 can you hear me now or do I need to speak no higher I guess maybe I can hold it is it better now can you hear me in the back no okay okay maybe I can just hold it let me try to put it somewhere maybe center it's not very hard what about now is it better okay I will try also to speak a bit louder than I generally speak and hopefully you really cannot hear me just make signs and I will I will try to be a bit louder so in the these last two lectures of the school I'm going to introduce some new capabilities that are implemented in EPW and this capability is allowed to look at superconducting properties with an isotropic resolution since I suspect that not everybody has worked with superconductivity I will start the presentation with an overview of basically what is superconductivity and what is the field of superconductivity at this point then I will overview the BCS theory of superconductivity and briefly mentioned the Macmillan-Alandais formula for critical temperature that you have already seen in one of the tutorials and the central part of the talk will be the Migdal-Eliashben approximation and the Nambu-Gorkos formalism that put together allowed to derive what is called the Migdal-Eliashben equation and finally I will just mention briefly another method to investigate superconductors from principle which is a density functional theory for superconductors so superconductivity is basically a quantum mechanical phenomena occurring in certain materials below a characteristic critical temperature and at the macroscopic scale there are two very intriguing manifestations that one can observe so this so one of them is this zero resistivity so basically below the critical temperature the electric current can flow without any resistance and this was discovered in 1911 by ONCE and then the second manifestation is for example the perfect diamagnetic so basically the magnetic field is expelled from a superconducting state and this was discovered in 1933 by Meisner and Oceantfeld so since basically superconductors can carry very high current densities and can generate very high magnetic field they can find numerous applications and currently the largest market for superconductors are MRI machines but for example the large hydron-halider in Europe would have not been possible without using superconducting magnets and also for over a decade now we have magnetic levitating trains the reason why superconductors are not still not widely used is that because they operate at very low temperature so ideally we would like to have superconductors that can function at room temperature working at nitrogen temperature would still be fine since nitrogen is is cheap and abundant but unfortunately as I said the most the widely used commercial superconductor they operate at helium temperature and this is not sustainable in the long run because helium is scarce and also expensive yeah so we are gonna run out of helium at some point so traditionally the the story the the progress in superconductivity is shown by a plot like this of critical temperature versus time so as you can see here until 80s the highest temperature critical temperature that was known was around 20 kelvin and I should mention that all these superconductors are what we call conventional superconductors so the mechanism is due to the its phonon mediated to be due to the interaction between electrons and phonons the lattice vibrations then in the in 1986 that was the first major breakthrough in superconductivity research when it was discovered that the superconductivity in oxides was discovered so very quickly the TC in the oxides exceeded basically 70 kelvin and it was hoped that this is gonna bring a lot of changes in the application of superconductivity research unfortunately these oxides are ceramic materials and they are very hard to be manufactured into wires I'm not gonna go over all this plot but I am just kind of mentioned main breakthrough in in this area so the next I should have also mentioned that these oxides have an unconvent they are called unconventional superconductors and the mechanism of superconductivity in these materials is not phonon mediated yeah it's not really known but it's known that it's not phonon mediated the next breakthrough in the area was in 2001 when a well-known compound in this case magnesium diboride so it was compound that was in a lot of on the shelf of a lot of experimentalists was found to be super to have a TC of 39 kelvin and this is a phonon mediated superconductor yeah so this has been a had a phonon major improvement in compared to the superconductors that were known in the 80s the next significant milestone was the discovery in I believe 2006 of iron-based superconductors these are also unconventional but still this unlike the oxides and finally just recently superconductivity have been discovered in sulphur hydride under pressure and in fact compressed sulphur hydride and this is also a phonon mediated superconductive yeah so this is just a general overview of where the superconductivity research is standing at this this point I guess I should move also on this side so now as I said the next step we are going to talk about the BCS BCS theory of superconductivity and this is the and this is the the first microscopic theory of superconductivity so the BCS relies or the main assumption the main assumption in the BCS theory is that there is some kind of attraction between electrons that can overcome the Cooper the Coulomb interaction and to see this very easily we can look to see this very easily we can look at two electrons in a system so for example if we have a crystal and we have an electron moving in one direction then the electron is going to create a deformation of the latins with a net accumulation of a positive charge as a result of this latins deformation another electron moving in the opposite direction will be attracted in this region of this highest positive charge as the result the two electrons become correlated and they form what is called a cooper pair yeah so I should mention that in the BCS theory the this pairing like the type of the potential that creates this pairing is not really it can be general but in the case of conventional superconductor the pairing is due to the interaction between electrons and this phonon vibration yeah