- Email: [email protected]
Extracting physically interpretable data from electron energy-loss spectra
Ultramicroscopy 110 (2010) 1390–1396
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Extracting physically interpretable data from electron energy-loss spectra C. Witte a, N.J. Zaluzec b, L.J. Allen a, a b
School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia Electron Microscopy Center, Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 6 November 2009 Received in revised form 12 May 2010 Accepted 8 July 2010
Principal component analysis is routinely applied to analyze data sets in electron energy-loss spectroscopy (EELS). We show how physically meaningful spectra can be obtained from the principal components using a knowledge of the scattering of the probe electron and the geometry of the experiment. This approach is illustrated by application to EELS data for the carbon K edge in graphite obtained using a conventional transmission electron microscope. The effect of scattering of the probe electron is accounted for, yielding spectra which are equivalent to experiments using linearly polarized X-rays. The approach is general and can also be applied to EELS in the context of scanning transmission electron microscopy. & 2010 Elsevier B.V. All rights reserved.
Keywords: Electron energy loss spectroscopy (EELS) Principal component analysis Theory of inelastic electron scattering
1. Introduction The fine structure observed in core-loss spectra arises from the interaction between the ejected electron and its local environment. The energy-loss near-edge structure (ELNES), in the range of approximately 0–50 eV above threshold, gives information about the local bonding environment [1] and the unoccupied density of states of the ionized atom [2]. Further from threshold the fine structure relates to the location and number of atoms and the direction and distance to the nearest neighbours of the ionized atom [3]. ELNES also allows us to extract additional information about the specimen, not accessible using just elastic scattering, such as the atomic number and valence state of the ionized atom, as well as bonding states and the nature of conduction bands [4,5]. Electron energy-loss spectroscopy (EELS) in the context of scanning transmission electron microscopy (STEM) utilizes a fine probe which is raster scanned across the specimen, recording an energy-loss spectrum for each probe position. This approach has been used to image single atom impurities within the bulk [6], based on core-loss EELS. More recently two-dimensional chemical mapping at atomic resolution has been demonstrated [7–11]. STEM EELS has the potential, by recording the change in fine structure at atomic resolution, to elucidate changes in structure, chemistry and bonding states at that length scale. This has obvious importance for the study of crystal interfaces [12]. High-angular-resolution electron-channelling electron-spectroscopy (HARECES) uses a plane wave incident on the specimen
Corresponding author. Tel.: + 61 3 8344 7402; fax: + 61 3 9347 4783.
E-mail address: [email protected] (L.J. Allen). 0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.07.003
and spectra are recorded as a function of scattering angle [13]. For dichoroic materials this leads to a change in fine structure as a function of momentum transfer. As in STEM EELS, this method yields a wealth of data but extracting meaningful information from such large data sets can be difficult. Interpretation of spectra is aided by being able to decompose a given spectrum into physically meaningful component spectra. Principal component analysis (PCA) provides a set of orthogonal eigenspectra, of which the principal components can be truncated (essentially without loss of relevant information) to provide a compact mathematical representation of the data. However, the principal components are not usually open to direct physical interpretation. Previous methods for extracting information from EELS data have used the experimental geometry to determine the physically meaningful spectra [14] but were limited to only using two recorded spectra and excluded the effects of dynamical scattering. In Ref. [12], it is shown that physical spectra can be obtained by a linear transformation of the principal components. We will show how this transformation can be determined in a physically meaningful way using electron scattering theory. This approach allows many recorded spectra to be used and can account for the effects of dynamical scattering. We will use an HARECES study on graphite as a case study to demonstrate this approach.
2. Experiment In Fig. 1(a) and (b) we can see HARECES measurements of the carbon K edge in graphite. Using an FEI Tecnai F20 TEM/STEM coupled to a Gatan Model 2001 spectrometer and custom written software, a nearly parallel (semi-angle o0:05 mrad) 200 keV probe was used to measure the angular dependence of inelastic
Intensity (arbitrary units)
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1391
285 eV peak 292 eV peak
-2
-1
0 Orientation
1
2
Fig. 2. Profiles of the first and second peaks of the experimental data shown in Fig. 1, at 285 and 292 eV, respectively, of the EELS carbon K shell in graphite as a function of orientation. A value of 1 on the orientation axis indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction.
Fig. 1. (a) Surface plot and (b) contour plot of experimental carbon K-shell EELS data taken on graphite. The intensity scale is arbitrary. An orientation of 0 indicates the exact [0 0 0 1] zone axis. A value of 1 on an orientation axis indicates that the incident beam has been tilted one Bragg angle or 5.9 mrad in the ð1 0 1 0Þ direction.
scattering from the specimen about the [0 0 0 1] zone axis. In the exact [0 0 0 1] orientation the crystal c-axis (perpendicular to the graphite sheets) is aligned parallel with the electron-optical axis of the experimental system. The probe was tilted in the ð1 0 1 0Þ direction, recording spectra between approximately 714.2 mrad. The detector, with acceptance semi-angle of 0.112 mrad, was fixed in the symmetric orientation. A value of 1 on an orientation axis in Fig. 1 indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction. The data shown in Fig. 1 have dark current and gain corrections applied. Background subtraction based on a power law has been applied to the preedge data to remove a small background in the data. The first two peaks in the energy-loss spectra, at approximately 285 and 292 eV, are, respectively, the transitions to the p and s antibonding orbitals lying above the Fermi level, and have quite different behaviour as a function of orientation [15]. As a function of orientation, the 285 eV peak decreases much quicker than the 292 eV peak, as can be seen in the line profiles shown in Fig. 2. Clearly we have a change in the fine structure as a function of incident beam tilt. To better understand the origin of this change we will perform principal component analysis of this data set and then carry out a linear transform to get physically meaningful spectra.
3. Principal component analysis The experimental data set shown in Fig. 1 contains a large volume of data and the challenge is to extract meaningful information from it. PCA can be used to reduce the dimensionality of large data sets while preserving most of the significant information.
Fig. 3. The first 10 largest eigenvalues of the matrix X T X . These eigenvalues quantify the amount of variance in the data set that is accounted for by the corresponding principal component. The first two principal components contain most of the variance of the original data set.
To perform PCA we first construct a p q matrix X , where each row is a spectrum of q elements and each column is the variation in the spectra as a function of incident tilt for a fixed energy. (No further processing of the data, beyond that discussed in the first paragraph of the previous section, has been carried out.) Using singular value decomposition (SVD) the data matrix X can be decomposed into component matrices as follows: X ¼ UDV T YV T :
ð1Þ
Here U is a p q matrix which satisfies U T U ¼ I and the q q matrix V satisfies V T V ¼ VV T ¼ I (the superscript T denotes a transpose). The matrix D is diagonal and contains the singular values of X . There is a strong connection between PCA and SVD [16] and we can identify the rows of V T as the principal components of X . These principal components constitute a set of orthogonal eigenspectra of X . The matrix Y ¼ UD gives us the weighting of these eigenspectra as a function of the probe tilt. In addition the squares of the singular values are the eigenvalues of the matrix X T X . These eigenvalues quantify the amount of variance in the data set that is accounted for by the corresponding principal component. For the data set in Fig. 1, these eigenvalues are shown in Fig. 3. If we have uniform, uncorrelated errors then
1392
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
Fig. 4. The first six principal components of the HARECES graphite experimental data. The first two principal components contain most of the signal while higher components contain progressively more noise. The intensities are all plotted using the same arbitrary units.
the spectroscopic information will be localized in the first few principal components, as appears to be the case here. PCA then provides a model where the spectral information is captured in the smallest possible number of components. The first six principal components of the experimental data are plotted in Fig. 4. The first two principal components appear to encapsulate the essential information in the signal. Noise increases as we go to higher principal components, and the sixth principal component appears to consist almost entirely of noise. We can then reconstruct the experimental data set from the first r principal components and their weightings to find an approximation to the original data, X ¼ YV T YuVuT ,
ð2Þ
where Yu is the p r truncation of Y and VuT is the r q truncation of V T . The approximation of the data by a PCA model, with say two factors, is good if we capture most of the structure of the original data set in our truncated set of two principal components. In Fig. 5 we display the reconstruction of the original data set using the first one, two and three principal components and the residuals for the different reconstructions. As can be seen, with just the first two principal components we have reliably reproduced most of the features in the original data. These reconstructions do not included pre-weighting for Poisson statistics, as described in Ref. [17], since this is not appropriate after background subtraction, which is necessary for us to compare the theoretical developments in Section 4 with the PCA of the data as described in Section 5. In Fig. 6 we can see the weighting of the first four principal components as a function of orientation. While the weighting of the first two principal components is roughly symmetric about the center orientation, the third is clearly asymmetric but only has a small weighting. To examine why these data have only two significant component spectra we now turn our discussion to the theory of ELNES.
the propagation of the elastic wave function in Fourier space as X CG ðK,z2 Þ ¼ S G,H ðK,z ¼ z2 z1 ÞCH ðK,z1 Þ, ð3Þ H
where the elastic scattering matrix element S G,H ðK,zÞ can be interpreted as the transition matrix element for elastic scattering from beam K+ H to K + G through a thickness z. The Fourier coefficients at the entrance surface of the crystal, CH ðK,0Þ, can be found using continuity of the wave function at the surface of the crystal. For plane wave illumination CH ðK,0Þ ¼ dH,0 and therefore X CG ðK,zÞ ¼ S G,H ðK,zÞdH,0 ¼ S G,0 ðK,zÞ: ð4Þ H
In STEM the wave function in Eq. (3) would depend on probe position and the boundary conditions are modified [19]. We will continue assuming a plane wave is incident on the specimen. The energy differential inelastic cross section can be written as Z X dsðK,tÞ 2m X ¼ NV c 2 dE ‘ K n a 0 G,H,Gu,Hu Z 1 t S G,0 ðK,zÞS H,0 ðK,zÞS 0,Gu ðKu,tzÞS 0,Hu ðKu,tzÞ dz t 0 Ku n0 XHHu,GGu ðK,Ku,EÞ dOKu : ð5Þ 2 Here S G,0 ðK,zÞ is the transition matrix element for elastic scattering from beam K + 0 to K +G through a depth z of crystal, S 0,Gu ðKu,tzÞ is the transition matrix element for elastic scattering from Ku þ Gu to Ku þ 0 in a thickness t z and similarly for the complex conjugates. The inelastic scattering matrix n0 XHHu,GGu ðK,Ku,EÞ connects these two elastic scattering processes. The inelastic scattering matrix is defined by n0 XHHu,GGu ðK,Ku,EÞ
¼ F site
2 ðq þHHu,EÞ Fn0 ðq þGGu,EÞ 4‘ Fn0 , ma20 4p2 jq þHHuj2 4p2 jq þ GGuj2
ð6Þ
4. Theory of ELNES Following previous work [18,19], consider a fast electron incident on a slab-like specimen. Let r? be a vector in the plane parallel to the surface of the specimen and let z increase as the fast electron propagates through the specimen. We can describe
where the atomic transition-matrix element for a particular orbital in the dipole approximation is X Z Fn0 ðq,EÞ ¼ 2pi qa un ðj,rÞra u0 ðrÞ dr, ð7Þ a
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
1393
Fig. 5. (a) The HARECES graphite data reconstructed from the first principal component and (b) the residual when subtracted from the full data set. Note that the reconstruction and the residual are each plotted on their own color scale. In (c) and (d) we show similar results using two principal components and in (e) and (f) three components. The orientation of the incident beam is plotted on a scale where 1 corresponds to a tilt through one Bragg angle or 5.9 mrad in the ð1 0 1 0Þ direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where u0(r) is the initial bound state wave function of the target electron and un ðj,rÞ denotes its final state with wave vector j. Furthermore ra and qa are the x, y or z components of r and q, respectively. (We note that the theory described here can easily be extended beyond the dipole approximation.) Substituting this into Eq. (6) we find n0 XHHu,GGu ðK,KuÞ
¼ F site
‘2 mp
X ðqa þHa Hua Þðqb þ Gb Gub Þ Rab ðEÞ , jq þ HHuj2 jq þ GGuj2
2 a2 0 a, b
where the 3 3 matrix Z XZ Rab ðEÞ ¼ ½un ðE,rÞra u0 ðrÞ dr ½un ðE,ruÞrb u0 ðruÞ dru:
ð8Þ
ð9Þ
na0
It has been shown in magic-angle studies, where aperture combinations are used which make the recorded spectra independent of specimen orientation, that relativistic corrections to the transition-matrix elements are important [20,21]. To apply the correction we replace q and r in Eq. (8) by
q-q
b2 ðq vÞ v
v^
ð10Þ
and
b2 ðr vÞ
r-r
v
^ v,
ð11Þ
where v is the magnitude of the velocity of the fast electron, v, in direction v^ and b ¼ v=c is the speed of the fast electron relative to the speed of light c. The net effect of this correction is a contraction of q and r in the direction of motion of the fast electron. This correction is most significant for detectors with a small acceptance angle, as is the case here. From Eqs. (5) and (8), for a fixed thickness so we no longer show the explicit t dependence, dsðKÞ X ¼ Bab ðK,EÞRab ðEÞ, dE a, b
ð12Þ
where Bab ðK,EÞ ¼ NV c 1 t
Z
2m 2
‘ K
Z
Ku X 2 G,H,Gu,Hu
t 0
F site
S G,0 ðK,zÞS H,0 ðK,zÞS 0,Gu ðKu,tzÞS 0,Hu ðKu,tzÞ dz
‘2
ðqa þ Ha Hua Þðqb þGb Gub Þ
mp2 a20
jq þ HHuj2 jq þ GGuj2
dOKu ,
ð13Þ
1394
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
Fig. 6. Weighting of the first four principal components as a function of orientation. All four are plotted on an arbitrary but common scale. A value of 1 on an orientation axis indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction.
