Kinetic Monte Carlo Simulation for Understanding Epitaxial Growth
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Miller, Wolfram
Kinetic Monte Carlo is a nice tool to study the growth kinetics of epitaxial processes on an atomistic scale. However, energy barriers for the various surface processes must be defined and thus either data from ab initio calcalutions or from dedicated experiments are required. Furthermore, processes can be handled only successively so that there is no natural way for parallization. We show two application cases: homoepitaxy of Ga2O3 and epitaxy of AlN or AlGaN on AlN. The specific numerical issues for the computation will be addressed.
 You Yes, thank you. Yeah, so it was two parts. We heard about the properties of the collect properties of materials or better to say are the numerical calculations of properties. And there was already some questions okay how they are grown. Okay, so now I want to speak how they are grown and also I want to speak in particular about modeling, and this is kinetic Monte Carlo simulation for understanding the epitaxial grows. And the outline is okay, some words and epitaxial grows in general, when the main topic, the KMC I also want to strongly describe the basics of KMC. And I will show two examples, which were calculated as the epitaxy of gallium oxide homo epitaxy, and the other is many also homo epitaxies or no strain. Except if you use aluminum gallium light right but this is not really was not really the topic of the strain and this computational in this research. So, and then I am some up there epitaxy grows in principle you have experimentally three different methods to do, or a deposition of your material on a surface, that is, either you use molecular beam epitaxy. Basically you have a source of your atomic ingredients. And so you put your atoms, while your atoms going to the surface. And you need there typically a low pressure. And because the atoms have to travel, or it's to the surface. It's a quite good method for fundamental investigations. If you go to industrial production, it's typically not used because it's not so scalable. You use this metal organic vapor face epitaxy so you provide the metals as gallium or aluminum by organic compared and a solution. And then you transported, we are gas, typically argon into the system, which is typically quite good for scalability. Nevertheless, in detail, you often you don't understand the process but if it's work, then it's fine for the industry. I'm making another method that is a pulse laser deposition, where you have already this kind of, or let's say more complex materialist, or let's be aluminum nitride you already have aluminum nitride at the source. And then you make a laser application of this, you heated this one on, and then you have clusters of aluminum nitride arriving at the surface. The problem is you don't know how much the cluster is typically the nice thing is that when you have new material are then you can just use a source of that and hope that something of this is going to the surface and you get a layer of the material you want to mitigate. So I will just stick to the metal organic vapor face epitaxy because this is a method we use for that as we are also in Leibniz Institute are interested going then towards industry process in the end. So that if your flavor of the machine, I think I can have that yeah so this is a machine as this is an academic machine where you see the chamber which is well quite small here in this case. So of course industrial or application it's much larger but you also sees, see a lot of gas wells and the gas system here which is in the heart of this kind of method and you see the sketch how it is working. And then organic precursor so to get a net lane for instance, you have the urban as a target of the gas. And then, while you're doing this is solution here you in the bubble or you transport your metallic compound organic compound towards the surface. And of course, and all the case or another and cases you need either nitrogen or oxygen what I will present as the this is an example of the same oxide so you need an oxygen source. There's a pressure in this chamber as much as much higher or could be much higher than the MBE, which is quite good for such systems where you need an access of oxygen. And here in the bottom you see already, okay that might be quite complicated says precursor, and this is even a simple one, but the problem there with the systems of course. And the decision of this is not so well known where it happens, need to the surface or far away or. So that is some point which is the problem with this kind of simulation. In the end we use and they are expected only the gallium is arriving at the surface and the, the oxygen as a molecule for this case. And when you want to understand it in more detail are you should consider then the different processes which is, of course, adoption, then you have the diffusion on the surface. And you have often these option, and the case of gallon oxide you have gallium sub oxide which can these are. And then you want to start with a different effects what could happen that is, for instance, first, if you have a flat surface nucleation and 3D grows, that's typically not what, what you want to do in a different version, because you want to have a nice flat or well defined layers afterwards you want to go to the set gross mode. But even if you have a step grows mode, it might happen that you have in the end step branching which is also something which you want to avoid the all these things on what to like to study and we use an atomistic method for that. So, what does it mean. So, think about that we have a certain surface configuration of atoms on the surface. So, you think about it, new state, one atom is diffusing or one atom is absorbing, though you have a new surface configuration, and then you have a transition from one surface configuration to the other one with a probability which is given by the energy barrier for this process, and your surface temperature. So you don't have only one new configuration but but many many many so we