Atom probe trajectory mapping using experimental tip shape measurements

Authors


D. Haley. Department of Materials, University of Oxford, Parks Road, Oxford OXI 3PH. Tel: +44 (0) 1865 283658; Fax: +44 (0) 1865 273789; E-mail: daniel.haley@materials.ox.ac.uk

Summary

Atom probe tomography is an accurate analytical and imaging technique which can reconstruct the complex structure and composition of a specimen in three dimensions. Despite providing locally high spatial resolution, atom probe tomography suffers from global distortions due to a complex projection function between the specimen and detector which is different for each experiment and can change during a single run. To aid characterization of this projection function, this work demonstrates a method for the reverse projection of ions from an arbitrary projection surface in 3D space back to an atom probe tomography specimen surface. Experimental data from transmission electron microscopy tilt tomography are combined with point cloud surface reconstruction algorithms and finite element modelling to generate a mapping back to the original tip surface in a physically and experimentally motivated manner. As a case study, aluminium tips are imaged using transmission electron microscopy before and after atom probe tomography, and the specimen profiles used as input in surface reconstruction methods. This reconstruction method is a general procedure that can be used to generate mappings between a selected surface and a known tip shape using numerical solutions to the electrostatic equation, with quantitative solutions to the projection problem readily achievable in tens of minutes on a contemporary workstation.

Introduction

Atom probe tomography (APT) is capable of combining atomic level spatial resolution, chemical identification and 3D tomographic data in a single experiment. Thus, this technique is well poised to answer critical materials science questions on the nanometre scale. It has been successfully employed in fields such as light alloy design (Ringer & Hono, 2000) and semiconductor analyses (Kelly et al., 2007). However, it is well known that atom probe data require careful interpretation by the analyst in order to avoid artefacts from the various complex atomic and ion-optical processes that influence the image projection and ion depth reconstruction. For example, apparent density fluctuations can arise in reconstructed datasets due to local specimen curvature changes throughout a sequence of field desorption events (Marquis et al., 2010). Such changes can occur from preferential evaporation of distinct phases or evolving tip faceting, resulting in local magnification effects or projection distortions (Miller et al., 1996) and creating unresolved challenges for the tomographic reconstruction.

In the past few decades, there has been a considerable research impetus for refining electron tomography (Markoe, 2006) to accurately measure the 3D structure of solid state materials on the nanoscale (Midgley & Weyland, 2003). Furthermore, instrumental advances in APT have enabled greatly increased acquisition rates and hence larger acquired volumes of data are collected in any given experiment. APT researchers have highlighted the potential for the coupling of transmission electron microscopy (TEM) and APT data as a source of improved data quality in the face of these developments (Larson et al., 2006; Arslan et al., 2008; Gorman et al., 2008; Thompson et al., 2009). Although it is relatively common to utilize TEM and APT information sources together in a single investigation in a correlative fashion, there exists no widely accepted method of simultaneously merging the high-quality spatial information obtainable by bright field TEM, with the chemical and atomistic information obtainable from APT in an automatic, quantitative fashion.

Prior to the advent of wide field-of-view atom probe microscopes, small-angle approximations were sufficient to interpret point projection field desorption images, which permitted the assumption of a linear increment for the reconstructed depth coordinate as a function of the ion evaporation sequence. In current generations of larger field-of-view instruments, estimating the specimen tip shape evolution to minimize lattice plane distortions, whilst simultaneously reproducing constituent nanoparticle morphologies in heterogeneous materials, is more challenging.