after making the relativistic corrections. We can then think of the effect of multiple scattering, described by Bab , as being a weighting for the different spectra, Rab , exhibited by the specimen. This weighting factor, Bab , is determined by the details of the multiple scattering and the momentum transfer needed to scatter into the detector. If we look at a small energy window above threshold, Bab ðK,EÞ does not vary strongly with energy. Therefore we can take a representative value of energy loss, say the threshold energy, such that Bab ðK,EÞ Bab ðK,E0 Þ
ð14Þ
and dsðKÞ X ¼ Bab ðK,E0 ÞRab ðEÞ: dE a, b
5. Orthogonal to non-orthogonal coordinates ð15Þ
If we measure the differential cross section at p tilts of K and we have q elements in each spectrum then we can rewrite Eq. (15) as a matrix product, S ¼ BR:
absorbing atom is located on a three, four or sixfold main rotation axis, the EELS cross section is dichroic [15,20–25]. In this case, if the main rotation axis is in the z-direction, then Rab is diagonal and Rxx ¼ Ryy a Rzz , i.e. the specimen exhibits two spectra, one associated with the plane of rotation and another associated with the axis of rotation. For still lower symmetries the specimen can exhibit trichroism [26–28] which has three, four or six inequivalent spectra. The carbon K edge in graphite is dichroic and as such the matrix R contains two spectra Rzz and Rxx +Ryy. The Rzz spectrum is due to transitions from an initial 1s state to an unoccupied 2pz final state while the Rxx + Ryy spectrum is due to transitions to an unoccupied 2px or 2py final state. These unoccupied atomic orbitals can be related to antibonding s and p molecular orbitals [15]. The 2s, 2px and 2py hybridize to form three s orbitals and the remaining 2pz orbital forms the p orbital. For a transition from a 1s state to a s state the 2s parts of these orbitals cannot contribute due to the dipole selection rule. For simplicity we will continue our discussion in terms of the atomic orbitals. Using a point detector aligned with the incident beam and ignoring multiple scattering, all of the momentum transfer is in the z direction. In this case we are only sampling the Rzz spectrum or transitions to pz states [29]. As the incident beam is tilted away from the detector, a greater transverse momentum is needed to scatter into the detector and we begin to sample the Rxx and Ryy spectra or transitions to px and py states. The different weighting of these two spectra as a function of incident tilt leads to the change in fine structure seen experimentally. From this we can speculate that the 285 eV peak and the 292 eV peak in Figs. 1 and 2 are associated with different admixtures of the spectra. To simplify interpretation of experiment it would be ideal to look at these spectra in isolation. In any actual experiment we will never be able to sample just one of the spectra due to the finite size of the detector, i.e. there is more than one vector q that will scatter into the detector. This fact has led to techniques that allow the extraction of these spectra from two measurements with different convergence and collection angles [14]. Similar techniques could be applied to this data but we would only be able to use a fraction of the entire data set, 2 of the 80 recorded spectra. To overcome this limitation we return to the PCA.
ð16Þ
Here each row of the p q matrix S is an EELS spectrum recorded at a specific orientation of K. Each row of the ða bÞ q matrix R is a spectrum and each column of the p ða bÞ matrix B is the weighting of the corresponding spectrum, for every orientation K. For STEM each row of S would correspond to a different probe position rather than an orientation. We have separated the orientation dependence (probe position in STEM), described by B, from the energy dependence, described by R. We can calculate B assuming the specimen consists of neutral atoms and ignoring bonding effects to a very good approximation using scattering theory. The information about bonding and valency is contained in R. Depending on the symmetries of the specimen, not all of the nine elements of Rab will be non-zero and distinct. When the
We have extracted the two most significant principal components, shown in Fig. 4(a) and (b), but interpreting them physically is problematic. We would like to be able to relate them back to the physical spectra R. PCA generates orthogonal eigenspectra, but there is no requirement that the physical spectra be orthogonal. Comparing Eqs. (2) and (16) we can see the similarity between the construction of the theoretical cross section and the reconstruction of the data in terms of the important principal components. This suggests we can make a linear transformation into a nonorthogonal basis to extract the physical spectra from the principal components. Referring back to Eq. (2), let us write X YuTT 1 VuT ¼ Y~ uV~ uT ,
ð17Þ T
T
where Y~ u ¼ YuT and V~ u ¼ T Vu . A similar technique has been utilized to extract information from spatially resolved EELS data across an interface in STEM [12]. In that case the spectra in the bulk away from the interface were used to calculate the transformation matrix rather than any theoretical calculations. To calculate the transformation matrix T we compare Eq. (17) with Eq. (16) and require that Y~ u behaves like the theoretical 1
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
weighting matrix B, i.e. Y~ u ¼ YuT ¼ B:
ð18Þ
We then find T ¼ YuT B:
ð19Þ
The inverse matrix T
1
can then be calculated and hence
V~ uT ¼ T 1 V T ,
ð20Þ
1395
number of beams used was found to be sufficient as carbon is a weak scatterer, as can be seen by the small Bragg peaks at orientations of 2 and 2 in Fig. 1. These theoretical calculations were then used to calculate the transformation of the eigenspectra, generated using PCA of the experimental data set, from an orthogonal to non-orthogonal basis. In Fig. 8 we can see the physical spectra extracted from the experimental data. These results show good agreement with other
which can be identified with R in Eq. (16). So knowing the matrix B in Eq. (16), which contains information about the dynamical scattering of the fast electrons, we can construct the physical spectra with the effects of dynamical scattering removed. We can calculate B without explicitly calculating R (apart from assumptions about the symmetry of R). This allows us to compare an ELNES spectrum with a XANES spectrum, a fact which we shall exploit in the next section.
6. Results Scattering calculations, taking into account the multiple elastic scattering of the probe electron both before and after the ionization event, were performed for the carbon K edge in graphite. In Fig. 7 we can see the weighting of the two different spectra, Rzz and Rxx + Ryy, as a function of orientation. The simulations were performed in zone axis conditions using five beams and a detector semi-angle of 0.112 mrad. The small
1 Intensity (arbitrary units)
Rzz weighting Rxx+Ryy weighting
0
-2
-1
0 Orientation
1
2
Fig. 7. Theoretical angular distribution of the Rzz and Rxx + Ryy spectra. A value of 1 on the orientation axis indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction.
Fig. 8. Physical spectra extracted from experimental data using PCA and transforming into a non-orthogonal basis.
Fig. 9. Comparison of ELNES and polarization-dependent X-ray data. (a) The Rxx + Ryy spectrum from Fig. 8 compared to the corresponding spectrum taken from Ref. [31], where the incident photon is parallel to the graphite crystal c-axis and the polarization vector is in the plane of the graphite sheets. (b) The linear 2 combination sin ð303 ÞRzz þ cos2 ð303 ÞðRxx þ Ryy Þ of the spectra in Fig. 8 is compared to the case where the incident photons make an angle of 301to the c-axis. (c) Lastly sin2 ð603 ÞRzz þ cos2 ð603 ÞðRxx þ Ryy Þ is compared to the case where the incident photons make an angle of 601to the c-axis.
1396
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
orientation dependent ELNES studies [30] and also with polarization-dependent X-ray studies [31]. The latter agreement can be seen in Fig. 9, where the appropriate combinations of the physical spectra shown in Fig. 8 are compared with three experimental spectra from Fig. 1 in Ref. [31]. This confirms directly the conventional wisdom that, despite the different physical processes involved, these two experimental situations deliver the same information about the unoccupied states.
