Kinetic Monte Carlo Simulation for Understanding Epitaxial Growth
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Leibniz MMS Days 20223 / 22
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Computer animation
00:29
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Transcript: English(auto-generated)
00:29
Yeah, I'd like to introduce the next speaker, who will be another one of the largest statistical consultants in the room, or a consumer, or a architect.
00:41
For gradation, and yeah, I would say it's my role to talk about the presentation. Yes, thank you, yeah, so in those two talks, we heard about the properties, optical electronic properties of materials, or better to say are the numerical calculations of these properties.
01:01
And there was already some questions, okay, how they are grown. Okay, 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 growth. So the outline is, okay,
01:21
some words on epitaxial growth in general. Then the main topic, the KMC, I also want to strongly describe what the basics of KMC. And then I will show two examples, which were calculated as the epitaxy of gallium oxide,
01:43
homoepitaxy, and the other is many other homoepitaxis or no strain so far, except if you use aluminum gallium nitride, but this was not really the topic, the strain in this computational in this research.
02:03
So, and then I sum up. So epitaxy growth 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.
02:20
So typically 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. So you can, because the atoms have to travel or through the surface.
02:42
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. And they use this metal organic vapor phase epitaxy. So you provide the metals as gallium or aluminum by organic compared in a solution.
03:03
And then you transport it via gas, typically argon into your system. And that 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.
03:23
And there is another method that is a pulse laser deposition, where you'll have already this kind of, let's say more complex materials, or let's be aluminum nitrate. You already have aluminum nitrate at the source. And then you make a laser ablation of this,
03:41
you needed this one. And then you have clusters of aluminum nitrate arriving at the surface. The problem is you don't know how large the cluster is typically. The nice thing is that when you have new material, then you can just use the source of it and hope that something of this is going to the surface and you get a layer of the material
04:03
you want to investigate. So I will just stick to the metal organic vapor phase epitaxy because this is a method we use for that as we are also in Leibniz Institute and we are interested going then towards industrial process in the end.
04:21
So if you're a flavor of the machine, I think I can, I thought, yeah, so this is a machine as this is an academic machine. So you see the chamber, which is well, quite small here in this case. So of course, industrial application, it's much larger, but you also see a lot of gas wells and the gas system here,
04:40
which is in the heart of this kind of method. And you see the sketch how it is working. You have organic precursor. So gallium ethylene, for instance, you have the argon, it's a target of the gas. Then, well, during this solution here,
05:03
in the bubble you transport your metallic compound, organic compound towards the surface. And of course, in the cases you need either nitrogen or oxygen, what I will present or this is an example of gallium oxide.
05:21
So you need an oxygen source. So the pressure in this chamber is much higher or could be much higher than the MDE, which is quite good for such systems where you need an excess of oxygen. And here in the bottom you see already, okay, that might be quite complicated. Precursor, this is even a simple one,
05:41
but the problem there with these systems, of course, the dissociation of this is not so well known where it happens near to the surface or far away. So that is some point which is the problem with this kind of simulation. In the end, we expect that only the gallium
06:05
is arriving at the surface and the oxygen as a molecule for this case. So now when you want to understand it in more detail, you should consider then the different processes,
06:22
which is, of course, adsorption, then you have the diffusion on the surface and you have often desorption. In the case of gallium oxide, you have gallium suboxide which can desorb. And then you want to start with the different effects, what could happen, that is, for instance, if you have a flat surface nucleation and 3D growth,
06:43
that's typically not what you want to do in a production because you want to have a nice flat or well defined layers afterwards so you want to go to the step growth mode. But even if you have a step growth mode,
07:01
it might happen that you have in the end step branching, which is also something which you want to avoid. But all these things on what to like to study and we use an atomistic method for that, it's a kinetic Monte Carlo method. What does it mean? So think about that we have a certain surface
07:23
configuration of atoms on the surface. And now you think about a new state, one atom is diffusing or one atom is absorbing, so you have a new surface configuration. And then you'll have a transition from one surface configuration to the other one
07:42
with a probability which is given by the energy barrier for this process and your surface temperature. So of course, you don't have only one new configuration, but many, many, many. So we have a lot of from the original initial configuration,
08:01
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. And so what you have to do is in the end, in one iteration or time step, you have to pick up randomly one of these processes
08:26
or what you do is you make a list of all this potential processes. And then, and here are the rates, there are normalized rates because our random number is in the end between zero and one.
08:42
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 here. So that might be a very long vector indeed, if you have very many processes.
09:02
And just to mention that when you do this event process, then there is a correlated time, which is given by an independent random number. So, and then this is your time step. So in every iteration, in every step, your time might be completely different
09:22
or it might be different depending on the sum of all your events or the rates of the events. Then it's the problem from the moment point is that KMC is a sequential process. That means I cannot paralyze it per se.
