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The rise of computational techniques in atom probe microscopy

Anna V. Ceguerra a,b,⇑, Andrew J. Breen a,b, Leigh T. Stephenson a,b, Peter J. Felfer a,b,Vicente J. Araullo-Peters a,b, Peter V. Liddicoat a,b, XiangYuan Cui a,b, Lan Yao c, Daniel Haley d,Michael P. Moody e, Baptiste Gault e, Julie M. Cairney a,b, Simon P. Ringer a,b,⇑a Australian Centre for Microscopy & Microanalysis, The University of Sydney, NSW 2006, Australiab School of Aerospace, Mechanical & Mechatronic Engineering, The University of Sydney, NSW 2006, Australiac Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USAd Max-Planck-Institut für Eisenforschung, Max-Planck-Str. 1, 40237 Düsseldorf, Germanye Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom

a r t i c l e i n f o

Article history:Available online 7 October 2013

Keywords:Atom probe microscopyAtom probe tomographyMaterials informaticsComputational materials science

a b s t r a c t

Much effort has been devoted to the development of computational techniques in atom probe micros-copy over the past decade. There have been several drivers for this effort. Firstly, there has been effortdevoted to addressing the challenges of discerning information from the increasingly large size of thedata, and capturing the opportunities that this large data presents. Secondly, there has been significantnew effort devoted to the simulation of atom probe data so that pristine datasets that contain microstruc-tural features of increasing complexity can be generated in-silico, and subjected to complex data-miningalgorithms. This has enabled the benchmarking of various algorithms, guided the setting of parametersfor particular analyses, and exposed the effects of instrumentation parameters such as detector efficiencyand aberrations in ionic flight path. The authors are especially interested in the prospects of convergingatomic-scale microscopy with atomic-scale materials modelling via first principles approaches. Thisinvolves excising parts of the APM data and using these as super-cell inputs to calculations of materialsproperties via density functional theory. It is our opinion that this represents a major advance for mate-rials science because it enables microscopy to advance microstructure–property relationships to thedirect mapping of such relationships based on many-body interactions. As such, this approach has greatpotential for materials design and development.

The final part of this paper focuses on how cloud-based computing represents an exciting frontier ofthe computational aspects of atom probe microscopy. We discuss the opportunities and the barriersfor conducting new materials science through the analysis and visualisation of atom probe data vianew generation tools that are cloud-based, and which are managed, curated and governed with signifi-cant user-community input and integrated with contemporary electronic laboratory notebooktechnology.

Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.

1. In LEAPs and bounds

Atoms were first observed experimentally when Erwin Müllerinvented the field ion microscope during the middle of the 20thcentury [1]. Coincidentally, this was around the very same timethat the first commercial computer became available [2]. It is thecombination of the two that makes the atom probe the uniquelypowerful tool that it is today. Computation has long played a role

in atom probe—not only enabling automated instrumental control(see e.g. [3]), but also for the analysis of atom probe data (see e.g.[4]).

Atom probe microscopy (APM) has grown rapidly in the lastdecade or so, the beginning of which was marked by the construc-tion of the first prototype Imago Local Electrode Atom Probes(Imago LEAP™) [5], and the first commercial LEAP™ installationsat Oak Ridge National Laboratories and the University of Sydneyin 2003/4. The arrival of the LEAP™ and similar instruments [6] be-gan a new era in APM, which has seen a dramatic rise in the sophis-tication in computational analysis. The post-experimental APManalysis workflow is summarised in Fig. 1 and it is clear that com-putational methods are now pervasive across each phase of thisworkflow. We discuss each of these in turn. The actual workingsof the instrument are described elsewhere [7–9], including in

1359-0286/$ - see front matter Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cossms.2013.09.006

⇑ Corresponding authors. Present/permanent address: Australian Centre forMicroscopy & Microanalysis, The University of Sydney, Madsen Building F09,NSW 2006, Australia. Tel.: +61 2 9036 6416; fax: +61 2 9351 7682 (A.V. Ceguerra),tel.: +61 2 9351 2351; fax: +61 2 9351 7682 (S.P. Ringer).

E-mail addresses: anna.ceguerra@sydney.edu.au (A.V. Ceguerra), simon.ringer@sydney.edu.au (S.P. Ringer).

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articles presented in this special issue. We also anticipate thatcomputational methods will increasingly be part of sample prepa-ration, especially where site specificity is required and as focusedion beam microscopy becomes more sophisticated. Returning tothe workflow illustrated in Fig. 1, we note that the archetypalexperiment usually flows from an evaluation of the quality of thedata, through to an approach for reconstruction that will dependon the nature and quality of the data itself, through to data analysisand visualisation. In fact, there are numerous approaches to each ofthese aspects, and it is important that the researcher maintainsclarity around the questions and hypotheses that are being posed.

A typical volume of data for an atom probe experiment is�80 � 80 � 500 nm, though larger volumes have been recorded.This field-of-view coupled with the developments in nanotechnol-ogy mean that it is possible not only to sample larger volumes of anengineering component at atomic resolution, as is usually the case,but that it is also possible to capture entire components within thedata, such has recently been achieved for the analysis of transistors[10–13].

