Nuclear Instruments and Methods in Physics Research B 268 (2010) 1731–1735
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Energy and depth resolution in elastic recoil coincidence spectrometry E. Szilágyi * KFKI Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary
a r t i c l e
i n f o
Article history: Available online 25 February 2010 Keywords: Coincidence spectrometry Simulation Multiple scattering FSS ERDA ESDSR
a b s t r a c t Elastic recoil coincidence spectrometry was implemented into the analytical ion beam simulation program DEPTH. In the calculations, effective detector geometry and multiple scattering effects are considered. Mott’s cross section for the identical, spin zero particles is included. Spectra based on the individual detector signal and summing the energy of the recoiled and scattered particles originating from the same scattering events can also be calculated. To calculate this latter case, the dependency of the energy spread contributions had to be reconsidered. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Coincident detection of elastically recoiled and scattered particles originating from the same collision event is a powerful method to obtain the depth profile of light elements in relatively thick foil targets. At first it was developed for hydrogen profiling [1] using ±45° geometry for the detection of the two protons, and later it was extended to the detection of heavier elements [2]. Various detector geometries and effects of kinematics were considered by Hofsäss et al. [3]. They used a 2 MeV He beam to demonstrate profiling of carbon and oxygen in 2 lm thick self-supporting polycarbonate foils (C16O3H14) and boron and the surface oxygen of thin B doped Si crystal were also investigated. Since at least the recoiled particles have to be detected in forward direction, the coincident method can be applied only for foil samples with thicknesses of the order of a few microns. Recently, a new method of carbon depth profiling based on the elastic recoil coincidence spectrometry has been suggested [4]. Recoiled carbon atoms as well as scattered carbon ions from the primary beam are detected by two solid-state detectors placed symmetrically at 45° relative to the beam direction. Recording the energy sum of the detected scattered and recoiled (ESDSR) carbon atoms originating from the same scattering event, the depth and energy resolution of the coincident spectra are significantly better than collecting spectra using single detector (as shown Fig. 2 in Ref. [4]). The reason of the better resolution is really simple, some of the energy spread contributions cannot be considered as independent contributions.
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[email protected] 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.02.048
To determine the three-dimensional profiling of light elements the ESDSR method was further extended using microbeam [5]. Reichart et al. used the coincident proton–proton elastic scattering at a 17 MeV microprobe to investigate 3D-hydrogen distributions inside freestanding samples up to some 100 lm thickness with subppm detection limits [6,7]. Although the coincident spectrometry is practically as old as other ion beam techniques the most used analytical simulation programs are not able to interpret the spectra of coincidence experiments. There is only a Monte Carlo code named CORTEO [8] that can be used to simulate coincidence spectra [4,9]. In this paper the feature of evaluating coincidence experiments will be considered for implementation into an analytical code DEPTH. The results will be compared to the experimental spectra published in Ref. [4] and CORTEO simulations. 2. Special features of coincidence spectrometry The energy and depth resolution of ion beam analysis, i.e., forward scattering spectrometry (FSS) or elastic recoil detection analysis (ERDA), can be calculated by the DEPTH code (all versions) [10], which is based on a proper treatment of energy spread contributions from both extrinsic (energy and angular spread of the beam, geometric spread caused by a finite beam spot and detector solid angle, the effect of the absorber foils (if any), energy resolution of the detection method) and intrinsic origins (energy straggling and small-angle multiple scattering effects in the sample, Doppler effect). The fluctuations of physical and experimental origin can be handled as additive and independent random variables. The only dependence is that between the angular deviation and the lateral displacement of the same particle along its trajectory: these random variables are strongly correlated and the distribution of
