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M. S. S. Khan, S. F. Mao, Y. B. Zou, Y. G. Li, [B. Da](https://orcid.org/0000-0002-0785-8662), Z. J. Ding

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Uncertainty evaluation of Monte Carlo simulated line scan profiles of a critical dimension scanning electron microscope (CD-SEM)ViewOnlineExportCitationCrossMarkRESEARCH ARTICLE |  JUNE 30 2023Uncertainty evaluation of Monte Carlo simulated line scanprofiles of a critical dimension scanning electronmicroscope (CD-SEM) M. S. S. Khan  ; S. F. Mao  ; Y. B. Zou  ; Y. G. Li   ; B. Da  ; Z. J. Ding  J. Appl. Phys. 133, 245303 (2023)https://doi.org/10.1063/5.0153379 13 December 2023 02:50:02https://pubs.aip.org/aip/jap/article/133/24/245303/2900707/Uncertainty-evaluation-of-Monte-Carlo-simulatedhttps://pubs.aip.org/aip/jap/article/133/24/245303/2900707/Uncertainty-evaluation-of-Monte-Carlo-simulated?pdfCoverIconEvent=citehttps://pubs.aip.org/aip/jap/article/133/24/245303/2900707/Uncertainty-evaluation-of-Monte-Carlo-simulated?pdfCoverIconEvent=crossmarkjavascript:;https://orcid.org/0000-0001-7723-2887javascript:;https://orcid.org/0000-0002-2370-1585javascript:;https://orcid.org/0000-0002-0572-1726javascript:;https://orcid.org/0000-0003-2820-2424javascript:;https://orcid.org/0000-0002-0785-8662javascript:;https://orcid.org/0000-0001-5767-1145javascript:;https://doi.org/10.1063/5.0153379https://servedbyadbutler.com/redirect.spark?MID=176720&plid=2281281&setID=592934&channelID=0&CID=837567&banID=521596642&PID=0&textadID=0&tc=1&scheduleID=2201582&adSize=1640x440&data_keys=%7B%22%22%3A%22%22%7D&matches=%5B%22inurl%3A%5C%2Fjap%22%5D&mt=1702435802262581&spr=1&referrer=http%3A%2F%2Fpubs.aip.org%2Faip%2Fjap%2Farticle-pdf%2Fdoi%2F10.1063%2F5.0153379%2F18025785%2F245303_1_5.0153379.pdf&hc=76d2058683ee997ee3d3ff5f4bdf0cf9d63e91e9&location=Uncertainty evaluation of Monte Carlo simulatedline scan profiles of a critical dimension scanningelectron microscope (CD-SEM)Cite as: J. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379View Online Export Citation CrossMarkSubmitted: 7 April 2023 · Accepted: 7 June 2023 ·Published Online: 30 June 2023M. S. S. Khan,1,2 S. F. Mao,3 Y. B. Zou,4 Y. G. Li,2,a) B. Da,5 and Z. J. Ding1,6,a)AFFILIATIONS1Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China2Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei, Anhui 230031,People’s Republic of China3Department of Nuclear Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026,People’s Republic of China4School of Physics and Electronic Engineering, Xinjiang Normal University, Urumchi, Xinjiang 830054, People’s Republic of China5Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan6Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei,Anhui 230026, People’s Republic of Chinaa)Authors to whom correspondence should be addressed: ygli@theory.issp.ac.cn and zjding@ustc.edu.cnABSTRACTIn recent years, precision and accuracy for a more precise critical dimension (CD) control have been required in CD measurement tech-nology. CD distortion between the measurement by a critical dimension scanning electron microscope (CD-SEM) and a reference tool isthe most important factor for a more accurate CD measurement. CD bias varies by a CD-SEM and a pattern condition. Therefore, it isurgently needed to identify, characterize, and quantify those parameters that may or may not affect the CD measurement by a CD-SEM.The sensitivity of the Monte Carlo simulated CD-SEM images with multiple physical modeling components has been studied previously.In this study, we demonstrate that the work function and elastic scattering potential models have a significant impact on secondary elec-tron emission intensity, but their influence on the shape of the linescan profile is small, and other factors like the optical energy lossfunction and dielectric function models have even smaller effects. We have evaluated the uncertainty in the linescan profiles of Si linestructures with different sidewall angles due to several different physical factors. It is found that when the CD is evaluated by a peak/valley method, the uncertainty of the CD is negligible. Therefore, it is concluded that the CD value and its related uncertainty are notcritically related to the physical factors of the present Monte Carlo simulation model but rely dominantly on the line structure and elec-tron beam parameters.Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0153379I. INTRODUCTIONMonte Carlo simulation techniques in microbeam analysis,electron beam lithography, electron spectroscopy, and scanningelectron microscopy (SEM) have been in use for decades.1–10 Thedemand for Monte Carlo simulation study of scanning electronmicroscope (SEM) imaging11 mainly comes from the interpretationof the image contrast and imaging mechanisms in experiments. Inrecent years, the application of the Monte Carlo technique to criti-cal dimension scanning electron microscope (CD-SEM) images hasbecome an important topic for industrial application.12In the semiconductor industry, the CD-SEM is a specializedlength measuring instrument for the critical dimension (CD) mea-surement of nanostructures13 in the integrated circuit due to itshigh resolution, fast imaging speed, and non-destructive processing.The CD is the minimum geometrical feature size in the waferJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-1Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://doi.org/10.1063/5.0153379https://doi.org/10.1063/5.0153379https://pubs.aip.org/action/showCitFormats?type=show&doi=10.1063/5.0153379http://crossmark.crossref.org/dialog/?doi=10.1063/5.0153379&domain=pdf&date_stamp=2023-06-30https://orcid.org/0000-0001-7723-2887https://orcid.org/0000-0002-2370-1585https://orcid.org/0000-0002-0572-1726https://orcid.org/0000-0003-2820-2424https://orcid.org/0000-0002-0785-8662https://orcid.org/0000-0001-5767-1145mailto:ygli@theory.issp.ac.cnmailto:zjding@ustc.edu.cnhttps://doi.org/10.1063/5.0153379https://pubs.aip.org/aip/japlimited by photolithography technology used for the fabricationprocess; it is one of the important quantities in dimensionalmetrology. Accurate CD measurements are needed to meet thestrict dimensional control requirements in manufacturing. Theaccuracy of the measured CD values is a challenging issue with verytight specifications in process control for mask and wafer