B k k. Fig. 1: Energy-loss spectrum of BN, showing the how K-loss intensities I K (β, ) for boron and nitrogen are defined and measured.

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1 The accuracy of EELS elemental analysis The procedure of EELS elemental analysis can be divided into three parts, each of which involves some approximation, with associated systematic or statistical errors. (A) Algorithm of the method. Unlike EDX microanalysis, in which an experimentallybased sensitivity factor is used for each element, EELS normally uses a standardless method of estimating the concentration ratio of two elements (A and B), based on [1]: A B CA Ik ( β, ) σ k ( β, ) (1) B A C I ( β, ) σ ( β, ) B k k The coreloss intensities I k (β, ) and cross sections σ k (β, ) depend on the range of integration of the scattering angle (i.e. the collection semiangle β) and of energy loss (i.e. the energy window ), as well as on the incident energy E 0. Fig. 1: Energy-loss spectrum of BN, showing the how K-loss intensities I K (β, ) for boron and nitrogen are defined and measured. Equation (1) is expected to be a good approximation if the specimen is sufficiently thin, such that the inelastic scattering recorded by the spectrometer contains only a small contribution from electrons that were also elastically scattered; see Fig.2. In practice the elemental ratio is found to be reliable if the thickness is less than about half the inelastic mean free path λ i (Zaluzec, 1983), where λ i is of the order of 100 nm for light-element samples. Small thickness is also beneficial in terms of core-loss signal/background ratio (SBR in Fig.2).

2 Fig.2. Location of the angle-limiting aperture (e.g. objective aperture) relative to the Bragg beams of electrons diffracted from a crystalline TEM specimen. Inelastically scattered electrons surround each Bragg beam but their angular distribution is narrow and (for a sufficiently thin specimen) their intensity low, so that relatively little inelastic scattering from non-central beams enters the aperture. Fig.3. Intensity ratio I K B (β, )/I K N (β, ) plotted against the thickness of a boron nitride specimen, together with the boron and nitrogen signal/background ratios, I K B (β, )/I b B (β, ) and I K N (β, )/I b N (β, ). Alternatively, an aberration-corrected spectrometer can be used to record all of the inelastic and elastic scattering but this requires a collection semiangle β exceeding 100 mrad and leads to poor edge/background ratio at most ionization edges. Strictly speaking, Eq.(1) is exact only if we integrate over all energy loss (large integration window ), which considerably increases the background-subtraction errors [3]. (B) Background subtraction. Ionization edges occur on a relatively large background and the core-loss intensity decays has an extended tail at higher energy loss. Good

3 background subtraction is therefore critical to obtaining accuracy. For this reason, increasingly complex (and time-consuming) methods have been devised to separate the core-loss intensity from the background. (1) Blind extrapolation involves fitting the background before the edge (energy loss E < edge threshold E k ) to a simple analytical function, usually AE -r where A and r are constants usually determined by least-squares fitting over a fitting window of chosen width (typically around 50 ev). The accuracy of the fit depends on the noise level in the background data; extrapolation results in a statistical error about h/2 times larger than if the background could be interpolated, h being typically in the range 4 to 14 [3]. In addition, systematic errors are possible if A and r vary with energy loss, which they do in practice because the power-law fit is only an approximation. Fig.3. Blind extrapolation involves least-squares fitting in a pre-edge window (mauve) and extrapolation beneath the ionization edge. For a tied background, the fitting takes place in two energy windows, below and above the ionization threshold, which is equivalent to interpolation of the background. Guided extrapolation ensures that the intensity in the upper fitting window (yellow) consistent with a power-law decay of the core-loss intensity. (2) Tied or guided extrapolation. If least-squares fitting is extended to two energy windows, before and after the ionization threshold (Fig.3), extrapolation is replaced by interpolation, reducing the statistical error. The necessary procedure can be carried out with most versions of Gatan s Digital Micrograph software. However, the backgroundsubtracted intensity now falls to zero at the upper energy window, which introduces a systematic error. Guided extrapolation involves assuming that the core-loss intensity has a similar E-dependence to the background, allowing its contribution in the upper window to be estimated. Matlab or C versions of the necessary software are available [4].