have a lot of from the original initial configuration, there are many possibilities for a new configuration. And so you have a whole bunch of transition rates which are given by the probability times the frequency. So here is in the end in one iteration or time step, you have to pick up randomly one of these processes. What you do is you make a list of all this put potential processes. And then, and here are the rates are normalized rates because our random number is in the end between zero and one. And then you have a certain value of the random number. And then you see, you would pick up this green event, the event, which belongs to this green color. And then you have a very long vector indeed, if you have very many processes. And just to mention, the, when you do this event process, then there is a correlated time, which is given by independent random number. So, and then this is your time step so in every iteration, as in every step, your time might be completely different, or I might be different, depending on the, on the sum of all your events, or all your rates of the events. The problem from the moment point is that came see as a sequential process. That means, I cannot, I cannot paralyze it per se. And if I have a large computational domain, of course I have many transitions, it's just for for for this large computational domain. And then this is large and you already see on the, on the side the time step will become very small from the equation from the top so this is made makes this computational happy. And then we go bring a little bit more into the practice. So, do we have been the notes where the atoms, Residate, are every note has, of course, coordinate, which is fixed. So we have fixed letters. We have on the other side we have cells, which are the unit cells of this material we want to compute. And here is the unit cell of aluminum nitrate. We have four atoms as a tune, aluminum and two nitrogen nitrogen atoms in a unit cell. And now what we do numerically is we store in every for every cell we store all the events for the atoms in this cell. So one of these atoms can move or as on the site that can be your adoption, and also we make a list per cell are for all the events. And then he in the iteration loop. So, okay, depends where we start here. Okay, let's start here. So first of all, we have to make this list, the list for the cells, and only for the cells, which have to be updated. There's no change in the rest of the cell from the previous iteration. I have to do nothing. But okay, so when I have all this events per cell, then I have to create this long list is one vector from all the cells. So just make this makes this list of all of all cells. And then I have to see when as I show. And then I have to see what happened here to make a new list of the cells, which have to be updated so it's in principle it's very simple. Let's see here when you have a movement of particle from a to B, and you have an interaction sphere like like the gray cells so you have to update all the gray cells are with respect to the events happen in this, or I might happen in this. And this is not really great but you see the, the violet boundary on it. So now, so the question remains how we compute the energy the energy barriers and they were what does that is what defines your probability. So if you compare two possibilities typically you make bond counting or it better better to say a neighbor counting. You look for for instance here for the aluminum atom sitting here so it's a few are on the top. And this show for how many neighbors it has or if you need to. So this outer shell we have to look for the atoms. Here are nitrogen atoms sitting that was the reason to go up to this shell here to look for the neighbors, and then you'll have an input energy for this atom atom interaction, but I better to say it's not really an input for the binding between the two atoms that is an input. And then you have the total and a bond energy is just a sum. You can also use like an empty potentials for this are the one who was was presenting the MD is not here. Okay. So you can, you can compute the energy by the distance as a function of the distance. The problem, what typically occur for all problems is that the potentials are good for bulk. So the parameter parameter potentials but parameters, but often not not for this for the surface or that problem that you don't have the right energy so what what I show was everything was computed by the first approach, not by potentials. Yeah, and now we have the bond energies and there is of course we want to have the barrier so I just showed how we do it for the aluminum nitride. So, expect a diffusion from this nitrogen at oh sorry, from this nitrogen atom are from this position to that one. It might have the initial one such energy computed just by the other method, and the final one that one. So we have some difference here, plus a diffusion energy and we define that in a way that we define it fixed diffusion energy given for the situation that the initial and final energy is the same. And if it is not, then we then it is the energy barrier is computed by this expression. And the problem is then to get any the values from for for these energies on you can do it. Well, wait, and get it by DFT calculations but typically it's, you cannot get everything from that because it's a huge number and then DFT people would say oh that's taking very long time. And other ways to get it from experimental evidence. So this is, this is still the, the main problem for the current become Monte Carlo methods was a gallium oxide. There was no DFT calculation at all for this, or for diffusion on the surface. And we get it from some, some estimate for it from from experimental experiments and from a very simple theory for one D diffusion so one get some idea. So, what the, what an overall diffusion coefficient might be, but this gives you a flavor of gallium oxide, as it is in the experiment with it. So, the solution applies to have on a surface on a substrate surface of it to the position with this already much more complicated than aluminum nitride