It is because APM reconstructions contain between tens andhundreds-of-millions of atoms that it can be difficult to examinethe dataset on an individual atom-by-atom basis with the compu-tational power typically available in a personal computer. As a re-sult, a key question that researchers need to address early in theiranalysis is which part of the data to include, and which to excludefrom certain calculations. Moreover, the raw data does not possessperfect spatial or chemical resolution, and the instruments do nothave perfect detector efficiency [7,14–17]. These issues haveplayed a key role in driving new computational approaches. Onthe other hand, the counting statistics for the analysis of micro-structure are now in the realm of ‘big data’, alongside that recordedin the fields of neuroscience, geoscience and others, and it is thisaspect that has required the use of advanced computational tech-niques to rigorously define spatial resolution [14,18–21], mass res-olution [4,22] and to conduct thorough analyses.

2. Atom probe microscopy

Just as there are a flourishing range of imaging and analysistechniques incorporated into transmission electron microscopy(TEM), scanning electron microscopy, scanned probe microscopy,and other forms of microscopy, the same can also be said ofAPM. For this reason, we have chosen to mainstream the nomen-clature of the technique, consistent with other microscopies, toproperly recognise the value of the various techniques and dataformats available in a typical APM experiment. These include massspectrometry, field desorption imaging, field ion microscopy (FIM),and atom probe tomography (APT). Computational developmentsin several of these constituent techniques are now discussed.

2.1. Mass spectrometry in APM

The ideal atom probe experiment measures the time-of-flightand thus mass-to-charge ratio, of field evaporated ions that are lib-erated from the needle-shaped specimen one-at-a-time via indi-vidual pulses of energy. This produces a frequency distribution ofmass-to-charge ratios that is commonly referred to as a mass spec-trum, and which has uniquely identifiable peaks based on thechemical species in the sample. Computational approaches to theanalysis of these spectra are important topics that have receivedrelatively little attention, and most analyses are still done rela-tively manually. Our opinion is that better materials science willbe possible when the APM community converge on a standardpractice for the interpretation of these peaks. The sorts of mathe-matical and computational techniques that are already availableand used by the microanalysis communities undertaking energydispersive X-ray spectroscopy or electron energy loss spectroscopymust be brought to bear on the mass spectra derived from atomprobe microscopy experiments [23]. The absence of a standardpractice around interpretation of the mass spectra means that dif-ferent analysts can record significantly different analysis resultsfrom the same data, which is unsatisfactory. Practices for massspectrometry in APM are discussed in [24–26].

Additionally, there is information in this spectral data that isnot usually analysed. This stems from the fact that one individualpulse may not always translate into one ion being field evaporated.In reality, multiple detector events can occur during one pulse cy-cle, and the propensity for this will depend on the nature of theatomic bonding in the material and the heterogeneity of evapora-tive field throughout the specimen. In unfavourable circumstances,a single pulse will frequently induce the field evaporation of sev-eral ions simultaneously, and thus the detection of multiple ionsis attributed to a single pulse [27]. This and related complex evap-oration phenomena can cause various artefacts in the mass spectraand these include raising the noise floor, the introduction of spuri-ous peaks, and peak convolutions. Saxey deduced correlations inmultiple hit events [28], and Yao et al. introduced a spectral anal-ysis technique for APM data that is based on the multiplicity ofdetector events for each atom [29]. This allows for multi-dimen-sional mass spectra that can be used to differentiate signal fromnoise. More recently, these multiple detector events were furtherinvestigated by correlating temporal, chemical and spatial infor-mation from raw detector data [30]. Analysing ions that evapo-rated simultaneously was shown to be a useful way to identifyand visualise preferred solute–solute interactions. The time-chem-ical correlation introduced in that work represents an altogetherdifferent concept to conventional spatial–chemical based analysesfor the investigation of phenomena such as clustering and precip-itation, and was shown to be effective in measuring subtle atomicarrangements within materials.

Even if detector events are well separated in both space andtime, molecular/complex ion detection can pose a limitation tothe analysis. This occurs when a group of atoms are evaporatedfrom the tip and remain together until they are detected. Eachmolecular ion is then subsequently represented as a single entitywithin the resultant reconstruction rather than each constituentatom being positioned separately. This affects the accuracy of thevisualisation of the reconstruction, since atoms contained withinthe molecular ions are essentially hidden, and this degrades theaccuracy of many statistical analyses such as the investigationsof fine-scale clustering phenomena or frequency distributions ofparticular atomic species. It also represents a barrier to APM everfully achieving atomic scale tomography (AST) since this would re-quire each atom within the analysed material to be representedindividually [31]. A practical approach to alleviating this issuehas been proposed by Breen et al. [32], in which molecular ions

Fig. 1. Atom probe microscopy workflow, post-experiment.

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within the reconstruction are separated so that each atom has itsown unique position. Algorithms have been developed to simulatethe expected distribution of the constituent ions contained withinthe molecular ions. Fig. 2 demonstrates this concept for a recon-struction of a commercially available Nb-microalloyed strip caststeel containing NbN molecular ions.

2.2. Field desorption images

Field desorption images are one of several important types ofdata available from atom probe microscopy. These are 2D mapsof the counts of field-desorbed ions hitting specific sites on thedetector. For some APM experiments it is possible to resolve crys-tallographic poles, which are flat regions (usually low-indexplanes) exposed on the surface of the hemisphere-shaped speci-men tip. Poles may be observed as low-density regions on the fielddesorption image, because of field differences between the terraceedges and the flat plane region, causing trajectory aberrations ofthe ions. Alternatively, certain poles such as {111} in body-cen-tred-cubic crystal are regions of high density. Detailed indexedfield desorption images have recently been compiled for variouscrystal systems [7]. It is important to note that most of the pub-lished field desorption images are from pure metals, or near purematerials and there is the possible connotation that these sortsof images are difficult or impossible to attain in multicomponentengineering alloys. While it is certainly easier to obtain the imagesfrom pure metals, in fact, clear images that are readily indexed arepossible, and we have attained images such as this in a wide rangeof commercial ferrous and non-ferrous alloys as well as doped sil-icon semiconductor materials [33]. This is important because itemphasises the importance of these images in APM as a step-ping-stone towards APT where, if the images are of sufficient qual-ity, they play a key role in correctly setting the parameters in thetomographic reconstruction. Finally, we note that high qualityFIM images have been recorded for various ceramics, superconduc-tors, and semiconductors (see e.g. [4] for detailed references) andthat this imaging mode in APM has a diverse capacity for recordingcrystallographic information.