Conventional atom probe reconstruction methods assume the tip to have a spherical apex, with a conical shank, which maintains this geometry and progressively blunts throughout the evaporation process (Bas et al., 1995). Point projections are used to construct reverse projections to invert the field induced magnification, with the projection centre empirically offset from the sphere origin to compensate for trajectory compression due to the influence of the shank (Geiser et al., 2009). Key limitations of these methods are the necessary assumptions of analytical electrostatic surfaces for which closed or approximate solutions to the ion projection problem exist. Such solutions are often necessarily expressed as cylindrically symmetric objects around some projection axis, e.g. hyperboloidal, spherical or parabolic shapes, thus converting the 3D problem into a simpler 2D problem in cylindrical coordinates. For the sphere-on-cone tip shape analytical models, imaging parameters can be calibrated, yielding inter-planar spacings and crystallographic orientations for single-phase specimens (Gault et al., 2009). For more complex or asymmetrical tip shapes, such reconstruction approaches can still be used; albeit with a best-effort trade-off within the confines of the model. Global deviations from the analytical projection surface are then often visible in reconstructions as a ‘curvature’ of lattice planes, although errors in the estimation of shape evolution can create artificial density and particle morphology modulations, for which some iterative post-correction methods have been suggested (De Gueser et al., 2007).

Numerical methods have previously shown that the stereographic or modified stereographic projection laws for ion trajectories may be insufficient to represent the complex geometries involved in the field evaporation process (Smith et al., 1978; Vurpillot et al., 2001; Niewieczerzal et al., 2010; Oberdorfer & Schmitz, 2011). In principle, if the precise specimen shape could be known throughout an evaporation sequence, the lens action of the tip could be computed and used for reverse projection of the ion trajectories. More detailed shape information can be obtained using electron tomography, although the atom probe measurements must be interrupted to follow the shape evolution, which can cause detrimental degradation of some specimens (Marquis et al., 2010).

Recently a local electron tomography algorithm has been developed to measure specimen morphology, referred to as ‘STOMO’, which is well suited for estimating atom probe tip shapes in three dimensions (Petersen & Ringer, 2009). Hence there exists an opportunity to evaluate the lens action of atom probe specimens using classical electrostatics in three dimensions. In this work, we measure the nanoscale shape of several field-desorbed atom probe tips with STOMO and estimate ion trajectories from the tip surfaces using classical electrostatics computed with finite element modelling (FEM). The coupling of atom probe and electron tomography information can strengthen both techniques and ultimately provide improved information about the specimen (Arslan et al., 2008). Here we have endeavoured to study the effect of specimen shapes on the ion trajectories using such a coupling in a computationally efficient and feasible manner; it is expected such a method will aid construction of advanced reconstruction algorithms to enhance spatial resolution and dataset consistency. Such methods may require moving outside of analytical models with subsequent parameter optimization from user-guided experimental observations, and may instead need to more directly engage with experimental information in a best-effort fashion – indeed this work represents a preliminary effort in this direction.

Concept

Evaporation of an ion following a voltage pulse on a tip can be modelled as an initially stationary charge free particle obtaining all of its electrical potential energy due to a sudden acquisition of charge. If the geometry of the system is known, then the trajectory of the particle can be computed. This is possible due to the result that particles with no initial lateral velocity will follow the same path regardless of any scalar multiple of either their charge or the underlying electric field vector, a result that can be derived from equating acceleration to the electrostatic force (Smith & Walls, 1978). From the perspective of the atom probe microscope, this result is critical – the action of the specimen ‘lens’ in this model, post-evaporation, is independent of charge state and ion mass. Subsequently, given the geometry of the system it should be theoretically possible to compute the trajectories of the ions in a physically realistic manner, by solution of ion trajectories in the electrostatic equation – albeit with assumptions such as a steady-state potential field. These trajectories can be solved using numerical methods for arbitrary geometries.

Thus the remaining step is the generation of an appropriate domain for the numerical solution. TEM tomographic methods have been well explored and are regularly utilized in the generation of 3D scalar density fields from 2D images under angle-independent imaging density assumptions (Frank, 1992; Markoe, 2006). Recently, a surface reconstruction method which utilizes a feature tracking approach to link features between images in a TEM tilt series has been proposed (Petersen & Ringer, 2009), thus estimating their motion vectors and subsequently their position in 3D space. The advantage of this method is that this is more robust under non-attenuative imaging assumptions (such as the Radon transform) which are critical to back projection methods. This generates a 3D point cloud of data, which represents the position of interfaces within a TEM sample.

