Czochralski Growth of Ge Computed by Using the Finite Element Tool Elmer
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Transcript: English(auto-generated)
00:00
Okay, thank you for the introduction. So before I come to this particular topic, I would be a little bit more general on this. I'm from the Leibniz Institute for Crystal Growth in Berlin, so I will give a general overview what we are doing. And I just noticed that I can give you the link between the first talk today and the second talk today.
00:23
And so I added something here because from sand, well probably from the desert, towards quantum dots. So what is the sand? Sand contains silicon. Silicon is a major ingredient for our electronic devices today because you've got silicon crystals, you make the structures and so on. Okay,
00:43
so this is some basics to go in the end for quantum dots. There are many steps in between of course and yes, I will give you, that is the outline of the complete talk, so you will be more and more specific by time. This is what we do in our institute. So we grow
01:01
bulk crystals, maybe silicon, but this is gallium arsenide. For the oxide crystals, because they are always doped, they look this nice with the colors. So you see the applications for everything. There's optics, it's also electronics,
01:21
it's everything what you can expect from that for electronic industries. So what you do normally, you cut this into wafers and then on the wafers you do epitaxial growth. So we heard about epitaxial growth yesterday in the condensed matter session upstairs. That's also what we are doing and
01:43
okay, part is also the nanostructure, so also going to quantum dots in a certain way. So this is a very general outline what we are doing at ICAZ and we do also numerics and that's in the end what I will talk about later. But first of all, okay, this is a real silicon crystal as grown in a
02:05
siltronic algae in Bolkhausen, one of the major players for silicon crystal growth. This is just to have a look on it because okay, how we grow the crystals, it's very simple. You have the melt here and the crucible, and then you pull out the crystal.
02:23
This is done here, also you can see the melt a little bit there, you see the crystal. And yeah, that's a technique. So it's very simple, looks very simple, but in practice is of course a little bit more complicated because if you want to go to quantum dots in the end,
02:43
you don't want to have many defects or at least no defects in your crystal. So that is a challenge for the crystal growth. Whoop. So for the numerical modeling of this process here, you normally start with the entire furnace like this and
03:01
the grower is interested in the thermal field. That's the main point for him. So typically you use axisymmetric calculations for this purpose and then you get a color picture showing you the temperatures and that's what you learn. And then if you go more into detail because you have a melt,
03:21
you're interesting in the melt flow because this has all an impact on the thermal field, but of course also on the transport of dopants and so that's of major interest. Unfortunately, the melt flow is typically 3D and so you have to, if you want to model it correctly, you have to go to 3D calculations or
03:44
even though the geometry is axisymmetric. I will not talk about this today. So this is another topic and also here we have a collaboration with spheres partly on the on this part on the 3D calculation of the melt for certain systems.
04:05
So now I start with the topic I want to speak about. Once again, very general. Why we do this crystal growth? It's not silicon. It's not for such an industrial application. It is for
04:21
scientific application. It's for the projects called Gerda and Legend and the aim is to find a neutrino loss better to say this one. So why is it so interesting? One knows that neutrinos have a mass, but one knows only the relative masses, not the absolute one. So
04:40
they want to find the absolute mass and the first who find it, well, it's expected that it gives another Nobel Prize. So everyone is running to that. So what they do is they make an experiment currently at Grand Sasser. Well, that is, you see the size that is human being here and they are germanium detectors. So
05:02
germanium single crystals when used for the detectors. Currently there's 30 kilogram and you want to go to 200 kilogram. They are, here you see these detectors in more detail. So there's a lot of material you want to use to find in the end if there is a neutrino loss better to say.
05:23
And the germanium is worse. You see, this is the target and it is also the detector. So the challenge is to grow the single crystal. Okay, this is not really the challenge. You can grow single crystals. Oh, sorry. The main problem is you need a very high resistivity.
05:46
Sorry. So which means you need an extreme purity and that means in the end you are restricted with your furnace and you cannot use all this equipment you normally use.
