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Crystallographic structural analysis in atom probe microscopy via 3D Hough transformation
Ultramicroscopy 111 (2011) 458–463
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Crystallographic structural analysis in atom probe microscopy via 3D Hough transformation L. Yao a, M.P. Moody a, J.M. Cairney a, Daniel Haley a,b, A.V. Ceguerra a, C. Zhu a, S.P. Ringer a,n a b
Australian Centre for Microscopy and Microanalysis, The University of Sydney, NSW 2006, Australia Department of Materials, University of Oxford, 16 Parks Road, Oxford OX1 3PH, UK
a r t i c l e i n f o
a b s t r a c t
Available online 19 November 2010
Whereas the atom probe is regarded almost exclusively as a technique for 3D chemical microanalysis of solids with the highest chemical and spatial resolution, we demonstrate that the technique can be used for detailed crystallographic determinations. We present a new method for the quantitative determination of crystal structure (plane spacings and angles) using a Hough transformation of the reconstructed atom probe data. The resolving power is shown to be high enough to identify poorly established, discontinuous planes that are typical in semiconducting materials. We demonstrate the determination of crystal geometry around a grain boundary and the use of the technique for the optimisation of tomographic reconstruction. We propose that this method will enable automatic spatial analysis and, ultimately, automated tomographic reconstruction in atom probe microscopy. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: Tomography Atom probe microscopy 3D Hough transformation Tomographic reconstruction
1. Introduction Atom probe microscopy allows the study of materials in three dimensions (3D) with atomic-scale precision [1,2]. The spatial resolution of this 3D technique is anisotropic, being the highest in the depth direction, parallel to the direction of analysis, and very high index crystallographic planes may be observed, routinely [3,4]. Further, this spatial resolution is also specimen specific. While atom probe has long been used to provide tomographic information about the elemental distribution of atoms [5,6], the application of the technique to the determination of crystallographic information is less well developed. Recently, the characterisation of crystallographic structure within atom probe data has found several significant applications, including calibration of the parameters central to the creation of 3D reconstruction [7,8], quantifying the limits and behaviour of spatial resolution [3,9], site specific analyses of atomic distributions in ordered intermetallic materials [10], investigation of grain misorientation in nanocrystalline materials [11] and an extension to the reconstruction analysis restoring atoms to the perfect lattice configuration of the original specimen. Various approaches have been attempted to recover structural information from atom probe data. For example, Fourier transform and related autocorrelation techniques have been developed, though they are restricted because they can operate on limited sub-volumes of near perfect raw data, which is not always available [12–16]. Radial distribution functions (RDFs) have been computed
n
Corresponding author. E-mail address: [email protected] (S.P. Ringer).
from APT data [17] and Haley et al. [18] recently showed a computational approach that enabled RDF analysis to determine atomic-lattice information in a high quality pure Al reconstruction. However, a more informative technique for the analysis of crystallographic spacings is the recently developed spatial distribution maps (SDMs) [10,19,20]. SDMs are an efficient real-space technique that effectively splits an RDF analysis into two parts: a 1D atomic distribution analysis along a specific in-depth crystallographic direction and a corresponding 2D distribution map of atomic positions in the plane lateral to this direction. However, the application of SDM is not always straightforward. The analysis requires identification of regions containing crystallographic planes and subsequently determining the orientation of these planes with respect to the analysis direction. The ostensibly ‘manual’ method outlined in Ref. [20] is guided by the pattern of crystallographic poles often present in field desorption images, but in cases where poles are not readily evident or where multiple crystallographic directions are to be characterised, this approach is operator intensive, time-consuming and hence limiting. In this work, we present a robust new method for structural analysis of atom probe data that utilises 3D Hough transformations. In 2D form, Hough transformations have been widely applied to detect lines and circles in digital imaging [21] and are used in advanced microscopy to index electron backscatter diffraction (EBSD) patterns [22]. The 3D Hough transformation has been extended to extract planes from 3D data clouds, and is increasingly required in new detection technology such as laser sensing [23,24]. An advantage of the Hough transformation is the ability to detect planes in data, despite noise and discontinuities [25]. On the other hand, long computational times and storage requirements can be drawbacks, particularly for real-time applications. For the post-acquisition analysis required in
0304-3991/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.11.018
L. Yao et al. / Ultramicroscopy 111 (2011) 458–463
atom probe microscopy, the 3D Hough transformation can be implemented without prohibitive computing requirements and can automatically resolve structural information, including atomic-plane spacings and orientations, from a selected region of interest (ROI). Further, we simplify the process of interpreting the results of the Hough transform and the extraction of crystallographic information by the subsequent application of a Fast-Fourier transformation (FFT). Our approach of 3D Hough transformation coupled with FFT techniques makes for an effective automation of the analysis. We firstly demonstrate this approach by the determination of crystal structure within a high quality pure Al dataset and show how this can be used to tune the tomographic reconstruction parameters. We further demonstrate the approach by retrieving structural information from a Si dataset, where
Fig. 1. A 3D Hough transformation changes data from (a) point representation space to (b) plane representation space in normal form.
Fig. 2. Field desorption image from atom probe microscopy of pure Al. Dashed lines indicate the positions of the 3 cubic regions of interest (ROI) having dimension 5 nm 5 nm 10 nm. ROI-1, ROI-2 and ROI-3 were centred around the (0 0 2), (1 1¯ 3) and (1 1¯ 5) crystallographic poles, respectively.
