Low-energy electron microscopy intensity-voltage data - Factorization, sparse sampling and classification

Abstract

Low-energy electron microscopy (LEEM) taken as intensity-voltage (I-V) curves provides hyperspectral images of surfaces, which can be used to identify the surface type, but are difficult to analyse. Here, we demonstrate the use of an algorithm for factorizing the data into spectra and concentrations of characteristic components (FSC³ ) for identifying distinct physical surface phases. Importantly, FSC³ is an unsupervised and fast algorithm. As example data we use experiments on the growth of praseodymium oxide or ruthenium oxide on ruthenium single crystal substrates, both featuring a complex distribution of coexisting surface components, varying in both chemical composition and crystallographic structure. With the factorization result a sparse sampling method is demonstrated, reducing the measurement time by 1-2 orders of magnitude, relevant for dynamic surface studies. The FSC³ concentrations are providing the features for a support vector machine-based supervised classification of the surface types. Here, specific surface regions which have been identified structurally, via their diffraction pattern, as well as chemically by complementary spectro-microscopic techniques, are used as training sets. A reliable classification is demonstrated on both example LEEM I-V data sets.

Keywords: classification; hyperspectral analysis; low-energy electron microscopy; oxide films; praseodymia; rare-earth oxides; ruthenium dioxide; sparse sampling.

FIGURE 1

Sketch of the analysis method. LEEM I–V stacks are recorded, and factorized into non‐negative components and spectra by FSC³. Manually selected regions are used as a training set for a classifier on the resulting component concentrations. Sparse sampling retrieves the FSC³ concentrations from a smaller number of spectral points, down to the number of FSC³ components. The spectral positions are optimized for the smallest number of unclassified or misclassified points, resulting in a measurement speed‐up by an order of magnitude

FIGURE 2

FSC³ analysis of LEEM I–V data taken on PrO _x . (A) Concentrations of the components ${\mathcal{C}}_i$, $i=1\text{…}9$ are given in the images on a greyscale from $m=0$ to M as labelled, together with the spectral error E _s. For the latter, a logarithmic scale has been used between the given m and M. The concentration error E _c is shown on a colour scale of red (blue) hue for $E_{\mathrm{c}}>0$ ($E_{\mathrm{c}}<0$) as shown. The value is proportional to $\log (|E_{\mathrm{c}}|)$, with black indicating $|E_{\mathrm{c}}|\le 0.01$. Each spatial region used as training set for classification is shown in one of the images, and is encircled by coloured lines. (B) Component spectra S

FIGURE 3

(A) Classification of LEEM I–V data taken on PrO _x . The hue of the colour represents the assigned class i, with a saturation given by $\max (1-{\stackrel{\sim }{\sigma }}_i/5,0)$, indicating the assignment confidence. (B) Component concentrations with standard deviations taken from the training regions indicated in Figure 2 with the same colour coding as the classification results. (C) Class LEEM I–V spectra (solid lines) considering only points with $\stackrel{\sim }{\sigma }\le 3$, with the shaded region indicating plus and minus the standard deviation of the spectra classified into the class region. The separate $\stackrel{\sim }{\sigma }$ for each class are given in the Figure S1

FIGURE 4

Results of the classification using sparse sampling. (A) f _l versus the number of spectral points N _s for spectrally equidistant points (black), random (blue), random walk (red), gradient‐based (green) and surrogate (magenta) optimization and sequential selection (cyan). The checker board black fields are used to define factorization, classification and optimization of spectral points, while white fields serve to verify the method on an unseen data set. The empty symbols show the FODs during the optimization procedure (black fields), while the full symbols (horizontally offset) refer to the FOD obtained when applying the method in the white validation fields. The latter are slightly displaced on the horizontal axis for visibility. The dashed line shows $1/N_{\mathrm{s}}$. The probability threshold used was ${\stackrel{\sim }{\sigma }}_{\mathrm{t}}=3$. The classification images b‐f combine the results obtained using the full spectral information (in the black fields) with the results of the sparse sampling (in the white fields) obtained either using N _s= 9 equidistant points or one of the optimization methods as labelled. Colour code as Figure 3. The results for ${\stackrel{\sim }{\sigma }}_{\mathrm{t}}$ of 2 and 4 using the random and random walk sampling methods are given in Figure S6

FIGURE 5

Same as Figure 4, but for the minimization of f _m. The results for ${\stackrel{\sim }{\sigma }}_{\mathrm{t}}$ of 2 and 4 using the random and random walk optimization methods are given in Figure S7

FIGURE 6

(A–C) Spectral points (symbols) obtained by minimizing the FODs, f _l (A) and f _m (C) using the random (blue), random walk (red), surrogate (green) and sequential (orange) optimization. The solid line shows the spatially averaged spectrum of the data. (B,D) The thumbnails show the LEED intensity of a selected region at the energies of the sampled spectral points, in the same horizontal order. The colour frames identify the sampling method used to minimise f _l in (B) or f _m in (D). The results use ${\stackrel{\sim }{\sigma }}_{\mathrm{t}}=3$. The results for ${\stackrel{\sim }{\sigma }}_{\mathrm{t}}$ of 2 and 4 for the random methods are given in Figures S16 and S17, respectively

FIGURE 7

(A) LEEM image of the oxidized Ru(0001) surface taken at 19.0 eV with a field of view of 10 μm. (B) Mean I–V spectra extracted from the RuO₂ data by averaging over the spatial regions as indicated by encircling lines of corresponding colour in (A), used as training set for the classification

FIGURE 8

FSC³ analysis and classification of LEEM I–V data taken on a RuO₂/Ru(0001) surface. (A) Concentrations of the components ${\mathcal{C}}_i$, $i=1\text{…}6$ are given in the images on a greyscale from $m=0$ to M as labelled, together with the spectral error E _s, concentration error E _c and weight w calculated by the weighted factorization. E _s and w are shown on a logarithmic scale between m and M as indicated. The concentration error E _c is shown on a colour scale red (blue) hue for $E_{\mathrm{c}}>0$ ($E_{\mathrm{c}}<0$), with value proportional to $\log (|E_{\mathrm{c}}|)$, and black indicating $|E_{\mathrm{c}}|<0.01$. The spatial regions used as training sets for classification into five classes are indicated by lines coloured according to the associated class, in the component image where they are most pronounced. The inset of ${\mathcal{C}}_2$ shows an example of the regions used for the training of c₅, from m = 0.18 to M = 0.28. (B) Component spectra S. (C) Classification results, with the hue indicating the class, and the saturation the confidence, as in Figure 3. The component concentrations over the training regions and the class spectra are shown in Figure S4, and separate $\stackrel{\sim }{\sigma }$ for each class are given in Figure S5