fabrica-tion. Secondary electron imaging is the mode used in CD-SEMobservation, which allows one to observe the sample structure withhigh contrast. CD metrology based on secondary electron imagingconsists of two steps, namely, pixel-based secondary electron signalimaging and subsequent extraction of CD values from the intensityprofile of secondary electron signals for edge detection by anappropriate algorithm.14 A secondary electron image is apoint-by-point representation of the secondary electron intensityproduced by a raster scan of the primary electron beam across thesample surface. The secondary electron intensity at a given positionon the sample surface is contributed by the secondary electronsemitted from the nearby locations around the incident position inthe excited volume. In a secondary electron image of a line struc-ture, the line edges are usually brighter than their surroundingsdue to the edge effect resulting from the enhanced effective emis-sion area around the edge than elsewhere. Therefore, the secondaryelectron profile has a characteristic peak with a certain extension atan edge.15CD metrology relies on the understanding of the relationshipbetween the intensity profile of secondary electron signals and thesample structure. Although the CD-SEM has a good contrast andhigh resolution, it is still subjected to an unavoidable error in CDmetrology because the intensity of secondary electron signals is acomplicated function of the instrument setup, specimen geometricstructure, and material composition. An empirical method for CDmetrology is based on the artificially specified brightness thresholdor, in some special cases, is determined by using a referencesample.16 The methods described for CD measurement include aline-fit algorithm,13 linear regression to baseline, maximal deriva-tive, sigmoidal fit,17 and profile characterization of several parame-ters.18,19 These empirical algorithms have limitations, e.g., arbitraryCD definitions13 and single-valued CD; they might be useful onlyunder certain conditions but give incorrect estimates under otherconditions. Furthermore, dimensional metrology based on second-ary electron imaging has greater potential in 3D metrology, and theempirical algorithm cannot meet the needs in this area.The CD metrology would include significant uncertainty if itwere not having a solid physical background of a secondary elec-tron signal generation mechanism. A physical modeling basedmetrology algorithm should be the best way to meet the measure-ment tolerances and strict CD control requirements. Thus, theMonte Carlo simulation technique by modeling electron interactionprocesses in the sample has played an important role in developingthe CD metrology algorithm20–24 to derive an accurate method forlinewidth determination. Several simple algorithms have been pro-posed for determining linewidths based on the Monte Carlo simu-lation results in different areas.25–31 These CD determinationmethods are approximate and also have certain limitations for theapplication circumstances. To overcome the limitations, a model-based library (MBL) method was proposed28–36 and adopted by anISO standard.37 By this MBL method, a Monte Carlo simulationmethod that models the physical process of electron beam interac-tion with a specimen in the CD-SEM imaging is used to establish anumerical data library of simulated linescan profiles for variousconcerned experimental parameters. The Monte Carlo simulationmakes it possible to establish a one-to-one relationship between thesecondary electron linescan profile, the geometric structure, andthe beam condition. The MBL method could be an ideal tool forCD-SEM metrology at advanced nodes for its high accuracy, andits effectiveness has been verified experimentally.34,35Despite extensive theoretical and experimental works doneaiming for accurate estimation of the CD, however, the evaluationof the error sources and uncertainty has been rarely reported so far.The uncertainty in CD may depend on several experimentalfactors, including electron beam conditions (e.g., angle of inci-dence, beam diameter, and focusing), sample structure (structuralshape and related parameters, e.g., top and bottom line widths, lineheight, sidewall angles, top rounding, and footing for a trapezoidalline structure). Particularly, the uncertainty in CD determinationdue to the physical factors (work function, energy loss function,and elastic and inelastic mean free path calculation models) in aMonte Carlo modeling employed by the MBL method has also notbeen fully investigated. Only sensitivity analysis of the effects of, forexample, inelastic scattering cross section, model sensitivity, side-wall roughness, and surface potential on the linewidth measure-ments has been performed.38–41 In this study, we have performed aCD-SEM image simulation combined with some of the uncertain-ties to arrive at a more detailed characterization of CD-SEM mea-surements of the CD. It should be noted that certain uncertaintiesassociated with systematic errors, unknown beam size, andunknown edge angle are not explicitly addressed in this studybecause the focus of our analysis lies on evaluating and quantifyingtheoretical uncertainties within the context of the simulatedCD-SEM images.First, all error sources of uncertainties considered here in theMonte Carlo modeling of CD-SEM imaging are sorted into twocategories: (1) the input of material properties (i.e., work functionand optical constants of the material) and (2) modeling of elec-tron scattering cross sections (i.e., scattering potential for the cal-culation of electron elastic scattering cross section and dielectricfunction model for the calculation of electron inelastic scatteringcross section). The measurement uncertainty is then evaluatedaccording to the principle of a Monte Carlo uncertainty quantifi-cation approach,42,43 whose standard procedure consists of thefollowing steps: (a) definition of the measurand and inputs,(b) modeling, (c) estimation of probability density functions forthe inputs, (d) setup and execution of the Monte Carlo simula-tions, and (e) summary and presentation of the results. In the fol-lowing, the chief elements in this uncertainty quantificationprocess are described.II. THEORETICAL MODELWe have adopted our up-to-date simulation code, ClassicalTrajectory Monte Carlo simulation of Scanning ElectronMicroscopy (CTMC-SEM),44,45 for this calculation. The simulationof the generation and emission processes of secondary electronsignals is performed by tracing incident electron trajectories insideJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-2Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japthe bulk of a sample with a random sampling technique for elec-tron elastic and inelastic scattering events.A. Geometric representationThe 3D sample structure modeling and the optimization for afast simulation are the two main aspects of a Monte Carlo simula-tion of the CD-SEM image of a complex geometric structure. Mostof the samples in actual observations have various complicatedsurface morphologies or 3D structures. In this study, for thepurpose of uncertainty quantification of the CD, we have consid-ered only the simple trapezoidal-like structure. This is because weaim to evaluate the uncertainty of simulation while ignoring theuncertainties due to geometric structure parameters. Here, a linestructure with a rectangle-shaped cross section is considered, andall the surfaces are smooth by ignoring the surface roughness. A3D structure can be constructed by several approaches, e.g., asimple constructive solid geometry approach has been employedfor SEM imaging.46,47 The geometric structure used in this study isbased on our previous work by using a finite element triangularmesh.48–51 The triangular mesh of a line structure is constructedwith the aid of Gmsh, a freely available GNU (General PublicLicense) meshing software.52 Figure 1 shows a constructed siliconline structure, made of three lines in a 50 × 50 nm rectangle-shapedcross section corresponding to a 90° sidewall angle and placed on asilicon substrate (the density of 2.329 g/cm3). We have alsoemployed the space subdivision method to accelerate our MonteCarlo simulation. This technique involves dividing the simulatedmaterial volume into smaller regions or cells with a 3D finiteelement mesh of the sample surface to improve the computationalefficiency of electron trajectory simulation in the 3D target. Pleaserefer to Ref. 48 for further details on the space subdivision methodused in our study.B. Elastic scattering modelThe sensitivity of the simulated secondary electron signals tothe elastic scattering cross section models has been less investigatedin the literature. Elastic scattering happens when an electron isdeflected by the nuclear potential of an atom without a change ofkinetic energy. The exact probability of the scattering events andthe probability distribution of the deflected angles are described byMott’s cross sections,53 whose formulation was derived from thesolution of the Dirac equation for a kinetic electron moving in anatomic potentialdσedΩ¼ j f (θ)j2 þ jg(θ)j2, (1)where σe is the elastic cross section, f (θ) and g(θ) are the scatteringamplitudes that can be calculated with the partial wave expansionmethodf (θ) ¼ 12iKX1l¼0��‘þ 1��e2iδþ‘ � 1�þ ‘�e2iδ�‘ � 1��P‘(cosθ), (2)g(θ) ¼ 12iKX1‘¼1��e2iδþ‘ þ e2iδ�‘�P1‘ (cosθ), (3)where �hK is the momentum of the electron; δþl and δ�l are spin upand spin down phase shifts of the ‘th partial wave, respectively;P‘(cosθ) and P1‘ (cosθ) are the Legendre and the first order associ-ated Legendre functions, respectively.The phase shifts are calculated from a radial equation of theelectron motion in a potential field V(r), where r represents thedistance from the atomic nucleus of a target atom placed at theorigin of the coordinate system. The effective interaction potentialfor a projectile electron at a given distance r can be described bymeans of an optical potential modelV(r) ¼ Vst(r)þ Vex(r)þ Vcp(r)� iWabs(r), (4)where Vst(r) is the electrostatic potential, Vex(r) is the exchangepotential, Vcp(r) is the correlation-polarization potential, andWabs(r) is the absorption potential (Wabs is taken zero in thisstudy). There are various potential models, which may significantlyaffect the calculated Mott’s cross section and, hence, the scatteringpotential model plays the role of an uncertainty factor in a MonteCarlo simulation.We have used Salvat’s latest version of Fortran 90 code, ElasticScattering of Electrons and Positrons by Atoms (ELSEPA),54 to cal-culate Mott’s cross sections for 384 different scattering potentialmodels in the whole interested electron kinetic energy region fromprimary energy down to the potential barrier for the secondaryFIG. 1. 3D finite element triangularmesh of the line structure.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-3Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japelectron emission. In all these Mott’s cross sections, we have con-sidered four nuclear charge distribution models, four electron dis-tribution models, four electron exchange potential models, andthree correlation-polarization potential models. For each of thesepotential models, we have considered both free atomic potentialand the muffin-tin potential, resulting in a total of 384(=4 × 4 × 4 × 3 × 2) different scattering potentials.55 In the followingexamples, “No. 42001” elastic cross section is used where “4”stands for Helm’s nuclear charge distribution model, “2” for theThomas–Fermi–Dirac electron distribution model, “0” for the non-exchange potential, “0” for the non-correlation-polarization poten-tial, and “1” for the muffin-tin atomic potential model.C. Inelastic scattering modelsUnlike elastic scattering events, an electron changes bothkinetic energy and momentum in an inelastic scattering event.These events play a critical role in secondary electron generation.Energy loss processes include electron interactions with valenceelectrons (single particle and plasmon excitations), inner-shell elec-trons (interband transitions and ionizations), and the solid lattice(longitudinal optical phonon excitations). These energy loss mech-anisms are described by the optical constants or the energy lossfunction according to the dielectric functional theory, and theseenergy loss channels for a kinetic electron are simulated discretelywith the use of the energy loss function. The differential inelasticscattering cross section for an electron moving in a solid of thedielectric function, ε(q, ω), is expressed in terms of the differentialinverse inelastic mean free path56d2λ�1indqdω¼ �hπa0EIm�1ε(q, ω)� �1q, (5)where λin is the electron inelastic mean free path, the average dis-tance that an electron can travel in the material before losingenergy; a0 is the Bohr radius, �hq and �hω are the momentum trans-fer and the energy loss, respectively. The energy loss function,defined as Im{�1/ε}, is a vital quantity to describe the response ofthe medium to the disturbance of an external electron and, thus,completely determinesthe behavior of electron inelastic scatteringin a specific material. However, except for a free electron gas (Alis a typical example of a nearly free electron metal) whose dielec-tric response can be well described by the Lindhard dielectricfunction,57 for other materials, there are several dielectric functionmodels. Even though all these models employ the experimentalmeasured optical constants, which largely specify the behavior ofthe dielectric response at a long wavelength limit (q ! 