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Strange reflectance from periodic structure

  1. Jan 26, 2014 #1
    Hi, last semester I did a project with two fellow students. We made a numerical model to calculate the reflectance from a periodic V-shaped structure of silicon similar to this:


    In the course of doing so, we came across something we could not explain. I have placed an image of it below:


    What you should notice, is that at a specific point it is only the 0'th order reflectance, which propagates. The wavelength at this point matches exactly the height of the structure. We saw this for any structure we modelled no matter the height.

    Our model was quite good as our professor assisted us a lot with it, so we have no reason to doubt our results, and they were in good agreement with our measurements, so no problems there.
    So far nobody at the university was able to explain this phenomenon, so now I am trying here, to see if someone might know of it or possibly have an idea, of how it could be explained?

    The model was made for a period structure, which was infinitely deep (so that no light could be reflected from the bottom of the structure). In general it gave a realistic view of how these V-shaped grooves could reduce the reflectance from the surface of a silicon wafer.
    Last edited: Jan 26, 2014
  2. jcsd
  3. Jan 26, 2014 #2

    Simon Bridge

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    If nobody at the university could account for it - what did they try?
    Since the height is about a wavelength, I would suspect an interference effect.

    I don't see any place where either curve hits zero reflectance though there is a sharp drop at about 450nm (about half the structure spacing).

    There seems to be some mismatch in terminology though:
    ... so what is the difference between height and depth here?
    Do you mean that the finite-height triangle structure is supposed to model an infinitely deep structure?

    Note: being assisted by a professor says nothing about how good the model is.
    How good it is depends on how it accounts for different effects or handles the inevitable artifacts in the calculation process... ultimately on how it matches an experiment.

    Nitpick: the label on the graph looks odd to me. A plot of the difference between these would be a single curve, rtot-r0, which is not what is plotted - the graph shows two plots on the same axis: comparing the zeroth-order and total reflectance vs wavelength curves.
    Also "period" is for time, rather than distance... but well...
  4. Jan 27, 2014 #3
    Well, I don't think they tried much, they looked through some articles to see if someone had plotted something similar. They talked to some colleges to see if somebody knew, what it might be. We discussed doing a "test" using a laser with a specific wavelength matching that height and see what would happen to the diffraction orders, but even doing that would not explain it, just validate, what we saw.

    Maybe I should note, that this is for s-polarised light only, and the angle of incidence is 90 degrees. I don't think it's supposed to hit zero reflectance, but we do have some graphs, where the 0'th order reflection almost hits zero.

    The point of it was for instance usage in solar cells. What I meant about depth of the structure, was below the grooves, my bad with the formulation. The point is no light is reflected backwards from underneath the grooves, so no interference is caused in this manner. The grooves are as deep as the pyramids are high.

    We kind of used our professors program to model and calculate the reflectance. It's the kind of model, where you don't make too many approximations (and it's some good approximations), and you calculate the electric field inside the structure in a quite accurate manner. Some of our calculations took very long (like 24 hours with a normal computer). For instance the model could work for structures with a period of 5 ┬Ám or more, it would just take long... but it could also work for small structures of say 100 nm period. Faster models are somewhat limited to the size of the structures, and thus less accurate.

    Well we plotted both, because we were interested in seeing the difference between them. We could only measure the 0'th order reflectance, so when we compared model to results, we also wanted to say "this is not the entire reflectance", as there may be quite a difference between them.
    You do of course have a good point, because we could show the same effects in a "more appropriate" manner, I will remember that for another time. Also you can see that when the wavelength becomes larger than the period, there is no longer any reflected diffraction, which was also expected and as thus is yet another validation of the model. We also used it to for instance plot the reflectence from a planar surface, just to do some more validation. We also looked at cut-off effects, which show when a new diffraction order propagates, and we also saw this as peaks in a graph at the spots we calculated.