2.3. Atom probe tomography

The reconstruction of the raw microscope data to generate atomogram is the crucial first step towards 3D data analysis, and

various approaches to the tomographic reconstruction have re-cently been discussed [7,34]. These approaches are based on amodel of ionic projection that is based on a spherical tip geometryas first proposed by Bas et al. [35] rather than the cylindrical geom-etry initially proposed by Cerezo et al. [36], which had been effec-tive for small fields of view. The Bas et al. approach [35] uses thesequence of individual ions hitting the position-sensitive detector,and has a significant number of free parameters that the user musteither know, or assume. The recent development of incorporatingthe analysis of field evaporation images into the determination ofthe parameters required for the tomographic reconstruction [37]requires only a knowledge of the crystal structure of the specimenmaterial. This practice provides an accurate and rigorous basis forthe reconstruction and effectively achieves a significant improve-ment in the spatial resolution of the technique. ‘Dynamic recon-struction’ [38] is an extension of this approach whereby groupsof atoms are assigned different parameter values as the experimentprogresses, rather than forcing all atoms into a single set of aver-aged parameters.

Despite these advances, the fact remains that the reverse-pro-jection algorithm for tomographic reconstruction proposed byBas et al. [35] oversimplifies critical aspects such as the tip shape,the potential for variations in evaporation field within the speci-men [39], and the possibility of preferential evaporation and reten-tion [40]. It is also our view that this reconstruction approach is notthe obvious starting point for the tomographic analysis of amor-phous materials via APM, since it is clear that the process of fieldevaporation is much less uniform in amorphous materials thanthat which occurs in crystals [41]. Notwithstanding the eleganceof that initial treatment [35], it is our view that the resolution ofthe atom probe is now at a point where a more advanced tomo-graphic physics must be developed.

3. Frontiers of data analysis

3.1. Atom probe crystallography

Traditionally, atom probe is synonymous with the notion ofvery high resolution microanalysis and it remains true that thistechnique offers the highest spatial resolution microanalysis tech-nique available. Recent computational methods have revealedmore clearly the depth of information that is available within thedata, and it is clear that this is firmly in the realm of microscopy

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Fig. 2. separating molecular ions from a commercially available Nb-microalloyed strip cast steel containing NbN molecualr ions. (a) Before and (b) after NbN molecular ionshave been separated.

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and, more specifically, crystallography. Indeed, atom probe crystal-lography is now a burgeoning area of research and many chal-lenges and opportunities lie ahead [33,42,43]. The materialsscience driver here is the fundamental need to correlate localstructural information with local chemical information, sincemany transport processes in materials are driven by a combinationof the local lattice mechanics and local chemical potential. Thesetwo types of measurements are usually quite separate processes.Usually, structural information might be derived from a diffractionexperiment whilst chemical information might be derived from X-ray or electron energy loss spectroscopy. It represents a great chal-lenge to integrate this data from the same region of interest, and aneven greater challenge to achieve meaningful counting statistics ofthese types of data compilations. The rigorous integration of thesedata represents an important new microstructural frontier thatcombines the morphological shape, chemistry and crystallographyof a feature or interface in a 3D context. This approach is especiallyimportant for nanocrystalline materials where the grain sizes aretypically less than 100 nm. In this case, transmission-based ap-proaches (e.g., transmission electron microscopy, X-ray diffraction,or small-angle X-ray scattering) generate scattering data that oftencontain irrecoverable convolutions from multiple grains. In thecase of TEM, recent developments have enabled the extraction ofthe 3D aspect of the data on a nanometre scale [44], but none ofthe above approaches is suitable to facilitate a companion chemicalanalysis on the same length scale.

Before describing the state-of-the-art in atom probe crystallog-raphy, it is important to discuss briefly the various computationaltools that are being developed and/or adapted for this purpose.These tools include Fourier transformations, Hough transforma-tions, spatial distribution maps (SDMs), and radial distributionfunctions (RDFs). These may be viewed as a set of computationaltechniques that are designed to operate on the APM data so as todeduce the crystal orientation with a self-consistent reference.

The adaptation of the Fourier transformation was one of the earli-est signal processing techniques applied to APT data [16]. The useof this transformation was primarily to detect solute clustersrather than crystallographic analysis, and the implementationwas limited to very small regions of crystal via an adaptation ofthe Discrete Fourier Transformation algorithm, where the sparsematrix of 3D points is represented as Dirac delta functions, centredon each point. This approach is computationally intensive, requir-ing the integration of complex trigonometric functions. In the fu-ture, we anticipate that this approach could be adapted as aviable method for atom probe crystallography, given the wide-spread use of central processing unit (CPU) parallelisation, as wellas floating point accelerators such as dedicated graphics processingunits (GPUs).

