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Spatial decomposition of molecular ions within 3D atom probe reconstructions
Ultramicroscopy 132 (2013) 92–99
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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Spatial decomposition of molecular ions within 3D atom probe reconstructions Andrew Breen a,b, Michael P. Moody c, Baptiste Gault d, Anna V. Ceguerra a,b, Kelvin Y. Xie e, Sichao Du b,f, Simon P. Ringer a,b,n a
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia Australian Centre for Microscopy and Microanalysis, Madsen Building F09, The University of Sydney, NSW 2006, Australia c Department of Materials, University of Oxford, Parks Road, OX13PH, Oxford, UK d Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ont. L8S4L8, Canada e Johns Hopkins University, Department of Mechanical Engineering, Baltimore, MD 21218, USA f School of Physics, The University of Sydney, NSW 2006, Australia b
a r t i c l e i n f o
a b s t r a c t
Available online 7 March 2013
Two methods for separating the constituent atoms of molecular ions within atom probe tomography reconstructions are presented. The Gaussian Separation Method efficiently deconvolutes molecular ions containing two constituent atoms and is tested on simulated data before being applied to an experimental HSLA steel dataset containing NbN. The Delaunay Separation Method extends separation to larger complex ions and is also tested on simulated data before being applied to an experimental GaAs dataset containing many large (4 3 atoms) complex ions. First nearest neighbour (1NN) distributions and images of the reconstruction before and after the separations are used to show the effect of the algorithms and their validity and practicality are also discussed. & 2013 Elsevier B.V. All rights reserved.
Keywords: Atom probe tomography Complex molecular ions Delaunay tessellation Nearest neighbour analysis
1. Introduction In the course of an atom probe tomography experiment, atoms are ideally individually ionised and field evaporated from the specimen. However, very often this is not the case. Molecular ions (also known as complex ions or cluster ions) are routinely observed in atom probe mass spectra [1] and are usually represented as a single point in the resulting reconstruction rather than each constituent atom being individually located — a clearly unphysical model which has so far largely been conveniently ignored. This poses a problem, not only for the sake of visual clarity in the resulting reconstruction, an obvious yet important issue, but also for many statistical analysis tools at the heart of atom probe analysis. For some of the current statistical analysis tools available, atoms contained within complex ions are essentially invisible to the same species of atom that were evaporated as singular ions and this will skew the results of the analysis — the severity of which is dependent on the size and number of complex ions detected. A partial solution to this problem is decomposing the molecular ions into their constituent atoms at the same point and indeed this is already being done. The commercially available
n Corresponding author at: School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia. Tel.: þ61 2 9351 2351; fax: þ 61 2 9351 7682. E-mail address: [email protected] (S.P. Ringer).
0304-3991/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2013.02.014
IVASs software allows ‘atom-based’ rather than ‘ion-based’ analysis to be chosen for many of its analysis tools. However, positioning multiple atoms at a single point can lead to a significant bias in statistical analysis of the results. Of particular concern is the nearest neighbour (NN) analysis which histograms the distance between neighbouring atoms of chosen atomic species [2–4]. Even if an atom-based approach is used, some atoms will have a 1NN separation of 0 nm. While an NN distribution may not be particularly useful in itself, other than to indicate to segregation or anti-segregation of a particular atomic species, it is a fundamental component of many other higher order analysis techniques. Clustering algorithms [5,6], for instance, commonly use NN distributions for parameter selection, when applied to isolate and characterise nanoscale segregation phenomena [3,7,8]. The results of such an analysis are therefore inextricably linked to the quality of the NN analysis. The visualisation, cluster morphology and cluster size distribution are some aspects of this analysis that could be misleading if current molecular ion representations are used. NN distributions have also been used for phase composition measurements [4,9]. One such method derives a protocol based on the first nearest neighbour (1NN) distribution of a particular atomic species, to deconvolute the contributions from the different phases and thus determine the phase composition, but these measurements will be inaccurate if the solute species is contained within molecular ions.
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Radial distribution functions (RDFs) also suffer similar drawbacks to an NN analysis when the reconstruction being analysed contains complex ions. RDFs histogram the radial distance between chosen atomic species and can be used for observing crystallographic information as well as segregation phenomena [10,11]. However, it follows that if some atomic species are occupied within molecular ions, any measured RDF distribution concerning these atomic species will be skewed. Grid based analysis techniques that divide the reconstruction into voxels represent yet another group of data mining tools particularly susceptible to inaccuracies due to conventional molecular ion representation. One important example is the frequency distribution analysis algorithms [12–14] that divide the reconstruction up into a grid of blocks that all contain the same number of ions. Histograms of the number of ions of a particular species contained within each block are then created. Current available data analysis software generally do not offer an atom-based frequency distribution analysis and consequently molecular ions are counted as a single contribution to the size of the block. This not only causes the number of actual atoms in each block to vary but also renders atomic species tied up in molecular ions invisible to those that are not. Previously, Gault et al. used their own atom-based technique whereby the molecules within a reconstruction of a thermoelectric material were simply split into single atoms occupying the same spatial coordinates, facilitating species-specific frequency distribution analyses of Zn and Sb atoms [15]. This is acceptable except in cases where there are large molecular ions and fine voxel sizes such that there is increased chance of multiple atoms contributing to one voxel when they should be spread over neighbouring voxels. Other grid based analysis tools such as proxigrams [16], contingency table analysis [17] as well as density and concentration profiles will have similar issues. There are still many other analytical tools that stand to benefit from representing constituent atoms within molecular ions individually — lattice rectification techniques [10,18], that use the crystallographic information contained within an atom probe reconstruction to restore the original crystal structure of the sample is a notable example. Molecular ions inevitably complicate this process, even if they are decomposed into their constituent atoms at the same point, but by separating the constituent atoms into their likely locations, the rectification procedure is facilitated particularly in cases where datasets contain large molecular ions. As atom probe tomography (APT) is applied to an increasing array of sophisticated multicomponent materials, complex ion detection is becoming more common and hence, potentially more of an issue. A prominent example is the analysis of steels, where the detection of complex ions such as C2, NbN and FeN in various charge states are commonly observed [19]. Semiconductors, in particular III–V such as GaSb, are another material group that face similar challenges where Sb2, Sb3 and even larger molecular ions in various charge states are routinely detected [20]. It is therefore apparent that the representation of complex molecular ions within atom probe reconstructions should be avoided. Minimising the detection of complex ions in the experiment in the first place may seem like a reasonable approach, however, it is often inconvenient to change the experimental parameters to minimise complex ion detection, as other aspects related to the quality of data will be adversely affected [21]. Still, research has been undertaken studying the physics governing the field evaporation of complex molecular ions [1,20,22,23], information that could be used to control the severity of their detection. The experimental conditions have been found to play a critical role in molecular ion evaporation — higher laser pulse energies for instance have often been attributed to higher
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proportions of molecular ion detection, as well as more massive molecular ions themselves. It should also be noted that the field evaporation behaviour of metals is significantly different to that of semiconductors and other material groups. Much remains unknown, however, about the formation of these molecular ions on an atom probe tip. A more practical approach may be to spatially separate complex ions within the reconstruction so that each atom has its own unique position within the reconstruction. Fig. 1 illustrates this concept. By considering the crystallography and evaporation behaviour of the specimen as well as observed atomic distributions within the reconstruction itself, the spatial separation of complex ions within the reconstruction can be simulated accurately. At present it is impossible to precisely locate each individual atom from the molecular ions but the approach suggested could restore expected distributions on average and hence help to get better results from the analysis techniques aforementioned. For small complex molecular ions of 2 atoms, a method to separate the constituent atoms based on the measured 1NN distribution of singular ions occupying the same phase is proposed. The separation distance and orientation of the molecular pair are selected so that they follow a Gaussian curve fitting the 1NN distribution. For larger complex ions, this approach becomes analytically cumbersome and a technique that considers the available local volume surrounding the complex ion is more appropriate. In this case the use of Delaunay tessellations is proposed. Delaunay tessellations have many practical applications including pattern recognition techniques and finite element methods, and their implementation to a range of analytical studies related to APT is starting to be realised [24–26]. A Delaunay tessellation is formed by joining all pairs of points belonging to the set P¼{p1, p2, y, pn} where n Z4 such that no point in P is inside the circumsphere of any tetrahedra of the tessellation [27]. By measuring the volume of the tetrahedra surrounding the detected co-ordinates of a molecular ion, positions for the spatial redistribution of the constituent atoms can be determined that minimise any localised density fluctuations. Here, these algorithms are first applied to simulated reconstructions for proof of concept before being demonstrated on experimental data — first to a low-alloyed steel containing NbN complex ions and then to a GaAs nanowire containing very large molecular ions. Focus is placed on the effect of this reconstruction artefact to the 1NN distribution because these analyses represent a simple and directly comparable way of testing the performance of the algorithms. Further, 1NN distributions are an important component of other analysis techniques and improvement in the accuracy of these distributions will have flow-on benefits to the clustering algorithms, phase compositional measurements and
Fig. 1. Schematic demonstrating the splitting of complex molecular ions within an atom probe reconstruction: (1) before separation when molecular ions are represented at a single point and (2) after constituent atoms have been separated.
