 Prej, ne. Prej, sem zelo, da ne biš tudi pošeljena za moment. Ne, če sem... Se pa da se pošeljena. Tak. Svolj ne pravi. Tukaj, da sem se pošeljena. Tukaj. To se pošeljena. The next speaker for today is Giovanni Pizzi from Laboratory of Tourismulation Materials at the PFL of Switzerland. And National Centre for Computational Design and Discovery of Noble Materials. He will give us a lecture and gets a short hands on on thermoelectric and electronic transport properties with many a 90 in bolstven. So Giovanni is actually also director of the workshops, he will join us in presence se jaz se na delarjare, dovolj se, da se predajamo na vložo. Poz budu. Poz jaz sem tudi, da sem ne vsakčen in ne bo o tega, je se vzelo, da se pomečim, da se vzelo, da se časem, da se početi, da se početi, da se pomečim, tudi zčati, da se vse početi, da se početi, da se početi, da se početi, da se početi, oči najbolj obosno generična interpolatija proportovačne in operatore v zonu, vzvega obašne stručje. In, zelo, da se glasba bila umrežovati pri termalitričnih proportov, nekaj nekaj je razgled, če je tudi termalitrično zelo zelo zelo zelo zelo zelo zelo. Selo ki je tudi nekaj je tebačno zelo zelo zelo, načo zelo začal, nekaj nekaj nekaj je to zelo zelo zelo zelo zelo, kroz, nekaj pomečite je, na začnek začine je. Pa najšlišom energije nekaj ispe in da si je brda zteroma v tezvosnjen, kaj je objživot, da najšliš jazv, je nekaj pa življan več. Vz affecting in električnih. Rav nračte je, da imaš nekaj radži odlast. Pozipaj na zelo, da ne čekajjš, nekaj všeč jazva nekaj nekaj, prej vrv v��k tezvosnih, njewo je svar nr! elektricična in energi. And you can work out the equations that define essentially the efficiency of such a material. And it turns out the efficiency is related, it's not proportional. It's the function of this z t quantity you see here, which is a linear... it's a combination, it's a form rule, which depends on an electric conductivity of the material, na zelo, na temperativnih, na zelo, da je vse težko, kaj je težko, vse težko, da je vse težko, in na elektronikovih, vse termokondaktivih. Vse termokondaktivih je vse materijalne. In nekaj ne zelo, kaj je materijalno, in nekaj je zelo, da je vse materijalne, boj, da se počutimo, nekaj je zelo, da je vse materijalne,pping, jo sem preti iz SmackDown putra. O咸de zdaj ne nataj jednak. Well, the problem. The problem, there are very high models in they actually stand out for me, but this point when we look at your planets. But these are no matter h slaughter hupventaj. The correlation is, that the io is just very unilateral, very sense of stability when I was in원이. I carried a specific impression, vse je taj odličen krokem. Oi, ga je to prišlega o sacredi, tačan, tako prišlega, nekaj je v vsem kontekstu. A zato sem ideja ekipnije. Elite v sovami kako smo ko gleda, se neseljili ones Prišlih, tako pa je bil je kresi. Silo večjama radi neselj, kjer bi sem prišli, cocoa iz pomečjovutama,เอnikovu, noperativnu, in bil je to prevši vse. Dakle, to je vse o prevši. In zato. Mjelaj na razrednji je, da vse dobro se sklepno imelajo materijali. Vse nekaj imaš materijali, z vse zašaj na database, začelimo, da je počkodnje. Na razrednji se počkodnje, si si pričeš, da se zvok je materijali, in si kaj smo potrebat materijali, nekaj ne se počkodnje. Na razrednji se to pričkodnje, je to začkodnje otvar. Tak je pričkodnje je na komputerne, ki bomo počkodnje, izvršal bi bilo, sklepno, elektrika kondaktivitaja, ko zač citu to izrednji. I v Ed SE z tudi bolo da je joškosne. Kaj je režite, kuji je, da je jeva? Tako prišlo, da sem bozda se z tudi pozorni neko so tez, da je na OKG po njal večضog nekaj zelo. Vse, da so nekaj zelo, da bojte v tudi pozorni. A SCD, da je, da se zelo, da je, da je bilo, kaj je, da je. Ale malo pri tako, da je AutoS, a nekaj se je, da je, da je zaprave. Ogledaj, ki vidiš, bo zač tudi, da to zač, pa nekaj pasač jaz kaj je prav, kot prijena, ki se rešelje tega in da mi lep nablok na dan, na kako bi je odličovo unijene na tj Tea. If you get something where you really manage to to send this quantity to a very low or zero, that's great, then what we compute is kind of the best we can get, but in real matter, it wouldn't be worse. To niste, ki se stresijo, KL is actually very important in applications it will be computing mostly this part because this requires computing electron form. It's scattering and we see this query in the afternoon. Zato je zelo vanje spole, halj smo predmjanje poslenili, na kontekstu vanje functionalne. Zatim je, razložijo, že v zelo potrede, da vših svoje tudi izgledaj, je začala, bako se čutite, posledaj, štak bienicu, je kapital sigma letr, ki je tensor, deprej na i in j, so two bands, and it depends on the energy. It's kind of a density of state. You see here you have a delta over the energy, which typically then you smir out with some motion. So it's very similar to density of states. If you want, actually, if you remove this part, it would be exactly the density of states, but it's actually a weighted density of states, which contains the velocities and scattering time. In particular, this tau is what's called a relaxation time. In general, it's a function of the band in the k-point. It tells you a typical time in which an electron in a given band would be scattered by an electron phonoskeptering or anything else into something else, something in picosecond in seconds. Then the other important ingredient is, and this will be one of the reasons why we use many functions, is the band velocity. And the band velocity is relatively simple. It's just essentially the derivative of your band structure in k-space. So this will be your k-axis. This is your energy. This will be your bands. If you take a point and you depart the derivatives, apart from a factor h-bar, this gives you something which is the units of a velocity. And in a quasi-classical approximation, you can really think to it as the velocity an electron has if it is exactly the specific point in the band. And the other important thing to remember, as I