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the field evaporation characteristics are not as uniform as Al and so the data possesses inferior spatial resolution and limited crystallographic information is available in the field desorption image. We then show how this technique can be used to determine grain boundary angles in APT datasets, which is particularly useful for the reconstruction and characterisation of polycrystalline and nanocrystalline materials.
2. A 3D Hough transformation method for extracting crystallographic information from atom probe data 3D Hough transformation operates on 3D point cloud datasets, represented by x, y and z values in a Cartesian coordinate system and transforms these to a 3D plane representation system denoted by the angles y and ø and the spacing d. The geometry of the transformation is illustrated in Fig. 1 and the algorithm used is the one described by Tarsha-Kurdi et al. [24]. In principle, it predicts and records all possible planes that can go through each single point from any possible orientation. In atom probe, only planes with a certain range of elevated angle, ø, can exist. For efficiency, we narrowed the prediction of ø between 901 and 451 and scanned y from 01 to 3601. When applied in this way, the Hough transformation algorithm creates a series of histograms that plot the atomic distribution within the ROI along the d-axis, as defined by every possible combination of the angles, y and ø. Effectively, the 3D Cartesian coordinates are transformed into a 3D cube of data, in which each entry is in the form of a histogram. Further, by applying a 1D FFT along d-axis, the periodic features of each distance histogram can be easily extracted. This new approach is firstly demonstrated on the atom probe data from pure Al. Three cubic regions of interest of dimensions 5 nm 5 nm 10 nm, designated as ROI-1, ROI-2 and ROI-3, were selected such that they were centred around the (0 0 2), (1 1¯ 3) and (1 1¯ 5), respectively, in the field desorption image provided in Fig. 2. The results of applying a 3D Hough transformation to ROI-3 are provided in Fig. 3, which shows four of these distance histograms from four different sets of angles, together with their corresponding FFT-frequency spectra. Fig. 3(a)–(c) shows three different periodic distance histograms extruded from ROI-3’s Hough transformation matrix at certain y and ø angles at which three different sets of planes were easily resolved. For comparison, Fig. 3(d) shows a distance histogram at a random orientation where no periodicity was visible. The FFT frequency spectra derived from the periodic distance histograms show clear frequency peaks that correspond to unique d-spacings, Fig. 3(a)–(c), whereas the non-periodic distribution does not exhibit any peak in the frequency domain (Fig. 3(d)).
Fig. 3. Four 1D distance histograms and corresponding FFT frequency spectra at (a) y ¼ 111 and ø¼861, (b) y ¼ 1241 and ø ¼71, (c) y ¼3411and ø¼ 791 and (d) y ¼2691 and ø¼ 751 obtained from a Hough transformation applied to ROI-3 in Fig. 2.
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2.1. Calculation of crystal plane angles An intensity map depicting the periodicity of the atomic distribution in 3D space with respect to crystal plane angles can be constructed using the value of the largest-magnitude peak detected in each FFT frequency spectrum. The angles at which periodic distributions of atoms are detected (i.e.), the orientation of the crystallographic planes, are revealed as higher intensity regions on the map. Fig. 4(a), (b) and (c) shows intensity maps indicating the atomic periodicity detected at all angles within ROI-2, ROI-3 and ROI-1 in Fig. 2, respectively. In both ROI-1 and ROI-2, just one significant peak was observed, representing the presence of the crystallographic plane corresponding to the predominant local poles (0 0 2) and (1 1¯ 3), respectively. However, three different sets of crystal planes were detected in ROI-3. The positions of high intensity peaks indicate the orientation of the planes with respect to the analysis direction in the atom probe experiment. One of these sets of planes corresponds to the predominant local pole, viz. (0 0 2) and (1 1¯ 5). However, the other two sets of crystalline planes revealed in the intensity map of ROI-3 are oriented in the same direction as those observed in ROI-1 and ROI-2. This self-consistency indicates that the atomic spatial distribution corresponding to the lower index pole regions extends much further than the local pole and serves to mask the extent to which the crystalline structure of adjacent high-index poles can be distinguished. In the same way, the intensity of the peaks in the FFT-frequency spectrum is representative of the relative sharpness of the observable planes. For example, ROI-1 and ROI-2 have strong intensity peaks, indicating that these low-index regions have more established crystallographic planes along one direction. On the other hand, the relatively high-index region, ROI-3, in which multiple sets of planes can be simultaneously observed, is more useful in representing the overall original crystal structure. The origin of this effect is due to a combination of the nature of field evaporation in the original experiment and the manner in which the 3D reconstruction is built. This point was discussed recently in detail by Gault et al. [3,4]. What is important here is that this approach appears to be highly robust and can fully quantify the crystalline structure in the atom probe dataset.
major pole with the corresponding experimental ones. This is done for the (1 1 5) pole in Fig. 4(d), where the measured angles are plotted together with the theoretically expected angles on the same map. The excellent fit between the measured and expected results is promising. Nevertheless, some angular error exists in the experimentally determined angles of the crystallographic planes. It is significant that the angle measurement by this new approach is an absolute method for the determination of plane orientations based on each atom’s spatial coordinates in the local ROI. These angles can serve as a more reliable approach for the determination of crystallographic orientation of the tip and hence avoid the ambiguities associated with indexing poles on field desorption maps [7]. After pinpointing the crystalline angles from the color map, the plane spacings can be easily measured by referring back to the 3D cubic result within known angles y and ø and these have been shown in Fig. 5. The measurements are very close to the theoretical inter-planar spacings.