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their sum can be calculated separately [11]. In the calculations, the various contributions to the energy spread can be composed in a way that takes into account the peculiar shapes of the corresponding probability densities [10]. DEPTH (version 1) was developed originally only to calculate the energy and depth resolution of ion beam methods using a single target layer [12,13]. Later, calculation for multilayer and spectrum simulation were also implemented (version 2), at first for ERDA [14,15], then for other methods [16]. From version 3, besides surface barrier detectors, a magnetic spectrograph can be also used [17]. The simulation quality of DEPTH (version 3) was shown in the intercomparison of ion beam analysis software sponsored by the International Atomic Energy Agency (IAEA) [18–20]. For any type of coincident experiments, the energy spread calculation published in [10] cannot be used anymore without modification. Even the spectrum based on the individual detector of the coincidence experiments can be significantly different from the spectrum taken with the same detector but without coincidence. If the shapes of the two detector apertures used in coincident detection are not the same, which is often the case when the beam and target particles are different, effective shapes and solid angles have to be calculated [3] using the kinematics. In fig. 1, the schematics of the scattering geometry are shown. It can be shown that the following relation has to be fulfilled between the recoil angle of u and scattering angle of #:
3. Results and discussion
sin u cos u tg# ¼ M2 2 1 M2
1þ2 sin u
ð1Þ
if M 1 ¼ M 2 ! tg# ¼ ctg u; where M1 and M2 are the mass number of the scattered and recoiled particles. This equation can be used to calculate the effective shape of the detectors, and thus the effective solid angles. If the scattered and recoiled particles have the same mass, they will be scattered at 90° relative to each other and if a ±45° geometry is used with similar detector shapes and distances, the detector shape correction due to kinematics can be disregarded. If the foil is thick enough, i.e., the effect of angular spread of the outgoing particles cannot be neglected, it will also influence the shape of the spectra. In fact, the ions scattered/recoiled directly to the detectors can change their direction due to multiple scattering, which can be characterised by the angular spread. The full
front and back side of the sample
DF
ϑ ϕ α
width at half maximum (FWHM) and the shape of the angular spread will depend on the path length, i.e., the depth where the collision event takes place. If one of the particles suffers higher angular deviation from the initial direction than the size of the detector allows, this particle will be not detected, and therefore it will be missing from the coincident spectra resulting a yield loss due to angular spread. The probability of the detection can be determined from the angular spread distribution and the opening angle of the detector. DEPTH calculates the depth dependent angular spread distributions both for the recoiled and the scattered particles, which are considered to be independent from each other and determines the joint probability of the detection of both particles. To calculate ESDSR spectra, the dependency in the energy spread calculation has to be reconsidered, too. In Table 1 the energy spread contributions are given for the individual methods (FSS and ERDA) calculated as in Ref. [10], and for ESDSR spectra are given, this latter case being also calculated using identical particles in ±45° geometry. These contributions are already independent from each other, so adding them is equivalent to the convolution of their probability functions. In the simulations of ESDSR spectra both the correct calculation of the energy spread contributions and the yield loss due to angular spread have to be taken into account.
βR βF DR
Fig. 1. Schematic of the coincident spectrometry of FSS and ERDA. In IBM, transmission geometry the scattering angle, # and recoil angle, u are defined as # ¼ ða þ bF Þ 180 and u ¼ ða þ bR Þ 180, where a is the incident angle, bF and bR are the outgoing angles for FS and ERDA, respectively. Using this definition # is positive, while u negative, indicating that the two detectors are on different sides of the beam.
Based on above considerations a new part was included in DEPTH to calculate the energy and depth resolution as well as the energy spectra of the coincident method. Both the signal of an individual detector and ESDSR spectra can be calculated. To determine spectra of identical, spin zero particles Mott’s cross section also included [21]. The foil samples are usually not homogeneous, so it is necessary to describe their thickness inhomogeneity. This is also included in the calculation similarly to the inhomogeneity of the absorbers. For Mylar foil typically 3– 5% inhomogeneity was found in Ref. [13]. To check the quality of DEPTH calculations experimental spectra of 12C–12C scattering at normal incidence of 12 MeV 12C ions and in ±45° geometry were used from Ref. [4]. In the simulations the experimental parameters taken of Ref. [4] were used: beam energy spread 0.1%, the detector distance and diaphragm were set to 80 and 5 mm, respectively, which results a solid angle of 3.07 msr and an angular distribution of 45 ± 2° and the detector resolution was 150 keV. Because the dead layer of the detectors was unknown, the pulse height effect can not be determined as precisely as Pascual-Izarra and Barradas did [22]. However, the energy loss of C ions in a thin dead layer is nearly constant in the detected energy range of 1–6 MeV. Therefore, the pulse height effect can be considered through E0 of the linear energy calibration of the spectra. For pure Coulomb scattering of 12C–12C, Mott’s cross section has to be used. Fig. 2 shows an energy spectrum of the coincidence spectrometry using only one of the detector signals