mentioned, this is numerically very close to computing in the state. It's sum over all bands of delta functions over the energy. And so you really need a very high and dense match of k-points in order to get something smooth. As you see here, I have a little animation where you start from something where you have a lot, you have not enough k-points. You get something very noisy. And really you have to go to maybe 100 by 100 by 100 k-points, really have something smooth. So all of this makes it very exciting and interesting to use band functions for two reasons. As I'm going to mention in a moment, on one side you can compute these velocities analytically. Analytically means that typically if you have a band structure, you can compute the differences, but in a finite difference approach. You would take the band here, the band in a closed by point, to a finite difference with a finite velocity, which means that a, you have to put multiple points, but also b, you have a crossing making mistakes. You can maybe get a velocity, which is wrong, because you are taking this band, and this band, in reality, you want to follow this band. Analytically means you really give, when in 90, a k-point. And it will not only tell you, OK, these are the energies, but in the same calculation, it will also tell you, these are the velocities in this point. We will tell you to look at the neighboring points. And the other thing, as you will now, by now know, is a deficient interpolation that a function would give you. So you can really do, I don't know, an 8 by 8 by 8 k-point mesh in reciprocal space with DFT. Then once you have a converged vanille function basis that you have a real space of a Newtonian, your interpolated band structure can be done very fast in a very, very dense mesh. Putting these things together, this makes them exactly very attractive to use this, to compute this point. This is in a very converged way. Of course, one caveat is that you need to first find the vanille function. So this is something else to think about. Count, and very often this, as you saw before, requires human time, requires to understand what the vanille functions are. Sorry. Even if you saw that especially on Wednesday, there are nowadays a number of methods to make this process as automated as possible. I want to briefly give you an idea of theory, how we get to these formulas, how we get to this transport distribution function. You will see then, after you understand this, the actual code, the actual simulation of to run is very, very simple. It's essentially a standard organization with one more run with three, five lines. It's very simple. So the tutorial will be almost me showing you and you will see it's not complicated, but I want to tell you a bit more about theory. So you start from a function which gives you the distribution in phase space of your electron. We'll tell you what's probability of finding an electron with a given position and a given k vector, essentially velocity in a sense, at a given time. So this quantity is the number of electrons at a given time in a little r decay in phase space. And as soon as you put some perturbing effect, this distribution is not anymore the equilibrium distribution, which we call f of zero. Like a term of distribution of electrons. Because if you put an electric field, for instance, you might shift electrons on one side of the sample. If you put a local heat, this local heat will increase some population on one side of the sample and reduce it. So we need to remember that there are these two quantities. We call f zero in the equilibrium one. I remove all fields, I remove everything in the equilibrium one and then a local perturbation of it. The other thing to remember is that in a theorem, you will use theorem, which tells you that essentially if there are no collisions, you have in total the volume in phase space will be conserved. Your particle will move, so it will have a given position and a given velocity. After sometime it will go away. You change the velocity, so you change the position and this will move out. But the volume which is occupied will be conserved in time. However, if you have collisions, you have a contribution which changes the actual number of electrons in space. And you can imagine, you have a particle with a given velocity in position, it will move. At some point it hits something, it will jump. The position will stay the same, but the velocity will jump to something else, it will revert the velocity and go away. So really if you look at what happens in a given position in phase space, at some point it's some particle which essentially disappears because it goes somewhere else. And it's given by a rate of collisions. More often this collision can happen, the more the number of electrons on average will change. And you can write out explicitly this derivative as a partial derivatives. Okay, you have the actual change of f over time plus contribution which is the fact that you have a velocity, so particles will move away, plus a fact you have some forces and so this force will change the velocity of your particles. And so what you're saying is that the rate you have would be related to a number of components which depend on the velocity and the forces in your system. And this is what is called the Boltzmann transport equation which would be the basic equation we will use now to get a numerical expression we can use to compute transport coefficients. So let me write it again. That's a full Boltzmann transport equation but it's very generic. Now we don't know yet we didn't say what the forces are on the system, we didn't say anything about the f and we didn't say anything about how collisions work in the material. So now we start doing some approximations to try to get an equation which you can describe our problem and we can compute. The first thing is we will assume that any perturbation you put temperature gradient or electric field in the system will be small. So it will just perturb your system and put an electric field