06:03
Let's see back story for that. And the main part is also here that you need inductive heating and not resistant heat. That is all because it would be too dirty. So that's the story. And now you can see
06:20
the real world. This is the furnace where we are growing germanium and so it looks inside. Once again, the main part is here. You have a melt in red, and a crystal in blue, and it's a Czochralski method as mentioned previously. You pull out this crystal in that direction and then hopefully you get a crystal of good quality.
06:44
Okay, that already we achieved but the high purity is really the challenge. But I will not tell about a high purity. I will tell about how to model this process. So now, okay, we go further down to the process. Once again, it's an axisymmetric calculation.
07:03
You see all the parts which are going into the simulation and what we need to do is, okay, we want to calculate the temperature and in addition also later the stress field. That means the thermal stress field and the crystal.
07:21
So we're just going to temperature. Okay, what we have in physics. We have a heat conduction. That's clear. That's simple. But we have also radiation, wall-to-wall radiation between all the parts here, which is a little bit more complicated in a way because it goes to temperature to the force power. So it makes all numerical kind of problems maybe, but we are in a good
07:45
situation because germanium is not a high melting material. Oops, it's always up. So then again, we also need to compute the melt convection in a simpler way here. You might also be interested in a gas convection.
08:04
Okay, so this is this part and now comes the specific part on crystal growth. What to do? Okay, and you compute the inductive heating here, but it gives you some heat input in the graphite. But in the experiment, you know, okay, the crystal has this shape as a blue one.
08:23
What you do in the numeric is you have a three-phase point. Three-phase because it's crystal melt and gas. Exactly at this point, you know, that should be the melting point temperature here on the experiment. What you do, you make a control by worse,
08:42
by a PID controller to tune the power that you gathered melting point temperature here. The next step is, of course, you all the want to have a solid melt interface shape computed and easily we say here it is just a melting point temperature. So you go looking for the temperature isolein.
09:07
Well, that's the way we compute. Then now you see this is one situation at one time of the of the growth. The growth is rather slow, it's at some millimeter per hour, so it's a slow process.
09:20
But it's only one time short. So if you want to make a transient calculation, you have two possibilities. Either you make a really transient calculation and then also squeezing on elongates the grid. So that's something here. Then it becomes very dense here after a while and
09:40
well, why are there? That's one possibility. I will not go to this kind of calculation. I will speak about a series of steady-state calculations. So, but a series. So that in the end you can have the same kind of video. So now comes a solving strategy exactly for that problem I'm facing here.
10:04
So first step is always that we try to validate the experimental data. It comes later back what I mean by experimental data. So therefore we use exactly the shape as observed in the experiment. So after the experiment you got this
10:20
crystal and you see it's not optimal. The radius was not constant here. You see some buckling, but okay we put in this shape of the crystal into the calculations. Of course you can give by hand an optimal one and see what to do with with this one, but I will only
10:44
tell you results for this one. So now comes a little bit more complicated story because, okay, oops-a-la. We say this is a crystal length zero and then we start first to compute how much melt I need. So the melt height, you have your crucible and it's a certain melt height
11:06
and because you know the crystal weights, I think in that case it was 900 gram, so you know how much melt should be in the beginning. And then you start your computation. Okay, red is a melt gas interface and this is a small crystal.
11:25
So we are at length Lm, so now in the next time step, the time step is not really a time step, but a length step. It's always delta L. So the crystal is longer by delta L.