7. Conclusion Using HARECES of graphite as a case study we have presented a general method for extracting physically meaningful information from EELS spectra. Using PCA we reduced the dimensionality of the large experimental data set by extracting the two most significant principal components. We observed that the principal components have little physical meaning. We presented the theory of dynamical scattering in the context of EELS and found that the outcome of dynamical scattering is to weight the different spectra exhibited by the specimen. Using this theory we calculated a linear transformation of the principal components into a physically meaningful basis. This allowed us to extract the spectra due to transitions to specific final states, removing the complications due to dynamical scattering and experimental geometry. This enabled us to demonstrate an equivalence to experiments using linearly polarized X-rays. Use of PCA in this manner facilitated the use of the entire 80 spectra in the data set, in contrast to similar techniques that can only utilize two spectra [14]. This unique combination of scattering theory and statistical analysis greatly simplified this large data set and similar analysis can be used in STEM EELS provided the dynamical scattering is well understood. References [1] M.S. Moreno, K. Jorissen, J.J. Rehr, Practical aspects of electron energy-loss spectroscopy (EELS) calculations using FEFF8, Micron 38 (1) (2007) 1–11. [2] R.F. Edgerton, Electron Energy-loss Spectroscopy in the Electron Microscope, Plenum Press, New York, 1996. [3] O.L. Krivanek, M.M. Disko, J. Taftø, J.C.H. Spence, Electron-energy loss spectroscopy as a probe of the local atomic environment, Ultramicroscopy 9 (3) (1982) 249–254. [4] P. Rez, J. Bruley, P. Brohan, M. Payne, L.A.J. Garvie, Review of methods for calculating near-edge structure, Ultramicroscopy 59 (1–4) (1995) 159–167. [5] V.J. Keast, A.J. Scott, R. Brydson, D.B. Williams, J. Bruley, Electron energy-loss near-edge structure—a tool for the investigation of electronic structure on the nanometre scale, Journal of Microscopy—Oxford 203 (2001) 135–175. [6] M. Varela, S.D. Findlay, A.R. Lupini, H.M. Christen, A.Y. Borisevich, N. Dellby, O.L. Krivanek, P.D. Nellist, M.P. Oxley, L.J. Allen, S.J. Pennycook, Spectroscopic imaging of single atoms within a bulk solid, Physical Review Letters 92 (9) (2004) 095502. [7] E. Okunishi, H. Sawada, J. Kondo, M. Kersker, Atomic resolution elemental map of EELS with a Cs corrected STEM, Microscopy and Microanalysis 12 (Supplement 02) (2006) 1150–1151.
[8] M. Bosman, V.J. Keast, J.L. Garcia-Munoz, A.J. D’Alfonso, S.D. Findlay, L.J. Allen, Two-dimensional mapping of chemical information at atomic resolution, Physical Review Letters 99 (8) (2007) 086102. [9] K. Kimoto, T. Asaka, T. Nagai, M. Saito, Y. Matsui, K. Ishizuka, Elementselective imaging of atomic columns in a crystal using STEM and EELS, Nature 450 (7170) (2007) 702–704. [10] D.A. Muller, L.F. Kourkoutis, M. Murfitt, J.H. Song, H.Y. Hwang, J. Silcox, N. Dellby, O.L. Krivanek, Atomic-scale chemical imaging of composition and bonding by aberration-corrected microscopy, Science 319 (5866) (2008) 1073–1076. [11] L.J. Allen, Electron microscopy—new directions for chemical maps, Nature Nanotechnology 3 (5) (2008) 255–256. [12] N. Bonnet, N. Brun, C. Colliex, Extracting information from sequences of spatially resolved EELS spectra using multivariate statistical analysis, Ultramicroscopy 77 (3–4) (1999) 97–112. [13] N.J. Zaluzec, M.G. Blackford, K.L. Smith, M. Colella, HARECES measurements of carbon K shell excitation in graphite, Microscopy and Microanalysis 11 (S2) (2005) 718–719. [14] N.D. Browning, J. Yuan, L.M. Brown, Real-space determination of anisotropic electronic-structure by electron-energy loss spectroscopy, Ultramicroscopy 38 (3–4) (1991) 291–298. [15] R.D. Leapman, P.L. Fejes, J. Silcox, Orientation dependence of core edges from anisotropic materials determined by inelastic-scattering of fast electrons, Physical Review B 28 (5) (1983) 2361–2373. [16] I.T. Jolliffe, Principal Component Analysis, Springer-Verlag, New York, 1986. [17] M.R. Keenan, P.G. Kotula, Accounting for Poisson noise in the multivariate analysis of ToF-SIMS spectrum images, Surface and Interface Analysis 36 (2004) 203–212. [18] L.J. Allen, S.D. Findlay, M.P. Oxley, C. Witte, N.J. Zaluzec, Channeling effects in high-angular-resolution electron spectroscopy, Physical Review B 73 (9) (2006) 094104. [19] C. Witte, S.D. Findlay, M.P. Oxley, J.J. Rehr, L.J. Allen, Theory of dynamical scattering in near-edge electron energy loss spectroscopy, Physical Review B 80 (18) (2009) 184108. [20] P. Schattschneider, C. He´bert, H. Franco, B. Jouffrey, Anisotropic relativistic cross sections for inelastic electron scattering, and the magic angle, Physical Review B 72 (4) (2005) 045142. [21] J.C. Le Bosse´, T. E´picier, B. Jouffrey, Polarization dependence in ELNES: influence of probe convergence, collector aperture and electron beam incidence angle, Ultramicroscopy 106 (6) (2006) 449–460. [22] R.D. Leapman, J. Silcox, Orientation dependence of core edges in electronenergy-loss spectra from anisotropic materials, Physical Review Letters 42 (20) (1979) 1361–1364. [23] M. Nelhiebel, P.H. Louf, P. Schattschneider, P. Blaha, K. Schwarz, B. Jouffrey, Theory of orientation-sensitive near-edge fine-structure core-level spectroscopy, Physical Review B 59 (20) (1999) 12807–12814. [24] B. Jouffrey, P. Schattschneider, C. He´bert, The magic angle: a solved mystery, Ultramicroscopy 102 (1) (2004) 61–66. [25] C. He´bert, P. Schattschneider, H. Franco, B. Jouffrey, ELNES at magic angle conditions, Ultramicroscopy 106 (11–12) (2006) 1139–1143. [26] C. Brouder, Angular-dependence of X-ray absorption-spectra, Journal of Physics: Condensed Matter 2 (3) (1990) 701–738. [27] C. Brouder, M.F. Ruiz Lo´pez, M. Benfatto, C.R. Natoli, R.F. Pettifer, Systematicapproach to the calculation of the polarization-dependent (and polarizationaveraged) general term of the curved-wave multiple-scattering series in the X-ray-absorption cross-section, Physical Review B 39 (3) (1989) 1488–1500. [28] J.C. Le Bosse´, T. E´picier, H. Chermette, Trichroism in energy-loss near-edge structure spectroscopy: polarization dependence of near-edge fine structures, Physical Review B 76 (7) (2007) 075127. [29] G. Radtke, T. Epicier, P. Bayle-Guillemaud, J.C. Le Bosse, N-K ELNES study of anisotropy effects in hexagonal AlN, Journal of Microscopy 210 (2003) 60–65. [30] M.M. Disko, O.L. Krivanek, P. Rez, Orientation-dependent extended finestructure in electron-energy-loss spectra, Physical Review B 25 (6) (1982) 4252–4255. [31] R.A. Rosenberg, P.J. Love, V. Rehn, Polarization-dependent C(K) near-edge Xray-absorption fine-structure of graphite, Physical Review B 33 (6) (1986) 4034–4037.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Extracting physically interpretable data from electron energy-loss spectra C. Witte a, N.J. Zaluzec b, L.J. Allen a, a b
School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia Electron Microscopy Center, Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 6 November 2009 Received in revised form 12 May 2010 Accepted 8 July 2010
Principal component analysis is routinely applied to analyze data sets in electron energy-loss spectroscopy (EELS). We show how physically meaningful spectra can be obtained from the principal components using a knowledge of the scattering of the probe electron and the geometry of the experiment. This approach is illustrated by application to EELS data for the carbon K edge in graphite obtained using a conventional transmission electron microscope. The effect of scattering of the probe electron is accounted for, yielding spectra which are equivalent to experiments using linearly polarized X-rays. The approach is general and can also be applied to EELS in the context of scanning transmission electron microscopy. & 2010 Elsevier B.V. All rights reserved.