In contrast to Fourier transformation techniques, radial distri-bution functions (RDFs) [41] are computationally simpler, sincethey are a direct vector calculation from a reference atom to itsnearest neighbours. Each atom in the dataset acts as a referenceatom for all other atoms, in turn. Unfortunately, raw atom probedata does not usually have sufficient spatial resolution to make thistechnique universally applicable. As a result, the computed distri-bution functions usually have unacceptably high noise and includenon-physical effects such as peaks at distance positions that do notcorrespond to nodal positions of the crystal lattice. This arises fromthe anisotropic spatial resolution of APT, which means that whenexamining a dataset, both poor and good spatial resolution direc-tions in the data are incorporated simultaneously, blurring theresultant 1D RDF. The 3D RDFs suffer alternatively from problemswith the available counting statistics, and need to be filtered tomerge the higher quality signal, as is implicitly done via SDMs(see below).Where the data are of high quality, RDF methods arevery effective and produce unique crystallographic solutions.

Spatial distribution maps (SDMs) were first introduced byGeiser et al. to inform data quality measurements [45], and these

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Fig. 3. (1D) SDMs within a HSLA steel reconstruction, clearly detecting the latent BCC ferrite lattice planes. (a) A 2D density map of the reconstruction with the 112, 013 and011 poles indexed, (b)–(d) show the resulting z0-SDMs calculated within regions surrounding these poles. The signal-to-background ratio has been optimised by rotating thereconstruction around the x- and y-axis by the given phi and theta values and this information can be used to determine the normal to the family of lattice planes beingdetected.

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were subsequently adapted by Moody et al. [46,47] for crystallo-graphic measurements and by Boll et al. for investigating site occu-pancy [48]. SDMs are histograms of the distance between atomsalong a specific crystallographic direction, within a specific latticeplane. As such, they are closely related to RDF methods. There isthe so-called z-SDM (or 1D) and xy-SDM (or 2D), based on the axisupon which the SDM is calculated. Figs. 3 and 4 provide examplesof 1D and 2D SDM, respectively, of various poles of a HSLA steel. Itmay be noted that these images were recorded from a commercialengineering steel alloy containing well over ten different elementalalloying additions. Therefore, we wish to point out that although itis usual to display SDMs only of pure metals [7,46], careful selec-tion of the parameters during the experimental acquisition, carefulcalibration of the reconstruction, and proper selection of signalprocessing parameters such as orientation and binning size makeit possible to observe crystallographic information in commerciallyimportant multi-component materials. By generating three non-parallel sets of SDMs from within a single grain, such as in Fig. 5,it is possible to determine the crystal orientation with respect toa self-consistent and independent reference axis.

Liddicoat et al. in 2010 [43] used manual measurements of thelattice plane spacings from multiple directions to compute individ-ual lattice (grain) orientations from a large nanocrystalline AA7075 APM dataset, after high pressure torsion experiments demon-strated remarkable strengthening in that alloy. The result is dis-played in Fig. 5, which represents a powerful data integration of

crystallographic information and solute cluster analyses. It isapparent from the data that segregation across grain boundariesis not uniform, but rather can occur in patches, and that this segre-gation depends both on crystal misorientation and local grainboundary curvature. There is a further aspect to the data integra-tion in Fig. 5 that is noteworthy: in recognising the importanceof grain boundary curvature, there is a need for a self-consistentapproach to map the boundaries between adjacent grains. In someinstances, this can be straightforward because there exists anextensive, uniform chemical segregation to the grain boundaries.In those cases, the boundaries are effectively chemically labelledand there is little ambiguity in their identification. However, whenthe grain boundary segregation is more subtle and a chemicalmapping approach is unavailable, such as the case of Fig. 5, densitymapping is effective. In fact, grain boundaries can be clearly iden-tified by measuring the atomic density variations of all elements inthe raw data [43]. These density variations exist because of trajec-tory aberrations that occur during the typically non-uniform fieldevaporation in the vicinity of grain boundaries. In this way, specificranges of atomic density can be used to discriminate the inter- andintragranular regions. In Fig. 5, the core-linkage algorithm (see be-low and [50]) was used to provide point-density measurements ofatomic density.

The approaches described above and encapsulated in Fig. 5 maybe thought of as an advance in microstructural characterisation inthat they integrate information about grain orientation, grain

Fig. 4. Examples of 2D SDMs from the same HSLA steel reconstruction shown in Fig. 3. The BCC crystal structure of ferrite is apparent. Left: The 2D SDMs are applied to the{112}, {013} and {011} plane families ((a), (c) and (e) respectively). Right: crystal models of single {112}, {013} and {011} planes made using the crystallographicvisualisation software VESTA� [49] ((b), (d) and (f) respectively) – inserts of the crystal planes within a 3D model of the BCC ferrite lattice are also shown. The inter-atomicconfiguration in the 2D SDMs directly matches that shown in the models. Each 2D-SDM was computed across the entire volume of a �1 million atom slice of thereconstruction.

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boundary curvature, and chemical segregation. Whilst serving as akey proof-of-concept for the possibility of extracting crystallo-graphic information in nanocrystalline materials, it must be saidthat the approaches used in [43] were highly ‘manual’, requiringmany repeated painstaking individual calculations. The develop-ment of an analysis method for any new microstructural topologymust be both accurate but also accessible and efficient if it is to bewidely used. Important progress has been made to address thesechallenging aspects and these are summarised briefly below.

