I. INTRODUCTION
X-ray emission spectroscopy (XES) has been used for chemical state analysis of materials (Lundquist, Reference Lundquist1925; Gohshi et al., Reference Gohshi, Kamada, Kohra, Utaka and Arai1982; Takahara et al., Reference Takahara, Shoji and Ito2022a). The spectral profile provides information about the electronic structure related to coordination, valence, and spin. The Si Kβ XES spectrum is a valence-to-core emission spectrum, so it shows higher sensitivity to chemical states of materials than the Si Kα core-to-core spectrum. Figure 1 shows Si Kβ XES spectra of crystalline Si (c-Si), amorphous Si (a-Si), crystalline SiO2, and amorphous SiO2 (a-SiO2), along with their schematic molecular structures. The peak energy is about 5 eV different between metallic Si and oxide. There is no difference between the amorphous and crystalline phases of silicon oxides because Si has tetrahedral coordination with four oxygen atoms in the local structure. However, the amorphous phase of metallic Si shows an asymmetric bonding profile compared to the crystalline phase due to the variation in the coordination of Si bonding.
The XES spectra and the schematic molecular structures of c-Si, a-Si, a-SiO2, and c-SiO2.
SiO is an amorphous material with a composition of Si and O of 1:1 and is used as a composite electrode with graphite for commercial lithium-ion batteries. Following the disproportionation reaction, it is understood that silicon nanoparticles precipitate within SiO2, assuming a chemical state conducive to optimal battery performance.
In this study, the spectral change of Si Kβ XES is investigated for SiO at different annealing temperatures during the disproportionation reaction. The Si Kβ spectrum of SiO consists of two peaks close to those from metallic Si and SiO2. However, the spectrum before annealing does not fully agree with the linear combination of amorphous Si and SiO spectra, suggesting the contribution of possible intermediate states. To study the disproportionation reaction quantitatively, in the first trial, linear combination fitting (LCF) is applied by assuming the intermediate-state spectrum from the residuals of a linear combination of a-Si and SiO2 in the spectrum before annealing.
Moreover, we have developed a method for spectral analysis that includes unknown chemical states, since the LCF needs the assumption of standard spectra. The matrix factorization (Paatero and Tapper, Reference Paatero and Tapper1994; Smaragdis et al., Reference Smaragdis, Raj and Shashanka2007) is currently used in the analysis of spectroscopic data to reduce the dimensionality of data. By matrix factorization, we can extract characteristic patterns from a large amount of measurement data. We will combine this method with the LCF to include intermediate states and their estimated spectra.
II. EXPERIMENTAL
A. Sample preparation
SiO powder for lithium-ion batteries, amorphous structures of a few micrometers in grain size, was heat-treated for 18 hours in the temperature range of 700–1,100°C under an Ar atmosphere. In the following, the measurement spectrum of the untreated sample is labeled as 1. The measurement spectrum at temperatures 700, 800, 900, 1,000, and 1,100°C are labeled as 2, 3, 4, 5, and 6, respectively. Amorphous Si (1-μm-thick film) and chemical reagent powders of crystalline Si, amorphous SiO2, and crystalline SiO2 (Kojundo chemical) were used as reference materials.
B. Measurement conditions
The samples before and after heat treatment were pasted on carbon tape, and Si Kα and Kβ XES measurements were performed. A wide-area double-crystal monochromator (Gohshi et al., Reference Gohshi, Kamada, Kohra, Utaka and Arai1982) made by Rigaku with a 4-kW Rh target X-ray tube, a gas-flow type proportional counter (F-PC), and InSb (111) monochromator crystals with the (++) configuration (anti-parallel configuration) were used. The sweep angle is from 128.7° to 131.7° (1,820–1,850 eV) with 0.01° step, and the integration time is 50 seconds per step. The analysis diameter was ϕ10 mm, and the measurements were performed under a vacuum atmosphere.
III. RESULTS AND DISCUSSION
A. Measurement result
Figure 2a shows the Si Kβ spectra of SiO during heat treatment. The spectral shape changes slightly and monotonically with increasing temperature. We can see a decrease in intensity at the locations indicated by black arrows as the temperature increases. Note that the spectrum on the SiO2 side also became narrower with increasing temperature, as shown in the zoomed window in Figure 2b.