11:43
But as I say you have a pulling rate, it's pulling out in that direction. But also because the crystal is growing, the melt height is going down, though the melt height is now in the next step here. From that you can get now an equation. Well, delta L, as I said, this is our time step,
12:06
so to say. But the real time is this one here and from this one, which I cannot compute because this is a given, this is a given parameter, this I can compute. And though I know the growth velocity of my crystal,
12:24
the growth velocity of this interface that I can get here. Why is it important? It is important because when the crystal is growing, then it releases the latent heat. That's the one which is going into a calculation because we have to
12:45
put in the latent heat, which is coming between that step and that step. So in a steady-state calculation, we let release this heat here according to this set of equations. The important point is, typically what people do is, they use
13:04
for computing the latent heat only that value of the pulling velocity. But of course, if the melt is going down, because it will be not the same and we get different results. So,
13:21
yeah, so this is done for, this computation for the velocity is done with smaller steps and then we can think, okay, when we want to make a real calculation of the temperature field, so we do it maybe every tenth or every second step, so at what we want to at that moment. So we perform, as I said, a steady-state calculation. We
13:45
compute the geometry. So from that geometry, we gather mesh by gmesh. So for every calculation, we get a new mesh, which is adopted to the new situation from crystal on melt. And then we compute by the open source software
14:02
package Elmer. Elmer had the advantage that the developers were also interested in crystal growth processes. So there's almost everything in what we need for describing crystal growth processes. Yeah, so it's, in principle, it's a multi-physic
14:22
simulation software based on finite elements, or you see where it was developed. Ten years later, they make it open to the public. This is also still, we had a discussion on this, it's still written in Fortran. And yes, you can put in some mathematical expressions for that via this kind of
14:44
MetC The Elmer contains already solvers written by some users, so you don't recompile everything, so you have a compiled program and just call the solvers if all the solvers you need are available. This looks just like this.
15:04
So this is for the melt. You can have this solver, this one, this one, that one. In other domains like crystal, I can have different solvers because I want to solve the stress also when I need a different one. So just, I put out here the heat equations. Okay, then they say it's, this is just the name,
15:20
but this is a procedure that really accords the right solver, and then you have the density up. The temperature is your variable here, and you can give, as you see, you can give your own names for the velocity field or for the pressure. So this is the story. You can write on
15:44
this as the input file and then start to calculate everything. So I think I have to start the video. Yeah, now it's a little slower. So this is now just a sequence of steady-state solutions. You see red and blue, which means different in the different domains. Here it just means temperature
16:07
in the melt part, and the crystal part has different meanings. You have the Van Mises stress on this side, so this is something which you compute directly in your solver. Red means you have high thermal stress,
16:25
which is bad because then you get dislocations or multiplication of dislocations. But okay, this is not the full story for the multiplication of dislocations because dislocations
16:41
multiplication depends also on the temperature by a Rhenius factor. This is some kind of pre-factor for the multiplication, and this looks slightly a little bit different, but of course most dangerous parts are near to the interface. But also at that point I have also to stress the point how
17:02
important is the shape of the interface in principle, because you see the stress here is the highest and also the Rhenius factor is the highest here, so that's very important how this shape is.
17:23
Yeah, so one more to the melt convection. For all these computations, I choose higher risk causities than the physical one. This is a physical one, but in the end I choose this one for the computations to be clear that this is still a steady-state motion.
17:47
And you can start, and this is once again how you can use the Mazzi expressions here in the beginning. If there is no flow, then you start with a higher risk causities. And you lower it, because you already have a solution as an input for the next iteration, and you can do it by these expressions.
18:05
This is a time variable here, and so if you're less than 20 iterations, you increase it linearly to get a stable solution to that problem. And now comes a point linked to the experiments. So what you can do in the experiments, you cannot look really into
18:23
we saw the growing crystals, but we cannot measure anything. You don't see the interface, but what you do, you cut afterwards you cut the crystal, and then using the so-called LPS method to visualize our differences in
18:42
the resistivity, and this corresponds then to the interface shape during growth. So those are these lines here, and then the overlaid yellow lines is on the calculated interface lines, and this agreed quite well, except at certain regions where the radius was changing dramatically.
19:08
So this gives a quite good result. So the other thing you can compare is the power, the input power, not directly, because in your experiment you have some outer loop for your
19:26
electricity, so you have to subtract here something to come to this curve. This one is the experimental one, the red one is the computed that it has these peaks, because when the radius was changing too sharply, then I have some peaks,
19:46
but in the end this follows the experimental line, and now you see what happened if you would choose the the growth velocity identical to the pulling velocity, you would get a completely different story even in the picture of power.
