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Mapping interfacial excess in atom probe data
Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Mapping interfacial excess in atom probe data Peter Felfer a,b,n, Barbara Scherrer b,c, Jelle Demeulemeester d, Wilfried Vandervorst d,e, Julie M. Cairney a,b a
School of Aerospace Mechanical and Mechatronic Engineering, The University of Sydney, Australia Australian Centre for Microscopy and Microanalysis, The University of Sydney, Australia c Eidgenossische Technische Hochschule Zürich, Switzerland d Imec vzw, Kapeldreef 75, Heverlee 3001, Belgium e Instituut voor Kern- en Stralingsfysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium b
art ic l e i nf o
a b s t r a c t
Article history: Received 31 August 2014 Received in revised form 5 May 2015 Accepted 1 June 2015
Using modern wide-angle atom probes, it is possible to acquire atomic scale 3D data containing 1000 s of nm2 of interfaces. It is therefore possible to probe the distribution of segregated species across these interfaces. Here, we present techniques that allow the production of models for interfacial excess (IE) mapping and discuss the underlying considerations and sampling statistics. We also show, how the same principles can be used to achieve thickness mapping of thin films. We demonstrate the effectiveness on example applications, including the analysis of segregation to a phase boundary in stainless steel, segregation to a metal–ceramic interface and the assessment of thickness variations of the gate oxide in a fin-FET. & 2015 Elsevier B.V. All rights reserved.
Keywords: Atom probe tomography Interface Grain boundary Interfacial excess mapping Data analysis
1. Introduction In recent years, atom probe tomography (APT) has proven to be a valuable tool for the analysis of interfaces in materials, thin film structures and micro-electronic devices [1,2]. The field of view of modern atom probes allows the capture of several 1000 s of nm2 of interfacial area within single datasets. This has increased the need for data analysis methods that enable the extraction of quantitative values from interfaces. The analysis of interfaces in APT data was so far largely limited to the analysis of concentration values towards an interface. This was facilitated by either manual cropping of the data to small box or cylinder shaped volumes [3], effectively reducing the analysis to a 1D problem, or the use of iso-surfaces [4]. These objects acted as reference coordinate systems for 1D concentration profiles, cumulative plots for the determination of interfacial excess [5,6] or concentration vs. distance plots (‘proximity histograms’ [7]). These methods are very useful for the analysis of phenomena where the distribution of certain species towards an interface is of interest, such as precipitation in metals [8–10]. However, for interfaces, surfaces and thin films (thicknesso10 nm) the distribution of the n Corresponding author at: School of Aerospace Mechanical and Mechatronic Engineering, The University of Sydney, Australia. E-mail address: [email protected] (P. Felfer).
elements in the plane of the interface or the thin film is of great interest. This demands the mapping of the distribution of the elements or chemical species across the feature, which can achieved by mapping either the concentration of the species, or it is interfacial excess. In most cases, the use of the interfacial excess is preferred, since it is much less sensitive to artefacts such as local magnification [11] and preferential retention [12] and can easily be applied to interfaces and thin films with varying thickness. The interfacial excess Γi of a species i describes the excess number of atoms per unit area that are caused by the presence of an interface [13]. In atom probe data, this is approximated by counting the number of atoms of a certain species in the vicinity of the interface, minus the extrapolated number of atoms that would be present without the segregation contribution of the interface. The interfacial excess was introduced by Gibbs [14], who defined it relative to a surface (Gibbs dividing surface), at which the interfacial excess Γkk of a species k with respect to itself as the reference species is 0 (the subscript denotes the species of interest, the superscript is the reference species). It entails that the excess Γik of species i is in reference to a selected species k and therefore not unique. Guggenheim [15] later showed that the interfacial excess for a given species is unique and therefore a thermodynamic quantity, if the dividing surface is replaced by an interfacial layer that is thick enough to incorporate any volume that is influenced
http://dx.doi.org/10.1016/j.ultramic.2015.06.002 0304-3991/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
P. Felfer et al. / Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
PB
ΓB [at/nm2]
2.3
0.4
z
z x
z y
y
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Fig. 1. IE mapping of B at a phase boundary (marked PB) in a super duplex stainless steel [16] with B atom positions overlayed. While he variation in IE is not obvious from visual inspection of the atomic position data shown in two perpendicular views in (a), the formation of enriched zones becomes apparent when overlaying the IE map (b).
by the presence of the interface. This ambiguity is small in data commonly encountered in APT analysis, where thicknesses of interfaces are often 10 nm and below, with a pronounced maximum. We will therefore use the Gibbs definition throughout. IE mapping provides both a quantification of the distribution of a selected species as well as a method for the detection of variations that are not apparent by simple visual inspection of the data. How large such variations can be is illustrated in Fig. 1. In this figure, we show the distribution of B at a curved phase boundary in nano-crystalline super-duplex stainless steel [16], together with the IE map. Although the variation in the atomic density of the B is significant ( 80%), it is not easily seen by a visual analysis of the atomic distribution (the phase boundary is shown in two perpendicular orientations in Fig. 1a). The IE map (Fig. 1b) however, picks these variations up and corresponds well with the distribution of the atoms close to the grain boundary (in Fig. 1b, only B atoms closer than 5 nm to the interface are displayed). In an earlier paper [17], we introduced a framework, based on computational geometry, to delineate features in atom probe data and then to use these features as a basis to partition the data by using a Voronoi decomposition [18]. This forms the basis of our analysis techniques. In the current paper, we focus on the specific technique of interfacial excess (IE) mapping. We show how the delineation of the dividing surfaces in atom probe data can be optimised and largely automated. This plays a central role in IE mapping, since the analysis statistics are dependent on the triangulated mesh that represents the interface. These analysis models can also be used to facilitate established analysis techniques such as proximity histograms [7]. We then show examples of how IE mapping can be applied to the analysis of various problem sets in interface and thin film analysis.
2. Experimental methods The data from the metal–ceramic interface was acquired using a Cameca LEAP 4000 Si instrument operated in pulsed laser mode at a temperature of 60 K, a pulse energy of 90 pJ and an evaporation rate of 1%. The fin-FET data was acquired on a Cameca LAWATAP in pulsed laser mode. The instrument is equipped with an S-pulse laser from Amplitude Systems, delivering 400 fs laser pulses at 10 kHz (spot size 4100 μm). The APT analysis was conducted using varying pulse energies (0.19, 0.24, 0.31 and 0.40 mJ) at
a wavelength of 515 nm, a base temperature of 80 K and an evaporation rate of 2%.