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radial distribution functions. Care should be taken in the interpretation of the results and spatial separation only performed when molecular ions are likely to bias the results from the analysis tools aforementioned.
2. Theory 2.1. The Gaussian separation method (GSM) The Gaussian separation method has been developed for the separation of molecular ions that consist of two constituent atoms (dimers). This approach is based on the assumption that the constituent atoms in the molecular ions were originally neighbouring atoms on the surface of the atom probe tip at the time of evaporation. It follows that the Dz component of the separations should be small while the overall separation distances should be similar to that of the atoms within the same phase on the surface of the atom probe tip. The separation distance distribution should match the 1NN distribution of the surface plane of atoms. For this to be accurate, the detector efficiency and a Dz limit should be taken into consideration. One approach for crystalline materials (and the one that has been used in this study) is to make a simulated reconstruction of the lattice in the same crystallographic orientation of the experimental reconstruction with 100% detector efficiency and apply atom probe like noise — then create a 1NN plot where Dz o0.05 nm of all the atoms. Information on determining the crystallographic orientation of APT reconstructions is described elsewhere [10,28]. The Dz cut-off value was selected because it is usually below the plane spacing commonly observed in atom probe data [29,30] and therefore creates a distribution more closely resembling that for the surface plane of atoms. The resulting curve can be closely fitted to a Gaussian distribution: ! ðxbÞ2 f ðxÞ ¼ a exp ð1Þ 2c2 where a is the amplitude, b is the mean and c is the standard deviation. The atoms composing the complex ions are then separated — a schematic of the process is presented in Fig. 2. For each dimer
Fig. 2. The separation of a dimer complex ion using the GSM: the dark blue atoms represent the constituent atoms of the complex ion where as the light blue atoms are surrounding atoms in the reconstruction. The dsep is the separation distance between the constituent atoms; the midpoint is kept as the original detected coordinates of the complex ion. d1NN_1 and d1NN_2 are the distances to the nearest neighbouring atoms after the separation and are used to determine whether the orientation of the separation will be accepted. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
complex ion in the reconstruction, the separation distance (dsep) is chosen at random from a set of distances matching the fitted Gaussian probability distribution. The original co-ordinates of the complex ion are kept as the midpoint/centre of mass between the two constituent p atoms andffi the orientation is also randomised ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such that dsep ¼ Dx2 þ Dy2 . The move is accepted only if the distances from the separated atoms to the nearest neighbouring atoms within the reconstruction are within 20% from a value selected at random from the Gaussian distribution — otherwise the orientation selection is repeated until this criteria is met. This results in the constituent atoms having a separation distribution similar to the theoretical 1NN distribution of atoms on the surface plane within the phase of interest. 2.2. The Delaunay separation method (DSM) The Delaunay separation method separates complex ions with two or more constituent atoms and uses the Delaunay tessellation of detected atom co-ordinates around a complex ion to determine the local volume available to distribute the atoms contained within the complex ion. It is based on the assumption that the constituent atoms were in close proximity to each other within the sample and that regions of larger empty volume surrounding the molecular ion are more likely to be where the constituent atoms were originally located. The algorithm identifies the simplexes (tetrahedra formed by Delaunay tessellation) that share a vertex with the detected coordinates of a complex ion, forming a convex hull within which the constituent atoms are distributed into the surrounding volume. The first constituent atom is placed at the original coordinates of the complex ion. The volume of each simplex is then calculated and the remaining constituent atoms are placed at the centroids of the largest tetrahedra. Fig. 3 is a schematic showing this process and clearly demonstrates how the constituent atoms are distributed into the available volume.
3. Simulated datasets 3.1. Method Simulated datasets of a face-centred cubic (FCC) binary alloy were used to test the performance of the GSM and DSM algorithms. The original simulated dataset contained 3 million atoms with composition Solvent (Sv) 1.0 at% Solute (Su), lattice parameter a0 ¼0.404 nm and Warren-Crowley parameter (which
Fig. 3. The separation of a large molecular complex ion using the DSM. Each vertex of the tetraheda represents the co-ordinates of an ion within the reconstruction — a molecular ion has been detected at the centre of the subvolume. One constituent atom is placed at the original reconstructed co-ordinates of the molecular ion and the rest are positioned at the centroids of the largest tetrahedra (highlighted in red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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controls the clustering/segregation of Su) a ¼0.01 [31]. 1% of random neighbouring Sv atoms on the (001) planes (i.e. perpendicular to the z-axis within the simulation) were converted to Su to simulate the true position of Su atoms contained within Su2 dimer complex ions. The resulting Su distribution is somewhat similar to that in the early stages of precipitate formation. 43% of all ions were then removed to simulate detector efficiency — if one of the Su ions within the Su–Su pairs was selected then the other Su ion was also removed. A complementary dataset was then created where approximately half of the converted Su–Su pairs remaining were replaced by Su2 complex ions at the centre of mass. A dithering function was then applied to the atom coordinates, with standard deviation sx ¼ sy ¼ sz ¼0.05 nm, to resemble APT like spatial noise. The GSM was then used to separate the simulated Su2 complex ions before comparing the Su distribution between the datasets. To simulate the true atomic distribution of larger complex ions containing 4 atoms, 0.25% of Sv atoms were selected at random from the original simulation along with 3 of their first nearest neighbours and were converted to Su. 43% of all ions were again removed to simulate detector efficiency — if one of the Su ions within a cluster of 4 Su atoms was selected then the other Su ions within the cluster were also removed. Another dataset was also created where the clusters of 4 Su atoms remaining were removed and replaced with Su4 complex ions. As before, a dithering function was then applied to simulate atom probe noise. The DSM was then used to separate Su4 and the Su distribution was compared between the datasets.
