Symmetries in TEM imaging of Semiconductor nanostructures with strain
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Leibniz MMS Days 20236 / 23
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Transkript: Englisch(automatisch erzeugt)
00:00
I want to talk to you about symmetries that occur because of strain, let's say, in transmission electron microscopy images. This is the latest work with Tomasz Kopruki and Alexander Milchian.
00:22
As an example of semiconductor nanostructures, I will use quantum dots mainly. So quantum dots, as you probably already know, they are semiconductor nanostructures with interesting optoelectronic properties. And their properties are determined by their geometry.
00:40
Usually transmission electron microscopy is used to image this material with a goal that we can extract some information on the structure from the image. But this is difficult because basically it's a nil-post problem in the sense that two different geometries might give you the same experimental image.
01:03
So you cannot really extract the information you want. And the reason is mainly because of the strain. So the strain comes from the fact that the quantum dot is embedded in another material with different elastic constants. And this mismatch will cause strain around the structure.
01:22
Now what you usually do in this kind of problems is that you use simulations. And that was the idea of Tomasz Kopruki and Carsten Tabello when they applied for a project called Model-Based Geometry Construction of Quantum Dots. And it was successful and it was funded by EC-MATH first and then MATH+.
01:42
So the idea in this project is that first we will simulate as many quantum dot configurations as possible. And then somehow we hope that we can use this database to solve the inverse problem.
02:02
Now in the simulation part the important thing is that you have to mimic what happens during the experiment so that you get realistic images. The last part, the inverse problem is something we work on. So I will not even go there today, but I will talk about the simulation.
02:25
And this is done, we have finished this. And then there is a step in between, I would say, where you have your data, you finish the simulation and you don't go to the reconstruction yet, but you take a break and you observe your data either because you want to see if the simulation was successful and the images
02:45
are indeed close to the experimental ones, or because you see things on the images that could help you in the reconstruction, or they will just give you a better insight of the physical phenomenon. So it was at this stage where we observed the symmetries.
03:02
And what was weird, let's say, is that the images were a bit too symmetric in the sense that they had more symmetries than the object. So this motivated us to study these symmetries on a mathematical level to understand where
03:21
do they come from. Do they come because of the imaging process, do they come because of the material, the geometry of the material, or a combination? So before going to the symmetries, I will talk a bit about the simulation. And for this I need to say just a few things about the microscope, how it works.
03:41
So how does STEM work? Well, we have a specimen that is a crystalline object, and we use electron beams. Now because of the periodic structure, these beams are diffracted in discrete directions. Then what we need to know from physics is Bragg's law, which basically gives us these directions.
04:00
So it says that if you have an incoming beam with wave vector k0, then a diffracted beam k prime may occur given by this sum, where g is a vector in the reciprocal lattice. Then you also have to use what is called the Ewald sphere, which basically it's a geometric
04:24
construction that takes these diffracted beams, and it says that, well, from these you only keep the ones that are on this sphere. OK, so with these two things in mind, you kind of reduce the set of beams that are diffracted. And then in practice, how do you do this?
04:42
You use the objective aperture, and you choose the specific beam you want to image. Now if the image includes the incoming beam that is denoted by 0 usually, always, it's called the bright field image. If it includes a diffracted beam, it's called the dark field image.
05:03
OK, so this is a brief description of how the microscope works. So we want to mimic this when we do the simulation. How do we simulate them? So we start with the Darwin-Hang-Willem equations that are a system of first order ODE.
05:23
Now to be honest, you don't start from DHW, you start from Schrodinger equation. But you end up with a DHW because you want to solve something easier. If you go from Schrodinger to DHW, if you do it step by step, you end up with an infinite
05:42
system because you have this for every vector in the reciprocal lattice. But in practice, in the simulation, you don't use an infinite system. You use very few beams. So these few beams are this lambda star M. So M is the number of beams. I will even use two beams in most of my examples.
06:01
And that's good enough. OK so, but if you don't trust me that it's good enough, you can go to our previous paper where we give error estimates on the solution based on the choice of these finite sets. OK, so what does the equation say? The equation is describing how the amplitude changes with depth in the crystal.
06:22
And it says that these changes depend on how close we are to the Ewald sphere that is described by this SG parameter. So this SG is called excitation error. And when it's zero, it means that you are on the Ewald sphere. And the good thing is that it's an experimental parameter, so you can choose its value.
06:42
And you want it to be small. The equations also say that the changes depend on the interactions with the other beams that is described by this term here, where UG are the Fourier coefficients of the potential. So the idea is, so in time you have high energy electrons, which means that if an incoming
07:06
beam at a specific point is diffracted, this diffraction angle will be small. And then you can assume that the beam will not be diffracted very far away, or it will
07:22
stay within a column centred around the incoming point. And you also assume, so you divide your specimen in different columns of equal size, and you assume that there is no interaction of electrons between columns. So in the end, you just solve the equation for every column in turn.
