Try XRR, r-ray reflectivity.

As to the title:

It all depends on what you mean by precise.In principle an epitaxial film thickness may be estimated from broadening of the diffraction peak using the Scherrer equation http://dx.doi.org/10.1103/PhysRev.56.978. But be careful! Several factors should be taken into account (e.g. instrumental function, lattice distortion, microstrain).

1) correct.

2) I think that you are trying to estimate the film thickness by absorption. In this case , the substrate works as a mirror with known reflectivity. This should in principle work, but since you are using a superlattice where some of the reflexes might coincide with the substrate, this is a very challenging way to do the job. The peak shape distortions you are seeing might be related to peak broadening on the detector resulting in crosstalk between pixels (spacial and angular information is mixed).

Therefore, the method might have application in some cases, and proper deconvolution should help a lot, but precision might differ strongly from case to case.

I hope I get the question correctly. Your SL peaks (important: their integrated intensity) are supporting points for the envelope function. This envelope function (for the theta-2theta scan) corresponds to the diffraction signal from your InAs/InAsSb bilayer (convolution theorem for Fourier Transforms). However to extract the relative film thicknesses the information may be difficult to extract from one Bragg peak only. The 200 in addition has better contrast between InAs and InAsSb. In any case the fit of such an envelope has to take into account strain and thickness of both layers. Strain gradients maybe neglected as pseudomorphic growth can be expected in your materials with (hopefully) no significant plastic relaxation effects

So SL peaks for two symmetric reflections would carry more information. If I understand correctly, your substrate peak falls exactly on one SL peak. In this case I see no way how this SL peak can be used for data analysis and it should thus be omitted. It is very important to take the integrated intensities as supporting points of the envelope as the peak intensities are affected by SL imperfections (as drifts etc..).

I have fitted SLs in this paper :

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.69.195307

and as you can see in Fig3, I included all kind of imperfection parameters in the SL simulation in order to fit the real data with the only goal of extracting the relative thicknesses AND strains. The effort (writing a ab initio simulation for each superlattice) could have been significantly shortened would I have extracted only the integrated intensity of every SL peak and fitted the envelope.

Or here:

http://www.schuelli.com/ in PhD thesis pages 50-56 and 81-92.

again the effortwas disproportional and the key figure in the data is not to sample at very high resolution the SL (in this particular case maybe, as the SL was perfect) but to get integrated intensities of as many SL peaks as possible.

For your point 1): I would say yes, as your lateral beam size on the sample seems to be much larger than the SL thickness one can consider the illuminated volume to be constant.

Point 2) I hope this is explained above

Wow! Thanks for your diligence in the responses!

1. XRR - How would the XRR signal look like in 2D from a SL specimen such as in this case? I must and will check this, the next opportunity I get.

2. I knew I was choosing the correct forum in RG to ask this question. The expertise exists in RG to "skin teeth" on this subject. Thanks a million for your insight and contributions.

• 10 "InAs/InAsSb bilayers" making the 0.5um film thickness.
• "The (200) in addition has better contrast between InAs and InAsSb". Noted, will try!
• You are correct in that the substrate Bragg peak and the 0th order reflection from the SL epi are "near coincident". However, a closer examination of the symmetric 2nd order reflection since they have good SNR, reveals a slight "misfit". and the "misfit" is non-uniform, center to periphery.  You may download the original image and inspect with MS Paint for measurements, when you please. The comparison of the GaSb substrate with the GaAs data shows the "deviation" which in this case is principally from the 0th order epi reflection. However, tough to deconvolute without the substrate data prior to epi deposition.
• You are also correct that the more harmonics that are included in the RCP (rocking curve profile) the better the precision.
• We are intended in using just one epi 2nd order peak to evaluate "relative" epi thickness directly from the integrated intensity of that one well resolved epi peak with NO CONTRIBUTIONS from the substrate. Would this work to precisely estimate the local film thickness at individual pixel/VOXEL?
• Once we calibrate the technique with "known" standards, then we may even be able to determine absolute values of film thickness.
• Pixel "cross-talk" - We've attempted to address this issue using age old image processing methods. In fact a 3X3 pixel integration seems to work well. We assume there is such "cross talk" and examine data accordingly. There seems to be good match with raw data and the image resolution seems to be improved significantly with higher SNR as a result of such integration.
• What seems to be more interesting is the "cross-talk" between lateral VOXELS (3D defect sub-grain structure). We are able to image and detect this despite the poorer SNR towards the periphery of the image. I'll post some new & better analyses of the old data soon to illustrate my point.
• The key to precision (repeatability) seems to be: (i) proper ID of "background" signal, (ii) correctly "normalizing" the data topographically in order to be able to compare pixel-to-pixel data from a non-homogeneous incident beam and any potential non-homogeneous detector response, and (iii) an abundance of signal (synchrotron better than rotating anode better than sealed tube generator :-).