Keywords: Electron energy loss spectroscopy (EELS) Principal component analysis Theory of inelastic electron scattering
1. Introduction The fine structure observed in core-loss spectra arises from the interaction between the ejected electron and its local environment. The energy-loss near-edge structure (ELNES), in the range of approximately 0–50 eV above threshold, gives information about the local bonding environment [1] and the unoccupied density of states of the ionized atom [2]. Further from threshold the fine structure relates to the location and number of atoms and the direction and distance to the nearest neighbours of the ionized atom [3]. ELNES also allows us to extract additional information about the specimen, not accessible using just elastic scattering, such as the atomic number and valence state of the ionized atom, as well as bonding states and the nature of conduction bands [4,5]. Electron energy-loss spectroscopy (EELS) in the context of scanning transmission electron microscopy (STEM) utilizes a fine probe which is raster scanned across the specimen, recording an energy-loss spectrum for each probe position. This approach has been used to image single atom impurities within the bulk [6], based on core-loss EELS. More recently two-dimensional chemical mapping at atomic resolution has been demonstrated [7–11]. STEM EELS has the potential, by recording the change in fine structure at atomic resolution, to elucidate changes in structure, chemistry and bonding states at that length scale. This has obvious importance for the study of crystal interfaces [12]. High-angular-resolution electron-channelling electron-spectroscopy (HARECES) uses a plane wave incident on the specimen
Corresponding author. Tel.: + 61 3 8344 7402; fax: + 61 3 9347 4783.
E-mail address: [email protected] (L.J. Allen). 0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.07.003
and spectra are recorded as a function of scattering angle [13]. For dichoroic materials this leads to a change in fine structure as a function of momentum transfer. As in STEM EELS, this method yields a wealth of data but extracting meaningful information from such large data sets can be difficult. Interpretation of spectra is aided by being able to decompose a given spectrum into physically meaningful component spectra. Principal component analysis (PCA) provides a set of orthogonal eigenspectra, of which the principal components can be truncated (essentially without loss of relevant information) to provide a compact mathematical representation of the data. However, the principal components are not usually open to direct physical interpretation. Previous methods for extracting information from EELS data have used the experimental geometry to determine the physically meaningful spectra [14] but were limited to only using two recorded spectra and excluded the effects of dynamical scattering. In Ref. [12], it is shown that physical spectra can be obtained by a linear transformation of the principal components. We will show how this transformation can be determined in a physically meaningful way using electron scattering theory. This approach allows many recorded spectra to be used and can account for the effects of dynamical scattering. We will use an HARECES study on graphite as a case study to demonstrate this approach.
2. Experiment In Fig. 1(a) and (b) we can see HARECES measurements of the carbon K edge in graphite. Using an FEI Tecnai F20 TEM/STEM coupled to a Gatan Model 2001 spectrometer and custom written software, a nearly parallel (semi-angle o0:05 mrad) 200 keV probe was used to measure the angular dependence of inelastic
Intensity (arbitrary units)
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1391
285 eV peak 292 eV peak
-2
-1
0 Orientation
1
2
Fig. 2. Profiles of the first and second peaks of the experimental data shown in Fig. 1, at 285 and 292 eV, respectively, of the EELS carbon K shell in graphite as a function of orientation. A value of 1 on the orientation axis indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction.
Fig. 1. (a) Surface plot and (b) contour plot of experimental carbon K-shell EELS data taken on graphite. The intensity scale is arbitrary. An orientation of 0 indicates the exact [0 0 0 1] zone axis. A value of 1 on an orientation axis indicates that the incident beam has been tilted one Bragg angle or 5.9 mrad in the ð1 0 1 0Þ direction.
scattering from the specimen about the [0 0 0 1] zone axis. In the exact [0 0 0 1] orientation the crystal c-axis (perpendicular to the graphite sheets) is aligned parallel with the electron-optical axis of the experimental system. The probe was tilted in the ð1 0 1 0Þ direction, recording spectra between approximately 714.2 mrad. The detector, with acceptance semi-angle of 0.112 mrad, was fixed in the symmetric orientation. A value of 1 on an orientation axis in Fig. 1 indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction. The data shown in Fig. 1 have dark current and gain corrections applied. Background subtraction based on a power law has been applied to the preedge data to remove a small background in the data. The first two peaks in the energy-loss spectra, at approximately 285 and 292 eV, are, respectively, the transitions to the p and s antibonding orbitals lying above the Fermi level, and have quite different behaviour as a function of orientation [15]. As a function of orientation, the 285 eV peak decreases much quicker than the 292 eV peak, as can be seen in the line profiles shown in Fig. 2. Clearly we have a change in the fine structure as a function of incident beam tilt. To better understand the origin of this change we will perform principal component analysis of this data set and then carry out a linear transform to get physically meaningful spectra.
3. Principal component analysis The experimental data set shown in Fig. 1 contains a large volume of data and the challenge is to extract meaningful information from it. PCA can be used to reduce the dimensionality of large data sets while preserving most of the significant information.
Fig. 3. The first 10 largest eigenvalues of the matrix X T X . These eigenvalues quantify the amount of variance in the data set that is accounted for by the corresponding principal component. The first two principal components contain most of the variance of the original data set.
To perform PCA we first construct a p q matrix X , where each row is a spectrum of q elements and each column is the variation in the spectra as a function of incident tilt for a fixed energy. (No further processing of the data, beyond that discussed in the first paragraph of the previous section, has been carried out.) Using singular value decomposition (SVD) the data matrix X can be decomposed into component matrices as follows: X ¼ UDV T YV T :
ð1Þ
Here U is a p q matrix which satisfies U T U ¼ I and the q q matrix V satisfies V T V ¼ VV T ¼ I (the superscript T denotes a transpose). The matrix D is diagonal and contains the singular values of X . There is a strong connection between PCA and SVD [16] and we can identify the rows of V T as the principal components of X . These principal components constitute a set of orthogonal eigenspectra of X . The matrix Y ¼ UD gives us the weighting of these eigenspectra as a function of the probe tilt. In addition the squares of the singular values are the eigenvalues of the matrix X T X . These eigenvalues quantify the amount of variance in the data set that is accounted for by the corresponding principal component. For the data set in Fig. 1, these eigenvalues are shown in Fig. 3. If we have uniform, uncorrelated errors then
1392
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
Fig. 4. The first six principal components of the HARECES graphite experimental data. The first two principal components contain most of the signal while higher components contain progressively more noise. The intensities are all plotted using the same arbitrary units.
