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Accurate automated non-resonant NRA depth profiling: Application to the low 3He concentration detection in UO2 and SiC
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 258 (2007) 471–478 www.elsevier.com/locate/nimb
Accurate automated non-resonant NRA depth profiling: Application to the low 3He concentration detection in UO2 and SiC G. Martin a, T. Sauvage a
a,*
, P. Desgardin a, P. Garcia b, G. Carlot b, M.F. Barthe
a
CNRS/Centre d’Etudes et de Recherches par Irradiation, 3A rue de la Fe´rollerie, 45071 Orle´ans Cedex 2, France b CEA, CE de Cadarache, DEN/DEC/SESC/LLCC, Baˆt. 151, 13108 Saint-Paul-lez-Durance Cedex, France Received 28 November 2006; received in revised form 23 January 2007 Available online 15 February 2007
Abstract An automated method was developed to extract elemental depth profiles from non-resonant nuclear reaction analyses (NRA), which involves a two-stage procedure. The first stage enables the determination of the number of layers to be used in the final depth profile determination along with the thicknesses of each of the layers. To this end, the RESNRA program, which relies on the SIMNRA 5.0 simulation software to calculate a multilayer target, was designed at CERI. A definition of the depth resolution based on statistical considerations is proposed. In the second stage of the fitting process, a depth profile and corresponding error bars are extracted from the experimental spectrum by running a generalized reduced gradient (GRG2) algorithm using the previously calculated multilayer target. The one-to-one correspondence between the experimental spectrum and the depth profile demonstrates the objectivity of the method. The method is then applied to determining low concentration 3He depth profiles in implanted UO2 and SiC samples using the 3 He(2H, 4He)1H non-resonant nuclear reaction. The results clearly demonstrate the relevance and potential of the method. 2007 Elsevier B.V. All rights reserved. PACS: 82.20.Wt; 61.18.Bn; 61.72.Ss Keywords: Automation; NRA; Depth profiling; Depth resolution; Helium; SIMNRA
1. Introduction Nuclear reaction analysis (NRA) is a well known technique for depth profiling light elements in materials. As nuclear reactions are isotope specific, NRA is a powerful method in a wide range of fields requiring isotopic tracer measurements. In non-resonant NRA, the incident particles have a prescribed energy which they lose as they penetrate the material. The incoming particles are then liable to undergo a nuclear reaction with target nuclei. If reaction products have a high enough kinetic energy to recoil out of the target, they are detected at different energies depending on where they appeared in the sample. Non-resonant NRA
*
Corresponding author. Tel.: +33 238 255 419; fax: +33 238 630 271. E-mail address: [email protected] (T. Sauvage).
0168-583X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.01.288
involves analysing the energy distribution of those reaction products, from which a depth profile is deduced. Generally, the analysed target is modelled as a multilayer material where the layer thickness and the relative elemental concentrations are manually adjusted, so that the spectrum calculated using the SIMNRA 5.0 software [1] fits the experimental data available. Therefore, depth profiles extracted by different users may vary, despite all leading to acceptable simulations. This can cause some imprecision in the depth profile determination which in turn can induce a misinterpretation or over-interpretation of the experimental data. The aim of this work is to develop an automated method to determine low concentration depth profiles in a one-toone correspondence with experimental non-resonant NRA spectra. The analysis method relies on a two-stage process, the first of which involves determining the thickness of each
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478
of the material layers to be used in the final profile determination (second stage). To this end, the target is modelled as a multilayer material in which the analysed element concentration is constant and assumed to be low (as a consequence of this latter assumption, the stopping powers of incoming or outgoing particles are not altered). The thickness of each consecutive material layer is then chosen to correspond to the depth resolution at a given distance from the sample surface, which is derived from statistical criteria applied to RESNRA calculation results. This newly developed program relies on SIMNRA simulations to generate energy spectra corresponding to homogeneous distributions of 3He atoms in the material. In the second stage, the concentration of the probed isotope in each of the layers, the thickness of which is determined in the first stage of the analysis process, is evaluated using a generalized reduced gradient (GRG2) algorithm [2]. No constraints are assumed to fit the experimental energy spectrum. An automated method was previously proposed [3] to extract depth profiles from NRA spectra. Some aspects of that method differ from the present work: in this paper, the depth resolution is an essential part of the fitting process. Our calculation procedure was used to extract 3He depth profiles from a-spectra obtained from the 3He(2H, 4He)1H non-resonant nuclear reaction in UO2 and SiC implanted samples. This method provides a single solution to the problem of the determination of the concentration profile and in addition provides error bars associated with the elemental concentrations determined in each of the modelled layers. This is obviously essential when analysing the experimental data and comparing two depth profiles. 2. Depth resolution and layer thickness determination 2.1. Statistical definition of the depth resolution The depth resolution in backscattering analysis – e.g. NRA – is usually defined as the minimum detectable depth difference dx related to the minimum detectable scattered particle energy variation dE [4]. The energy difference between detected particles originating at different depths within the material is the result of the energy loss of incident and outgoing particles. Sources of the broadening of the measured energy distribution are detector resolution, initial energy spread of the incident beam, straggling of the outgoing and incoming particles, geometrical energy spread and energy spread coming from the multiple scattering [5]. Since energy straggling increases as the ion beam penetrates the sample, the depth resolution deteriorates at increasing distances from the sample surface. To calculate the depth resolution at any depth, the minimum detectable energy variation has to be defined. In non-resonant backscattering analysis, dE can be correlated with the help of a useful criterion based on statistical considerations. For a given set of experimental conditions, we consider the NRA analysis of a thick material containing a homoge-
xi Ln Counts
472
...
L i+1
Surface
Li
L i-1
...
L1
wi
Si (E) spectrum
Sj (E) spectra with j ≠ i Sum of all spectra
E i+1
Ei
Energy E
Fig. 1. Representation of spectra from the analysis using a non-resonant reaction of a multilayer material.
neous concentration of an element whose concentration one wants to determine. With the SIMNRA software, this material is modelled as a multilayer target and the spectral response of this analysis is the sum of n spectra generated from each of the modelled layers. A layer Li of thickness wi centred at depth of xi corresponds to a detected energy distribution Si(E). Fig. 1 shows the contribution of each of the layers to the total energy spectrum of particles emitted from nuclear reactions in the material. To be clear in the understanding of Fig. 1, the cross section of the nuclear reaction is assumed to be constant in the laboratory frame and the thicknesses of the n layers identical. The Si(E) spectrum is intersected by other spectra at energies Ei+1 and Ei. The probability pi that the particles detected in the [Ei+1, Ei] energy range are emitted from the layer Li is the counts in Si(E) divided by total number of counts in this energy range R Ei S i ðEÞ dE E i : pi ¼ R E hiþ1P ð1Þ i S ðEÞ dE j j Eiþ1 We now assume the only particles to contribute to the spectrum in the [Ei+1, Ei] energy range are those produced in layers adjacent to layer Li. Therefore, a first order estimate pi can be calculated from Eq. (1) by considering three successive material layers Li1, Li and Li+1 only, and their resulting consecutive spectra: R Ei S ðEÞ dE Eiþ1 i p i R Ei ; ð2Þ ½S ðEÞ þ S i ðEÞ þ S iþ1 ðEÞ dE Eiþ1 i1 pi takes into account the variations of the cross section in the vicinity of xi, along the primary ion trajectory. Probability pi is directly related to the [Ei+1, Ei] energy range and
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478
xi Li+1
Surface
Li
Counts
to thickness wi of layer Li. Conversely, in given experimental conditions and for a given probability pi, the minimum detectable energy variation dE is given by (Ei+1 Ei) difference and the associated depth resolution dx by the corresponding thickness wi of layer Li at depth xi.