that your electron will have a velocity try to drift in one direction you put a heat difference in the two ends of a bar they will have a slightly different equilibrium thermodynamic distribution but this would be always small. And if they are small f is close to zero the equilibrium one and so the derivative can be essentially approximated with essentially the derivative be minus the difference in distribution over a time. And this time we will even stronger approximation which in general is not very justified but it works fine in most cases where we say this time doesn't depend on anything except there is a constant essentially. Whatever you are in space whatever you are in the bands is a time which is only on average if you have an electron how quickly will be discarded and actually what you to interpret this time you can say OK let's take a system I remove any forces I remove any electric field I remove any gradient of f essentially I remove any temperature gradient I just take a constant temperature bar no fields what happens? It happens that the solution of the equation now becomes the distribution of electrons f is nothing else that the equilibrium distribution whatever we get if I leave it there forever plus if I have a change I make a little perturbation I heat locally something this will go down the equilibrium distribution with an exponential and the typical time is tau imagine you take a bar you locally heat the sample you remove the heat this will the distribution of electrons we go back to the thermodynamic f0 with this tip characteristic time constant means it doesn't depend on space it doesn't depend on the band structure in reality this will see this afternoon this is actually a function of k so you will want to consider it in this today for what we discussed now we assume as a constant and we see it is relatively easy once you have a way to compute this tau with electron for non scattering for instance or any other scattering you have in the system to put it into other equations so now we have the full equation we already said how we can write this kind of very generic term let's do a few more approximations first of all we assume the material is homogeneous you don't have differ material so all the gradients are zero over space and you assume the reason electric field but it's steady it's not an oscillating field over time so the explicit partial derivative of f over t is zero we have a field we create something but the partial derivative is zero and of course we decide explicitly to put an electric field so the force is just an electric field times the charge a constant one because in the end it's what we have we want to see if by having a heat difference we create a constant field in the sample we write the current now it's a standard equation to write the current as the sum over all electrons you have it's a normalization times their velocity times their distribution it's nothing really fancy it's just a current is a number of particles times charge, times velocity and we know that the current is connected to actually as we found is a definition of the conductivity of a material the conductivity sigma is nothing else as the tensor connecting the current with the field and once you have this you just work it out you put all the pieces together one with explicitly here but essentially you get an expression by just replacing things which gives you an expression of the conductivity tensor sigma ij as a function of a number of things will be a function of some constants the constant for transition time being constant without the sum if you were dependent on the band it would have been here a sum over all bands of the velocity of the bands and a derivative of the distribution function in general the derivative of the fermic direct distribution function because we are speaking of electrons and the final step is to rewrite this expression in terms of what I defined before the transport distribution function and formally it's trivial in the sense I'm just taking this part and multiply by a delta over the energy and integrate over all energies so it's very simple if I replace this in here integral over delta will go away we're just giving back the same formula but it's very convenient because what I will do this is I will compute this quantity not anymore as a function of k-points but as a function of energy of a single variable like a density of states and then I can just integrate this extended density of states with some shape with some function and I get my transport coefficient and as you will see we see that actually in the next slide once you compute it once you can easily compute all other points you can use instead of using as before j equal to sigma e you can start imagining you also have a gradient in temperature you can write a more general expression for the current and for the charge to the heat current this is essentially the definition of sigma the definition of the silver coefficient the definition of the thermal conductivity you put them in you use the definition of sigma and you see that also not only the definition of sigma but also the definition of s or actually sigma s and the definition of k thermal conductivity are a very simple expression actually the zero, one and second momentum of the sigma with some prefactor so practically speaking as long as I am able to compute this compute the velocities multiply by some constant time do a dense interpolation I get this function as a function of energy and then it's just a matter of doing one-dimensional integrals so something everybody can do essentially numerically to get all these properties but there is one important thing as I said, the ingredients are the bound energies because you need it in order to do this delta and this we know how to compute this is a bound structural interpolation of any functions we need the bound velocities as I said, this is something we can actually do analytically in the next slide it should be balanced but also there is this factor which we didn't