09:45
And if I have a large computational domain, of course I have many transitions for this large computational domain. And then this is large and you already see on that side,
10:00
the time step will become very small from the equation from the top. So this makes this computational happy. Just to go a little bit more into the practice. So we have the nodes where the atoms resonate. Our every node has of course, coordinate, which is fixed.
10:23
So we have the fixed letters and we have, on the other side, we have cells, which are the unit cells of material we want to compute. And here is the unit cell of aluminum nitrate. So you have four atoms as a two aluminum
10:42
and two 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. One of these atoms can move
11:00
or on the sides that can be adsorption. And also we make a list per cell for all the events. And then the, 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,
11:22
the list for the cells and only for the cells, which have to be updated. Well, 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,
11:40
this one vector from all the cells. We'll just make this list of all cells and then choose event 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.
12:02
Let's see here, when you have a movement of a particle from A to B and you have an interaction sphere like the gray cell. So you have to update all the gray cells with respect to the events happened in this, or it might happen in this. Oh, sorry, this is not really gray, but you'll see the violet boundary on that.
12:26
So now still the question remains how we compute the energies, the energy barriers and the one that defines your probability. So there are two possibilities typically, you make a bond counting or better to say a neighbor counting.
12:42
You look for, for instance, here, oops, for the aluminum atom sitting here. So it's a few on the top. And then you look in this shelf or how many neighbors it has, or if you need, indeed we need a larger correlation area here.
13:01
You have to look for all the, in this shell up to this outer shell, we'll 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
13:21
for this atom-atom interaction, but better to say it's not really an interaction, but for the binding between these two atoms that is an input, and then you have the total bond energy is just a sum. You can also use like an empty potentials for this,
13:43
or the one who was presenting the empty is not here. Okay. And then you can compute the energy by the distance or the function of the distance. The problem, what typically occur for all problems
14:02
is that the potentials are good for bulk. So the parameters are not the potentials, but the parameters, but often not for the surface. So that's the problem that you don't have the right energy. So what I show was everything was computed by the first approach and not by potentials.
14:24
Yeah, and now we have the bond energies and there is, of course, we want to have the barrier, though I just showed how we do it for the aluminum nitride. So expect a diffusion from this nitrogen atom, sorry, from this nitrogen atom
14:42
or 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 you have already some difference here plus the diffusion energy. And we define that in a way that we define a fixed diffusion energy
15:02
given for the situations that the initial and final energy is the same. And if it is not, then it is, the energy barrier is computed by this expression. The main problem is then to get any, the values from, for these energies.
15:22
And you can do it 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 a very long time. The other way is to get it from experimental evidences. So this is, this is still the main problem
15:43
for the kinetic Monte Carlo method. So the gallium oxide, there was no DFT calculation at all for this or for diffusion on the surface. And we get it from some estimate for it from experimental experiments and from a very simple theory for 1D diffusions
16:04
or one gets some idea what the, what an overall diffusion coefficient might be. But it just gives you a flavor of gallium oxide as it is in the experiment with it, or these two lines you typically have on a surface,
16:22
on a substrate surface of a two deposition with this is much more complicated than aluminum nitride. I will not go into details in total 20 atoms and it 12 oxygen and eight gallium. So quite complicated situation.
16:42
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, on this surface here, a cluster of small cluster diffusion.
17:04
So this is the growth on a, on a flat surface, but it would be considered like the first point of glacial and 3D growth. This dimension in the KMC or the other direction.
17:22
And you'll see what happened here. There is growing this nucleation to one cluster, but you also see it's not really 3D growth. This is formed and then the layer is filled up. And then is the next one, new nucleation, new growth.
17:42
This corresponds to the experimental evidence as this was also observed in experiments that you don't have complete 3D growth. And also what we can qualitatively corresponds with the experiment is the size of the clusters
18:01
as a function of the temperature, of the surface temperature. So the next point is of course what's more interesting is in what I say the Z-flow growth mode, because that is that what you want to achieve in the end for growing a nice layer.
18:22
And okay, so in the numerical simulation, it really does work. You get a step growth mode. You see one step up that the other is growing. Let me know in the numerics, it's very easy to trigger the different parameters.
18:44
So for instance, desorption rate, so we can manipulate the desorption rate. And you see that you get a different structure if you're low desorption rate or a high desorption rate. So the video was for the middle or case.