3. Data treatment 3.1. The principle of IE mapping and surface concentration mapping IE mapping is based on the determination of interfacial excess values in small, elongated volumes along their long axis, as defined by the surface normals of the interface. These volumes are produced if the Voronoi cells of the vertices of a suitable, triangulated mesh are determined. In Fig. 2, the process of IE mapping is shown for an interface between Ni and yttria-doped zirconia (YDZ) in a nanostructured Ni–YDZ anode for a solid oxide fuel cell (SOFC). At the anodes of SOFCs, H is oxidised to form water and the released electrons can be used to power an electrical device. The nanostructured interface was generated via an in operando reverse-current treatment described elsewhere [19]. This is reported to reduce the polarisation resistance of the electrochemical reaction at the anode by 40% at 700 °C compared to the same anode before reverse current treatment. This is attributed to the properties of the interface between metal and ceramic [20]. Atom probe experiments have revealed that O is present in the Ni at the interface between metal and oxide, observed as NiO ions, raising the question of the influence of the O on the properties of the interface. Indeed, O in the metallic Ni phase was almost exclusively observed as NiO, making it possible to separate it from the O in the YDZ, detected as ZrOx. While some aspects of the NiO segregation, such as the large agglomeration of NiO in the top left corner, are immediately apparent in the atom map, smaller variations easily go overlooked. In Fig. 2a, a Voronoi filter [21] was applied to the data for visual clarity. The algorithm calculates the volume of the Voronoi cell of each atom of one or more species that are associated with the feature (Fig. 3 a). Atoms of these species that belong to highdensity regions in the vicinity of the feature are separated from the ‘bulk’ atoms by picking a threshold for the volume of the Voronoi cell of each atom. This threshold can automatically be determined by comparing a histogram of the individual volumes to a histogram of volumes from spatially random data. For a comparison of filtered and unfiltered data see supplementary movie. The IE map (Fig. 2b, included as a *.ply file in the
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
P. Felfer et al. / Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
cA
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z
cB
cA
y x
NiO
0
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20nm nm
Σ C at.
w [nm]
Σ at. (w)
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Fig. 2. Principle of interfacial excess mapping. (a) Segregated NiO after Voronoi filtering (see Supplementary movie). (b) IE map of the NiO (a movie is provided in the online version, along with the 3D data). (c) Analysis mesh and a single Voronoi volume (to see all volumes, see supplementary movie). (d) Voronoi volume and atoms with local w coordinate. (e) the resulting concentration curve and cumulative curve.
Supplementary materials) gives a much more comprehensive view on the distribution of the NiO, showing a dependence of the IE on the local interface properties (interface curvature, convex vs. concave areas). Supplementary material related to this article can be found online at 10.1016/j.ultramic.2015.06.002. After an interface model has been created, which is discussed in the next section of the paper, the first step of the analysis is to divide the analysis volume into subvolumes, based on the analysis mesh. In Fig. 2c, one such volume is displayed (for an overview see the Supplementary movies). This means that each atom is allocated to its nearest analysis vertex. Alternatively, Yao et al. [22] have used cylinders of constant size to determine the local IE, but this has the downside that self-intersections can easily occur and the overall amount of excess atoms will not be calculated correctly due to overlaps and omissions. The interfacial excess counts nIE for each vertex are calculated from the individual, per-vertex volumes that have been transformed into the vertice’s tangent coordinate space (Fig. 2d), as defined by the local surface normal of the vertex. This reduces the problem to a single coordinate axis (“w”). Based on this transformation into 1D, cumulative diagrams (Fig. 2e), previously used in 1D atom probe studies [5,6] or the resulting concentration profile can be used to determine the IE count. In the cumulative diagram, the cumulative amount of a certain atomic species (NiO in the example) is related to the cumulative number of all atoms within this volume along the w axis. In the infinitesimal limit, this represents the integral of the concentration profile across the volume in the w direction, meaning both approaches are equivalent. However, cumulative diagrams tend to display clearer in the case of low atom counts. To derive the Gibbs IE, the far-field concentration on each side of the boundary is approximated using linear regression in the cumulative plot and the IE count of atoms nIE is determined at the location of the interface. In the concentration profile, the IE is obtained by determining the number of atoms of the species of interest within a concentration bin and subtracting the number of atoms that would be present without segregation, based on a linear regression of the bulk concentration (the green area in the plot). Using the concentration profile is advantageous if there is a concentration gradient towards the interface, since a linear fit can be displayed more easily than for a cumulative diagram. However, in many cases the far field concentration (atomic density of the species) can be approximated as constant for the dataset or constant in the vicinity of the boundary. In this case, the
per-vertex excess determination is trivial. All atoms within a specified distance of the interface are counted (the reference concentration and distance can be determined by using a proximity histogram), and the amount of non-excess atoms is calculated based on the bulk concentration and the volume that was used. If such an assumption can be made, it has the additional benefit that the low counting statistics otherwise present on the concave side of highly curved interfaces do not lead to statistical uncertainties in the IE determination. The next step is to compute the area occupied by each vertex, which is the area of the analysis mesh that is within the Voronoi cell of the vertex. This defines a mesh, composed of polygonal patches, commonly referred to as a ‘dual mesh’. It is the geometric dual of the triangulation, much like the Voronoi decomposition is the geometric dual of the Delaunay decomposition. The dial mesh can be calculated by placing vertices on the centre of mass (barycenter) of each mesh triangle, if the mesh triangulation is a Delaunay triangulation [18]. Alternatively, one can count the number of atoms within a given distance to the interface and calculate the area via the material's density [7,17]. This latter approximation is advised for datasets where, due to density fluctuations, a geometric calculation of the interface area leads to errors. Having determined both the interfacial excess count and area per vertex Av, the interfacial excess value Γi of the vertex is obtained by Eq. (1), corrected by the atom probe's detection efficiency η.
Γi =
nie Aυ η
(1)
3.2. Automated mesh generation from open surfaces Since the analysis is based on the position of the mesh vertices within the interface, the mesh generation algorithms are of great importance to IE mapping. The methods we describe here are able to extract surfaces with a vertex distribution suitable for IE mapping from interfaces with segregation, where the atomic positions of the segregated species are used for the model creation. In other cases, where the partitioning of an element allows an interface to be defined by an iso-surface, the marching-cubes algorithm [4] can be used. However, it produces a vertex distribution that is not suitable for IE mapping in its raw form, but the algorithms below can be used to modify the meshes from marching cubes iso-surfaces to produce an appropriate vertex distribution. The initial models can also be created manually with standard 3D
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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4
largest object (Vthresh = 1.78 nm3)
20 nm
5nm voxels, rDCOM = 5nm, 5% energy conv. rdupl = 2.5 nm
projection tessellations
extention to dataset boundary
CVT (final mesh)
Fig. 3. Algorithms for mesh generation. The atoms that are associated with the feature (a) are filtered using a Voronoi filter [21] (b). A voxelisation of the atomic positions is used as initial mesh vertices The vertices were moved onto the feature along estimated normals using DCOM [17] (b) and triangulations are calculated (c). The mesh is then tangentially extended to the dataset boundary (d) and a centroidal Voronoi tessellation is used to achieve an even vertex distribution (e).