discuss explicitly yet but as I said, f is your fermitira which will look like this as a function of energy goes from one to zero with some spread which is essentially kT the temperature at which you are if you do a derivative you can imagine it will be at zero, zero, zero and here is zero and only in here here is flat so the derivative is zero essentially the derivative is not a Gaussian but looks like a Gaussian, let's say is something which is very peaked around the chemical potential of a width of kT the width of the temperature at which you are which means that in reality this integral is doing nothing else the sampling sigma the transport distribution function or sigma times some moments only very close to the fermi energy so what actually matters it doesn't matter what you have very high energy or very low energy so it probably matters what you have kT range around your chemical potential ok, let's go quick so now in y-manifunctions this will go very quickly in one slide just to remember what you already saw this week so you have a way to convert your block states into some unitary transformation and some iterative procedure to get the unitary matrix into local-manifunctions and then you get something which is localized in space and probably this is already so when you have the abinitiovan structure this is a material which will be starting because it's an interesting thermometric material it would compute your band structure with DFT only on certain points ok, only a 4x4x4 3 this will be your green circles here if you don't have any functions it's very hard to know what is, if the band is going like this or is going flat and this of course changes a lot when you do velocities if you just want to go to the density of states maybe it's ok but as soon as you need the velocities you know if the band starts like this your velocity will be negative instead of positive which completely changes the behavior but if you use many functions you get a very fast and efficient interpolation which actually you can check with DFT so this is the black one abinitiovan functions interpolated constructor and the red is very expensive DFT and SCF calculations and you see this reproduzor exactly your constructor so you have a you reproduzor anti-crossing reproduzor anti-crossing and so on and this is very efficient actually if you do in this material the math you have to take a big node to do the k-point in 30 seconds but if you do any functions every k-point would take tens of milliseconds so it's really orders of magnitude faster once you have any functions and as I said it's actually a paper written by Jonathan, David and Ivo and coworkers you can compute analytically the derivatives and the reason is that if you have your many functions what you get in the procedure is essentially these matrix elements which are the Hamiltonian matrix elements the hopping between two many functions into different sites and to get an interpolated bond structure what you do is essentially to take these hopings do a fully transform essentially it's a fully interpolation and you get the Hamiltonian k-point then you analyze it and you get your balance but in reality you can interpolate essentially any operator and in particular the fully transform on the derivative is multiplied by r in reciprocal space so if you actually do the free interpolation not of the hopping but the hopping times i, r and you are clever in the way you analyze this you could be a U matrix what you actually get is exactly the derivative so in this sense this is what it means making analytical ordinary analytical velocities it means that you don't have to do finite differences but once you have these matrix elements which you have in Vanja not only you can interpolate the bands regularize and get the energies but you can also multiply by r do it again and what you get directly at the k-point are the velocities for each band this is implemented in Vanja 90 actually you can get this just the velocities of a module which is called gen interp for general interpolator is part of positive 90 but what we will do today is to use it as part of of the both van kode which is also a sub module of positive 90 to obtain directly the transport distribution function and transport coefficients very briefly I wanted to show that the theory is over I wanted to show a bit an example of two slightly complex but scientifically relevant cases which actually showed in this paper where we present both the code and these examples just to give a sense of what we can do and then we go into the tutorial where we do only silicon something very simple because so the point of the tutorial we just want to understand how it works but anyway can allow us to think a bit and ask questions and try to understand better so one very interesting material where no material is this covalent tridentimonite is a material which at DFT has a number of valence bands and conduction bands is a very tiny gap at least in the FDA here is a structure and some information on the structure and essentially here the bottom one are very down as you remember we look at something very close to fermi energy so we really care about states here we don't care about states very down because this derivative of the fermi dirac will send a contribution to zero so we will exclude this in the calculation for many functions there is a s orbital on the entimonite and this is the variabilization we can do we use this frozen window we use this outer window and essentially we use p-like when a function on SB d-like on a function on covalent and this is the kind of interpolation we get which is quite good because we get very well the odd valence bands and very well also the bottom part of the conduction band which is more than enough because this is like two electron volts and room temperature we care on about a few kT which is a few hundreds of electron volts so this is more than enough even if these bands are accurate we don't care and these are bit the numbers which we use the way you see essentially