1. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 1 Bayesian Analysis of Ellipsometry Measurements Udo v. Toussaint and Thomas Schwarz-Selinger Ellipsometry Example Bayesian Analysis Results and Conclusion

2. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 2  immer noch komplexer Teilchenzoo! Detection of the change of polarization of linearly polarized light due to the reflection at the sample surface shutter Principles of Ellipsometry here: single wavelength rotating analyzer ellipsometer

3. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 3 Camera Laser polarizer analyzer controller ellipsometer power supplies high pressure lamp monochromator sample stage Jobin Yvon PZ 2000 Ellipsometer wavelength: 632 nm (400 -800 nm) spot size: 10 x 30  m (10  m) motorized xyz sample stage positioning accuracy: 30  m sample thickness: 2 mm measurement range: Å - 30  m measurement accuracy: > 0. 1 Å (1 nm) Principles of Ellipsometry

4. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 4 Principles of Ellipsometry: Reflection of light I solid sample: Fresnel equations complex index of refraction: defined by: : extinction coefficient

5. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 5 Principles of Ellipsometry: Reflection of light II multilayer system: Snell’s law:

6. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 6 Principles of Ellipsometry Measured data: detection of the change of polarization of linearly polarized light due to the reflection at the sample surface in fact we measure and (ellipsometric angles):  each measurement delivers only 2 pieces of information butdepends on: incident angle  n i and  i and d i of each medium i  for a single measurement result is ambiguous if neither n i nor  i is known!

7. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 7  immer noch komplexer Teilchenzoo! Principles of Ellipsometry the plane: Use of empirical models:

8. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 8 Duoplasmatron (Ivan Bizyukov): a-C:H flux probe (bombardment by 1 keV D + ) But sometimes…

9. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 9 Duoplasmatron (Ivan Bizyukov): a-C:H flux probe (bombardment by 1 keV D + ) But sometimes… ? ?

10. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 10 Duoplasmatron (Ivan Bizyukov): a-C:H flux probe (erosion by 1 keV D + ) But sometimes… measurement model for the plasma deposited a-C:H film

11. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 11 Surface reconstruction from interference images Interference images from ellipsometry: 2 data values (angles) per measurement point

12. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 12 Bayesian Model Likelihood: Gaussian likelihood Ill-posed problem: no. of parameters larger than no. of data Use prior-information: optical properties vary on a different length scale Two-scale approach: Nested grids for d and n,  Prior: Bounded, flat :

13. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 13 Bayesian Model Posterior: Bayes theorem Model specifications: 4 layers, 6 unknowns (in 2 layers) domain size inner grid: 3x3 -5x5 Optimization with respect to the parameters: Results were disappointing Why?

14. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 14 Surface reconstruction from interference images Ambigous solutions possible Important: Stay on correct branch of solution Virtually indistinguishable solutions: identical

15. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 15 Surface reconstruction from interference images Interference images from ellipsometry

16. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 16 Surface reconstruction from interference images And what about the edge?

17. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 17 Conclusions & Outlook Ellipsometry is a great non-perturbing surface analytical tool - but ML - evaluation of data may not be straightforward or even misleading  Prior information is essential Derived parameter estimation algorithm works reliable Outlook: Model comparison for number of layers Improved consideration of correlations Conclusions:

18. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 18 experimental setup ICP

19. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 19 film properties like - hydrogen content - density - refractive index are closely correlated T.Schwarz-Selinger, A. von Keudell, W.Jacob, J.Appl. Phys. 86, 3988 (1999) quantification of ellipsometry data

20. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 20 tender spot in general detection of atomic hydrogen in the plasma environment needed: hydrogen sensor 12 scans and measuring the erosion depth with ex-situ-ellipsometry is complicted in the plasma environment erosion of a dense a-C:H-film at 650 K  T. Schwarz-Selinger, W. Jacob, A. von Keudell, JVST A. 18 (3), 995 (2000)

21. udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 21 Profilometry versus Ellipsometry/Reflectometry Profilometry: mechanical contact with the sample  topography Ellipsometry/Reflectometry: optical response of the sample  thickness x refractive index general take home message: