back to this expression on any point in the blue and zero. Okay. And this is something that is very efficient. So just to show you something that is very vintage. So like from, again, 15 years ago, this was the first calculation on diamond. And you can see that the matrix settlement here are very, very highly localized. So they decay exponentially as you move away these kind of blobs in the magnetic settlement. And because of that, you can exploit them to essentially follow the, reproduce the ablution of its element that you would obtain from density function perturbation theory, but much lower computational cost. So just to give you an idea of how these things look like for more systems, I want to show you something more recent. This was calculated by Samuel Ponce, it was published last year. So what it needs to study the current mobility in many semiconductors. And one part of the paper was to look at how the magnetic settlement, the conformal magnetic settlement decay in the systems. So these are essentially the log plots of the decay in real space of this magnetic settlement. And you can see that the decay pretty fast. For example, let's look at this plot in the middle. So that purple are for the impedance bands and you can see that the decay exponentially here. So that's actually corresponds to what we expect. For the conduction band here, since you need to use this entanglement, you don't have this kind of concept of exponential decay and you have more something like exponential decay, but still the decay is good enough for practical applications. And this is pretty much as, you know, something to find across many compounds. So how this is done in practice. So what happens is that we use a code called EPW that is actually distributed as one of the modules of the quantum espresso suite. And the way this works is essentially it takes the FPT input and the conchamp states from quantum espresso. It takes information about one functions from 1.090 and then, you know, puts them together to realize this interpolation I described and then to calculate physical properties, such as those that you will see with Roxana Margin and Samuel Concey later, okay? So just to tell you a little bit about this code. So this is something that started many years ago and it was a very small operation, but now there is, you know, the team is growing and there are several people involved. So apart from us, there is, you know, Roxana Margin have been in turn, there is Samuel Concey at the CoalPolydifficient Reven, there is Manuscript Agnes in Michigan, there is Nikola Bonini in the King's College in London and then we also have some new additions. We find a displacement set up by Mario Sakurai at the University of REN. So it's a nice team that is growing. It's not as big as the 1.080 team, but, you know, we are getting there slowly. And just to give you a couple of updates, you know, there's been a lot of work in the code that's happening during the past three years to push it towards exascade computing. So usually here at UT Austin has been refactoring the code to essentially incorporate multi-level open MP and PI prioritization and demonstrate the scaling, you know, to the full, inside the full machine of from Terra. So it's basically, you know, the, it's a, you know, 40 petaflops super computer, so 500,000 cores. So this actually is very encouraging and hopefully this will become the standard going forward. And now I want to, if I have, you know, two, three other minutes, maybe five minutes, I want to tell you about some research that we are doing if I'm allowed. Antimo, can I take another five minutes? Yeah, sure, go ahead. Okay, so that basically was an introduction. And what I want to tell you now is just something we are much really very interested in personally and some activity we have been, you know, working on. And it's about tolerance, okay? So I see this as a new frontier of electron-phonon physics because it's really challenging the way we think about electrons and then maybe the most striking concept that we're still trying to kind of understand is that, you know, sometimes when you add the removal electrons to crystals, they may want to localize. So that's actually kind of in contrast to our concept that everything is, you know, block periodic and then, you know, we have this block here and that helps us. So polar, you know, intuitive fashion is basically just a very simple concept using any textbook or, you know, under the textbook where you take a crystal that is an insulator, you add an electron or you add an electron and maybe the interaction with ions is going to be so strong that there's going to be a local distortion and this distortion may actually pull in the electron in a way that they will actually trap in that place. Okay? And this may lead to all sorts of phenomena. For example, there is a change in, maybe in the potential of the landscape and this could lead to some kind of, you know, interesting ships in absorption of photo or luminescence. One could obtain the creation of some defect states in the band up and things like that. Okay? So I'm not going to say much about this because this has been reviewed by Chesa Refrantini and co-workers in this very, you know, extensive, very nice review on this topic. So if you might be interested in polar ones, I would recommend to read this article. And, you know, what we wanted to do is to try to understand whether we can calculate these things. We're using methods like the ones I described earlier. And so the first question is, you know, okay, what is that we would like to calculate? So in this cartoon, what I imagine is that we have a kind of an insulator or a semiconductor flight with valence bands completely filled and then you have a conduction band that is empty. So the dots will be the atoms and that they generate a potential is periodic. So everything is not periodic. Then I add an electron in the conduction. And if I do a calculation in DFT, what