1. COSIRES 2004 © Matej Mayer Bayesian Reconstruction of Surface Roughness and Depth Profiles M. Mayer 1, R. Fischer 1, S. Lindig 1, U. von Toussaint 1, R. Stark 2, V. Dose 1 1 Max-Planck-Institut für Plasmaphysik, EURATOM Association, Garching, Germany 2 University of Munich, Section Crystallography, München, Germany Introduction to Bayesian data analysis Improvement of the detector energy resolution by deconvolution of apparatus function Reconstruction of depth profiles of elements Reconstruction of surface roughness profiles with RBS

2. COSIRES 2004 © Matej Mayer MeV Ion Beam Analysis Sample MeV ions E Elemental composition and depth profiles of elements  Quantitative without reference samples Overlap of mass- and depth-information  Complicated data analysis Limited energy resolution of solid state detectors  Limits to mass- and depth resolution

3. COSIRES 2004 © Matej Mayer “Classical” IBA Data Analysis Parameters  (layer thickness, layer composition,...) Forward calculation p(d , I) = f(  2 ) “Classical” data analysis  fitting: 1. Assume sample parameters  2. Perform forward calculation, calculate  2 3. Vary  until  2 is minimal Sample

4. COSIRES 2004 © Matej Mayer Bayesian Data Analysis Parameters  Forward calculation p(d , I) Backward calculation, inverse problem p(  d, I) Bayes’ theorem p(  I) : Prior probability I : Additional background information Sample

5. COSIRES 2004 © Matej Mayer Bayesian Data Analysis (2) How to choose the prior probability p(  I) ?  Most uninformative prior for spectra is the entropic prior J. Skilling 1991  Solution with maximum information entropy  Additional previous information about  can be included Many solutions with identical entropy  Select simplest model consistent with the data  Adaptive kernels R. Fischer et al., 1996  Favours smooth solutions Marginalization  Allows to eliminate uninteresting variables

6. COSIRES 2004 © Matej Mayer Bayesian Data Analysis (3) The resulting distribution p(  |d, I) contains the complete knowledge of   mean value, most probable value of   error interval for  Note that  2 - minimising (fitting) will find most probable value

7. COSIRES 2004 © Matej Mayer Deconvolution of the Apparatus Function < 1 keV 10-15 keV Detector Sample straggling Measured spectrum: A : Apparatus function f(E) : Spectrum for “ideal” detector Discrete spectrum: Direct inversion: Does not work in presence of noise  MeV

8. COSIRES 2004 © Matej Mayer Deconvolution of the Apparatus Function (2) Example: Mock data set blurred with Gaussian apparatus function noise added with Poisson statistics R. Fischer et al., NIM B136-138 (1998) 1140

9. COSIRES 2004 © Matej Mayer Deconvolution of the Apparatus Function (3) Example: Cu on Si Apparatus function measured with ultra-thin Co layer Initial resolution: 19 keV FWHM After deconvolution: 3 keV FWHM  better by factor 3 than theoretical limit of 8 keV for solid state detectors 2.6 MeV 4 He, 165° R. Fischer et al., Phys. Rev. E55 (1997) 6667 R. Fischer et al., NIM B136-138 (1998) 1140

10. COSIRES 2004 © Matej Mayer Deconvolution of the Apparatus Function (4) Example: Cu on Si Error of apparatus function is taken into account Error bars, confidence intervals are obtained R. Fischer et al., Phys. Rev. E55 (1997) 6667 R. Fischer et al., NIM B136-138 (1998) 1140

11. COSIRES 2004 © Matej Mayer Deconvolution of the Apparatus Function (5) Example: Co-Au multilayer Apparatus function for Co and Au from ultra-thin films R. Fischer et al., Phys. Rev. E55 (1997) 6667 R. Fischer et al., NIM B136-138 (1998) 1140