I'll humbly share more of our endeavor with this data. Thanks for your guidance!

https://www.flickr.com/photos/85210325@N04/9314468316/in/set-72157645018820696

https://www.flickr.com/photos/85210325@N04/9311681125/in/set-72157645018820696/

Tobias! After just sifting through your PhD thesis, I realize that the MBE film quality in our sample may be high enough to create "anomalous scattering effects" as well.

A couple of observations  that I'm unable to account for:

A. The differences in the intensity of the harmonics on the "+" versus the "-" side. Your data seems to be more evenly spread.

B. The enormously high relative intensity of the +2 harmonic (RHS) despite the film thickness of only 0.5um compared with the estimated extinction depth of 10-15um for the (004) reflection. I'd have expected 97% of the signal coming from the substrate. It doesn't seem so? II - Integrated Intensity over 6X12 pixels almost 1mm2 sampling area at the center of the reflection with identical good SNR (>12); IISUB & 0th SL= 526043 au; IIEPI +2 SL=130908 au; Ratio of IIEPI:IISUB=24.8854 %. HOW DO I EXPLAIN THIS? Doesn't seem to make sense. https://www.flickr.com/photos/85210325@N04/7810133504/in/set-72157645018820696

C. The shape of the GaSb substrate (004) is much akin to the GaAs (004) from the same instrument under identical conditions, Pseudo-Voigt type. https://www.flickr.com/photos/85210325@N04/10647827636/in/set-72157648319384526 Whereas, the +2 SL peak profile is near PERFECT Lornetzian fit. What does the change in profile shape between substrate and epi SL +2 imply in terms of Nano structure of the subject VOXEL?  https://www.flickr.com/photos/85210325@N04/15533480889/

BTW what detector were you using for the rocking curve data and the RSM in your PhD work? How long did each take on the synchrotron? What was the size of slit used, if the detector was a 0D Geiger counter?

http://pd.chem.ucl.ac.uk/pdnn/peaks/loren.htm

My understanding is that in the case of a reflection topograph "spatial and angular information is mixed" in the 2D signal. The rocking curve technique however is able to deconvolute the two, "spatial and angular information". That is the beauty of the rocking curve method, it isolates (deconvolutes) the effect of the rocked crystal alone on the topograph. Excluding factors like instrumental effects, beam conditioners, monochromator, slits, detector non-uniformity, etc. So in essence a Berg-Barrett topograph is a convolution of all of the above mentioned factors while the relative Bragg Peak position map and relative FWHM map are parameters measured in the reciprocal space and are devoid the effect of all those factors. In my opinion, based on experimental data, devoid of the effect of pixel-to-pixel "cross-talk" as well. I have the evidence but have to figure out how to present it correctly. Please review the attached image of a series of rocking curve profiles (nearly 300 RCPs, 400 Arc Sec range,

The general problem is to be able to determine the homogeneity of such SL MBE epi films rapidly and in a manufacturing environment. We would like to be able to monitor the following Nano structural parameters for films and substrates non-destructively and in situ:

• Relative micro lattice strain state.
• Relative defect morphology, density and distribution.
• Relative Epi film thickness.
• Relative Epi film periodicity (SL).

Modeling is certainly a method used most often to decipher the Nano structure. Instrument calibration with "known standards" is an alternative or supplementary technique.

Taras! Thanx for all the details. Gave me lots of food for thought.

"since you are using a superlattice where some of the reflexes might coincide with the substrate, this is a very challenging way to do the job". I agree but the technique should have the resolution and dynamic range to accomplish it.