taken on a thin carbon film. To get the correct shape of the measured sample 10% inhomogeneity has to be used in the simulation. The ESDSR (recording the energy sum of the detected scattered and recoiled particles) spectra of the same carbon foils are shown in Fig. 3. For comparison, the convolution of the two individual detector signals is also shown. The ESDSR spectrum is sharper than the convoluted curves, which clearly shows that the correlation between the energy spread contributions is important in interpreting the results. DEPTH calculation using the same inhomogeneity as before failed to describe the experimental spectrum. To get the correct shape 250% inhomogeneity is necessary. This value is extremely high, and it overestimates the spread in the spectrum of single detection (see Fig. 2). The reason of this huge difference is probably
Energy spread contribution
Individual methods
Coincidence spectrometry, ESDSR for
FSS
ERDA
FSS and ERDA
Identical particles, ±45° geometry
DEb C in kF C out;F F @# ctg aK in kF þ @k @# @ a C out;F Da
DEb C in kR C out;R R @u ctg aK in kR þ @k @ u @ a C out;R Da
DEb C in ðkF C out;F þ kR C out;R Þ @kF F ctg aK in kF þ @k @# C out;F DaF þ ctg aK in kR þ @ u C out;R DaR
DEb C in C out
^S;in Energy straggling, inward, DE ^MS;in Multiple scattering, inward, DE
DES;in kF C out;F
DES;in kR C out;R
sin;F DUin
sin;R DUin
DES;in ðkF C out;F þ kR C out;R Þ in;F s DUin;F þ sin;R DUin;R
^Doppl Doppler-effect, DE ^b Geometric spread, DE
DEDoppl kF C out;F F @# ctgbF K out;F þ @k @# @b C out;F DbF 2 g wF 2 bF ðDbF Þ2 ¼ d;F þ gLbD;Fd sin LD;F sin a
DEDoppl kR C out;R R @/ ctgbR K out;R þ @k @/ @b C out;R DbR 2 g wR 2 d sin bR ðDbR Þ2 ¼ d;R þ gLbD;R LD;R sin a
^S;out Energy straggling, outward,DE
DES;out;F
DES;out;R
^MS;out Multiple scattering, outward, DE Detection system, DED
sout;F DUout;F
sout;R DUout;R
DED;F
DED;R
^b Beam energy, DE ^a Beam angular spread, DE
Where
DEDoppl kF C out;R þ kF C out;R @kF R ctgbF K out;F þ @# C out;F DbF þ ctgbR K out;R þ @k @/ C out;R DbR g w 2 g b d sin bF F 2 ðDbF Þ2 ¼ d;F LD;F Þ þ ð LD;F sin a g w 2 2 d sin bR R ðDbR Þ2 ¼ d;R þ gLbD;R LD;R sin a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDES;out;F Þ2 þ ðDES;out;R Þ2 sout;F DUout;F þ sout;R DUout;R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDED;F Þ2 þ ðDED;R Þ2
0 DaF ¼ DaR DES;in C out 0 DUin;F ¼ DUin;R DEDoppl C out 0
DbF ¼ DbR
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðDES;out Þ2 0, DUout;F ¼ DUout;R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðDED Þ2
E. Szilágyi / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1731–1735
Table 1 Energy spread contributions are calculated for FSS and ERDA from Ref. [10], for the ESDSR spectra of FSS and ERDA and for ESDSR of identical particles using ±45° geometry. All the contributions are calculated when the collision products leave the samples, i.e., using all corrections introduced in [10]. The correction caused by the reaction here is given by kF and kR, i.e., the kinematic factors of the forward scattered and recoiled particles. Cin and Cout are the thick target corrections for the incident and outgoing paths. Kin and Kout denote the energy loss for inward and outward paths calculated with corrections for thick targets. The geometric spread arises from the finite beam size d, and detector aperture w. LD is the detector distance. The factors gd and gb take into account the shapes of the beam and detector aperture (for circular detector or circular beam with uniform current density g = 0.59, while for rectangular shapes g = 0.68). The indices F and R denote if the quantity is different for forward scattered and recoiled particles. The relationship between the reaction angle, and incident and outgoing angles (see Fig 1) is # ¼ ða þ bF Þ 180 and @u u ¼ ða þ bR Þ 180 for FSS and ERDA, respectively. Consequently @@#a ¼ 1, @@ua ¼ 1, @b@#F ¼ 1 and @b ¼ 1. In the calculation of the MS contribution, the statistical dependence between the angular and lateral spread are considered. The angular R spread both for the inward, DUin and outward DUout directions result in a spread in the reaction angles of # and u, therefore the corresponding energy spreads are connected through the reaction like in the beam angular or geometric spreads. Sin contains the corrections for the outgoing path; therefore it is slightly different for FS and ERDA. In coincident spectrometry, all the energy spread contributions of FSS and ERDA originating from before the scattering are fully correlated. The collision event connects the scattering and recoil angles according to Eq. (1) (the changes in the reaction angles are anticorrelared, i.e., d# and du have different signs). This relationship has to be taken into account for the energy spreads arising from the spread in the outgoing angles. The effective detector shapes, w* and g* have to be calculated for both detectors, all other energy spreads in the sample can be considered as independent contributions. For coincident spectrometry used for identical particles in ±45° geometry the following thus simplify further the formalism: k = kF = kR, kF + kR = 1, Cout = Cout,F = Cout,R. Moreover, # þ u ¼ 90 therefore d# ¼ d/. In this case the two detectors are considered with similar detector shapes and distances. For perpendicular incident beam, = 90°, ctg is zero.