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Fig. 5. Showing the effect of the GSM on the measured Su–Su 1NN distribution: (black) correct distribution — no complex ions present, (red) when Su2 ions are simulated but only Su ions are used in the calculated distribution, (light blue) When Su and Su2 ions are used in the calculated distribution and (dark blue) after Su2 ions have been separated using the GSM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.2. Results The GSM was shown to separate dimer complex ions into their constituent atoms very close to the expected spatial distribution. Fig. 4 shows fitting a Gaussian curve to the 1NN of all atoms (Dzo0.05 nm) within the original simulation after APT like noise has been added. The resulting probability distribution was then used to model the separation distances and orientation of the constituent atoms using the GSM. Although there is some difference between the curves (R-square¼ 0.9742), the approximate fit is deemed appropriate for the algorithm to achieve sufficiently accurate results. The 1NN distribution is markedly more accurate after separating the simulated Su2 molecular ions with the GSM algorithm. Fig. 5 shows a range of 1NN distributions involving Su atoms
Fig. 6. Showing the effect of the DSM on the measured Su–Su 1NN distribution: (black) correct distribution — no complex ions present, (red) when Su4 ions are simulated but only Su ions are used in the calculated distribution, (light blue) Su4 and Su ions are used in the calculated distribution and (dark blue) after Su4 ions have been separated using the DSM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
before and after the GSM is used. The Su–Su 1NN before Su2 is separated (red) is significantly different to the correct distribution (black). Even if all Su containing ions are considered (light blue), essentially the same as an ‘‘atom-based’’ approach, the correct 1NN distribution is still not restored. However, after using the GSM, the resulting 1NN Su–Su distribution (dark blue) is much closer to the correct distribution. The DSM is also shown to improve the measured 1NN Su–Su distribution in the presence of larger molecular ions. Fig. 6 provides a similar range of 1NN distributions to the previous figure to show the effect of the DSM algorithm and the results are essentially the same — after using the DSM the resulting 1NN Su– Su distribution (dark blue) is much closer to the correct distribution (black).
4. Experimental datasets 4.1. Method Fig. 4. A Gaussian curve (red) is fitted to the 1NN distribution of all atoms using a depth threshold of Dz o 0.05 nm (blue) within a simulated dataset. The mean (b¼ 0.26 nm) and standard deviation (c ¼ 0.072 nm) of the Gaussian curve are used as inputs to the separation algorithm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The GSM was then applied to experimental data to separate NbN complex ions in a high strength low alloy (HSLA) steel produced using the CASTRIPs process [32]. The dataset was
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collected on a Cameca LEAP 3000 under voltage pulsing at 25% pulse fraction and 90 mm flight path at 25 K. A simulated BCC ferrite reconstruction with atom probe like noise orientated in a similar crystallographic direction to that observed in the experimental reconstruction was then used to produce the expected distribution of neighbouring atoms on the surface of the tip and was fitted with a Gaussian curve that was used when separating the Nb and N atoms. The mean and standard deviation of this distribution were found to be 0.204 nm and 0.127 nm respectively. Similarly, the DSM was applied to an experimental dataset collected from a GaAs nanowire containing many molecular ions such as As2, As3, As4, As5 and GaAs4. This dataset was collected on a Cameca LEAP 3000 Si under 0.05 nJ laser pulsing, 90 mm flight path at 25 K. Each dataset was reconstructed using the commercially available IVASs software. Significant trajectory aberrations were observed on the edges of the nanowire and were therefore cropped from the analysis region.
4.2. Results A significant difference in the atomic distributions of the Nb and N atoms contained within the HSLA steel dataset was observed after applying the GSM. The visual aspects of this separation are shown in Fig. 7. Larger-scale spatial features of the reconstruction have been conserved such as the noticeable segregation of the Nb and N atoms before and after using the GSM algorithm (Fig. 7 a, b). A close up of individual NbN positions (Fig. 7 c, d), clearly reveals how the individual atoms have been separated with varying dsep and orientation along the x-y plane relative to the detector, facilitating statistical analysis of Nb and N atomic distributions. Fig. 8 reveals how separating the complex ions affect the 1NN distribution of N and Nb atoms. The Nb–Nb 1NN distribution before the complex ions are split is significantly affected by a large fraction of Nb atoms within NbN complex ions — this was measured to be 23%. After separating the complex ions, the resulting distribution is a much more realistic representation of the true Nb–Nb 1NN distribution. No N was ranged in the original data since the N þ peaks overlapped with Si2 þ peaks in the mass spectrum but, after separating the complex ions, individual N atoms can be represented in the reconstruction. Their 1NN distribution after the split is almost identical to the original 1NN distribution of the NbN complex ions.
After applying the DSM to the GaAs nanowire, As representation within the reconstruction became much clearer. Fig. 9 shows the reconstruction before (a) and after (b) the DSM algorithm was applied. In particular, an overwhelming majority of the As atoms (99%) are contained within complex molecular ions. After separation, the complex ion species are replaced by a noticeably denser distribution of As atoms. This facilitates further statistical analysis of atomic distributions and architecture — particularly anything concerning As. Fig. 10 shows the 1NN distribution for any ions containing As before and after applying the DSM model. The frequency has been normalised so that the volume under each curve is equal to 1, this enables a better comparison between the distributions since such a large fraction of the As atoms were contained within molecular ions. This essentially transforms the curves into probability density functions. It is clear the true As–As 1NN distribution is severely affected by the large complex ions — before the separation, the mean 1NN As distance between As containing ions was 0.398 nm with a standard deviation of 0.158 nm and after the separation the mean was 0.136 nm with a standard deviation of 0.048 nm. The theoretical As–As 1NN distribution for the reconstruction is also given. This distribution was based on equations given by Philippe et al. [4]. The probability density function of finding a 1NN distance of r for any pair of atoms in a random solid solution
Fig. 8. The effect of the GSM on the measured Nb–Nb and N–N 1NN distributions in the HSLA steel sample: Nb–Nb 1NN distributions before and after using the GSM (purple), NbN–NbN 1NN before using the GSM (blue) and N–N 1NN after using the GSM (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Separating NbN complex ions within an experimental HSLA steel reconstruction: (a) original reconstruction showing the Nb and NbN complex ions and (b) after Nb and N atoms have been separated using the GSM — segregation and larger-scale features are still conserved after the separation. (c) Small sub-volume of reconstruction before separation and (d) after separation shows how the GSM separates individual NbN complex ions.
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Fig. 9. Separating complex ions from a GaAs nanowire reconstruction: (a) sub-volume of reconstruction before complex ions were split showing Ga and As containing ions. Approximately 292,000 As atoms are contained within only 80,000 reconstructed ions and (b) reconstruction after As containing complex ions were split.
Fig. 10. The As–As 1NN distribution before (blue) and after (red) the complex ions in the GaAs reconstruction were separated. The theoretically predicted distribution is also shown (dotted black). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is given by 4 P ðr Þ ¼ 4pr 2 Q C 0 exp pQ C 0 r 3 3
ð2Þ
where Q is the detector efficiency and C0 is the real concentration of the atoms. The maximum of P(r) can be found by making dP=dr ¼ 0: 1=3 1 r0 ¼ ð3Þ 2pQ C 0 Therefore the measured concentration of As within the reconstruction was substituted for Q C 0 in Eq. (2). The dotted line in Fig. 10 shows Pðr As Þ vs. r As and is very similar to the resulting As–As 1NN distribution suggesting that the DSM provides a valid representation of As within the reconstruction.