However, if I move away from the substrate and focus on the "known" epi SL +2 harmonic approximately 960 arc sec on the RHS of the GaSb substrate peak, then I should be able to relate the integrated intensity below that epi peak to the diffracting volume (film thickness) somehow, shouldn't I? Obviously, the background ID must be precise in this case besides good SNR.

As far as using the Scherrer type approach is concerned, I'm unfamiliar with its use for a single crystal wafer sample and in the rocking curve technique. It may perhaps provide an average cell size of dislocations in the VOXEL? Do you have any literature I may study on this subject? Thanx!

The one relationship (similar to the Scherrer formula) that I'm familiar with since 1979 @ Rutgers is, Dislocation Density, ρ=β2/9b2 for a Gaussian distribution, b=Burger’s Vector, ρ-Dislocation Density,  integral breadth, β, is related to the FWHM peak width, H, by β=0.5H(π / loge2)1/2. I've misplaced the original 1950's reference. I'll post when I find it. Besides, it is clear from our observations that there is more in the XRD rocking curve analysis than just the FWHM conventionally used. The relative shape of the RCP topographically has other Nano structural details that can be extracted (deconvoluted) to advantage as well.

The motivation behind all the "curve fitting" was to improve the precision topographically of INTEGRATED INTENSITY measurements with varying system generated SNR. Our objective was to minimize the effect of "system shot noise".

By "system shot noise", I mean generated from various sources:

• 2D Detector
• 2D Incident Beam
• Beam path scattering
• Fixed Optics slits
• Random cosmic scintillations

I went back and looked at the data again. Since I'm always working with the original 3D reciprocal space data,  I keep making exciting new discoveries :-) I'll post very soon. Really excited about the new analyses results from "old data" yet. I'm sure there is even more to discover. Stay tuned! I'll post grey scale and pseudo-color images soon. Contrast perception in pseudo color is so much easier than grey scale. You review and decide for yourselves. I'm even closer to the solution for the epi film thickness challenge. Mono-layers should be no problem. The super lattice structure complicates it just a bit.

BASICS: (Just checking with the "Master Mind" RG membership:-)

The diffracted intensity Id is a convolution of the incident beam Ii and Iµ a spatial function related to the Nano structure of the sample VOXEL being examined. Let us take the example of SiC 6H wafer standard, (0006) reflection: https://www.flickr.com/photos/85210325@N04/15565276200/

Ii - Incident Beam intensity function - f(x,y)

Iµ - Micro-structure (Nano structure) related and function of f(x,y, & ω)

Id - Diffracted Beam Intensity f(x,y, & ω)

Id = Ii * Iµ      * - Convolution

In the XRD rocking curve method we deconvolute the Iµ by rocking only the subject crystal about the diffractometer axis while maintaining a direction coplanar (preferably collinear) at least with the rocking/diffractometer axis as well ("optimized"). The relative reciprocal space parameter maps (like Bragg Peak Position, FWHM, Integrated Intensity) are devoid of all factors other than inherent instrumental shape determinable using a "known sample".

I'll add more notes to further describe the results of the analyses shown below. These are the first steps towards being able to detect the differences due to epi film thickness variation on the integrated intensity from the epi alone fully deconvoluted from the substrate signal below it. In this case, we would select the area (to compute integrated intensity) under the +2 SL epi (004) peak located 960 Arc Sec on the RHS (high 2ɵ side) to avoid any substrate related signal.

The integrated intensity for the SiC (0006) diffracted beam shows apparent spatial intensity modulations. These intensity modulations prompted us to examine all the data again. Using pseudo color images we were able to detect similar pattern in the incident beam and the diffracted beam. See image below for more details. The pseudo color scale was stretched from minimum value to the maximum value to maximize contrast. It was only after such image enhancements (pseudo color) that the intensity modulations in the original incident beam were discovered many years after the original data collection at BNL for this SiC 6H 75mm CVD grown production wafer. Excited, like a kid in a toy shop!

The enhancement to the contrast in the reciprocal space images is significant as a result of proper "back-ground" ID, 3X3 spatially weighted integration (" box blurring") and data "normalization" w.r.t. the integrated intensity for individual topographic pixel.