0), theirextensions into the finite momentum transfers vary. Therefore,the dielectric function model plays another role of uncertaintyfactor in a Monte Carlo simulation. In this study, we haveemployed three dielectric function models: (1) the Levine andLouie model (LLM),58 (2) the full Penn’s algorithm (FPA),57 and(3) the super-extended Mermin algorithm (SMA).59 The FPA andSMA are useful for metals, while the LLM is an extension to semi-conductors and insulators having a bandgap. The Drude-typeoscillator terms used in the SMA were obtained by fitting tothe experimental optical energy loss function data at the opticallimit q ! 0,55 where the Mermin-type oscillator terms agree withthe Drude-type oscillators. While for both the FPA and LLM, theoptical energy loss function data are directly used without the aidof oscillator terms.When an inelastic scattering occurs, the energy loss �hω andmomentum transfer �hq are determined by a random samplingfrom the double differential inverse inelastic mean free path,6 Eq.(5). The cascade electron is assumed to be excited from the valanceband; the excitation probability is proportional to the joint densityof states of a free electron material, i.e., p(E0, �hω)/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE0(E0 þ �hω)p,where E0 is the energy of the electron in the valance band measuredfrom the bottom of the valence band. Then the kinetic energy ofthe generated secondary electron is Es ¼ E0 þ �hω� (Ev þ Eg),where Ev is the width of the valence band and Eg is the bandgap.The energy reference level of the generated secondary electron inthe material is set at the bottom of the conduction band. The direc-tion of the cascade electron is along the direction of momentumtransfer �hq.D. Optical energy loss function dataThe linescan profile is entirely related to the secondary elec-tron signals, while the cascade secondary electrons are generatedin the inelastic scattering events. Therefore, the optical energy lossfunction data, Im{�1/ε(q ¼ 0, ω)}, is an essential input to aMonte Carlo simulation of the linescan profile. The optical dielec-tric function, ε(ω) ¼ ε1(ω)þ iε2(ω), can be derived from theoptical constants, i.e., the refractive index n(ω) and extinctioncoefficient k(ω): ε1 ¼ n2 � k2, ε2 ¼ 2nk. Palik has compiled themeasured optical constants by optical methods from low to highphoton energies, �hω, for some elements and compounds frommany sources into a database.60 However, even in the samephoton energy range, the measured optical energy loss functiondata by different researchers may differ. Therefore, the opticalenergy loss function data as a necessary input to a Monte Carlosimulation may act as an uncertainty factor to impact the simu-lated linescan profile.For Si, as the material considered in this study, we haveattained three combined optical energy loss function datasets(Fig. 4 in Ref. 55). They are named “crystal (Palik),” “crystal(Yang),” and “doped (Palik)” by combining several sources fromPalik,60 Henke et al.,61,62 and Yang et al.63,64 For the Sb-doped (atconcertation of ∼0.002) n-type Si, the energy loss function is muchenhanced below the bandgap as compared with that of pure Si.Limited by the energy resolution of the electron beam technique,the energy loss function in the “crystal (Yang)” dataset below 2 eVwas approximated by a straight line such that the sum rules are sat-isfied. Therefore, all the energy loss functions for crystalline silicon(c-Si) are very close in value above the bandgap, and their differ-ence is mainly at the phonon excitation part below the bandgap. Tofind the accuracy of these energy loss functions, we have used twosum rules, called the oscillator strength sum rule (f-sum rule) andperfect screening sum rule (ps-sum rule), for the verification of thethree energy loss function datasets.64 Each sum rule reinforces therespective energy region of importance and, therefore, these sumrules present a simple overall estimation for the accuracy of theenergy loss function.55Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-4Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japE. Work function dataAfter certain elastic and inelastic collisions inside the sample,an electron may arrive at the surface for emission, and the refrac-tion or reflection of the electron from the surface will be judged bya quantum mechanical transmission coefficient65 asT(E, β) ¼4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� U0/E cos2 βp1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� U0/E cos2 βph i2 , if E cos2 β . U0,0, otherwise,8>><>>:(6)where β is the angle between the direction of a moving electronand surface normal and U0 is the surface potential barrier. An elec-tron gains or loses its kinetic energy by the potential barrier whenpenetrating the surface from the vacuum side or the bulk side,respectively. Hence, the surface barrier plays a significant role insecondary electron emission. When the energy reference level inthe material is set at the bottom of the valence band, U0 ¼ EF þWfor a metal6,44,66 and U0 ¼ Ev þ Eg þ χ for a semiconductor or aninsulator, where EF is the Fermi energy, W is the work function, Evis the width of the valence band, Eg is the bandgap, and χ is theelectron affinity. While in the present study, the energy referenceFIG. 2. Flowchart for estimating uncertainty of the linescan profiles by a Monte Carlo simulation method, where the dielectric function models used are the LLM, FPA, andSMA.