Stopping power of different ions in Si measured with a bulk
sample method and Bayesian inference data analysis
N.P. Barradas1,2, E. Alves1,2, Z. Siketić3, I. Bogdanović Radović3
(1)
(2)
Instituto Tecnológico e Nuclear, E.N. 10 , Sacavém 2686-953, Portugal
Centro de Física Nuclear da Universidade de Lisboa, Av. Prof. Gama Pinto 2, Lisboa, Portugal
(3)
Ruđer Bošković Institute, P.O. Box 180, Zagreb 10002 , Croatia
Abstract. The accuracy of ion beam analysis experiments depends critically on the stopping power values available.
While for H and He ions accuracies normally better than 5% are achieved by usual interpolative schemes such as SRIM,
for heavier ions the accuracy is worse. One of the main reasons is that the experimental data bases are very sparse, even
for important materials such as Si. New measurements are therefore needed. Measurement of stopping power is often
made with transmission in thin films, with the usual problems of film thickness homogeneity. We have previously
developed an alternative method based on measuring bulk spectra, and fitting the yield by treating the stopping power as
a fit parameter in a Bayesian inference Markov chain Monte Carlo procedure included in the standard IBA code NDF.
We report on improvements of the method and on its application to the determination of the stopping power of 7Li in Si.
To validate the method, we also apply it to the stopping of 4He in Si, which is known with 2% accuracy.
Keywords: electronic stopping, RBS, silicon
PACS: 06.20.Fn, 02.50.Ga, 34.50.Bw, 61.18.Bn
INTRODUCTION
Quantification of the depth scale in ion beam
analysis techniques such as Rutherford backscattering
(RBS), Elastic Recoil Detection Analysis, or Nuclear
Reaction Analysis depends on knowledge of the
stopping powers involved. The accuracy of the
experiments thus depends crucially on the stopping
power values available. Given that not all ion/target
pairs have been measured, interpolative schemes such
as SRIM1 or MSTAR2 are often used.
The accuracy of SRIM is 4.2% for H and 4.1% for
He ions. However, for Li and heavier ions the
accuracy is only 5.1% and 6.1%, respectively. 3 These
numbers are averages, and are not the uncertainty for a
given system. For instance, for heavy ions 42% of
stopping power values calculated with SRIM2008 are
off from the experimental ones by more than 5%.
One of the main reasons is that the experimental
data bases are sparse, even for important materials
such as Si. Therefore, new measurements are needed
to improve experimental data bases. Stopping power
measurements are most often performed using thin
transmission films, with the usual problems of film
thickness homogeneity.
We have previously developed an alternative
method based on measuring bulk spectra, and fitting
the yield by treating the stopping power as a fit
parameter in a Bayesian inference Markov chain
Monte Carlo procedure included in the standard IBA
code NDF.4,5 We report on improvements of the
method and on its application to the determination of
the stopping power of 7Li ions in Si. To validate the
method, we first apply it to the stopping of 4He in Si,
which is known with a 2% accuracy6, and with 0.6%
accuracy at a 1.5 MeV beam energy.7
EXPERIMENTAL DETAILS
We self-amorphised Si by implanting Si at liquid
nitrogen temperature, with 5×1015 at/cm2 at 340 keV
and 5×1015 at/cm2 at 100 keV. This leads to an
amorphous Si layer 300 nm thick (see Figure 1a).
Then an Au marker layer around 20 nm thick was
deposited.
The 4He RBS experiments at 0.5, 1.0, 1.5, 2.0 and
2.4 MeV were done at ITN at an 179.9º scattering
angle using an annular detector. The 7Li RBS
experiments at 1.0, 1.5, 2.0, 2.5 and 3.0 MeV were
done at the Ruđer Bošković Institute using three
detectors simultaneously, at 118º, 150º and 165º
scattering angles. The energy calibration was made
using the narrow 991.88 keV resonance in
27
Al(p,γ)28S, and the neutron threshold reaction
7
Li(p,n)7Be at 1880.6 keV.
The exact Au areal density was determined to be
116.8(1.1)×1015 at/cm2 from the 1.5 MeV 4He
experiment (see Figure 1b). At this energy, a careful
analysis
of
the
certified
standard
IRMM0001/BAML003 released by IRMM, Geel8,9,
has shown that SRIM2008 stopping power of 4He in Si
is correct with an uncertainty of 0.6%.7 The total
0.97% error in the Au content includes errors from
statistics, background determination (namely pile-up),
and from the uncertainty in the angle of incidence.
20000
4
Au
a) 2 MeV He in Si - channelled
Yield (counts)
10000
6000
c-Si
a-Si
samples.14 We have also used these methods to analyse
the ambiguity of ellipsometry data.15
Basically, the method consists in treating the
stopping power function as a free parameter, and
fitting the energy spectra. In our case, the a1-a8
parameters of the ZBL85 parameterisation16 are the
free parameters, ensuring that the shape of the
stopping curve is realistic. A large number of different
simulations are made, all with a different stopping
power curve, but all leading to simulations consistent
with the data within the experimental uncertainties.