I will not go into the details of in total 20 atoms in it. So, we have a diffusion and eight gallium so quite complicated situation. So we consider here adsorption of gallium and oxygen molecule. We have the desorption as I mentioned, of the sub oxide. We have diffusion of the atoms and also we allow on this. This is a cluster of small cluster diffusion. So, this is the growth on a on a flat surface, but it would be considered like the first point nucleation and 3D grows. This dimension in the game C or the other direction. And you see what happened here there is growing. This nucleation to one cluster, but you also see, it's not really 3D grows. This is form and then the layer is filled up. And then the next one, new nucleation, you grows, this corresponds to the experimental evidence as I was also so observed an experiment that you don't have complete 3D grows, and also what we can qualitatively corresponds with experiment is the, is the size of the clusters as a function of the temperature of the surface temperature. And the size of course, what's more interesting is in what I say the step flow goes mode, because that is that what you want to achieve in the end for growing a nice layer. And okay, so in the numerical simulation, it really does work. You get a step growth mode, you see one step after the other is growing. And I know in the, in the numeric it's very easy to trigger the different parameters. So, for instance, a desorption rate, so we can manipulate the option right. And you see that you get a different structure if you load this option right, or a high desorption rate. So the video was for the middle, or a case. And you see the situation after 10 seconds, and after 20 seconds that looked like that. So here you have this nice phenomenon of step one step is faster than the other. And then you have a double step in the, in the end. And you also have here a button step up to this. But in between, it is, it's gross on the terrace, rather than a step launching. So the, you end up in the final picture would be the same, but the growth mechanism just from the came see would be different. And that's the difference between the two. In the experiment, you can also manipulate the desorption right by, by manipulating the pressure in the chamber. So one expects and the desorption rate was higher pressure is lower and vice versa. And then you see in between. And was a high pressure. So at night, they got this step punching structure in the middle. They're really got these step grows. So the message is here okay we can, qualitatively we can get the same result. And with the came see we can look much deeper into this system than just analyzing the final results from the experiments. So that's, so I'll let me know my tried. Well, it looks similar with the steps right here. It's different. It's much simpler, as you know, and just to mention, I put here always the characteristics why this materials of interest of elliom oxide is for high power electronics and here it's a little in the same direction. And I'm not here is of high interest. So I wanted to say, from the point of application, other D are, let's say the dynamics a little bit simpler. And another thing here, what I want to mention, what is known for for aluminum nitrate if you have a nitrogen atmosphere you have you provide nitrogen and access. And this is the surface only was a little bit. And this is this red nitrogen is now the surface with a minimum energy which was our computed by DFT, or very by several groups. And at KMC, we have to manage this in a dynamic way, because when this nitrogen atoms at the red ones from the adsorb, then we have always adsorb as we have always impinging of the nitrogen on the surface, and we have these option. And this should then cancel out in the mean that we have a dynamic balance or, or for the system. And now you see here, okay, you've got, we get the two by two reconstruction with the right positions as shown before. And also you have also additional are atoms absorbing at the surface, which are the same as the position differences. Also, some calculations as seen here on the structure. And the first point is okay with the energy parameters we put in the model, we are able to revisit this structure. So, and if you go now to a once again to a step for all growth. And then the system is not knowing how to arrange this two by two reconstruction because they have this terraces or fine widths. And so, I think it's very, very time steps on to see on the right side, the gross philosophy will be broke for four months later, and the entire time was two seconds for this was a system. And the challenge is typically with the systems as I already mentioned, you have large areas many steps, but if you want to resolve many steps. Later we want also to have an influence of dislocation circle. You need to be sure that the influence is then the areas the domain is largely not enough influences on the local. I did not, I did not show here the results for my position. But this is also of course, are in this future plan to do what we do. And then, very, very important is now how much time you spend and the single update. And then you see checking neighbor ones seem to be at the moment, the point which costs most of the CPU time so there's still some room for for improvement on the other reasons. So on the other one, this already the other is is interesting problem was from the, from the physics or from the system is said you could have something like this. And then it's, it stays or similar energy or very energy very is not very large. And then it's hoping for and back and forth back before something happened. Okay, this is very well, you stay for a long time in the situation, but the system is changing. So and there, they are possible solutions, first of all, you don't know what happened for a new system so you have to detect it. And once you know, then detect it then you can circumvent this problem so there are methods to do this. So that although I come to the end, or forces came see as a valuable tool to study the growth kinetics and to understand it in more detail. So, two