Such a data form is useful, as a body of research has been performed with the objective of optimizing surface reconstruction from point clouds, as generated by 3D laser range scanning methods and LiDAR data for more than a decade (Leberl et al., 1996). Such experimental methods are regularly utilized in fields such as aerial surveys and digital cultural heritage preservation, and extensive research has been performed in surface extraction from such unstructured datasets.

The extraction of a surface from a point cloud (surface reconstruction) is a complex problem for which multitude solutions exist. There are geometrical methods, such as the cocone/tight-cocone (Dey & Goswami, 2003) ball pivoting (Bernardini et al., 1999) or tangent plane methods, e.g. Singular Value Decomposition minimal eigenvalue (Klasing et al., 2009).

The primary problem is that there are an infinite number of possible surfaces that could be defined by any given point cloud. Any given realization of a specific surface must be based upon some form of reasonable assumptions, such as local smoothness, continuity or false positive rate within the point cloud. For the work here, local tangent plane, ball- and α-shape methods were empirically found to perform poorly, due to spurious outlier points, and radial basis methods produced a ‘cratering’ effect that seemed unreasonable.

One recently proposed method for such is the generation of surfaces using a technique known as ‘Poisson reconstruction’, where the surface is extracted by construction of a potential field that minimizes the Euclidean norm between the vector field observed by the points, and the field generated by application of a gradient operator to some nominated indicator function (Khazdan, et al., 2006). Subsequent mesh extraction is performed by extraction of an implicit contour from the resultant fitted indicator function, such as via a marching cubes algorithm. This method has the advantage of performing a global fit to the constrained point data, whilst simultaneously adapting to the local sampling density.

This Poisson method was found to produce surfaces that retained features of the TEM data, such as consistent specimen ridges, whilst minimizing spurious data points through a smooth interpolation. The cost of the interpolation is a limitation in the minimally resolvable feature size for any given point cloud, based upon interspacing. In the Poisson surface reconstruction method, this is tunable via octree depth, limited by the intrinsic density of the point cloud. As the initial STOMO image scales are purely microscope dependant, this does not affect the implicit imaging capabilities of this method.

By combining the STOMO method with existing surface reconstruction methods, the components required to generate the specimen geometry in a robust fashion are supplied. Subsequently, all that is required is to combine this information with ion trajectory calculations to provide a method for computation of mapping of ion trajectories from the atom probe detector, back on to the tip in 3D space. This can be done by integration of the solution to the electrostatic field as obtained from the specimen geometry.

Materials and methods

Experimental

Two different specimens are used in this work, each being fabricated from 99.99% aluminium wire prepared by standard electro-polishing in 10% perchloric acid/2-butoxyethanol. The first specimen was utilized in a previous work to generate a raw tilt series with angular range ±60° (Petersen & Ringer, 2010), and is re-examined here using a modified version of the STOMO algorithm. The second specimen was separately prepared for this work, using a larger angular range of ±80°.

For the first specimen, two distinct tilt series of 121 over ±60° were collected on a JEOL 3000F (JEOL U.S.A. Inc., Peabody, MA, U.S.A.), operating at 300 kV. Between the two tilt series, an APT experiment was conducted at ∼70 K using voltage pulsing, removing approximately 5 million ions from the tip. During transfer to the atom probe, a brief exposure of the specimen to atmosphere was unavoidable. For both TEM experiments the images were collected on a rectangular Orius SC1000 charge-coupled device camera of size 4008 by 2672 pixels (Gatan Inc., Pleasanton, CA, U.S.A.), with the tip axis running along the longest dimension. Note that in the previous work, only a square subset of the original data was used due to program limitations (Petersen & Ringer, 2010). Here we utilize the entire dataset to improve the quality of the image registration step.

