nucleon

Nuclear Instruments and Methods in Physics Research B 239 (2005) 135–146 www.elsevier.com/locate/nimb

Experimental stopping forces for He, C, O, Al and Si ions in Al2O3 in the energy range of 40–1250 keV/nucleon C. Pascual-Izarra a

b

a,b,*

, N.P. Barradas c, G. Garcı´a a, A. Climent-Font

a,b

CMAM, Centro de Microanalisis de Materiales, Acelerador, Campus Cantoblanco, Universidad Auto´noma de Madrid, E-28049 Madrid, Spain Dep. de Fı´sica Aplicada, Universidad Auto´noma de Madrid, E-28049 Madrid, Spain c Instituto Tecnolo´gico e Nuclear, E.N. 10, 2686-953 Sacave´m Codex, Portugal Received 4 August 2004; received in revised form 1 April 2005 Available online 31 May 2005

Abstract The stopping forces (a.k.a. stopping powers) for 4He, 12C, 16O, 27Al and 28Si ions in alumina (Al2O3) have been experimentally determined in the energy range of 40–1250 keV/nucleon. Two independent analysis methods based on Simulated Annealing and Bayesian Inference techniques (as implemented in the Hotstop and Data Furnace codes) have been used to extract the continuous stopping curves from sets of backscattering spectra. Bayesian analysis has been found to be very adequate for providing systematic estimations of the confidence level for the results. The results of both methods are reported and compared with other available data.
2005 Elsevier B.V. All rights reserved. PACS: 34.50.Bw; 82.80.Yc; 02.70.Lq; 02.50.Ph Keywords: Stopping power; Stopping force; Rutherford backscattering; RBS; Monte carlo; Simulated annealing; Bayesian inference; Hotstop; Data furnace; Aluminum oxide; Al2O3; Alumina; Pulse height defect

1. Introduction

*

Corresponding author. Address: CMAM, Centro de Microanalisis de Materiales, Acelerador, Campus Cantoblanco, Universidad Auto´noma de Madrid, E-28049 Madrid, Spain. Tel.: +34 91 4973632; fax: +34 91 4973623. E-mail address: [email protected] (C. Pascual-Izarra). URL: http://hotstop.sourceforge.net (C. Pascual-Izarra).

A precise knowledge of the stopping of ions in matter is desirable for fields such as ion beam analysis and modification of materials (IBA and IBMM respectively), radiation therapy or nuclear physics. Even after a century of stopping measurements, some combinations of ion and targets have not

0168-583X/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.04.068

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been studied with sufficient detail, specially in the case of heavy ions and/or compound targets [1]. In this work, we demonstrate the possibility of using Bayesian data analysis of backscattering spectra to obtain reliable continuous stopping force curves for heavy ions in compounds using only a standard experimental set-up. Two computer codes, called Hotstop and Data Furnace, have been used to obtain the stopping force for several ions in alumina (Al2O3). Each code has a different approach to the problem, which has been described in [2,3] for Hotstop and in [4,5] for Data Furnace. The main strength of the Bayesian analysis is that it provides a natural and systematic estimation of uncertainties associated with the results. Aluminum oxide is a common material found in IBA and IBMM samples due to its importance in optics and optoelectronics. In contrast to this importance, the available experimental stopping values are almost limited, to our knowledge, to light ions [2,6–12]. This work intends to contribute in filling this gap of heavier ions data, specially considering that the large binding energy in Al2O3 may produce deviations from the Bragg-calculated values [10], which have not been studied for the ions treated here. We would also like to remark the importance of the determination of the stopping force for oxygen and aluminum in Al2O3 since they are fundamental for elastic recoil detection (ERD) analysis of alumina samples.

