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Atsushi Machida, [Kenji Nagata](https://orcid.org/0000-0001-9894-4461), [Ryo Murakami](https://orcid.org/0000-0001-8585-9268), [Hiroshi Shinotsuka](https://orcid.org/0000-0001-5147-1396), Hayaru Shouno, [Hideki Yoshikawa](https://orcid.org/0000-0002-7389-8865), Masato Okada

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[Bayesian inference method utilizing SESSA in quantitative layer structure estimation from XPS data](https://mdr.nims.go.jp/datasets/a7136889-1aaa-48a8-95f9-e30132a984ca)

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Bayesian inference method utilizing SESSA in quantitative layer structure estimation from XPS dataJournal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A0Contents lists available at ScienceDirectJournal of Electron Spectroscopy and Related Phenomenajournal homepage: www.elsevier.com/locate/elspecBayesian inference method utilizing SESSA in quantitative layer structureestimation from XPS dataAtsushi Machida a, Kenji Nagata b, Ryo Murakami c, Hiroshi Shinotsuka c, Hayaru Shouno d,Hideki Yoshikawa b,c, Masato Okada c,e,∗a Graduate School of Science, The University of Tokyo, Bunkyo, Japanb Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Japanc Materials Data Platform, Research Network and Facility Services Division, National Institute for Materials Science, Tsukuba, Japand Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu, Japane Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, JapanA R T I C L E I N F OKeywords:X-ray photoelectron spectroscopyBayesian estimationExchange Monte Carlo methodSESSAA B S T R A C TX-ray photoelectron spectroscopy (XPS) is a surface analysis technique for the nondestructive identificationof elemental species and chemical states of solid samples, and the measured spectra are affected by not onlysample-specific information but also factors dependent on the measurement environment. This feature makesit difficult to analyze the data for the chemical state identification of mixed samples when referring to thedata measured with different models or in different environments. In a previous study, Bayesian inferencewas successfully applied to the analysis of XPS narrow-scan spectra, but the challenge was to apply Bayesianinference to XPS spectra of samples that are nonuniform in the depth direction. We propose a method toinfer the layer structure of a sample from XPS spectra by incorporating Bayesian inference into the simulationof electron spectra for surface analysis (SESSA). SESSA can simulate XPS spectra of samples with specifiedcomposition and microstructure, and is already in use as a simulator with highly reproducible results. Byutilizing the proposed method, one can estimate the layer structure of a sample from XPS data on thebasis of the posterior probability distribution. In a typical XPS measurement, wide-scan data are acquired toqualitatively identify elemental species, and narrow-scan data are acquired to the estimate detailed compositionand chemical state information of a sample. In this study, we have shown that given wide-scan or narrow-scandata without angle resolution, Bayesian inference can be applied to quantitatively analyze the layer structureinformation.1. IntroductionX-ray photoelectron spectroscopy (XPS) is a surface analysis tech-nique to nondestructively identify elemental species and their chemicalstates in solid samples through of the measurement of the energyspectrum of photoelectrons emitted by a sample irradiated with X-rays.XPS is characterized by the fact that not only the intrinsic sample-specific information but also extrinsic factors such as charging andenergy resolution, which depend on the measurement environment,affect the actual measurement spectrum. This feature of XPS makesit difficult to analyze the data when identifying the chemical statescontained in a mixed sample by referring to the individual chemicalstate XPS spectra measured in different environments in the litera-ture. To solve this problem, Machida et al. studied the application ofBayesian inference to spectral data analysis in XPS, and they succeededin automating data analysis for chemical state identification and inquantitatively determining its accuracy [1]. In this study, in the XPS∗ Corresponding author.E-mail address: okada@edu.k.u-tokyo.ac.jp (M. Okada).data generation process, the sample is assumed to be spatially uniformand the background due to inelastic photoelectron scattering is assumedto be generated by the approximate formula of the Shirley method,and the usefulness of this model is demonstrated by analyzing artificialdata. On the other hand, this generative model could not analyzethe microstructural structure of a sample that is nonuniform in thedepth direction, such as a sample with having a layered structure.Angle-resolved XPS is commonly used for the nondestructive analysisof the microstructural structure of samples that are nonuniform in thedepth direction [2–4]. On the other hand, as is well known to XPSexperts, it is possible to estimate the depth-directed microstructuralinformation of a sample from only its wide-scan spectrum withoutangle resolution with a wide energy range [5]. Wide-scan spectra areoften limited to qualitative analysis owing to the many and complexfactors that generate them, but they are rich in depth information.In a previous study [1], we applied Bayesian inference methods tovailable online 29 May 2024368-2048/© 2024 The Authors. Published by Elsevier B.V. This is an open access ahttps://doi.org/10.1016/j.elspec.2024.147449Received 9 February 2024; Received in revised form 30 April 2024; Accepted 22 Mrticle under the CC BY license (http://creativecommons.org/licenses/by/4.0/).ay 2024https://www.elsevier.com/locate/elspechttps://www.elsevier.com/locate/elspecmailto:okada@edu.k.u-tokyo.ac.jphttps://doi.org/10.1016/j.elspec.2024.147449https://doi.org/10.1016/j.elspec.2024.147449http://creativecommons.org/licenses/by/4.0/Journal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.tssde𝐵2esst𝐵wvathe analysis of XPS narrow-scan spectra of the inner-shell levels ofeach element in the mixed samples that are uniform in the depthdirection and confirmed their validity for the component identificationof the samples. In this study, we applied Bayesian inference methodsto XPS wide-scan spectra of mixed samples of nonuniform systemsconsisting of even more complex factors, enabling the data analysisof wide-scan spectra, which is often limited to qualitative analysis inthe past, leading to quantitative analysis. The background of spectraldata is correlated with the layer structure of the sample as it coversa wide energy range from the peak to a high binding energy [6], andtherefore in this study, the background of a wide-scan spectrum wasincluded in the analysis target of Bayesian inference. However, theShirley method used in our previous study [1] is an approximate forthe background of a narrow-scan spectral model and cannot reproducethe background over a wide-scan, which means that it is difficult to esti-mate the layer structure of a sample. To overcome the above-mentionedissues, we turned to the simulation of electron spectra for surfaceanalysis (SESSA), developed by Smekal and coworkers [7,8]. SESSAcontains the physical data necessary to quantitatively interpret XPSspectra of samples with a specified composition and microstructure,and it can simulate the XPS spectra for user-defined sample structuresand measurement conditions, and the simulated XPS spectra can becompared with the measured spectra. It also simulates not only peaksof photoelectrons and Auger electrons but also the background due tothe inelastic scattering of signal electrons; thus, it has been used asa simulator with highly reproducible results of analyses that use XPSdata.In this study, we propose a Bayesian inference method that employsSESSA as a generative model. Bayesian inference is a framework forprobabilistically inferring variables, assuming that model parametersbehave probabilistically. Also, by probabilistically modeling the datageneration process, we can calculate the likelihood probability of ob-taining data based on certain parameters, and by applying Bayes’theorem, we can get the posterior probability of the parameters whencertain data is obtained [9,10]. By combining this framework withSESSA, it is possible to estimate sample information with reliabilityevaluation from a variety of sample candidates. To incorporate SESSAinto Bayesian inference, the graphical user interface-based softwarewas replaced by a virtual window system that can be run from thecommand line. Exchange Monte Carlo methods, which are widely usedin statistical mechanics and machine learning to efficiently sampleprobability distributions in high-dimensional spaces by running multi-ple Markov chains simultaneously and exchanging states, were used toderive the posterior probability distribution of the parameters relatedto the layered structure of a sample from the given data. By using themethod proposed in this study, one can estimate the layer structureof a sample from XPS data via the posterior probability distributionwith confidence intervals. In a typical XPS measurement, wide-scanspectra are acquired for the qualitative identification of elementalspecies, including checking for the presence of impurities, and narrow-scan spectra are acquired to estimate detailed information on thecomposition and chemical state of the sample. We have confirmed theusefulness of the proposed method by conducting simulations in whichBayesian inference is applied to situations in which wide-scan andnarrow-scan data without angle-resolved measurements are given sep-arately, and by examining the extent to which the quantitative analysisof layer structure information is possible from both approaches.2. Methods2.1. Data generative modelIn this study, artificial data from SESSA are used for the analysis.In this section, we assume a layered sample and explain the processof data generation from the assumed sample. For a layered sample,three assumptions are made. First, the interface of each layer is steep.2pSecond, the substate is pure Si and the number of layers on the Sisubstrate is known in advance, but not the thicknesses of layers. Third,the elemental species composing each layer are also known in advance,but not the composition. From these assumptions, the parameter set forthe sample to be estimated is 𝜽𝑠𝑎𝑚𝑝𝑙𝑒 = {𝑇𝑗 , {𝑥𝑗,𝑚}𝑀𝑗𝑚=1}𝐿𝑗=1, where 𝐿 ishe number of layers excluding the Si substrate and 𝑀𝑗 is the numberof constituent elements in each layer 𝑗. 