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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 - PowerPoint PPT Presentation
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COSIRES 2004 © Matej Mayer
Bayesian Reconstruction of
Surface Roughness and Depth Profiles
M. Mayer1, R. Fischer1, S. Lindig1, U. von Toussaint1, R. Stark2, V. Dose1
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
MeV Ion Beam Analysis
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500
1000
Experimental data Simulation
Co
un
ts
Channel
SampleMeV 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
200 400 600 8000
500
1000
Experimental data Simulation
Co
un
tsChannel
“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
COSIRES 2004 © Matej Mayer
200 400 600 8000
500
1000
Experimental data Simulation
Co
un
tsChannel
Bayesian Data Analysis
Parameters
Forward calculation
p(d, I)
Backward calculation, inverse problem
p(d, I)
),|()|(),|( IdpIpIdp Bayes’ theoremp(I): Prior probability
I: Additional background information
Sample
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
dIdpIdp )|,()|(Marginalization
Allows to eliminate uninteresting variables
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
17%17% 66%
Mean
p()
Mostprobable
Note that 2-minimising
(fitting) will find most probable
value
COSIRES 2004 © Matej Mayer
Deconvolution of the Apparatus Function
< 1 keV
< 1 keV10-15 keV
Detector
Sample
straggling
EdEfEEAEf )( )()(~
Measured spectrum: A: Apparatus functionf(E): Spectrum for “ideal” detector
fAf ~
Discrete spectrum:
Direct inversion: fAf~1 Does not work in presence of noise
MeV
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
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 4He, 165°
R. Fischer et al., Phys. Rev. E55 (1997) 6667R. Fischer et al., NIM B136-138 (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. E55 (1997) 6667R. Fischer et al., NIM B136-138 (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. E55 (1997) 6667R. Fischer et al., NIM B136-138 (1998) 1140
COSIRES 2004 © Matej Mayer
0
1
Co
nce
ntr
atio
n
Depth
Reconstruction of depth profiles
COSIRES 2004 © Matej Mayer
0
1
N...21
Co
nce
ntr
atio
n
Depth
Layer
Reconstruction of depth profiles
Cou
nts
Energy
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 2min)
p(|d, I) remains unknown no error bars or confidence intervals
COSIRES 2004 © Matej Mayer
0
1
N...21
Co
nce
ntr
atio
n
Depth
Layer
Reconstruction of depth profiles (2)
Cou
nts
Energy
p(|d, I)
Bayesian data analysis:
• Calculate p(|d, I) using maximum entropy prior
• : Concentrations of elements in the layers
COSIRES 2004 © Matej Mayer
Reconstruction of depth profiles (2)
12C
13C
D 12C
Plasma
Mixture of 13C/12C due to
plasma exposure
Depth profiles from
Bayesian data analysis
Energy of backscattered 4He [keV]
Cou
nts
before
after(scaled)
13C
12C
U. von Toussaint et al., New Journal of Physics 1 (1999) 11.1
COSIRES 2004 © Matej Mayer
Reconstruction of depth profiles (3)
Depth [1015 atoms/cm2]
Con
cent
ratio
nC
once
ntra
tion
before
after
ChannelC
ount
s
DataSimulation
after
O
13C
12C12C
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
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 4He, 165°
COSIRES 2004 © Matej Mayer
Reconstruction of surface roughness distributions (2)
= + + + ...
Energy Energy Energy Energy
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)
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)
200 400 6000
500
1000
Experiment Simulation: Smooth Simulation: Rough
Co
un
ts
Channel
1.5 MeV 4He, Ni on C
200 400 6000
50
100
150
200
250
300
C
O
Al
Ni
Co
un
tsChannel
2 MeV 4He, Ni/Al/O on C
Distribution p(d)? -distribution is successful in many casesM. Mayer, NIM B194 (2002) 177
COSIRES 2004 © Matej Mayer
Reconstruction of surface roughness distributions (4)
200 nm In on Si
SEM
2 m
AFM
2 m
RBS2 MeV 4He, 165°
Reconstruction ofp(d) from RBS
COSIRES 2004 © Matej Mayer
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0.005
0.010
0.015
0.020
Dis
trib
uti
on
fu
nct
ion
Indium thickness [nm]
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)
2 m
backscattered electrons25 keV, normal incidence
secondary electronstilt 70°
Intensity of backscattered electrons depends on In thickness Thickness distribution from grey-values
COSIRES 2004 © Matej Mayer
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0.005
0.010
0.015
0.020 SEM AFM RBS
Dis
trib
uti
on
fu
nct
ion
Indium thickness [nm]
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
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