examples, both for both first the flat surface, and then, then the typical surface which you use for growing layers are for the gallon oxide. We have a good qualitative agreement with the experimental results are the aluminum nitride is now still in, in, in, in research. And I say numerical challenges are due to the sequential algorithm, one should have no as a normal focus to accelerate the setup because it should be very, very fast to, to allow, or then many, many iterations or larger systems. And of course the other point is to use our algorithm for the super local super person. And let's say more. So, I'm not going to do with America. But one plan is to incorporate this integrated in this atomistic framework and there is this pirate framework and just as an hour. So that, that was my presentation. So essentially you described the situation just a small patch of the. Exactly, I just, yeah, you can you can ask a question and I tried to make an image or show an image. So, you have the surface, the real surface practices longer. And you have some, probably some assumption on how these issues are transported to the surface. And there is no assumption that the Mali for the came see there's an impinging rate as a particles. And that's not a reference on the surface structure. But what happened in the guest is not considered. It's not considered in practice. This is kind of homogeneous. And I say, we have galley. And for instance, on the, on the, when the oxygen molecule arrives at the surface is some is this helpful to this dissociation. There's something like like OH group in between which makes the splitting of the oxygen are less energetic or energy barrier smaller know these effects are not included. Okay, if you know that then you might also include it, but okay it will be more complicated, the model will be more complicated and then it's difficult to set the parameters. But for your other question, I wanted to show this one. This white area is a complication. This is from a year. It's much larger of course the FM has also this and bunch is not a domestic really on the TAM, you can see the effects, but you see what, what the problem is. It's quite small. The main is rather small. No, that's, that's, that's a problem. This is to give you a flavor of how small it is in comparison to the structure they saw in the in the in this experiment. So I need some suggestions on this with content you on this afterwards. Okay. So question from my side. So what you have shown is basically, and kind of planar structures right we have seen that you had information but also need like the first step of a layer as far as I understood. What happens, certainly can go to systems that would facilitate a 3D. So, very possibly we did it and you can consider a strain in it for for Germanium silicon systems, and you can also have 3D growth here that's just here. Okay, we do this applies as KMC for the systems which are of interest of my interest at the moment so definitely for the for the for the systems which is really in house as a taxi process in house, you know that there's no 3D growth so far. So, you will not consider your happy that the numerics is the same as in the experiment. But basically, but basically you, you, of course, you can have a 3D growth here for other systems that might happen for the aluminium nitride yeah we that might be another. Yeah, it is different. But it was not the main focus at the moment. People are looking for the principle they were looking for the dislocation. And from that, we do not eliminate the aluminium nitride epitaxy in our house. And by the Martin Albrecht's and to the issues which were mentioned in the title and the authors, they do not make element or epitaxy but they analyze it, and though they were meant questions come from that from the atomistic. So, this was galium oxide. This is galium oxide. So it was a gallium ethylene. And now there's not in the industry though. I don't know if I have this. Oh, the other picture which is more from the experimental. Yeah, now I have a, I was just wondering in the experimental. Yeah. Okay, this is not to do with the numerics or only hard fit, but this was experimental work here. So you see, we're all seeing. See, but this is a gross rate, and you principle what you say is you want to increase the gross rate. And you see what happened is, although experimentally, you go from here, we have staff floor is hey okay I increase the gross rate. And I have step punching whatever I'm not quite sure which that bunching they really have here. And because I said in the end, it looks the same. Or if you have just two mechanisms. So and then they, they, they change the miscut angle or the terrorist bits, and then we're able to go to a situation where they still have flat step flow. And you see then that happens once again they increase the gross rate. And then, so the increase the gallium flux here just you see, you see this, but they do, but it's not unique, but the increase the gallium flux here and here's the increase the chamber pressure and the way that they're also the, the argon flux was increased by this. And you see this is a different methods measures in the experiment where the gross rate is then, as in result different. But you see what happened is okay you can try this why I know okay they are here. And it's already one and a half two years old. So that's the way. Unfortunately, I was not yet able because we stopped this calculation score for gallium oxide some time to really follow this quantitatively with a KMC, but this is all difficult because the energy parameters are not so well known to really fit it exactly. So, this is 3D and it's two plus one. So you have to be careful with that expressions you're completely right you do not, you do not have overhanging right. So what you have is, let's see, as a meaning you have these two periodic directions, which can be stepped or not stepped because of the years of course, what is going here. And it's up by the same, right, that this is a very common, but in this sense it's 3D but you are not a lot of overhanging. You see what I mean that, that there are empty spaces between layers. If this is what you mean by 3D, otherwise it's, it's, as I said, two plus one, one typically what one can name it. So the question is, is there any, is there any, any more questions?