3. Some initial applications of the 3D Hough transformation method 3.1. Crystallographic analyses of materials exhibiting poor field desorption images A good demonstration of the robustness of our 3D Hough transformation method is via the application to atom probe data from pure Si. Indeed, Si is the most important semiconductor material, and its analysis at the atomic scale is of great scientific and technological interest. However, owing to the limited resistivity of
2.2. Measurement of crystal plane spacings To compare the crystal plane spacings obtained from the experimental atom probe data to the known crystallographic lattice structure of the specimen, the Hough transform technique was used. The theoretical angles were obtained by aligning one
Fig. 5. Interspacing measurements of (0 0 2), (1 1¯ 3) and (1 1¯ 5) from each ROI.
Fig. 4. Periodicity intensity map of (a) ROI-2, (b) ROI-1, (c) ROI-3 in Fig. 2 and (d) the detected plane angles compared to the corresponding theoretical predictions.
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Si, its field evaporation is not as uniform as that observed in other, higher resistivity metals under voltage pulsing. These field evaporation characteristics may be regarded as intrinsic or fundamental behaviour in an atom probe experiment and they manifest as rather bland field desorption images that lack clear crystallographic information associated with poles and zone lines, such as in Fig. 6. Hence, the poles that are imaged are more difficult to identify unequivocally and this reduces the ease and confidence by which an accurate tomographic reconstruction can be undertaken using the methods proposed in Ref. [7]. Instead of indexing poles and estimating the angles between crystallographic directions from poor field desorption images such as Fig. 6, it is possible to use the outputs of a 3D Hough transformation to determine the structure and compare it with the theoretical one. Not surprisingly, the detection of planes from the periodicity intensity map is less straightforward in the case of the Si data. The planes existing in this dataset are less sharp, resulting in a much lower signal-to-noise ratio in the frequency domain used to detect periodicity (Fig. 7(a)). We have overcome this problem and improved the detection of planes by narrowing the range of frequencies investigated. Fig. 7(a) shows the relationship between frequency, inter-planar spacings and relative crystallographic index. By limiting the frequencies investigated to a range that covers most of the known theoretical Si d-spacings for lower-index planes, it is possible to achieve a significant improvement in the signal-to-noise ratio. Fig. 7(b) and (c) shows periodicity intensity maps of ROI-1 in Fig. 6. Fig. 7(b) was taken using the whole frequency spectrum, whereas Fig. 7(c) was generated using a more limited range of frequencies. If the expected d-spacings are known, the contrast of the periodicity intensity map can be further sharpened by setting the detection frequency band over just a small range, resulting in the type of map as seen in Fig. 7(d). Using such a narrowly focused frequency range, crystalline planes in all four ROI can be detected, as shown in Fig. 8(a)–(d). The measured plane values are compared to theoretical angles in Fig. 8(e). This example demonstrates the power of the 3D Hough transformation method for characterising blurred and discontinuous plane features, in the same way that a 2D Hough transformation is able to detect dim and discontinuous lines [21] 3.2. Crystal geometry around grain boundaries The chemical composition and crystallographic structure of grain boundaries often control the mechanical properties of materials and so these microstructural characteristics are of great scientific and technological interest. Unfortunately, methods that are effective at measuring one of these characteristics are usually ineffective or limited at measuring the other, and this is also true of atom probe. It is noteworthy that in the case of certain nanocrystalline materials
Fig. 6. A field desorption image from atom probe analysis of a Si microtip specimen. Five cylindrical ROIs of diameter 5 nm and depth 10 nm are indicated.
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Fig. 7. (a) The relationship between frequency, d-spacings and index for Si. Example of (1 1 3), where ROI-3 in Fig. 6 is taken, shows corresponding projections from the curve by a red dotted line. Three different frequency bandwidths were tested and the resulting periodicity intensity maps are shown in (b)–(d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where the average grain size is less than 50 nm, the atom probe provides very accurate 3D information about the grain boundary composition and such information is difficult or perhaps impossible to attain by any other technique. The desire to simultaneously measure the crystal geometry around each grain boundary is more difficult to address. Recently, methods were devised based on the manual identification of grain orientation from lattice plane spacings identified in each crystal in a nanocrystalline 7075 alloy [26]. Our 3D Hough transformation method presented here is a high throughput approach to the analysis of crystal geometry around grain boundaries and can be easily coupled with local compositional information. Fig. 9(a) shows a field desorption image from an atom probe analysis of a microalloyed steel specimen that contains a grain boundary. The grain boundary is delineated by a white dotted line. The two ROI indicated in Fig. 9(a) are each centred on (0 0 1) crystallographic poles identified in each of the respective grains. By applying the 3D Hough transformation method to the data and comparing the angles of the two (0 0 1) planes, the angle between two grains can be determined. Fig. 10(a) and (b) shows the periodicity intensity maps of ROI-1 and ROI-2 in Fig. 9(a), respectively. The different positions of the peaks on the color map quantitatively show the misorientation between the two grains. By taking the coordinate information of the peak, the grain boundary angle can be calculated. To ensure the reconstruction volume is precise to be taken into account, Fig. 10(c) and (d) shows distance histograms from the high-intensity regions in ROI-1 and ROI-2, in which the d-spacings are matched to theoretical ones. This allows us to calculate that the misorientation between the two grains is 5.461, around a reference /0 0 1S direction, using the formalism usually applied to grain boundary texture analyses.
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Fig. 8. Periodicity intensity maps (a)–(d) from each ROI identified in the reference field desorption image provided and (e) the detected plane angles compared to the corresponding theoretical predictions.