so this picture that I'm showing here is when the pairing is due to the to vibration the coupling between electrons and phonons and so as I said that the two electrons become correlated and there are many such pair in a in a material so since this electron pairs overlap over each other they form what is called a highly collective condensate so at very low temperature this electron the electron can stay paired and they can resist any kicks from the lattice so in other words the electrons can carry current without any energy also yes we will have this the resistivity in the superconductor is going to be zero so we can see also this pair formation we can also present it in a simple I think the this has stopped working okay finally so we can also look at the mechanism at the electron per pair formation in this simple diagrammatic representation so in this case let's say that we have an electron in a state k that emits a photon and as a result of this emission it moves to a new state with momentum k prime the photon will then be absorbed by another electron with momentum minus k and and this electron will move in a new state with momentum minus k prime so effectively what happens is that two electrons scattered on each other by exchanging a photon and since the new since for this basically scattering to occurs we need to have that the state with momentum k prime and minus k prime need to be unoccupied yeah and this can only happen if this state are in the vicinity of the Fermi energy so basically the electrons that form the cooper pairs are within a layer or another word this active electrons are only in a thin shell in the vicinity of the Fermi surface and the thickness of this shell is determined by the character the characteristic phonon frequency of the material so this is basically the of the order of the largest phonon frequency in your system so it's only of the order of milli-electron volts yeah for example magnesium diboride the highest phonon frequency is around 200 milli-electron volts just to to get a rough idea of what energy scale we are talking about so they so I should pause now and say that basically this collective state that is formed in is a key thing to bring in superconductivity so if we don't have this cooper pair formation we are not going to have a superconducting state in a metal and finally only also general teacher okay we can we can compare the density of states in a normal metal and the density of states in a superconductor yeah so in a normal metal or we have that the electronic bands below the Fermi level or electronic bands below the Fermi level are fully filled while all energy bands or energy states above the Fermi level are fully empty on the other hand in the ground state of the in the BCS ground state we have cooper pairs so in this case while single electrons are fermions and they should obey power exclusion principle electrons in cooper pairs behave are boson like particles so in other words they can condensate and they can occupy the same energy level yeah so as a result what happens is that a gap opens in the single particle excitation spectrum of a superconductor yeah and this gap is this two delta and it depends on temperature so we can see that we will see the behavior in the in the next in the next plot and basically this two delta is the binding energy the energy required to break a cooper pair yeah so now again if you want somehow to compare to to the to a semiconductor here this energy gap it's of the order of milli electron volts yeah well well in a semiconductors you talk about band gaps which are order of fraction of electron volts so it's again a different energy scale and you should also notice that here there is a peak in the density of state so this is because we are now moving these electrons from the states that were initially this part or occupied this part of the of the valence band that right now is is empty okay so we can move a bit further and now it will become start becoming more formal with the theory and introducing some equation but basically this superconducting gap can be solved sorry can be obtained if one solve what is called the bcs gap equation so in this this formula we will see that there are a number of indices but it's quite simple so I see ah maybe here even the sound is better for the so in this formula we can try to understand what each term represents so delta sorry it's the superconducting gap and v is the the pairing potential so this is the pairing potential as we said I mentioned earlier when I was describing the feature and then e and k is the quasi particle excitation which is expressed in terms of of energy eigenvalues single single particle energy eigenvalues and the superconducting gap and then you can see that we need this has basically an anisotropic form so you can solve it for any index band n and momentum k and then here you need to sum over all of of forms yeah so will be the q-point so if you solve if you were to solve this equation which I'm not going to go through through the math this time you will get a function like this yeah this is the behavior of delta as a function of temperature and what you you can see is that at t equal to zero the gap has a maximum and the t equal to the critical temperature the gap becomes zero vanishes yeah so once you pass that your temperature goes beyond the critical temperature you are in a normal state so if you were to look at this previous feature you will end up again with a metallic system yeah so your band gap is going will close so just to summarize briefly the bfs theory this is a this theory describes in detail the phenomenology of superconductivity however we should still keep in mind that this is a descriptive theory in other words it's materials independent and in fact it predicts this relationship between the superconducting gap at zero kelvin and the critical critical temperature and most importantly the the method the the model the theory does not account for the retardation of the electron phonon interaction so what does that mean if you if you remember the the previous feature when sorry for going back and forth when if you remember this feature i said that when electrons an electron moves to the crystal the ions are going to also move and they are going to perform a deformation so in other