we use a 6x6x6 for each other density only a 4x4x4 nScf to develop the functions but then we can quickly interpolate with 1N on a 40x40x40 kMesh to do the transport distribution functions and these are the kind of plots you produce here you see the sigma the electrical conductivity the severe coefficient and the electrical thermal conductivity as a function of the chemical potential where if you look at it again I didn't put it at zero sorry but the gap is here on 8.0 so the gap is here and so this will allow you also to investigate it not only what happens in the neutral material where essentially you have zero conductivity because you have a semiconductor so unless you include another temperature you have all the valence bands filled all the conduction bands empty so we have a very low conductivity but if you start to dope the material you move the thermi level into the conduction band which is an end-oping into the valence band which is a p-doping so increase the conductivity and you have interesting effects also in the function of the chemical potential at a given temperature on s and thermal conductivity and you see here that curves at a function of the temperature and what we did in this paper was a comparison of our results with a reference code which is called Boltz-Trap which is well known here you see a reference and it does some tricks to the derivatives numerically on the band structure while using valent functions so it doesn't require valent functions but of course requires a very expensive n-s-c-f vibration or not a relative dense free so this was mostly for the paper but just to show it our code works fine and it's comparable to other codes just as curiously there's some different material it's very similar but it's something cobald, antimonad, cobald, germanium and s and so forth and again a check to show it the two codes give you similar results one final thing I want to show is that here you see things as a function of the chemical potential which is something that theoretically is easy to compute is one of your variables if you remember in equations here we have the chemical potential as a variable but in experiments you don't know the chemical potential you know the doping of your material and so what you can do you can get the density of states you can convert a doping level in centimeters to the minus 3 into the position the chemical potential will have at a temperature that's something that can be easier to integrate revert the integral of the density of states and then you can convert you can make a plot where instead of having an x axis the chemical potential you have something experimentally measurable which is the doping of the material so essentially that's the kind of plot you can get is a functional temperature you can fix a doping level based on the experiment sample and then you can give an expert even a doping and a temperature you can get what is the chemical potential go back in the plot before and show it and what you see here the lines are what we get through this approach from the Boltzmann calculation in a very simple constant black section approximation the dots are data from experiments and you see it in the end is not perfect again we are using a very naive approximation concentration of time so a high temperature doesn't work very well but in the end is not bad and I can really get a bit qualitative behavior of this curves and also quantitatively is not completely wrong and one thing to remember is that since we are looking only at kT close to the chemical potential it doesn't matter so much in the end if the LDA gap is incorrect as long as you open it a bit you would typically be looking if you have a high end doping with mostly be looking only the conduction band only the valence band so in the end you can essentially imagine to assistor correction only look at those the error on the gap is also important what will matter actually will matter only here high temperature where you stop having transitions directly with the valence and the conduction but what will matter is the shape of the valence of the band structure and of course if you have advanced material of GW metals or advanced metals to get better band structure it will be more accurate with the same approach so let me summarize a bit this theory and then we just spend a few minutes to introduce it to the tutorial Vary90 has a model which is called Bolzvan which is part of POSTAG90 which allows you to compute electronic and thermal conductivity see with coefficient in the constant relaxation time approximation actually it was I didn't mention also for spring chloride systems and even if it's a constant relaxation time the code, the time only appears in one place so if you have a model for your time as a function of the band structure it's easy to adapt it and as I showed before the code works well and can be cross verified against another code and advantage of using so the Bolzvan doesn't use variable functions so the idea was to implement the same theory in a vaniya-based approach and the advantage is that once you have the vaniya functions interpolation is super fast band velocity is accurate analytically and since it's implemented as a post-processing of vaniya 90 essentially it works like transparently essentially any DFT code which is interface with vaniya 90 you don't need to have one implementation for each DFT code you just implemented it for vaniya 90 as long as you can get the vaniya functions in any folder you can then compute the structure of the functions now this concludes the theory let me briefly have a few slides to tell you how it works in practice and my goal is not to not to give you a to show how it works because you will see it's really really straightforward you have the PDF so you can go through it in the next 25 minutes you need 5 more minutes but you will see the first part we are going to use silicon it's the simplest system so you know exactly how to v vaniya 90 we find all the files but you know I guess you did it already a few