software (e.g. Blender [23]) and then be iteratively fitted to the feature using the distance-to-centre-of-mass (DCOM) approach [17]. In this approach, the vertices are iteratively fitted to the feature by moving the vertex along it is normal to the centre of mass of all atoms of interest or other points generated by e.g. a density gradient filter [24], within a specified radius rDCOM. Manual creation or editing is often necessary in the case of interfaces with very high local curvatures and/or areas with low excess. The first step for mesh generation is to find atoms that can be attributed uniquely to the feature to be delineated, by filtering out atoms that belong to the ‘bulk’ of the dataset. For the NiO example, this was done using a Voronoi filter [21] and is shown in Fig. 3a. This data was used as the basis for the mesh generation. An initial approximation of the vertex locations of the mesh can be achieved by picking a sample of the filtered points (Fig. 3b). A better initial mesh approximation is obtained if the points are sampled as uniformly as possible. We have used a simple voxelisation algorithm where a vertex is created in every voxel that contains filtered data. Importantly, this step defines the vertex density/mesh spacing. These tentative vertices are then moved onto the feature using DCOM [17], along their surface normals, which are estimated by a moving least squares fit of a plane to all points of the filtered data that are closer than a radius rMLS to the vertex [25]. From experience, this radius should be similar to rDCOM. This leaves a very smooth distribution of the vertices in the direction normal to the feature upon which a triangulation can be carried out. After the DCOM step, duplicate vertices (vertices closer to each other than a specified distance, usually ½ the desired mesh spacing) are removed. In order to determine the number of iterations of DCOM to be carried out, we are using a displacement energy criterion. The relaxation energy of a DCOM step is defined as the sum of the square of all displacements within a step. DCOM steps are carried out until this displacement energy is smaller than a certain percentage (usually 5%) of the displacement energy of the first DCOM step. For the triangulation, we divided the vertices into groups that are ‘flat’, i.e. for each of which a projection into 2D exists that is non-self-intersecting. For each group of vertices, a simple triangulation method can be used, where the average interface normal is estimated using a least square plane fit to the interface vertices. The vertices are then projected into 2D along this normal. A Delaunay triangulation can easily be performed in 2D. Since the boundary of the mesh is often not convex, we use alpha-shapes [26] in the 2D projection to define the boundary of the mesh.
Alpha shapes are boundaries to point distributions that allow for concavities up to a radius of alpha. An appropriate alpha value is about twice the mesh spacing. For features that are generally relatively flat, one triangulation can be carried out for all vertices. In the present example, where multiple sections of the mesh have been created, these sections need to be manually connected in a mesh editing programme such as Blender. 3.3. Mesh ‘clean-up’ for improved vertex distribution After these steps, the mesh already delineates the feature very well and could be used to facilitate any analysis that is concerned with the distance of the atoms to the feature such as proximity histograms. For any assessment of the in-plane distribution, we need to create an even distribution of the vertices and make sure that the boundary of the mesh extends all the way to the boundary of the dataset (Fig. 3d). The boundary of the dataset can be obtained by calculating the concave hull (alpha hull, [27]) of the dataset, or from co-reconstructed detector outlines (to be published elsewhere). Each vertex that belongs to the boundary of the mesh is then projected tangentially onto the boundary mesh. For the purpose of producing an even distribution of vertices on the mesh we use an adapted version of the centroidal Voronoi tessellation (CVT, [28], Fig. 3e), which is regularly used to create volume meshes for finite element analysis. In the CVT, the centroid of the Voronoi cell of each vertex is calculated and the vertex is then moved to this location, producing a locally even vertex distribution. In order to use this algorithm for 2D manifolds in 3D, local mapping into 2D is required. This was traditionally achieved by mapping the entire object [29] into 2D space, which requires prior knowledge about the topology and therefore involves manual steps. We use a method where we project each vertex and its nearest neighbours into 2D, defined by the tangent space (“null space” [30]) spanned by its vertex normal. In tangent space, we carry out a 2D CVT resulting in the ‘relaxed’ vertex location, the centre of the vertice’s 2D Voronoi cell. The new 2D centre is then transformed back into 3D. This process is repeated until an energetic convergence criterion is reached. The CVT energy is again defined as the sum of the squares of the length of all displacement vectors. We often used 5–10% of the CVT energy of the first CVT step as the limit. Since after the first step, the mesh slightly shifts towards the concave side of the feature, another DCOM step can be carried out after the CVT. For all boundary vertices (part of edges that only
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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belong to one triangle), an equivalent 1D process is employed where each vertex is moved to halfway between its neighbouring vertices, projected onto the original boundary line. We have added the option to ‘pin’ certain vertices at the boundary if the angle between their adjacent line segments is larger than a certain angle (usually 60°). This is used if the mesh is obtained from e.g. a boxshaped volume clipped from larger datasets, leading to feature meshes with close to rectangular corners which otherwise would be rounded out by the algorithm. The combination of all these algorithms leads to the final mesh, which is displayed in Fig. 3e. Although the process may appear complex, remarkably few variables are required to achieve the final result. Firstly, the object must be selected from the objects that are detected by the Voronoi filter (Fig. 3b). Secondly, it is necessary to define the average vertex area to determine how many vertices should be created. This can be adapted later by re-meshing the object. This leaves the MLS and DCOM radii as the main parameters, since the convergence limits for DCOM and CVT do not usually need to be changed. In all cases we have encountered so far, these radii were chosen to be approximately the thickness of the feature.
5
mesh spacing should be chosen so that the number of excess atoms per sample point exceeds these numbers. We can then estimate the required mesh spacing amesh using Eq. (3) from the overall area AI of the interface, the amount of atoms that are close to the interface nat (obtained after Voronoi filtering the original data) and the desired number of excess atoms per vertex nV, by assuming that each vertice's dual mesh cell (see Fig. 2f) can be approximated by a regular hexagon. We also assume a variation factor fvar, by which the IE is expected to vary, which is 2–3 in most cases where no variation is obvious. This estimate disregards the fraction of boundary vertices, which usually have only half the area associated with them, compared to internal vertices (see Fig. 2f). The mesh density obtained by this method may be adjusted later, once the interfacial excess of the entire interface is known, so that nat is the number of excess atoms. After an initial IE map is created, the mesh can also be refined in areas with good counting statistics and made coarser in areas with low excess.
