Bayesian Reconstruction of Surface Roughness and Depth Profiles
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 COSIRES 2004 © Matej Mayer
Me. V Ion Beam Analysis Sample Me. V 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 COSIRES 2004 © Matej Mayer
“Classical” IBA Data Analysis Forward calculation Sample p(d q, I) = f( 2) Parameters q (layer thickness, layer composition, . . . ) “Classical” data analysis fitting: 1. Assume sample parameters q 2. Perform forward calculation, calculate 2 3. Vary q until 2 is minimal COSIRES 2004 © Matej Mayer
Bayesian Data Analysis Forward calculation Sample p(d q, I) Parameters q p(q d, I) Backward calculation, inverse problem p(q I): Prior probability Bayes’ theorem COSIRES 2004 © Matej Mayer I: Additional background information
Bayesian Data Analysis (2) How to choose the prior probability p(q I)? Most uninformative prior for spectra is the entropic prior J. Skilling 1991 Solution with maximum information entropy Additional previous information about q 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 COSIRES 2004 © Matej Mayer
Bayesian Data Analysis (3) The resulting distribution p(q|d, I) contains the complete knowledge of q mean value, most probable value of q error interval for q Note that 2 -minimising (fitting) will find most probable value COSIRES 2004 © Matej Mayer
Deconvolution of the Apparatus Function < 1 ke. V Sample Me. V Detector straggling 10 -15 ke. V Measured spectrum: < 1 ke. V A: Apparatus function f(E): Spectrum for “ideal” detector Discrete spectrum: Direct inversion: COSIRES 2004 © Matej Mayer Does not work in presence of noise
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 B 136 -138 (1998) 1140 COSIRES 2004 © Matej Mayer
Deconvolution of the Apparatus Function (3) Example: Cu on Si 2. 6 Me. V 4 He, 165° • Apparatus function measured with ultra-thin Co layer • Initial resolution: 19 ke. V FWHM • After deconvolution: 3 ke. V FWHM better by factor 3 than theoretical limit of 8 ke. V for solid state detectors R. Fischer et al. , Phys. Rev. E 55 (1997) 6667 R. Fischer et al. , NIM B 136 -138 (1998) 1140 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. E 55 (1997) 6667 R. Fischer et al. , NIM B 136 -138 (1998) 1140 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. E 55 (1997) 6667 R. Fischer et al. , NIM B 136 -138 (1998) 1140 COSIRES 2004 © Matej Mayer
Reconstruction of depth profiles COSIRES 2004 © Matej Mayer
Reconstruction of depth profiles D 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) R 97 Fast + reliable, sufficient for many applications But: Not a full solution of the inverse problem Exactly one result (with 2 min) p(q |d, I) remains unknown no error bars or confidence intervals COSIRES 2004 © Matej Mayer
Reconstruction of depth profiles (2) D p(q|d, I) Bayesian data analysis: • Calculate p(q|d, I) using maximum entropy prior • q : Concentrations of elements in the layers COSIRES 2004 © Matej Mayer
Reconstruction of depth profiles (2) Plasma D 12 C 13 C after 13 C (scaled) Counts 12 C before Mixture of 13 C/12 C due to plasma exposure Depth profiles from Bayesian data analysis Energy of backscattered 4 He [ke. V] U. von Toussaint et al. , New Journal of Physics 1 (1999) 11. 1 COSIRES 2004 © Matej Mayer
before 12 C Counts Concentration Reconstruction of depth profiles (3) Data Simulation 13 C Concentration after 12 C O Channel Depth [1015 atoms/cm 2] U. von Toussaint et al. , New Journal of Physics 1 (1999) 11. 1 Note asymmetric confidence intervals COSIRES 2004 © Matej Mayer
Reconstruction of surface roughness distributions Substrate roughness Layer roughness In on Si, 2 Me. V 4 He, 165° Distribution p(j) Other types of roughness: N. Barradas et al. , NIM B 217 (2004) 479 COSIRES 2004 © Matej Mayer Distribution p(d)
Reconstruction of surface roughness distributions (2) L Distribution p(d) = + +. . . + M. Mayer, NIM B 194 (2002) 177 Correlation effects are neglected valid, if lateral variation L > d for typical RBS angles of 160°-170° COSIRES 2004 © Matej Mayer
Reconstruction of surface roughness distributions (3) 1. 5 Me. V 4 He, Ni on C 2 Me. V 4 He, Ni/Al/O on C Distribution p(d)? G-distribution is successful in many cases M. Mayer, NIM B 194 (2002) 177 Can we use RBS for measuring p(d) without prior knowledge of the distribution function? COSIRES 2004 © Matej Mayer
Reconstruction of surface roughness distributions (4) SEM AFM 2 mm RBS 2 Me. V 4 He, 165° 200 nm In on Si COSIRES 2004 © Matej Mayer Reconstruction of p(d) from RBS
Reconstruction of surface roughness distributions (5) Film thickness distribution RBS spectrum Simulation How well does this compare with other methods? COSIRES 2004 © Matej Mayer
Reconstruction of surface roughness distributions (6) secondary electrons tilt 70° backscattered electrons 25 ke. V, normal incidence Intensity of backscattered electrons depends on In thickness Thickness distribution from grey-values COSIRES 2004 © Matej Mayer 2 mm
Reconstruction of surface roughness distributions (7) 2 mm • Good agreement for large blobs (around 200 nm) • Small blobs are only visible with RBS and SEM, but not AFM 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 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 COSIRES 2004 © Matej Mayer
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