5. Discussion 5.1. The validity of the seperation algorithms It is important to note that the separation algorithms developed simulate the likely distribution of constituent atoms within the complex molecular atoms routinely seen in many materials but they do not restore the actual relative coordinates of the constituent atoms. Such a representation however, improves visibility as well as facilitates more accurate statistical analysis
concerning the spatial distribution of atoms contained within molecular ions including atom-based frequency distribution, nearest neighbour analysis and more advanced analysis tools such as clustering algorithms. However, it is up to the researcher to gauge whether or not the separation is necessary for the analysis they are performing. The GSM makes the assumption that the species contained within a dimer complex ion are neighbouring atoms on the surface of the atom probe tip at evaporation so that Dzij E0, where i and j are the pair of atoms contained within the complex ion, and that the separation distance follows the 1NN distribution of atoms on the surface of the tip. While much remains unknown about the formation of complex molecular ions before field evaporation, one would assume that evaporation would usually occur from the terrace edges, similar to that of singular ions and that it would be more energetically favourable for molecular ions to form with neighbouring atoms on the same terrace rather than between terraces. The DSM similarly assumes that constituent atoms were originally in close proximity to each other but relaxes the restriction that they were all neighbouring atoms on the same lattice terrace at evaporation. This was primarily used because it becomes difficult to distribute a large number of constituent atoms within a small area of the same x-y plane without causing localised density fluctuations. It seems more reasonable to distribute the constituent atoms within the available volume of a convex hull around the detected co–ordinates of the complex ion. It seems logical to assume that a large molecular ion would have a larger empty volume surrounding it than singular ions and would also make the approach of the DSM more valid. This was investigated for the GaAs reconstruction by comparing the average volume contained within the convex hull surrounding a singular ion with no molecular ion neighbours to the convex hull surrounding a molecular ion. Interestingly, it was found that the average volume surrounding the molecular ions was 0.0425 nm3 where as the average volume surrounding singular ions was 0.0398 nm3, which confirms the initial prediction. This finding also suggests that the molecular ions detected within the region of interest did not succumb to significantly worse trajectory aberrations than singular ions. It should be noted, however, that there are many other factors that influence the relative spatial distributions of atomic species within atom probe data and that these affects should be taken into consideration with any subsequent statistical analysis. The local magnification affect [33] and surface migration [34] are just two examples. Local magnification of the NbN molecular ions is
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known to have occurred in the HSLA steel sample used in this study. Some surface migration of N may also have occurred but this is considered negligible due to the N being strongly held within Nb, C and N rich clusters [35]. The GaAs data was also affected by these evaporation phenomena. The original reconstruction showed significant segregation of As containing ions to the side walls of the hexagonal GaAs nanowire and is a strong indication that surface migration occurred. The shape of the tip was also skewed around the edges suggesting either trajectory aberrations or the limitations of the reconstruction algorithm that assumes a tip in the shape of a hemispherical cap on a truncated cone — not a hexagonal prism. It was for these reasons that only the central region of the reconstruction was cropped for analysis — the observation of lattice planes within this region suggests that at least a moderate level of spatial integrity was maintained here. The algorithms developed are only appropriate for complex ion detection and if the accuracy of various statistical analyses was of concern due to other effects, further data treatment may be necessary. Still, the algorithms developed have been shown to improve the spatial distribution of atomic species contained within complex ions. This was clearly shown in Figs. 5 and 6 when the algorithms separated complex ions within simulated data and improved the resulting 1NN distribution of solute species. It should be noted however that both algorithms seem to slightly under-estimate the true separation distances i.e. the measured Su–Su 1NN distribution after separation is slightly shifted to the left for low 1NN values. This suggests that there is slightly more randomness or noise added to the positioning of constituent atoms relative to their actual positions after separation — the mean 1NN distribution tends to decrease as more randomness is added to the system which is suggested from Eq. 3 where r0 is always less than the theoretical 1NN distribution for a perfect crystal. However, some extra randomness is expected since the exact positioning of the constituent atoms cannot be known. The 1NN distribution does show a marked improvement however as compared to that when the molecular ions are not spatially separated which suggests a valid distribution of the atoms contained within the molecular ions has been achieved. This has significant implications — results from statistical tools employing the NN analyses and similar type algorithms can provide valuable information on the structure–property relationships of materials and can potentially be significantly improved for any material where complex ion detection poses a limitation such as the HSLA steel or GaAs nanowire used to demonstrate the algorithms. 5.2. Computational performance The DSM and GSM can be performed within a practical amount of time for millions of atoms on most personal computers. Both algorithms voxelise the dataset into 1 nm3 cubes so that any measurements only occur between neighbouring voxels, not over the entire dataset, significantly reducing computation time. The GSM was coded in C and was able to separate 1426 Su2 complex ions in the simulated dataset containing 2, 872,034 detected ions in 14 s. The DSM was coded in Matlabs and took 23 mins 60 s to separate 1426 Su4 atoms from 2867, 756 total detected ions. The processing time increases almost linearly with number of complex ions detected. The increased computer time for the DSM is largely due to the Delaunay tessellation of each voxel of atoms. All processing was done using a single 2.3 GHz Intel Core i7 processor with 8 GB memory. Further improvements to performance could be achieved through parallel computing, GPU processing and compiling the DSM in C/Cþþ code. While the DSM is necessary for complex ions containing more than 2 atoms, the GSM is much faster. A programme that uses
both algorithms, depending on the size of the complex ion could be implemented in the future.
6. Conclusion In summary,
Molecular ions are typically represented as single points
within atom probe reconstructions rather than each constituent atom being represented individually. This has negative implications to the visualisation of the reconstruction as well as many statistical analysis tools concerning the spatial distribution of atoms contained within molecular ions. Two techniques were developed in the study to separate complex molecular ions existing within atom probe tomography data: 1. The GSM was developed to split complex ions containing two constituent atoms. 2. The more computationally intensive DSM was developed to split larger complex ions.
A series of simulated datasets were used to test the perfor-
mance of the algorithms. 1NN plots were used to show that current molecular ion representation skews the observed distribution of atoms contained within the molecular ions and that the separation algorithms can be used to significantly improve this. The GSM was then used on a HSLA steel dataset containing NbN complex ions and facilitated Nb–Nb and N–N 1NN distribution measurements significantly. The DSM was applied to a GaAs dataset containing many large complex ions, significantly improving the visualisation of the reconstruction and facilitating more realistic measurements on atomic distributions — particularly the As–As 1NN distribution. The algorithms can be implemented on most personal computers within a practical amount of time.
Acknowledgements The authors would like to thank Peter Felfer and Leigh Stephenson for the Australian Centre for Microscopy and Microanalysis (ACMM) for fruitful discussions. The authors acknowledge the funding support from the Australian Research Council, scientific and technical support from the Australian Microscopy and Microanalysis Research Facility (AMMRF) at The University of Sydney. References [1] T.T. Tsong, Physical Review B 30 (1984) 4946–4961. [2] A. Shariq, T. Al-Kassab, R. Kirchheim, R.B. Schwarz, Ultramicroscopy 107 (2007) 773–780. [3] L.T. Stephenson, M.P. Moody, P.V. Liddicoat, S.P. Ringer, Microscopy and Microanalysis 13 (2007) 448–463. [4] T. Philippe, F. De Geuser, S. Duguay, W. Lefebvre, O. Cojocaru-Miredin, G. Da Costa, D. Blavette, Ultramicroscopy 109 (2009) 1304–1309. [5] E.A. Marquis, J.M. Hyde, Materials Science and Engineering: R: Reports 69 (2010) 37–62. [6] D. Vaumousse, A. Cerezo, P.J. Warren, Ultramicroscopy 95 (2003) 215–221. [7] T. Philippe, O. Cojocaru-Miredin, S. Duguay, D. Blavette, Journal of Microscopy 239 (2010) 72–77. [8] R.K.W. Marceau, L.T. Stephenson, C.R. Hutchinson, S.P. Ringer, Ultramicroscopy 111 (2011) 738–742. [9] F. De Geuser, W. Lefebvre, Microscopy Research and Technique 74 (2011) 257–263.
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[10] M.P. Moody, B. Gault, L.T. Stephenson, R.K.W. Marceau, R.C. Powles, A.V. Ceguerra, A.J. Breen, S.P. Ringer, Microscopy and Microanalysis 17 (2011) 226–239. [11] D. Haley, T. Petersen, G. Barton, S.P. Ringer, Philosophical Magazine 89 (2009) 925–943. [12] M.G. Hetherington, M.K. Miller, Journal De Physique 48 (1987) 559–564. [13] M.P. Moody, L.T. Stephenson, A.V. Ceguerra, S.P. Ringer, Microscopy Research and Technique 71 (2008) 542–550. [14] M.K. Miller, Atom Probe Tomography: Analysis at the Atomic Level, Kluwer Academic/Plenum Publishers, New York; London, 2000. [15] B. Gault, E.A. Marquis, D.W. Saxey, G.M. Hughes, D. Mangelinck, E.S. Toberer, G.J. Snyder, Scripta Materialia 63 (2010) 784–787. [16] O.C. Hellman, J.A. Vandenbroucke, J. Rusing, D. Isheim, D.N. Seidman, Microscopy and Microanalysis 6 (2000) 437–444. [17] M.P. Moody, L.T. Stephenson, P.V. Liddicoat, S.P. Ringer, Microscopy Research and Technique 70 (2007) 258–268. [18] F. Vurpillot, L. Renaud, D. Blavette, Ultramicroscopy 95 (2003) 223–229. [19] W. Sha, L. Chang, G.D.W. Smith, C. Liu, E.J. Mittemeijer, Surface Science 266 (1992) 416–423. [20] M. Muller, D.W. Saxey, G.D.W. Smith, B. Gault, Ultramicroscopy 111 (2011) 487–492. [21] F. Tang, B. Gault, S.P. Ringer, J.M. Cairney, Ultramicroscopy 110 (2010) 836–843. [22] B.P. Gorman, A.G. Norman, Y. Yan, Microscopy and Microanalysis 13 (2007) 493–502.