For the second specimen, a single dataset was collected. A 200 kV JEOL 2100 transmission electron microscope was employed to allow for a higher angular tilt range of ±80°. Using field evaporation in the atom probe, 4.8 million ions were initially removed to smooth the tip end form, remove oxide layers and residual electro-polishing contaminants. After exposure to air for ∼24 h, examination in the transmission electron microscope (single image, not shown here) revealed a thin oxide layer of only ∼2.5 nm and a very smooth end form. The tip was then quickly inserted back into the atom probe and a further 6 million field evaporation events were observed. Subsequently the sample was withdrawn and a tilt series of 161 images were collected using a tilt increment of 1° between each image.

All three tilt series were carefully aligned using Fourier transform based phase-correlation algorithms (Klugin & Hines, 1975) as described elsewhere (Petersen & Ringer, 2009). In this work, we also included corrections for small relative rotations between tilt images using an in-house script to implement robust correlation-based (De Castro & Moriandi, 1987) peak finding, which operated on 20 rotations between each image pair in increments of 0.05 ° for the in-plane angle.

Data processing

Tomographic construction

Point clouds were thresholded to remove statistical outliers using the explicit standard errors of every data point, as estimated using singular value decomposition throughout the reconstruction. Conventional back projection suffers from reconstruction artefacts, such as shape distortions, which arise from the limited angular range or ‘missing wedge’ (Midgley & Weyland, 2003). The STOMO algorithm, being a local reconstruction approach, creates no such global distortions. However, for all but ±90° reconstructions, the missing wedge still exists, in its local form, and creates missing regions in the STOMO point cloud. Such gaps would create large systematic errors in 3D FEM calculations. For atom probe tips, with typically smooth cylindrical perimeters, the wedge can be however be interpolated by measuring the slow angular variation of the specimen radius in cross-section to the tip axis. Here we have developed an algorithm to estimate the Taylor series coefficients of the smooth angular variation of each section of the tip up to second order. The algorithm was implemented on all three point clouds with ∼500 cross-sections for each and fitting point data ±15° from the angular limits using singular value decomposition (Press, 1992). For each point cloud, the ∼1500 polynomial coefficients were then used to solve ∼500 quadratic equations, to obtain the angular dependence of the cross-sectional radius, which was used to accurately place points between the angular extremes and thus fill in the missing wedge. Figure 1 shows the registered ±60° before and after point clouds with the missing wedges filled in this manner. The datasets were registered in six dimensions (three simultaneous translations and rotations) using an in-house script which implements a graphical user interface for visual alignment for three mutually orthogonal projections of the overlapping point clouds.

Figure 1.

Point clouds generated by STOMO, as derived from before and after APT run TEM observations (±60° case). Data were subsequently used for construction of finite element domain. Smoothly varying sections due to interpolation by locally fitted polynomials ‘wedge filling’ along tip axis.

To prepare the wedge-filled point clouds for FEM meshing, residual outliers were removed using a nearest-neighbour density threshold (third nearest neighbour); some manual removal was also performed near the tip apex and was achieved using planar clipping. The STOMO software has been modified to compute surface normals using eigenvalues of image gradient operators and surface tangent vector directions. Normal vector fields are required for most automated meshing and or rendering algorithms that operate on point clouds. Despite this, in this work, it was found that the implicit convolution-based vector field estimation in STOMO was far too localized and hence noisy to produce meshes of sufficient quality for accurate FEM computation. Hence the surface normals were replaced with a synthetic and approximate set derived by assuming perfect cylindrical geometry, in conjunction with the estimated spherical tip end forms. Theoretically, such noise estimation issues can be countered using an extension of the wedge-filling approach for every point of each tip to evaluate smooth surface differentials, thus smoothing the normal vector estimation, though such procedures were not implemented here.

