how to calculate unpolarized reflectance from PyElli Result

[paola-dececco]

Hello,

I wanted to use PyElli to analyze data from a normal incidence reflectometer. None of the field returned in Result seem to contain unpolarized, normal reflectance (Result.R does not match the result from other solvers)

Help!

Paola

I tested it on a simple stack: Air - 20A,TaucLorentz oscillator sum - Si. below is the code

NSi=sidata['N'] # complex refractive index n+ik
Si_table = elli.Table(TLdata['Wvl'],NSi)
Layer = [elli.Layer(DM_SnO.get_mat(), 20)]
exp = elli.Experiment(elli.Structure(elli.AIR, Layer, Si_table.get_mat()),TLdata['Wvl'],0)
Result=exp.evaluate(solver=elli.Solver2x2)
Rdata['PyElli Reflectance'] = Result.R
Rdata['PyElli WVL'] = exp.lbda

[domna]

Hey Paola,

at first glance everything looks fine to me, the R field is correct to get the reflectance of the material stack. We did, however, mainly benchmark against ellipsometry calculations, so it could also be that there is still a small error in the extraction of this value (Maybe @MarJMue has more experience with calculating reflectance and an example?). I will try to construct a small example later on to see whether we can reproduce literature results correctly.

If you are able and willing to share your model data (mainly the NSi an DM_Sn0 data) and if you have experimental data you want to compare to, then I'll have a look at this as well and see if I can spot any problems.

[domna]

I now had time to have a look at this.
I reproduced your example and this is the curve I get when I extract R (with a Silicon substrate):

And this is how the dielectric function of the SnO (generated with DM_SnO.get_dielectric_df().plot()) looks like:

Is this in accordance with your findings?

However, I think it would be helpful if you could share your reference data for this if this is possible (I don't have a wvase program available, so I cannot test it myself). Maybe there is some transformation of the data in wvase we do not do in pyElli, because as you explained the dielectric functions from wvase and from pyElli match as long as you set all Eg's in wvase to one fixed value, right? IMO the reflectance should also match in this case.

[paola-dececco]

I am putting together the data and we'll upload them soon. Sorry for the delay

[paola-dececco]

So as for the Eg [on for all oscillators or one per oscillators] I agree with you that one EG for all oscillators makes more sense. On the other end - Wvase offer both versions, and historically my company uses the one with many Eg, might be hard to convince people to move their baseline - even if it would make sense. I am attaching Wvase manual with details description page 8 - 23 talks about the 2 Tauc-Lorentz implementations
WVASE_Manual.pdf
most likely I am making a mistake setting up PyElli - but if you think PyElli could be improved with access to WVase, I can generate any other test data you'd like me to and share it.

[domna]

Yes, I see. From how I understand it is that they are just adding different Tauc-Lorentz functions together with completely separated parameters. You can achieve this in pyElli by just adding completely separated Tauc-Lorentz dispersions with only one oscillator together, like so:

DM_SnO_multi = elli.TaucLorentz(Eg = params["SnO1_Eg"]).add(A=params["SnO1_A"], E=params["SnO1_E"], C=params["SnO1_C"])
DM_SnO_multi += elli.TaucLorentz(Eg = params["SnO2_Eg"]).add(A=params["SnO2_A"], E=params["SnO2_E"], C=params["SnO2_C"])
DM_SnO_multi += elli.TaucLorentz(Eg = params["SnO3_Eg"]).add(A=params["SnO3_A"], E=params["SnO3_E"], C=params["SnO3_C"])

DM_SnO_multi += params['Einf'].value

You may use this for testing now. If you find it useful we can add a TaucLorentzMulti dispersion or so.