PS - Relative Bragg Peak Shift

FWHM - Relative FWHM of Bragg profile

II - Relative  Integrated Intensity per pixel

TOPO - (0006) diffraction topograph both sample nano structure and the beam conditioner (monochromator, collimator) are convoluted in each image frame.

Y-ω Map - A series (nearly 300) of RCPs (Rocking Curve Profile, Bragg Profile) along a column of pixels on a vertical cursor line approximately at the center of the topograph.

Grey - 8Bit Grey scale images demonstrating lack of discernible contrast compared with the pseudo color alternative "stretched to the max" in color range (256).

11, 14, 17 - number of image frames integrated to create the image @

Here is another example of the underlying sub structure we found with an old data set from the early 2000's kindly provided by Panalytical for a ZnSe (224) asymmetric reflection rocking curve scan using a PIXcel 3D area detector with 256x256 pixels 55um/pixel.

Raw Data: https://www.youtube.com/watch?v=OQ2feymYzAA

Pay attention to image frames from minutes 1:10 to 1:14 into the video. Now we'll compare this raw data from the PIXcel 3D to pseudo color images using "background reduction", stretching the image dynamic range (DR) across the entire pseudo color range and we've also used a 3X3 pixel integration ("blurring"). As a result, the contrast perceivable in reciprocal space is AMAZING! Judge for yourself, when I post the comparison soon.

Here is just a taste:

FWHM Map: https://www.flickr.com/photos/85210325@N04/15747701901/

Integrated Intensity Map: https://www.flickr.com/photos/85210325@N04/15751131322/

Here is the comparison between grey scale image, pseudo color image and the image with BG (background) subtraction and pseudo color (PC) "stretch" over the dynamic range (DR) of raw data (minimum to maximum). https://www.flickr.com/photos/85210325@N04/15136173733/

(1). It is nearly impossible to detect the Nano structural details (checker board pattern) seen at 1:11 minutes into the raw data video.

(2). Once the data is viewed as pseudo color the image clarity (contrast) improves.

(3). Minus BG plus PC "stretch" shows contrast best.

These details were clearly overlooked in the prior analyses due to lack of perceived image contrast.

https://www.youtube.com/watch?v=OQ2feymYzAA

This is the same analysis repeated for the data from our favorite Super Lattice (SL epi) sample. I've had this data for a couple of years now and had no clue what was hidden in it. I'm glad I didn't neglect the real time 2D XRD data but recorded it at the best possible spatial resolution, dynamic range & sensitivity available at that time.

SL sample contrasted with SiC standard and GaAs standard. Data acquired through the help of relationships at BNL NY and AFRL OH. Thanks! https://www.flickr.com/photos/85210325@N04/15745207681/in/photostream/lightbox/

I had an enlightening conversation about this data with my friend in OH. Thanks Kurt! I feel a lot more confident about this approach now.

Original relative reciprocal space data compared with "normalized" data: https://www.flickr.com/photos/85210325@N04/15577802698/

Here is more excitement!

Data from way back in 2010 with one of our first prototype Bragg XRD Microscopes attached to a Rigaku. I had no idea the monochromated incident beam without soller slits had so much "character" in pseudo-color. My eyes were virtually blind to the variations in the grey scale images until I was able to review the pseudo color images. See for yourselves!

Rigaku experimental set-up:  https://www.flickr.com/photos/85210325@N04/15588381800/

Incident Beam Characteristics: https://www.flickr.com/photos/85210325@N04/15773182865/in/photostream/

After a whole lot more digging into the old data! Here is clear evidence that the relative integrated intensity maps for "optically flat" samples have direct correlation to the incident beam relative intensity profile (2D): https://www.flickr.com/photos/85210325@N04/15602951869/

The incident beam image (PC - pseudo color) clearly shows the topograph of the combined effects of all the components in the Hybrid Beam Conditioner plus soller slit.

A Panalytical X'Pert/MRD with a hybrid Bartel (220) Ge beam conditioner and a sealed tube system was used to acquire this data utilizing the AXIS12257608 2D real time Bragg XRD Microscope attachment: https://www.flickr.com/photos/85210325@N04/8001095415/in/set-72157632729013664