19:00
You see the situation after 10 seconds and after 20 seconds, that looked like that. So here you have this nice phenomenon of step function. One step is faster than the other. And then you have a double step in the end. Here is what you also have here, a double step after this,
19:21
but in between it's a growth on the terrace rather than a step launching. So you add up in the final picture would be the same, but the growth mechanism just from the KMC would be different in this case in that case. In the experiment you can also manipulate the desorption
19:42
rate by manipulating the pressure in the chamber. So one expects and the desorption rate with higher pressure is lower and vice versa. And you really observe all of this situation that for those, they call it a step punch because if they saw one of the final result
20:00
and not see in between, and was at a high pressure desorption rate, they got this step punching structure and in the middle, they really got this step growth. So the message is here, okay, we can, qualitatively we can get the same result. And with the KMC we can look much deeper
20:20
into this system than just analyzing the final results from the experiments. Yeah, so that's, so aluminum nitride, well, it looks similar with the steps, right? Here the system is of course different. It's much simpler as you know. And just to mention, I put here always
20:42
the characteristics why this material is of interest to gallium oxide is for high power electronics and here it's a little bit in the same direction. So these are materials of high interest. So what I wanted to say from the point of application. Also the, or let's say the dynamics
21:02
are a little bit simpler, but you have another thing here what I want to mention, what is known for aluminum nitride. If you have a nitrogen atmosphere, you provide nitrogen in excess and you don't have this flat surface
21:23
only with aluminum at top. Like when you have a nitrogen in, these are the hole positions in the middle of the cells of the hexagonal cells. And this red nitrogen is now the surface
21:43
with a minimum energy, which was computed by DFT already by several groups. So in the KMC, we have to manage this in a dynamic way because when this nitrogen atoms excerpt here this red ones from the excerpt,
22:03
then we have always absorbed as we have always impinging of the nitrogen on the surface and we have desorption. So in the end, this should then cancel out in the mean that we have a dynamic balance or for the system.
22:23
And now you see here, okay, you've got two by two reconstruction as the right positions as shown before. But of course you have other additional atoms at absorbing at the surface,
22:42
which are sitting at the up positions here for instance, sometimes you have also some calculations as seen here on the structure. But the first point is, okay, with the energy parameters we put in the model, we are able to revisit this structure.
23:03
So, and if you go now to a, once again, to a step world growth, as a first you see, there are many calculations on the terrace because of this system is disturbed by the steps. Of course, it's not a nice flat surface. And then the system is not knowing how to arrange this two by two reconstruction
23:22
because they have this terraces of finite widths. But you see all of them in time, the steps you see on the right hand side, the growth velocities will be both four and a half later. And the entire time was two seconds was a system.
23:47
So the challenge is typically with the systems, as I already mentioned, you have notch areas, many steps. But if you want to resolve many steps, later we want also to have an influence,
24:00
to study the influence of dislocations there. You need to be sure that the influence is then the areas, the domain is large enough that the influence is only local. And, okay, I did not show here the results for a little bit of my project position. But this is also, of course,
24:21
are in this future plan, what to do. Many, what you can do. Very, very important is now how much time you spend and a single update of the iteration. And you've just shown what this aluminum nitride, why you stay for a longer time or when you need some more stability time.
24:41
It's not so critical at one point. You see checking neighbor ones seem to be at the moment, the point which caused most of the CPU time. So there's still some room for improvement on the algorithms. So on the other one, ah, this is already, the other is intrinsic problem works
25:02
from the physics or from the system is that you could have something like this. So you have a huge space which are similar in energy or with barrier energy barrier is not very large. And then it's hopping forward and back and forth before something happened.
25:21
Okay, this is very, well, you stay for a long time in a situation when the system is changing. So there 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.
25:40
So there are methods to do this. So whether though I come to the end or forces KMC is a valuable tool to study the gross kinetics and to understand it in more detail. I showed you two examples, both for both first the flat surface and then the typical surface,
26:02
which you use for growing layers or for the gallium oxide, we have a good quality of our agreement with the experimental results or the aluminum nitride is now still in research. So numerical challenges are due to the sequential algorithm
26:24
and should I've known as a normal focus to accelerate the set up because it should be very, very fast 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.
26:44
A more, let's say more, yeah, they don't agree to do with America, but one plan is to incorporate this integrated in this atom into mystic framework. And there is this pyran framework
27:04
just as our aim on mid time then would be good to have such a calculation over in such a framework into that general framework. So that was my presentation.
27:34
So I have a question. Okay, so essentially you described the situation
27:41
just a small patch of the surface there. Exactly. I just, yeah, you can ask a question and I try to make an image or show an image. Ah, yes. So of course the surface, the real surface in practice is larger
28:02
and you have some probably some assumption on how the species are transported to the surface. No, there's no assumption for the KMC. There's an impinging rate as a particles. As the impinging rate particles hits the surface
28:22
then there is some sticking coefficient calculation that depends on the surface structure. But what happened in the gas is not considered. It's not considered. It's not considered. In practice, this is kind of homogeneous in your experiment also. It's probably homogeneous and more of the problem is what really arrived.