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20 μg C film
Normalized yield
60 50
160
E1 DEPTH with layer inhomogeneity 10 % 250%
140
DEPTH ZBL 95 DEPTH SRIM 03 DEPTH SRIM 06 CORTEO SRIM 06
120
Yield (a.u.)
70
180
40
100 80
30
60
20
40 20
10
0 4000
0 5000
6000
6000
7000
8000
10000
Energy (keV)
Energy (keV) 12
12
Fig. 2. Energy spectra of C– C scattering at normal incidence of 12 MeV 12C ions and in ±45° geometry taken on a 20 lg/cm2 thick carbon layer using one of the individual detector signal of the coincident detectors. Symbols and lines denote the experimental and simulated spectra. In the simulation the experimental parameters taken of Ref. [4] were used. Beam energy spread 0.1%, the detector distance and diaphragm were set to 80 and 5 mm, respectively, which results a solid angle of 3 msr and an angular distribution of 45 ± 2°, detector resolution 150 keV.
Fig. 4. Effect of various stopping powers calculated by DEPTH for ESDSR spectra using Mylar samples, without angular spread correction. Using the same experimental conditions a similar calculation was performed by the CORTEO program.
200
2.75 μm Mylar, 10 % inhomogeneity
20 μg C foil
Normalized yield
100
back side
150
Yield (a.u.)
150
measured CORTEO DEPTH
(E1*E2) (E1+E2) DEPTH with layer inhomogeneity 10 % 250 %
front side 100
50
0
50
0
2000
4000
6000
8000
10000
12000
Energy (keV)
0 10500
11000
11500
12000
12500
13000
Energy (keV) Fig. 3. Measured (solid symbol) and simulated (lines) ESDSR spectra of the 20 lg/ cm2 thick carbon layer. The result of the convolution of the two individual measured spectra (open symbols) is also shown. If the energy spread fluctuations would be independent from each other, the result of the sum energy would look like this. The spectra are normalized to the maximal height. According to the DEPTH calculation, the spectrum using an inhomogeneity of 10% (Fig. 2) will be too sharp. To describe the shape properly a 250% inhomogeneity in layer thickness has to be used. The experimental parameters are the same as in the case of Fig. 2.
that only a very simple inhomogeneity model is used in DEPTH. In case of very thin foils, where the foil thickness is comparable to the correlation length, the correlation length of the surfaces cannot be neglected. The effects of surface roughness and correlation lengths were already met in IBA practice [23]. The coincident technique is even more sensitive to the correlation lengths. Analysing thicker foils, the accuracy of the stopping power may become crucial, as it usually is in spectrometry using heavy ions. In Fig. 4 ESDSR spectra of a thick Mylar layer are shown simulated by CORTEO and DEPTH using the same experimental conditions and SRIM 06 stopping powers [24]. The yield loss due to angular spread was not taken into account in the simulations. The shapes of the
Fig. 5. Experimental ESDSR spectrum of a Mylar foil together with CORTEO and DEPTH calculations. The curves are normalized to peak area. The CORTEO curve was calculated using 2.85 lm Mylar, with a roughness of 285 nm and stopping power of SRIM06. In DEPTH 2.75 lm Mylar, 10% inhomogeneity and stopping power of ZBL95 was used. In both cases the effect of angular spread was taken into account. The experimental parameters are the same as in the case of Fig. 2.