473
Li-1
wi
2.2. Depth resolution calculation A system of equations can be set up to determine the depth resolution at a chosen probability Pc by applying expression (2). However, such a system is impractical to solve. Therefore, expression (2) has been simplified by making the three following assumptions:
These considerations allow the calculation of pi considering a system of two consecutive layers, analogous to the system of three layers used to calculate pi according to relation (2). Fig. 2 compares both analyses and their resulting spectral responses. Probability pi can be determined by calculating the hatched areas represented in Fig. 2. In the system of three layers (a) and according to the previous assumptions, relation (2) can be written as 1
pi 1 2
R E
R min½S i1 ðEÞ; S i ðEÞ dE þ 12 E min½S i ðEÞ; S iþ1 ðEÞ dE : R Ei ½S ðEÞ þ S i ðEÞ þ S iþ1 ðEÞ dE Eiþ1 i1
ð3Þ
It is worth noting that (1 pi) represents the uncertainty on the detected element position at depth xi. In the twolayer system (b) where both layers have the same thickness wi, the probability pi is calculated from the formula (4). According to the previous assumptions, the numerators as well as the denominators in relation (3) and in relation (4) can be equalled R min½S i ðEÞ; S iþ ðEÞ dE pi 1 1E R : ð4Þ ½S ðEÞ þ S iþ ðEÞ dE 2 E i
min[Si+1(E), Si(E)]
Si(E)
Ei+1
Si-1(E)
Ei
min[Si(E), Si-1(E)]
Energy E
xi
Counts
• The energy straggling of detected particles does not locally depend on their mean energy and therefore on the depth at which they are emitted. Indeed spectra resulting from three consecutive layers Li1, Li and Li+1 should have similar shapes. • Stopping powers and energies of reaction product particles are a locally linear function of depth. Therefore the mean energy of detected particles is locally proportional to the depth at which they are emitted. This implies that the energy differences between spectra peaks resulting from three consecutive layers Li1, Li and Li+1 are proportional to the material thicknesses that separate them. • The cross section of the reaction is a linear function of the depth at which incident particles react with target nuclei. The number of counts inside spectra resulting from three consecutive layers Li1, Li and Li+1 should be then a linear function of their respective depth location xi1, xi and xi+1.
Si+1(E)
Surface
Li+
Li-
wi
wi
Si+(E)
Si-(E)
min[Si+(E), Si-(E)]
Energy E
Fig. 2. Representation of the signals from the simulation of three consecutive material layers (a) and of two consecutive material layers of identical thickness wi (b). Both systems are equivalent.
Using this last equation, it is possible to converge pi to a given probability Pc by adjusting only the thickness wi of both considered layers. This enables the simple calculation using a simulation program of the depth resolution wi at a depth xi inside the material, defined for a given probability Pc. 2.3. Multilayer target determination The software RESNRA was developed at CERI to determine the thickness of the successive material layers to be used in the depth profile calculation. This software communicates with SIMNRA 5.0. which is used to calculate NRA energy spectra. It takes into account all the contributions to energy spread of NRA spectra, notably the straggling associated with incident and emitted particles. Both nuclear and electronic energy straggling are taken into account in the calculations through the application of Bohr’s theory, and use of the Chu correction [6] with respect to electronic energy straggling. The energy distribution is calculated from a multilayer material. This multilayer structure of the target enables the simulation of elemental gradients through variations in relative concentrations from one layer
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to another. Quantitative information about the target elemental composition is obtained by comparing the calculated energy distribution with that obtained experimentally. Our contribution lies in the calculation of the depth resolution R(x) as a function of depth x, the estimate of the thicknesses of the successive layers based on function R(x), and the automation of the depth profile determination. For a given probability Pc, the RESNRA program can calculate from relation (4) the depth resolution R(x) by considering a system of two layers of identical thickness (Fig. 2) at different depths. At each depth xi, the thickness wi is chosen so that Eq. (4) is verified. A simple root-finding algorithm (bisection method) is used so that the right hand side of Eq. (4) converges to a prescribed value of pi. When pi = Pc, wi is then the calculated depth resolution of the method at depth xi. The calculation stops when the reaction cross section becomes null in the depth range including the layers Li and Li+, which in practical terms corresponds to the depth range beyond which the material cannot be analysed under the applied experimental conditions. Finally, RESNRA generates the multilayer target for which layer thicknesses correspond to the depth resolution R(x) defined for a given probability Pc. Thicknesses wi and central positions xi of consecutive layers Li are successively calculated by RESNRA by solving the following system of Eq. (5): 8 i1 > < x ¼ wi þ P w ; i j 2 ð5Þ j¼1 > : wi ¼ Rðxi Þ: The second equation of system (5) is solved numerically by approximating R(x), evaluated at discreet positions from the sample surface, with an exponential function. Although this has always provided excellent results, it is indeed conceivable that other functions may provide better results for different systems. 3. Depth profile calculation One then chooses arbitrarily a charge Q0 to simulate the spectra Si(E) corresponding to each of the layers used for modelling the material response. The concentration C0, which is kept constant in the first stage, should ideally equal half of the maximum concentration of the probed element so as to minimise the variations of the stopping power of the particles used for the analysis. The position and thickness of the layers are determined from the procedure detailed in the preceding section. Let Ci be the unknown elemental concentration in material layer Li and Qexp the total charge of the analysis that results in the experimental spectrum Sexp(E). The simulated helium profile Ssim(E) can be expressed as a linear combination of the energy spectra Si(E) Qexp X C i S i ðEÞ: ð6Þ S sim ðEÞ ¼ C 0 Q0 i
The problem is then reduced to determining the set of values for Ci which minimises the following error function: R 2 ½S exp ðEÞ S sim ðEÞ dE : ð7Þ v2 ¼ E R ½ E S exp ðEÞ dE2 The minimisation problem is solved by using a GRG2 type algorithm [2]. An essential part of the depth profile determination is to evaluate the error bar associated with each of the concentrations resulting from the minimisation procedure. There are several terms that can contribute to the total uncertainty on measured concentrations in ion beam analysis (IBA) such as uncertainties on the detector energy calibration, on the charge measurement . . . [7]. Within the framework of this study, the counting uncertainty is assumed to be the main source of error. The error bar relative to the elemental concentration in the material layer Li is therefore determined from the total number of counts Ni associated with the corresponding spectrum Z Qexp C i S i ðEÞ dE: ð8Þ Ni ¼ E Q0 C 0 If the signal follows a Gaussian distribution, then according to the Laplace–Liapounoff theorem [8], the relative standard deviation ri for this distribution is given by 1 ri ¼ pffiffiffiffiffi : Ni
ð9Þ
By definition, there is a 68.3% chance that the true concentration C corresponding to layer Li is such that C i ð1 ri Þ < C < C i ð1 þ ri Þ:
ð10Þ