An especially intensive part of crystallographic analyisis viaAPM is the local computation of grain orientation, and this is anarea that we have devoted particular attention to. A computationalapproach that does show promise to enable the rapid generation ofthe sort of textural information described above is the 3D Houghtransformation technique [51]. This approach is an extension ofthe 2D feature extraction technique that is usually used to detectlines in 2D images. The application of the 3D Hough transformationto APM data allows the detection of planes in the dataset, resultingin a two-dimensional plot that maps the periodicity of the atomicarrangements in a specific direction defined by elevation and azi-muthal angles with respect to the direction of analysis. An exampleis provided in Fig. 6. The high intensity regions in these plots cor-respond to the directions normal to the orientation of the crystal-lographic planes. Normally, the Hough transformation is applied to

a number of different regions within the dataset, to detect a rangeof different planes, as shown in Fig. 6a and b. Knowledge of threeplanes allows the calculation any fourth direction by simple linearalgebra in cubic systems, enabling the grain orientation to be accu-rately calculated. This information can be used to generate 3D ori-entation maps, such as that provided in Fig. 6c, where it is clearthat all degrees of freedom that define these polycrystalline inter-faces are determinable [52].

In addition to computational developments geared to computeEuler angles and identify grain orientation, there have been a num-ber of key developments for identifying grain boundaries, andthese advances bring us much closer towards the possibility of via-ble, automated chemico-textural analysis via APM. For example,Samudrala et al. have shown that grain boundaries may alterna-tively be identified by voxelising the data and searching for localdensity gradient extrema [53]. Other recent developments in ap-proaches for grain network extraction are discussed in Section 3.3.Further detailed discussions of atom probe crystallography areprovided in [7,33]. This is an exciting frontier topic in APM andwe anticipate major growth in this area as the local chemical nat-ure of texture aids understanding of materials processing.

3.2. Cluster analyses

Atom probe research has been at the vanguard of discerningstructure within solid solutions, and the notion that solute atomsor dopants may be clustered in distinctly non-random arrange-ments. These clusters change the local free energy landscape andso change the properties of materials, and/or create alternate phasetransformation pathways. Maximum separation (MaxSep) is analgorithm that is identical in operation to friends-of-friends, sin-gle-link or cluster-search [54–57] approaches. The most recentversion of MaxSep is called GraPTop [58] and usefully formulatesthe approach from a graph-theoretic perspective. In general, themethod associates solute atoms with a cluster if they are withina distance d_max of at least one other solute atom within that clus-ter. The distance d_max is a user-selected parameter. It is one ofthe easiest algorithms to implement and there are other associatedmethods to analyse the chemistry of atoms [59].

DBSCAN is termed a ‘‘density-based’’ algorithm for the reasonthat density is naturally associated with the high order kth nearestneighbour distances upon which the calculations are based [60].This method associates solute atoms with a cluster if it is withina distance d_max of a ‘‘core’’ atom that has a kth nearest neighbour(kNN) distance less than or equal to d_max. Maximum separationis a special case of DBSCAN (k = 1). Using DBSCAN is slightly moreinvolved for the sole reason that calculating high order nearestneighbours is more time and memory intensive.

Core-linkage separately addresses two steps that are implicit inMaxSep and DBSCAN [50]. It first identifies ‘‘core’’ solute atomsthat have a kNN distance less than d_max. Following this, it thenlinks any solute atom to a cluster if it is with d_link of a core soluteatom. Because of this flexibility, it can notionally be more effectivethan either MaxSep or DBSCAN. However, this accuracy comeswith the burden of choosing an additional parameter.

Other cluster detection approaches include the 3D Markov Field[61], Delaunay triangulation [62], and Fourier Transformation [63].3D Markov Field considers the crystallography of the sample in cal-culating the nearest neighbours. It includes single atoms in thecluster counts, implying that even a sample with a random distri-bution of solute will contain clusters. This method includes a wayto determine the standard deviation of the number of solute atomsclassified as being part of clusters of a particular size, using the Ber-noulli trial method.

While the application of any clustering algorithm is subject toheuristic parameter choice or some optimisation technique, the

Fig. 5. Atom probe characterisation of a nanostructured 7075 Al alloy [32] (a) grainboundary identification and misorientation (angles on boundaries) measured fromatomic density variations and crystal lattice planes (example plane inset in top-right corner), (b) tomographic reconstruction of key features – grain boundaries andintergranular line and node solute structures.

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following is emerging as best practice in the analysis of fine-scaleclusters. Firstly, it is clear that using random labelling is a valuablemethod to determine the margins of error in the analyses, forparameter selection, and to establish the degree by which the ob-served clustering occurs over and above that which may have beenexpected. Secondly, the use of simulations is extremely valuable inthe establishment of parameters for clustering algorithms and forbenchmarking the algorithms [64]. Finally, there are now methodsavailable that adjust the observed solute cluster frequency distri-butions to accommodate the statistical errors associated with thefact that the detector efficiency is less than 100%. Combining theseapproaches represents the state-of-the-art in cluster analysis.

3.3. Feature analyses

The raw reconstructed APM data allows us to map atomic dis-tributions in 3D space (viz. APT). In the APT reconstruction, manyfeatures such as grain boundaries and dislocations can directly beobserved. These structures nevertheless need to be extracted fromthe data based on the concentration and density variations thatthey cause [39,43,53,65,66]. The extraction of the structures andanalysis of the atomic distribution relative to them may be referredto as mapping methods, as they attempt to map the informationthat is contained in the 3D distribution of atoms into a local coor-dinate system that is specific to a feature or its boundary.