2. Experiments The experimental backscattering spectra used were obtained at the Centro de Micro-Ana´lisis de Materiales (CMAM) [13,14] with the new 5 MV Tandetron. The energy calibration of the accelerator was performed using the 27Al(p,c)28Si reaction at 992 keV as well as ten resonances in the a backscattering cross sections of O, C, N and Si at energies in the range from 3034 keV to 8139 keV [15] (i.e. the calibration points for terminal voltages are in the 0.5–4.0 MV range). A remarkable stability and low voltage ripple at the level of 10
5 has been observed [13]. The samples were mounted on a high-precision three axis goniometer and ori-

ented by channeling measurements on a Si h1 0 0i crystal. The sample-holder was biased with +180 V and connected to a current integrator to have an estimation of the beam fluence (with only a limited accuracy of 10%). A particle detector was placed at 170.6 ± 0.5 scattering angle (as determined by optical measurements). The detector dead-layer thickness was determined by measuring the energy shift of the main a-emission peak of a 241Am source as a function of the tilt angle of the detector. The rather high value obtained (equivalent to 1018 Si/cm2) is attributed to the combination of the contact electrode as well as to a possible surface contamination of the detector. In order to avoid the possible dependence of the detector dead-layer with the applied field [16], the same detector bias voltage was maintained throughout all the experiments as well as during the dead-layer characterization. Two alumina samples, which have been already used in the experiments reported in [6] were prepared by sequential pulsed laser deposition and consist of layers of Au nano-crystals, used as markers, embedded in an amorphous Al2O3 matrix deposited over a graphite substrate. In situ optical reflectometry characterization has been performed during growth to control the film thickness. The calibration of the reflectometry parameters was performed by ellipsometric experiments. The density of the alumina matrix (q = 8.9 · 1022 at/cm3) was obtained as described in [6] and is in good agreement with the results of [17], where the preparation set-up is described in more detail. One of the alumina samples (S1 in the following) consists, nominally, of five 100 nm-thick layers of Al2O3, each one having a Au marker layer on top. The other sample (S2) consists on a single 200 nm-thick Al2O3 film, with Au markers located at the substrate–alumina interface. Note that each of the Au marker layers is formed by 1015 at/cm2 distributed in isolated nano-crystals and, hence do not constitute an uniform layer. The optical characterization of the thickness as well as the areal density measurements presented in [6] were performed in a central spot of the sample. Unfortunately, these samples present lateral inhomogeneity – evident to naked eye as interference color patterns – as a consequence of the

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

preparation method used: the deposition rate decreases with the distance to the center of the sample. Since some of the spectra sets used in this work were taken at slightly off-centered positions, a possible deviation towards lower thickness values of up to 5% may occur for some of the ions studied. This possibility as well as the 2% uncertainty of the central position thickness measurements from [6] have been taken into account when analyzing the data. Apart from the thickness characterization, other aspects of the samples have also been analyzed. In the first place, the possible hydrogen contamination in the alumina has been studied by ERD and found to be less than 0.5 at.%. The stoichiometry of the aluminum oxide was checked by Rutherford backscattering spectrometry (RBS) with 2 MeV He and these measurements also showed the presence of a thin layer of hydrocarbons contamination on the samplesÕ surface (equivalent to 13 · 1015 C/cm2). Finally, these RBS measurements were performed before and after heavy ion irradiation of the samples, in order to check for any hypothetic ion induced transport of the Au in Al2O3 [18], and no relevant effect was detected. In order to obtain sufficient amount of data for the calculations, a set of backscattering spectra were taken for each ion with beam energies in the range of 0.25–1.25 MeV/nucleon using both S1 and S2 samples. More precisely, the following beams were used: 1, 2, 3, 4 and 5 MeV of 4He; 9, 12 and 15 MeV of 12C; 4, 8, 12, 16 and 18.7 MeV of 16O; 6.75, 13.5 and 20.25 MeV of 27Al and, finally, 7, 14, 21, 28 and 35 MeV of 28Si. Note that, in order to produce all the mentioned beams with our accelerator, different charge states of the ions were needed. Since our samples are relatively thick, no measurable effect was expected to occur in the backscattering spectra due to the initial charge of the ions. Nevertheless, we checked that this was the case by comparing oxygen – RBS spectra taken with the same beam energy (8 MeV) but different charge states (O2+ and O3+). In addition to the backscattering spectra of the S1 and S2 samples, spectra were also taken from a third sample for most of the beams. This sample – produced specifically for energy calibration pur-

137

poses – consists on a very thin layer (1016 at/ cm2) of a compound containing Bi, W, Sr, Cu, Ca and O deposited on a C substrate. The advantage of using this calibration sample is that we obtain several clean surface signals – uniformly distributed along the energy range of interest – from each single calibration spectrum.