words the time it takes for ions to move from to displace from their equilibrium position to their maximum displacement it's inverse proportional to the to the characteristic phonon frequency so in other words smaller the characteristic phonon frequency of a material larger longer the time that it takes for an ion to reach this displacement yeah so longer the retardation and this this retardation effect is missing in the bcs model yeah so it's considered that the electron phonon interaction is basically instantaneous questions this far so far yes so you can then average over the Fermi surface yeah and you will get a simple gap like this in principle you can solve it like this as well yes you can see an anisotropy in the superconducting gap so we'll see that later also in in the amygdala-diaspora formulas and i'm going to show some results that will show you how the gap looks in when you calculate with all these factors with the anisotropy this will become more clear when i will show a few examples but it depends on the fermi surface yeah so the depends on the form of your fermi surface so right now let's see how tc is in fact calculated in practice you're in nowadays beyond the bcs model and as i said i will briefly go over the macmillan allen dines formula that you have already seen in i think on on on tuesday so in this in this approach we basically can evaluate or we can predict the the critical temperature if i stay on this side somehow i can realize it i speak louder i think it's the microphone it's on the side sorry i will have to point to this to this slide so in order to calculate to predict the superconducting temperature we can do it once we know the electron phonon coupling strength yeah and this we can see we saw that we can calculate ab initio and the other parameter that enters in this formula is what we call the semi empirical coulomb pseudo potential and this omega log is just an average phonon frequency so many in fact the majority of ab initio calculation that exists in the literature rely on this formula and the reason is that well it's quite i mean all you need is basically a value for for lambda so it's easy to calculate and it's been around for many years in quantum expression and i believe probably since you can estimate lambda this can be done also with ab initio and this formula works reasonably well for isotropic superconductors and the reason why it works for isotropic superconductors is because it's been determined or it's been derived based on the isotropic parametrization from results for isotropic migdal eliasberg equations however you should keep in mind that in order to go to get a converged value for lambda and as hopefully you have realized in this school is that you require very dense k and q meshes another thing where this critical temperature or sorry this formula fails is that the critical temperature that we predict for an isotropic or multi-gap superconductors will be wrong yeah so it doesn't capture any anisotropy also you only predict the tc you don't really have any information about the superconducting gap and then it also approximates the coulomb interaction to a semi empirical parameter yeah so this is usually chosen between point one and point two for most materials now we can move on and finally start talking about the migdal eliasberg theory and this theory was developed in 1960 by eliasberg that generalized basically the bcs theory of superconductivity by introducing the time dependent electron phonon interaction that was developed by migdal for the normal state so this formula for the pairing self-energy which is the central part where we start with the in the migdal eliasberg theory should have all should already be familiar since this has been shown in lecture a fellow channel's lecture on on wednesday yeah so this is nothing else but the fun migdal self-energy where the vertex gamma vertex was eclectic yeah so this is the d is the dress phonon propagator g are the electron phonon matrix elements that we can we can calculate and this capital g this operator this is the interacting greens function while this term here describes the electron-electron interaction and basically it can be in principle calculated uh is the gw self-energy and can be obtained from gw calculation yeah and the fact that we are neglecting we are only keeping the first order terms in the electron phonon self-energy if i my diagram for electron for uh self-energy in superconductivity is called the migdal's theorem it's referred to as the migdal's theorem and this based on the observation that neglecting terms of the order of the square root of the electron mass to the iron mass and this can be shown that is proportional to the characteristic phonon frequencies this will be the divide frequency divided by the fermi energy so we talk about the ratio of mev to electron volts yeah now you need to keep in mind if electron if the phonon energy is of the order of the fermi energy this approximation may break down because this is based on the adiabatic approximation as far as i know there are no studies that have investigated uh at which point the migdal's theorem is uh uh breaks breaks down so uh now we can the next slide i don't remember four or five slides i are are quite intense we met so if you don't get them please don't get desperate it's if you later on you can go through them and you can work out and you are you are going to realize that the formalism is not that that complicated yeah so this is the sorry this is the again the pairing self-energy for the electron and what we can do is the first step we can replace the uh the phonon propagator uh or we can rewrite the phonon propagator in its spectral representation yeah and that's already again if you go uh yesterday's lecture by by feliciano uh he has already given this expression for the phonon uh dress phonon propagator if i remember correctly and uh and uh and then uh another trick that we can do here we can now introduce a new quantities that we uh that couples g with the dress phonon propagator and this we called the anisotropic electron phonon coupling strength yeah so this we can see that it has the electron phonon matrix element and um and uh compared the uh we can see later on that this is basically uh the the same electron phonon coupling strength