The simplest and most common way of extracting features suchas phase boundaries from APT data is the definition of a scalar fieldby voxelization and the extraction of iso-surfaces from this scalarfield. Voxelization methods divide space into small, regular shapedvolumes, usually cubes, for each of which a property such as con-centration or density of a species is calculated. This translates the3D point distribution of the atom probe reconstruction into a scalarfield. Specifically in APT, the use of an additional, smoothing trans-fer function is common [65]. On this scalar field, objects are de-fined as boundaries to volumes within which the scalar field hasspecific, thresholded values (iso-surface). Since all centres of thevoxels are lying on a regular grid, a triangulation of an iso-surfacecan easily be obtained via the ‘‘marching cubes’’ algorithm [67].

This is however only suitable for features that are characterisedby uniform properties or that occupy a significant volume. Forother types of features, methods based on extremal gradient values[66], or based on local density fluctuations [53], have recently beenintroduced. The ability of the former to distinguish features is gi-ven in Fig. 7, where the network of grain boundaries in a nanocrys-talline Al alloy has been delineated. The strength of latter is that itis very effective at distinguishing microstructural features (such asgrain boundaries) from atom probe artefacts (such as poles orlarge-scale density fluctuations).

Based on the objects extracted from the data, the distribution ofthe atoms is transformed into the local coordinate system of theobjects for analysis. This yields, for example, the pseudo-1D histo-gram of atomic concentrations based on the distance to the object.This histogram is commonly referred to as a proximity histogram(‘‘proxigram’’) [68] and, in combination with iso-surfaces, is oneof the most common methods for analysing atom probe data.Based on the distance to an object, the interfacial excess of a spe-cies can also be calculated either based on cumulative curves[69] or the proximity histogram [70]. As a more recent develop-ment, Felfer et al. [66] have also mapped properties such as theinterfacial excess across features into pseudo 2D propertyhistograms.

3.4. Lattice rectification

Lattice rectification techniques use the aforementioned crystal-lographic information contained within an APM data to restore theoriginal crystal structure of the sample. Initial attempts at restor-ing the lattice of crystalline atom probe samples used FourierTransformation analysis techniques [16,71], however, this wascomputationally expensive and could only be applied to severalthousand atoms in practical amounts of time. Recently, Moodyet al. [47] developed a real-space approach to lattice rectificationthat was computationally faster and easily applicable to large vol-umes of data. This method involves characterising lattice planeswithin atom probe reconstructions using 1D or z-SDMs aroundpole regions where in-depth resolution is highest. By rotating the

a b

c

Fig. 6. Crystal orientation mapping of a nanocrystalline Al sample using a Hough transformation. (a) is a 2D density map showing a number of poles in three separate grains,(b) is a periodicity intensity map generated by a Hough transformation from the three poles specified by the arrows and (c) a 3D orientation map coloured according to theinverse pole figure provided. The disorientations (minimum misorientations) between the three grains are shown. The rotation unit vectors for the disorientations displayedare 48� about [0.75, �0.64, 0.17], 11.4� about [�0.62, �0.51, �0.59], 24.7� about [0, �0.67, 0.75] and 9.7 � about [�0.17, 0.13, 0.98].

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reconstruction around the x- and y-axis, the signal-to-backgroundratio of the 1D SDMs can be optimised. The transformed z-directionthen corresponds to the normal to the set of lattice planes beingdetected. If the orientation of at least three families of latticeplanes can be known, the triangulation of these sets of latticeplanes can be used to determine the lattice structure. The co-ordi-nates of each atom can then be expressed as the number of latticeplanes away from a reference point for each family of lattice planesdetected. Atoms are then returned to their closest lattice position.Typical results are provided in Fig. 8. The figure is reproduced from[31], and contrasts some of the challenges in achieving 3D atomic-

scale microscopy. The excellent results achieved using aberration-corrected TEM of a monolayer of BN are reproduced here as a mile-stone in microscopy. Yet the challenges to achieve this sort ofresult in 3D, or bulk materials remains difficult to address via scat-tering techniques such as TEM due to convolutions in the scatteredintensity from different chemical species both along and betweenindividual atomic columns. It is in relation to this challenge thatAPM is uniquely powerful. The lattice rectification technique re-veals detail of the atomic distributions and enables the calculationof rigorous partial radial distribution functions and, as will be dis-cussed below, bridges APM with computational materials science.

Computationally, this approach to lattice rectification is veryfast. The limitations to this technique are as follows. Firstly, thetechnique described in [47] only works for Bravais lattices, asnon-Bravais lattices introduce additional plane intersections thatare not lattice points. Also, the technique assumes the data is a per-fect single crystal, and so it does not account for vacancies, disloca-tions, stacking faults or twins and the like. In addition, thetechnique relies upon to manual identification of the poles andthe information contained therein, and automated approachesare an area of current research. Another frontier in this area is tomeasure the accuracy of the technique, via the use of simulateddatasets (see below) with a transformation that emulates the atomprobe noise and detector efficiency. New approaches to lattice rec-tification have recently been developed that address some of theselimitations and will be published in full details soon [72,73].

4. Materials simulation and modelling via APM

Here, we set out some developments and opportunities for APMdata to serve as a quantitative input to materials modelling. Sincethe APM data does contain intrinsic aberrations, recent effort hasbeen devoted to the simulation of atom probe-like data and wesuggest that image simulations must play a key role if APM datais to serve as inputs for materials modelling. The generation of thissimulation data, in-silico, is an important development that enablesthe rigorous setting of analysis parameters, and a quantitative

Fig. 7. Ga atoms at a grain boundary (GB) network in a nanocrystalline Al thin film.The grain boundary network (grey) analysis model has been created by minimisingthe local distance to the centre of mass (DCOM) of the Ga atoms [66]. Only Ga atomscloser than 5 nm to the grain boundaries are shown.