amesh =
nvfvar
2AI 3 3 nat
(3)
3.4. Errors, statistical scattering and spatial binning
d/IE = 0.67/sqrt ( nE )|α= 50%
(2)
If we assume that the relative confidence interval d/IE is less than 20% of the value, we get a minimum excess count of 12. If we assume it is less than 10%, we get a minimum excess count of 45. As a result, in order to keep counting statistics significant, the
4. Results and discussion 4.1. IE mapping of a phase boundary The counting statistics for the IE map presented in Fig. 2, where these princliples have been applied, are shown in Fig. 4 together with the significance limits. Fig. 4a shows the dual mesh of the analysis model, coloured according to the per-vertex area. While most vertices occupy areas larger than 20 nm2, inevitably many of the boundary vertices occupy areas of less than 10 nm2. This is reflected in the counting statistics in Fig. 4b, where the green histogram, representing the interior vertices, shows that all interior vertices have more than 12 excess atoms associated with them, and only 3% of them have an excess of less than 45. The situation is naturally a bit less favourable for the boundary vertices, but still no boundary vertices an excess of less than 12. However 23% of boundary vertices have less than 45 excess atoms associated with them. 4.2. IE mapping for thickness quantification of thin films
12 45
One very common use of atom probe tomography is the investigation of thin films. In the case of films that are only a few nm
regular boundary
60 frequency
For IE mapping, the factors influencing the overall error are the global distortions of the atom probe reconstruction, leading to an error in the determination of the interface area and an error in determining the number of excess atoms associated with each vertex, which is mostly dependent on counting statistics. The error in area determination was discussed previously [31]. It depends on the error in reconstruction parameters, their absolute value and the orientation of the interface with respect to the measurement axis. It is generally lower for interfaces that are roughly perpendicular to the measurement axis. However, in Ref. [31], where the IE values of entire interfaces were calculated, the counting statistics did not play a significant role due to the large number of atoms involved (at least several hundred per value). This needs to be considered for IE mapping. The calculation of an IE map, being a histogram on a 2D surface (manifold), represents an estimator of a density function. This leads to the problem that any histogram depends on the choice of bin width and starting point. This has been the topic of a large body of work (see e.g. [32]), also specific to 3D histograms (voxelisations) of atom probe data [33]. For non-flat surfaces, the situation is slightly different as it is geometrically impossible to have one single bin size. Therefore, the relative uncertainties caused by counting statistics governs the choice of local bin size (area per vertex). Since an appropriate bin size can only be determined if the density function is known a priori, we determine a lower limit for the bin size. This lower limit describes the maximum number of excess atoms per sample point. However, the choice of bin size is always a compromise between smoothing out insignificant bumps and avoiding smoothing that might smear out real variations. If we assume the measured count of excess atoms is a firstorder approximation of the distribution function (IE varies only linearly within one Voronoi cell), Poisson statistics apply and the width of the confidence interval d is given by the χ2 distribution [34]. The confidence interval for a confidence level of 50% then is calculated after Eq. (2).
40 20 0
vertex area [nm2] 5 30
0
500 atoms per vertex
Fig. 4. Counting statistics of IE map creation from the data in Fig. 2. (a) Number of excess atoms per vertex for regular and boundary vertices. (b) Per-vertex area of the analysis model.
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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Si Hf Ti O N
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distance [nm]
z
z
equivalent HfO2 thickness [nm]
y x z y
apparent oxide thickness [nm] 4 5 6
x y
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100n 100nm 50nm Fig. 5. Quantification of the gate oxide thickness variation in a fin-FET. (a) A 4 nm thick slice through the transistor. The atomic distribution of Hf is displayed in perspective in (b) with the corresponding analysis model and dataset hull in (c). (d) Corresponding thickness measurement of gate oxide using atomic positions. (e) Thickness measurement using interfacial excess mapping.
thick and and/or have field evaporation properties that are very different to those of the surrounding material, a quantification of film thickness through direct measurement of the atomic positions in the atom probe reconstruction is unreliable [35,36] and the values obtained depend on the experimental parameters. This is a challenge for data from gate oxides in transistors, which are thin layers of oxides deposited by atomic layer deposition. This problem can be circumvented by making use of the fact that the total amount of atoms per unit area of thin film will be reproduced correctly in the atom probe data (as long as the detector efficiency is accounted for). This value can be used to determine the local thickness of the film. Fig. 5 shows such a quantification of the thickness of a HfO2 gate oxide in a fin-FET as depicted in Fig. 5a. As described above, the first step is to select the target species (Hf in this case, Fig. 5b), from which the 3D model of the fin is created (Fig. 5c, shown together with the data hull). Using this model (average per vertex area is 60 nm2) the IE map displayed in Fig. 5d is calculated. The result shows that the interfacial excess on average is 74.4 at/nm2, with a total variation of 74.4 at/nm2, excluding boundary vertices and the top of the dataset, where the oxide layer is partially outside the field of view. While the raw data from the atom probe only quantifies the amount of Hf per unit area ΓHf, this value can easily be converted into a thickness by using the atomic density of Hf in HfO2, 27.7 at/nm3. This is equivalent to a gate oxide thickness of 2.7 nm (the nominal thickness is 37 1 nm) with a total variation of 70.16 nm, or around 6%. More importantly, this is considerably smaller than one lattice constant of HfO2 ( 0.52 nm [37]) demonstrating the sub-monolayer accuracy of this method. There also is no variation in measured thickness caused by the change of the laser energy (0.19, 0.24, 0.31 and 0.40 mJ) during the experiment. We have compared this result to the determination of the oxide thickness by a direct thickness measurement from the 3D reconstruction as shown in Fig. 5e. For this purpose, we calculated the distance from each Hf atom to the centre of the oxide layer.
This gives a 2D distribution of distance against z. The oxide thickness vs. z was then determined by calculating the standard deviation of this distribution for 10 nm thick slices of the data in the z direction. This thickness measurement, aside from overestimating the oxide thickness due to reconstruction artefacts, varies significantly over the course of the measurement. While close to the top of the fin, a thickness of 4.3 nm is measured, this thickness increases by 30% to 5.6 nm further down the fin. This is also consistent with the observations by Koelling et al. [35].
5. Conclusions Interfacial excess (IE) mapping is a powerful method to investigate the distribution of species of interest within the plane of an interface or surface. It allows quantification of the number of excess atoms per unit area of interface (Gibbsian IE) or the local thickness of a thin film. We have introduced a variety of methods that can be used to delineate the interface. For interfaces characterised by segregation and very thin films (less than 10 nm), an initial mesh can be produced from filtered data, either manually or by creating points at the interface by voxelisation and triangulation. This mesh is then fitted via a distance-to-centre-of-mass (DCOM) approach [17], which is extended to the dataset boundary and given a locally uniform vertex distribution using a centroidal Voronoi tessellation (CVT). By basing the sampling on the mesh used to delineate the interface, a great deal of control over the counting statistics of the analysis is obtained. The mesh density can be adjusted, to ensure that relative uncertainties caused by small samples can be avoided. We have identified the lower limits of per-vertex excess to be 12 for a relative error of 20% and 45 for a relative error of 10%. Following the path chosen by the authors to facilitate the mapping using a Voronoi decomposition of the mesh, the counting statistics of the method can be controlled by adjusting the surface mesh
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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density locally. This optimises the sampling efficiency for interfaces with large variations in IE [38]. An application of the technique to an interface in a ceramic– metal composite electrode was used to demonstrate the effectiveness of IE mapping for both the quantification of the segregation but also the detection of distributions that would go unnoticed purely upon visual inspection of the data. We have also shown how IE mapping is an invaluable tool for the quantification of thickness variations in thin films. Using IE maps, the variation e.g. of the thickness of the gate oxide in fin-FET transistors can be analysed with lattice level accuracy.
Acknowledgements We would like to thank Saritha Samudrala for the dataset of the grain boundary in Fig. 1 and Sachin Shrestha for the dataset of the grain boundary in Figs. 2 and 3. The sample used to acquire the data is Fig. 5 is courtesy of E. Evers – Tiffee (KIT, Karlsruhe). The authors acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy and Microanalysis Research Facility at the University of Sydney.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ultramic.2015.06. 002.