20:03
So that makes it important for this particular problem to take into account the real velocity. I think this was just a video to visualize our some flow here, and that it is changing in time.
20:24
So the main contribution to flow is the buoyancy convection, but you also have a Marangoni convection on the surface, and you also see, well, you have this line here between the gas and melt. This is a fixed line, which is computed beforehand to have an, I don't know if it, I have it here, no,
20:46
I saw it, I have it somewhere, okay, sorry, because this angle between the crystal and this line is 13 degrees. This is known from experiments, so the line is just computed to match this,
21:02
and there's a question by Harlow to do this, so this is done before I start the steady-state calculation for every step this is computed. The other thing, what you might be interested in, because of the, well, we need this high purity, and so there's a gas convection, it's a different story, you can also compute it
21:20
in this framework. Here there's two different, you see two different configurations, and in one you get a gas convection near to the crystal, and the other not. I don't want to go into detail here. Ah, here, here, sorry, here was this exactly the picture I was looking for.
21:42
This curve is, yeah, this is given, as I said, by the input, and this angle here is 13 degrees, Laura. And, okay, I mentioned now where I have the problem with Elmer, because, okay, this is the, this is the boundary condition for our Marangoni convection, as I already mentioned that you have some flow here from the,
22:05
from the, on the hot to the cold, and, and, okay, for principle I like to separate the melt flow and the gas flow, but it means I have
22:20
to solve the Navier-Stokes equation, the red part here, with this boundary condition, and separately I want to solve the Navier-Stokes equation and the rest where the gas is, yeah, just with another boundary condition. So, the problem is, with the, of course I can write a new function and recompile everything, but with the
22:43
recompiled system, it is, I can use two Navier-Stokes solvers, just rename it, even the developer was not aware of that, that it is really, that one can do it. The only problem is, the, the thermal field, I can, as of the, the Navier-Stokes, the velocity field cannot enter the thermal field, because the thermal field I want to, to
23:05
compute globally. It's just one heat equation over the entire domain. So, in principle you want to do, what you want to do is, to give the, our advection term, or in the, in the melt domain is a melt velocity, and here it is a gas velocity,
23:23
and I didn't find any way to do it, and since the developer, Peter Rabeck, was even astonished that I can use two, two Navier-Stokes solver in parallel, okay, he didn't also know an answer for that question. Yeah, but this is only one part, okay, the other one is,
23:43
to be more precise with the computation of this gross velocity is, that we don't take into account the actual shape of the interface. So, what we do, what you saw is a flat interface, and we should, we look for the difference in the, of the two flat interfaces, and then compute the
24:03
the overall gross velocity. So, of course, it would be more accurate if you compute that locally, because of the change, or sometimes there's a drastic change in the shape you saw between one step and the next step, and of course this has also
24:21
the release of latent heat, or other way around, the consumption of latent heat. And, of course, the main point is also the computation of dislocation density, which is an extra step, so you, to, to, you, you put in all this, our results, and this, either the thermal stress field and the temperature, or your, this function,
24:46
the pre-function, refactor function, as shown before, into another calculation, where you compute the evolution of the dislocation density. But you can also, from a mathematical point, think about other
25:01
things, because everything is tunable with that open source card, one can think about it, some kind of optimization, optimization loop outside, around whatever we are looking for, but maybe can define something, what we are looking for, or want to look for, or the, maybe also over the entire process, or a minimum of the
25:25
dislocation stress over the entire run, or something else, and then you can parameterize that in the way that you change something in the system, to, to be better for that, or to achieve this,
25:41
or to minimize it. So, to end on with that, though the main part, of course, in this work, where, where the experimental one, you see all the people involved in it, and yes, this is, the SCADA project, is although supported by BMPF. And thank you for staying here to the almost end, or the end of the talk,
26:06
so there will be discussion afterwards, but to the end of the talk. Thank you.