Fig. 8. (a) ADF STEM image of the location of different atoms in a monolayer of BN taken in a Nion UltraSTEM [31]. Part of a density functional theory (DFT) simulation of asingle BN layer containing the experimentally observed substitutional impurities is overlaid on the corresponding part of the experimental image. Red, B; yellow, C; green, N;blue, O. atoms. Courtesy Krivanek et al., 2010, (b) http://www.nature.com/nature/journal/v464/n7288/, (c) HRTEM image of atomic columns in an Al–Mg–Cu structural alloy,(d) corresponding lattice-rectified APT images showing the distribution of the Mg and Cu solute atoms in 3D [47].

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assessment of the impact of certain instrumental constraints, suchas the limitations in detection efficiency, on particular analyses.We then discuss the opportunity for using atom probe data asthe direct input to atomistic modelling, which is another burgeon-ing area of current research. All of these are highly relevant to con-temporary approaches to solid-state materials science research,where real-world observations are critical for the development ofpredictive structure-property relationships.

4.1. Simulations of atom probe data

Our primary focus here is on simulations of the tomographicdata, post-acquisition. However, it is noteworthy that there areseveral important contributions that have sought to simulate theion optics of field evaporation [74–76], some of which incorporatethe effects of a reflectron lens [77]. Indeed, there is renewed inter-est in such ion optical simulations [78] as approaches to accuratelymeasure and parameterise the needle-like shape of the specimenbecome available [79,80].

4.1.1. Random labelling of experimental dataA simple but important computational operation on experimen-

tal atom probe data is to randomise the assignment of species inthe dataset. This preserves the spatial coordinates of the data,but imposes a random chemical distribution. The operation of clus-tering algorithms on these so-called random labelled data is highlyeffective for quantifying errors and for tuning the selection of anal-ysis parameters. Importantly, this technique serves as a rigorousbasis for comparing the solute-solute interactions observed inexperimental data with those that are to be expected in a randomdistribution.

4.1.2. Microstructural simulations: From short-range order toprecipitation

Having established a generalised theory for short-range order(SRO) in multicomponent systems [81], it is possible to describereasonably complex microstructures in terms of numerical orderparameters that assign the relative proximity of the chemical spe-cies with respect to each other. These so-called generalised multi-component short-range parameters (GM-SRO) can either behypothesised so as to generate microstructures ab initio, or theycan be measured from experimental APM data. As a result, we havedevised a Monte Carlo (MC) algorithm [81,82], that is capable ofgenerating a wide range of microstructures that include short-range order, or even precipitation. Once the GM-SRO parametersare established, either from experimental data, or theoretically, asimulation dataset is created with the appropriate crystal structure

and composition, with the chemical species randomized. The algo-rithm then proceeds to swap atoms until they reach the target GM-SRO parameters. Therefore, this MC approach does not use interac-tion energies directly such as has been done previously [83,84], butrather the target GM-SRO parameters.

It is possible to further operate on these simulation microstruc-tures and bring them closer to the nature of experimental APMdata. For example, it is possible to remove a certain fraction ofthe atoms from the data to introduce the detector efficiency (seefor example, [82]). Moreover, it is possible to apply a noise filteron the spatial positions of the atoms to approximate the aberra-tions that occur experimentally when reconstructing the spatialarrangement of the atoms. The value in this is in assessing the ef-fect on analyses of parameter selection [85] and in assessing themissing data from the finite detector efficiency.

4.2. Combining APM data and density functional theory

APM experimental data can be compared with atomistic simu-lations, such as the method based on a Monte Carlo technique[86,87]. The atomic-scale information from APM further offers anopportunity to bridge the gap between experimental microscopyand theoretical modelling at the atomic scale. This represents anew paradigm in the exploration of structure-property relation-ships in materials. In particular, first principles materials simula-tions based on density functional theory (DFT) are intrinsicallycomplementary to APM measurements. This is because APM datacan inform directly the starting geometrical models for DFT calcu-lations, thus enabling the study of technologically important mate-rials. This is significant because, as a self-consistent theory, DFTresults rely heavily on the initial input structures. Indeed, theselection of the starting atomistic configuration for DFT is a majorresearch endeavour in its own right [88,89]. Secondly, DFT calcula-tions do provide valuable insights into the origins, as well as the ef-fects of the atomic distribution detected from APM.Fundamentally, the atomic arrangement, interaction and theresulting physical properties stem from the corresponding elec-tronic structure. Such correlations can be schematically shown inFig. 9.

Recently, we have implemented these concepts in three casestudies. We implemented the traditional method of comparingAPM results with DFT calculations [90]. More innovatively, ourwork utilised APM measurements as direct input into DFT simula-tions [91,92]. In all cases, APM combined with DFT exposed therole of certain elements in phenomena ranging from ferromagne-tism to superconductivity.

Fig. 9. Materials engineering using atom probe microscopy and first principles atomic simulations. (1) Atom probe tomography as the starting point for (2, 3) theoreticalcalculations at the atomic level, gaining the (4) chemical structure, (5) electronic structure, and finally (6) the energetics, atomic interaction, and physical properties.