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-5Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japlevel of an electron is set as the vacuum level in the vacuum, andthe lowest unoccupied state inside the material [i.e., the Fermi levelfor metals (U0 ¼ W) or the bottom of the conduction band forsemiconductors and insulators (U0 ¼ χ)67]. Since the Fermi level ofa semiconductor is at the mid of energy gap at room temperature,the electron affinity measured from the bottom of the conductionband to the vacuum level is χ ¼ W � Eg /2. For Si, Eg ≃ 1:1 eVand Ev ≃ 12:5 eV.There are several methods for the experimental determinationof the work function, i.e., the thermionic method, the photoelectricmethod, the field-emission method, the effusion method, thecontact potential difference method, and the calorimetric method.For Si, the available data of the work function and electron affinityχ measured by different researchers under different experimentalconditions are listed elsewhere.55 Experimentally, the biggest sourceof the measured secondary electron yield data comes from surfacecontamination and surface roughness. In our present approach, wehave considered part of the contamination effect with the factor ofwork function. The reported range of the measured W is from 3.59to 5.4 eV,68–81 and we used this range for our uncertainty evalua-tion in the simulation.III. MONTE CARLO UNCERTAINTY QUANTIFICATIONThe combined uncertainty in a simulation due to the uncer-tain variables can be derived analytically if a functional expressionof these variables is given. For example, for a functiony ¼ f (x1, x2, . . . , xN ), the combined standard uncertainty for themeasurand y can be determined as u2y ¼PNi¼1(@f /@xi)2u2xi , where uxiis the uncertainty due to the ith input quantity.82 However, in thepresent case, the functional dependence of electron emission yieldson the uncertainty variables (i.e., scattering potential model, dielec-tric function model, energy loss function data, and work functiondata) cannot be explicitly given. Therefore, we have adopted aMonte Carlo uncertainty quantification approach to evaluate thepropagation of uncertainties. For this, one has to first set the proba-bility density functions for the input quantities and then execute aMonte Carlo simulation process for a certain number of times. Thesimulation will provide the distribution of the output and, hence,the uncertainty of the output. In this study, the Monte Carlo uncer-tainty procedure is just the Monte Carlo simulation process of sec-ondary electron emission with different combinations of theselected uncertainty variable values.FIG. 3. Scattering trajectories of 50 primary electrons (black) and their generated cascade secondary electrons (red) nearby a rectangle Si line structure. The black andred arrows indicate the emitted backscattered and secondary electrons. The electron beam of the size of FWHM = 1 nm is normally incident onto (a) and (b) the mid of thetop surface, (c) and (d) the top edge and (e) and (f ) the mid of the bottom substrate surface. Whereas primary energy is (a), (c) and (e) 1 keV and (b), (d), and (f ) 5 keV.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-6Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japFigure 2 demonstrates the flow chart for the present theoreticalcalculation of secondary electron emission and the related uncer-tainty quantification. Uncertain inputs of the material propertiesinvolve work function and energy loss function, while uncertaintyfor the theoretical modeling includes the elastic scattering potentialand the dielectric function model. Here, we have considered that allthe uncertain inputs are equally important and uniformly distribu-ted. Then, several Monte Carlo trials, M ¼ 17280 (a combination of384 scattering potentials, 3 dielectric function models, 5 work func-tion values, and 3 energy loss function datasets) for each incidentenergy are chosen priori. We have employed a parallel computer toperform the Monte Carlo calculation, which is very suitable for ourpresent purpose of uncertainty quantification. For each simulation,we have traced 1 × 105 incident electron trajectories and severaldozens of cascade secondary electron trajectories.IV. RESULTS AND DISCUSSIONFigure 3 demonstrates the simulated electron trajectories,including incident electrons and their generated cascade electronsnear a rectangle Si line structure surface. The primary beam at thebeam size (measured by the full width at half maximum, FWHM)of 1 nm and the energy of 1 and 5 keV is incident onto either thetop surface, the edge, or the substrate of the Si line structure. Onecan observe that at a higher energy, 5 keV [Figs. 3(b), 3(d), and3(f )], the incident electrons can penetrate much deeper inside thestructure than at a lower energy, 1 keV [Figs. 3(a), 3(c), and 3(e)].While secondary electrons are produced more widely in the lateraldirection at the lower energy, representing the stronger neighboringeffect. When the primary beam is incident at an edge side of the Siline structure [Figs. 3(c) and 3(d)], more secondary electrons areemitted from the edge and nearby surfaces, while in the case ofthe beam incident at the substrate position [Figs. 3(c) and 3(f )],the secondary electron emission is more local and depressed due tothe absorption by the nearby structure.Figure 4 compares the spatial density distribution of the inter-nal population of cascade secondary electrons at their birth sitesbetween the different incident positions of an electron beam of1 keV. The top panel shows the distributions for the emitted sec-ondary electrons and the bottom panel for the spatial distributionsof all the generated secondary electrons. This figure more clearlyillustrates the varied local emission properties of secondary elec-trons when the beam is incident at different positions. At the topand bottom substrate surfaces [Figs. 4(a) and 4(d)] the secondaryelectrons are only emitted from the region around the incidentposition within a depth of ∼1 nm. When the incident position is atthe top edge [Fig. 4(b)], many secondary electrons can be emittednot only from the top surface and the side surface but also somefrom the bottom substrate surface far away from the incident loca-tion; hence, the so-called edge effect for the edge blooming inten-sity is produced. When the incident position is at the bottom edge[Fig. 4(c)], the fraction of emission from the line structure (the leftpart in the figure) is attenuated compared to the planar surfacecase, producing the edge bottom dip in the linescan profile of sec-ondary electron intensity.Figure 5 shows the effect of the variation of work functionvalue on the calculated linescan profile of the Si rectangle linestructure. Here, we deal with a rather ideal experimental condition,FIG. 4. Simulated spatial density distributions of the internal population of cascade secondary electrons at their birth sites in a rectangle Si line structure for different inci-dent positions of 1 keV incident electron beam at normal incidence. The top panel (a)–(d) is for the emitted secondary electron signals in SEM imaging and the bottompanel (e)–(h) is for all the generated secondary electrons. The beam incident positions are (from left to right): the mid of the top surface, the top edge, the bottom edge,and the mid of the bottom substrate surface.