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[23] A. Cerezo, C.R.M. Grovenor, G.D.W. Smith, Journal De Physique 47 (1986) 309–314. [24] B. Gault, M.P. Moody, F. de Geuser, G. Tsafnat, A. La Fontaine, L.T. Stephenson, D. Haley, S.P. Ringer, Journal of Applied Physics 105 (2009). [25] R.A. Karnesky, D. Isheim, D.N. Seidman, Applied Physics Letters 91 (2007). [26] W. Lefebvre, T. Philippe, F. Vurpillot, Ultramicroscopy 111 (2011) 200–206. [27] L. Guibas, D. Knuth, M. Sharir, Algorithmica 7 (1992) 381–413. [28] V.J. Araullo-Peters, B. Gault, S.L. Shrestha, L. Yao, M.P. Moody, S.P. Ringer, J.M. Cairney, Scripta Materialia 66 (2012) 907–910. [29] B. Gault, M.P. Moody, F. de Geuser, D. Haley, L.T. Stephenson, S.P. Ringer, Applied Physics Letters 95 (2009). [30] B. Gault, M.P. Moody, F. De Geuser, A. La Fontaine, L.T. Stephenson, D. Haley, S.P. Ringer, Microscopy and Microanalysis 16 (2010) 99–110. [31] A.V. Ceguerra, R.C. Powles, M.P. Moody, S.P. Ringer, Physical Review B 82 (2010) 132201. [32] K.Y. Xie, T.X. Zheng, J.M. Cairney, H. Kaul, J.G. Williams, F.J. Barbaro, C.R. Killmore, S.P. Ringer, Scripta Materialia 66 (2012) 710–713. [33] T. Philippe, M. Gruber, F. Vurpillot, D. Blavette, Microscopy and Microanalysis 16 (2010) 643–648. [34] B. Gault, M. Muller, A. La Fontaine, M.P. Moody, A. Shariq, A. Cerezo, S.P. Ringer, G.D.W. Smith, Journal of Applied Physics 108 (2010). [35] K.Y. Xie, A.J. Breen, L. Yao, M.P. Moody, B. Gault, J.M. Cairney, S.P. Ringer, Ultramicroscopy 112 (2012) 32–38.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Spatial decomposition of molecular ions within 3D atom probe reconstructions Andrew Breen a,b, Michael P. Moody c, Baptiste Gault d, Anna V. Ceguerra a,b, Kelvin Y. Xie e, Sichao Du b,f, Simon P. Ringer a,b,n a
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia Australian Centre for Microscopy and Microanalysis, Madsen Building F09, The University of Sydney, NSW 2006, Australia c Department of Materials, University of Oxford, Parks Road, OX13PH, Oxford, UK d Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ont. L8S4L8, Canada e Johns Hopkins University, Department of Mechanical Engineering, Baltimore, MD 21218, USA f School of Physics, The University of Sydney, NSW 2006, Australia b
a r t i c l e i n f o
a b s t r a c t
Available online 7 March 2013
Two methods for separating the constituent atoms of molecular ions within atom probe tomography reconstructions are presented. The Gaussian Separation Method efficiently deconvolutes molecular ions containing two constituent atoms and is tested on simulated data before being applied to an experimental HSLA steel dataset containing NbN. The Delaunay Separation Method extends separation to larger complex ions and is also tested on simulated data before being applied to an experimental GaAs dataset containing many large (4 3 atoms) complex ions. First nearest neighbour (1NN) distributions and images of the reconstruction before and after the separations are used to show the effect of the algorithms and their validity and practicality are also discussed. & 2013 Elsevier B.V. All rights reserved.
Keywords: Atom probe tomography Complex molecular ions Delaunay tessellation Nearest neighbour analysis
1. Introduction In the course of an atom probe tomography experiment, atoms are ideally individually ionised and field evaporated from the specimen. However, very often this is not the case. Molecular ions (also known as complex ions or cluster ions) are routinely observed in atom probe mass spectra [1] and are usually represented as a single point in the resulting reconstruction rather than each constituent atom being individually located — a clearly unphysical model which has so far largely been conveniently ignored. This poses a problem, not only for the sake of visual clarity in the resulting reconstruction, an obvious yet important issue, but also for many statistical analysis tools at the heart of atom probe analysis. For some of the current statistical analysis tools available, atoms contained within complex ions are essentially invisible to the same species of atom that were evaporated as singular ions and this will skew the results of the analysis — the severity of which is dependent on the size and number of complex ions detected. A partial solution to this problem is decomposing the molecular ions into their constituent atoms at the same point and indeed this is already being done. The commercially available
n Corresponding author at: School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia. Tel.: þ61 2 9351 2351; fax: þ 61 2 9351 7682. E-mail address: [email protected] (S.P. Ringer).
0304-3991/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2013.02.014
IVASs software allows ‘atom-based’ rather than ‘ion-based’ analysis to be chosen for many of its analysis tools. However, positioning multiple atoms at a single point can lead to a significant bias in statistical analysis of the results. Of particular concern is the nearest neighbour (NN) analysis which histograms the distance between neighbouring atoms of chosen atomic species [2–4]. Even if an atom-based approach is used, some atoms will have a 1NN separation of 0 nm. While an NN distribution may not be particularly useful in itself, other than to indicate to segregation or anti-segregation of a particular atomic species, it is a fundamental component of many other higher order analysis techniques. Clustering algorithms [5,6], for instance, commonly use NN distributions for parameter selection, when applied to isolate and characterise nanoscale segregation phenomena [3,7,8]. The results of such an analysis are therefore inextricably linked to the quality of the NN analysis. The visualisation, cluster morphology and cluster size distribution are some aspects of this analysis that could be misleading if current molecular ion representations are used. NN distributions have also been used for phase composition measurements [4,9]. One such method derives a protocol based on the first nearest neighbour (1NN) distribution of a particular atomic species, to deconvolute the contributions from the different phases and thus determine the phase composition, but these measurements will be inaccurate if the solute species is contained within molecular ions.