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5. The future: atom probe microscopy and the cloud

An exciting challenge of modern microscopy is the ability toanalyse and interpret the large amounts of data being collected.In this way, APM will play a key role in materials informatics[93–95]. APM data is highly amenable to this type of treatment gi-ven the digital nature of the data being generated [42,93,95,96]. Animportant opportunity exists to increase the efficiency of experi-mental workflows from specimen preparation through to the anal-ysis and visualisation of the tomographic reconstructions. Withsignificant advancements in computational methods over the pasthalf century, a standardised approach to atom probe analysis maysoon be possible. It is our view that this further ‘mainstreaming’ ofAPM data analysis and interpretation will bring the technique intoline with developments in kindred microscopy and microanalysistechniques, and will be an important development in solid-statematerials science.

So, what are the hallmarks of the future atom probe microscopyexperiment, given the opportunities associated with the rise ofcomputational techniques? Below, we discuss what we considerto be these hallmarks, and describe initial efforts to implement sys-tems capable of managing this workflow.

1. The Cloud: Firstly, we anticipate that cloud-based computingwill be one of the hallmarks of the future APM experiment.The cloud may serve as a repository for data, analysis toolsand workflow management. While an all-encompassingdefinition of cloud computing is still evolving, we think ofthis as simply a shared virtual computer resource thatmay be accessed remotely by a global community of usersto process their data based on a service agreement [97,98].Users can log onto the cloud, upload their data and remotelyperform on the fly data treatment. The results may then beexported back to the user’s local computer for any furtherpost-processing. This captures the advantage that the hard-ware is shared, potentially enabling a much more cost effec-tive, powerful and efficient means of processing data.

2. Community: We believe that coordinated, community-basedefforts to the development of analysis and visualisationtechniques will be a critical factor in advancing this science.Just as occurs in the neuroscience, astronomy and geosci-ence communities, to name just a few examples wherethere are great challenges and opportunities around theexploration of ‘big-data’, so too is there an opportunity forthe materials science APM community to coordinate indi-vidual efforts on an opt-in basis. This could herald an erawhere subscribers might able to access and use a suite ofhighly sophisticated analysis tools created by their peersso as to advance the quality and consistency of atom probeanalyses. This would also serve as a forum for scientific dis-cussion on analysis and a governance process that wouldidentify community-based priorities for development.

3. Open Source: Consistent with the community aspectdescribed above would be the need for an open source likeparadigm, where there is transparency of the computationalprocesses, together with opportunities for ongoing develop-ment in a way that properly protects the moral rights ofdevelopers. We envisage a process whereby registered usersfrom anywhere would be entitled to submit analysis soft-ware for consideration to the portfolio of tools. The softwarewould then be subjected to a validation process, as agreedby the community, before being available for general use.

4. Standards and automation: We envisage that the future ofAPM experimentation, enabled by cloud-based infrastruc-ture will facilitate an easier implementation of standardisedmethods for reconstructing and analysing data. It remains

unsatisfactory that there is no standard way of ranging themass spectrum or calibrating the dimensions of the recon-struction. These processes can, in fact, be highly automatedand the comparison of analysis results between differentresearchers made robust as is the case with many other crit-ical metrology techniques.

5. Collections and digital curation: Our vision of a future APMexperiment incorporates the use of large collections of dataand workflows. The cloud will provide opportunities for thestorage and curation of both experimental and simulationdata that will be widely available for analysis by the com-munity. The prospect of a data wharehouse that has highlycurated datasets from specific materials in specific condi-tions will lead to more standardised, rigorous analysis. Sim-ilarly, this will provide a means to curate the progressivedevelopment of subsequent generations of analysis tools,meaning data recorded at different times can be compared.It may even be possible to extend the workflow to incorpo-rate specimen preparation, such as downloading scripts toperform semi-automated FIB milling and APM instrumentalparameters for optimised data acquisition.

6. Electronic Laboratory Notebooks: It is common that when anew student or researcher joins our laboratory, one of thefirst induction events is to provide them with a new lab-oratory notebook and to explain the rules and expecta-tions around the use of these notebooks. We anticipatethat electronic laboratory notebooks (ELNs) will fulfil asimilar function. We envisage that new users would regis-ter to use the cloud-based APM workbench and be issuedwith access to an ELN. They would upload their own rawdata into a new ‘page’ of their ELN, and use the work-bench to perform a variety of analyses and visualisations.The numerical, graphical and tomographical outputs ofthese calculations would be uploaded into their personalELN in presentation-quality, coupled with a flexible rangeof export functions.

A major project at the University of Sydney1 is focussed on thedevelopment of a workbench for cloud-based microscopy researchthat includes APM. Some of the tools that will realise the potentialof atom probe microscopy in the cloud are listed in the following ref-erences [99–101]. The years ahead will take us closer towards theultimate in microscopy: to determine the position, identity, andchemical nature of every atom in a structure with high precisionand for large volumes [31,42,96]. It is our view that the challengestherein are not only in instrumentation, but also in data processingand analytics.

Acknowledgements

The authors are grateful for the scientific and technical inputand support from staff of the Australian Microscopy and Micro-analysis Research Facility (AMMRF) at the University of Sydney(Sydney Microscopy & Microanalysis). The data in Fig. 6c wasobtained by F. Tang, from a sample that was prepared by D.Gianola at Johns Hopkins University. The sample in Fig. 7 iscourtesy of D. Gianola, now at University of Pennsylvania andK. Hemker, Johns Hopkins University. BG acknowledges that heis a full time employee of Elsevier Ltd. but declares no conflictof interest as his contribution to this article corresponds to hisactivity out of office hours.

1 http://www.nectar.org.au/characterisation-virtual-laboratory (accessed on 4 July2013)

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