More generally, assuming a Gaussian distribution for the element concentration C, for any probability F between 0% and 100%, there exists a positive real k such that F is the probability that C is known to within ±k Æ ri. In a similar way, as seen in the previous sections, the probability that the position of a detected element belongs to the interval [xi wi/2, xi + wi/2] is Pc. In order that the analysis be consistent, the probabilities associated with the confidence intervals relative to the concentration and location of detected elements must be equal. In other words F must be equal to Pc. In Table 1, coverage factors k are given for corresponding Pc values according to this last condition. Therefore for a given probability Pc, the relative expanded uncertainty on concentration Ci is given by DC i ¼ k ri : Ci
ð11Þ
The resulting depth profile can then be represented as a set of boxes the widths of which are the expanded uncertainties relative to location and the heights of which are the expanded uncertainties on element concentrations. When the probed element was introduced by implantation, its total fluence / in at cm2 in the analysed sample
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478 Table 1 Coverage factors k, corresponding probability Pc and subsequent number of layers necessary to depth profile 3He in UO2 and SiC Coverage factor k
Probability Pc (%)
Number of layers used for modelling the spectrum in UO2 (625 keV NRA)
Number of layers used for modelling the spectrum in SiC (750 keV NRA)
1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5
86.6 83.8 80.6 77 72.9 68.3 63.2 57.6 51.6 45.1 38.3
8 10 12 14 17 20 23 27 32 39
10 13 16 19 22 26 30 35 42 51 63
is given from layer thicknesses wi in at cm2 and atomic concentrations Ci by X wi C i : ð12Þ U¼ i
As will be shown in Section 4 that v2 is negligible and the correlation between experimental and simulated data is excellent, the total number of counts Nsim. in Ssim(E) represents a physical measure of the implanted fluence. Therefore the relative standard uncertainty on the fluence is given by DU 1 ¼ pffiffiffiffiffiffiffiffiffi : U N sim
ð13Þ
4. Application of our calculation technique to 3He depth profiling in UO2 and SiC 4.1. Sample preparation A sintered uranium dioxide disk and a sintered silicon carbide sample were polished on one side. The UO2 disk thickness is approximately 300 lm and its diameter 8.2 mm. The SiC sample size is 7 · 7 · 0.3 mm3. The densities of sintered UO2 and SiC, determined by immersion in pure water and in pure ethanol, are respectively 10.7 g cm3 and 3.18 g cm3. The UO2 and SiC samples were implanted at room temperature with 0.5 MeV 3He+ ions to fluences of respectively 9 · 1015 at cm2 and 1015 at cm2 using the 3.5 MV Van de Graaff accelerator at CERI Orle´ans. Implanted fluences were chosen to obtain a maximum 3He concentration below the value of 1 at.% in both cases. 4.2. 3He analysis conditions The 3He(2H, 4He)1H NRA method was used to determine helium concentrations at depths up to a few microm-
475
eters. The measurements were performed at the 3.5 MV Van De Graaff accelerator at CERI Orle´ans with an experimental set-up based on the detection in coincidence of both reaction products, alpha particles and protons. a-Particles emitted from the sample at angles between 177 to 179 are detected by an annular 100 mm2 PIPS (passivated implanted planar silicon) detector. The protons transmitted through the sample are detected by a 1200 mm2 detector placed at 0 behind the sample. Its large solid angle allows the detection of all the protons emitted in coincidence with the detected alpha particle which ensures a coincidence yield of 100%. This experimental set-up provides alpha particle spectra free from parasite signals in the region of interest. Helium depth profiles, respectively in UO2 and in SiC, were measured with a 500 · 500 lm2 D+ beam at energies of 625 keV and 750 keV. The charge used for the analyses was 300 lC for UO2 and 2820 lC for SiC. Total numbers of counts of UO2 and SiC spectra are respectively 335 and 1370. Series of backscattered deuterons measurements with the annular detector show a relative uncertainty of charge measurements of ±3%. Further details pertaining to the experimental set-up or the coincidence technique are described in [9]. SIMNRA is configured to use Ziegler’s stopping power [10] in order to simulate spectra. Ziegler’s stopping power is roughly in agreement with the experimental stopping power of alpha particles in UO2 at energies up to 9 MeV [11]. Regarding SiC, a micrograph of a cross sectional specimen implanted with a five times greater fluence (5 · 1015 at cm2) and annealed at 1300 C/30 min reveals helium bubble nucleation in the grain at a depth of 1.2 lm. As is well known, this region corresponds to the nuclear cascades of primary ions. The SRIM simulation of the defect peak depth gives a value of 1.24 lm. Ziegler’s stopping power of helium in SiC is then in excellent agreement with experiments [12]. The angular spread of the incident beam is less than 0.1 and has negligible effect on SIMNRA simulations. The energy resolution of 12 keV of the PIPS detector was taken into account. The cross section used for the 3He(2H, 4 He)1H nuclear reaction was determined at CERI [13]. 4.3. Importance of the layer thickness determination procedure To show the importance of determining the thickness of each of the target layers, the a-particle energy spectrum relative to the UO2 experiment was fitted using 30 consecutive layers all 70 nm thick. The GRG2 algorithm that minimises the v2 indicator was then applied to solving the problem. No relevant depth profile was obtained from running the minimisation algorithm. In particular, He concentrations appeared to change from negative to positive values in consecutive layers. In other words, if the target layers are not carefully calculated, then the minimisation procedure may come up with physically irrelevant concentration values.
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4.4. Depth resolution and multilayer target determinations Several sets of multilayer targets were generated to extract depth profiles for the SiC and UO2 experiments based on different probabilities in the 38.3–86.4% range. Table 1 shows the number of layers contained in each multilayer target corresponding to a chosen probability Pc. As is expected, the number of layers which have to be modelled increases as the probability decreases (i.e. as the probability that two consecutive layers interact increases). Fig. 3 shows the depth resolutions R as a function of depth for different probabilities Pc in the case of the 3He implanted UO2 analysis. All calculated depth resolutions are found to be exponential functions of the depth in this study (correlation coefficients r2 > 0.99). Fig. 4 shows a set of 20 spectra deduced from RESNRA simulations of consecutive layers
in UO2 that contains 1 at.% of 3He with a probability Pc of 68.3%. For this value of Pc, the coverage factor k is equal to 1 and the maxima associated with two consecutive spectra are separated by twice their standard deviation (strictly true if spectra were Gaussians). All spectra appear to intersect at the same relative height, which was to be expected since they all have a similar shape. It was also checked that the sum of all elementary spectra is consistent with a spectrum generated from the simulation of one large UO2 layer containing 1 at.% of 3He. This demonstrates that a linear combination of these elementary spectra can fit our experimental data thus providing an estimate of a 3He depth profile in UO2. At low helium concentrations below 1 at.%, small variations in material composition have no influence on the stopping powers of incident and outgoing particles. 4.5. Helium depth profiles
1200 - 86.6 - 83.3 800 - 77
600
- 68.3
400
- 57.6 - 45.1
200 0 0.0
Probability Pc (%)
Depth resolution (nm)
1000
0.5
1.0 1.5 Depth in UO2 (µm)
2.0
2.5
Fig. 3. Depth resolutions for several probabilities in the case of an analysis involving 3He implanted UO2 with a 625 keV deuteron beam.