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Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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Mapping interfacial excess in atom probe data Peter Felfer a,b,n, Barbara Scherrer b,c, Jelle Demeulemeester d, Wilfried Vandervorst d,e, Julie M. Cairney a,b a
School of Aerospace Mechanical and Mechatronic Engineering, The University of Sydney, Australia Australian Centre for Microscopy and Microanalysis, The University of Sydney, Australia c Eidgenossische Technische Hochschule Zürich, Switzerland d Imec vzw, Kapeldreef 75, Heverlee 3001, Belgium e Instituut voor Kern- en Stralingsfysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium b
art ic l e i nf o
a b s t r a c t
Article history: Received 31 August 2014 Received in revised form 5 May 2015 Accepted 1 June 2015
Using modern wide-angle atom probes, it is possible to acquire atomic scale 3D data containing 1000 s of nm2 of interfaces. It is therefore possible to probe the distribution of segregated species across these interfaces. Here, we present techniques that allow the production of models for interfacial excess (IE) mapping and discuss the underlying considerations and sampling statistics. We also show, how the same principles can be used to achieve thickness mapping of thin films. We demonstrate the effectiveness on example applications, including the analysis of segregation to a phase boundary in stainless steel, segregation to a metal–ceramic interface and the assessment of thickness variations of the gate oxide in a fin-FET. & 2015 Elsevier B.V. All rights reserved.
Keywords: Atom probe tomography Interface Grain boundary Interfacial excess mapping Data analysis
1. Introduction In recent years, atom probe tomography (APT) has proven to be a valuable tool for the analysis of interfaces in materials, thin film structures and micro-electronic devices [1,2]. The field of view of modern atom probes allows the capture of several 1000 s of nm2 of interfacial area within single datasets. This has increased the need for data analysis methods that enable the extraction of quantitative values from interfaces. The analysis of interfaces in APT data was so far largely limited to the analysis of concentration values towards an interface. This was facilitated by either manual cropping of the data to small box or cylinder shaped volumes [3], effectively reducing the analysis to a 1D problem, or the use of iso-surfaces [4]. These objects acted as reference coordinate systems for 1D concentration profiles, cumulative plots for the determination of interfacial excess [5,6] or concentration vs. distance plots (‘proximity histograms’ [7]). These methods are very useful for the analysis of phenomena where the distribution of certain species towards an interface is of interest, such as precipitation in metals [8–10]. However, for interfaces, surfaces and thin films (thicknesso10 nm) the distribution of the n Corresponding author at: School of Aerospace Mechanical and Mechatronic Engineering, The University of Sydney, Australia. E-mail address: [email protected] (P. Felfer).
elements in the plane of the interface or the thin film is of great interest. This demands the mapping of the distribution of the elements or chemical species across the feature, which can achieved by mapping either the concentration of the species, or it is interfacial excess. In most cases, the use of the interfacial excess is preferred, since it is much less sensitive to artefacts such as local magnification [11] and preferential retention [12] and can easily be applied to interfaces and thin films with varying thickness. The interfacial excess Γi of a species i describes the excess number of atoms per unit area that are caused by the presence of an interface [13]. In atom probe data, this is approximated by counting the number of atoms of a certain species in the vicinity of the interface, minus the extrapolated number of atoms that would be present without the segregation contribution of the interface. The interfacial excess was introduced by Gibbs [14], who defined it relative to a surface (Gibbs dividing surface), at which the interfacial excess Γkk of a species k with respect to itself as the reference species is 0 (the subscript denotes the species of interest, the superscript is the reference species). It entails that the excess Γik of species i is in reference to a selected species k and therefore not unique. Guggenheim [15] later showed that the interfacial excess for a given species is unique and therefore a thermodynamic quantity, if the dividing surface is replaced by an interfacial layer that is thick enough to incorporate any volume that is influenced
http://dx.doi.org/10.1016/j.ultramic.2015.06.002 0304-3991/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
P. Felfer et al. / Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
PB
ΓB [at/nm2]
2.3
0.4
z
z x
z y
y
30nm
Fig. 1. IE mapping of B at a phase boundary (marked PB) in a super duplex stainless steel [16] with B atom positions overlayed. While he variation in IE is not obvious from visual inspection of the atomic position data shown in two perpendicular views in (a), the formation of enriched zones becomes apparent when overlaying the IE map (b).
by the presence of the interface. This ambiguity is small in data commonly encountered in APT analysis, where thicknesses of interfaces are often 10 nm and below, with a pronounced maximum. We will therefore use the Gibbs definition throughout. IE mapping provides both a quantification of the distribution of a selected species as well as a method for the detection of variations that are not apparent by simple visual inspection of the data. How large such variations can be is illustrated in Fig. 1. In this figure, we show the distribution of B at a curved phase boundary in nano-crystalline super-duplex stainless steel [16], together with the IE map. Although the variation in the atomic density of the B is significant ( 80%), it is not easily seen by a visual analysis of the atomic distribution (the phase boundary is shown in two perpendicular orientations in Fig. 1a). The IE map (Fig. 1b) however, picks these variations up and corresponds well with the distribution of the atoms close to the grain boundary (in Fig. 1b, only B atoms closer than 5 nm to the interface are displayed). In an earlier paper [17], we introduced a framework, based on computational geometry, to delineate features in atom probe data and then to use these features as a basis to partition the data by using a Voronoi decomposition [18]. This forms the basis of our analysis techniques. In the current paper, we focus on the specific technique of interfacial excess (IE) mapping. We show how the delineation of the dividing surfaces in atom probe data can be optimised and largely automated. This plays a central role in IE mapping, since the analysis statistics are dependent on the triangulated mesh that represents the interface. These analysis models can also be used to facilitate established analysis techniques such as proximity histograms [7]. We then show examples of how IE mapping can be applied to the analysis of various problem sets in interface and thin film analysis.
2. Experimental methods The data from the metal–ceramic interface was acquired using a Cameca LEAP 4000 Si instrument operated in pulsed laser mode at a temperature of 60 K, a pulse energy of 90 pJ and an evaporation rate of 1%. The fin-FET data was acquired on a Cameca LAWATAP in pulsed laser mode. The instrument is equipped with an S-pulse laser from Amplitude Systems, delivering 400 fs laser pulses at 10 kHz (spot size 4100 μm). The APT analysis was conducted using varying pulse energies (0.19, 0.24, 0.31 and 0.40 mJ) at
a wavelength of 515 nm, a base temperature of 80 K and an evaporation rate of 2%.