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Radial distribution functions (RDFs) also suffer similar drawbacks to an NN analysis when the reconstruction being analysed contains complex ions. RDFs histogram the radial distance between chosen atomic species and can be used for observing crystallographic information as well as segregation phenomena [10,11]. However, it follows that if some atomic species are occupied within molecular ions, any measured RDF distribution concerning these atomic species will be skewed. Grid based analysis techniques that divide the reconstruction into voxels represent yet another group of data mining tools particularly susceptible to inaccuracies due to conventional molecular ion representation. One important example is the frequency distribution analysis algorithms [12–14] that divide the reconstruction up into a grid of blocks that all contain the same number of ions. Histograms of the number of ions of a particular species contained within each block are then created. Current available data analysis software generally do not offer an atom-based frequency distribution analysis and consequently molecular ions are counted as a single contribution to the size of the block. This not only causes the number of actual atoms in each block to vary but also renders atomic species tied up in molecular ions invisible to those that are not. Previously, Gault et al. used their own atom-based technique whereby the molecules within a reconstruction of a thermoelectric material were simply split into single atoms occupying the same spatial coordinates, facilitating species-specific frequency distribution analyses of Zn and Sb atoms [15]. This is acceptable except in cases where there are large molecular ions and fine voxel sizes such that there is increased chance of multiple atoms contributing to one voxel when they should be spread over neighbouring voxels. Other grid based analysis tools such as proxigrams [16], contingency table analysis [17] as well as density and concentration profiles will have similar issues. There are still many other analytical tools that stand to benefit from representing constituent atoms within molecular ions individually — lattice rectification techniques [10,18], that use the crystallographic information contained within an atom probe reconstruction to restore the original crystal structure of the sample is a notable example. Molecular ions inevitably complicate this process, even if they are decomposed into their constituent atoms at the same point, but by separating the constituent atoms into their likely locations, the rectification procedure is facilitated particularly in cases where datasets contain large molecular ions. As atom probe tomography (APT) is applied to an increasing array of sophisticated multicomponent materials, complex ion detection is becoming more common and hence, potentially more of an issue. A prominent example is the analysis of steels, where the detection of complex ions such as C2, NbN and FeN in various charge states are commonly observed [19]. Semiconductors, in particular III–V such as GaSb, are another material group that face similar challenges where Sb2, Sb3 and even larger molecular ions in various charge states are routinely detected [20]. It is therefore apparent that the representation of complex molecular ions within atom probe reconstructions should be avoided. Minimising the detection of complex ions in the experiment in the first place may seem like a reasonable approach, however, it is often inconvenient to change the experimental parameters to minimise complex ion detection, as other aspects related to the quality of data will be adversely affected [21]. Still, research has been undertaken studying the physics governing the field evaporation of complex molecular ions [1,20,22,23], information that could be used to control the severity of their detection. The experimental conditions have been found to play a critical role in molecular ion evaporation — higher laser pulse energies for instance have often been attributed to higher
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proportions of molecular ion detection, as well as more massive molecular ions themselves. It should also be noted that the field evaporation behaviour of metals is significantly different to that of semiconductors and other material groups. Much remains unknown, however, about the formation of these molecular ions on an atom probe tip. A more practical approach may be to spatially separate complex ions within the reconstruction so that each atom has its own unique position within the reconstruction. Fig. 1 illustrates this concept. By considering the crystallography and evaporation behaviour of the specimen as well as observed atomic distributions within the reconstruction itself, the spatial separation of complex ions within the reconstruction can be simulated accurately. At present it is impossible to precisely locate each individual atom from the molecular ions but the approach suggested could restore expected distributions on average and hence help to get better results from the analysis techniques aforementioned. For small complex molecular ions of 2 atoms, a method to separate the constituent atoms based on the measured 1NN distribution of singular ions occupying the same phase is proposed. The separation distance and orientation of the molecular pair are selected so that they follow a Gaussian curve fitting the 1NN distribution. For larger complex ions, this approach becomes analytically cumbersome and a technique that considers the available local volume surrounding the complex ion is more appropriate. In this case the use of Delaunay tessellations is proposed. Delaunay tessellations have many practical applications including pattern recognition techniques and finite element methods, and their implementation to a range of analytical studies related to APT is starting to be realised [24–26]. A Delaunay tessellation is formed by joining all pairs of points belonging to the set P¼{p1, p2, y, pn} where n Z4 such that no point in P is inside the circumsphere of any tetrahedra of the tessellation [27]. By measuring the volume of the tetrahedra surrounding the detected co-ordinates of a molecular ion, positions for the spatial redistribution of the constituent atoms can be determined that minimise any localised density fluctuations. Here, these algorithms are first applied to simulated reconstructions for proof of concept before being demonstrated on experimental data — first to a low-alloyed steel containing NbN complex ions and then to a GaAs nanowire containing very large molecular ions. Focus is placed on the effect of this reconstruction artefact to the 1NN distribution because these analyses represent a simple and directly comparable way of testing the performance of the algorithms. Further, 1NN distributions are an important component of other analysis techniques and improvement in the accuracy of these distributions will have flow-on benefits to the clustering algorithms, phase compositional measurements and
Fig. 1. Schematic demonstrating the splitting of complex molecular ions within an atom probe reconstruction: (1) before separation when molecular ions are represented at a single point and (2) after constituent atoms have been separated.
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radial distribution functions. Care should be taken in the interpretation of the results and spatial separation only performed when molecular ions are likely to bias the results from the analysis tools aforementioned.
2. Theory 2.1. The Gaussian separation method (GSM) The Gaussian separation method has been developed for the separation of molecular ions that consist of two constituent atoms (dimers). This approach is based on the assumption that the constituent atoms in the molecular ions were originally neighbouring atoms on the surface of the atom probe tip at the time of evaporation. It follows that the Dz component of the separations should be small while the overall separation distances should be similar to that of the atoms within the same phase on the surface of the atom probe tip. The separation distance distribution should match the 1NN distribution of the surface plane of atoms. For this to be accurate, the detector efficiency and a Dz limit should be taken into consideration. One approach for crystalline materials (and the one that has been used in this study) is to make a simulated reconstruction of the lattice in the same crystallographic orientation of the experimental reconstruction with 100% detector efficiency and apply atom probe like noise — then create a 1NN plot where Dz o0.05 nm of all the atoms. Information on determining the crystallographic orientation of APT reconstructions is described elsewhere [10,28]. The Dz cut-off value was selected because it is usually below the plane spacing commonly observed in atom probe data [29,30] and therefore creates a distribution more closely resembling that for the surface plane of atoms. The resulting curve can be closely fitted to a Gaussian distribution: ! ðxbÞ2 f ðxÞ ¼ a exp ð1Þ 2c2 where a is the amplitude, b is the mean and c is the standard deviation. The atoms composing the complex ions are then separated — a schematic of the process is presented in Fig. 2. For each dimer
Fig. 2. The separation of a dimer complex ion using the GSM: the dark blue atoms represent the constituent atoms of the complex ion where as the light blue atoms are surrounding atoms in the reconstruction. The dsep is the separation distance between the constituent atoms; the midpoint is kept as the original detected coordinates of the complex ion. d1NN_1 and d1NN_2 are the distances to the nearest neighbouring atoms after the separation and are used to determine whether the orientation of the separation will be accepted. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
complex ion in the reconstruction, the separation distance (dsep) is chosen at random from a set of distances matching the fitted Gaussian probability distribution. The original co-ordinates of the complex ion are kept as the midpoint/centre of mass between the two constituent p atoms andffi the orientation is also randomised ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such that dsep ¼ Dx2 þ Dy2 . The move is accepted only if the distances from the separated atoms to the nearest neighbouring atoms within the reconstruction are within 20% from a value selected at random from the Gaussian distribution — otherwise the orientation selection is repeated until this criteria is met. This results in the constituent atoms having a separation distribution similar to the theoretical 1NN distribution of atoms on the surface plane within the phase of interest. 2.2. The Delaunay separation method (DSM) The Delaunay separation method separates complex ions with two or more constituent atoms and uses the Delaunay tessellation of detected atom co-ordinates around a complex ion to determine the local volume available to distribute the atoms contained within the complex ion. It is based on the assumption that the constituent atoms were in close proximity to each other within the sample and that regions of larger empty volume surrounding the molecular ion are more likely to be where the constituent atoms were originally located. The algorithm identifies the simplexes (tetrahedra formed by Delaunay tessellation) that share a vertex with the detected coordinates of a complex ion, forming a convex hull within which the constituent atoms are distributed into the surrounding volume. The first constituent atom is placed at the original coordinates of the complex ion. The volume of each simplex is then calculated and the remaining constituent atoms are placed at the centroids of the largest tetrahedra. Fig. 3 is a schematic showing this process and clearly demonstrates how the constituent atoms are distributed into the available volume.