20 elementary spectra Sum of the 20 spectra
Counts
Simulation of a large UO2 layer that contains 3 He
1600
1700
1800
1900
2000
2100
2200
2300
Energy (keV)
Fig. 4. Sets of elementary spectra from RESNRA simulations of successive UO2 layers containing 1 at.% 3He, calculated with a probability of 68.3%. Their sum is overlapped to the simulation of a large layer of the same chemical composition.
Both helium depth profiles in UO2 and SiC defined for a probability of 68.3% are presented in Fig. 5, with their corresponding experimental spectra. The numerical algorithm converges to provide a relevant depth profile and the correspondence between experiment and simulation is excellent at this value of Pc, with low values of v2 inferior to 0.01. Each profile calculation lasted approximately 10 min using a computer equipped with a 2.4 GHz processor and with 512 Mo of random access memory (RAM). Helium fluences in UO2 and SiC calculated from the relation (12) are 9.2 ± 0.5 · 1015 at cm2 and 0.97 ± 0.03 · 1015 at cm2, respectively. The estimated values of fluences are in excellent agreement with the implantation ones. In Fig. 6, depth profiles resulting from automated simulations of experimental spectra for different values of Pc are presented. When Pc increases, the helium depth profile in UO2 or SiC broadens and the indicator of the quality of the fit v2 increases to an unacceptable value. For low Pc values, helium concentrations derived from the fitting procedure of the UO2 experiment (a) appear to oscillate. The error bars that represent counting uncertainties on concentration are smaller than the oscillations of the concentration for Pc = 51.6%. This indicates that the fitting process is likely to introduce uncertainties larger than counting uncertainties. As several depth profiles are expected to minimise v2 when Pc is too low, the fitting algorithm can converge towards a profile that is not physically relevant. In SiC, the fact that helium is concentrated in a very narrow layer prevents oscillations in the concentration profile even for the lowest values of Pc. Nevertheless, for both SiC and UO2, a probability Pc of 68.3% – i.e. k = 1 – appears to constitute the best compromise. Error bars on the SiC profile are small because of the statistics (number of counts) of the corresponding a-spectrum.
0.30
12
0.25
10
Counts / keV
Helium concentration (at.%)
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478
0.20 0.15 0.10 0.05
UO2 exp. UO2 sim.
8 6 4 2
0.00 0.0
0.5
1.0
1.5
2.0
0 2300
2.5
2200
2100
Depth in UO 2 (µm)
c
2000
1900
1800
1700
Energy (keV)
d
0.06 0.05
50
SiC exp. SiC sim.
40
Counts / keV
Helium concentration (at.%)
477
0.04 0.03 0.02
30 20 10
0.01 0.00 0.0
0.5
1.0
1.5
2.0
0 2000
2.5
1900
1800
1700
1600
Energy (keV)
Depth in SiC (µm)
Fig. 5. 3He depth profiles (a and c), experimental spectra and simulated spectra (b and d) of as-implanted UO2 and SiC. Depth profiles are presented with error bars at a probability of 68.3% (standard uncertainties).
0.06
Pc = 51.6 % Pc = 68.3 % Pc = 83.8 %
0.3
Helium concentration (at.%)
Helium concentration (at.%)
0.4
0.2
0.1
0.0
Pc = 38.3 % Pc = 68.3 % Pc = 83.8 %
0.05 0.04 0.03 0.02 0.01 0.00
0.0
0.5
1.0
1.5
2.0
2.5
Depth in UO 2 (µm)
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Depth in SiC (µm)
Fig. 6. Helium depth profiles in UO2 (a) and SiC (b) at different probabilities Pc.
5. Conclusion A definition of the depth resolution based on statistical considerations has been proposed as a first step towards extracting elemental depth profiles from non-resonant NRA spectra. The advantages of this useful definition have been demonstrated through the treatment of NRA data obtained from implanted UO2 and SiC samples containing low 3He concentrations. A program (RESNRA) that relies on SIMNRA 5.0 has been developed at CERI to calculate the depth resolution as a function of the distance from the sample surface, for a given expanded uncertainty. By generating a multilayer target from the depth resolution calcu-
lation, a RESNRA simulated dataset is introduced in a GRG2 algorithm that allows the depth profile extraction from experimental spectra. Counting uncertainties for each of the inferred 3He concentrations are also determined. It is shown that if one takes into account the variations of the depth resolution with depth, then there is a unique set of helium concentrations that fits the experimental data. When the probability on the helium depth location is not adapted, the depth profile simulation provides unsatisfactory results. However, the algorithm converges without calculation constraints towards a single and relevant depth profile when layer thicknesses are calculated according to a depth resolution calculated for a probability of 68.3%.
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The correspondence between experimental and simulated spectra is then excellent. Furthermore, error bars on obtained depth profiles have also been calculated. The accuracy of this method for profile extraction has encouraged us to apply it to data recently generated and will be used in the future. Furthermore, it should be applicable to other areas of IBA, such as RBS (Rutherford backscattering spectroscopy) depth profiling. Acknowledgements Jean-Charles DUMONT is thanked for his help in developing the RESNRA software. The authors are grateful to their collaborators of the ACTINET network and to the PRECCI programme, funded by CEA and EDF. References [1] M. Mayer, SIMNRA User’s Guide, Report IPP 9/113, Max-PlanckInstitut fu¨r Plasmaphysik, Garching, Germany, 1997.
[2] L.S. Lasdon, A.D. Waren, Comput. Chem. Eng. 7 (1983) 159. [3] N.P. Barradas, R. Smith, J. Phys. D: Appl. Phys. 32 (1999) 2964. [4] J.R. Tesmer, N. Nastasi, Handbook of Modern Ion Beam Materials Analysis, MRS, Pittsburgh, 1995, p. 46. [5] J.R. Tesmer, N. Nastasi, Handbook of Modern Ion Beam Materials Analysis, MRS, Pittsburgh, 1995, p. 144. [6] W.K. Chu, Phys. Rev. 13 (1976) 2057. [7] K.A. Sjo¨land, F. Munnik, U. Wa¨tjen, Nucl. Instr. and Meth. B 161– 163 (2000) 275. [8] G. Charlot, Statistiques applique´es a` l’exploitation des mesures, Masson, Paris, 1978. [9] T. Sauvage, et al., in preparation. [10] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [11] Hj. Matzke, J. Nucl. Mater. 270 (1999) 49. [12] T. Sauvage, G. Carlot, G. Martin, L. Vincent, P. Garcia, M.F. Barthe, E. Gentils, P. Desgardin, Nucl. Instr. and Meth. B, in press. [13] T. Sauvage, H. Erramli, S. Guilbert, L. Vincent, M.F. Barthe, P. Desgardin, G. Blondiaux, C. Corbel, J.P. Piron, F. Labohm, J. Van Veen, J. Nucl. Mater. 327 (2004) 159.