12. COSIRES 2004 © Matej Mayer Reconstruction of depth profiles

13. COSIRES 2004 © Matej Mayer Reconstruction of depth profiles Forward calculation “Classical” data analysis: Minimise  2 by varying elemental concentrations in layers Many parameters (  100)  simulated annealing C. Jeynes et al., J. Phys. D: Appl. Phys. 36 (2003) R97  Fast + reliable, sufficient for many applications But: Not a full solution of the inverse problem  Exactly one result (with  2 min )  p(  |d, I) remains unknown  no error bars or confidence intervals 

14. COSIRES 2004 © Matej Mayer Reconstruction of depth profiles (2) p(  |d, I) Bayesian data analysis: Calculate p(  |d, I) using maximum entropy prior  : Concentrations of elements in the layers 

15. COSIRES 2004 © Matej Mayer Reconstruction of depth profiles (2) 12 C 13 C D 12 C Plasma Mixture of 13 C/ 12 C due to plasma exposure  Depth profiles from Bayesian data analysis Energy of backscattered 4 He [keV] Counts before after (scaled) 13 C 12 C U. von Toussaint et al., New Journal of Physics 1 (1999) 11.1

16. COSIRES 2004 © Matej Mayer Reconstruction of depth profiles (3) Depth [10 15 atoms/cm 2 ] Concentration before after Channel Counts Data Simulation after O 13 C 12 C U. von Toussaint et al., New Journal of Physics 1 (1999) 11.1 Note asymmetric confidence intervals

17. COSIRES 2004 © Matej Mayer Reconstruction of surface roughness distributions Other types of roughness: N. Barradas et al., NIM B217 (2004) 479 Layer roughness Distribution p(d) Substrate roughness Distribution p(  ) In on Si, 2 MeV 4 He, 165°

18. COSIRES 2004 © Matej Mayer Reconstruction of surface roughness distributions (2) = + + +... Correlation effects are neglected  valid, if lateral variation  > d for typical RBS angles of 160°-170° M. Mayer, NIM B194 (2002) 177  Distribution p(d)

19. COSIRES 2004 © Matej Mayer Can we use RBS for measuring p(d) without prior knowledge of the distribution function? Reconstruction of surface roughness distributions (3) 1.5 MeV 4 He, Ni on C 2 MeV 4 He, Ni/Al/O on C Distribution p(d) ?   -distribution is successful in many cases M. Mayer, NIM B194 (2002) 177

20. COSIRES 2004 © Matej Mayer Reconstruction of surface roughness distributions (4) 200 nm In on Si SEM 2  m AFM 2  m RBS 2 MeV 4 He, 165° Reconstruction of p(d) from RBS

21. COSIRES 2004 © Matej Mayer Reconstruction of surface roughness distributions (5) Film thickness distributionRBS spectrum Simulation How well does this compare with other methods?

22. COSIRES 2004 © Matej Mayer Reconstruction of surface roughness distributions (6) 2  m backscattered electrons 25 keV, normal incidence secondary electrons tilt 70°  Intensity of backscattered electrons depends on In thickness  Thickness distribution from grey-values

23. COSIRES 2004 © Matej Mayer Reconstruction of surface roughness distributions (7) 2  m Good agreement for large blobs (around 200 nm) Small blobs are only visible with RBS and SEM, but not AFM

24. COSIRES 2004 © Matej Mayer Disadvantages of Bayesian Data Analysis Computational: Complicated (and sometimes scaring) mathematics Longer computing times, compared to fitting Experimental: High quality experimental data required – apparatus function with good statistics – reliable energy calibration –...  longer experimental time

25. COSIRES 2004 © Matej Mayer Conclusions Bayesian data analysis provides a consistent probabilistic theory for the solution of inverse problems  Determines sample parameters plus confidence intervals  Uncertainties of input parameters can be taken into account Deconvolution of apparatus function: Resolution improvement by factor 6 Depth profiles of elements with confidence intervals Surface-roughness distribution from RBS  New method for surface roughness measurements