Fig. 9. (a) A field desorption image recorded from a microalloyed steel specimen containing a grain boundary. The grain boundary region is between white dotted lines. White lines indicate ROI centred on adjacent (0 0 1) poles for 3D Hough transform analysis and (b) an atom map viewed edge on, revealing C segregation to the grain boundary.
approach is more automated. Indeed, the information obtained from the Hough transform could be seeded into a subsequent SDM analysis to provide complementary insights into the atomic distribution on the lattice. Moreover, we can envisage further automation. Commonly used image processing techniques such as 2D Gaussian filtering and local maximum detection could be used to extract the positions of the major peaks from the FFT periodicity intensity maps and hence discern the orientation of the plane. Once the plane angles, y and ø, are determined, it would be possible to refer back to the 3D data-cube and obtain the 1D distance histogram from that particular angle within the volume. The techniques of 1D FFT and 1D maximum detection could then be used to determine the maximum frequency and derive the plane spacing from its reciprocal, creating a highly automated approach to plane identification. 4.2. Towards an automatic tomographic reconstruction procedure Fig. 10. Periodicity intensity maps of (a) ROI-1 and (b) ROI-2, along with their corresponding distance histograms, (c,d), from the angles where intensity is the highest.
4. Additional outlooks for the 3D Hough transformation analyses 4.1. Towards an automatic analysis of crystal structure Unlike SDM analyses, which can require extensive searching and calibration to identify lattice planes and characterise their orientation within an atom probe dataset, the 3D Hough transform
The atomic planes information described by y, ø and d provide valuable information for the estimation of reconstruction parameters using the methods proposed in Ref. [7]. Assuming that the analysis of crystal structure can be automated via the 3D Hough transformation, as described above, we envisage an algorithm for the automatic indexing of the observed planes. Measured crystal structure could then be compared to the known crystal structure, providing feedback in the selection of reconstruction parameters, and hence an automated 3D reconstruction of APT experiment would be achieved. The subjective manner in which much of the current tomographic reconstruction parameters are selected was
L. Yao et al. / Ultramicroscopy 111 (2011) 458–463
discussed in Ref. [23] and is a concern for the technique. An automated, objective reconstruction approach will improve the efficiency, consistency and accuracy of atom probe microscopy.
5. Conclusion
An efficient method for crystallographic structural analysis
atom probe data, utilising a 3D Hough transformation coupled with a FFT approach has been developed and demonstrated. This method can resolve blurred and discontinuous planes, such as those that are present in atom probe experiments where the field desorption images are poorly resolved. This common situation occurs in (e.g.) the analysis of Si-based materials. Similarly, the method provides a straightforward measurement of the crystal geometry around grain boundaries. The method is amenable to automation and requires less user interaction than corresponding SDM analyses. Notwithstanding this, SDM analysis will benefit from the results of 3D Hough transform techniques and can subsequently be applied to provide a more complete characterisation of the existing lattice structure
Acknowledgement This research was supported by the Australian Research Council and BlueScope Steel Pty. Ltd. (BSL). The authors thank Frank Barbaro, Jim Williams and Chris Kilmore of BSL for useful discussions. Technical and scientific support from the Australian Microscopy & Microanalysis Research Facility (AMMRF-ammrf.org.au) node at the University of Sydney is also gratefully acknowledged, with particular thanks to Baptiste Gault, Gang Sha, Tim Peterson, Alex La Fontaine, Andrew Breen and Kelvin Xie. References [1] M.K. Miller, A. Cerezo, M.G. Hetherington, G.D.W. Smith, Atom Probe Field Ion Microscopy, Oxford University Press, Oxford, 1996. [2] M.K. Miller, Atom Probe Tomography: Analysis at the Atomic Level, Kluwer Academic/Plenum, New York, 2000. [3] B. Gault, M.P. Moody, F.D. Geuser, A.L. Fontaine, L.T. Stephenson, D. Haley, S.P. Ringer, Spatial resolution in atom probe tomography, Microsc. Microanal. 16 (2010) 99–110. [4] B. Gault, M.P. Moody, F.D. Geuser, D. Haley, L.T. Stephenson, S.P. Ringer, Origin of the spatial resolution in atom probe microscopy, Appl. Phys. Lett. 95 (2009) 034103.
