Bayesian Reconstruction of Surface Roughness and Depth Profiles

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


MeV Ion Beam Analysis

200 400 600 8000

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


200 400 600 8000

500

1000


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


200 400 600 8000

500

1000


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


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


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


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


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



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



Example: Cu on Si

• Error of apparatus function is

taken into account

• Error bars, confidence intervals

are obtained




Example: Co-Au multilayer

• Apparatus function for

Co and Au from ultra-thin films



0

1

Co

nce

ntr

atio

n

Depth

Reconstruction of depth profiles


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


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



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



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


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°


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)


Can we use RBS for measuring p(d) without prior knowledge of the distribution function?


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



200 nm In on Si

SEM

2 m

AFM

2 m

RBS2 MeV 4He, 165°

Reconstruction ofp(d) from RBS


0 100 200 3000.000

0.005

0.010

0.015

0.020

Dis

trib

uti

on

fu

nct

ion

Indium thickness [nm]


Film thickness distribution RBS spectrum

Simulation

How well does this compare with other methods?



2 m

backscattered electrons25 keV, normal incidence

secondary electronstilt 70°

Intensity of backscattered electrons depends on In thickness Thickness distribution from grey-values


0 100 200 3000.000

0.005

0.010

0.015

0.020 SEM AFM RBS

Dis

trib

uti

on

fu

nct

ion

Indium thickness [nm]


2 m

• Good agreement for large blobs (around 200 nm)

• Small blobs are only visible with RBS and SEM, but not AFM


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


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

Bayesian Reconstruction of Surface Roughness and Depth Profiles

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