3. Data treatment 3.1. The principle of IE mapping and surface concentration mapping IE mapping is based on the determination of interfacial excess values in small, elongated volumes along their long axis, as defined by the surface normals of the interface. These volumes are produced if the Voronoi cells of the vertices of a suitable, triangulated mesh are determined. In Fig. 2, the process of IE mapping is shown for an interface between Ni and yttria-doped zirconia (YDZ) in a nanostructured Ni–YDZ anode for a solid oxide fuel cell (SOFC). At the anodes of SOFCs, H is oxidised to form water and the released electrons can be used to power an electrical device. The nanostructured interface was generated via an in operando reverse-current treatment described elsewhere [19]. This is reported to reduce the polarisation resistance of the electrochemical reaction at the anode by 40% at 700 °C compared to the same anode before reverse current treatment. This is attributed to the properties of the interface between metal and ceramic [20]. Atom probe experiments have revealed that O is present in the Ni at the interface between metal and oxide, observed as NiO ions, raising the question of the influence of the O on the properties of the interface. Indeed, O in the metallic Ni phase was almost exclusively observed as NiO, making it possible to separate it from the O in the YDZ, detected as ZrOx. While some aspects of the NiO segregation, such as the large agglomeration of NiO in the top left corner, are immediately apparent in the atom map, smaller variations easily go overlooked. In Fig. 2a, a Voronoi filter [21] was applied to the data for visual clarity. The algorithm calculates the volume of the Voronoi cell of each atom of one or more species that are associated with the feature (Fig. 3 a). Atoms of these species that belong to highdensity regions in the vicinity of the feature are separated from the ‘bulk’ atoms by picking a threshold for the volume of the Voronoi cell of each atom. This threshold can automatically be determined by comparing a histogram of the individual volumes to a histogram of volumes from spatially random data. For a comparison of filtered and unfiltered data see supplementary movie. The IE map (Fig. 2b, included as a *.ply file in the
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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cA
cB
w
cC [at%]
3
nIE
z
cB
cA
y x
NiO
0
ΓNiO [at/nm2]
20nm nm
Σ C at.
w [nm]
Σ at. (w)
25
Fig. 2. Principle of interfacial excess mapping. (a) Segregated NiO after Voronoi filtering (see Supplementary movie). (b) IE map of the NiO (a movie is provided in the online version, along with the 3D data). (c) Analysis mesh and a single Voronoi volume (to see all volumes, see supplementary movie). (d) Voronoi volume and atoms with local w coordinate. (e) the resulting concentration curve and cumulative curve.
Supplementary materials) gives a much more comprehensive view on the distribution of the NiO, showing a dependence of the IE on the local interface properties (interface curvature, convex vs. concave areas). Supplementary material related to this article can be found online at 10.1016/j.ultramic.2015.06.002. After an interface model has been created, which is discussed in the next section of the paper, the first step of the analysis is to divide the analysis volume into subvolumes, based on the analysis mesh. In Fig. 2c, one such volume is displayed (for an overview see the Supplementary movies). This means that each atom is allocated to its nearest analysis vertex. Alternatively, Yao et al. [22] have used cylinders of constant size to determine the local IE, but this has the downside that self-intersections can easily occur and the overall amount of excess atoms will not be calculated correctly due to overlaps and omissions. The interfacial excess counts nIE for each vertex are calculated from the individual, per-vertex volumes that have been transformed into the vertice’s tangent coordinate space (Fig. 2d), as defined by the local surface normal of the vertex. This reduces the problem to a single coordinate axis (“w”). Based on this transformation into 1D, cumulative diagrams (Fig. 2e), previously used in 1D atom probe studies [5,6] or the resulting concentration profile can be used to determine the IE count. In the cumulative diagram, the cumulative amount of a certain atomic species (NiO in the example) is related to the cumulative number of all atoms within this volume along the w axis. In the infinitesimal limit, this represents the integral of the concentration profile across the volume in the w direction, meaning both approaches are equivalent. However, cumulative diagrams tend to display clearer in the case of low atom counts. To derive the Gibbs IE, the far-field concentration on each side of the boundary is approximated using linear regression in the cumulative plot and the IE count of atoms nIE is determined at the location of the interface. In the concentration profile, the IE is obtained by determining the number of atoms of the species of interest within a concentration bin and subtracting the number of atoms that would be present without segregation, based on a linear regression of the bulk concentration (the green area in the plot). Using the concentration profile is advantageous if there is a concentration gradient towards the interface, since a linear fit can be displayed more easily than for a cumulative diagram. However, in many cases the far field concentration (atomic density of the species) can be approximated as constant for the dataset or constant in the vicinity of the boundary. In this case, the
per-vertex excess determination is trivial. All atoms within a specified distance of the interface are counted (the reference concentration and distance can be determined by using a proximity histogram), and the amount of non-excess atoms is calculated based on the bulk concentration and the volume that was used. If such an assumption can be made, it has the additional benefit that the low counting statistics otherwise present on the concave side of highly curved interfaces do not lead to statistical uncertainties in the IE determination. The next step is to compute the area occupied by each vertex, which is the area of the analysis mesh that is within the Voronoi cell of the vertex. This defines a mesh, composed of polygonal patches, commonly referred to as a ‘dual mesh’. It is the geometric dual of the triangulation, much like the Voronoi decomposition is the geometric dual of the Delaunay decomposition. The dial mesh can be calculated by placing vertices on the centre of mass (barycenter) of each mesh triangle, if the mesh triangulation is a Delaunay triangulation [18]. Alternatively, one can count the number of atoms within a given distance to the interface and calculate the area via the material's density [7,17]. This latter approximation is advised for datasets where, due to density fluctuations, a geometric calculation of the interface area leads to errors. Having determined both the interfacial excess count and area per vertex Av, the interfacial excess value Γi of the vertex is obtained by Eq. (1), corrected by the atom probe's detection efficiency η.
Γi =
nie Aυ η
(1)
3.2. Automated mesh generation from open surfaces Since the analysis is based on the position of the mesh vertices within the interface, the mesh generation algorithms are of great importance to IE mapping. The methods we describe here are able to extract surfaces with a vertex distribution suitable for IE mapping from interfaces with segregation, where the atomic positions of the segregated species are used for the model creation. In other cases, where the partitioning of an element allows an interface to be defined by an iso-surface, the marching-cubes algorithm [4] can be used. However, it produces a vertex distribution that is not suitable for IE mapping in its raw form, but the algorithms below can be used to modify the meshes from marching cubes iso-surfaces to produce an appropriate vertex distribution. The initial models can also be created manually with standard 3D
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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largest object (Vthresh = 1.78 nm3)
20 nm
5nm voxels, rDCOM = 5nm, 5% energy conv. rdupl = 2.5 nm
projection tessellations
extention to dataset boundary
CVT (final mesh)
Fig. 3. Algorithms for mesh generation. The atoms that are associated with the feature (a) are filtered using a Voronoi filter [21] (b). A voxelisation of the atomic positions is used as initial mesh vertices The vertices were moved onto the feature along estimated normals using DCOM [17] (b) and triangulations are calculated (c). The mesh is then tangentially extended to the dataset boundary (d) and a centroidal Voronoi tessellation is used to achieve an even vertex distribution (e).