 You  Yes, thank you. Yeah, so it was two parts. We heard about the properties of the collect  properties of materials or better to say are the numerical calculations of properties.  And there was already some questions okay how they are grown. Okay, so now I want to speak how they are grown and also I want to speak in particular about modeling, and this is kinetic Monte Carlo simulation for understanding the epitaxial grows.  And the outline is okay, some words and epitaxial grows in general, when the main topic, the KMC I also want to strongly describe the basics of KMC.  And I will show two examples, which were calculated as the epitaxy of gallium oxide homo epitaxy, and the other is many also homo epitaxies or no strain.  Except if you use aluminum gallium light right but this is not really was not really the topic of the strain and this computational in this research.  So, and then I am some up there epitaxy grows in principle you have experimentally three different methods to do, or a deposition of your material on a surface, that is, either you use molecular beam epitaxy.  Basically you have a source of your atomic ingredients. And so you put your atoms, while your atoms going to the surface.  And you need there typically a low pressure.  And because the atoms have to travel, or it's to the surface. It's a quite good method for fundamental investigations. If you go to industrial production, it's typically not used because it's not so scalable.  You use this metal organic vapor face epitaxy so you provide the metals as gallium or aluminum by organic compared and a solution. And then you transported, we are gas, typically argon into the system, which is typically quite good for scalability.  Nevertheless, in detail, you often you don't understand the process but if it's work, then it's fine for the industry.  I'm making another method that is a pulse laser deposition, where you have already this kind of, or let's say more complex materialist, or let's be aluminum nitride you already have aluminum nitride at the source.  And then you make a laser application of this, you heated this one on, and then you have clusters of aluminum nitride arriving at the surface.  The problem is you don't know how much the cluster is typically the nice thing is that when you have new material are then you can just use a source of that and hope that something of this is going to the surface and you get a layer of the material you want to  mitigate. So I will just stick to the metal organic vapor face epitaxy because this is a method we use for that as we are also in Leibniz Institute are interested going then towards industry process in the end.  So that if your flavor of the machine, I think I can have that yeah so this is a machine as this is an academic machine where you see the chamber which is well quite small here in this case.  So of course industrial or application it's much larger but you also sees, see a lot of gas wells and the gas system here which is in the heart of this kind of method and you see the sketch how it is working.  And then organic precursor so to get a net lane for instance, you have the urban as a target of the gas.  And then, while you're doing this is solution here you in the bubble or you transport your metallic compound organic compound towards the surface. And of course, and all the case or another and cases you need either nitrogen or oxygen what I will present as the this is an example of  the same oxide so you need an oxygen source. There's a pressure in this chamber as much as much higher or could be much higher than the MBE, which is quite good for such systems where you need an access of oxygen.  And here in the bottom you see already, okay that might be quite complicated says precursor, and this is even a simple one, but the problem there with the systems of course.  And the decision of this is not so well known where it happens, need to the surface or far away or.  So that is some point which is the problem with this kind of simulation. In the end we use and they are expected only the gallium is arriving at the surface and the, the oxygen as a molecule for this case.  And when you want to understand it in more detail are you should consider then the different processes which is, of course, adoption, then you have the diffusion on the surface.  And you have often these option, and the case of gallon oxide you have gallium sub oxide which can these are. And then you want to start with a different effects what could happen that is, for instance, first, if you have a flat surface nucleation and 3D grows, that's typically not what, what you want to do in a  different version, because you want to have a nice flat or well defined layers afterwards you want to go to the set gross mode.  But even if you have a step grows mode, it might happen that you have in the end step branching which is also something which you want to avoid the all these things on what to like to study and we use an atomistic method for that.  So, what does it mean.  So, think about that we have a certain surface configuration of atoms on the surface.  So, you think about it, new state, one atom is diffusing or one atom is absorbing, though you have a new surface configuration, and then you have a transition from one surface configuration to the other one with a probability which is given by the energy barrier for this  process, and your surface temperature.  So you don't have only one new configuration but but many many many so we have a lot of from the original initial configuration, there are many possibilities for a new configuration.  And so you have a whole bunch of transition rates which are given by the probability times the frequency.  So here is in the end in one iteration or time step, you have to pick up randomly one of these processes.  What you do is you make a list of all this put potential processes.  