Fluctuation Component Analysis-based K-Means Clustering in 4D-STEM of Heterogeneous Materials

Fluctuation analysis as embodied in fluctuation electron microscopy (FEM) is a powerful method for the study of disordered materials [1]. While FEM examines diffraction intensities Ik in the reciprocal space, similar analysis can be applied to the Fourier transform of log(Ik) in real space. The result of Fourier transform is known as cepstrum and the cepstral peaks can be related to interatomic distances. By analyzing fluctuations of cepstral signals, Pidaparthy et al. have demonstrated that structural heterogeneity in a Si anode material could be separated by the difference in interatomic lengths of amorphous and crystalline phases, and thus provide a much-needed new approach for the study of disordered materials, which is named fluctuation cepstral scanning transmission electron microscopy (FC-STEM) [2]. However, identifying different phases from FC-STEM images can be challenging. Here, we introduce Fluctuation Component Analysis (FCA) as clustering technique for FC-STEM. The basic idea is that fluctuation of a certain cepstral region can reflect the uniqueness of a material such as crystallinity and atomic distances. A set of well-defined area fluctuation signals can then be used to represent each phase or crystal grains as a vector of major components. Figure 1 shows an example. The annular fluctuation profile obtained from the identified silicon region is peaked near 0.44nm, while the fluctuation profile of the carbon region is peaked near 0.28nm and 0.52nm. These characteristic peaks in the fluctuation spectrum can be used to identify silicon and carbon uniquely in this case. For the general phase analysis, we extract finite number of regional fluctuations in cepstral patterns as phasic parameters using annular masks of different radii. Each cepstral pattern can be represented by a vector including a set of fluctuation components. These vectors after additional transformation, such as weight balance, form the FCA space vectors. K-means clustering define k number of groups and mandatorily separate the all samples into one of the k groups by counting the Euclidean spacing or other geometric spacing [3]. Here, we apply k-means clustering to the FCA extracted vectors by considering each vector as a virtual spatial position. Patterns with similar features have similar fluctuations, and therefore could be grouped into the same cluster. Figure 2.(c-d), show the clustering results of the Silicon lithium ion battery (LIB) anode sample with 3 and 6 groups, respectively. As the Figure shows, the results compare well to the EDS maps of silicon and carbon signals from Figure 2.(a-b). The FCA based analysis with 3 groups identifies the spatial distributions of three phases: silicon (purple), graphic carbon (yellow), and (oxy)fluorophosphate (blue) as shown in Figure 2.(c). Figure 2.(d) shows a more detailed separations using of groups, with the appearance of compositional boundaries information. Therefore, our approach provides possibility for detecting subtle features embedded in a disordered material. The FCA analysis is general and can be extended to any spectral signals collected in a STEM. a) the average fluctuation profile of silicon and carbon regions from annular masks with inner outer radius gap of 10 pixels, b) FC-STEM reconstruction with silicon region highlighted in red and carbon region highlighted in blue, also corresponding to the regional average fluctuation profile in a). EDS maps of Silicon a), and carbon b); k-mean clustering with FCA vectors with k=3 c), with k=6 d).