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[5] D.N. Seidman, Three-dimensional atom-probe tomography: advances and applications, Annu. Rev. Mater. Res. 37 (2007) 127–158. [6] T.F. Kelly, M.K. Miller, Invited review article: atom probe tomography, Rev. Sci. Instrum. 78 (2007) 031101. [7] B. Gault, F.d. Geuser, L.T. Stephenson, M.P. Moody, B.C. Muddle, S.P. Ringer, Estimation of the reconstruction parameters for atom probe tomography, Microsc. Microanal. 14 (2008) 296–305. [8] B. Gault, M.P. Moody, F.D. Geuser, G. Tsafnat, A.L. Fontaine, L.T. Stephenson, D. Haley, S.P. Ringer, Advances in the calibration of atom probe tomographic reconstruction, J. Appl. Phys. 105 (2009). [9] B. Gault, A. La Fontaine, M.P. Moody, S.P. Ringer, E.A. Marquis, Impact of laser pulsing on the reconstruction in an atom probe tomography, Ultramicroscopy 110 (2010) 1215–1222. [10] T. Boll, T. Al-Kassab, Y. Yuan, Z.G. Liu, Investigation of the site occupation of atoms in pure and doped TiAl/Ti3Al intermetallic, Ultramicroscopy 107 (2007) 796–801. [11] P.V. Liddicoat, X.-Z. Liao, S.P. Ringer, Novel grain boundary solute architecture in a nanostructured ultra-high strength 7075 aluminium alloy, in: Proceedings of the fourth International Light Metals Technology Conference, 2009. [12] P.P. Camus, D.J. Larson, T.F. Kelly, A method for reconstructing and locating atoms on the crystal lattice in three-dimensional atom probe data, Appl. Surf. Sci. 87–88 (1995) 305–310. [13] F. Vurpillot, L. Renaud, D. Blavette, A new step towards the lattice reconstruction in 3DAP, Ultramicroscopy 95 (2003) 223–229. [14] F. Vurpillot, G.D. Costa, A. Menand, D. Blavette, Structural analyses in threedimensional atom probe: a Fourier transform approach, J. Microsc. 203 (2001) 295–302. [15] F. Vurpillot, F.D. Geuser, G.D. Costa, D. Blavette, Application of Fourier transform and autocorrelation to cluster identification in the three-dimensional atom probe, J. Microsc. 216 (2004) 234–240. [16] A. Cerezo, M. Abraham, H. Lane, D.J. Larson, M. Thuvander, K. Seto, P.J. Warren, G.D.W. Smith, Three-dimensional atomic scale analysis of interfaces, Electron Microsc. Anal. (1999) 29–34. [17] C.K. Sudbrack, R.D. Noebe, D.N. Seidman, Direct observations of nucleation in a nondilute multicomponent alloy, Phys. Rev. B 73 (2006) 212101. [18] D. Haley, T. Petersen, G. Barton, S.P. Ringer, Influence of field evaporation on radial distribution functions in atom probe tomography, Philos. Mag. 89 (2009) 925–943. [19] B.P. Geiser, T.F. Kelly, D.J. Larson, J. Schneir, J.P. Roberts, Spatial distribution maps for atom probe tomography, Microsc. Microanal. 13 (2007) 437–447. [20] M.P. Moody, B. Gault, L.T. Stephenson, D. Haley, S.P. Ringer, Qualification of the tomographic reconstruction in atom probe by advanced spatial distribution map techniques, Ultramicroscopy 109 (2009) 815–824. [21] P.V.C. Hough, Method and means for recognizing complex patterns, US Patent 3.069.654, 1962. [22] N.C.K. Lassen, D.J. Jensen, K. Conradsen, Image processing procedures for analysis of electron back scattering patterns, Scan. Microsc. 6 (1992). [23] G. Vosselman, S. Dijkman, 3D building model reconstruction from point clouds and ground plans, in: Proceedings of the International Archives of the Photogrammetry Sciences, 34, 2001, pp. 37–44. [24] F. Tarsha-Kurdi, T. Landes, P. Grussenmeyer, Hough-transform and extended RANSAC algorithms for automatic detection of 3D building roof planes from Lidar data, in: ISPRS proceedings, vol. XXXVI Part 3, 2007, p. W52. [25] F. Karsli, O. Kahya, Building extraction from laser scanning data, in: Proceedings of the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. XXXVII, Beijing, Part B3b, 2008. [26] P.V. Liddicoat, X.Z. Liao, Y.H. Zhao, Y. Zhu, M.Y. Murashkin, E.J. Lavernia, R.Z. Valiev, S.P. Ringer, Nat. Commun., accepted for publication.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Crystallographic structural analysis in atom probe microscopy via 3D Hough transformation L. Yao a, M.P. Moody a, J.M. Cairney a, Daniel Haley a,b, A.V. Ceguerra a, C. Zhu a, S.P. Ringer a,n a b
Australian Centre for Microscopy and Microanalysis, The University of Sydney, NSW 2006, Australia Department of Materials, University of Oxford, 16 Parks Road, Oxford OX1 3PH, UK
a r t i c l e i n f o
a b s t r a c t
Available online 19 November 2010
Whereas the atom probe is regarded almost exclusively as a technique for 3D chemical microanalysis of solids with the highest chemical and spatial resolution, we demonstrate that the technique can be used for detailed crystallographic determinations. We present a new method for the quantitative determination of crystal structure (plane spacings and angles) using a Hough transformation of the reconstructed atom probe data. The resolving power is shown to be high enough to identify poorly established, discontinuous planes that are typical in semiconducting materials. We demonstrate the determination of crystal geometry around a grain boundary and the use of the technique for the optimisation of tomographic reconstruction. We propose that this method will enable automatic spatial analysis and, ultimately, automated tomographic reconstruction in atom probe microscopy. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: Tomography Atom probe microscopy 3D Hough transformation Tomographic reconstruction
1. Introduction Atom probe microscopy allows the study of materials in three dimensions (3D) with atomic-scale precision [1,2]. The spatial resolution of this 3D technique is anisotropic, being the highest in the depth direction, parallel to the direction of analysis, and very high index crystallographic planes may be observed, routinely [3,4]. Further, this spatial resolution is also specimen specific. While atom probe has long been used to provide tomographic information about the elemental distribution of atoms [5,6], the application of the technique to the determination of crystallographic information is less well developed. Recently, the characterisation of crystallographic structure within atom probe data has found several significant applications, including calibration of the parameters central to the creation of 3D reconstruction [7,8], quantifying the limits and behaviour of spatial resolution [3,9], site specific analyses of atomic distributions in ordered intermetallic materials [10], investigation of grain misorientation in nanocrystalline materials [11] and an extension to the reconstruction analysis restoring atoms to the perfect lattice configuration of the original specimen. Various approaches have been attempted to recover structural information from atom probe data. For example, Fourier transform and related autocorrelation techniques have been developed, though they are restricted because they can operate on limited sub-volumes of near perfect raw data, which is not always available [12–16]. Radial distribution functions (RDFs) have been computed
n
Corresponding author. E-mail address: [email protected] (S.P. Ringer).