3. Calculations The Bayesian inference methods rely on comparing simulated and experimental data. It is, therefore, fundamental to have an accurate model that allows to simulate the experiments, given the relevant parameters. The main difference between the two methods used in this work (Hotstop and Data Furnace) is, precisely, the type of experimental data that is chosen for simulation and, consequently, the model used. In Data Furnace, the backscattering spectra are simulated (i.e. yield as a function of energy) and compared, in chosen regions, with the experimental spectra [4,5]. In the case of Hotstop, only the channel number at which the relevant features (either peaks or signal edges) of the spectra appear are simulated [2,3]. The relative advantages and limitations of each approach have been already summarized in [2,6]. 3.1. Pulse height defect correction In both Hotstop and Data Furnace calculations, a correction for the pulse height defect (PHD) [19] has been performed. The PHD occurs with solid state particle detectors and can be described as the deficit in the detected energy with respect to the energy of the ion that reaches the detector. Since the PHD is a function of energy as well as of the atomic number of the ion, Z, it is a major cause of nonlinearities in the energy calibration of the detection system. The PHD is often neglected when dealing with light ions, but should be considered when heavier ions are detected with solid state particle detectors [16,20]. Note that failing to consider the PHD can yield to systematic errors in the results, specially in the low energy ranges. In this work, the utmost care was taken to minimize such a situation by a combination of

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

where Einc is the energy of the incoming particle just before reaching the detector, NtDL is the thickness of the dead layer and SDL(E) is the stopping force in the dead layer material (typically, Si or equivalent). Fig. 1(a) shows the calculated DEDL corrections for each ion, assuming NtDL = 1157 · 1015 Si/cm2, as suggested by the energy calibration of the He spectra (see Section 3.2). During the slowing down of an ion in the detector, most of the energy is transferred directly to electrons (ionizing) but some is transferred to recoils (DErcl). These recoils may themselves ionize or produce other recoils, and so on. At the end, some of the energy ends up producing phonons or vacancies in the detector material (this is the non-ionizing contribution to the PHD, DENI). This chain of collisions – and hence DENI – may in principle be calculated using a Monte Carlo code such as SRIM [11] but the calculation takes relatively long time, specially if it has to be done for many initial energies and considering full cascades of recoils. To speed up calculations, we propose an ad-hoc model based on assuming that DENI is approximately proportional to DErcl for the full range of incident energies considered: DENI ’ gDErcl ; where g is the scaling constant.

ð2Þ

The energy transferred from the primary ion directly to recoils, DErcl, can be easily calculated as that corresponding to the nuclear energy loss in the detector: DErcl ðE Þ ¼

Z

E

0

S n;det ðEÞ dE; S det ðEÞ

ð3Þ

where E* = Einc
DEDL is the energy of the ion when entering the active volume of the detector, and Sn,det, Sdet are, respectively, the nuclear and total stopping force in the detector material (usually silicon). Once DErcl is known for a continuous range of incoming energies, a single – carefully calculated – value of DENI can be used to scale the whole curve. In the present work, the scaling was done independently for each ion by comparing the obtained DErcl with the value calculated by SRIM2003 DENI,SRIM for 1 MeV/nucleon incoming ions. The scaling constants found are: gHe = 0.66, gC = 0.62, gO = 0.64, gAl = 0.50 and gSi = 0.46. Fig. 1(b) shows the obtained DENI values, which are in very good agreement with single points calculated with SRIM for all the energy range.