software (e.g. Blender [23]) and then be iteratively fitted to the feature using the distance-to-centre-of-mass (DCOM) approach [17]. In this approach, the vertices are iteratively fitted to the feature by moving the vertex along it is normal to the centre of mass of all atoms of interest or other points generated by e.g. a density gradient filter [24], within a specified radius rDCOM. Manual creation or editing is often necessary in the case of interfaces with very high local curvatures and/or areas with low excess. The first step for mesh generation is to find atoms that can be attributed uniquely to the feature to be delineated, by filtering out atoms that belong to the ‘bulk’ of the dataset. For the NiO example, this was done using a Voronoi filter [21] and is shown in Fig. 3a. This data was used as the basis for the mesh generation. An initial approximation of the vertex locations of the mesh can be achieved by picking a sample of the filtered points (Fig. 3b). A better initial mesh approximation is obtained if the points are sampled as uniformly as possible. We have used a simple voxelisation algorithm where a vertex is created in every voxel that contains filtered data. Importantly, this step defines the vertex density/mesh spacing. These tentative vertices are then moved onto the feature using DCOM [17], along their surface normals, which are estimated by a moving least squares fit of a plane to all points of the filtered data that are closer than a radius rMLS to the vertex [25]. From experience, this radius should be similar to rDCOM. This leaves a very smooth distribution of the vertices in the direction normal to the feature upon which a triangulation can be carried out. After the DCOM step, duplicate vertices (vertices closer to each other than a specified distance, usually ½ the desired mesh spacing) are removed. In order to determine the number of iterations of DCOM to be carried out, we are using a displacement energy criterion. The relaxation energy of a DCOM step is defined as the sum of the square of all displacements within a step. DCOM steps are carried out until this displacement energy is smaller than a certain percentage (usually 5%) of the displacement energy of the first DCOM step. For the triangulation, we divided the vertices into groups that are ‘flat’, i.e. for each of which a projection into 2D exists that is non-self-intersecting. For each group of vertices, a simple triangulation method can be used, where the average interface normal is estimated using a least square plane fit to the interface vertices. The vertices are then projected into 2D along this normal. A Delaunay triangulation can easily be performed in 2D. Since the boundary of the mesh is often not convex, we use alpha-shapes [26] in the 2D projection to define the boundary of the mesh.
Alpha shapes are boundaries to point distributions that allow for concavities up to a radius of alpha. An appropriate alpha value is about twice the mesh spacing. For features that are generally relatively flat, one triangulation can be carried out for all vertices. In the present example, where multiple sections of the mesh have been created, these sections need to be manually connected in a mesh editing programme such as Blender. 3.3. Mesh ‘clean-up’ for improved vertex distribution After these steps, the mesh already delineates the feature very well and could be used to facilitate any analysis that is concerned with the distance of the atoms to the feature such as proximity histograms. For any assessment of the in-plane distribution, we need to create an even distribution of the vertices and make sure that the boundary of the mesh extends all the way to the boundary of the dataset (Fig. 3d). The boundary of the dataset can be obtained by calculating the concave hull (alpha hull, [27]) of the dataset, or from co-reconstructed detector outlines (to be published elsewhere). Each vertex that belongs to the boundary of the mesh is then projected tangentially onto the boundary mesh. For the purpose of producing an even distribution of vertices on the mesh we use an adapted version of the centroidal Voronoi tessellation (CVT, [28], Fig. 3e), which is regularly used to create volume meshes for finite element analysis. In the CVT, the centroid of the Voronoi cell of each vertex is calculated and the vertex is then moved to this location, producing a locally even vertex distribution. In order to use this algorithm for 2D manifolds in 3D, local mapping into 2D is required. This was traditionally achieved by mapping the entire object [29] into 2D space, which requires prior knowledge about the topology and therefore involves manual steps. We use a method where we project each vertex and its nearest neighbours into 2D, defined by the tangent space (“null space” [30]) spanned by its vertex normal. In tangent space, we carry out a 2D CVT resulting in the ‘relaxed’ vertex location, the centre of the vertice’s 2D Voronoi cell. The new 2D centre is then transformed back into 3D. This process is repeated until an energetic convergence criterion is reached. The CVT energy is again defined as the sum of the squares of the length of all displacement vectors. We often used 5–10% of the CVT energy of the first CVT step as the limit. Since after the first step, the mesh slightly shifts towards the concave side of the feature, another DCOM step can be carried out after the CVT. For all boundary vertices (part of edges that only
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belong to one triangle), an equivalent 1D process is employed where each vertex is moved to halfway between its neighbouring vertices, projected onto the original boundary line. We have added the option to ‘pin’ certain vertices at the boundary if the angle between their adjacent line segments is larger than a certain angle (usually 60°). This is used if the mesh is obtained from e.g. a boxshaped volume clipped from larger datasets, leading to feature meshes with close to rectangular corners which otherwise would be rounded out by the algorithm. The combination of all these algorithms leads to the final mesh, which is displayed in Fig. 3e. Although the process may appear complex, remarkably few variables are required to achieve the final result. Firstly, the object must be selected from the objects that are detected by the Voronoi filter (Fig. 3b). Secondly, it is necessary to define the average vertex area to determine how many vertices should be created. This can be adapted later by re-meshing the object. This leaves the MLS and DCOM radii as the main parameters, since the convergence limits for DCOM and CVT do not usually need to be changed. In all cases we have encountered so far, these radii were chosen to be approximately the thickness of the feature.
5
mesh spacing should be chosen so that the number of excess atoms per sample point exceeds these numbers. We can then estimate the required mesh spacing amesh using Eq. (3) from the overall area AI of the interface, the amount of atoms that are close to the interface nat (obtained after Voronoi filtering the original data) and the desired number of excess atoms per vertex nV, by assuming that each vertice's dual mesh cell (see Fig. 2f) can be approximated by a regular hexagon. We also assume a variation factor fvar, by which the IE is expected to vary, which is 2–3 in most cases where no variation is obvious. This estimate disregards the fraction of boundary vertices, which usually have only half the area associated with them, compared to internal vertices (see Fig. 2f). The mesh density obtained by this method may be adjusted later, once the interfacial excess of the entire interface is known, so that nat is the number of excess atoms. After an initial IE map is created, the mesh can also be refined in areas with good counting statistics and made coarser in areas with low excess.