Nuclear Instruments and Methods in Physics Research B 258 (2007) 471–478 www.elsevier.com/locate/nimb
Accurate automated non-resonant NRA depth profiling: Application to the low 3He concentration detection in UO2 and SiC G. Martin a, T. Sauvage a
a,*
, P. Desgardin a, P. Garcia b, G. Carlot b, M.F. Barthe
a
CNRS/Centre d’Etudes et de Recherches par Irradiation, 3A rue de la Fe´rollerie, 45071 Orle´ans Cedex 2, France b CEA, CE de Cadarache, DEN/DEC/SESC/LLCC, Baˆt. 151, 13108 Saint-Paul-lez-Durance Cedex, France Received 28 November 2006; received in revised form 23 January 2007 Available online 15 February 2007
Abstract An automated method was developed to extract elemental depth profiles from non-resonant nuclear reaction analyses (NRA), which involves a two-stage procedure. The first stage enables the determination of the number of layers to be used in the final depth profile determination along with the thicknesses of each of the layers. To this end, the RESNRA program, which relies on the SIMNRA 5.0 simulation software to calculate a multilayer target, was designed at CERI. A definition of the depth resolution based on statistical considerations is proposed. In the second stage of the fitting process, a depth profile and corresponding error bars are extracted from the experimental spectrum by running a generalized reduced gradient (GRG2) algorithm using the previously calculated multilayer target. The one-to-one correspondence between the experimental spectrum and the depth profile demonstrates the objectivity of the method. The method is then applied to determining low concentration 3He depth profiles in implanted UO2 and SiC samples using the 3 He(2H, 4He)1H non-resonant nuclear reaction. The results clearly demonstrate the relevance and potential of the method. 2007 Elsevier B.V. All rights reserved. PACS: 82.20.Wt; 61.18.Bn; 61.72.Ss Keywords: Automation; NRA; Depth profiling; Depth resolution; Helium; SIMNRA
1. Introduction Nuclear reaction analysis (NRA) is a well known technique for depth profiling light elements in materials. As nuclear reactions are isotope specific, NRA is a powerful method in a wide range of fields requiring isotopic tracer measurements. In non-resonant NRA, the incident particles have a prescribed energy which they lose as they penetrate the material. The incoming particles are then liable to undergo a nuclear reaction with target nuclei. If reaction products have a high enough kinetic energy to recoil out of the target, they are detected at different energies depending on where they appeared in the sample. Non-resonant NRA
*
Corresponding author. Tel.: +33 238 255 419; fax: +33 238 630 271. E-mail address: [email protected] (T. Sauvage).
0168-583X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.01.288
involves analysing the energy distribution of those reaction products, from which a depth profile is deduced. Generally, the analysed target is modelled as a multilayer material where the layer thickness and the relative elemental concentrations are manually adjusted, so that the spectrum calculated using the SIMNRA 5.0 software [1] fits the experimental data available. Therefore, depth profiles extracted by different users may vary, despite all leading to acceptable simulations. This can cause some imprecision in the depth profile determination which in turn can induce a misinterpretation or over-interpretation of the experimental data. The aim of this work is to develop an automated method to determine low concentration depth profiles in a one-toone correspondence with experimental non-resonant NRA spectra. The analysis method relies on a two-stage process, the first of which involves determining the thickness of each
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478
of the material layers to be used in the final profile determination (second stage). To this end, the target is modelled as a multilayer material in which the analysed element concentration is constant and assumed to be low (as a consequence of this latter assumption, the stopping powers of incoming or outgoing particles are not altered). The thickness of each consecutive material layer is then chosen to correspond to the depth resolution at a given distance from the sample surface, which is derived from statistical criteria applied to RESNRA calculation results. This newly developed program relies on SIMNRA simulations to generate energy spectra corresponding to homogeneous distributions of 3He atoms in the material. In the second stage, the concentration of the probed isotope in each of the layers, the thickness of which is determined in the first stage of the analysis process, is evaluated using a generalized reduced gradient (GRG2) algorithm [2]. No constraints are assumed to fit the experimental energy spectrum. An automated method was previously proposed [3] to extract depth profiles from NRA spectra. Some aspects of that method differ from the present work: in this paper, the depth resolution is an essential part of the fitting process. Our calculation procedure was used to extract 3He depth profiles from a-spectra obtained from the 3He(2H, 4He)1H non-resonant nuclear reaction in UO2 and SiC implanted samples. This method provides a single solution to the problem of the determination of the concentration profile and in addition provides error bars associated with the elemental concentrations determined in each of the modelled layers. This is obviously essential when analysing the experimental data and comparing two depth profiles. 2. Depth resolution and layer thickness determination 2.1. Statistical definition of the depth resolution The depth resolution in backscattering analysis – e.g. NRA – is usually defined as the minimum detectable depth difference dx related to the minimum detectable scattered particle energy variation dE [4]. The energy difference between detected particles originating at different depths within the material is the result of the energy loss of incident and outgoing particles. Sources of the broadening of the measured energy distribution are detector resolution, initial energy spread of the incident beam, straggling of the outgoing and incoming particles, geometrical energy spread and energy spread coming from the multiple scattering [5]. Since energy straggling increases as the ion beam penetrates the sample, the depth resolution deteriorates at increasing distances from the sample surface. To calculate the depth resolution at any depth, the minimum detectable energy variation has to be defined. In non-resonant backscattering analysis, dE can be correlated with the help of a useful criterion based on statistical considerations. For a given set of experimental conditions, we consider the NRA analysis of a thick material containing a homoge-
xi Ln Counts
472
...
L i+1
Surface
Li
L i-1
...
L1
wi
Si (E) spectrum
Sj (E) spectra with j ≠ i Sum of all spectra
E i+1
Ei
Energy E
Fig. 1. Representation of spectra from the analysis using a non-resonant reaction of a multilayer material.
neous concentration of an element whose concentration one wants to determine. With the SIMNRA software, this material is modelled as a multilayer target and the spectral response of this analysis is the sum of n spectra generated from each of the modelled layers. A layer Li of thickness wi centred at depth of xi corresponds to a detected energy distribution Si(E). Fig. 1 shows the contribution of each of the layers to the total energy spectrum of particles emitted from nuclear reactions in the material. To be clear in the understanding of Fig. 1, the cross section of the nuclear reaction is assumed to be constant in the laboratory frame and the thicknesses of the n layers identical. The Si(E) spectrum is intersected by other spectra at energies Ei+1 and Ei. The probability pi that the particles detected in the [Ei+1, Ei] energy range are emitted from the layer Li is the counts in Si(E) divided by total number of counts in this energy range R Ei S i ðEÞ dE E i : pi ¼ R E hiþ1P ð1Þ i S ðEÞ dE j j Eiþ1 We now assume the only particles to contribute to the spectrum in the [Ei+1, Ei] energy range are those produced in layers adjacent to layer Li. Therefore, a first order estimate pi can be calculated from Eq. (1) by considering three successive material layers Li1, Li and Li+1 only, and their resulting consecutive spectra: R Ei S ðEÞ dE Eiþ1 i p i R Ei ; ð2Þ ½S ðEÞ þ S i ðEÞ þ S iþ1 ðEÞ dE Eiþ1 i1 pi takes into account the variations of the cross section in the vicinity of xi, along the primary ion trajectory. Probability pi is directly related to the [Ei+1, Ei] energy range and
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478
xi Li+1
Surface
Li
Counts
to thickness wi of layer Li. Conversely, in given experimental conditions and for a given probability pi, the minimum detectable energy variation dE is given by (Ei+1 Ei) difference and the associated depth resolution dx by the corresponding thickness wi of layer Li at depth xi.