that we have seen before only that now we keep the full anisotropy in in in the expression yeah so it depends on the initial state uh nk and the final state mk plus q so if these two equations are plugged in the uh so uh if we now use lambda in the self-energy expression we can rewrite it in a bit more compact form where lambda can be easily calculated once we know the electron phonon matrix elements and as I said uh the Coulomb interaction can be in principle calculated through a gw approach so what we are left right now in order to uh to have to estimate the pairing self-energy we need to find this uh non-interacting uh green function uh uh green function yeah um so in the migdal or in the superconductivity uh state uh this is um uh i would say it's formally done or it's it's more easily done if we use what is called the uh nambu gorko's uh formalism so uh um in within formalism we can describe the propagation of the electron quasi particle and of superconducting cooper pairs using a generalized green function yeah so this is the generalized green function and again this looks quite familiar or it you have uh to what we've seen on wednesday yeah so this tau it's again it's an imaginary time this t tau is the vix time ordering uh time ordering operator and the only difference while uh on wednesday this was just a single operator now is a two component field operator so it has basically two components one component destroys uh an electron at the state uh nk and spin up and the other electrons uh sorry the other operator creates an electron at the state uh n uh minus k and spin down i i forgot to mention but uh in the bcs model it can also be shown that the cooper pairs are formed not only between electrons with opposite momentum but also with electrons with opposite spins yeah so this will become uh basically this feature can be reduced to the to the bcs uh bcs model so now if we replace if we plug in this field operator in this uh in this expression you can see that we have uh a column vector multiplied by a line vector yeah so this is just the complex conjugate so what you are going to get is a two by two matrix uh element yeah and now let's pause for a moment and try to let's try to understand the the elements of this matrix so now if we look on the diagonal what you can see is that you can you create and destroy or here you destroy and create uh electrons but what is important they are on the same state yeah so this uh should tell you that we are now talking about normal state green functions and this basically they describe the single particle uh electronic expectation that we will have in a normal uh metal now if you look at the non of diagonal elements you can see here that you this you either destroy or you either either create particles but these are in uh state different states yeah so one state is with momentum k the other is with momentum minus k and then the these are spin up and spin down and as i said this is basically what it what we have when we form a cooper pair so this of diagonal elements are called anomalous green functions and describe cooper pairs amplitude so basically if you if these uh of diagonal elements are zero as i said you will just have the behavior of a of a normal a normal metal and uh when these elements so and these elements are basically non zero only in the super constructive uh superconducting uh state now since uh this operator this generalized operator in the imaginary time is periodic in tau it can be expanded in a Fourier series and mathematically this is convenient because now we need to calculate this uh this matrix elements what we call on the matubara frequencies that frequency axis yeah so uh these are just discrete values they are integer of j and they depend on temperature yeah so this is how temperature effects are introduced through uh through uh through uh through green's function and we can see here that larger the temperature farther apart this uh imaginary frequency points will be on the are on the imaginary frequency axis and in fact this can cause issues with convergence when you start doing numerical uh calculations when you have fewer and fewer uh points in this sum because at some point you need to drink k b some and fewer points you have harder will be for the algorithm to converge so uh what we are left with right now as I said we now need to estimate to to to to find this generalized green function uh in terms of this matubara frequency and this can be done by evaluated by solving the Dyson equation for electrons uh so in in this expression g not minus one this is the non-interacting green's function this is our electron self-energy and this is the our interacting green's function and now it becomes uh a bit uh another formalism what we do we can okay we will do two steps first we are writing the uh non-interacting green's function and uh this only uh has basically a diagonal form you can see these are powerly uh matrices um and then we can also express uh the self uh pairing self-energy in terms of a linear combination of uh powerly matrices as a function of three scalar variables and this the the name of these variables has kind of a historical reason but for example if we just ignore this term in the beginning we can see that this term is diagonal this is also diagonal and uh this comes from the normal state and now this z this pass renormalization is nothing it can be shown if you do a bit of math that it's related to the master renormalization that we saw in the normal state so this is the inverse of one minus derivative of the real part of the self-energy with respect to the frequency yeah so and the second term in this is basically the energy shift so this is just the real part the eigenvalue plus the real part of the of the self-energy again what we've seen for the normal state and this second term is the superconducting uh the part that is related to the superconductivity and this we can see that is non uh this is off diagonal yeah so formally now we just introduce this expression for the pairing self-energy and we in uh we are replacing them in the expression for the non-interacting rings function so remember that our goal was to find an expression for g that we basically we can plug back in the self-energy so we will go we go kind of around the problem yeah we