3. Simulated datasets 3.1. Method Simulated datasets of a face-centred cubic (FCC) binary alloy were used to test the performance of the GSM and DSM algorithms. The original simulated dataset contained 3 million atoms with composition Solvent (Sv) 1.0 at% Solute (Su), lattice parameter a0 ¼0.404 nm and Warren-Crowley parameter (which
Fig. 3. The separation of a large molecular complex ion using the DSM. Each vertex of the tetraheda represents the co-ordinates of an ion within the reconstruction — a molecular ion has been detected at the centre of the subvolume. One constituent atom is placed at the original reconstructed co-ordinates of the molecular ion and the rest are positioned at the centroids of the largest tetrahedra (highlighted in red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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controls the clustering/segregation of Su) a ¼0.01 [31]. 1% of random neighbouring Sv atoms on the (001) planes (i.e. perpendicular to the z-axis within the simulation) were converted to Su to simulate the true position of Su atoms contained within Su2 dimer complex ions. The resulting Su distribution is somewhat similar to that in the early stages of precipitate formation. 43% of all ions were then removed to simulate detector efficiency — if one of the Su ions within the Su–Su pairs was selected then the other Su ion was also removed. A complementary dataset was then created where approximately half of the converted Su–Su pairs remaining were replaced by Su2 complex ions at the centre of mass. A dithering function was then applied to the atom coordinates, with standard deviation sx ¼ sy ¼ sz ¼0.05 nm, to resemble APT like spatial noise. The GSM was then used to separate the simulated Su2 complex ions before comparing the Su distribution between the datasets. To simulate the true atomic distribution of larger complex ions containing 4 atoms, 0.25% of Sv atoms were selected at random from the original simulation along with 3 of their first nearest neighbours and were converted to Su. 43% of all ions were again removed to simulate detector efficiency — if one of the Su ions within a cluster of 4 Su atoms was selected then the other Su ions within the cluster were also removed. Another dataset was also created where the clusters of 4 Su atoms remaining were removed and replaced with Su4 complex ions. As before, a dithering function was then applied to simulate atom probe noise. The DSM was then used to separate Su4 and the Su distribution was compared between the datasets.
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Fig. 5. Showing the effect of the GSM on the measured Su–Su 1NN distribution: (black) correct distribution — no complex ions present, (red) when Su2 ions are simulated but only Su ions are used in the calculated distribution, (light blue) When Su and Su2 ions are used in the calculated distribution and (dark blue) after Su2 ions have been separated using the GSM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.2. Results The GSM was shown to separate dimer complex ions into their constituent atoms very close to the expected spatial distribution. Fig. 4 shows fitting a Gaussian curve to the 1NN of all atoms (Dzo0.05 nm) within the original simulation after APT like noise has been added. The resulting probability distribution was then used to model the separation distances and orientation of the constituent atoms using the GSM. Although there is some difference between the curves (R-square¼ 0.9742), the approximate fit is deemed appropriate for the algorithm to achieve sufficiently accurate results. The 1NN distribution is markedly more accurate after separating the simulated Su2 molecular ions with the GSM algorithm. Fig. 5 shows a range of 1NN distributions involving Su atoms
Fig. 6. Showing the effect of the DSM on the measured Su–Su 1NN distribution: (black) correct distribution — no complex ions present, (red) when Su4 ions are simulated but only Su ions are used in the calculated distribution, (light blue) Su4 and Su ions are used in the calculated distribution and (dark blue) after Su4 ions have been separated using the DSM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
before and after the GSM is used. The Su–Su 1NN before Su2 is separated (red) is significantly different to the correct distribution (black). Even if all Su containing ions are considered (light blue), essentially the same as an ‘‘atom-based’’ approach, the correct 1NN distribution is still not restored. However, after using the GSM, the resulting 1NN Su–Su distribution (dark blue) is much closer to the correct distribution. The DSM is also shown to improve the measured 1NN Su–Su distribution in the presence of larger molecular ions. Fig. 6 provides a similar range of 1NN distributions to the previous figure to show the effect of the DSM algorithm and the results are essentially the same — after using the DSM the resulting 1NN Su– Su distribution (dark blue) is much closer to the correct distribution (black).
4. Experimental datasets 4.1. Method Fig. 4. A Gaussian curve (red) is fitted to the 1NN distribution of all atoms using a depth threshold of Dz o 0.05 nm (blue) within a simulated dataset. The mean (b¼ 0.26 nm) and standard deviation (c ¼ 0.072 nm) of the Gaussian curve are used as inputs to the separation algorithm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The GSM was then applied to experimental data to separate NbN complex ions in a high strength low alloy (HSLA) steel produced using the CASTRIPs process [32]. The dataset was
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collected on a Cameca LEAP 3000 under voltage pulsing at 25% pulse fraction and 90 mm flight path at 25 K. A simulated BCC ferrite reconstruction with atom probe like noise orientated in a similar crystallographic direction to that observed in the experimental reconstruction was then used to produce the expected distribution of neighbouring atoms on the surface of the tip and was fitted with a Gaussian curve that was used when separating the Nb and N atoms. The mean and standard deviation of this distribution were found to be 0.204 nm and 0.127 nm respectively. Similarly, the DSM was applied to an experimental dataset collected from a GaAs nanowire containing many molecular ions such as As2, As3, As4, As5 and GaAs4. This dataset was collected on a Cameca LEAP 3000 Si under 0.05 nJ laser pulsing, 90 mm flight path at 25 K. Each dataset was reconstructed using the commercially available IVASs software. Significant trajectory aberrations were observed on the edges of the nanowire and were therefore cropped from the analysis region.
4.2. Results A significant difference in the atomic distributions of the Nb and N atoms contained within the HSLA steel dataset was observed after applying the GSM. The visual aspects of this separation are shown in Fig. 7. Larger-scale spatial features of the reconstruction have been conserved such as the noticeable segregation of the Nb and N atoms before and after using the GSM algorithm (Fig. 7 a, b). A close up of individual NbN positions (Fig. 7 c, d), clearly reveals how the individual atoms have been separated with varying dsep and orientation along the x-y plane relative to the detector, facilitating statistical analysis of Nb and N atomic distributions. Fig. 8 reveals how separating the complex ions affect the 1NN distribution of N and Nb atoms. The Nb–Nb 1NN distribution before the complex ions are split is significantly affected by a large fraction of Nb atoms within NbN complex ions — this was measured to be 23%. After separating the complex ions, the resulting distribution is a much more realistic representation of the true Nb–Nb 1NN distribution. No N was ranged in the original data since the N þ peaks overlapped with Si2 þ peaks in the mass spectrum but, after separating the complex ions, individual N atoms can be represented in the reconstruction. Their 1NN distribution after the split is almost identical to the original 1NN distribution of the NbN complex ions.
After applying the DSM to the GaAs nanowire, As representation within the reconstruction became much clearer. Fig. 9 shows the reconstruction before (a) and after (b) the DSM algorithm was applied. In particular, an overwhelming majority of the As atoms (99%) are contained within complex molecular ions. After separation, the complex ion species are replaced by a noticeably denser distribution of As atoms. This facilitates further statistical analysis of atomic distributions and architecture — particularly anything concerning As. Fig. 10 shows the 1NN distribution for any ions containing As before and after applying the DSM model. The frequency has been normalised so that the volume under each curve is equal to 1, this enables a better comparison between the distributions since such a large fraction of the As atoms were contained within molecular ions. This essentially transforms the curves into probability density functions. It is clear the true As–As 1NN distribution is severely affected by the large complex ions — before the separation, the mean 1NN As distance between As containing ions was 0.398 nm with a standard deviation of 0.158 nm and after the separation the mean was 0.136 nm with a standard deviation of 0.048 nm. The theoretical As–As 1NN distribution for the reconstruction is also given. This distribution was based on equations given by Philippe et al. [4]. The probability density function of finding a 1NN distance of r for any pair of atoms in a random solid solution
Fig. 8. The effect of the GSM on the measured Nb–Nb and N–N 1NN distributions in the HSLA steel sample: Nb–Nb 1NN distributions before and after using the GSM (purple), NbN–NbN 1NN before using the GSM (blue) and N–N 1NN after using the GSM (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Separating NbN complex ions within an experimental HSLA steel reconstruction: (a) original reconstruction showing the Nb and NbN complex ions and (b) after Nb and N atoms have been separated using the GSM — segregation and larger-scale features are still conserved after the separation. (c) Small sub-volume of reconstruction before separation and (d) after separation shows how the GSM separates individual NbN complex ions.