    Well I think in this study area, everyone use period. It's kind of like a translational vector though, which might have been more correct. It comes from Bloch conditions, which you usually use to define a periodic structure. Whenever you move one "period" you arrive the same spot in a new but identical groove. And as we don't model every atom, we can't really use lattice vectors and so on.
    Last edited: Jan 27, 2014
  5. Jan 27, 2014 #4


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    It's a sad fact but you really have to learn to nag them and nag them until they are prepared to give your problem more than 5 seconds of thought. The main thing is to sound verrrrry enthusiastic yourself and that may shame one of them into getting interested in your problem. Don't act all helpless and clueless. Make them think that their input can make a difference.
    Many teaching staff in University are only there because they want to do their own research (which may be far more interesting than most of the problems they hand out to students). Study your 'people skills' to get what you want out of the lazy sods haha.
  6. Jan 27, 2014 #5

    Simon Bridge

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    Me neither.
    " nobody at the university was able to explain this phenomenon,..." sounds really good until you realize that nobody actually tried hard, if at all. All you really have is that the immediate circle of people who were asked (guessing: order of a half-dozen?) did not think they'd seen a graph like that in recent memory after a few seconds thought. That's not really anything: it's not unusual for something to turn up that defies immediate and definitive explanation.

    Such a test is not supposed to validate the model, but to refute it.
    If you do not get the result predicted in the model, then the effect is an artifact of the model - or the software/hardware combination the math is run on.

    If nobody else has been getting these graphs off the same type of model, but high agreement between the model and experiment, then the chances are good that the effect is just something gone wrong in the calculation.

    I've seen this happen: a computer simulation shows an odd resonance, experiments do not show it, nobody else has seen it, further investigation showed that there was a glitch in the way the software handled a particular part of the calculation... which is often why people are reluctant to go into it when it happens at the student level.

    Finding that took a lot of very boring work. It's more usual to rewrite the simulation in some (in principle) transparent form ... for people I worked with that usually meant using gcc and writing in c or c++. If the model behaves itself in that form, then the error is put down to some unknown artifact in the previous setup and left at that. Maybe alert the software developer to the glitch. It's not your job to debug someone elses code.

    If the error sticks around then the model can confidently be adjusted to better match reality: congratulations, you may have discovered something :)

    But there remains not enough information right now to say anything beyond: "looks like an interference effect".
    Unless you get lucky and someone here has seen it before, and investigated. The only other thing I can think of is that sometimes there are features in low-order approaches which disappear when higher orders are taken into account. You could probably figure that out by doing a simplified calculation by hand.
  7. Jan 27, 2014 #6
    That is what we concluded as well, and it's also what our professor said, :-)

    That was what I was hoping for, at least giving it a try would not cause any harm. I doubt we will be doing the laser experiment, as it is not too simple to perform, especially since the structures we synthesised weren't perfectly periodic, and it would require a wavelength of several microns due to the sizes of the structures we synthesised. Thanks anyway!
  8. Jan 29, 2014 #7

    Andy Resnick

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    I'm a little confused- are you simply modeling a reflection grating?
  9. Jan 29, 2014 #8


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    Have you tried different blaze angles?
  10. Jan 30, 2014 #9
    Well I don't think it is simple to do, but I modolled the reflectance from a specific grating as the one shown in the beginning.

    No we only made pyramidal structure as the one shown.

    The structure was locked by an etching relationship between the (100) and (111) crystallographic planes, because we could etch in the (100) direction 100 times faster than the (111) direction, thus forming the structures as seen in the post. This alsmo meant, that the relationship between width and depth of a groove was locked due to the angle between the (100) and (111) planes of 54.7 degrees.
  11. Jan 31, 2014 #10

    Andy Resnick

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    I didn't mean to imply that the simulation was trivial, only that there is a single, clear, goal. A few questions:

    1) what was the polarization of incident light?
    2) how is 'reflectance' defined, exactly- over a whole hemisphere, a particular scatting angle, etc.?
    3) have you compared your results with that already reported, or against validated simulation code? the grating geometry looks 'standard' enough that someone has probably already done this.
  12. Jan 31, 2014 #11
    Ah yeah it was quite clear, what the goal was :-)

    1) It was s-polarised light.