(a) Change of Si Kβ spectral shape of SiO with the heat treatment temperature. Around the energy region about 1834 eV and 1838 eV (indicated by the black arrows in Figure 2(a)), the intensity decreases as the heat treatment temperature increases. The rectangle region is zoomed up in Figure 2(b). Around the energy region about 1830 eV (indicated by the black arrow in Figure 2(b)), the spectrum became narrower with increasing temperature.
B. Linear combination fitting
For the quantification of the XES spectra, the previous study considered the LCF method using some known spectra as the reference spectra (Takahara et al., Reference Takahara, Shoji, Yoshimoto and Kobayashi2022b). The LCF tries to reconstruct a sample spectrum from a linear combination of standard spectra. Suppose we have a spectrum f of a sample and area-normalized spectra {fi} (i = 1, 2, …, S), where S is the number of the reference spectra. Then, the LCF is expressed as follows:
$$ f=\sum \limits_{i=1}^S{K}_i{f}_i. $$
The coefficients {Ki} of the reference spectra are to be optimized by, for example, a least square method with nonnegative constraint (Gilbert et al., Reference Gilbert, Luttrel, Stout and Vogt2008). From the coefficients, we can quantify the weight fraction wj of a spectrum fj (j = 1, 2, …, S) by the fraction as
$$ {w}_j=\frac{K_j}{\sum \limits_{i=1}^S{K}_i}. $$
If we can assume that the X-ray intensity per Si atom is the same, the weight fraction corresponds to the molar ratio.
The three reference spectra of c-Si, a-Si, and a-SiO2 and the result of the LCF are shown in Figure 3. The black line represents the measured spectrum of the untreated sample, and the gray line represents the fitting result of LCF using the three spectra. We can see a large residual intensity between the measured spectrum and the fitting result, which suggests the possible intermediate states such as SiO x.
The three reference spectra of the LCF. The spectrum of c-Si, a-Si, and SiO2 is shown in Figures 3(a), 3(b), and 3(c), respectively. The fitting results for the spectrum of the untreated sample are shown in Figure 3(d). The black line represents the measured spectrum of untreated sample, and the gray line represents the fitting result of LCF using the three reference spectra. The weighted reference spectra are shown. The residue is the difference between measured spectrum and the fitting result.
To include intermediate states in the LCF, we consider an additional standard called “Peak 1,” as shown in Figure 4a. “Peak 1” is broader than the spectrum of a-SiO2. Since there is no reference material for it, “Peak 1” was obtained as an unknown asymmetric Lorentzian function in the fit of data 1, which had the most residuals. “Peak 1” is added as a reference spectrum for the LCF analysis.
(a) The fitting result for data 1. The black line represents the measured spectrum of data 1, and the red line represents the fitting result of LCF using the three reference spectra and “Peak 1”. The weighted spectra are shown. The difference is the difference between measured spectrum and the fitting result. (b) The quantification results of the LCF for data 1 to 6.
Figure 4 shows the LCF results using c-Si, a-Si, a-SiO2, and “Peak 1” as the reference spectra. Figure 4a is the fitting result for data 1. We can see that the residual at lower energy is suppressed. Figure 4b is the quantification result as a function of the heat treatment temperature. As the temperature increases, the weight fraction of “Peak 1” decreases, whereas that of a-SiO2 increases. This reaction causes a decrease in the intensity around 1,834 eV and a narrowing of the peak shape around 1,830 eV, as shown in Figure 2. Then, the contribution of a-Si decreases, and that of c-Si increases at higher temperatures. This reaction causes a decrease in the intensity around 1,838 eV, as shown in Figure 2. The quantification results suggest that the intermediate states could disappear in the lower temperature treatment.