473
Li-1
wi
2.2. Depth resolution calculation A system of equations can be set up to determine the depth resolution at a chosen probability Pc by applying expression (2). However, such a system is impractical to solve. Therefore, expression (2) has been simplified by making the three following assumptions:
These considerations allow the calculation of pi considering a system of two consecutive layers, analogous to the system of three layers used to calculate pi according to relation (2). Fig. 2 compares both analyses and their resulting spectral responses. Probability pi can be determined by calculating the hatched areas represented in Fig. 2. In the system of three layers (a) and according to the previous assumptions, relation (2) can be written as 1
pi 1 2
R E
R min½S i1 ðEÞ; S i ðEÞ dE þ 12 E min½S i ðEÞ; S iþ1 ðEÞ dE : R Ei ½S ðEÞ þ S i ðEÞ þ S iþ1 ðEÞ dE Eiþ1 i1
ð3Þ
It is worth noting that (1 pi) represents the uncertainty on the detected element position at depth xi. In the twolayer system (b) where both layers have the same thickness wi, the probability pi is calculated from the formula (4). According to the previous assumptions, the numerators as well as the denominators in relation (3) and in relation (4) can be equalled R min½S i ðEÞ; S iþ ðEÞ dE pi 1 1E R : ð4Þ ½S ðEÞ þ S iþ ðEÞ dE 2 E i
min[Si+1(E), Si(E)]
Si(E)
Ei+1
Si-1(E)
Ei
min[Si(E), Si-1(E)]
Energy E
xi
Counts
• The energy straggling of detected particles does not locally depend on their mean energy and therefore on the depth at which they are emitted. Indeed spectra resulting from three consecutive layers Li1, Li and Li+1 should have similar shapes. • Stopping powers and energies of reaction product particles are a locally linear function of depth. Therefore the mean energy of detected particles is locally proportional to the depth at which they are emitted. This implies that the energy differences between spectra peaks resulting from three consecutive layers Li1, Li and Li+1 are proportional to the material thicknesses that separate them. • The cross section of the reaction is a linear function of the depth at which incident particles react with target nuclei. The number of counts inside spectra resulting from three consecutive layers Li1, Li and Li+1 should be then a linear function of their respective depth location xi1, xi and xi+1.
Si+1(E)
Surface
Li+
Li-
wi
wi
Si+(E)
Si-(E)
min[Si+(E), Si-(E)]
Energy E
Fig. 2. Representation of the signals from the simulation of three consecutive material layers (a) and of two consecutive material layers of identical thickness wi (b). Both systems are equivalent.
Using this last equation, it is possible to converge pi to a given probability Pc by adjusting only the thickness wi of both considered layers. This enables the simple calculation using a simulation program of the depth resolution wi at a depth xi inside the material, defined for a given probability Pc. 2.3. Multilayer target determination The software RESNRA was developed at CERI to determine the thickness of the successive material layers to be used in the depth profile calculation. This software communicates with SIMNRA 5.0. which is used to calculate NRA energy spectra. It takes into account all the contributions to energy spread of NRA spectra, notably the straggling associated with incident and emitted particles. Both nuclear and electronic energy straggling are taken into account in the calculations through the application of Bohr’s theory, and use of the Chu correction [6] with respect to electronic energy straggling. The energy distribution is calculated from a multilayer material. This multilayer structure of the target enables the simulation of elemental gradients through variations in relative concentrations from one layer
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to another. Quantitative information about the target elemental composition is obtained by comparing the calculated energy distribution with that obtained experimentally. Our contribution lies in the calculation of the depth resolution R(x) as a function of depth x, the estimate of the thicknesses of the successive layers based on function R(x), and the automation of the depth profile determination. For a given probability Pc, the RESNRA program can calculate from relation (4) the depth resolution R(x) by considering a system of two layers of identical thickness (Fig. 2) at different depths. At each depth xi, the thickness wi is chosen so that Eq. (4) is verified. A simple root-finding algorithm (bisection method) is used so that the right hand side of Eq. (4) converges to a prescribed value of pi. When pi = Pc, wi is then the calculated depth resolution of the method at depth xi. The calculation stops when the reaction cross section becomes null in the depth range including the layers Li and Li+, which in practical terms corresponds to the depth range beyond which the material cannot be analysed under the applied experimental conditions. Finally, RESNRA generates the multilayer target for which layer thicknesses correspond to the depth resolution R(x) defined for a given probability Pc. Thicknesses wi and central positions xi of consecutive layers Li are successively calculated by RESNRA by solving the following system of Eq. (5): 8 i1 > < x ¼ wi þ P w ; i j 2 ð5Þ j¼1 > : wi ¼ Rðxi Þ: The second equation of system (5) is solved numerically by approximating R(x), evaluated at discreet positions from the sample surface, with an exponential function. Although this has always provided excellent results, it is indeed conceivable that other functions may provide better results for different systems. 3. Depth profile calculation One then chooses arbitrarily a charge Q0 to simulate the spectra Si(E) corresponding to each of the layers used for modelling the material response. The concentration C0, which is kept constant in the first stage, should ideally equal half of the maximum concentration of the probed element so as to minimise the variations of the stopping power of the particles used for the analysis. The position and thickness of the layers are determined from the procedure detailed in the preceding section. Let Ci be the unknown elemental concentration in material layer Li and Qexp the total charge of the analysis that results in the experimental spectrum Sexp(E). The simulated helium profile Ssim(E) can be expressed as a linear combination of the energy spectra Si(E) Qexp X C i S i ðEÞ: ð6Þ S sim ðEÞ ¼ C 0 Q0 i
The problem is then reduced to determining the set of values for Ci which minimises the following error function: R 2 ½S exp ðEÞ S sim ðEÞ dE : ð7Þ v2 ¼ E R ½ E S exp ðEÞ dE2 The minimisation problem is solved by using a GRG2 type algorithm [2]. An essential part of the depth profile determination is to evaluate the error bar associated with each of the concentrations resulting from the minimisation procedure. There are several terms that can contribute to the total uncertainty on measured concentrations in ion beam analysis (IBA) such as uncertainties on the detector energy calibration, on the charge measurement . . . [7]. Within the framework of this study, the counting uncertainty is assumed to be the main source of error. The error bar relative to the elemental concentration in the material layer Li is therefore determined from the total number of counts Ni associated with the corresponding spectrum Z Qexp C i S i ðEÞ dE: ð8Þ Ni ¼ E Q0 C 0 If the signal follows a Gaussian distribution, then according to the Laplace–Liapounoff theorem [8], the relative standard deviation ri for this distribution is given by 1 ri ¼ pffiffiffiffiffi : Ni
ð9Þ
By definition, there is a 68.3% chance that the true concentration C corresponding to layer Li is such that C i ð1 ri Þ < C < C i ð1 þ ri Þ:
ð10Þ