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-7Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japthat is, the maximum edge effect with the sharpest edge shape andthe smallest electron beam size by considering that the experimen-tal probe size for a nanowire LaB6 field-emission electron gun is∼2 nm.83 In Monte Carlo simulations, the number of incident tra-jectories, Np ¼ 1� 105, is used for each incident position of eachlinescan profile. Therefore, the statistical fluctuation of simulatedsecondary electron signal intensity can be estimated as1/ffiffiffiffiffiNsp ¼ 1/ffiffiffiffiffiffiffiffiNpδp � 0:6%, where Ns is the number of emitted sec-ondary electrons with energies less than 50 eV and δ is the second-ary electron yield (which represents the ratio of the number ofemitted secondary electrons to the number of incident primaryelectrons), where δ � 0:3 at 1 keV.55 This fluctuation for the MonteCarlo computation uncertainty is so small and can be omitted. Thelinescan profiles were simulated 10 000 times by changing the workfunction value according to the uniform distribution and for all theFPA, SMA, and LLM. In Fig. 5 the black line represents the meanlinescan profile and the red-shaded region represents the standarddeviation of linescans due to the change of work function. Theedge blooming and the edge bottom dip are clearly seen in the line-scan profile for the reason stated above. All three dielectric functionmodels produce a similar linescan profile shape, except theintensities are somewhat different. The work function also changesthe emission intensity [Figs. 5(a)–5(c)] but not the shape of thenormalized linescan profile [Figs. 5(d)–5(f )] because the lowenergy secondary electrons are sensitive to the value of the workfunction when passing through the potential barrier of the surfacewhere they lose energy. One may observe that the standard devia-tion of the intensity is the vastest in the middle region of the linestructure.The CD estimation in this study is performed accordingly asfollows: we define CD1 to be the distance between the two peakmaxima, and CD3 to be the distance between the two valleyminima in a linescan profile. Then CD2 is taken as the mean ofCD1 and CD3. It should be noted that in the simulation, the linestructure is specified and the structure size is given; what we mea-sured as the CD value here is the size from the simulated linescanprofile. For a rectangle line structure, CD2 should be quite close tothe CD value in the line structure modeling. In the following, wehave also considered other sidewall angle cases: 80° and 100°, whilein structure modeling, we keep the CD at the mid height of thesetwo line structures to be the same value, 50 nm, as in the rectanglecase. However, it should be noted that this simple peak/valley esti-mation method of the CD cannot be accurate as it ignores theFIG. 5. Simulated linescan profiles of a rectangle Si line structure on a Si substrate for a variation of work function in the range of 3.59–5.41 eV by using the (a) and (d)LLM, (b) and (e) FPA, and (c) and (f ) SMA. Whereas the bottom panel [(d)–( f )] shows the mean and the standard deviation of the normalized linescans by the peakmaximum. The black line represents the mean profile, and the red-shaded region represents standard deviations. The elastic scattering potential = “No. 42 001” and energyloss function = “crystal (Palik).” In these Monte Carlo simulations, an electron beam of 1 keV at the beam size (FWHM) of 0 nm is normally incident and 1 × 105 incident tra-jectories are used for each incident position.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-8Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japlinescan shape, which contains much more information on the linestructure than the peak/valley position. A reasonable estimationshould be done with the MBL method.37 However, for our presentpurpose of uncertainty estimation, we focus on the CD variation(i.e., the standard deviation) value, but not the absolute CD value;therefore, this definition of mean CD (i.e., CD2) is convenient forthe evaluation of uncertainty.Figure 6 shows the effect of the elastic scattering potential onthe simulated linescan profiles. Here, 384 potential models are usedand the corresponding elastic scattering cross section files are pro-vided as input to the Monte Carlo simulation. All of these elasticscattering potentials are considered to be equally important. Fromthe figure, one may find that the elastic scattering cross section alsoinfluences, to some extent, the secondary electron emission inten-sity, but, not the linescan shape. For this rectangle line structure ofthe side length of 50 nm, CD1 and CD3 estimated are 48 and52 nm, respectively, which are the same for the three dielectricfunction models. Hence, CD2 is obtained as the modeling value,50 nm.Figure 7 illustrates the influence of the energy loss functiondataset on the linescan profiles. It is evident from Fig. 7(a) that theLLM dataset has a negligible effect on the emission yield becauseall the differences in the different energy loss functions below thebandgap have been removed in the LLM. While in the FPA and theSMA, the energy loss function dataset has a small effect. In addi-tion, the LLM produces more secondary electrons than the othertwo models because the electrons cannot have energy loss belowthe bandgap, i.e., phonon excitation is omitted. Nevertheless,neither the CD measurement is found to be influenced by the elec-tron inelastic mean free path calculation method via the dielectricfunction model nor by the calculated electron inelastic mean freepath values via the energy loss function dataset.Figures 8(a)–8(c) show the final evaluated mean linescan pro-files for the line structures with different sidewall angles, i.e., 80°,90°, and 100°, from the Monte Carlo simulations together with theuncertainty (standard deviation),uI ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXMi¼1(Ii � �I)2/(M � 1)vuut , (7)where Ii represents the intensity of secondary electrons emittedfrom the ith scanning position, and �I represents the mean value. AtFIG. 