07:40
So you choose an x and y that will be your pixels, and then you only solve in z. And the solution will give you the intensity of the corresponding point. Okay, so, but, if I solve this equation, I will not get a realistic image. And the reason is that I didn't include the strain.
08:01
So if I solve only this, this for a perfect crystal, I will get something like that, which is not useful. So I have to include the strain in my model. I do this by modifying the Fourier coefficients. And then these are basically the equations that I want to solve, and I see that it's basically the projection of the displacement on the beam that I'm choosing to image that
08:24
influences the contrast. Okay, and this is what we did with Thomas and Carsten, and we also worked for this with Timo Streckenbach from Vias, and Tora Nyron from the Oberlin. So what we did is we created a tool chain where we coupled the elasticity problem with
08:45
a dynamical electron scattering problem. And we start by having as an input a specific geometry and concentration. This enters the VSPD lip, that it's a solver that solves the elasticity problem. And the solution from this will enter Python, that it's a software that solves the DHW equations,
09:06
and will take as input the elasticity result plus the excitation conditions that we want. And then we get our simulated image. So with this tool chain, we created a database of indium gallium arsenide quantum dots for
09:22
different shapes. So this is a small example of the database where we have four different geometries and the corresponding dark fields. So now this is the time to pause and just observe your data. A first observation is that if I compare in the 0-4-0 reflection, if I compare this with
09:48
an experimental image, I don't get any information really, I cannot distinguish. But if I go to the 0-4-0-0-4 reflection, then I can get some result.
10:03
So I can basically say that this is not a pyramid, this is not a pyramidal quantum dot. I still don't know what I'm looking at, but I know that it's not a pyramid. So a result from this observation is that the 0-4-0 reflection is not the right reflection
10:21
to compare simulation and experiments. Instead you should use the 0-0-4 reflection. So at this stage, at the observation stage, we also notice that images are symmetric. And what I mean by symmetric, I mean pixel-wise symmetry. So two different pixels in the image, they have the same intensity.
10:44
Okay. Here, so as I said, the image is influenced by the strain. So when you notice something on the image, the first reaction is to go back to the strain or the displacement. I will use this kind of arbitrary somehow.
11:03
So if you go, if you look at this image, we see that it is symmetric. If you go to the strain, to the displacement, you see that it has the same value in this pixel. So this is not weird. It makes sense that it's symmetric. If you look at the 0-4-0 reflection, symmetry here, here we have an opposite value.
11:24
So why is it symmetric? So the first question was, okay, is it, is the imaging process somehow invariant if you change the sign of the strain? Or is there something else? But here the same symmetry happens to all the structures.
11:43
So the goal will be to distinguish what is a property of the imaging process and what comes from the object. Before going to the analysis, we wanted to see more images. So we contacted our collaborators at Tew Berlin.
12:02
And Laura Nerman, in her thesis, had some very nice simulations where we could make more observations. Okay, so what do we see here? We have, here we have a quantum well and the spherical quantum dot on the other side. And here we have the intensities at different depths, at different positions, and for different
12:23
excitation errors. And the same in the second example. Now I have the displacement of the strain here because we should always keep in mind what happens in the strain. And what do we see? We see in the first here, we see that if we shift from the centre, we get the same pixel,
12:44
no matter what the choice of the excitation error is. So this indicates that the shifting from the centre is a symmetry, or might be a symmetry. But then if you go to the spherical quantum dot, you see that shifting from the centre doesn't give you the same pixel.
13:01
So shifting is not a symmetry. But even here, if you choose an excitation error of zero, which will usually be the case in the experiments, shifting is again a symmetry. Okay, so the question again, as before, is this observation property of the imaging process?
13:23
Or is it because of the strain? Because the strain in both cases has a very specific profile here. It's either an even or odd function. Cool. Then we also observed some symmetries between images. So now, when I talk about symmetries, I mean that every pixel on the first image is the same as every pixel on the second image.
13:42
Again, same observation with shifting, if the excitation error is zero. If the excitation error is not zero, I have to shift and change the sign of the excitation error to get the same image. And if I only shift, I get these mirror-like images. Okay, so these were the observations we made, and then we were curious if we can explain
14:05
this by analysing the equations only. So let's go back to the equations. I rewrite them in a matrix form. What is important here is my structure, especially the fact that V, V will be the influence of
14:23
the potential, so V is supposed to be Hermitian. And then sigma, sigma and f are diagonal matrices. So the Hermitian property is crucial for all the proofs. But this also means for physicists that I'm only talking about elastic scattering.
14:42
But for the moment, I'm satisfied if I can have something for elastic scattering. So then I ask the question in the following way. What transformations of a and f of these matrices will give me the same pixels? And it's these transformations that we will call symmetries. Now for the mathematical analysis, we defined symmetry more specific.