𝑇𝑗 is the thickness of layer 𝑗and 𝑥𝑗,𝑚 is the ratio of element 𝑚 in layer 𝑗. The elemental ratio isconstrained to be ∑𝑀𝑗𝑚 𝑥𝑗,𝑚 = 1 for all 𝑗. Two points should be notedhere. First, the output directly obtained from SESSA has an arbitraryunit on the vertical axis and does not reflect the actual intensityof the measurement. Therefore, in this study, an intensity correctionparameter ℎ was introduced and multiplied directly to the output fromSESSA to adjust it. Second, the background of wide-scan spectra wascalculated by combining peak backgrounds and matrix backgroundbecause SESSA has an upper limit of the total number of energy pointsin a simulated spectrum, and wide-scan and narrow-scan spectra withbackground were calculated separately. Unlike the previous study [1],SESSA also simulates the backgrounds due to inelastic scattering frompeak components within a specified energy range. However, whensimulating over a specified energy range, the effect of the matrixbackground from peaks at energies lower than the specified range isnot taken into account. Therefore, another simulation was performedand this was added as a correction. A constant background of a fewcounts, 𝑐, was also added as a correction based on the assumption thatnoise is generated by the detector, assuming the experimental case. Insummary, the spectral intensity 𝑓 (𝑥;𝜣) at energy 𝑥 under the givenparameter set 𝜣 can be formulated as𝑓 (𝑥;𝜣) ={𝑆(𝑥;𝜽𝑠𝑎𝑚𝑝𝑙𝑒) + 𝐵(𝑥;𝜽𝑠𝑎𝑚𝑝𝑙𝑒, 𝑐)}× ℎ, (1)where ℎ is an intensity correction parameter for adjusting SESSA outputto the measurement data and varies depending on the irradiated X-ray intensity and measurement time. The parameter set in the datagenerative model is 𝜣 = {𝜽𝑠𝑎𝑚𝑝𝑙𝑒, ℎ, 𝑐}. However, in the case of narrow-can data, the noise due to the detector is assumed to be sufficientlymall and 𝑐 = 0, and only background correction by SESSA is applied togenerate the data. In the case of wide-scan data, all peaks are includedin the range, i.e., the entire matrix background is taken into account,and no constant background correction is performed by SESSA. Thespectral intensity defined in Eq. (1) corresponds to the expected valuein the measurement and does not take into account the statistical noisefrom the observations. Since the XPS data are usually count data underthe experimental condition of pulse counting detection, the intensity 𝑦at the energy 𝑥 under the given parameter set 𝜣 follows the Poissondistribution shown below:𝑝(𝑦|𝜣) =𝑓 (𝑥;𝜣)𝑦 exp (−𝑓 (𝑥;𝜣))𝑦!. (2)Fig. 1 shows the actual process used to generate the dataset . Theataset  is generated from a Poisson distribution with 𝑓 (𝑥;𝜣) as thexpected value, where 𝑓 (𝑥;𝜣) consists of two components, 𝑆(𝑥;𝜣) and(𝑥;𝜣)..2. Estimation modelIn the wide-scan case, the data generation process assumed in thestimation employs the model described in Section 2.1. In the narrow-can case, however, the generation of the spectrum requires two SESSAimulations, which is computationally time-consuming. Therefore, inhe estimation, we assumed the following matrix background:𝑛𝑎𝑟𝑟𝑜𝑤(𝑥;𝜣) = 𝑙 +𝑥 − 𝑥𝑚𝑖𝑛𝑥𝑚𝑎𝑥 − 𝑥𝑚𝑖𝑛(𝑟 − 𝑙), (3)here 𝑥𝑚𝑖𝑛 and 𝑥𝑚𝑎𝑥 are respectively the minimum and maximumalues of energies in the range and {𝑙, 𝑟} is the background intensityt the endpoints. Note that when estimating using this model, thearameter set is 𝜣 = {𝜽 , ℎ, 𝑙, 𝑟}, with no true value for {𝑙, 𝑟}. Given𝑠𝑎𝑚𝑝𝑙𝑒Journal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.Fig. 1. Generation process for the data used in this study. For clarity, the background used in Fig. 1(b) is shown at a higher intensity than that used in the estimation experiment,also such a large detector-dependent background is not realistic in actual XPS measurements. Also, ℎ is the intensity parameter corresponding to the measurement time.a dataset  with 𝑁 data, the likelihood of the data given the parameterset 𝜣 is as follows:𝑝(|𝜣) =𝑁∏𝑖=1𝑝(𝑦𝑖|𝜣) =𝑁∏𝑖=1𝑓 (𝑥𝑖;𝜣)𝑦𝑖 exp(−𝑓 (𝑥𝑖;𝜣))𝑦𝑖!. (4)Bayes’ theorem yields the following posterior distribution for the pa-rameters when using this likelihood:𝑝(𝜣|) =𝑝(|𝜣)𝑝(𝜣)𝑝(). (5)However, 𝑝(𝜣) is a prior distribution with respect to the parameters,and Bayesian estimation can incorporate a priori information withrespect to the parameters in the form of such a probability distributionin the estimation. 