6. (Top panel) Simulated linescan profiles of a rectangle Si line structure on a Si substrate for a variation of elastic scattering potential, and using the (a) LLM,(b) FPA, and (c) SMA. (bottom panel) the mean and standard deviation of the normalized linescans by the peak maximum for a variation of elastic scattering potential byusing the (d) LLM, (e) FPA, and (f ) SMA. The black line represents the mean profile and the red-shaded region represents standard deviations. The work function is4.61 eV and the energy loss function = “crystal (Palik).” In these Monte Carlo simulations, an electron beam of 1 keV at the beam size (FWHM) of 0 nm is normally incidentand 1 × 105 incident trajectories are used for each incident position.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-9Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japeach scanning point, the calculations are performed over a totalnumber of M ¼ 17280 Monte Carlo simulations by changing aninput. Figures 8(d)–8(f ) show all these 17 280 linescan profiles sim-ulated for the line structures with different sidewall angles, i.e., 80°,90°, and 100°, respectively, from which we have estimated CD1,CD3 and, hence, CD2 for each linescan profile. The associated CDuncertainty (standard deviation) is given similarly byud ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXMi¼1(di � �d)2/(M � 1)vuut , (8)where d represents the estimated CD1, CD2, or CD3 and �d repre-sents the mean value. Two x-grid intervals are used to reduce com-putation: a coarse grid of 2 nm for intensity uncertainty evaluationwith Eq. (7) as shown in the main frame of Figs. 8(a)–8(f ), and afine grid of 0.1 nm for the CD uncertainty evaluation with Eq. (8)as shown in the insets of Figs. 8(g)–8(l). Figures 8(a)–8(f ) and8(g)–8(l) emphasize on the intensity change and the shape changeof the linescan profiles, respectively. It is shown in Figs. 8(a)–8(c)that the relative intensity uncertainties at the mid of the topsurface, uI(0 nm) /I(0 nm), are 0.30, 0.26, and 0.29, respectively, for thesidewall angles of 80°, 90°, and 100°, and the ones at an edge,uI(25 nm) /I(25 nm), are 0.21, 0.22, and 0.24, respectively. For normalizedlinescan profiles, they are uI(0 nm) /I(0 nm) ¼ 0:13, 0:10, and 0.14 forthe sidewall angles of 80°, 90°, and 100°, respectively. Therefore, itis seen that although the secondary electron intensity does changewith physical parameters in a Monte Carlo modeling, the shape ofthe linescan profile varies very slightly.From Table I, one can see that the standard deviation of CD1is always greater than that of CD3 and, hence, that of CD2 has avalue in between. The averaged CD2 for the line structures withsidewall angles of 80°, 90°, and 100° are, respectively, 49.9, 49.6,and 57.7 nm with the standard derivation of 0.06, 0.15, and0.24 nm. Then in this limited investigation of simple structuregeometry, the estimated CD2 is very close to the modeling value of50 nm with a negligible standard deviation for sidewall anglessmaller than 90°. But, for sidewall angles greater than 90°, the esti-mated CD2 is far from the true value. In addition, by consideringother factors like the edge angle, the estimated CD2 may furtherdiffer from the modeling value of the line width; this is exactly thereason to use the MBL method other than the simple peak/valleymethod. However, even though the CD evaluation may beFIG. 7. (Top panel) Simulated linescan profiles of a rectangle Si line structure on a Si substrate for a variation of energy loss function dataset and using the (a) LLM,(b) FPA, and (c) SMA. (bottom panel) The mean and standard deviation of the normalized linescans by the peak maximum for a variation of energy loss function datasetby using the (d) LLM, (e) FPA, and (f ) SMA. The black line represents the mean profile and the red-shaded region represents the standard deviation. The work function is4.61 eV and the elastic scattering potential = “No. 42 001.” In these Monte Carlo simulations, an electron beam of 1 keV at the beam size (FWHM) of 0 nm is normally inci-dent and 1 × 105 incident trajectories are used for each incident position.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-10Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japFIG. 8. (a)–(c) The mean (black line) of the simulated linescan profiles together with the standard deviation (red-shaded region) for a Si line structure on a Si substrate.(d)–(f ) All the simulated 17 280 linescan profiles by taking every considered theoretical uncertainty into account, where the red, yellow, and green lines represent three indi-vidual linescans for visual clarity. (g)–(i) The mean (black line) of the normalized linescan profiles by the peak maximum together with the standard deviation (red-shadedregion). ( j)–(l) All the normalized 17 280 linescans. The inset shows the linescan profiles near an edge. The x-grid intervals are 0.1 and 2 nm for the inset and the mainframe, respectively. The sidewall angle of the Si line structure is (a), (d), (g), and ( j) 80°; (b), (e), (h), and (k) 90°; (c), (f ), (i), and (l) 100°. In these Monte Carlo simula-tions, an electron beam of 1 keV at the beam size (FWHM) of 0 nm is normally incident and 1 × 105 incident trajectories are used for each incident position.