Mesh generation

Structured datasets were reconstructed from the point clouds using the Poisson surface reconstruction methods. The Poisson implementation used was that of Kazhdan (Kazhdan et al., 2006) (Version 2, running on a desktop PC, 2.8 GHz x86–64), using the default octree depth of 8 on the TEM-generated input points. The resultant reconstructed surfaces are shown in Figs 2 and 3. These meshes consisted of 129 286, 170 964 and 169 672 triangles for the initial, final ±60° and the ±80° datasets, respectively. For each dataset, only the largest connected mesh component was retained, and to speed tetrahedral removal computations (Jordan curve testing), a quadric mesh approximation method (Garland & Heckbert, 1998) was subsequently applied using the software Meshlab (1.2.2–2) to reduce each mesh to 30 000 triangles. A close up of the ±80° mesh prior to quadric approximation is shown in Fig. 4 (terracing effects normal to tip axis are non-physical).

Figure 2.

Generated tip meshes for before and after ±60° case, and close-ups of subsequently generated FEM domains (after subtraction), with overlaid tip geometries, to show tetrahedral approximation. Left hand images are to scale, right hand are not.

Figure 3.

Poisson reconstructed ±80° mesh (reduced mesh) and full associated finite element domain.

Figure 4.

Close-up of ±80° Poisson reconstructed surface mesh (prior to mesh approximation). Faceting is due to sampling in the acquired TEM image. Small ridges have been preserved, running along the upper side of the sample and onto the face.

To generate the finite element domain, a cylindrical volume was created using Gmsh (Geuzaine & Remacle, 2009), with an aspect ratio of 2: 1 (length: diameter). In order to increase the number of nodes and spatial sampling in the vicinity of the simulated tip, the mesh element size was tapered with a relative element length scale of 0.2:1 from start to end. Subsequently, the reconstructed tip mesh was placed within the cylindrical domain at the element-dense end, aligned along the cylinder axis. The final mesh contained approximately 4.95 million tetrahedral elements, prior to tip tetrahedra removal. To generate an approximation to the triangular tip surface mesh in the tetrahedral domain, partially intersecting or wholly contained tetrahedral elements within the tip mesh were removed. The newly generated boundary, which approximated the tip mesh, was set to a potential of 1 V. The potential fields for the meshes investigated are shown in Fig. 5; despite some near-field distortions due to the tetrahedral subtraction algorithms, the potential field rapidly smooths due to the diffusive nature of the Poisson equation.

Figure 5.

Comparison of the potential fields in analysed tip FEM simulations. Potential isosurfaces for Φ= 0.9 to 1 are shown.

Trajectory computation

The electrostatic solution was computed using the finite element software ElmerFEM (6.1, rev 4668), using the electrostatics module, and a particle tracking module1 was used to insert particles in the computed potential field. The trajectories of the particles were computed using the electrostatic force equation with a charged particle solved with a second-order Runge–Kutta integrator, and the trajectories terminated upon leaving the tetrahedral simulation mesh. Running on a single processor, solution times for the electrostatic equation were typically 2 to 3 min. Roughly 200 trajectories per second were thus computed from the electrostatic potential.

Initially, the particles were launched from random points distributed over the tip mesh, and scaled around the mesh's bounding box centroid by 5%, to avoid numerical concerns with field computation at the bounding surface. These particles were then integrated forwards in time, to compute their projection to the domain exit. Subsequently, a surface was fitted to each velocity component (with respect to the position of the particles). An interpolated velocity vector field was then constructed to initialize a second set of particle trajectories in a region farthest from the tip apex.

Particle positions (near the domain exit) from the forwards integration step were interpolated using a second-order polynomial fit to each velocity component. This interpolation was performed upon a rectilinear grid normal to the Z-axis at 99.5% of the cylinder length placed symmetrically around the cylinder centre axis, to emulate an atom probe detector. The initial energy for the particles was renormalized to that given by the potential applied to the particle's initial position, and the velocity vector was computed using the previous fitted surface.

After resampling along the Z-axis plane at this position, integration with a negative timestep was used to reverse the projection of the ions back to the tip. Providing the particle with insufficient energy will result in the particle being unable to approach the tip, and the integration will result in the particle being reflected from the tip's field; too high an energy will result in the particle being only minimally affected by the tip's electrostatic field.