times by now so you have to do this under SCF and SCF without vaniya 90 is an entanglement vaniya 90 as you saw we typically might might be interested also in knowing what happens in the conduction but that's why we have to do a decent technique we need to have a good description also of the both of the conduction but especially if you want to do end-doping in a material and so since we want this v vaniya functions as you probably already saw for silicon we want to use a SP3 initial projections on the two silicon atoms in the inside this you know just redo it when it's really pressing a button the final thing is running a final executable called POSSIB 90 which was already introduced on Monday we'd use the same input file as v vaniya 90 but we've taken all between 30 seconds and 2 minutes with the inputs we gave you if you do in serial it's going to be 2 or 3 minutes if you go in parallel it should take 30 seconds and you will get a machine where we tell exactly please use both of them please compute these quantities the input is the same it just has an additional part which I'm showing here and you will have the files in the github which contains the flags for both of them so what you have to say is first of all you want to say I want to use both of them so you want to enable this module because POSSIB 90 allows to use a lot of modules if you don't put it or you put it to false this variable is all ignored and POSSIB 90 will do nothing and once you put it through you have to say a few things first thing as we discussed once you compute on a very dense grid energies and velocities computing the density of states is for free so even if there are modules to compute dots already in POSSIB 90 we have a flag to ask both of them to compute the directed dots because it's a zero cost so it doesn't make sense to have another module which we do with some computation and so if you want it it's not a compulsory but to put it to true and then you have to say the energy grid on which you compute the density of states in this case you can go in the documentation we tell you all the units but these are electron volts so it means a grid of 10 million electron volts under the density of states you want to do a Gaussian smearing this is a feature which maybe I mentioned in a moment you can do even an adaptive smearing but we don't do it we do a fixed smearing of 30 million electron volts and use a 40 by 40 by 40 mesh is what it means you can do also 40 by 50 by 60 if you want to but since this is a system which is 40 by 40 by 40 whereas there is a shortcut you can just say 40 it's a cubic 40 by 40 by 40 so these six lines are about density of states one thing to I wanted to just to mention if you go in the paper of Jonathan and for workers which I mentioned before they also introduce a concept of adaptive smearing which is essentially the fact that if you have a band which is almost flat if you go from one k point to the next one you need a very small smearing because in the end your band will be essentially flat if your band is very dispersive is very high derivative so as a high velocity you might want to use in the point a much higher smearing because in the end you know going from one point to the next one will have essentially change a lot of the energy and so the adaptive smearing is something which has been described in the paper and is implemented in Bolton with the library we have smearing amplitude which depends point by point and allows it to essentially have with a relatively coarse grid a relatively smooth density of states if you want to try if the example you can just try to play with the smearing width and adaptive smearing for false look at the distance states but that's not the main point of this tutorial so I will just mention it the other important part is the actual parameters for Bolton and first of all the same k mesh will be used to compute the transport distribution function so we are computing this on a 40 by 40 by 40 mesh you can put as smearing but I suggest actually not to unless you are a very low temperature because it will give you artificial effects and since you don't want to show sigma but you want to integrate sigma it's better not to put any smearing to have the quantitative correct quantity but in case there is also a Bolts to DF smearing parameter instead of what you want to say is in which range of chemical potentials in which range of temperatures you want to compute and these are specified by a minimum a maximum value and a step so this will compute between 5 and 8 electron volts every 10 electron volts for the chemical potential and you need to know where the thermal level is you want to be around with thermal level of course and in this case I want to get only a 300 Kelvin so what I do is I say min and max is 300 Kelvin so this step is useless but if I wanted to do 300 and 500 and 700 I could have said 300 here maximum 700 step of 200 so that was 300, 500, 700 in finally you need of course a relaxation time tau in 50 seconds and this optional is commented as you see you want to say how many states you have on each bandwidth essentially this is typically understood by default by 1A90 if you have a spin ampullarized system is 2, you have spin up and spin down if you do spin orbit you want to put 1 because you have explicitly both bands let's say all the inputs essentially apart from activating it if you want to compute in a dose and saying which range of mu and temperature you want to compute a relaxation time let's say input you run it and what you get in the output are files which essentially allow you to plot the state of the material this is a silicon and the state let's say the gap and for instance here I'm plotting the civil coefficient as a function of the e but in reality this is a chemical potential you will see so you know where it is maybe I show it quickly but it's in the usual github there so you go in a 2022.05 we go in day 5 morning bolstron you will find the usual file so SCF and SCF the L2090 and the vane input which contains