the spectroscopic information will be localized in the first few principal components, as appears to be the case here. PCA then provides a model where the spectral information is captured in the smallest possible number of components. The first six principal components of the experimental data are plotted in Fig. 4. The first two principal components appear to encapsulate the essential information in the signal. Noise increases as we go to higher principal components, and the sixth principal component appears to consist almost entirely of noise. We can then reconstruct the experimental data set from the first r principal components and their weightings to find an approximation to the original data, X ¼ YV T YuVuT ,
ð2Þ
where Yu is the p r truncation of Y and VuT is the r q truncation of V T . The approximation of the data by a PCA model, with say two factors, is good if we capture most of the structure of the original data set in our truncated set of two principal components. In Fig. 5 we display the reconstruction of the original data set using the first one, two and three principal components and the residuals for the different reconstructions. As can be seen, with just the first two principal components we have reliably reproduced most of the features in the original data. These reconstructions do not included pre-weighting for Poisson statistics, as described in Ref. [17], since this is not appropriate after background subtraction, which is necessary for us to compare the theoretical developments in Section 4 with the PCA of the data as described in Section 5. In Fig. 6 we can see the weighting of the first four principal components as a function of orientation. While the weighting of the first two principal components is roughly symmetric about the center orientation, the third is clearly asymmetric but only has a small weighting. To examine why these data have only two significant component spectra we now turn our discussion to the theory of ELNES.
the propagation of the elastic wave function in Fourier space as X CG ðK,z2 Þ ¼ S G,H ðK,z ¼ z2 z1 ÞCH ðK,z1 Þ, ð3Þ H
where the elastic scattering matrix element S G,H ðK,zÞ can be interpreted as the transition matrix element for elastic scattering from beam K+ H to K + G through a thickness z. The Fourier coefficients at the entrance surface of the crystal, CH ðK,0Þ, can be found using continuity of the wave function at the surface of the crystal. For plane wave illumination CH ðK,0Þ ¼ dH,0 and therefore X CG ðK,zÞ ¼ S G,H ðK,zÞdH,0 ¼ S G,0 ðK,zÞ: ð4Þ H
In STEM the wave function in Eq. (3) would depend on probe position and the boundary conditions are modified [19]. We will continue assuming a plane wave is incident on the specimen. The energy differential inelastic cross section can be written as Z X dsðK,tÞ 2m X ¼ NV c 2 dE ‘ K n a 0 G,H,Gu,Hu Z 1 t S G,0 ðK,zÞS H,0 ðK,zÞS 0,Gu ðKu,tzÞS 0,Hu ðKu,tzÞ dz t 0 Ku n0 XHHu,GGu ðK,Ku,EÞ dOKu : ð5Þ 2 Here S G,0 ðK,zÞ is the transition matrix element for elastic scattering from beam K + 0 to K +G through a depth z of crystal, S 0,Gu ðKu,tzÞ is the transition matrix element for elastic scattering from Ku þ Gu to Ku þ 0 in a thickness t z and similarly for the complex conjugates. The inelastic scattering matrix n0 XHHu,GGu ðK,Ku,EÞ connects these two elastic scattering processes. The inelastic scattering matrix is defined by n0 XHHu,GGu ðK,Ku,EÞ
¼ F site
2 ðq þHHu,EÞ Fn0 ðq þGGu,EÞ 4‘ Fn0 , ma20 4p2 jq þHHuj2 4p2 jq þ GGuj2
ð6Þ
4. Theory of ELNES Following previous work [18,19], consider a fast electron incident on a slab-like specimen. Let r? be a vector in the plane parallel to the surface of the specimen and let z increase as the fast electron propagates through the specimen. We can describe
where the atomic transition-matrix element for a particular orbital in the dipole approximation is X Z Fn0 ðq,EÞ ¼ 2pi qa un ðj,rÞra u0 ðrÞ dr, ð7Þ a
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
1393
Fig. 5. (a) The HARECES graphite data reconstructed from the first principal component and (b) the residual when subtracted from the full data set. Note that the reconstruction and the residual are each plotted on their own color scale. In (c) and (d) we show similar results using two principal components and in (e) and (f) three components. The orientation of the incident beam is plotted on a scale where 1 corresponds to a tilt through one Bragg angle or 5.9 mrad in the ð1 0 1 0Þ direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where u0(r) is the initial bound state wave function of the target electron and un ðj,rÞ denotes its final state with wave vector j. Furthermore ra and qa are the x, y or z components of r and q, respectively. (We note that the theory described here can easily be extended beyond the dipole approximation.) Substituting this into Eq. (6) we find n0 XHHu,GGu ðK,KuÞ
¼ F site
‘2 mp
X ðqa þHa Hua Þðqb þ Gb Gub Þ Rab ðEÞ , jq þ HHuj2 jq þ GGuj2
2 a2 0 a, b
where the 3 3 matrix Z XZ Rab ðEÞ ¼ ½un ðE,rÞra u0 ðrÞ dr ½un ðE,ruÞrb u0 ðruÞ dru:
ð8Þ
ð9Þ
na0
It has been shown in magic-angle studies, where aperture combinations are used which make the recorded spectra independent of specimen orientation, that relativistic corrections to the transition-matrix elements are important [20,21]. To apply the correction we replace q and r in Eq. (8) by
q-q
b2 ðq vÞ v
v^
ð10Þ
and
b2 ðr vÞ
r-r
v
^ v,
ð11Þ
where v is the magnitude of the velocity of the fast electron, v, in direction v^ and b ¼ v=c is the speed of the fast electron relative to the speed of light c. The net effect of this correction is a contraction of q and r in the direction of motion of the fast electron. This correction is most significant for detectors with a small acceptance angle, as is the case here. From Eqs. (5) and (8), for a fixed thickness so we no longer show the explicit t dependence, dsðKÞ X ¼ Bab ðK,EÞRab ðEÞ, dE a, b
ð12Þ
where Bab ðK,EÞ ¼ NV c 1 t
Z
2m 2
‘ K
Z
Ku X 2 G,H,Gu,Hu
t 0
F site
S G,0 ðK,zÞS H,0 ðK,zÞS 0,Gu ðKu,tzÞS 0,Hu ðKu,tzÞ dz
‘2
ðqa þ Ha Hua Þðqb þGb Gub Þ
mp2 a20
jq þ HHuj2 jq þ GGuj2
dOKu ,
ð13Þ
1394
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
Fig. 6. Weighting of the first four principal components as a function of orientation. All four are plotted on an arbitrary but common scale. A value of 1 on an orientation axis indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction.