For this reverse projection step, a grid of ±12%, ±12% of the cylinder diameter with 400 × 400 points was placed for the reverse integration step, and the particle velocities for each grid point were interpolated and used for the seed positions for reverse integration. This ±12% grid was constructed to match the angular range observed in the initial crystallographic desorption image, as measured between the (002) and (115) poles. To visualize this mapping 500k lateral ion coordinates detected in the corresponding atom probe experiments were summed across the tip surfaces in three dimensions to yield greyscale intensities, with lighter shades of grey indicating greater areal densities of ions at a particular point on a tip apex. It should be noted that, like the tip orientation, the lateral positioning of these mappings were also set in an arbitrary manner.

These three mappings are shown in Fig. 6 for the before and after case of the ±60° tip, and in Fig. 7 for the ±80° tip. For a tip shaped as a perfect sphere, this reverse mapping from the flat detector surface to part of the spherical surface would constitute a radially symmetric point projection, and should intersect at a single point, provided that the trajectories are artificially extended inside the body of this spherical ‘tip’. Hence, by extension of the lines connecting the start and the initial points in the simulations here, the trajectory maps can be used to study the lensing action induced by the shape of the atom probe tips. This allows for comparisons to be made with analytical results for image projection, such as the modified stereographic projection functions.

Figure 6.

Possible 3D mappings to tip shape of 2D greyscale desorption map in the ±60° case. Note that the desorption map translation in detector space is uncalibrated.

Figure 7.

Possible 3D mapping onto ±80° tip, front and side views. The detector scaling used is as calibrated from the initial ±60° case.

A simpler alternative approach for describing the mapping of trajectories is to examine directions of simulated ions far from the tip apex. To this end, the manually fitted apex sphere centres were assigned as reference points, using which the final direction of trajectories could be compared against. For every point on the simulated detector plane, the angle of each trajectory was computed using the dot product between Z-axis unit vector and a trajectory vector defined with point of application corresponding to the apex sphere centre and point of termination at the detector surface, as shown in Fig. 8.

Figure 8.

Forward mapping estimations in detector space for constant tip impact to tip-axis angle for before (left) and after (right) tip states in ±60° mesh. Note the change in angular magnification from the differing geometries.

Trajectory mapping

Point projection models of atom probe ion trajectories describe the specimen as a lens with a single projection centre or focal point from which ions appear to have emanated. Having computed ion trajectories as entire maps from the tip to detector surfaces, as described in the previous section, it is also possible to study the gross lensing action of the experimental atom probe tips. One of the most intuitive aspects to study is the effective projection centre implied by the calculated ion trajectory map for each tip. To this end, we have computed best-fit models for point projection via an iterative least-squares method. Although the least-squares construction is not strictly necessary for this work, it provides a compact representation for the problem solution.

To describe the average implied centre of projection, we compute the solution to the problem of the closest point of approach for lines from the initial launch surface to their final position. Subsequent linear extrapolation beyond this final position is used to follow reversed ion trajectories inside each tip. The distribution of ‘closest point of approach’ for a cloud of points locked on to these rays is analogous to computing a minimal ‘disc of confusion’ for a lens, if we were to constrain ourselves to purely point projection methods.

This closest point of approach can be structured as a least-squares problem (Fig. 9), i.e. inline image, where inline image is the mean of inline image overall i, ti is a free parameter (length along line), Qi and Pi are the final and initial positions for the ith ion. The component inline image can be structured as a matrix (Eq. 1), operated upon by the vector inline image with the homogeneous component being inline image.

Figure 9.

Iterative solution to best-fit projection point using electrostatic mapping. Matrix M is constructed from displacement vectors and extended with t to update estimate of Sbest. Vector t is computed at each iteration with least squares, thus updating the estimate of S.

Figure Equation 1: .

Figure Equation 1: .

Least-Squares coefficient matrix for closest point on lines problem.