from APT data [17] and Haley et al. [18] recently showed a computational approach that enabled RDF analysis to determine atomic-lattice information in a high quality pure Al reconstruction. However, a more informative technique for the analysis of crystallographic spacings is the recently developed spatial distribution maps (SDMs) [10,19,20]. SDMs are an efficient real-space technique that effectively splits an RDF analysis into two parts: a 1D atomic distribution analysis along a specific in-depth crystallographic direction and a corresponding 2D distribution map of atomic positions in the plane lateral to this direction. However, the application of SDM is not always straightforward. The analysis requires identification of regions containing crystallographic planes and subsequently determining the orientation of these planes with respect to the analysis direction. The ostensibly ‘manual’ method outlined in Ref. [20] is guided by the pattern of crystallographic poles often present in field desorption images, but in cases where poles are not readily evident or where multiple crystallographic directions are to be characterised, this approach is operator intensive, time-consuming and hence limiting. In this work, we present a robust new method for structural analysis of atom probe data that utilises 3D Hough transformations. In 2D form, Hough transformations have been widely applied to detect lines and circles in digital imaging [21] and are used in advanced microscopy to index electron backscatter diffraction (EBSD) patterns [22]. The 3D Hough transformation has been extended to extract planes from 3D data clouds, and is increasingly required in new detection technology such as laser sensing [23,24]. An advantage of the Hough transformation is the ability to detect planes in data, despite noise and discontinuities [25]. On the other hand, long computational times and storage requirements can be drawbacks, particularly for real-time applications. For the post-acquisition analysis required in
0304-3991/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.11.018
L. Yao et al. / Ultramicroscopy 111 (2011) 458–463
atom probe microscopy, the 3D Hough transformation can be implemented without prohibitive computing requirements and can automatically resolve structural information, including atomic-plane spacings and orientations, from a selected region of interest (ROI). Further, we simplify the process of interpreting the results of the Hough transform and the extraction of crystallographic information by the subsequent application of a Fast-Fourier transformation (FFT). Our approach of 3D Hough transformation coupled with FFT techniques makes for an effective automation of the analysis. We firstly demonstrate this approach by the determination of crystal structure within a high quality pure Al dataset and show how this can be used to tune the tomographic reconstruction parameters. We further demonstrate the approach by retrieving structural information from a Si dataset, where
Fig. 1. A 3D Hough transformation changes data from (a) point representation space to (b) plane representation space in normal form.
Fig. 2. Field desorption image from atom probe microscopy of pure Al. Dashed lines indicate the positions of the 3 cubic regions of interest (ROI) having dimension 5 nm 5 nm 10 nm. ROI-1, ROI-2 and ROI-3 were centred around the (0 0 2), (1 1¯ 3) and (1 1¯ 5) crystallographic poles, respectively.
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the field evaporation characteristics are not as uniform as Al and so the data possesses inferior spatial resolution and limited crystallographic information is available in the field desorption image. We then show how this technique can be used to determine grain boundary angles in APT datasets, which is particularly useful for the reconstruction and characterisation of polycrystalline and nanocrystalline materials.
2. A 3D Hough transformation method for extracting crystallographic information from atom probe data 3D Hough transformation operates on 3D point cloud datasets, represented by x, y and z values in a Cartesian coordinate system and transforms these to a 3D plane representation system denoted by the angles y and ø and the spacing d. The geometry of the transformation is illustrated in Fig. 1 and the algorithm used is the one described by Tarsha-Kurdi et al. [24]. In principle, it predicts and records all possible planes that can go through each single point from any possible orientation. In atom probe, only planes with a certain range of elevated angle, ø, can exist. For efficiency, we narrowed the prediction of ø between 901 and 451 and scanned y from 01 to 3601. When applied in this way, the Hough transformation algorithm creates a series of histograms that plot the atomic distribution within the ROI along the d-axis, as defined by every possible combination of the angles, y and ø. Effectively, the 3D Cartesian coordinates are transformed into a 3D cube of data, in which each entry is in the form of a histogram. Further, by applying a 1D FFT along d-axis, the periodic features of each distance histogram can be easily extracted. This new approach is firstly demonstrated on the atom probe data from pure Al. Three cubic regions of interest of dimensions 5 nm 5 nm 10 nm, designated as ROI-1, ROI-2 and ROI-3, were selected such that they were centred around the (0 0 2), (1 1¯ 3) and (1 1¯ 5), respectively, in the field desorption image provided in Fig. 2. The results of applying a 3D Hough transformation to ROI-3 are provided in Fig. 3, which shows four of these distance histograms from four different sets of angles, together with their corresponding FFT-frequency spectra. Fig. 3(a)–(c) shows three different periodic distance histograms extruded from ROI-3’s Hough transformation matrix at certain y and ø angles at which three different sets of planes were easily resolved. For comparison, Fig. 3(d) shows a distance histogram at a random orientation where no periodicity was visible. The FFT frequency spectra derived from the periodic distance histograms show clear frequency peaks that correspond to unique d-spacings, Fig. 3(a)–(c), whereas the non-periodic distribution does not exhibit any peak in the frequency domain (Fig. 3(d)).
Fig. 3. Four 1D distance histograms and corresponding FFT frequency spectra at (a) y ¼ 111 and ø¼861, (b) y ¼ 1241 and ø ¼71, (c) y ¼3411and ø¼ 791 and (d) y ¼2691 and ø¼ 751 obtained from a Hough transformation applied to ROI-3 in Fig. 2.