𝑝() = ∫ 𝑝(|𝜣)𝑝(𝜣)d𝜣 is a normalization constantindependent of 𝜣. In the Bayesian estimation employed in this study,parameters are estimated probabilistically by obtaining this posteriordistribution.2.3. Exchange Monte Carlo methodIf the parameters are multidimensional, it is difficult to obtainEq. (5) analytically. Therefore, in this study, we prepared replicasgiven by the following equations and performed exchange Monte Carlosampling [11]:𝑞(𝜣; 𝛽) = 𝑝(𝜣) exp −𝛽𝐸(𝜣) , (6)3( )𝐸(𝜣) = − log (𝑝(|𝜣)) . (7)In Eq. (6), 𝛽 is an auxiliary variable called the inverse temperature,which is used to relate the energy of the system to the probabilitydistribution of states. In exchange Monte Carlo, multiple Markov chainsare run at different inverse temperatures and the states are exchangedbetween the chains, resulting in efficient sampling. 𝐸 is an errorfunction, and 𝑞(𝜣; 𝛽 = 1) ∝ 𝑝(𝜣|). The exchange Monte Carlomethod enables global sampling without falling into local solutionsby preparing the inverse temperature sequence 0 = 𝛽1 < 𝛽2 < ⋯ <𝛽𝜏 = 1 and performing Monte Carlo sampling for each replica, as wellas exchanging between replicas. Exchange Monte Carlo methods areused to estimate the posterior distribution by sampling parameters thatfollow the posterior distribution.2.4. Set upIn this study, the parameters related to peak width and peak positionnot explained above have the same as those referenced by default inSESSA (see Table 1).We used the computer that has Intel(R) Xeon(R) CPU E5–2697A v4at 2.60 GHz with 2 cores and 251 GB memory. Monte Carlo calculationswere performed for 6000 steps for each replica, 3000 steps of whichwere rejected as burn-in. Thirty-two replicas were prepared, and theinverse temperatures were set as follows:𝛽 = 𝛾 𝑖−32 (𝑖 = 2, 3,… , 31). (8)𝑖Journal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.Table 1Parameters related to peak position and peak width referenced by defaultin SESSA.Peak Type Position [eV] Width [eV]Si2s (Substrate) Gauss 149.7 1.00Si2p1∕2 (Substrate) Gauss 100.3 0.58Si2p3∕2 (Substrate) Gauss 99.7 0.59Si2s (2nd layer) Gauss 149.7 1.00Si2p1∕2 (2nd layer) Gauss 103.7 1.39Si2p3∕2 (2nd layer) Gauss 103.7 1.39O1s (2nd layer) Gauss 533.0 1.70O2s (2nd layer) Gauss 41.6 1.00C1s (1st layer) Gauss 285.7 1.24O1s (1st layer) Gauss 534.7 1.80O2s (1st layer) Gauss 41.6 1.00Fig. 2. Assumed sample: SiO2 layer on a Si substrate with a thickness of 10 Å andCO2 adsorbed on the surface with a thickness of 5 Å.3. Experiments and discussionIn this study, we generated artificial data from a hypotheticalsample as shown in Fig. 2, and conducted two experiments to estimatesample information using the proposed method. One is an estimationexperiment from one-set wide-scan data, and the other is an estimationexperiment from narrow-scan data focusing on three atomic orbitals,O1s, C1s, and Si2p. The parameter 𝜽𝑠𝑎𝑚𝑝𝑙𝑒 and its true value 𝜽∗𝑠𝑎𝑚𝑝𝑙𝑒 areas follows:𝜽𝑠𝑎𝑚𝑝𝑙𝑒 = {𝑇1, 𝑇2, 𝑥1,C, 𝑥1,O, 𝑥2,Si, 𝑥2,O}, (9)𝜽∗𝑠𝑎𝑚𝑝𝑙𝑒 = {𝑇 ∗1 , 𝑇∗2 , 𝑥∗1,C, 𝑥∗1,O, 𝑥∗2,Si, 𝑥∗2,O}, (10)𝑇 ∗1 = 10Å, 𝑇 ∗2 = 5Å, (11)𝑥∗1,C = 0.333333, 𝑥∗1,O = 0.666666, (12)𝑥∗2,Si = 0.333333, 𝑥∗2,O = 0.666666. (13)3.1. Estimation from one-set wide-scan data3.1.1. Artificial data and parameter setFirst, estimation was performed from the wide-scan data shown inFig. 