Other parameters such as the energy calibration or the
detector dead layer thickness (which leads to changes
in the calculated yield17,18), are also varied during the
procedure. In the end, a consistent estimate of the
errors can be made taking the average and standard
deviation of all stopping curves that led to a good fit.
Note that the Au areal density was fixed during the
BI/MCMC at the value determined from the 1.5 MeV
4
He experiment. The corresponding 0.97% error was
then added to the BI/MCMC error.
4000
RESULTS
2000
0
50
100
150
200
250
300
350
400
Channel
150000
Au
4
b) 1.5 MeV He in Si - SRIM2008
Yield (counts)
100000
50000
For the 4He data, we constructed a Markov chain
with 60000 elements, that is, 60000 stopping power
data sets in the energy range up to 3 MeV. Each of
these data sets leads to RBS spectra that
simultaneously fit the five experimental spectra well,
i.e. within experimental errors. The final results are
shown in Figure 2, where the BI/MCMC is given as
the two curves, corresponding to the average plus
minus one standard deviation.
20000
10000
80
0
200
250
4
He in Si
300
Channel
FIGURE 1. a) Channelled 4He spectrum. The
amorphised region is clearly seen. b) 1.5 MeV 4He
spectrum. The simulations use SRIM2008 stopping.
2
150
60
15
100
[S] (eV / 10 at/cm )
50
40
SRIM2008
KKKNS
BI/MCMC
20
BAYESIAN INFERENCE
0
Full details of the Bayesian inference (BI) method
with Markov chain Monte Carlo (MCMC) integration
as applied to stopping power determination is given
in.4 The improvements reported in Ref. 10 were also
used. We have previously reported a similar Bayesian
approach to the determination of the error of the depth
profiles extracted from IBA spectra11 for high
resolution data12, RBS/ERD data13 and to extract
scattering cross sections from bulk and thin film
0
500
1000
E (keV)
1500
2000
FIGURE 2. Confidence limits (± 1) obtained for the
4
He in Si stopping power with BI/MCMC. SRIM2008 1
and KKKNS6 stopping are also shown.
The results agree very well both with SRIM2008
and with KKKNS6 stopping values, which validates
the present approach. Note that at low energies, below
The final results are shown in Fig. 4, together with
SRIM2008 and other experimental measurements. 22-27
The results indicate a slightly lower maximum than
given by SRIM2008, and slightly higher stopping at
low energies, but always within three standard
deviations. This shows that SRIM2008 is
fundamentally correct.
120
7
Li in Si
100
2
[S] (eV / 10 at/cm )
80
15
500 keV, the deviation seen is within two standard
deviations, so it is statistically not significant. The
deviation might be due to the notorious difficulties in
simulating the yield at low energies19, even when
double scattering20 and pile-up21 were included in the
simulations.
For the 7Li data, we constructed a Markov chain
with 15000 elements. The smaller length of the
Markov chain is justified because 15 spectra (3 per
beam energy, corresponding to the three detectors)
were collected, leading to more information in the
data. We show three of the spectra in Fig. 3, together
with simulations using SRIM2008 stopping and the
average value determined in this work. An
improvement in the simulation is seen particularly at
low energies. Note that the minimum energy probed is
around 0.3 MeV, corresponding to the way out in the 1
MeV experiment.
60
Ap70 [22]
Hof76 [23]
Sa84 [24]
Liu96 [25]
Jia99 [26]
Aze02 [27]
40
SRIM2008
BI/MCMC
20
0
0
20000
Au
7
Yield (counts)
10000
500
1000
1500
E (keV)
2000
2500
3000
Li 2.5 MeV, scatt=150º
4000
FIGURE 4. Confidence limits (± 1) obtained for the 7Li in
Si stopping power with BI/MCMC. SRIM2008 1 and
different experimental22-77 stopping values are also shown.
c-Si a-Si
2000
DS+PUP
0
100
20000
200
740
760
780
7
Li 1.5 MeV, scatt=150º
10000
Yield (counts)
300
800
Au
c-Si a-Si
4000
2000
DS+PUP
0
50
100
150
400
420
440
460
20000
7
Yield (counts)
ACKNOWLEDGMENTS
4000
a-Si
We would like to thank the International Atomic
Energy Agency for support under the Research
Contracts No. 14635 and 14580.
2000
DS+PUP
0
60
We reported on a method to determine the stopping
of ions in matter using bulk samples. We validated the
method by determining the stopping of 4He in Si and
comparing the results with the highly accurate current
knowledge. We then applied to the determination of
the stopping power of 7Li in Si in the 0.3-3 MeV
range.
Au
Li 1 MeV, scatt=150º
10000
SUMMARY
80
100
240
Channel
260
280
300
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FIGURE 3. Three of the 15 spectra collected with the
7
Li beam. Simulations using SRIM2008 stopping and
the average value found in this work are also shown.
The vertical dashed lines indicate, for 1.5 and 2.5
MeV, where the a-Si layer begins. For 1 MeV, only
the a-Si layer is probed, and the vertical dashed line
indicates the ROI for the fit. DS+PUP indicate the
calculated double scattering and pile-up.
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