will go back and forth more or less so once if we introduce this uh to expression in in the g minus and you do a inversion now it's this is a matrix I remember that everything is basically a two by two matrix so you can do a matrix inversion and what you find in this case what you find in this case is this g where I just that these denominators in otherwise the expression would have looked they've been too long I just wrote it separately so now we have what we are looking for yeah so we have our expression for g so we can go back to our expression for the self-energy we plug it in and now what we are left with you can see that g uh is still a linear combination in terms of the power matrices here we have uh uh also power matrices so we can do we just simply do uh multiplication of uh three term uh two by two matrices and the first term to tau three times tau naught times tau three gives us tau naught again so here we are going to have for this first term it's going to be sorry this first term is going to be tau uh sorry tau see here it's going to be tau naught here it's going to remain uh I don't know here sorry this was was tau naught this is going to be tau three and this if we do tau three times tau one tau star three this is again minus tau one yeah so if you do this matrix multiplication what you get is this final expression for the self uh self-energy so uh but basically what we are looking for is this superconducting get function so we so if we just stop here we still don't really know uh sigma we just have a formal expression for for sigma so we need to still use our uh other expression for sigma where we just wrote it as a linear combination and what we are we can do right now we can equate the scalar coefficient coefficients of the two expression for sigma and this finally leads to our migdal anisotropic migdal eliasberg equation yeah so basically what we are we have three equation and you can see that they are two of them are coupled and they are also if you were to solve this you will need to do it self consistently because the terms z chi and delta they appear both on the left and on the right side yeah so so this is quite complicated and in fact if you were to solve that this three equation you also need to supplement the equation with a with another equation that I have not written down but this is for the total number of electrons in order to to evaluate to find the the Fermi energy so in this expression I think there is one term that I didn't mention this nf this is the density of states at the at the Fermi level yeah but luckily we right now we can do some some approximation so we are not really going to solve these three equations so um and there are a few standard standard approximation that one can do and the first approximation is that only of diagonal contribution to the Coulomb self-energy will be retained in order to avoid double counting of the Coulomb effects and we are also going to work in the static screening approximation so what this means is that the Coulomb contribution to the self-energy is only given by the tau one component of g which by definition is already of diagonal so it means that in this expression so if this was the expression for z that I had on the previous slide basically the contribution the Coulomb contribution in the expression for the mass renormalization function vanishes yeah so this is one simplification second we are only working if you remember in the first picture when I described the bcs theory I said that only uh we are only interested in the electrons in the vicinity of the Fermi surface so in other words all quantities are going to be evaluated around the Fermi surface so if you if you do a bit of math you can show that chi the expression for chi vanishes when you do an integral when integrated of the Fermi surface because this is an odd function yeah so you are going to have the pulse you will see that this this function is going to be equal to zero so what this means it means that out of three equations we are now down only to two equations and another assumptions that we are going to do is that the electron density of states is assumed to be constant again this may not be the case and there are not too many studies there are no studies that shows how this really affects the final result but it's it's quite an important approximation and finally what we or at least the current implementation in the epw is is is based on the on the following approximation that the dynamically screened interaction it will be embedded in this semi empirical so the potential new start however you can still calculate as I will show in a you know it was the end of the talk you in principle we can still calculate this new star outside dpw abinish but it is provided in the in the equation through to a parameter so to summarize what we are left this will be our two anisotropic gap equations sorry anisotropic mcdalelyard equations and that we are left to it and this is what is implemented in inside the code so z as I said this is the mass renormalization function and d and k is the superconducting gap the same as the is in the bcs model and again we can see that the two equations are coupled and they and if you want to solve and to you to get the superconducting gap or to solve them the the key parameter that we have right now or the key quantity that enters right now is this anisotropic electron phonon coupling strength that is basically is estimated based on the electron phonon matrix element and you can formally also write this lambda anisotropic lambda in terms of anisotropic elias spectral function this is just to make a bit the connection with what you've saw when you use the make me an expression when you estimated lambda from the from elias spectral function yeah so to get to effectively get the superconducting gap you will need to solve these equations at different temperatures so you don't solve it just once you will need to to solve it for different t yeah so this can it can be quite time consuming since they will need to be solved every time for every temperature that you choose you will need to solve these two equations self-consistency. Second now you notice that we have this anisotropy and this is the key point so the equation will have to be solved on dense electron meshes but at