Solutions were computed for the grid of 400 × 400 (±12% of domain, at Z= 9900) points previously mentioned, randomly sub-sampled to 300 lines to reduce least-squares computational time. The subsequent problem could be solved in tens of seconds (per tip, dependent upon solution tolerances) on the previously described workstation. Smaller numbers of lines were used to approximate the mean t, prior to final solution in order to speed up convergence.

Figure 10 shows the distribution of offset vectors from the projection centre for the solution to the closest point on the line, in tip radius normalized coordinates. The best-fit projection centres were found to lie very close to the tip's Z-axis, at approximately 3.6 tip radii along the Z-axis for the ±80° dataset with 6.3 and 3.3 tip radii for the before and after cases in the ±60° dataset, respectively. Note that the tip radius for the after case is 1.65 times greater than that of the before case, such that if normalized by the final tip radius alone, the Z-axis offset for the ±60° dataset become 3.8 and 3.3, respectively.

Figure 10.

Distribution of offset vectors around fitted centre of projection (repeated least squares on ion detector and tip position) for ±80° tip case. Units are dimensionless (offset/tip radii).

Because both atom probe tips were field desorbed in the atom probe, one final interesting measure of the trajectory maps is to partially reconstruct the positions of experimentally detected ion positions on the tip surface, using only very short desorption sequences, so that the atom depth and apex shape evolution need not be considered. Between the atom probe and transmission electron microscope, the azimuth of the tip orientation is difficult to register so the positions of poles in APT desorption maps cannot be reconciled with the explicit tip shape. For this work, an arbitrary azimuth orientation was hence chosen. In principle, if potential faceting or internal features of the tip could be construed from electron tomograms and compared with the APT data, then the apex orientation about the tip axis could indeed be experimentally assigned. Alternatively, the counting of rings in desorption patterns can be used to estimate local radii of curvature (Drechsler & Wolf, 1958), which could compared with the STOMO surface. However, the experimental and computational prospects of such orientation registration procedures are outside the scope of this work.

Discussion

Figure 8 shows the resultant angular distribution of ion trajectories at the tip as mapped back to the detector for the initial (left) and final (right) tip shapes. The drift in the centre function is due to computational alignment errors. Such errors could be overcome by repeating the tip alignment step; however, this was not required to demonstrate the method outlined here.

Using the ion mapping, each point in the desorption map can be mapped back to the tip's 3D approximation, such as in Figs 5 and 6. However, it should be noted that these mappings are somewhat tentative, because it is not possible to determine the correct detector-to-tip rotation, without additional information.

The desorption mapping from a prospective ‘detector plane’, back to a proposed tip is not a new concept – indeed analytical solutions to the projection equation do this implicitly. The modification that is of interest here, is that it has been demonstrated that tomographic methods providing tip geometry can be coupled with atom probe ion optics using a combination of existing algorithms. The result is sufficient, due to the trajectory invariance on mass, charge and voltage, to provide a single tip surface mapping, in a time scale on the order of tens of minutes, which could be readily utilized on a per-run basis to generate 3D distorted surfaces to aid the analyst in generation of reconstructions.

The method utilized here, whilst successfully generating a possible mapping, could be further refined in several respects; firstly, the use of a direct mesh subtraction step results in a coarse tetrahedral approximation on the tip surface, which could be improved by the use of dedicated mesh generation systems, or more advanced domain removal, rather than by simple tetrahedral subtraction. Additionally, for the hyperboloidal emitters, Smith provides a log–log plot that shows that too short a distance between the tip and the detector can introduce compression distortions in the final result (Smith & Walls, 1978). Interpretation of this suggests that for small angles, 24° launch angles, only >5 tip radii are required to obtain a reasonable approximation to this angular range, which is achieved here. However, for larger angular ranges, this increases dramatically, up to 100 tip radii may be required for a ±45° angular range to be computed. The limitations on manipulable domain sizes, and the ability to generate appropriate domains with FEM techniques places an upper bound on domains that can be handled in a contiguous manner (as opposed to piecewise domains).