15:06
So it's the transformations that will give me the same pixel for the bright field. So this f zero means that I'm only talking about the bright field. And also, only for the mathematical analysis, I have a strong symmetry if this holds for
15:23
every z, and I have a weak symmetry if this holds only at the exit plane. So here the first two is a strong symmetry, and the last two is a weak symmetry. This is only for the proofs because in reality, like in the image, it doesn't matter. What you see is what happens at the exit plane. Okay, so I keep this structure in mind, this definition of symmetry, and I answer this
15:45
question which transformations give me the same pixel. So the results, I will give the results that answer the questions that we made. So the first result is sign change that says if I'm under strong beam conditions, which means that the excitation area is zero, then I can change the sign of the strain and I
16:02
get the same pixel. And this immediately answers the first observation because it was indeed the simulations were under strong beam conditions. Then a second result is a mid-plane reflection which doesn't require strong beam conditions, it just says that the mid-plane reflection here is a symmetry.
16:23
Now this can explain why in the quantum well we have this shifting as a symmetry. The reason is that if you combine, if you do a mid-plane reflection in this function, it's the same as shifting because it is an even function. So a combination of mid-plane plus the shape gives you, explains this observation.
16:44
Now both properties explain why for the spherical quantum dot we have symmetry if the SG is zero. The reason is that SG zero means that I can apply both properties. And if I apply mid-plane plus sign change in a node function, then it's the same as
17:03
shifting. And then we have a last result here that says that I can do both without strong beam conditions if I change the sign of the excitation error. So this would explain also these green images here. Now the red are explained by the same as before.
17:24
How much time do I have? Okay. Now for the blue, maybe it's easier if I explain it here. So for the blue what happens is that I'm just shifting the quantum dot. So the first one is in this position, the other one is in a symmetrical position to
17:42
the centre. And I notice that let's say this pixel A here is the same as the pixel B here. So why is this? Because if I go and look at the strain, I see that this is connected to this by shifting first and then changing the sign.
18:02
And then if you do this in a node function, it's the same as mid-plane deflection. So it's property two. Cool. Then we manage to explain all the observations, but we also manage to have more results. So basically we have more symmetry properties that come out of the mathematical analysis
18:20
that they also involve changes in this matrix A, which is basically the influence of the potential plus the excitation error. So it has a physical meaning, the matrix A. The results here, they just came out of the mathematical analysis, but I suspect I'm sure that they must have some physical interpretation. I don't know what it is though.
18:41
So if there are physicists here and you have any idea, come and talk to me. And finally, the difficult part. So everything was nice with the proofs because we had the Hermitian property. What happens if we don't? What happens if we include absorption? The problem is that I will lose this Hermitian property that was first important for the
19:03
proofs, but also it gave me conservation of the Euclidean norm. So any results I was having for the bright field, they would also hold for the dark field. So here, even if something survives, I can only talk about bright field. I cannot make any assumptions for the dark field.
19:21
Okay. So what happens? How do we model absorption? We include an imaginary part. And we do this by introducing a positive semi-definite Hermitian matrix, this D. Then we are lucky because there are two different kinds of absorption. It's the small angle absorption, which means that this matrix D will be basically a diagonal
19:42
matrix with the same value on the diagonal, which is great because then I can transform my system to a Hermitian form and everything still holds. But then there is also the white angle absorption, which means that matrix D is a full matrix. So I have no hope there.
20:01
So we will definitely lose something, but something could survive. So before doing any... The proofs were not working. We couldn't do the proofs the same way. So before giving up, we went back to simulations to see if there is some indication there. And what we saw is that for the bright field, it looks as if the mid-plane reflection holds.
20:28
And for the dark field, it looks as if there is a symmetry if you do a mid-plane reflection and sign change. So these were the observations. And then what we managed to do is that we actually proved that indeed mid-plane reflection
20:43
is strong and it survives absorption. For the bright field, I cannot go to the dark field, but we still don't know what happens here. So there is a symmetry in the dark field with mid-plane reflection and sign change that
21:01
we couldn't prove why. We do know from the simulations that it will break if the excitation error is not zero. But somehow in strong beam conditions, there is this weird symmetry that I have no idea where it comes from. Okay. So now I'm done basically. I tried to give you a summary of our latest paper that it's about symmetries in 10 images.
21:26
And there are more examples in the paper that I didn't show here that have to do with general strain profiles, not even rod. And what questions are left to answer from this work is the physical interpretation of
21:40
the other symmetries that we proved. This weird symmetry for the dark field, I would really like to know where it comes from. And also, I would like to see what would happen if I include thickness and composition variations, which would make my Fourier coefficients of the potential z-dependent. And still we have the big project of solving the inverse problem.
22:04
And with this, I'm done and I thank you for your attention.
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