the same time you also will need to do on dense q meshes to properly describe anisotropic effects. Third you also have this sum over the Matsubara frequency and in in principle you should go to infinity but in practice we can do that so we'll need to be to truncate this somewhere and typically we set this frequency up to the point where let's say omega j prime is about this is a convergence criteria that we need to check but it's about four to ten times the largest quantum frequency in your system and I think I have one more point is that then you also need to pay attention that this z and delta are only meaningful for states near the Fermi surface yeah so if you are to calculate this for in fact you can calculate them for energy states far away but you can see that you have a delta function here yeah so once you go away from the Fermi surface the delta function is zero so you can take advantage of this and only use a Fermi window when you evaluate this this equation around near near the Fermi surface and I will give in the practical in the technical lecture I will give a bit more details about how this is done in practice. So to simplify this yeah sorry so this is the form the final expression for the anisotropic case but in principle you can also right now perform a second average on the Fermi surface and you can get what is called the isotropic at Milik-Dalleli-Aschberg equation yeah so these are the same but now you see that there is no dependence on the k or q and this is directly connected with the expression that Macmillan expression because right now this lambda is basically your electron phonon coupling state so in the Macmillan expression you just have this mu j equals zero and and now you can record you should be able to recognize the electron phonon coupling state and the same here this is just the isotropic Eliashberg spectral function yeah so as I said in the beginning that Macmillan tc and then later refined by Elandine was found as a solution or basically fitting through data obtained for by solving the isotropic equation for some specific materials yeah so it's just an empirical fit through through data obtained by solving these equations. So I think I am almost approaching the end with this part of the topic about the theory as I just to say a few things about the Coulomb interaction so as I showed in this expression I said that the if you look even here we have replaced the Coulomb interaction by this parameter new star but in principle this can be calculated ab initio because the screen Coulomb interaction can be calculated within the random phase approximation so if we do that once we have this matrix element we can estimate we can do a double average over the Fermi surface and we can calculate mu and then the mu star can be defined in the model moral Anderson model and it has this expression yeah so here omega L this is a characteristic electronic plasmon frequency so this is of the order on electron volts and this is again a characteristic phonon frequency this is the order of milli-electron volts yeah and we have done a couple of studies in which we have estimated this value using this this approach and then we have used basically an ab initio estimated the first principle estimated semi empirical Coulomb so the potential that then that we then plugged in the migda-lelly-yashberg equations so as a summary for for the theory can say that we can say that migda-lelly-yashberg theories has predictive power so it's material dependent yeah I should have put an accent but I forgot that it that is in fact I mentioned the beginning to the migdal approximation it accounts for the retardation of the electron phonon interaction if you solve the equation in the anisotropic form you can use it for anisotropic superconductors in other words for multiband superconductors generally it just uses approximates is the Coulomb interaction as mu star in principle the theory can be worked out to have this V calculated inside the migda-lelly-yashberg equation but it's not currently done and it will require when you want to solve it in the anisotropic case it will require this k and q matches in order to to reach to reach convergence so I will also just briefly describe another method that can be used to calculate to solve the superconducting gap from first principle and this is called the density functional theory of superconductivity and it has been developed in Hardis Gross group so the central part I would say of the SCFDFT is a superconducting gap equation and this formally if you go back at the beginning of my talk this looks like very close to the BCS gap equation so you can see that here we have just an extra term that did not appear in the BCS gap but this delta is again the superconducting gap function z is something similar to this mass renormalization function that we have in the migda-lelly-yashberg theory and it accounts for electron phonon interaction while this kernel k accounts for both electron phonon and electron electron interaction so so this can be solved also self consistently but if you can see that in this case we only have one equation that needs to be solved so not two equations and as in the migda-lelly-yashberg formulas so the method is again is an ab initio method so it has predictive power it's material independent the way that it accounts for the retardation effect is basically through exchange correlation functionals since again we have an isotropy if you can solve it if you solve it in an isotropic form it works for multi-band superconductors the nice thing about this method this is that it treats it really treats everything both the electron phonon interaction and the electron electron interaction are equal footing yeah so it has no parametrization as as I show for the implementation of the an isotropic equation in epw however it requires development of new functional to describe the electron phonon interaction so if you take current functional I guess and if you use them I suppose that they are not working they want to work and again it will require dense k and q matches yeah so I think I will I can take a break at this point and then we will see a few examples for calculation in the second part and then we will go with the more practical implementation of these equations in the in the code so questions