3. The energy step in this data set is 1.0 eV. However, the truevalue of the intensity correction parameter was set to ℎ∗ = 1.0 × 109and the true value of the background correction parameter was set to𝑐∗ = 2.0. Since there is only one set of data, the dataset and parameterset for this experiment are as follows: = {𝑥𝑖, 𝑦𝑖}𝑁𝑖=1, (14)𝜣 = {𝜽 , ℎ, 𝑐}, (15)4𝑠𝑎𝑚𝑝𝑙𝑒where 𝑁 is the number of data points for the wide-scan data, and ℎ and𝑐 are the intensity and background correction parameters, respectively.The prior distributions for each parameter were set as follows:𝑝(𝜣) = 𝑝(𝑇1)𝑝(𝑇2)𝑝(𝑥1,C, 𝑥1,O)𝑝(𝑥2,Si, 𝑥2,O)𝑝(ℎ)𝑝(𝑐), (16)𝑝(𝑇1) = 𝐺𝑎𝑚𝑚𝑎(𝑇1; 𝑛𝑇 , 𝜇𝑇 ) ≡1𝛤 (𝑛𝑇 )𝜇𝑛𝑇𝑇𝑇 (𝑛𝑇 −1)1 exp(−𝑇1𝜇𝑇), (17)𝑝(𝑇2) = 𝐺𝑎𝑚𝑚𝑎(𝑇2; 𝑛𝑇 , 𝜇𝑇 ), (18)𝑝(𝑥1,C, 𝑥1,O) = 𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(𝑥1,C, 𝑥1,O; 𝛼) ≡𝛤 (2𝛼)𝛤 (𝛼)2𝑥𝛼−11,C 𝑥𝛼−11,O , (19)𝑝(𝑥2,Si, 𝑥2,O) = 𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(𝑥2,Si, 𝑥2,O; 𝛼), (20)𝑝(ℎ) = 𝐺𝑎𝑚𝑚𝑎(ℎ; 𝑛ℎ, 𝜇ℎ), (21)𝑝(𝑐) = 𝐺𝑎𝑚𝑚𝑎(𝑐; 𝑛𝑐 , 𝜇𝑐 ), (22)𝑛𝑇 = 6, 𝜇𝑇 = 1.5, 𝛼 = 1, (23)𝑛ℎ = 𝑛𝑐 = 11, (24)𝜇ℎ = 1.0 × 108, (25)𝜇𝑐 = 2.0 × 10−1. (26)Since all parameters are non-negative, a gamma distribution was used.However, the Dirichlet distribution was used for the elemantal ratiosbecause of the limitation that the sum of the elemantal ratios must be1 in each layer.3.1.2. ResultsFig. 4 shows the posterior distributions and maximum a posteriori(MAP) estimates of each sample parameter obtained by Bayesian es-timation. MAP estimation is a method for estimating the value thatmaximizes the posterior probability.Fig. 4 also shows that sampling for both elemental ratio and thick-ness is concentrated around the true value. The MAP estimates donot always agree with the true values, but this can be attributed tothe effect of noise in the data. On the other hand, the distributionof the elemental ratios of the CO2 layer is more extensive than thatof other parameters, indicating that the accuracy is low. The peak ataround 290 eV in Fig. 3 corresponds to that derived from the C1sorbital, indicating that the data intensity is low. It is considered thatthe insufficient intensity here makes it difficult to accurately estimatethe elemental ratio of the C elements. Below are the results of fittingthe data using the MAP estimates (see Fig. 5). It can be seen that thefitting can be performed correctly on the data.3.2. Estimation from narrow-scan data3.2.1. Artificial data and parameter setNext, an estimation experiment was conducted using the three setsof narrow-scan data shown in Fig. 6. The energy step in these datasets is 0.10 eV. Since there are intensity parameter ℎ and backgroundcorrection parameters 𝑙 and 𝑟 for each set of narrow-scan data, theparameter set 𝜽𝑠𝑎𝑚𝑝𝑙𝑒 in this experiment is as shown in Eq. (27).𝜣 = {𝜽𝑠𝑎𝑚𝑝𝑙𝑒, {ℎ𝑘, 𝑙𝑘, 𝑟𝑘}𝑘=O1s,C1s,Si2p} (27)Note that there are no true values for the background correction param-eters 𝑙 and 𝑟, and the true values of the intensity correction parametersare ℎ∗O1s = 1.0×109, ℎ∗C1s = 5.0×109 and ℎ∗Si2p = 2.5×109. Since the estima-tion is performed simultaneously from three sets of data, the likelihoodis as in Eq. (28). The dataset is  = {{𝑥𝑖, 𝑦𝑖}𝑁𝑘𝑖=1}3𝑘=1, where 𝑘 is the labelcorresponding to the three atomic orbitals O1s, C1s, and Si2p, and 𝑁𝑘 isthe number of data points in the narrow-scan data for atomic orbital 𝑘.