after making the relativistic corrections. We can then think of the effect of multiple scattering, described by Bab , as being a weighting for the different spectra, Rab , exhibited by the specimen. This weighting factor, Bab , is determined by the details of the multiple scattering and the momentum transfer needed to scatter into the detector. If we look at a small energy window above threshold, Bab ðK,EÞ does not vary strongly with energy. Therefore we can take a representative value of energy loss, say the threshold energy, such that Bab ðK,EÞ Bab ðK,E0 Þ
ð14Þ
and dsðKÞ X ¼ Bab ðK,E0 ÞRab ðEÞ: dE a, b
5. Orthogonal to non-orthogonal coordinates ð15Þ
If we measure the differential cross section at p tilts of K and we have q elements in each spectrum then we can rewrite Eq. (15) as a matrix product, S ¼ BR:
absorbing atom is located on a three, four or sixfold main rotation axis, the EELS cross section is dichroic [15,20–25]. In this case, if the main rotation axis is in the z-direction, then Rab is diagonal and Rxx ¼ Ryy a Rzz , i.e. the specimen exhibits two spectra, one associated with the plane of rotation and another associated with the axis of rotation. For still lower symmetries the specimen can exhibit trichroism [26–28] which has three, four or six inequivalent spectra. The carbon K edge in graphite is dichroic and as such the matrix R contains two spectra Rzz and Rxx +Ryy. The Rzz spectrum is due to transitions from an initial 1s state to an unoccupied 2pz final state while the Rxx + Ryy spectrum is due to transitions to an unoccupied 2px or 2py final state. These unoccupied atomic orbitals can be related to antibonding s and p molecular orbitals [15]. The 2s, 2px and 2py hybridize to form three s orbitals and the remaining 2pz orbital forms the p orbital. For a transition from a 1s state to a s state the 2s parts of these orbitals cannot contribute due to the dipole selection rule. For simplicity we will continue our discussion in terms of the atomic orbitals. Using a point detector aligned with the incident beam and ignoring multiple scattering, all of the momentum transfer is in the z direction. In this case we are only sampling the Rzz spectrum or transitions to pz states [29]. As the incident beam is tilted away from the detector, a greater transverse momentum is needed to scatter into the detector and we begin to sample the Rxx and Ryy spectra or transitions to px and py states. The different weighting of these two spectra as a function of incident tilt leads to the change in fine structure seen experimentally. From this we can speculate that the 285 eV peak and the 292 eV peak in Figs. 1 and 2 are associated with different admixtures of the spectra. To simplify interpretation of experiment it would be ideal to look at these spectra in isolation. In any actual experiment we will never be able to sample just one of the spectra due to the finite size of the detector, i.e. there is more than one vector q that will scatter into the detector. This fact has led to techniques that allow the extraction of these spectra from two measurements with different convergence and collection angles [14]. Similar techniques could be applied to this data but we would only be able to use a fraction of the entire data set, 2 of the 80 recorded spectra. To overcome this limitation we return to the PCA.
ð16Þ
Here each row of the p q matrix S is an EELS spectrum recorded at a specific orientation of K. Each row of the ða bÞ q matrix R is a spectrum and each column of the p ða bÞ matrix B is the weighting of the corresponding spectrum, for every orientation K. For STEM each row of S would correspond to a different probe position rather than an orientation. We have separated the orientation dependence (probe position in STEM), described by B, from the energy dependence, described by R. We can calculate B assuming the specimen consists of neutral atoms and ignoring bonding effects to a very good approximation using scattering theory. The information about bonding and valency is contained in R. Depending on the symmetries of the specimen, not all of the nine elements of Rab will be non-zero and distinct. When the
We have extracted the two most significant principal components, shown in Fig. 4(a) and (b), but interpreting them physically is problematic. We would like to be able to relate them back to the physical spectra R. PCA generates orthogonal eigenspectra, but there is no requirement that the physical spectra be orthogonal. Comparing Eqs. (2) and (16) we can see the similarity between the construction of the theoretical cross section and the reconstruction of the data in terms of the important principal components. This suggests we can make a linear transformation into a nonorthogonal basis to extract the physical spectra from the principal components. Referring back to Eq. (2), let us write X YuTT 1 VuT ¼ Y~ uV~ uT ,
ð17Þ T
T
where Y~ u ¼ YuT and V~ u ¼ T Vu . A similar technique has been utilized to extract information from spatially resolved EELS data across an interface in STEM [12]. In that case the spectra in the bulk away from the interface were used to calculate the transformation matrix rather than any theoretical calculations. To calculate the transformation matrix T we compare Eq. (17) with Eq. (16) and require that Y~ u behaves like the theoretical 1
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
weighting matrix B, i.e. Y~ u ¼ YuT ¼ B:
ð18Þ
We then find T ¼ YuT B:
ð19Þ
The inverse matrix T
1
can then be calculated and hence
V~ uT ¼ T 1 V T ,
ð20Þ
1395
number of beams used was found to be sufficient as carbon is a weak scatterer, as can be seen by the small Bragg peaks at orientations of 2 and 2 in Fig. 1. These theoretical calculations were then used to calculate the transformation of the eigenspectra, generated using PCA of the experimental data set, from an orthogonal to non-orthogonal basis. In Fig. 8 we can see the physical spectra extracted from the experimental data. These results show good agreement with other
which can be identified with R in Eq. (16). So knowing the matrix B in Eq. (16), which contains information about the dynamical scattering of the fast electrons, we can construct the physical spectra with the effects of dynamical scattering removed. We can calculate B without explicitly calculating R (apart from assumptions about the symmetry of R). This allows us to compare an ELNES spectrum with a XANES spectrum, a fact which we shall exploit in the next section.
6. Results Scattering calculations, taking into account the multiple elastic scattering of the probe electron both before and after the ionization event, were performed for the carbon K edge in graphite. In Fig. 7 we can see the weighting of the two different spectra, Rzz and Rxx + Ryy, as a function of orientation. The simulations were performed in zone axis conditions using five beams and a detector semi-angle of 0.112 mrad. The small
1 Intensity (arbitrary units)
Rzz weighting Rxx+Ryy weighting
0
-2
-1
0 Orientation
1
2
Fig. 7. Theoretical angular distribution of the Rzz and Rxx + Ryy spectra. A value of 1 on the orientation axis indicates that the incident beam has been tilted one Bragg angle (5.9 mrad) in the ð1 0 1 0Þ direction.
Fig. 8. Physical spectra extracted from experimental data using PCA and transforming into a non-orthogonal basis.
Fig. 9. Comparison of ELNES and polarization-dependent X-ray data. (a) The Rxx + Ryy spectrum from Fig. 8 compared to the corresponding spectrum taken from Ref. [31], where the incident photon is parallel to the graphite crystal c-axis and the polarization vector is in the plane of the graphite sheets. (b) The linear 2 combination sin ð303 ÞRzz þ cos2 ð303 ÞðRxx þ Ryy Þ of the spectra in Fig. 8 is compared to the case where the incident photons make an angle of 301to the c-axis. (c) Lastly sin2 ð603 ÞRzz þ cos2 ð603 ÞðRxx þ Ryy Þ is compared to the case where the incident photons make an angle of 601to the c-axis.
1396
C. Witte et al. / Ultramicroscopy 110 (2010) 1390–1396
orientation dependent ELNES studies [30] and also with polarization-dependent X-ray studies [31]. The latter agreement can be seen in Fig. 9, where the appropriate combinations of the physical spectra shown in Fig. 8 are compared with three experimental spectra from Fig. 1 in Ref. [31]. This confirms directly the conventional wisdom that, despite the different physical processes involved, these two experimental situations deliver the same information about the unoccupied states.