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2.1. Calculation of crystal plane angles An intensity map depicting the periodicity of the atomic distribution in 3D space with respect to crystal plane angles can be constructed using the value of the largest-magnitude peak detected in each FFT frequency spectrum. The angles at which periodic distributions of atoms are detected (i.e.), the orientation of the crystallographic planes, are revealed as higher intensity regions on the map. Fig. 4(a), (b) and (c) shows intensity maps indicating the atomic periodicity detected at all angles within ROI-2, ROI-3 and ROI-1 in Fig. 2, respectively. In both ROI-1 and ROI-2, just one significant peak was observed, representing the presence of the crystallographic plane corresponding to the predominant local poles (0 0 2) and (1 1¯ 3), respectively. However, three different sets of crystal planes were detected in ROI-3. The positions of high intensity peaks indicate the orientation of the planes with respect to the analysis direction in the atom probe experiment. One of these sets of planes corresponds to the predominant local pole, viz. (0 0 2) and (1 1¯ 5). However, the other two sets of crystalline planes revealed in the intensity map of ROI-3 are oriented in the same direction as those observed in ROI-1 and ROI-2. This self-consistency indicates that the atomic spatial distribution corresponding to the lower index pole regions extends much further than the local pole and serves to mask the extent to which the crystalline structure of adjacent high-index poles can be distinguished. In the same way, the intensity of the peaks in the FFT-frequency spectrum is representative of the relative sharpness of the observable planes. For example, ROI-1 and ROI-2 have strong intensity peaks, indicating that these low-index regions have more established crystallographic planes along one direction. On the other hand, the relatively high-index region, ROI-3, in which multiple sets of planes can be simultaneously observed, is more useful in representing the overall original crystal structure. The origin of this effect is due to a combination of the nature of field evaporation in the original experiment and the manner in which the 3D reconstruction is built. This point was discussed recently in detail by Gault et al. [3,4]. What is important here is that this approach appears to be highly robust and can fully quantify the crystalline structure in the atom probe dataset.
major pole with the corresponding experimental ones. This is done for the (1 1 5) pole in Fig. 4(d), where the measured angles are plotted together with the theoretically expected angles on the same map. The excellent fit between the measured and expected results is promising. Nevertheless, some angular error exists in the experimentally determined angles of the crystallographic planes. It is significant that the angle measurement by this new approach is an absolute method for the determination of plane orientations based on each atom’s spatial coordinates in the local ROI. These angles can serve as a more reliable approach for the determination of crystallographic orientation of the tip and hence avoid the ambiguities associated with indexing poles on field desorption maps [7]. After pinpointing the crystalline angles from the color map, the plane spacings can be easily measured by referring back to the 3D cubic result within known angles y and ø and these have been shown in Fig. 5. The measurements are very close to the theoretical inter-planar spacings.
3. Some initial applications of the 3D Hough transformation method 3.1. Crystallographic analyses of materials exhibiting poor field desorption images A good demonstration of the robustness of our 3D Hough transformation method is via the application to atom probe data from pure Si. Indeed, Si is the most important semiconductor material, and its analysis at the atomic scale is of great scientific and technological interest. However, owing to the limited resistivity of
2.2. Measurement of crystal plane spacings To compare the crystal plane spacings obtained from the experimental atom probe data to the known crystallographic lattice structure of the specimen, the Hough transform technique was used. The theoretical angles were obtained by aligning one
Fig. 5. Interspacing measurements of (0 0 2), (1 1¯ 3) and (1 1¯ 5) from each ROI.
Fig. 4. Periodicity intensity map of (a) ROI-2, (b) ROI-1, (c) ROI-3 in Fig. 2 and (d) the detected plane angles compared to the corresponding theoretical predictions.
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Si, its field evaporation is not as uniform as that observed in other, higher resistivity metals under voltage pulsing. These field evaporation characteristics may be regarded as intrinsic or fundamental behaviour in an atom probe experiment and they manifest as rather bland field desorption images that lack clear crystallographic information associated with poles and zone lines, such as in Fig. 6. Hence, the poles that are imaged are more difficult to identify unequivocally and this reduces the ease and confidence by which an accurate tomographic reconstruction can be undertaken using the methods proposed in Ref. [7]. Instead of indexing poles and estimating the angles between crystallographic directions from poor field desorption images such as Fig. 6, it is possible to use the outputs of a 3D Hough transformation to determine the structure and compare it with the theoretical one. Not surprisingly, the detection of planes from the periodicity intensity map is less straightforward in the case of the Si data. The planes existing in this dataset are less sharp, resulting in a much lower signal-to-noise ratio in the frequency domain used to detect periodicity (Fig. 7(a)). We have overcome this problem and improved the detection of planes by narrowing the range of frequencies investigated. Fig. 7(a) shows the relationship between frequency, inter-planar spacings and relative crystallographic index. By limiting the frequencies investigated to a range that covers most of the known theoretical Si d-spacings for lower-index planes, it is possible to achieve a significant improvement in the signal-to-noise ratio. Fig. 7(b) and (c) shows periodicity intensity maps of ROI-1 in Fig. 6. Fig. 7(b) was taken using the whole frequency spectrum, whereas Fig. 7(c) was generated using a more limited range of frequencies. If the expected d-spacings are known, the contrast of the periodicity intensity map can be further sharpened by setting the detection frequency band over just a small range, resulting in the type of map as seen in Fig. 7(d). Using such a narrowly focused frequency range, crystalline planes in all four ROI can be detected, as shown in Fig. 8(a)–(d). The measured plane values are compared to theoretical angles in Fig. 8(e). This example demonstrates the power of the 3D Hough transformation method for characterising blurred and discontinuous plane features, in the same way that a 2D Hough transformation is able to detect dim and discontinuous lines [21] 3.2. Crystal geometry around grain boundaries The chemical composition and crystallographic structure of grain boundaries often control the mechanical properties of materials and so these microstructural characteristics are of great scientific and technological interest. Unfortunately, methods that are effective at measuring one of these characteristics are usually ineffective or limited at measuring the other, and this is also true of atom probe. It is noteworthy that in the case of certain nanocrystalline materials
Fig. 6. A field desorption image from atom probe analysis of a Si microtip specimen. Five cylindrical ROIs of diameter 5 nm and depth 10 nm are indicated.