is gonna happen is that this electron will land into the, so in the ground state will be the conduction by minimum. Since it's a band state, it's also a block state. So it's block periodic. So the density will be periodic, right? So that's all we're used to. Now I can change the perspective and say, okay, what is, I do the same calculation, not in a unit cell, but in a large supercell. And on top of that, I move the atoms a little bit to create a lamp of atomic charge. So basically I move the atoms a little bit and then I add the electron. So if I keep the atoms fixed, the electron will, maybe if the distortion is strong enough, the electron might relax around this atom and perform some kind of an effect, okay? Now what I can do is to let the atoms go, essentially relax them and check whether we go back to this scenario or we stay in a localized scenario, okay? So if we stay in a localized scenario, we say we have a problem, right? And the question is, okay, we calculate this with DFT and they know how to do that efficiently. So in DFT, there are a couple of issues. So there is one issue which is practical and that's the fact that suppose your polarity is maybe five nanometers or two nanometers, you might need a supercell with maybe 10 or 20 or 30,000 atoms, okay? So that becomes obviously impractical. So one can do it for one system, one's in a while, but it's not something you can do routinely. There is a second problem which is more important is that you know that in density functional theory, the heart rate contains the interaction of the electron with itself. So one can prove that that interaction tends to kill the polarity, meaning that it tends to delocalize it, okay? So that's essentially the self-interaction problem of DFT that becomes particularly complex and challenging because of polarity. So to overcome these things, what we did is to devise a simplified format that starts from density functional theory, removes the self-interaction using a self-interaction corrector functional and then essentially concentrates on the polarity functions and gets rid of all the valence states in order to simplify the problem, okay? So I'm not gonna go into the details but essentially the bottom line is that what you're looking at here is the energy difference between a system with an extra electron and the neutral system with valence band field and conduction band empty. And the other thing that we did is to truncate essentially to expand all atomic displacements or positions to second order displacement. And this is useful because when you work to second order you can use concepts from more forums, okay? So you can talk about forums and the interaction with electrons. So this is a new energy function that essentially depends on the way function of the electron and the next electron and the position of the space of the nuclei. And what you do when you have a function you try to find the ground state. So to do so you do a minimization, a variation of the derivative with respect to size star and the derivative with respect to u sub k and this leads to two equations. The first one is a modified constant equation where there is an extra term that corresponds to the linear electron for interaction. And the second one tells me the relation between the electron wave functions where it's sitting and then the atomic displacements. So this kind of formulation fixes the problem of the self-interaction, but it does not address the problem that I might need a very large supersize to perform this calculation. So to also overcome this problem what we did is something very simple is to observe that the block functions and the formalization methods form a complete basis for the real space functions. So you can always write down your polar on function as a combination of block states and you can write any atomic displacement as a linear combination of the form of a displacement. And if you use these two expressions and replace them in the previous equations you obtain something where we don't need anymore a supercell because everything is expressed in the reciprocal space. So in particular we have a non-linear problem with two sets of unknowns, the A coefficients that describe the electron wave function and the B coefficients that describe the atomic distortions. And the ingredients are the electron bands, the formal dispersions and the electron form of the experiments. So every day you can calculate using density function perturbation theory is used here. And then from this we can perform a solution and check what happens. So I'm just going to give you two quick examples because I don't want to hold your too long. So this is an example calculation for the electron polar in lithium fluoride. So this is just a rock salt ionic compound. What we need is to add one electron to the conduction band and you can see that this is the electron wave function. It is localized, but this is something one will call a large polar which spans maybe 10 unit cells. And if you were to calculate this you would need at least 10,000 atoms in your supercell if you want to run its crazy calculation. So that's why using a supercell space formulation is advantageous. In the same system, if instead of adding an electron you remove an electron, so you create a full polar, the scenario is completely different. What you find is that these objects it spans only a couple of unit cells. So this is essentially a linear combination of two p orbitals. So that's a very small polar and what is interesting in this context is that in the same crystal you just look at the valence for the conduction you find completely different behavior and this is reflecting the energetics. So if you look for example at the energy of formation of the electron polar this is about 200 millivolve below the conduction of momentum and in the visible polar is about 10 times larger. So this also means that there is a potentially a significant renormalization of band gaps and the band