I used this parameterset and get the exact same result from above:

params = ParamsHist()
params.add("SnO1_A", value=100, min=-100, max=100, vary=False)
params.add("SnO1_E", value=7, min=-40000, max=40000, vary=False)
params.add("SnO1_C", value=3, min=-40000, max=40000, vary=False)

params.add("SnO2_A", value=75, min=-100, max=100, vary=False)
params.add("SnO2_E", value=5, min=-40000, max=40000, vary=False)
params.add("SnO2_C", value=0.5, min=-40000, max=40000, vary=False)

params.add("SnO3_A", value=25, min=-100, max=100, vary=False)
params.add("SnO3_E", value=3, min=-40000, max=40000, vary=False)
params.add("SnO3_C", value=0.5, min=-40000, max=40000, vary=False)

params.add("Eg", value=0, min=0, max=40000, vary=False)
params.add("Einf", value=1.5, min=-40000, max=40000, vary=True)

params.add("SnO1_Eg", value=0, min=0, max=40000, vary=False)
params.add("SnO2_Eg", value=0, min=0, max=40000, vary=False)
params.add("SnO3_Eg", value=0, min=0, max=40000, vary=False)

[MarJMue]

Hi Paola,
as @domna mentioned we both have no access to woollam devices and it is always helpful if someone benchmarks our code agianst other software. So we are thankful for all requests like this.
I will try to find out more, how WVASE calculates reflection.

[paola-dececco]

ReflectanceComparison.zip
Again sorry for the delay, I am not exactly the best python programmer - switch from MATlab just a month ago, and wanted to write a notebook to send you with data as well. In the zip folder you should find:

1. SiliconDM: standard Si dispersion table 200-800nm range
2. LorentzDM = Dispersion table generated by Wvase as 2 Lorentz oscillators with einf=1.5 in the 200-800nm range
3. Excel file with Reflectance calculated by Wvase for Thickness=20A (nominal), 2A, 68A, 134A and 200A for a film stack of Air/LorentzDM [Thickness]/SiliconDM

The notebook attempt to

1. Compute in PyElli the same Lorentz dispersion - results are close but not exactly the same. Since e2 is the same, but e1 is slightly different I wonder if it is a Cauchy integral convergence issue
2. Compute in PyElli reflectance for all the thicknesses mentioned above and compare with WVase - here we are really far.

[paola-dececco]

Unfortunately my company 'security' does not allow me to access the NOMAD site. I will have to escalate a request for it - which will take few days...

[domna]

No worries, it's not a necessity. It just was convenient in another case we tried out some things. So feel free to join or not to join, whatever you prefer. It's also no problem for me to share code snippets with you.

[paola-dececco]

GOOD news!
I did find the discrepancy. your Lorentz energy assume einf=1. mine assume einf=0, so for eing=1.5 in PyElli I need to set einf=0.5, and it matches Wvase with Einf=1.5
So once again I can get the dispersion model to match exactly.
Now the only difference is in the normal unpolarized reflectance... looking at the calcs Result.R=Rss/Rpp I think what I need is sqrt(Rss Rss_conj+RppRpp_conj) let me try that

[paola-dececco]

EVEN BETTER news!
I think all discrepancy is resolved. When I look into result.py to change the R definition - I realized it is the documentation which is confusing (Rss/Rpp) but the code is correct that is where it hit me... I work in A because the largest layer I see is <50A nut pyElli works... in nm! now everything matches perfectly. So I will have to be extra careful with the parameterization of dispersion model, or decide to add some to pyElli - let me know if you'd be interested in additional flavors of dispersion model if I go that route. Thank you Thank you!

[domna]

This is indeed great news 🎉

Yes, sorry for the wrong documentation. I spotted this as well while checking this but found the underlying code correct. Anyways, we will correct this (and thank you for spotting this!).
Indeed, we do use einf=1 for the Lorentz functions. Maybe we should drop this to make the building of the models via summing cleaner and clearer.

Thank you for your feedback. It is very valuable for us to check this against other programs and to find common pitfalls while working with pyElli. It would be really nice and helpful if you let us now your experience with different models and what we could do to make it easier for people coming form other ellipsometry software. We definitely are interested in flavours of dispersion models.

[paola-dececco]

Erratum_Jellison Modine Tauc Lorentz.pdf
Maybe you are aware - in the original paper there was an error in the formula for the Tauc-Lorentz. probably not a problem in pYElli - sharing just in case

[domna]

Thank you. Yes we are aware of this Erratum and our model is build according to the corrected formula.