amesh =
nvfvar
2AI 3 3 nat
(3)
3.4. Errors, statistical scattering and spatial binning
d/IE = 0.67/sqrt ( nE )|α= 50%
(2)
If we assume that the relative confidence interval d/IE is less than 20% of the value, we get a minimum excess count of 12. If we assume it is less than 10%, we get a minimum excess count of 45. As a result, in order to keep counting statistics significant, the
4. Results and discussion 4.1. IE mapping of a phase boundary The counting statistics for the IE map presented in Fig. 2, where these princliples have been applied, are shown in Fig. 4 together with the significance limits. Fig. 4a shows the dual mesh of the analysis model, coloured according to the per-vertex area. While most vertices occupy areas larger than 20 nm2, inevitably many of the boundary vertices occupy areas of less than 10 nm2. This is reflected in the counting statistics in Fig. 4b, where the green histogram, representing the interior vertices, shows that all interior vertices have more than 12 excess atoms associated with them, and only 3% of them have an excess of less than 45. The situation is naturally a bit less favourable for the boundary vertices, but still no boundary vertices an excess of less than 12. However 23% of boundary vertices have less than 45 excess atoms associated with them. 4.2. IE mapping for thickness quantification of thin films
12 45
One very common use of atom probe tomography is the investigation of thin films. In the case of films that are only a few nm
regular boundary
60 frequency
For IE mapping, the factors influencing the overall error are the global distortions of the atom probe reconstruction, leading to an error in the determination of the interface area and an error in determining the number of excess atoms associated with each vertex, which is mostly dependent on counting statistics. The error in area determination was discussed previously [31]. It depends on the error in reconstruction parameters, their absolute value and the orientation of the interface with respect to the measurement axis. It is generally lower for interfaces that are roughly perpendicular to the measurement axis. However, in Ref. [31], where the IE values of entire interfaces were calculated, the counting statistics did not play a significant role due to the large number of atoms involved (at least several hundred per value). This needs to be considered for IE mapping. The calculation of an IE map, being a histogram on a 2D surface (manifold), represents an estimator of a density function. This leads to the problem that any histogram depends on the choice of bin width and starting point. This has been the topic of a large body of work (see e.g. [32]), also specific to 3D histograms (voxelisations) of atom probe data [33]. For non-flat surfaces, the situation is slightly different as it is geometrically impossible to have one single bin size. Therefore, the relative uncertainties caused by counting statistics governs the choice of local bin size (area per vertex). Since an appropriate bin size can only be determined if the density function is known a priori, we determine a lower limit for the bin size. This lower limit describes the maximum number of excess atoms per sample point. However, the choice of bin size is always a compromise between smoothing out insignificant bumps and avoiding smoothing that might smear out real variations. If we assume the measured count of excess atoms is a firstorder approximation of the distribution function (IE varies only linearly within one Voronoi cell), Poisson statistics apply and the width of the confidence interval d is given by the χ2 distribution [34]. The confidence interval for a confidence level of 50% then is calculated after Eq. (2).
40 20 0
vertex area [nm2] 5 30
0
500 atoms per vertex
Fig. 4. Counting statistics of IE map creation from the data in Fig. 2. (a) Number of excess atoms per vertex for regular and boundary vertices. (b) Per-vertex area of the analysis model.
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
P. Felfer et al. / Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Si Hf Ti O N
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distance [nm]
z
z
equivalent HfO2 thickness [nm]
y x z y
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x y
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100n 100nm 50nm Fig. 5. Quantification of the gate oxide thickness variation in a fin-FET. (a) A 4 nm thick slice through the transistor. The atomic distribution of Hf is displayed in perspective in (b) with the corresponding analysis model and dataset hull in (c). (d) Corresponding thickness measurement of gate oxide using atomic positions. (e) Thickness measurement using interfacial excess mapping.
thick and and/or have field evaporation properties that are very different to those of the surrounding material, a quantification of film thickness through direct measurement of the atomic positions in the atom probe reconstruction is unreliable [35,36] and the values obtained depend on the experimental parameters. This is a challenge for data from gate oxides in transistors, which are thin layers of oxides deposited by atomic layer deposition. This problem can be circumvented by making use of the fact that the total amount of atoms per unit area of thin film will be reproduced correctly in the atom probe data (as long as the detector efficiency is accounted for). This value can be used to determine the local thickness of the film. Fig. 5 shows such a quantification of the thickness of a HfO2 gate oxide in a fin-FET as depicted in Fig. 5a. As described above, the first step is to select the target species (Hf in this case, Fig. 5b), from which the 3D model of the fin is created (Fig. 5c, shown together with the data hull). Using this model (average per vertex area is 60 nm2) the IE map displayed in Fig. 5d is calculated. The result shows that the interfacial excess on average is 74.4 at/nm2, with a total variation of 74.4 at/nm2, excluding boundary vertices and the top of the dataset, where the oxide layer is partially outside the field of view. While the raw data from the atom probe only quantifies the amount of Hf per unit area ΓHf, this value can easily be converted into a thickness by using the atomic density of Hf in HfO2, 27.7 at/nm3. This is equivalent to a gate oxide thickness of 2.7 nm (the nominal thickness is 37 1 nm) with a total variation of 70.16 nm, or around 6%. More importantly, this is considerably smaller than one lattice constant of HfO2 ( 0.52 nm [37]) demonstrating the sub-monolayer accuracy of this method. There also is no variation in measured thickness caused by the change of the laser energy (0.19, 0.24, 0.31 and 0.40 mJ) during the experiment. We have compared this result to the determination of the oxide thickness by a direct thickness measurement from the 3D reconstruction as shown in Fig. 5e. For this purpose, we calculated the distance from each Hf atom to the centre of the oxide layer.
This gives a 2D distribution of distance against z. The oxide thickness vs. z was then determined by calculating the standard deviation of this distribution for 10 nm thick slices of the data in the z direction. This thickness measurement, aside from overestimating the oxide thickness due to reconstruction artefacts, varies significantly over the course of the measurement. While close to the top of the fin, a thickness of 4.3 nm is measured, this thickness increases by 30% to 5.6 nm further down the fin. This is also consistent with the observations by Koelling et al. [35].
5. Conclusions Interfacial excess (IE) mapping is a powerful method to investigate the distribution of species of interest within the plane of an interface or surface. It allows quantification of the number of excess atoms per unit area of interface (Gibbsian IE) or the local thickness of a thin film. We have introduced a variety of methods that can be used to delineate the interface. For interfaces characterised by segregation and very thin films (less than 10 nm), an initial mesh can be produced from filtered data, either manually or by creating points at the interface by voxelisation and triangulation. This mesh is then fitted via a distance-to-centre-of-mass (DCOM) approach [17], which is extended to the dataset boundary and given a locally uniform vertex distribution using a centroidal Voronoi tessellation (CVT). By basing the sampling on the mesh used to delineate the interface, a great deal of control over the counting statistics of the analysis is obtained. The mesh density can be adjusted, to ensure that relative uncertainties caused by small samples can be avoided. We have identified the lower limits of per-vertex excess to be 12 for a relative error of 20% and 45 for a relative error of 10%. Following the path chosen by the authors to facilitate the mapping using a Voronoi decomposition of the mesh, the counting statistics of the method can be controlled by adjusting the surface mesh
Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i
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density locally. This optimises the sampling efficiency for interfaces with large variations in IE [38]. An application of the technique to an interface in a ceramic– metal composite electrode was used to demonstrate the effectiveness of IE mapping for both the quantification of the segregation but also the detection of distributions that would go unnoticed purely upon visual inspection of the data. We have also shown how IE mapping is an invaluable tool for the quantification of thickness variations in thin films. Using IE maps, the variation e.g. of the thickness of the gate oxide in fin-FET transistors can be analysed with lattice level accuracy.
Acknowledgements We would like to thank Saritha Samudrala for the dataset of the grain boundary in Fig. 1 and Sachin Shrestha for the dataset of the grain boundary in Figs. 2 and 3. The sample used to acquire the data is Fig. 5 is courtesy of E. Evers – Tiffee (KIT, Karlsruhe). The authors acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy and Microanalysis Research Facility at the University of Sydney.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ultramic.2015.06. 002.
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Please cite this article as: P. Felfer, et al., Mapping interfacial excess in atom probe data, Ultramicroscopy (2015), http://dx.doi.org/ 10.1016/j.ultramic.2015.06.002i