More generally, assuming a Gaussian distribution for the element concentration C, for any probability F between 0% and 100%, there exists a positive real k such that F is the probability that C is known to within ±k Æ ri. In a similar way, as seen in the previous sections, the probability that the position of a detected element belongs to the interval [xi wi/2, xi + wi/2] is Pc. In order that the analysis be consistent, the probabilities associated with the confidence intervals relative to the concentration and location of detected elements must be equal. In other words F must be equal to Pc. In Table 1, coverage factors k are given for corresponding Pc values according to this last condition. Therefore for a given probability Pc, the relative expanded uncertainty on concentration Ci is given by DC i ¼ k ri : Ci
ð11Þ
The resulting depth profile can then be represented as a set of boxes the widths of which are the expanded uncertainties relative to location and the heights of which are the expanded uncertainties on element concentrations. When the probed element was introduced by implantation, its total fluence / in at cm2 in the analysed sample
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478 Table 1 Coverage factors k, corresponding probability Pc and subsequent number of layers necessary to depth profile 3He in UO2 and SiC Coverage factor k
Probability Pc (%)
Number of layers used for modelling the spectrum in UO2 (625 keV NRA)
Number of layers used for modelling the spectrum in SiC (750 keV NRA)
1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5
86.6 83.8 80.6 77 72.9 68.3 63.2 57.6 51.6 45.1 38.3
8 10 12 14 17 20 23 27 32 39
10 13 16 19 22 26 30 35 42 51 63
is given from layer thicknesses wi in at cm2 and atomic concentrations Ci by X wi C i : ð12Þ U¼ i
As will be shown in Section 4 that v2 is negligible and the correlation between experimental and simulated data is excellent, the total number of counts Nsim. in Ssim(E) represents a physical measure of the implanted fluence. Therefore the relative standard uncertainty on the fluence is given by DU 1 ¼ pffiffiffiffiffiffiffiffiffi : U N sim
ð13Þ
4. Application of our calculation technique to 3He depth profiling in UO2 and SiC 4.1. Sample preparation A sintered uranium dioxide disk and a sintered silicon carbide sample were polished on one side. The UO2 disk thickness is approximately 300 lm and its diameter 8.2 mm. The SiC sample size is 7 · 7 · 0.3 mm3. The densities of sintered UO2 and SiC, determined by immersion in pure water and in pure ethanol, are respectively 10.7 g cm3 and 3.18 g cm3. The UO2 and SiC samples were implanted at room temperature with 0.5 MeV 3He+ ions to fluences of respectively 9 · 1015 at cm2 and 1015 at cm2 using the 3.5 MV Van de Graaff accelerator at CERI Orle´ans. Implanted fluences were chosen to obtain a maximum 3He concentration below the value of 1 at.% in both cases. 4.2. 3He analysis conditions The 3He(2H, 4He)1H NRA method was used to determine helium concentrations at depths up to a few microm-
475
eters. The measurements were performed at the 3.5 MV Van De Graaff accelerator at CERI Orle´ans with an experimental set-up based on the detection in coincidence of both reaction products, alpha particles and protons. a-Particles emitted from the sample at angles between 177 to 179 are detected by an annular 100 mm2 PIPS (passivated implanted planar silicon) detector. The protons transmitted through the sample are detected by a 1200 mm2 detector placed at 0 behind the sample. Its large solid angle allows the detection of all the protons emitted in coincidence with the detected alpha particle which ensures a coincidence yield of 100%. This experimental set-up provides alpha particle spectra free from parasite signals in the region of interest. Helium depth profiles, respectively in UO2 and in SiC, were measured with a 500 · 500 lm2 D+ beam at energies of 625 keV and 750 keV. The charge used for the analyses was 300 lC for UO2 and 2820 lC for SiC. Total numbers of counts of UO2 and SiC spectra are respectively 335 and 1370. Series of backscattered deuterons measurements with the annular detector show a relative uncertainty of charge measurements of ±3%. Further details pertaining to the experimental set-up or the coincidence technique are described in [9]. SIMNRA is configured to use Ziegler’s stopping power [10] in order to simulate spectra. Ziegler’s stopping power is roughly in agreement with the experimental stopping power of alpha particles in UO2 at energies up to 9 MeV [11]. Regarding SiC, a micrograph of a cross sectional specimen implanted with a five times greater fluence (5 · 1015 at cm2) and annealed at 1300 C/30 min reveals helium bubble nucleation in the grain at a depth of 1.2 lm. As is well known, this region corresponds to the nuclear cascades of primary ions. The SRIM simulation of the defect peak depth gives a value of 1.24 lm. Ziegler’s stopping power of helium in SiC is then in excellent agreement with experiments [12]. The angular spread of the incident beam is less than 0.1 and has negligible effect on SIMNRA simulations. The energy resolution of 12 keV of the PIPS detector was taken into account. The cross section used for the 3He(2H, 4 He)1H nuclear reaction was determined at CERI [13]. 4.3. Importance of the layer thickness determination procedure To show the importance of determining the thickness of each of the target layers, the a-particle energy spectrum relative to the UO2 experiment was fitted using 30 consecutive layers all 70 nm thick. The GRG2 algorithm that minimises the v2 indicator was then applied to solving the problem. No relevant depth profile was obtained from running the minimisation algorithm. In particular, He concentrations appeared to change from negative to positive values in consecutive layers. In other words, if the target layers are not carefully calculated, then the minimisation procedure may come up with physically irrelevant concentration values.
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4.4. Depth resolution and multilayer target determinations Several sets of multilayer targets were generated to extract depth profiles for the SiC and UO2 experiments based on different probabilities in the 38.3–86.4% range. Table 1 shows the number of layers contained in each multilayer target corresponding to a chosen probability Pc. As is expected, the number of layers which have to be modelled increases as the probability decreases (i.e. as the probability that two consecutive layers interact increases). Fig. 3 shows the depth resolutions R as a function of depth for different probabilities Pc in the case of the 3He implanted UO2 analysis. All calculated depth resolutions are found to be exponential functions of the depth in this study (correlation coefficients r2 > 0.99). Fig. 4 shows a set of 20 spectra deduced from RESNRA simulations of consecutive layers
in UO2 that contains 1 at.% of 3He with a probability Pc of 68.3%. For this value of Pc, the coverage factor k is equal to 1 and the maxima associated with two consecutive spectra are separated by twice their standard deviation (strictly true if spectra were Gaussians). All spectra appear to intersect at the same relative height, which was to be expected since they all have a similar shape. It was also checked that the sum of all elementary spectra is consistent with a spectrum generated from the simulation of one large UO2 layer containing 1 at.% of 3He. This demonstrates that a linear combination of these elementary spectra can fit our experimental data thus providing an estimate of a 3He depth profile in UO2. At low helium concentrations below 1 at.%, small variations in material composition have no influence on the stopping powers of incident and outgoing particles. 4.5. Helium depth profiles
1200 - 86.6 - 83.3 800 - 77
600
- 68.3
400
- 57.6 - 45.1
200 0 0.0
Probability Pc (%)
Depth resolution (nm)
1000
0.5
1.0 1.5 Depth in UO2 (µm)
2.0
2.5
Fig. 3. Depth resolutions for several probabilities in the case of an analysis involving 3He implanted UO2 with a 625 keV deuteron beam.