what I just showed you plus the standard things for silicon so the bottom part you should know exactly and this top part is what I just described and there is a tutorial you can just follow which explains exactly what you have to do so it's a few pages in principle in 10 minutes you should be able to get to this final plot what I wanted to tell you is that there are a number of ideas let me thank a lot for the tutorial and we will be there to help but there is a number of interesting things that the simulation is really putting in the numbers and pressing enter try to think a bit to what you can do and there are a few suggestions of what you can do which I put also here so this is my last slide first of all think looking back at the equations I showed maybe I will give a link to the slides why the civil coefficient in this approximation is independent of the relaxation time while sigma and k are the linear proportional essentially so that's one exercise more mathematical and question is this true even beyond the constant relaxation time second thing you can change the k mesh these are relatively fast calculations you can see 20, 40, 60 how much these functions change third thing you can try to learn how to plot as a function of temperature either multiple runs or a single run specify multiple temperatures as I mentioned the expensive part is typically the transport distribution function so if you take these inputs and you run it once and then you change here 500, 500, run it again it will take a lot of time it has to re-compute the transport distribution function so one trick would be it would relate 300, 700, 200 the time spent for it is written at the bottom the time spent for the transport distribution function is the largest part then interpolation on multiple temperature is very fast so it actually would be to try to do it in a single calculation and compare the time and finally as I mentioned before experimentally you have not you cannot measure the chemical potential but you measure the doping of the material so try to imagine how you would convert this plot so this not as a function of chemical potential but as a function of the doping as I showed before but this I stopped I will let you go through this short tutorial and of course you can look at it also later if you don't have time but you should be short and I am happy to take any questions that will be live now on the matrix chat so maybe I am going to look maybe through the chat thank you very much thanks Giovanni we should all thank our speakers thanks Giovanni for the very nice talk so I noticed I think a couple of questions online we can start with first one is there any possibility of having interband terms sum over N and M transfer coefficient definitions so I think let's go here this one right so I think no if you write it down the depends on which approximation you make on tau ok so here we made a very simple approximation where actually we put this as a constant so essentially wherever you are on the bands you have a constant time if you already now say ok what they can compute is given a band I have computed scattering on all other bands and they give a time for scattering out of the state in any other state still in lucky case where the sum will still be on one band and you have a tau which depends on the band you might have something more complex you really want a very advanced model in bands as you consider bands simple approximation this is the correct equation interband states there was another question why there is a maxima and a minima for the C-back coefficient versus energy plot for silicon no that's a typical shape of a C-back coefficient as a function again this is not the isn't energy in the sense is the position of the chemical potential we can make simple models actually an exercise I give you is to take a parabolic band very simple e equal to h-bar k-square over 2 m and you try to analytically derive what's the expression of sigma and s in both in one dimension maybe for sigma and s when your band is parabolic if you do it you will see the shape essentially as this one so you will have something which goes up and then goes down that's the kind of standard shape for this function and actually this is what makes complicated to optimize this product sigma s-squared over k because s and k already are kind of connected you see it here and also k is high when sigma is low so the ratio already is typically zero in a sense you cannot optimize it and you need to make an interplay because when this goes up your signal coefficient goes to zero so you really have a very narrow range of doping where you have non-zero sigma but also relatively larger s if you multiply these two you get a peak so hi thank you for the talk for topological materials we know that we have to correct the Boltzmann equations of motions with the corrective factor and also the transport properties takes this factor is the can we put this corrective factor into the formulas for the integral in the code thanks for the very nice question so I have to say I am not an expert so maybe other people in the room might answer better than me if there are such corrections these are not there is no correction in the code the equations which are shown here are the ones which are implemented but most of the code in the end is a bit optimized but in the code it is simple so if you have any correction it is easy to put it into the code but I have to be then not an expert of transport in topological materials so someone else can also be able to answer better than me we have another question hi thank you for the talk I would like to ask here in this code meantime is a parameter and how can you compute it you will see the disaster I think so the typical time it takes an electron to scatter out of a state into something else now it depends on your material on the purity of it you can of course scatter in impurities you have an electron and it has a doping itself for instance and so you can take out approximate you can work them out but