Somewhat critically, the algorithm is limited by the scale of the data maintained in the transmission electron microscope tomography – slightly higher than the atomic scale required to visualize the surface modification that are construed by the atomic surface. It is well known that local atomic arrangements in FIM can radically affect the local distortions in the resultant image (Larson & Stiller, 2007; Niewieczerzal et al., 2010), and this also extends to atom probe, with local density distortion an artefact which has previously been the subject of numerical research (Vurpillot et al., 2001). There is an implicit smoothing during the acquisition of the TEM image owing to the function of the instrument; and is one limit to the local roughness that can be estimated from the tip during STOMO reconstruction – assuming a clean in-vacuum movement of the tip from imaging to atom probe analysis. As previously discussed, techniques such as ring counting may prove necessary to overcome this limitation in order to enhance the TEM model.

Thus, currently the point clouds generated by the STOMO reconstruction here do not clearly exhibit atomic-level information; artificial terracing effects visible in the ±80° tip studied here are most visible normal to the tip axis (Fig. 4) and can be ascribed to the sampling in the acquired TEM images. However, the ridges observed, which are physically realistic, show that information is being maintained at the near-atomic level. Regardless, the precise measurement of atomic terracing is currently beyond the resolution of electron tomography. Starting from explicit atom models, simulation studies have shown that the electrostatic effects of terraces can accurately reproduce prominent local features in atom probe datasets such as zone lines (Vurpilot et al., 2001; Niewieczerzal et al., 2010). To incorporate such detail into the tip shape modelling described here, additional computational methods and theoretical models would be required, which are outside the scope of this work.

A further avenue of interest for extending the model would be a method for the interpolation of the observed surfaces between the initial and final tip state acquisitions. Currently STOMO reconstruction was only performed before and after the tip run – the surface during the intervening experiment remains unknown. In order to facilitate utilization of this method for actual atom probe reconstruction, the method would need to be extended to provide a continuous mapping between these two states. Such extensions to this method are currently under investigation.

Despite these limitations, the method is a useful point from which atom probe reconstructions may be revised. The detector to tip mapping may now be performed in a significantly less equivocal manner than before, by utilization of TEM data to augment decision making in the reconstruction process. Although the aluminium system under analysis here is relatively simple in nature, the STOMO method algorithm has been shown to function on more complex specimens, such as TEM data of highly diffracting samples like MgO cubes (Petersen & Ringer, 2009), it is expected that the method should be extensible to arbitrary systems, as it does not rely on the simple evaporation behaviour of aluminium tips.

Interestingly, as the field solution is independent of the scalar potential, if it is assumed that the potential equilibrates instantaneously across the solution domain, then one can simply adjust the potential during the integration procedure to simulate a time-varying potential field in the vicinity of an ion, allowing for the examination of time-varying potential fields without additional computational overhead.

Conclusions

A new method for the combination of data from APT and TEM experiments in a manner that allows for a data-driven approach to atom probe reverse projection has been outlined. This method utilizes ‘Poisson’ surface reconstruction methods commonly utilized in LiDar, computer graphics and art history fields, in conjunction with recent TEM tomographic work which make use of image analysis methodologies to improve edge extraction from TEM images for 3D point cloud construction.

This method was applied to combined APT/TEM experiments to yield high-quality surface approximations, which can then subsequently be utilized in FEM modelling methods to simulate the local tip electrostatic fields. This generates global mappings between the tip space and any other 2D space embedded in the 3D domain, such as a theoretical scaled ‘detector’. This information can be used to augment the decision making process when performing atom probe reconstructions. Further developments in this direction may allow for physically realistic reconstruction constraints, and the ability to combine both TEM and atom probe analyses into a single dataset in a physically realistic manner.

Such developments may provide a cornerstone for the construction of physically motivated reconstruction algorithms, with the objective of diminishing spatial distortions in atom probe reconstruction, particularly if combined with morphological evolution methods.

Footnotes

  • 1

    Particle tracking module developed specifically for this application by the Elmer FEM author, Peter Råback, Computational Centre for Science, Finland.

Ancillary