C. Semi-supervisor optimization
We develop a method called the semi-supervisor optimization (SSO) using the matrix factorization. The SSO assumes the existence of unknown states and estimates the spectral shapes. It is based on the nonnegative matrix factorization to the data matrix X where measurement spectrum is set in each row.
$$ {\displaystyle \begin{array}{l}X\left(n,m\right)={\sum}_{r=1}^RW\left(n,r\right)B\left(r,m\right)+{\sum}_{s=1}^S{W}^{\prime}\left(n,s\right){B}^{\prime}\left(s,m\right),\\ {}n=1,2,\dots, N,\\ {}m=1,2,\dots, M.\end{array}} $$
Here, N is the number of the measurement spectra and M is the number of points in the measurement spectrum. R is the number of unknown states, and S is the number of known spectra. We call the N × R matrix W and the N × S matrix W′ the weight matrices, and the R × M matrix B and the S × M matrix B′ the basis matrices. The elements in matrices W, B, and W′ are variables to be optimized, whereas those in matrix B′ are fixed. As shown in Figure 5, an unknown spectrum to be optimized and a known spectrum are set in each row of the basis matrices B and B′, respectively.
Schematic representation of the SSO. Measurement spectrum is entered in each row of data matrix X. An unknown spectrum to be optimized and a known spectrum is set in each row of the basis matrix B and B’, respectively. The coefficient of the spectrum is set in weight matrix W and W’.
The SSO tries to factorize the data matrix X into an unsupervised part (WB) and a semi-supervised part (W′B′) by minimizing the distance between the left-hand side and right-hand side of Eq. (3) with nonnegative constraint. The optimization was done using the majorization-minimization algorithm (Hunter and Lange, Reference Hunter and Lange2000; Kitamura et al., Reference Kitamura, Saruwatari, Yagi, Shikano, Takahashi and Kondo2014). The unsupervised part represents the superposition of unknown spectra. Only with this part, Eq. (3) becomes the equation of nonnegative matrix factorization and is solved by, for example, alternating least squares (Paatero and Tapper, Reference Paatero and Tapper1994). The supervised part represents the superposition of known spectra (supervisor data). Only with this part, Eq. (3) is reduced to Eq. (1) of the LCF. Therefore, the SSO assumes some known spectra and uses them during the factorization process. Because we use supervisor data in the factorization process, we call this method the SSO. If each row in the base matrices is normalized (i.e., the spectrum is area-normalized), we can quantify the measurement spectrum by using the value of the corresponding rows of weight matrices.
For the analysis of the XES spectra, we assume one unknown spectrum called “Component 1” to represent the intermediate state spectrum. Figure 6 shows the result of the SSO. Figure 6a is the optimization result of data 1, and the spectrum of “Component 1” obtained by the SSO is shown. Figure 6b is the quantification result as a function of the heat treatment temperature. For comparison, the spectra of “Peak 1” and “Component 1” are shown in Figure 7. The shape of “Component 1” is similar to that of “Peak 1,” while two higher-energy peaks are obvious in “Component 1.” They would suppress the residue and make the contribution of “Component 1” larger than that of “Peak 1” in the LCF result. Both quantification results showed a similar trend in the temperature dependence of the heat treatment, which demonstrates the effectiveness of the SSO. Since the SSO does not require all the spectra of the materials in the samples, it would be useful for analyzing samples that are expected to have unknown chemical states and intermediate states.
(a) The fitting result for data 1. The black line represents the measured spectrum of data 1, and the red line represents the fitting result of SSO using the three reference spectra and unknown spectrum called “Component 1”. The weighted spectra are shown. The residue is the difference between measured spectrum and the fitting result. (b) The quantification results of the SSO for data 1 to 6.
The spectra of “Peak 1” used in the LCF and “Component 1” obtained by the SSO.
IV. CONCLUSION
We have shown two methods for the spectral analysis of XES and applied them to lithium-ion battery materials. The LCF using an asymmetric Lorentzian has been introduced. Then, a new method called the SSO has been developed, giving the quantification result consistent with that of the LCF. Although the LCF requires all the spectra of the materials in the sample, the SSO assumes the inclusion of unknown spectra of some materials and enables quantitative analysis by estimating the spectra. Therefore, the SSO will be effective for quantification analysis when samples are expected to have unknown chemical states and intermediate states, such as the disproportionation reaction by the heat treatment of lithium-ion battery materials.
The analysis shown in this paper can be done using SmartLab Studio II. The SSO is implemented in the cluster analysis. The quantification result can be obtained using the Direct Derivation or DD method™ (Toraya, Reference Toraya2016, Reference Toraya2024). Although the DD method requires the chemical formula for each component in addition to the profile, we can use the same formula, such as Si, to obtain results.
CONFLICTS OF INTEREST
The authors declare no competing interests.