structures as well that comes from the localization effects. And I will not have time to show this here but we perform additional work to basically understand how this kind of renormalization relates to standard calculations like the alienino theory where you want to change look at the change of mass structure with temperature while the bottom line essentially that both effects have to be taken into account and this effect is particularly significant when you have small polars and the energy shifts is very large. So this is basically a submarine or the kind of things that I tried to tell you. So we have basically, I went through a few example to show you that one can calculate things like mass structure renormalization affecting masses lifetimes, effects like form satellites like people called them polar and satellites in our experiments. One can also look at how elections affect the inspections of phonons in a phonospectroscopy. We start having methods to look at kind of polarization and but these things are really in the beginning so there's a lot of work to be done here. And there is much more that one could do and for that I think I would refer you to two instructors of the school. So Professor Roxana Marginet will tell you about superconductivity and how to do these calculations in APW code. And then Dr. Samuel Ponce will tell you about a current transport, for example, mobility and things like that and how to do these calculations in APW. And with that, I will just put up a knowledge slide. So many people contribute to these slides that I mentioned over the years. So I mentioned here, so this is the current group and this is our funding. And with that, I'm happy to stop here and take questions if there are any. Thank you so much. Thank you very much Feliciano for the really wonderful talk. So are there any questions from participants here in Trieste in presence? Please raise your hand. Okay, could you come here and speak at the microphone? Thanks. Here. Hi. Okay, thank you for the nice talk. My question is about the RFS experiment of Polaron. So for example, say you consider an electron dope system then should I understand the RFS experiment as measuring electron polarance at the initial state or as measuring whole polarance in the final state? You mean this one? Yeah. Yeah, yeah, that's a very good point. So this is actually a, so this system is, so it's essentially it's, there is electrons doped into the conduction, okay? So this is really, if you want a whole Polaron in the conductor, okay? Is that what you're asking? Because basically this is the removal of an electron from the system after realizing the dope, right? Okay, there's another question here while Stepan goes to take the question. I ask if participants connected on Zoom want to write the question in the chat or either raise your hand and we will unmute you for asking the question. Okay. Try that, this one. Oh, great. Can you shout out to each of you? Yes, I think it should work. Okay. I have a question about your perovskite calculation and that you could see in the realization of the effective masters. So my question would be, does it help to explain the lifetime in this material, the long lifetime of fixations? So, sorry, which material you're referring to? Perovskite. Okay, let me see. Okay, so you're asking whether these lifetimes, much experiments, is that the question? How long is the long lifetime of optical expectations? Oh, yeah, that's a good point. So I think there are two different things. So when people talk about long-lived carriers in this compound, what they refer to is that when you create an electron or hole pair, it takes a long time before recombining, okay? While in this case, what these lifetimes are, is for example, imagine an injection in something like, so you're transporting an electron and you want to know how long it stays in that state. So it's more like a scattering lifetime. So there is no concept of recombination. So that requires essentially looking at the band to band combination and the short-lived hole, which goes through the facts. And that's basically another piece of physics. Okay, that was good. Thank you very much. Okay, there is another question here. Maybe just the last question here. And if people are on Zoom, no, I don't see any questions on Zoom. So please go ahead, Sarah. Hi, thank you for the talk. I was wondering if you have also considered, have you also calculated, for example, the optical properties, considering the electron point of interaction by way of, for example, the BSC? Oh, yeah, that's a very nice question. We have not, but we would like to. So there is a lot, so there are several papers. So the issue with that is that, I think at this point, a complete calculation with BSC is probably too complicated. So different groups have basically decided to approximate different things. And I think, yeah, there's still a lot of work to understand what needs to be done there, in my opinion. So what we are particularly interested in is to see whether, for example, now we've kind of not understand how to do polarons for electronic excitations. And the question is whether one can find similar objects in optical excitations, for example. And the other question is, for example, whether one can introduce lifetimes in this BSC calculation, this has been done by, you know, and by the people in periphery at some point. So there are several options, but there are two, I would say there are so many terms there that is not entirely clear which ones one has to keep in which one one can draw. So there are one that's really to discuss specific experiments in my view. But, you know, we haven't done anything yet in this area. So I cannot really answer two, you know, in very detail. Okay, thanks a lot. I think we should stop here. Let's thank again for the channel. Okay, thank you guys. So now there is the coffee break and then we will start in 10 plus four for the last talk of the day by Lin Lin.