    2) Hmm, as the structure was periodic we only modelled a single 'period' in the grating. We used Green's functions to solve the electric field inside the scatterer, and we made it periodic with Bloch conditions, so we only had to model one groove. We also made substrate beneath the bottom of the grooves, which the Green's function saw as infinitely deep, in other words no light would be reflected from the bottom of the structure (everything not reflected from the surface would eventually be absorbed). I made a quick sketch of the area we modelled:
    Perhaps I should note, that we only modelled in in 2-dimensions as above, because the 3rd dimension would just stretch out in a line to infinity. So we have long lines of grooves, which seen from the side looks like what I have shown above.

    At the particular graph I put here, it would be normal incidence light. We could, however, freely change the angle and did also model graphs doing this. For instance we couldn't measure reflectance at normal incidence, so when we compared our model to experiments we used a rather high angle of incidence.

    3) yeah, we are fairly certain there is nothing wrong with the results. Our professor also worked a lot with this and wrote several articles about it, though he used different models (he wanted to do it with something he hadn't already done), but the results matched quite well. For instance we took a reflectance vs wavelength graph from an article, we then chose the exact same size of structure and modelled the reflectance spectrum, and it matched.
    we also made other tests, such as using the model to plot reflectance from a flat interface, so we could see it was correct.
    We also did some kind of validation of the code, but that was more our professor doing it. I don't think I totally understood, how he did that, but it was like he plotted 2 different arbitrary color plots of different size, and then checked if small one could be found somewhere in the larger one. But that was more because of some problems that came along.
    And yeah it's quite standard, but we didn't find anyone that made that exact type of plot, which we did, where we could see this depth of structure = wavelength resulting in only 0'th order reflectance.

    Also our model actually made a lot of small square area elements where the electric field is assumed constant through a single element, it then calculates the scattering contribution from each element in a single element, and doing this for every element and then taking some arbitrary point outside the scatterer to calculate the electric field at this point.

    So my point is, when normal incident light is the case, perhaps the incoming light "sees" all parts of the structure as being a small flat surface, because it consists of a lot of small square elements. I don't know if that could explain some special case of interference perhaps?
    This was the case, when the wavelength became larger than the period, and we thus saw the reflected wave as being a plane wave, which makes sense, and you can also see that in the graph at a wavelength of 900 nm, as that equals the period.
    However, the depth of the structure would usually be less than the period, yet resulting in a plane wave.
    Last edited: Jan 31, 2014
  13. Jan 31, 2014 #12
    How did you verify that the angle of incidence was exactly 90 degrees?
  14. Jan 31, 2014 #13
    Uhm, I'm not sure I get what you mean. We just put it to whatever value in the program (Matlab).
  15. Jan 31, 2014 #14
    If the angle of incidence was possibly off in the experiment, it might interfere with itself and give you data that may have some coherence with the surface it was being reflected off of. Unless the values you are talking about are only from the model.
  16. Jan 31, 2014 #15
    Yeah, they are only from the model. We did not actually show this phenomenom I am wondering about experimentially, because well, we didn't have time. :-)

    But we used an ellipsometer for the experiments, it had an angle adjustment, and you have this type of "alignment"-camera-thing, which you use to align the stage beforehand. You aligned two things, I am sure one of them was focus, so I would guess the other, was the alignment of the stage, and thus the angle. So I think it was quite accurate.
  17. Jan 31, 2014 #16
    Okay, I was just throwing out ideas as to why this may have been happening.
  18. Jan 31, 2014 #17

    Claude Bile

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    Without details of how the model was actually calculated, it is terribly difficult offer a substantial critique.

    Judging from your comments and the fact that model matches experiment, You probably ought to (attempt to) publish and canvas some feedback from a proper reviewer.


    P.S. I would not assume that this problem has previously been modeled in-depth - "pointy" structures are notoriously difficult to model.
  19. Jan 31, 2014 #18
    Yeah I am aware of all this.

    My point was quite simply to see, if somebody had seen, or had an idea of what this interference pattern I saw in my model could be. So I just wanted people to assume, the model was working as it should, and then comment on the graph.
    Although it's probably hard without having and understanding the model.
  20. Feb 3, 2014 #19

    Andy Resnick

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    I agree with this- the 'blaze angle' is significantly larger than anything I'm familiar with. The Richardson Grating book discusses an 'anomaly' or "Wood's anomaly" in s-polarization scattering efficiency, but this is for much lower blaze angles.
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