28:43
And as I say, we have gallium. Probably you have also hydrogen on the surface because the hydrogen is coming with the gallium. And then there's a question which is not quite not yet complete and not really known what the hydrogen has an influence. There is an impact, for instance,
29:01
when the oxygen molecule arrives at the surface. Is this helpful to this dissociation? Is there something like the OH group in between which makes the splitting of the oxygen are less energetic or the energy area smaller?
29:23
No, these effects are not included because, 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,
29:41
I wanted to show this one. This white area is a complication to me. This is from an AFM. What do we have? I don't know. We'll use them on scale on it. Okay, it's much larger. Of course, the AFM has also disadvantage.
30:01
It's not atomistic really. On the TAM, you can see the bit of the effects, but you see what the problem is. This area is quite small. I think you get in the end rather large clusters. So you will not get, of course, the right type of cluster because it remains. This is rather small.
30:21
No, that's a problem. This is just to give you a flavor how small it is in comparison to the structures they saw in this experiment. Okay, so I have some suggestions on this, but I might contact you on this afterwards. Okay, yeah, yeah, yeah, yeah.
30:43
So a question from my side. So what you have shown is basically you end up with kind of planar structures, right? I mean, we have seen that you had item formation, but it was only like the first step of a layer as far as I understood. And what happens is you certainly can go through systems that would facilitate a 3D growth.
31:02
Yes, yes, okay. And in the very past, we did it and even consider a strain in it for germanium silicon systems. And you can also have 3D growth here. But that's here, okay. We do this, I apply this KMC for the systems which are of interest, of main interest at the moment.
31:22
So definitely for the gallium oxide, which is really also done in-house as an epitaxy process in-house, you know that there's no 3D growth so far. So you will not consider it, but you're happy that the numerics
31:40
is the same as in the experiment. But basically, of course, you can have a 3D growth. For other systems, it might happen. For the aluminium nitrite, yeah, that might be not as a fair, it is different. But it was not the main focus at the moment. So the people are looking for the,
32:01
in principle, they were looking for the dislocation and from the, we do not do aluminium nitrite epitaxy in our eyes. What we do is 10, by the Martin Albrecht's and Godier Schutz, which were mentioned in the title and the authors. They do not make aluminium or epitaxy,
32:21
but they analyze it. And so there were many questions coming from that, from the atomistic factorization. I have a question based on 3D's question. And the first question was, what is the end of this choice?
32:43
I don't know about the end of the decision, but this is gas metal that we... This was gallium oxide. This is gallium oxide. So it was a gallium metal in ethylene.
33:02
That's in the industry. And now there's not in the industry. I don't know if I have this or 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 part of it,
33:20
but it was experimental work here. And you see, well, let's see. So this is a growth rate, and in principle, what you say is that you want to increase the growth rate. And you see what happened is, although experimentally, you go from here, way of staff flow and say, okay, I increase the growth rate.
33:42
So, and then I have step bunching, whatever. I'm not quite sure which step bunchings they really have here. Because I said in the end, it looks the same, or if you have just two mechanisms. So, and then they changed the miscut angle or the terrace widths.
34:00
And then we are able to go to a situation where they still have step flow. You see then that happens once again, they increase the growth rate, and then they increase the gallium flux. Here just, you see, you see just what they do.
34:20
It's not unique. So they increase the gallium flux here, and here's the increase in chamber pressure and the way that the argon flux was increased by this. And you see that this is a different methods, measures in the experiment, where the growth rate is then, as in results different. But you see what happened is,
34:42
okay, you can try this way and now, okay, they are here. Oh, that is already one and a half, two years old. So that's the way. Unfortunately, I was not yet able because we stopped this calculations for gallium oxide some time to really follow this quantitatively with a KMC.
35:03
But this is all difficult because the energy parameters are not so well known to really fit it exactly. Can I actually consider 3D or can I split it? This is 3D, and it's two plus one.
35:23
Yeah, you have to be careful with that expressions. You're completely right. You do not have overhanging, right? So what you have is, let's see, at the meaning you have these two periodic directions, which can be stepped or not stepped because of the years, of course,
35:42
what is going here or this step, and it's up by the lower step, right? That is a very boundary condition. But in this sense, it's 3D, but you are typically not allowed in overhanging. You can see what I mean, that there are empty spaces between layers.
36:05
So if this is what you mean by 3D, otherwise it's, as I said, two plus one, or typically what one can name it.