800 (a) ∆EDL ( keV )

detailed treatment of the detector response (including in situ characterization of its dead-layer) and a very careful calibration process. The two main contributions to the PHD – i.e. detector dead layer and non-ionizing processes in the detector active area – are explicitly calculated in our codes and a simple stand-alone program which calculates both contributions is distributed with the Hotstop package. Any remaining residual contributions such as the so-called Plasma effect or the possible Z dependence of the ionization energy of Si [21,22] are handled by performing an independent non-linear energy calibration for each different ion species (see below). The dead layer contribution to PHD, DEDL, is calculated by solving the following integral equation: Z Einc dE ¼ NtDL ; ð1Þ S ðEÞ DL Einc
DEDL

Si Al

600 400

O C

200 He

0 (b) 200 ∆ENI ( keV )

138

Si Al

150 100

O C

50

He

0 0

200 400 600 800 1000 1200 1400 Energy (keV/nucleon)

Fig. 1. Pulse height defect corrections due to (a) 1157 · 1015 Si/ cm2 thick dead layer and (b) non-ionizing processes in the active area of the detector.

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

3.2. Energy calibration Another critical part of our calculations has to do with the energy calibration of the detection system. As mentioned before, in order to account for the Z dependence in the solid-state detector response, independent calibrations were performed for each ion species used. Also, we allow those calibrations to be non-linear, which is useful for heavy ions (we found that quadratic calibration is enough for this work, in consonance with [16], but any polynomial order can be used). As a drawback, any additional degree of freedom in the calibration curve needs to be compensated with an increase in the number of calibration points. In order to have enough points, we used the signals from the calibration sample spectra as well as the surface signals from the spectra of S1 and S2. In addition, we maintained unaltered the configuration of the spectroscopic amplifier during the acquisition of all the spectra for each given ion, therefore accumulating an average of 18 calibration points for each ion, covering the whole energy range of interest. The calibrations were performed with Hotstop, which allowed to introduce the PHD correction also during the calibration, as well as the effect of the surface hydrocarbon contamination in S1 and S2. A polynomial least square method was used to fit the calibration curve to the experimental calibration data for each ion separately. In the case of He, a Bayesian fit of the calibration data was also performed in which NtDL was allowed to vary simultaneously with the calibration coefficients in order to obtain an optimum value for NtDL which was adopted in all subsequent calculations. The choice of the He calibration data for this calculation was made due to the larger number of calibration points [29] and to the better accuracy of available stopping data for 4He in Si [23], since they are needed for the PHD correction. Note that the value for the dead layer obtained by Bayesian Inference (1157 ± 150 · 1015 Si/cm2), is compatible with that given by direct characterization with the 241Am source (1000 ± 200 · 1015 Si/cm2). Finally, it should also be noted that the nonlinear term of the calibration is systematically

139

Table 1 Improvement in the linearity of the calibration when applying both DENI and DEDL PHD corrections 4

12

16

27

28

56%

41%

67%

69%

93%

He

C

O

Al

Si

For each quadratic calibration performed, the relative reduction of the non-linear coefficient is shown.

reduced when making use of the PHD correction, confirming the convenience of such a correction (see Table 1). 3.3. Parameterization of the stopping curves Both Hotstop and Data Furnace make use of a parameterized expression for the stopping force. During the Bayesian inference, the parameters defining the curve are varied to get the best possible simulation of the experimental input data (i.e. of the spectra yield in the case of Data Furnace [4] or the signal positions in the case of Hotstop [2]). The functional form chosen for describing the stopping should not be relevant (from a theoretical point of view), as long as it is flexible enough to describe any stopping force. But, from a practical point of view, the parameterization is required to be fast to compute, and to have as few free parameters as possible – the number of steps needed in the Bayesian inference varies exponentially with the number of free parameters – while still being able to describe any typical stopping curve in the energy range of interest. Also it should be robust and free of artificial oscillations. Finally, since our goal is to describe the stopping for any material (compounds included) with a limited set of free parameters, the chosen parameterization should allow to describe stopping in compounds with the same number of parameters as in elemental targets. Three options have been tried: • A formula equivalent to the ZBL [24] for protons of medium energy, but applied to any ion in any material and at any energy. This approach was used in [6] but was afterwards discarded due to stability problems in the low energy range for heavy ions.