CORTEO and DEPTH simulations agree quite well with each other. The effect of various stopping powers (SRIM 03 and ZBL 95) was also calculated by DEPTH. Although, in this work better agreement is found between the calculated and experimental spectra of Figs. 2, 3, 5–7 using ZBL 95 than using SRIM 03 or SRIM 06, this experiment is not appropriate to decide which stopping power is the better. Moreover, any uncertainties in the experimental parameters (e.g., foil thicknesses, scattering angles, energy calibration of the accelerator, energy calibration of the detector, dead layer correction, etc.) have an effect to the evaluation. CORTEO and DEPTH simulation with yield loss due to angular spread calculations are also compared to each other and to an experimental ESDSR spectrum taken on a Mylar foil as shown in Fig. 5. Areas of the calculated curves are normalized to the measured spectrum. In both CORTEO and DEPTH simulations only a simple Mylar layer is used without taking into account any modification of the sample. There is a significant difference between the
E. Szilágyi / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1731–1735 180
measured simulated
160
back side
140
Sample: 11 nm CH 2180 nm Al 11 nm CH
Yield
120 100
Fig. 7 represents the measured and simulated ESDSR spectra taken on a 12C implanted foil with a fluence of 1.3 1016 at/cm2 [4], measured under the same experimental conditions as pure Al foil. In DEPTH simulation the total amount of implanted carbon is found to be 1.5 1016 at/cm2, and the hydrocarbon deposited layer is thicker at the front side compared to the pure Al foil indicating that same hydrocarbon deposit occurs also during the ion implantation. A foil inhomogeneity of 15% is used.
80
layer inhomogeneity 15%
4. Conclusions
60
front side 40 20 0 0
2000
4000
6000
8000
10000
12000
Energy (keV) Fig. 6. Measured ESDSR spectrum (symbols) and DEPTH simulation (line) taken on a pure Al foil of a thickness of 2 lm. The carbon peak at the front side of the sample is weaker and wider than that of at the back side because of the decreased probability of the coincident detection due to the angular spreads of the outgoing ions and the larger energy spread. The experimental parameters are the same as in the case of Fig. 2.
measured simulated
200
back side
160 nm Al1C0.003 670 nm Al 12 nm CH
I.B. Radovic´ is gratefully thanked for providing the spectra of C–12C coincidence experiments. The author is indebted to F. Schiettekatte and E. Kótai for the helpful discussions.
12
implanted carbon
250 nm Al1C0.006
100
front side
References
50
0 0
2000
The energy spread contributions are reconsidered for elastic recoil coincidence spectrometry. The correlations between FSS and ERDA are taken into account and included in the analytical ion beam simulation program DEPTH. In the calculations, effective detector geometry and multiple scattering effects are considered. Mott’s cross section for the identical, spin zero particles is included. Spectra based on the individual detector signal and containing the energy sum of the detected scattered and recoiled (ESDSR) particles originating from the same scattering events can be calculated. The DEPTH calculation is compared to experimental 12 C–12C spectra and calculations using similar conditions with a Monte Carlo code CORTEO. Although the first results are promising, further studies are necessary to elaborate the quantitative results. The coincident technique also looks very sensitive to the correlation length of the surface roughness, if the foil thickness is comparable to the correlation length. Acknowledgements
Sample: 16 nm CH 1080 nm Al 160 nm Al1C0.003
150
Yield
1735
4000
6000
8000
10000
12000
Energy (keV) Fig. 7. Measured and simulated ESDSR spectra taken on an implanted Al layer. The implantation was carried out with 1 MeV 12C ions with a fluence of 1.3 1016 12C/ cm2. The experimental parameters are the same as in the case of Fig. 2.
two simulations, however, it cannot be decided which code calculates the shape of the angular distribution better, because of the sample degradation under the beam. The slopes of the front and back sides of the DEPTH simulation agree quite well to the measured spectrum indicating that the energy spread of the ESDSR spectrum is calculated properly. Fig. 6 shows the measured and simulated ESDSR spectra taken on pure Al film with a nominal thickness of 2 lm. The spectrum was collected with a charge of 8 lC using a 12C3+ beam. The simulated spectrum by DEPTH reproduces the measured spectrum using a quite reasonable layer structure. The model layer is an Al foil with a thickness of 2.18 lm with deposited hydrocarbon layers with a thickness of 11 nm both on the front and back side (for the calculations a layer composition of CH and an atomic density of 11.4 1022 at/cm3 are used to describe the hydrocarbon deposit). The inhomogeneity of the foil is set to 15%.
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