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Fig. 7. (a) The relationship between frequency, d-spacings and index for Si. Example of (1 1 3), where ROI-3 in Fig. 6 is taken, shows corresponding projections from the curve by a red dotted line. Three different frequency bandwidths were tested and the resulting periodicity intensity maps are shown in (b)–(d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where the average grain size is less than 50 nm, the atom probe provides very accurate 3D information about the grain boundary composition and such information is difficult or perhaps impossible to attain by any other technique. The desire to simultaneously measure the crystal geometry around each grain boundary is more difficult to address. Recently, methods were devised based on the manual identification of grain orientation from lattice plane spacings identified in each crystal in a nanocrystalline 7075 alloy [26]. Our 3D Hough transformation method presented here is a high throughput approach to the analysis of crystal geometry around grain boundaries and can be easily coupled with local compositional information. Fig. 9(a) shows a field desorption image from an atom probe analysis of a microalloyed steel specimen that contains a grain boundary. The grain boundary is delineated by a white dotted line. The two ROI indicated in Fig. 9(a) are each centred on (0 0 1) crystallographic poles identified in each of the respective grains. By applying the 3D Hough transformation method to the data and comparing the angles of the two (0 0 1) planes, the angle between two grains can be determined. Fig. 10(a) and (b) shows the periodicity intensity maps of ROI-1 and ROI-2 in Fig. 9(a), respectively. The different positions of the peaks on the color map quantitatively show the misorientation between the two grains. By taking the coordinate information of the peak, the grain boundary angle can be calculated. To ensure the reconstruction volume is precise to be taken into account, Fig. 10(c) and (d) shows distance histograms from the high-intensity regions in ROI-1 and ROI-2, in which the d-spacings are matched to theoretical ones. This allows us to calculate that the misorientation between the two grains is 5.461, around a reference /0 0 1S direction, using the formalism usually applied to grain boundary texture analyses.
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Fig. 8. Periodicity intensity maps (a)–(d) from each ROI identified in the reference field desorption image provided and (e) the detected plane angles compared to the corresponding theoretical predictions.
Fig. 9. (a) A field desorption image recorded from a microalloyed steel specimen containing a grain boundary. The grain boundary region is between white dotted lines. White lines indicate ROI centred on adjacent (0 0 1) poles for 3D Hough transform analysis and (b) an atom map viewed edge on, revealing C segregation to the grain boundary.
approach is more automated. Indeed, the information obtained from the Hough transform could be seeded into a subsequent SDM analysis to provide complementary insights into the atomic distribution on the lattice. Moreover, we can envisage further automation. Commonly used image processing techniques such as 2D Gaussian filtering and local maximum detection could be used to extract the positions of the major peaks from the FFT periodicity intensity maps and hence discern the orientation of the plane. Once the plane angles, y and ø, are determined, it would be possible to refer back to the 3D data-cube and obtain the 1D distance histogram from that particular angle within the volume. The techniques of 1D FFT and 1D maximum detection could then be used to determine the maximum frequency and derive the plane spacing from its reciprocal, creating a highly automated approach to plane identification. 4.2. Towards an automatic tomographic reconstruction procedure Fig. 10. Periodicity intensity maps of (a) ROI-1 and (b) ROI-2, along with their corresponding distance histograms, (c,d), from the angles where intensity is the highest.
4. Additional outlooks for the 3D Hough transformation analyses 4.1. Towards an automatic analysis of crystal structure Unlike SDM analyses, which can require extensive searching and calibration to identify lattice planes and characterise their orientation within an atom probe dataset, the 3D Hough transform
The atomic planes information described by y, ø and d provide valuable information for the estimation of reconstruction parameters using the methods proposed in Ref. [7]. Assuming that the analysis of crystal structure can be automated via the 3D Hough transformation, as described above, we envisage an algorithm for the automatic indexing of the observed planes. Measured crystal structure could then be compared to the known crystal structure, providing feedback in the selection of reconstruction parameters, and hence an automated 3D reconstruction of APT experiment would be achieved. The subjective manner in which much of the current tomographic reconstruction parameters are selected was
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discussed in Ref. [23] and is a concern for the technique. An automated, objective reconstruction approach will improve the efficiency, consistency and accuracy of atom probe microscopy.
5. Conclusion
An efficient method for crystallographic structural analysis
atom probe data, utilising a 3D Hough transformation coupled with a FFT approach has been developed and demonstrated. This method can resolve blurred and discontinuous planes, such as those that are present in atom probe experiments where the field desorption images are poorly resolved. This common situation occurs in (e.g.) the analysis of Si-based materials. Similarly, the method provides a straightforward measurement of the crystal geometry around grain boundaries. The method is amenable to automation and requires less user interaction than corresponding SDM analyses. Notwithstanding this, SDM analysis will benefit from the results of 3D Hough transform techniques and can subsequently be applied to provide a more complete characterisation of the existing lattice structure
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