𝑝(|𝜣) =3∏𝑁𝑘∏𝑝(𝑦𝑖|𝜣) (28)𝑘=1 𝑖=1Journal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.Fig. 3. Wide-scan data generated from the assumed sample.Fig. 4. Results of posterior distributions of sample parameters. The vertical axis shows the sampling frequency, and the dotted lines represent the true values and MAP estimates.The vertical axis is on a logarithmic scale.The prior distributions for each parameter were set as follows:𝑝(𝜣) = 𝑝(𝑇1)𝑝(𝑇2)𝑝(𝑥1,C, 𝑥1,O)𝑝(𝑥2,Si, 𝑥2,O)3∏𝑘=1𝑝(ℎ𝑘)𝑝(𝑙𝑘)𝑝(𝑟𝑘), (29)𝑝(𝑇1) = 𝐺𝑎𝑚𝑚𝑎(𝑇1; 𝑛𝑇 , 𝜇𝑇 ), (30)𝑝(𝑇2) = 𝐺𝑎𝑚𝑚𝑎(𝑇2; 𝑛𝑇 , 𝜇𝑇 ), (31)𝑝(𝑥1,C, 𝑥1,O) = 𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(𝑥1,C, 𝑥1,O; 𝛼), (32)𝑝(𝑥 , 𝑥 ) = 𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(𝑥 , 𝑥 ; 𝛼), (33)52,Si 2,O 2,Si 2,O𝑝(ℎ𝑘) = 𝐺𝑎𝑚𝑚𝑎(ℎ𝑘; 𝑛𝑘, 𝜇ℎ𝑘 ) for 𝑘= O1s,C1s,Si2p, (34)𝑝(𝑙𝑘) = 𝐺𝑎𝑚𝑚𝑎(𝑙𝑘; 𝑛𝑘, 𝜇𝑙𝑘 ) for 𝑘= O1s,C1s,Si2p, (35)𝑝(𝑟𝑘) = 𝐺𝑎𝑚𝑚𝑎(𝑟𝑘; 𝑛𝑘, 𝜇𝑟𝑘 ) for 𝑘= O1s,C1s,Si2p, (36)𝑛O1s = 𝑛C1s = 𝑛Si2p = 11, (37)𝜇ℎO1s = 1.0 × 108, (38)𝜇 = 5.0 × 108, (39)ℎC1sJournal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.Fig. 5. Fitting results obtained using MAP estimates.Fig. 6. Three sets of narrow-scan data generated from the assumed sample.𝜇ℎSi2p = 2.5 × 108, (40)𝜇𝑙O1s = 𝜇𝑟O1s = 27, (41)𝜇𝑙C1s = 𝜇𝑟C1s = 1.6 × 102, (42)𝜇𝑙Si2p = 𝜇𝑟Si2p = 1.0. (43)3.2.2. ResultsFig. 7 shows the posterior distributions and estimated values of eachsample parameter obtained by Bayesian estimation.Fig. 7 also shows that the sampling concentrated around the truevalues for both elemantal ratio and thickness. The MAP estimatesdo not always agree with the true values, but this can be attributedto the effect of noise in the data. However, as in Section 3.1, theelemental ratios of the CO2 layer have wider distributions than theother parameters, and the accuracy is low. We will discuss the reasonsfor this.From the setup in this experiment, the main component contributingto the narrow-scan data for the C1s orbital is only the C element in theCO2 layer. However, the data intensity in Fig. 6(b) is dominated bythe matrix background, and the intensity from the C1s orbital is low.This makes it difficult to estimate the elemental ratio of the C elements.Below are the results of fitting the data using the MAP estimates (seeFig. 8). It can be seen that the fitting can be performed correctly on thedata.3.3. DiscussionTo discuss the results in Sections 3.1.2 and 3.2.2 quantitatively,Bayesian credible interval (BCI) [12,13] results for each experimentare presented in Tables 2 and 3. BCI is the interval of the posterior inwhich a parameter is included under a certain probability and is usedin Bayesian statistics to indicate parameter uncertainty. In this study,the 95% BCI was calculated from the sampling obtained.6Table 2True values, MAP estimates and BCI in estimation from wide scan data.parameter True value MAP value 95% BCIC/O ratio (1st layer) 0.500 0.410 [0.394 0.554]Si/O ratio (2nd layer) 0.500 0.601 [0.489 0.604]Thickness (1st layer) [Å] 5.0 5.3 [4.8 5.3]Thickness (2nd layer) [Å] 10.0 10.6 [10.0 10.6]From the results in Sections 3.1.2 and 3.2.2, it appears that thesampling is distributed around the true values and the fitting with theMAP estimates is successful. Similarly, it can be seen from Tables 2and 3 that the 95% BCI for all parameters in both results also containtrue values. From the results shown in Section 3.2.2, it was foundthat for the present three-layer model, information on compositionand film thickness in the depth direction can be obtained sufficientlyby analyzing narrow-scan spectra up to 30 eV away from the peakwithout angle-resolved measurement. Also these results indicates thatwide-scan data can be sufficiently used for the quantitative analysisby this method. Moreover, quantitative analysis of wide-scan datacan contribute to the efficiency of the comprehensive analysis of