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

• The ZBL parameterization [24], which uses a set of 8 parameters to define the stopping for protons in the target material and then does a scaling attending to various properties of the target. It works well but needs non-trivial adaptation to be able to describe compounds with a single set of parameters. Also, some artificial structures may appear for heavy ions at the energies where different scaling functions are linked. This parameterization has been used in [2,4] and is the one used by Data Furnace in this work. • The most promising parameterization tested to the moment, and the one used by Hotstop in this work is that of [25]. It is a simplified version (only 5 free parameters) of the expression given in [26]. The advantages of this parameterization are that it is compact and smooth over the whole energy range of interest in this work. It does not require any scaling and it is very robust. Moreover, it can easily be adapted to describe stopping in compounds [3]. This parameterization, which is already included in programs like RUMP and can be easily implemented in any code, can be expressed as: SðE : s;~ aÞ ¼

Es logðe þ bEÞ ; a0 þ a1 E1=2 þ a2 E þ a3 E1þs

ð4Þ

where E is in (MeV/amu), b ’ 219.49/Zt (amu/ MeV), Zt being the atomic number of the target material and s, a0, . . . ,a3 the free parameters. In the case of compound targets, we propose the use of the following value for the atomQ effective x Z =Z ic number: Z eff ¼ i Z i i i , where the i accounts for each element in the P compound, x is the atomic fraction and Z
¼ i xi Z i . It can be shown that this effective value reproduces the Bethe–Bloch limit of Eq. (4) when adding stoppings following the BraggÕs rule. To check for consistency, we performed identical analysis except that in one case we used the parameterization of [24] while in the other we used [25]. We found an important improvement in the speed and smoothness of the results with the latter while obtaining, essentially, the same results for all ions. Note that the mentioned parameterizations describe only the electronic stopping force. The

nuclear contribution is not fitted, and it is calculated as described in [24].

4. Results and discussion Figs. 2–6 show the results obtained with Hotstop and Data Furnace codes. Bayesian inference results are shown as confidence intervals defined by SðEÞ  rðEÞ, where SðEÞ is the bin-by-bin average of the curves obtained during the Bayesian process and r(E) is the corresponding standard deviation. Hotstop performed, for each ion, a Simulated Annealing pre-fit followed by a Bayesian inference calculation with 106 steps, using all the available experimental data. All (five) coefficients of the stopping parameterization were allowed to vary as well as the thickness of the alumina films, which were free to vary within an uncertainty level of
5% to +2% of the nominal values to account for the accuracy of the thickness determination and for the effect of lateral inhomogeneity of the sample. In the case of Data Furnace, Bayesian calculations with 5 · 105 steps were attempted using only the data from sample S1 and allowing the

Stopping for He in Al2O3

50 Stopping Force ( eV cm2/1015 at )

140

45 40 35 HS (Bayesian) DF (local fits) SRIM2003 Pascual04a Hoshino00 Bauer98 Santry86

30 25 20 10

2

10

3

Energy (keV/nucleon)

Fig. 2. Stopping force for 4He in Al2O3 as obtained with Hotstop (HS) and Data Furnace (DF). Predictions from SRIM2003 as well as data points from [6,8–10] are also plotted.

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146 Stopping for C in Al2O3

550 Stopping Force (eV cm2/1015 at)

Stopping Force (eV cm2 /1015 at)

200

180

160

140

HS (Bayesian) DF (Bayesian) DF (local fits) SRIM2003

120

141

Stopping for Al in Al2O3

500 450 400 350 300

HS (Bayesian) DF (local fits) SRIM2003

250 200 150

100 10

2

100

3

Energy (keV/nucleon)

Fig. 3. Stopping force for 12C in Al2O3 as obtained with Hotstop (HS) and Data Furnace (DF). Predictions from SRIM2003 are also plotted.

280

3

10

Fig. 5. Stopping force for 27Al in Al2O3 as obtained with Hotstop (HS) and Data Furnace (DF). Predictions from SRIM2003 are also plotted.