7. Conclusion Using HARECES of graphite as a case study we have presented a general method for extracting physically meaningful information from EELS spectra. Using PCA we reduced the dimensionality of the large experimental data set by extracting the two most significant principal components. We observed that the principal components have little physical meaning. We presented the theory of dynamical scattering in the context of EELS and found that the outcome of dynamical scattering is to weight the different spectra exhibited by the specimen. Using this theory we calculated a linear transformation of the principal components into a physically meaningful basis. This allowed us to extract the spectra due to transitions to specific final states, removing the complications due to dynamical scattering and experimental geometry. This enabled us to demonstrate an equivalence to experiments using linearly polarized X-rays. Use of PCA in this manner facilitated the use of the entire 80 spectra in the data set, in contrast to similar techniques that can only utilize two spectra [14]. This unique combination of scattering theory and statistical analysis greatly simplified this large data set and similar analysis can be used in STEM EELS provided the dynamical scattering is well understood. References [1] M.S. Moreno, K. Jorissen, J.J. Rehr, Practical aspects of electron energy-loss spectroscopy (EELS) calculations using FEFF8, Micron 38 (1) (2007) 1–11. [2] R.F. Edgerton, Electron Energy-loss Spectroscopy in the Electron Microscope, Plenum Press, New York, 1996. [3] O.L. Krivanek, M.M. Disko, J. Taftø, J.C.H. Spence, Electron-energy loss spectroscopy as a probe of the local atomic environment, Ultramicroscopy 9 (3) (1982) 249–254. [4] P. Rez, J. Bruley, P. Brohan, M. Payne, L.A.J. Garvie, Review of methods for calculating near-edge structure, Ultramicroscopy 59 (1–4) (1995) 159–167. [5] V.J. Keast, A.J. Scott, R. Brydson, D.B. Williams, J. Bruley, Electron energy-loss near-edge structure—a tool for the investigation of electronic structure on the nanometre scale, Journal of Microscopy—Oxford 203 (2001) 135–175. [6] M. Varela, S.D. Findlay, A.R. Lupini, H.M. Christen, A.Y. Borisevich, N. Dellby, O.L. Krivanek, P.D. Nellist, M.P. Oxley, L.J. Allen, S.J. Pennycook, Spectroscopic imaging of single atoms within a bulk solid, Physical Review Letters 92 (9) (2004) 095502. [7] E. Okunishi, H. Sawada, J. Kondo, M. Kersker, Atomic resolution elemental map of EELS with a Cs corrected STEM, Microscopy and Microanalysis 12 (Supplement 02) (2006) 1150–1151.
[8] M. Bosman, V.J. Keast, J.L. Garcia-Munoz, A.J. D’Alfonso, S.D. Findlay, L.J. Allen, Two-dimensional mapping of chemical information at atomic resolution, Physical Review Letters 99 (8) (2007) 086102. [9] K. Kimoto, T. Asaka, T. Nagai, M. Saito, Y. Matsui, K. Ishizuka, Elementselective imaging of atomic columns in a crystal using STEM and EELS, Nature 450 (7170) (2007) 702–704. [10] D.A. Muller, L.F. Kourkoutis, M. Murfitt, J.H. Song, H.Y. Hwang, J. Silcox, N. Dellby, O.L. Krivanek, Atomic-scale chemical imaging of composition and bonding by aberration-corrected microscopy, Science 319 (5866) (2008) 1073–1076. [11] L.J. Allen, Electron microscopy—new directions for chemical maps, Nature Nanotechnology 3 (5) (2008) 255–256. [12] N. Bonnet, N. Brun, C. Colliex, Extracting information from sequences of spatially resolved EELS spectra using multivariate statistical analysis, Ultramicroscopy 77 (3–4) (1999) 97–112. [13] N.J. Zaluzec, M.G. Blackford, K.L. Smith, M. Colella, HARECES measurements of carbon K shell excitation in graphite, Microscopy and Microanalysis 11 (S2) (2005) 718–719. [14] N.D. Browning, J. Yuan, L.M. Brown, Real-space determination of anisotropic electronic-structure by electron-energy loss spectroscopy, Ultramicroscopy 38 (3–4) (1991) 291–298. [15] R.D. Leapman, P.L. Fejes, J. Silcox, Orientation dependence of core edges from anisotropic materials determined by inelastic-scattering of fast electrons, Physical Review B 28 (5) (1983) 2361–2373. [16] I.T. Jolliffe, Principal Component Analysis, Springer-Verlag, New York, 1986. [17] M.R. Keenan, P.G. Kotula, Accounting for Poisson noise in the multivariate analysis of ToF-SIMS spectrum images, Surface and Interface Analysis 36 (2004) 203–212. [18] L.J. Allen, S.D. Findlay, M.P. Oxley, C. Witte, N.J. Zaluzec, Channeling effects in high-angular-resolution electron spectroscopy, Physical Review B 73 (9) (2006) 094104. [19] C. Witte, S.D. Findlay, M.P. Oxley, J.J. Rehr, L.J. Allen, Theory of dynamical scattering in near-edge electron energy loss spectroscopy, Physical Review B 80 (18) (2009) 184108. [20] P. Schattschneider, C. He´bert, H. Franco, B. Jouffrey, Anisotropic relativistic cross sections for inelastic electron scattering, and the magic angle, Physical Review B 72 (4) (2005) 045142. [21] J.C. Le Bosse´, T. E´picier, B. Jouffrey, Polarization dependence in ELNES: influence of probe convergence, collector aperture and electron beam incidence angle, Ultramicroscopy 106 (6) (2006) 449–460. [22] R.D. Leapman, J. Silcox, Orientation dependence of core edges in electronenergy-loss spectra from anisotropic materials, Physical Review Letters 42 (20) (1979) 1361–1364. [23] M. Nelhiebel, P.H. Louf, P. Schattschneider, P. Blaha, K. Schwarz, B. Jouffrey, Theory of orientation-sensitive near-edge fine-structure core-level spectroscopy, Physical Review B 59 (20) (1999) 12807–12814. [24] B. Jouffrey, P. Schattschneider, C. He´bert, The magic angle: a solved mystery, Ultramicroscopy 102 (1) (2004) 61–66. [25] C. He´bert, P. Schattschneider, H. Franco, B. Jouffrey, ELNES at magic angle conditions, Ultramicroscopy 106 (11–12) (2006) 1139–1143. [26] C. Brouder, Angular-dependence of X-ray absorption-spectra, Journal of Physics: Condensed Matter 2 (3) (1990) 701–738. [27] C. Brouder, M.F. Ruiz Lo´pez, M. Benfatto, C.R. Natoli, R.F. Pettifer, Systematicapproach to the calculation of the polarization-dependent (and polarizationaveraged) general term of the curved-wave multiple-scattering series in the X-ray-absorption cross-section, Physical Review B 39 (3) (1989) 1488–1500. [28] J.C. Le Bosse´, T. E´picier, H. Chermette, Trichroism in energy-loss near-edge structure spectroscopy: polarization dependence of near-edge fine structures, Physical Review B 76 (7) (2007) 075127. [29] G. Radtke, T. Epicier, P. Bayle-Guillemaud, J.C. Le Bosse, N-K ELNES study of anisotropy effects in hexagonal AlN, Journal of Microscopy 210 (2003) 60–65. [30] M.M. Disko, O.L. Krivanek, P. Rez, Orientation-dependent extended finestructure in electron-energy-loss spectra, Physical Review B 25 (6) (1982) 4252–4255. [31] R.A. Rosenberg, P.J. Love, V. Rehn, Polarization-dependent C(K) near-edge Xray-absorption fine-structure of graphite, Physical Review B 33 (6) (1986) 4034–4037.