And then, and here are the rates are normalized rates because our random number is in the end between zero and one. And then you have a certain value of the random number.  And then you see, you would pick up this green event, the event, which belongs to this green color.  And then you have a very long vector indeed, if you have very many processes.  And just to mention, the, when you do this event process, then there is a correlated time, which is given by independent random number.  So, and then this is your time step so in every iteration, as in every step, your time might be completely different, or I might be different, depending on the, on the sum of all your events, or all your rates of the events.  The problem from the moment point is that came see as a sequential process. That means, I cannot, I cannot paralyze it per se.  And if I have a large computational domain, of course I have many transitions, it's just for for for this large computational domain.  And then this is large and you already see on the, on the side the time step will become very small from the equation from the top so this is made makes this computational happy.  And then we go bring a little bit more into the practice. So, do we have been the notes where the atoms, Residate, are every note has, of course, coordinate, which is fixed.  So we have fixed letters.  We have on the other side we have cells, which are the unit cells of this material we want to compute. And here is the unit cell of aluminum nitrate.  We have four atoms as a tune, aluminum and two nitrogen nitrogen atoms in a unit cell. And now what we do numerically is we store in every for every cell we store all the events for the atoms in this cell.  So one of these atoms can move or as on the site that can be your adoption, and also we make a list per cell are for all the events.  And then he in the iteration loop.  So, okay, depends where we start here. Okay, let's start here. So first of all, we have to make this list, the list for the cells, and only for the cells, which have to be updated.  There's no change in the rest of the cell from the previous iteration. I have to do nothing. But okay, so when I have all this events per cell, then I have to create this long list is one vector from all the cells.  So just make this makes this list of all of all cells.  And then I have to see when as I show. And then I have to see what happened here to make a new list of the cells, which have to be updated so it's in principle it's very simple.  Let's see here when you have a movement of particle from a to B, and you have an interaction sphere like like the gray cells so you have to update all the gray cells are with respect to the events happen in this, or I might happen in this.  And this is not really great but you see the, the violet boundary on it.  So now, so the question remains how we compute the energy the energy barriers and they were what does that is what defines your probability.  So if you compare two possibilities typically you make bond counting or it better better to say a neighbor counting. You look for for instance here for the aluminum atom sitting here so it's a few are on the top.  And this show for how many neighbors it has or if you need to.  So this outer shell we have to look for the atoms. Here are nitrogen atoms sitting that was the reason to go up to this shell here to look for the neighbors, and then you'll have an input energy for this atom atom interaction, but I better to say it's not really an  input for the binding between the two atoms that is an input.  And then you have the total and a bond energy is just a sum.  You can also use like an empty potentials for this are the one who was was presenting the MD is not here. Okay.  So you can, you can compute the energy by the distance as a function of the distance.  The problem, what typically occur for all problems is that the potentials are good for bulk. So the parameter parameter potentials but parameters, but often not not for this for the surface or that problem that you don't have the right energy so what what I show was everything was  computed by the first approach, not by potentials.  Yeah, and now we have the bond energies and there is of course we want to have the barrier so I just showed how we do it for the aluminum nitride.  So, expect a diffusion from this nitrogen at oh sorry, from this nitrogen atom are from this position to that one. It might have the initial one such energy computed just by the other method, and the final one that one.  So we have some difference here, plus a diffusion energy and we define that in a way that we define it fixed diffusion energy given for the situation that the initial and final energy is the same.  And if it is not, then we then it is the energy barrier is computed by this expression.  And the problem is then to get any the values from for for these energies on you can do it. Well, wait, and get it by DFT calculations but typically it's, you cannot get everything from that because it's a huge number and then DFT people would say oh that's taking very long time.  And other ways to get it from experimental evidence. So this is, this is still the, the main problem for the current become Monte Carlo methods was a gallium oxide.  There was no DFT calculation at all for this, or for diffusion on the surface. And we get it from some, some estimate for it from from experimental experiments and from a very simple theory for one D diffusion so one get some idea.  So, what the, what an overall diffusion coefficient might be, but this gives you a flavor of gallium oxide, as it is in the experiment with it.  So, the solution applies to have on a surface on a substrate surface of it to the position with this already much more complicated than aluminum nitride I will not go into the details of in total 20 atoms in it.  So, we have a diffusion and eight gallium so quite complicated situation. So we consider here adsorption of gallium and oxygen molecule. We have the desorption as I mentioned, of the sub oxide.  We have diffusion of the atoms and also we allow on this.  This is a cluster of small cluster diffusion.  