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Fig. 9. Separating complex ions from a GaAs nanowire reconstruction: (a) sub-volume of reconstruction before complex ions were split showing Ga and As containing ions. Approximately 292,000 As atoms are contained within only 80,000 reconstructed ions and (b) reconstruction after As containing complex ions were split.
Fig. 10. The As–As 1NN distribution before (blue) and after (red) the complex ions in the GaAs reconstruction were separated. The theoretically predicted distribution is also shown (dotted black). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is given by 4 P ðr Þ ¼ 4pr 2 Q C 0 exp pQ C 0 r 3 3
ð2Þ
where Q is the detector efficiency and C0 is the real concentration of the atoms. The maximum of P(r) can be found by making dP=dr ¼ 0: 1=3 1 r0 ¼ ð3Þ 2pQ C 0 Therefore the measured concentration of As within the reconstruction was substituted for Q C 0 in Eq. (2). The dotted line in Fig. 10 shows Pðr As Þ vs. r As and is very similar to the resulting As–As 1NN distribution suggesting that the DSM provides a valid representation of As within the reconstruction.
5. Discussion 5.1. The validity of the seperation algorithms It is important to note that the separation algorithms developed simulate the likely distribution of constituent atoms within the complex molecular atoms routinely seen in many materials but they do not restore the actual relative coordinates of the constituent atoms. Such a representation however, improves visibility as well as facilitates more accurate statistical analysis
concerning the spatial distribution of atoms contained within molecular ions including atom-based frequency distribution, nearest neighbour analysis and more advanced analysis tools such as clustering algorithms. However, it is up to the researcher to gauge whether or not the separation is necessary for the analysis they are performing. The GSM makes the assumption that the species contained within a dimer complex ion are neighbouring atoms on the surface of the atom probe tip at evaporation so that Dzij E0, where i and j are the pair of atoms contained within the complex ion, and that the separation distance follows the 1NN distribution of atoms on the surface of the tip. While much remains unknown about the formation of complex molecular ions before field evaporation, one would assume that evaporation would usually occur from the terrace edges, similar to that of singular ions and that it would be more energetically favourable for molecular ions to form with neighbouring atoms on the same terrace rather than between terraces. The DSM similarly assumes that constituent atoms were originally in close proximity to each other but relaxes the restriction that they were all neighbouring atoms on the same lattice terrace at evaporation. This was primarily used because it becomes difficult to distribute a large number of constituent atoms within a small area of the same x-y plane without causing localised density fluctuations. It seems more reasonable to distribute the constituent atoms within the available volume of a convex hull around the detected co–ordinates of the complex ion. It seems logical to assume that a large molecular ion would have a larger empty volume surrounding it than singular ions and would also make the approach of the DSM more valid. This was investigated for the GaAs reconstruction by comparing the average volume contained within the convex hull surrounding a singular ion with no molecular ion neighbours to the convex hull surrounding a molecular ion. Interestingly, it was found that the average volume surrounding the molecular ions was 0.0425 nm3 where as the average volume surrounding singular ions was 0.0398 nm3, which confirms the initial prediction. This finding also suggests that the molecular ions detected within the region of interest did not succumb to significantly worse trajectory aberrations than singular ions. It should be noted, however, that there are many other factors that influence the relative spatial distributions of atomic species within atom probe data and that these affects should be taken into consideration with any subsequent statistical analysis. The local magnification affect [33] and surface migration [34] are just two examples. Local magnification of the NbN molecular ions is
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known to have occurred in the HSLA steel sample used in this study. Some surface migration of N may also have occurred but this is considered negligible due to the N being strongly held within Nb, C and N rich clusters [35]. The GaAs data was also affected by these evaporation phenomena. The original reconstruction showed significant segregation of As containing ions to the side walls of the hexagonal GaAs nanowire and is a strong indication that surface migration occurred. The shape of the tip was also skewed around the edges suggesting either trajectory aberrations or the limitations of the reconstruction algorithm that assumes a tip in the shape of a hemispherical cap on a truncated cone — not a hexagonal prism. It was for these reasons that only the central region of the reconstruction was cropped for analysis — the observation of lattice planes within this region suggests that at least a moderate level of spatial integrity was maintained here. The algorithms developed are only appropriate for complex ion detection and if the accuracy of various statistical analyses was of concern due to other effects, further data treatment may be necessary. Still, the algorithms developed have been shown to improve the spatial distribution of atomic species contained within complex ions. This was clearly shown in Figs. 5 and 6 when the algorithms separated complex ions within simulated data and improved the resulting 1NN distribution of solute species. It should be noted however that both algorithms seem to slightly under-estimate the true separation distances i.e. the measured Su–Su 1NN distribution after separation is slightly shifted to the left for low 1NN values. This suggests that there is slightly more randomness or noise added to the positioning of constituent atoms relative to their actual positions after separation — the mean 1NN distribution tends to decrease as more randomness is added to the system which is suggested from Eq. 3 where r0 is always less than the theoretical 1NN distribution for a perfect crystal. However, some extra randomness is expected since the exact positioning of the constituent atoms cannot be known. The 1NN distribution does show a marked improvement however as compared to that when the molecular ions are not spatially separated which suggests a valid distribution of the atoms contained within the molecular ions has been achieved. This has significant implications — results from statistical tools employing the NN analyses and similar type algorithms can provide valuable information on the structure–property relationships of materials and can potentially be significantly improved for any material where complex ion detection poses a limitation such as the HSLA steel or GaAs nanowire used to demonstrate the algorithms. 5.2. Computational performance The DSM and GSM can be performed within a practical amount of time for millions of atoms on most personal computers. Both algorithms voxelise the dataset into 1 nm3 cubes so that any measurements only occur between neighbouring voxels, not over the entire dataset, significantly reducing computation time. The GSM was coded in C and was able to separate 1426 Su2 complex ions in the simulated dataset containing 2, 872,034 detected ions in 14 s. The DSM was coded in Matlabs and took 23 mins 60 s to separate 1426 Su4 atoms from 2867, 756 total detected ions. The processing time increases almost linearly with number of complex ions detected. The increased computer time for the DSM is largely due to the Delaunay tessellation of each voxel of atoms. All processing was done using a single 2.3 GHz Intel Core i7 processor with 8 GB memory. Further improvements to performance could be achieved through parallel computing, GPU processing and compiling the DSM in C/Cþþ code. While the DSM is necessary for complex ions containing more than 2 atoms, the GSM is much faster. A programme that uses
both algorithms, depending on the size of the complex ion could be implemented in the future.
6. Conclusion In summary,
Molecular ions are typically represented as single points
within atom probe reconstructions rather than each constituent atom being represented individually. This has negative implications to the visualisation of the reconstruction as well as many statistical analysis tools concerning the spatial distribution of atoms contained within molecular ions. Two techniques were developed in the study to separate complex molecular ions existing within atom probe tomography data: 1. The GSM was developed to split complex ions containing two constituent atoms. 2. The more computationally intensive DSM was developed to split larger complex ions.
A series of simulated datasets were used to test the perfor-
mance of the algorithms. 1NN plots were used to show that current molecular ion representation skews the observed distribution of atoms contained within the molecular ions and that the separation algorithms can be used to significantly improve this. The GSM was then used on a HSLA steel dataset containing NbN complex ions and facilitated Nb–Nb and N–N 1NN distribution measurements significantly. The DSM was applied to a GaAs dataset containing many large complex ions, significantly improving the visualisation of the reconstruction and facilitating more realistic measurements on atomic distributions — particularly the As–As 1NN distribution. The algorithms can be implemented on most personal computers within a practical amount of time.
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