20 elementary spectra Sum of the 20 spectra
Counts
Simulation of a large UO2 layer that contains 3 He
1600
1700
1800
1900
2000
2100
2200
2300
Energy (keV)
Fig. 4. Sets of elementary spectra from RESNRA simulations of successive UO2 layers containing 1 at.% 3He, calculated with a probability of 68.3%. Their sum is overlapped to the simulation of a large layer of the same chemical composition.
Both helium depth profiles in UO2 and SiC defined for a probability of 68.3% are presented in Fig. 5, with their corresponding experimental spectra. The numerical algorithm converges to provide a relevant depth profile and the correspondence between experiment and simulation is excellent at this value of Pc, with low values of v2 inferior to 0.01. Each profile calculation lasted approximately 10 min using a computer equipped with a 2.4 GHz processor and with 512 Mo of random access memory (RAM). Helium fluences in UO2 and SiC calculated from the relation (12) are 9.2 ± 0.5 · 1015 at cm2 and 0.97 ± 0.03 · 1015 at cm2, respectively. The estimated values of fluences are in excellent agreement with the implantation ones. In Fig. 6, depth profiles resulting from automated simulations of experimental spectra for different values of Pc are presented. When Pc increases, the helium depth profile in UO2 or SiC broadens and the indicator of the quality of the fit v2 increases to an unacceptable value. For low Pc values, helium concentrations derived from the fitting procedure of the UO2 experiment (a) appear to oscillate. The error bars that represent counting uncertainties on concentration are smaller than the oscillations of the concentration for Pc = 51.6%. This indicates that the fitting process is likely to introduce uncertainties larger than counting uncertainties. As several depth profiles are expected to minimise v2 when Pc is too low, the fitting algorithm can converge towards a profile that is not physically relevant. In SiC, the fact that helium is concentrated in a very narrow layer prevents oscillations in the concentration profile even for the lowest values of Pc. Nevertheless, for both SiC and UO2, a probability Pc of 68.3% – i.e. k = 1 – appears to constitute the best compromise. Error bars on the SiC profile are small because of the statistics (number of counts) of the corresponding a-spectrum.
0.30
12
0.25
10
Counts / keV
Helium concentration (at.%)
G. Martin et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 471–478
0.20 0.15 0.10 0.05
UO2 exp. UO2 sim.
8 6 4 2
0.00 0.0
0.5
1.0
1.5
2.0
0 2300
2.5
2200
2100
Depth in UO 2 (µm)
c
2000
1900
1800
1700
Energy (keV)
d
0.06 0.05
50
SiC exp. SiC sim.
40
Counts / keV
Helium concentration (at.%)
477
0.04 0.03 0.02
30 20 10
0.01 0.00 0.0
0.5
1.0
1.5
2.0
0 2000
2.5
1900
1800
1700
1600
Energy (keV)
Depth in SiC (µm)
Fig. 5. 3He depth profiles (a and c), experimental spectra and simulated spectra (b and d) of as-implanted UO2 and SiC. Depth profiles are presented with error bars at a probability of 68.3% (standard uncertainties).
0.06
Pc = 51.6 % Pc = 68.3 % Pc = 83.8 %
0.3
Helium concentration (at.%)
Helium concentration (at.%)
0.4
0.2
0.1
0.0
Pc = 38.3 % Pc = 68.3 % Pc = 83.8 %
0.05 0.04 0.03 0.02 0.01 0.00
0.0
0.5
1.0
1.5
2.0
2.5
Depth in UO 2 (µm)
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Depth in SiC (µm)
Fig. 6. Helium depth profiles in UO2 (a) and SiC (b) at different probabilities Pc.
5. Conclusion A definition of the depth resolution based on statistical considerations has been proposed as a first step towards extracting elemental depth profiles from non-resonant NRA spectra. The advantages of this useful definition have been demonstrated through the treatment of NRA data obtained from implanted UO2 and SiC samples containing low 3He concentrations. A program (RESNRA) that relies on SIMNRA 5.0 has been developed at CERI to calculate the depth resolution as a function of the distance from the sample surface, for a given expanded uncertainty. By generating a multilayer target from the depth resolution calcu-
lation, a RESNRA simulated dataset is introduced in a GRG2 algorithm that allows the depth profile extraction from experimental spectra. Counting uncertainties for each of the inferred 3He concentrations are also determined. It is shown that if one takes into account the variations of the depth resolution with depth, then there is a unique set of helium concentrations that fits the experimental data. When the probability on the helium depth location is not adapted, the depth profile simulation provides unsatisfactory results. However, the algorithm converges without calculation constraints towards a single and relevant depth profile when layer thicknesses are calculated according to a depth resolution calculated for a probability of 68.3%.
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The correspondence between experimental and simulated spectra is then excellent. Furthermore, error bars on obtained depth profiles have also been calculated. The accuracy of this method for profile extraction has encouraged us to apply it to data recently generated and will be used in the future. Furthermore, it should be applicable to other areas of IBA, such as RBS (Rutherford backscattering spectroscopy) depth profiling. Acknowledgements Jean-Charles DUMONT is thanked for his help in developing the RESNRA software. The authors are grateful to their collaborators of the ACTINET network and to the PRECCI programme, funded by CEA and EDF. References [1] M. Mayer, SIMNRA User’s Guide, Report IPP 9/113, Max-PlanckInstitut fu¨r Plasmaphysik, Garching, Germany, 1997.
[2] L.S. Lasdon, A.D. Waren, Comput. Chem. Eng. 7 (1983) 159. [3] N.P. Barradas, R. Smith, J. Phys. D: Appl. Phys. 32 (1999) 2964. [4] J.R. Tesmer, N. Nastasi, Handbook of Modern Ion Beam Materials Analysis, MRS, Pittsburgh, 1995, p. 46. [5] J.R. Tesmer, N. Nastasi, Handbook of Modern Ion Beam Materials Analysis, MRS, Pittsburgh, 1995, p. 144. [6] W.K. Chu, Phys. Rev. 13 (1976) 2057. [7] K.A. Sjo¨land, F. Munnik, U. Wa¨tjen, Nucl. Instr. and Meth. B 161– 163 (2000) 275. [8] G. Charlot, Statistiques applique´es a` l’exploitation des mesures, Masson, Paris, 1978. [9] T. Sauvage, et al., in preparation. [10] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [11] Hj. Matzke, J. Nucl. Mater. 270 (1999) 49. [12] T. Sauvage, G. Carlot, G. Martin, L. Vincent, P. Garcia, M.F. Barthe, E. Gentils, P. Desgardin, Nucl. Instr. and Meth. B, in press. [13] T. Sauvage, H. Erramli, S. Guilbert, L. Vincent, M.F. Barthe, P. Desgardin, G. Blondiaux, C. Corbel, J.P. Piron, F. Labohm, J. Van Veen, J. Nucl. Mater. 327 (2004) 159.