4 (3) Multiple least-squares fitting requires ionization edges of standards containing the elements involved to be recorded under conditions similar to those used to acquire the spectrum recorded from the sample of interest. The latter is fitted to a weighted sum of a power-law background and the necessary core-loss profiles S n (E): F(E) = AE r + B 1 S 1 (E) +B 2 S 2 (E) +... (2) The weighting coefficients B n are then a measure of the elemental concentrations, Eq. (1) being used to ensure that the core-loss components are in the correct ratios [5]. Provided the core-loss edges of the standards are similar to those of the spectrum being analysed, the statistical error of background fitting is minimized. This fitting procedure is similar to a method employed in EDX spectroscopy [6]. (4) Multivariate statistical analysis is another procedure taken from other forms of spectroscopy. Its most common form, principal component analysis (PCA), is now available in the Gatan DM software for application to spectrum-image (SI) data. The SI is factorized into a score matrix and a loading matrix, whose rows contain independent spectral components (eigenspectra) and whose columns contain the spatial distribution of these components. The components are ranked according to their magnitude and plotted against component number in a so-called Scree plot, which exhibits a rapid decay followed by a more gradual tail representing noise components. By using only the principal components that precede the tail, noise is largely eliminated. The other advantage of PCA is that it provides an unbiased analysis of a large amount of data, which would be tedious to analyse on a pixel-by-pixel basis [7,8]. Fig.4. Principal component analysis of spectrum-image data recorded from thin flakes of BN on a holey carbon support [7]. The components of the SI data are ranked in a scree plot; each represents a property of the specimen or an artifact of the analysis. Higher-number components represent noise and are therefore discarded.

5 (C) Core-loss cross sections take the place of sensitivity-factors in Eq.(1). They can be calculated on the basis of atomic physics, ignoring the effects of interatomic interaction in a solid, which are calculated to be below 5% for > 20 ev [9]. For K-shells, a hydrogenic approximation is appropriate and the calculation can be done rapidly by a short computer program, with only the atomic number Z, incident energy E 0, and values of β and as input parameters [10]. For other shells, the Hartree-Slater method is preferable and the calculation is more elaborate; however the results have been parameterized [11]. Comparison with experimental EELDS and x-ray absorption measurements suggests that K-shell cross sections are mostly known to within 5% and L- shell cross sections to 10%, with errors around 20% possible for M- and N-shells [12]. EELS can also be used to measure an absolute elemental concentration n (atoms per unit volume) through use of the formula: I k (β, ) = n t I l (β, ) σ k (β, ) (3) where I l (β, ) is the intensity in the low-loss region, including the zero-loss peak and integrated up to an energy loss, the same value as used to measure the core-loss intensity I k (β, ) and to calculate the core-loss cross section σ k (β, ). The specimen thickness t can be obtained from analysis of the low-loss region [4]. Fig. 5. Low-loss spectrum and an ionization edge, showing how the measured core-loss intensity I k (β, ) contains a contribution (cross-hatched) from electrons that undergo both plasmon and core-loss scattering, in addition to the intensity I k 1 (β, ) that represents only core-loss scattering. I k (β, )/I k 1 (β, ) = I(β, )/I 0 is assumed in Eq.(3) and a similar assumption justifies Eq.(1).

6 References [1] R.F. Egerton, Ultramicroscopy 3 (1978) 243. [2] N.J. Zaluzec (1983) Proc. 41 st EMSA meeting (1983), p.388. [3] R.F. Egerton, Ultramicroscopy 9 (1982) 387. [4] R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3 rd edition (Springer, New York, 2011); see also [5] R.D. Leapman and C.R. Swyt, Ultramicroscopy 26 (1988) 393. [6] N.J. Zaluzec, Ultramicroscopy 18 (1985) 185. [7] M. Bosman et al., Ultramicroscopy 106 (2006) [8] S. Lozano-Perez et al., Ultramicroscopy 109 (2009) [9] X. Weng and P. Rez, Ultramicroscopy, 25 (1988) 345. [10] R.F. Egerton, Ultramicroscopy 4 (1979) 169. [11] Rez, P., Ultramicroscopy 9 (1982) 283. [12] R.F. Egerton, Ultramicroscopy 50 (1993) 13.