Stopping for O in Al2O3

550

260

Stopping for Si in Al2O3

500

240

450

15

Stopping Force (eV cm2/10 at)

Stopping Force (eV cm 2 /1015 at)

2

10 Energy (keV/nucleon)

10

220 200 180

HS (Bayesian) DF (local fits) SRIM2003

160 140 120

400 350 300

HS (Bayesian) DF (local fits) SRIM2003 Arstila00

250 200 150

100 10

2

10

3

Energy (keV/nucleon)

Fig. 4. Stopping force for 16O in Al2O3 as obtained with Hotstop (HS) and Data Furnace (DF). Predictions from SRIM2003 are also plotted.

variation of all (eight) coefficients from the stopping force as well as the alumina and Au film thicknesses, the beam fluence and the energy calibration coefficients for each spectrum. In Fig. 3 the results of such calculation for C ions are shown. It can be seen that the error bar is rather narrow and that it does not agree well with the Hotstop results. The

100 10

2

10

3

Energy (keV/nucleon)

Fig. 6. Stopping force for 28Si in Al2O3 as obtained with Hotstop (HS) and Data Furnace (DF). Predictions from SRIM2003 are also plotted as well as data points for 29Si ions from [27].

reason is that, even with 5 · 105 steps, convergence has not been reached, as it can be seen in Fig. 7 where the evolution during the Bayesian run of three of the stopping parameters is shown (while a3 and a5 seem to have reached stability, a7 has not). In this case, the Bayesian inference results

142

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

obtained with Data Furnace are not reliable. The reason is that the Bayesian search for the stopping parameters implemented in Data Furnace, while it is, in principle, of general application, has been developed and optimized for the analysis of bulk spectra as in [4]. In the case of complex multilayered samples, each transition of the search involves not only changes in the stopping coefficients, but also in the thickness of each layer, energy calibration, etc. The algorithm for varying each of these quantities during the Bayesian process does not take into account the existing correlations amongst them and, as a consequence, the efficiency of the search is decreased. In other words, mixing is very poor and convergence extremely slow. This situation may, in principle, also occur in the Hotstop calculations but three factors help to avoid it: in the first place, less free parameters are needed in Hotstop, reducing the dimensionality of the solution space and facilitating the search and the mixing. Secondly, the search is more efficient since the correlations between parameters associated to the yield are effectively ignored in Hotstop and, finally, the Hotstop simpler simulation model allows to perform considerably longer

Fig. 7. Evolution of three of the parameters defining the stopping curve during a 105 step Bayesian calculation with Data Furnace. The ZBL parameterization [24] is used in this case.

searches for a given computation time, helping to reach stability. Some work is under way in order to upgrade the Bayesian search algorithm in Data Furnace but, for the moment, we have chosen to use Data Furnace as a fitting-only code, using its simulated annealing and local minimization capabilities to find the stopping force curve that leads to the best fit to the data. As this depends on the thickness of the layers, we did fits for the nominal thickness values and for its upper and lower limits (nominal +2% and nominal
5%, respectively). These fits are shown in Figs. 2–6 as triplets of dash–dotted lines (note that the larger the value of the thickness assumed, the lower the stopping results). Since the average stopping curve (SðEÞ) resulting from the Bayesian process is the result of averaging many curves, it is not defined by a set of parameters. For convenience, a least squares fit can be done to SðEÞ using Eq. (4) in order to provide an easy way of reproducing it (the fitted curve reproduces the values of SðEÞ with accuracy much better than 1% in the energy ranges reported). The coefficients of such a fit are given in the first part of Table 2. A similar fit is performed to the results obtained with Data Furnace for the nominal sample thickness. The corresponding coefficients are shown in the second part of Table 2 (note that, in this case, the curves defined by the coefficients reproduce well the Data Furnace curves except for the artificial ‘‘bumps’’ associated to the original ZBL parameterization). Apart from the Hotstop and Data Furnace results, the predictions of SRIM2003 are plotted as a reference in Figs. 2–6. When available, previously published data (obtained by other methods) are also presented for comparison. For the case of He, data from [6,8–10] are shown. For the other ions, no previous experimental data have been found, except for Si, in which results for a different isotope (29Si) were presented by Arstila [27]. As can be seen, the Bayesian inference results from Hotstop and the fits of Data Furnace show good overall agreement for all ions. Comparing our results with SRIM2003 predictions, it can be seen that our data seem to indicate that SRIM2003 systematically overestimates the stopping forces in Al2O3 for energies immediately below the Bragg