very often it is easiest to take approximate function given a concentration of impurities you can have other scattering options one intrinsic one you cannot move with the scattering of electrons with phonons so you need a code which is able to compute the electron phonons scattering like EPW we see this afternoon which is able to compute this tau as a function of N and K so this would be a way to get the extra number so I am not sure if this is going to be part of the tutorial maybe it rocks on or someone can say but the EPW code so you can compute this actually yeah thanks that you mentioned that because rocks on I think is a theatrical in some wells tutorial you are going to see that the second tutorial so just now you see the easy one and then later we see the last one so Giovanni maybe you can go to the previous slide to answer this also this one no the one where you explained there are two components do two thermoclonge activities somehow that is also related if you want to hear at the bottom in reality have the sum of total thermoconductivity of the system you have really to sum all contributions you have a contribution which is phonon-phonon interaction and your contribution which is electron with something else and these are electrons which are just transferring heat because they are moving and as I said we are computing only this so electron will be hot and will move through the system will not only transfer charge but will also transfer some heat within it itself this KE is a weighted sum about the possible scattering contributions in the material you don't see it here electron impurity one could be electron boundary of the actual sample so when the electron phonon you see the doctrine is still part of this and then the KL is really phonon-phonon interaction great are there any other questions if not can I ask a follow up question on that yeah absolutely please so can you go to the slide with the tau k which is not the constant this one yeah so I'm wondering so if I have an electron with k let's say it's moving to the right then if that electron scatters off of something shouldn't you also keep track of where that electron scattered because if it continues going to the right then it still carries energy heat whatever yeah you're totally right and indeed the actual equation will take into account initial and final state and this kind of goes back to the previous question and matrix so in the case of phonon, electron phonon matrix elements between these two because you also have to have to be the rate will depend if you have a phonon with a k, a q vector, that energy difference so yes in general it's true but you know the point is can we make a simple model what kind of estimations we can make to get an approximate model and so you can go from a fully ab initio you compute all the phonons what you do in the afternoon given a state on average I know where it goes and get a 8 time which doesn't mean roughly how this can turn out 2 are constant which is very rough but gives you a qualitative sense of what in many cases is what's going on the heat you have to separate into this is even more important for phonon phonon we have a contribution which is elastic where in a sense you scatter but you continue transferring the heat so it doesn't matter, it doesn't affect we have the inelastic contribution where in a sense send it back if you want we have a reciprocal lattice vector which connects the initial and the final state and it really affects the thermal conductivity so you would basically average that out but the equation would still have the same form would only have to no, I think the question becomes more complicated because here you see I'm really making assumptions on the shape the derivative of f with collisions and this will allow me to simplify a lot everything but if in reality this is a complex thing then you do not necessarily get something which is relatively simple you will end up with also some initial and final states this tau enters here with some delta on the difference of k vector the difference in energy in the initial and final state and the g square coefficient is like a double sum double sum and more ok, thanks ok, so I guess there is 5-6 minutes left probably we should just let people work in the tutorial what do you think, Giovanni? I don't know what's the timing for the lunches but maybe we can let people at least give a look to these questions and you can ask questions directly in matrix, I would say we reused the Vanier 90 tutorial room since it's also part of the 90s 5 minutes doesn't make sense to make anything yeah, no exactly we have this matrix channel, please use that to ask technical questions this is especially true if you are connected through the zoom but if you are here in presence you can still raise your hand we will bring you the microphone so since there is a question in the chat maybe I read it out loud for everybody and try to answer from Igor so I mentioned and did the possibility to interpolate not only the Hamiltonians the Bans and the velocity but also other operators so this it's not described in the manual in the sense that it's not something where you can really pass a file and it's getting interpolated but it's very easy if you want to go in essentially as long as you can give matrix elements of that operator with respect to the states in the then it's very easy to use the same routines for interpolating for instance for all the spin property which are a lot of spin modules which you didn't show in the positive 90 these are all using the same approach so you interpolate also the spin matrix elements but you need the DFT code to compute also some additional matrix elements so you need but the advantage again is you do this on our course grid and then when you have it interpolate so if you are interested in a specific operator you should unless it's something simple like a derivative which means multiplying by arc you need to first implement something in the P90 and then also go to 90 that say in 90 in 90 in 90