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)Instituto Tecnológico e Nuclear, E.N. 10 , Sacavém 2686-953, Portugal(2) 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, siliconPACS: 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 27Al(p,γ)28S, and the neutron threshold reaction 7Li(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.

FIGURE 1. a) Channelled 4He spectrum. The amorphised region is clearly seen. b) 1.5 MeV 4He spectrum. The simulations use SRIM2008 stopping.

BAYESIAN INFERENCE

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

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 4He experiment. The corresponding 0.97% error was then added to the BI/MCMC error.

RESULTS

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.

FIGURE 2. Confidence limits (± 1) obtained for the 4He in Si stopping power with BI/MCMC. SRIM20081

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

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.

FIGURE 3. Three of the 15 spectra collected with the 7Li 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.

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.

FIGURE 4. Confidence limits (± 1) obtained for the 7Li in Si stopping power with BI/MCMC. SRIM20081 and different experimental22-77 stopping values are also shown.

SUMMARY

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.

ACKNOWLEDGMENTS

We would like to thank the International Atomic Energy Agency for support under the Research Contracts No. 14635 and 14580.

REFERENCES

1. J.F. Ziegler, Nucl. Instrum. Methods Phys. Res. B 219, 1027 (2004).

2. N.P. Barradas, C. Jeynes, R.P. Webb, E. Wendler, Nucl. Instrum. Methods Phys. Res. B 194, 15 (2002).

3. C. Pascual-Izarra, N. P. Barradas, G. García, A. Climent.Font, Nucl. Instrum. Methods Phys. Res. B 239, 135 (2005).

4. G. Konac, S. Kalbitzer, C. Klatt, D. Niemann, R. Stoll, Nucl. Instrum. Methods Phys. Res. B136-138, 159 (1998).

5. N.P. Barradas, K. Arstila, G. Battistig, M. Bianconi, N. Dytlewski, C. Jeynes, E. Kótai, G. Lulli, M. Mayer, E. Rauhala, E. Szilágyi, M. Thompson, Nucl. Instrum. Methods Phys. Res. B 262, 281 (2007).

6. K.H.Ecker, A.Berger, R.Grötzschel, L.Persson, U. Wätjen, Nucl. Instrum. Methods Phys. Res. B 175, 797 (2001).

7. K. H. Ecker, U. Wätjen, A. Berger, L. Persson, W. Pritzcow, M. Radtke, H. Riesemeier, Nucl. Instrum. Methods Phys. Res. B 188, 120 (2002).

8. N.P. Barradas, C. Jeynes, Nucl. Instrum. Methods Phys. Res. B 266, 1875 (2008).

9. N.P.Barradas, C.Jeynes, M.Jenkin, P.K.Marriott, Thin Solid Films 343-344, 31 (1999).

10. N.P.Barradas, A.P.Knights, C.Jeynes, O.A.Mironov, T.J.Grasby, E.H.C.Parker, Phys. Rev. B59, 5097 (1999).

11. N.P.Barradas, S.A.Almeida, C.Jeynes, A.P.Knights, S.R.P.Silva, B.J.Sealy, Nucl. Instrum. Methods Phys. Res. B 148, 463 (1999).

12. N.P. Barradas, A.R. Ramos, E. Alves, Nucl. Instrum. Methods Phys. Res. B 266, 1180 (2008).

13. N.P.Barradas, J.L.Keddie, R.Sackin, Phys.Rev.E 59, 6138 (1999).

14. J. F. Ziegler, J. P. Biersack, and U. Littmark, in Stopping and Ranges of Ions in Solids Publisher, Pergamon, New York, 1985.

15. W.N. Lennard, H. Geissel, K.B. Winterbon, D. Phillips, T.K. Alexander, J.S. Forster, Nucl. Instrum. Methods Phys. Res. A 248, 454 (1986).

16. C. Pascual-Izarra, N. P. Barradas, Nucl. Instrum. Methods Phys. Res. B 266, 1866 (2008).

17. N. P. Barradas, Nucl. Instrum. Methods Phys. Res. B 261, 418 (2007).

18. N.P. Barradas, Nucl. Instrum. Methods Phys. Res. B 225, 318 (2004).

19. N.P. Barradas, M. Reis, X-Ray Spectrometry 35, 232 (2006).

20. F.Apel, U.Mueller-Jahrreiss, G.Rockstroh, S.Schwabe, Phys.stat.solidi(a) 3, K173 (1970).

21. I.Hoffmann, E.Jäger, U.Müller-Jahreis, Rad.Effects 31, 57 (1976).

22. D.C.Santry, R.D.Werner, Nucl. Instrum. Methods Phys. Res. B 1, 13 (1984).

23. Jiarui Liu, Zongshuang Zheng, Wei-Kan Chu, Nucl. Instrum. Methods Phys. Res. B 118, 24 (1996).

24. W.Jiang, R.Grötzschel, W.Pilz, B.Schmidt, W.Möller, Phys.Rev. B59, 226 (1999).

25. G.de M.Azevedo, M.Behar, J.F.Dias, P.L.Grande, D.L.da Silva, Phys. Rev. B65, 075203 (2002).