So, this is the growth on a on a flat surface, but it would be considered like the first point nucleation and 3D grows.  This dimension in the game C or the other direction.  And you see what happened here there is growing. This nucleation to one cluster, but you also see, it's not really 3D grows.  This is form and then the layer is filled up.  And then the next one, new nucleation, you grows, this corresponds to the experimental evidence as I was also so observed an experiment that you don't have complete 3D grows, and also what we can qualitatively corresponds with experiment is the, is the size of the  clusters as a function of the temperature of the surface temperature.  And the size of course, what's more interesting is in what I say the step flow goes mode, because that is that what you want to achieve in the end for growing a nice layer.  And okay, so in the numerical simulation, it really does work. You get a step growth mode, you see one step after the other is growing.  And I know in the, in the numeric it's very easy to trigger the different parameters. So, for instance, a desorption rate, so we can manipulate the option right.  And you see that you get a different structure if you load this option right, or a high desorption rate. So the video was for the middle, or a case.  And you see the situation after 10 seconds, and after 20 seconds that looked like that.  So here you have this nice phenomenon of step one step is faster than the other.  And then you have a double step in the, in the end.  And you also have here a button step up to this. But in between, it is, it's gross on the terrace, rather than a step launching. So the, you end up in the final picture would be the same, but the growth mechanism just from the came see would be different.  And that's the difference between the two.  In the experiment, you can also manipulate the desorption right by, by manipulating the pressure in the chamber. So one expects and the desorption rate was higher pressure is lower and vice versa.  And then you see in between. And was a high pressure.  So at night, they got this step punching structure in the middle. They're really got these step grows.  So the message is here okay we can, qualitatively we can get the same result.  And with the came see we can look much deeper into this system than just analyzing the final results from the experiments.  So that's, so I'll let me know my tried. Well, it looks similar with the steps right here.  It's different.  It's much simpler, as you know, and just to mention, I put here always the characteristics why this materials of interest of elliom oxide is for high power electronics and here it's a little in the same direction.  And I'm not here is of high interest. So I wanted to say, from the point of application, other D are, let's say the dynamics a little bit simpler.  And another thing here, what I want to mention, what is known for for aluminum nitrate if you have a nitrogen atmosphere you have you provide nitrogen and access.  And this is the surface only was a little bit.  And this is this red nitrogen is now the surface with a minimum energy which was our computed by DFT, or very by several groups.  And at KMC, we have to manage this in a dynamic way, because when this nitrogen atoms at the red ones from the adsorb, then we have always adsorb as we have always impinging of the nitrogen on the surface, and we have these option.  And this should then cancel out in the mean that we have a dynamic balance or, or for the system.  And now you see here, okay, you've got, we get the two by two reconstruction with the right positions as shown before.  And also you have also additional are atoms absorbing at the surface, which are the same as the position differences.  Also, some calculations as seen here on the structure.  And the first point is okay with the energy parameters we put in the model, we are able to revisit this structure. So, and if you go now to a once again to a step for all growth.  And then the system is not knowing how to arrange this two by two reconstruction because they have this terraces or fine widths.  And so, I think it's very, very time steps on to see on the right side, the gross philosophy will be broke for four months later, and the entire time was two seconds for this was a system.  And the challenge is typically with the systems as I already mentioned, you have large areas many steps, but if you want to resolve many steps.  Later we want also to have an influence of dislocation circle. You need to be sure that the influence is then the areas the domain is largely not enough influences on the local.  I did not, I did not show here the results for my position.  But this is also of course, are in this future plan to do what we do.  And then, very, very important is now how much time you spend and the single update.  And then you see checking neighbor ones seem to be at the moment, the point which costs most of the CPU time so there's still some room for for improvement on the other reasons.  So on the other one, this already the other is is interesting problem was from the, from the physics or from the system is said you could have something like this.  And then it's, it stays or similar energy or very energy very is not very large.  And then it's hoping for and back and forth back before something happened. Okay, this is very well, you stay for a long time in the situation, but the system is changing.  So and there, they are possible solutions, first of all, you don't know what happened for a new system so you have to detect it.  And once you know, then detect it then you can circumvent this problem so there are methods to do this.  So that although I come to the end, or forces came see as a valuable tool to study the growth kinetics and to understand it in more detail.  So, two examples, both for both first the flat surface, and then, then the typical surface which you use for growing layers are for the gallon oxide.  We have a good qualitative agreement with the experimental results are the aluminum nitride is now still in, in, in, in research.  