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

143

Table 2 Parameters describing the average stopping curves found with Hotstop and Data Furnace Ion

s

a0

a1

7.421 · 10
3 3.343 · 10
3 2.132 · 10
4 3.204 · 10
3 2.178 · 10
3


1.199 · 10
3 4.302 · 10
3 3.410 · 10
3
3.616 · 10
3
3.871 · 10
4

Data Furnace (local fit for nominal thickness) 1.581 · 10
3 He 6.364 · 10
1
1 C 7.033 · 10 6.731 · 10
4 1.012 · 10
3 O 6.546 · 10
1
1 Al 6.984 · 10 7.809 · 10
4 Si 1.262 5.758 · 10
5

1.742 · 10
2 1.632 · 10
3 8.278 · 10
4 4.198 · 10
4
2.352 · 10
4

Hotstop (SðEÞ) He 5.142 · 10
1 C 3.068 · 10
1 O 7.678 · 10
1 Al 3.980 · 10
1 Si 4.580 · 10
1

a2

a3

b

3.350 · 10
3 6.030 · 10
3 2.063 · 10
3 7.391 · 10
3 2.060 · 10
3

1.109 · 10
1 4.539 · 10
3 7.288 · 10
3
1.173 · 10
4 2.349 · 10
3

21.31 21.31 21.31 21.31 21.31


1.885 · 10
2 8.259 · 10
3 7.480 · 10
3 4.916 · 10
3 3.496 · 10
3

1.219 · 10
1 8.298 · 10
3 3.751 · 10
3
7.233 · 10
6 2.850 · 10
3

21.31 21.31 21.31 21.31 21.31

Use them with Eq. (4) to obtain the electronic stopping force in eV cm2/1015at, introducing E in MeV/amu. Note that b is not a free parameter but a constant calculated, for alumina, as: b ’ 219.49/Zeff ’ 219.49/10.297 (see Section 3.3 for the definition of Zeff). Also note that due to the strong correlation between parameters, very different parameter sets may yield similar curves, as it is evident when comparing the Hotstop and Data Furnace curves of Figs. 2–6.

Yield (counts)

200 150

50

(a)

0

400

200

0 (b)

Exp. SRIM HS / DF

100

Yield (counts)

Peak for all the studied ions. This is in agreement with the results from Arstila [27], as shown in Fig. 6 (note that the effect of the difference of mass between 29Si and 28Si is expected to be below 5 eVcm2/1015at in this energy range). When comparing with the available experimental data for He ions, (Fig. 2) our results are intermediate between those from Bauer et al. [10] and those from Santry et al. [9] and Hoshino et al. [8] and agree with our previous results obtained by the STES technique [6] within their errors of 5%. Fig. 8 demonstrates the effect that the different stopping forces have in the simulation of backscattering spectra. Experimental spectra from both samples are shown together with simulations using the stopping curves obtained in this work as well as the SRIM2003 stopping. All the simulations were performed using Data Furnace, which allowed to include the PHD corrections and the non-linear energy calibration. Fig. 8(a) shows, actually three spectra of the sample S1, taken with 500, 750 and 1000 keV/nucleon 28Si ions. It can be seen that, while the Au surface peak is well simulated in all cases, the simulation using SRIM values becomes progressively worse for deeper signals (specially in the lower 500 keV). In Fig. 8(b), a spectrum of sample S2 for 750 keV/nucleon 12 C ions is shown to be well simulated with all

150 200 250 300 350 400 450 500

Exp. SRIM HS / DF

50 100 150 200 250 300 350 400 Channel

Fig. 8. Experimental RBS spectra and simulations using the curves reported in Table 2 for Hotstop and Data Furnace (thin solid line, both simulations are indistinguishable) as well as using the SRIM2003 stoppings (dashed line). The figure at the top (a) shows three spectra for sample S1 using 28Si ions with three different energies (500, 750 and 1000 keV/nucleon). The bottom figure (b) corresponds to 750 keV/nucleon 12C ions on sample S2.