XPSdata, because such analysis can lead to the immediate detection ofunexpected situations in actual samples (e.g., the presence of impurityelements).However, we believe that there is room for improvement withregard to the accuracy of the estimation, which can be attributed tothe small number of Monte Carlo steps. In this study, the number ofsteps was set to 6000 for the sake of computation time, but this maynot have been sufficient to converge to the posterior distribution. Theestimation accuracy can also be improved by increasing the intensityof the data used.Journal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.Fig. 7. Results of posterior distributions of sample parameters. The vertical axis shows the sampling frequency, and the dotted lines represent the true values and MAP estimates.The vertical axis is on a logarithmic scale.Fig. 8. Fitting results obtained using MAP estimates.Table 3True values, MAP estimates and BCI in estimation from narrow scan data.parameter True value MAP value 95% BCIC/O ratio (1st layer) 0.500 0.424 [0.422 0.613]Si/O ratio (2nd layer) 0.500 0.456 [0.454 0.522]Thickness (1st layer) [Å] 5.0 4.7 [4.7 5.1]Thickness (2nd layer) [Å] 10.0 10.2 [9.8 10.3]4. ConclusionIn this paper, we proposed a Bayesian estimation method utiliz-ing the simulator SESSA for XPS data analysis. In conventional XPSdata analysis, it has been considered difficult to obtain information7on the layer structure using only narrow-scan data, and analysis incombination with wide-scan data analysis has been necessary. It wasalso believed that wide-scan data alone could only provide qualitativefindings and not quantitative layer structure information. We usedBayesian inference to estimate the layer structure from artificial datafor these two situations. The analysis of two sets of synthetic data in thisstudy show that the proposed method can be used to estimate the layerstructure information independently from both wide-scan and narrow-scan data. However, we believe that there is room for improvement interms of estimation accuracy, which could be improved by increasingthe intensity of the data or by using a sufficiently large number ofMonte Carlo steps. This achievement is significant for the automatedanalysis of XPS data, and future prospects include the extension ofJournal of Electron Spectroscopy and Related Phenomena 273 (2024) 147449A. Machida et al.ODAKfRthis method and its application to real data. However, for computa-tional convenience, in this paper, we made some assumptions aboutthe sample in the estimation; thus, the estimation can be consideredsuccessful. It is necessary to confirm the validity of various sampleconditions by changing the assumptions to different sample conditionsand by conducting experiments. In addition, it is necessary to solve theheavy computation time of SESSA in order to conduct more generalexperiments. These issues will be addressed in the future.CRediT authorship contribution statementAtsushi Machida: Writing – original draft. Kenji Nagata: Projectadministration. Ryo Murakami: Writing – review & editing. HiroshiShinotsuka: Writing – review & editing. Hayaru Shouno: Projectadministration. Hideki Yoshikawa: Project administration. Masatokada: Project administration.ata availabilityData will be made available on request.cknowledgmentsThis work was supported by JSPS KAKENHI Grant Numbers JP23J0471 and JP23H00486, and CREST Grant Number JPMJCR1761rom the Japan Science and Technology Agency (JST).eferences[1] A. Machida, K. Nagata, R. Murakami, H. Shinotsuka, H. Shouno, H. Yoshikawa,M. 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