And I say numerical challenges are due to the sequential algorithm, one should have no as a normal focus to accelerate the setup because it should be very, very fast to, to allow, or then many, many iterations or larger systems.  And of course the other point is to use our algorithm for the super local super person.  And let's say more.  So, I'm not going to do with America. But one plan is to incorporate this integrated in this atomistic framework and there is this pirate framework and just as an hour.  So that, that was my presentation.  So essentially you described the situation just a small patch of the.  Exactly, I just, yeah, you can you can ask a question and I tried to make an image or show an image.  So, you have the surface, the real surface practices longer.  And you have some, probably some assumption on how these issues are transported to the surface.  And there is no assumption that the Mali for the came see there's an impinging rate as a particles.  And that's not a reference on the surface structure. But what happened in the guest is not considered. It's not considered in practice. This is kind of homogeneous.  And I say, we have galley.  And for instance, on the, on the, when the oxygen molecule arrives at the surface is some is this helpful to this dissociation.  There's something like like OH group in between which makes the splitting of the oxygen are less energetic or energy barrier smaller know these effects are not included.  Okay, if you know that then you might also include it, but okay it will be more complicated, the model will be more complicated and then it's difficult to set the parameters.  But for your other question, I wanted to show this one.  This white area is a complication.  This is from a year.  It's much larger of course the FM has also this and bunch is not a domestic really on the TAM, you can see the effects, but you see what, what the problem is.  It's quite small.  The main is rather small.  No, that's, that's, that's a problem. This is to give you a flavor of how small it is in comparison to the structure they saw in the in the in this experiment.  So I need some suggestions on this with content you on this afterwards. Okay.  So question from my side. So what you have shown is basically, and kind of planar structures right we have seen that you had information but also need like the first step of a layer as far as I understood.  What happens, certainly can go to systems that would facilitate a 3D.  So, very possibly we did it and you can consider a strain in it for for Germanium silicon systems, and you can also have 3D growth here that's just here. Okay, we do this applies as KMC for the systems which are of interest of my interest at the moment so definitely for the for the  for the systems which is really in house as a taxi process in house, you know that there's no 3D growth so far.  So, you will not consider your happy that the numerics is the same as in the experiment.  But basically, but basically you, you, of course, you can have a 3D growth here for other systems that might happen for the aluminium nitride yeah we that might be another.  Yeah, it is different. But it was not the main focus at the moment.  People are looking for the principle they were looking for the dislocation. And from that, we do not eliminate the aluminium nitride epitaxy in our house.  And by the Martin Albrecht's and to the issues which were mentioned in the title and the authors, they do not make element or epitaxy but they analyze it, and though they were meant questions come from that from the atomistic.  So, this was galium oxide. This is galium oxide.  So it was a gallium ethylene.  And now there's not in the industry though. I don't know if I have this. Oh, the other picture which is more from the experimental. Yeah, now I have a, I was just wondering in the experimental.  Yeah. Okay, this is not to do with the numerics or only hard fit, but this was experimental work here.  So you see, we're all seeing. See, but this is a gross rate, and you principle what you say is you want to increase the gross rate. And you see what happened is, although experimentally, you go from here, we have staff floor is hey okay I increase the gross rate.  And I have step punching whatever I'm not quite sure which that bunching they really have here. And because I said in the end, it looks the same.  Or if you have just two mechanisms. So and then they, they, they change the miscut angle or the terrorist bits, and then we're able to go to a situation where they still have flat step flow.  And you see then that happens once again they increase the gross rate. And then, so the increase the gallium flux here just you see, you see this, but they do, but it's not unique, but the increase the gallium flux here and here's the increase the chamber pressure and the way that they're also the,  the argon flux was increased by this. And you see this is a different methods measures in the experiment where the gross rate is then, as in result different.  But you see what happened is okay you can try this why I know okay they are here.  And it's already one and a half two years old. So that's the way. Unfortunately, I was not yet able because we stopped this calculation score for gallium oxide some time to really follow this quantitatively with a KMC, but this is all difficult because the energy parameters are not so well known to really fit it exactly.  So, this is 3D and it's two plus one.  So you have to be careful with that expressions you're completely right you do not, you do not have overhanging right. So what you have is, let's see, as a meaning you have these two periodic directions, which can be stepped or not stepped because of the years of course, what is going here.  And it's up by the same, right, that this is a very common, but in this sense it's 3D but you are not a lot of overhanging.  You see what I mean that, that there are empty spaces between layers.  If this is what you mean by 3D, otherwise it's, it's, as I said, two plus one, one typically what one can name it.  So the question is, is there any, is there any, any more questions?
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