three stopping curves in a very large energy region (the peak on the right corresponds to the Au marker layer at the substrate–alumina interface of

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

sample S2, while the signal on the left corresponds to the Al in the alumina layer). Two crosschecks can be done considering the results shown in Fig. 8. On the one hand, the simulations performed with the Hotstop results reproduce nicely not only the energy positions, but also the height of the signals. On the other hand, the Data Furnace results yield a very good fit of the spectrum of the sample S2, which was not included in the input data for the Data Furnace fits. An important question to answer about our results is which is the energy range in which they can be used. In other words, which energies have been studied, and to what extent. To give a reasonable estimation, consider an ion that is detected after being scattered from the sample at a certain depth. Let E0 be the energy before entering the sample, E 0 the energy just before being scattered, KE 0 the energy after scattering (K < 1 is the kinematic factor [28]) and E00 the energy when the ion exits the sample. Then, this ion will carry information about the stopping force in the ranges [E00 , KE 0 ] and [E 0 , E0]. For any signal of the input data it is possible to trace back all these quantities and, therefore, we can estimate its contribution to the determination of the stopping force to be: ( 1=d2 if E00 < E < KE0 or E0 < E < E0 xðEÞ ¼ ; 0 elsewhere

He

C

W(E) (normalized)

144

O

Al

Si

0

200

400

600

800 1000 1200

Energy (keV/nucleon)

Fig. 9. Histograms showing the energies that have been studied, and their relative weight in the results. Results shown here are for the Hotstop calculations.

Fig. 9 is useful to interpret the variation of the uncertainties of the results (r) with the energy for each ion. The uncertainties are, as expected, larger in the energy regions where W is lower (see, for example the range between 100 and 500 keV/nucleon for C, or the region above 750 keV/nucleon for Al). The W(E) histograms are also very helpful as a guideline when designing new experiments.

ð5Þ where d is the error associated to the input signal (i.e. the Poisson noise of the signal in the case of Data Furnace or the channel position uncertainty in the case of Hotstop). Summing the contributions by all the input data P signals, we can construct a histogram, W ðEÞ ¼ xðEÞ, which indicates, for each ion, the energy ranges that have been explored and their relative weight in the results. Fig. 9 shows W(E) for each ion (note, e.g. that for Al there is no information above 750 keV/nucleon, due to the unavailability of input data). However, note that this kind of histograms are mostly qualitative since Eq. (5) is only a convenient approximation (the functional form assumed for S(E) has the effect of ‘‘blurring’’ the range limits of Eq. (5), which is equivalent to smoothing the W(E) histogram).

5. Conclusions We have presented experimental results for the stopping of 4He, 12C, 16O, 27Al and 28Si in the energy range of interest for most IBA applications. This constitutes, to our knowledge, the first experimental result of this type for all the ions except for He. The results for 16O and 27Al are specially relevant considering their applicability to ERD analysis of alumina samples. It is also worth to note the interest of the present results and of the proposed method in relation with the recent developments in the theoretical prediction of heavy ion stoppings in compounds [29] using the binary stopping theory [30,31]. We have also demonstrated the applicability of the Bayesian inference methods to the experimental determination of stopping forces and we

C. Pascual-Izarra et al. / Nucl. Instr. and Meth. in Phys. Res. B 239 (2005) 135–146

hope to contribute to the stopping data production by providing to the ion beam community with analysis tools which allow to obtain reliable stopping force curves requiring only standard experimental facilities. Hotstop can be freely downloaded from http://hotstop.sourceforge.net and a demo of Data Furnace is available from http://www.ee.surrey.ac.uk/ibc/ndf/.

[11]

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[13]

Acknowledgements Thanks are due to J. Gonzalo and C.N. Afonso for providing the alumina samples. This work has been partially supported by the project DGICYT PB98-0065 from the Spanish Government.

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