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-11Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japinaccurate, the standard derivation is still negligible. This resultindicates that by taking every considered theoretical uncertaintyinto account, all the obtained CDs with the simple peak/valleymethod are quite the same within the accuracy, in comparison tothe grid precision, 0.1 nm. Hence, the estimated CD values are neg-ligibly related to the physical factors in a Monte Carlo modeling asthe present Monte Carlo theoretical framework is concerned; it isevident that CDs rely more critically on the line structure and elec-tron beam parameters. However, it should be also pointed out thatby considering the fact that the MBL method is shape-sensitive andthe insets of Figs. 8(g)–8(i) show that the shape of a normalizedlinescan curve varies only slightly with modeling, the use of theMBL method in CD evaluation may thus give slightly differentresults, which means that when an experimental linescan profile ismeasured, the use of different MBL database built with differentMonte Carlo physical modeling may result in slightly different CDvalues. Therefore, according to ISO-21466, the modeling parame-ters should be specified when building a MBL database.It is also necessary to mention that the qualitative conclusionfor the present results obtained for a Si line structure on a Sisubstrate should be also valid for a Si line structure on other sub-strates, e.g., TiN. This is because the effect of the substrate mostlycontributes only to the background intensity but cannot changesignificantly the shape of the linescan curve near the line edge.Considering the fact that TiN or other similar semiconductor mate-rials have a similar secondary electron yield curve as Si,84 then thesubstrate effect is even less. Furthermore, the present Monte Carlophysical modeling is limited to the up-to-date model, which is thecombination of Mott’s electron elastic scattering cross section anddielectric function formulation for electron inelastic scattering crosssection; the old-fashioned Monte Carlo physical modeling with theuse of Rutherford’s electron elastic scattering cross section andstopping power-based approach to electron inelastic scattering hasbeen known inaccurate to low energy secondary electrons. Othergeometrical factors, such as the aspect ratio and the spacingbetween the lines, have no obvious effect on the present observa-tions. Logically, modeling physical factors are independent of thegeometrical factors like the sidewall; therefore, the CD determina-tion, considering these geometrical factors, should present a stan-dard deviation no greater than that of the sidewall as given inTable I.According to the ISO-JCGM 100,82 the following parametersshould be reported as the measurement data: (i) an estimation ofthe output quantity, taken as the mean values generated; (ii) thestandard uncertainty, taken as the standard deviation of these gen-erated values; (iii) the chosen coverage probability or the level ofconfidence (usually 95%), and (iv) the endpoints corresponding tothe selected coverage interval. The last step of uncertainty quantifi-cation is about the expanded uncertainty at the k% level of confi-dence, i.e.,U% ¼ k%u, (8)where the values of k% at 95% and 75% confidence levels arek95 ¼ 1:96 and k75 ¼ 1:16, respectively. Figure 9 shows theexpanded uncertainty plots for the calculated linescan of the lineTABLE I. The evaluated CD values and their standard deviations for Si line struc-tures at different sidewall angles calculated with 17 280 linescan profiles.Sidewallangle CDMin(nm)Max(nm)Mean(nm)Standarddeviation (nm)80° CD1 40.0 41.2 41.0 0.13CD2 49.4 50.0 49.9 0.06CD3 58.6 58.8 58.8 0.0190° CD1 47.2 49.6 49.2 0.31CD2 48.6 49.8 49.6 0.15CD3 50.0 50.0 50.0 0.00100° CD1 54.0 57.6 56.5 0.48CD2 56.4 58.2 57.7 0.24CD3 58.8 60.2 58.8 0.02FIG. 9. Simulated mean linescan profile (black line) together with the expanded uncertainties at 75% (dark red) and 95% (light red) level of confidence for a Si line struc-ture having sidewall angles (a) 80°, (b) 90°, and (c) 100° placed on a Si substrate. The electron beam of 1 keV at the beam size (FWHM) of 0 nm is normally incident. Inthe Monte Carlo simulations, 1 × 105 incident trajectories are used for each incident position.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 133, 245303 (2023); doi: 10.1063/5.0153379 133, 245303-12Published under an exclusive license by AIP Publishing 13 December 2023 02:50:02https://pubs.aip.org/aip/japstructure having different sidewall angles at 95% (light red) and75% (dark red) levels of confidence. One may also observe that thecalculated uncertainty in the linescan profiles of the Si line struc-ture due to different factors hardly affects the CD measurement.Although the present study provides an insight into the influ-ence of theoretical input parameters on CD measurements with theCD-SEM by using the Monte Carlo simulation techniques, it isalso necessary to acknowledge the limitations of this study. First,we focused on the fixed experimental morphology and beamparameters throughout our simulation process, neglecting thepotential variations that occur in real-world scenarios. This simpli-fied approach allowed us to isolate the effects of theoretical inputsfrom the CD measurements, but may not fully reflect the complexi-ties in practical applications. Second, our study primarily examinedthe impact of simulation procedures and theoretical inputs, poten-tially overlooking other sources of uncertainty that could arise fromsample preparation, instrument calibration, and other systematicerrors. To address these limitations and provide a more compre-hensive understanding of the CD measurements, we plan toconduct an experimental study in the near future. This forthcom-ing study will involve utilizing a field-emission SEM instrumentequipped with a nanotube LaB6 electron gun to investigate theinfluence of beam parameters on CD measurements. By incorpo-rating the experimental data and considering a wider range ofuncertainty factors, we aim to further enhance the accuracy andreliability of CD-SEM metrology.Overall, our present study serves as a foundation for futureresearch, emphasizing the importance of both theoretical inputsand the related experimental factors in CD measurements. Wehope that our findings would inspire further investigations into theoptimization of CD measurement techniques, ultimately advancingdimensional metrology in the semiconductor industry and facilitat-ing more precise manufacturing processes.V. CONCLUSIONSIn this study, we have used the Monte Carlo uncertainty quan-tification procedure to investigate the uncertainty of CD measure-ment due to the involved uncertain theoretical modeling factors.We demonstrate that the uncertain theoretical factors, includingthe work function, elastic scattering potential model, dielectricfunction model, and energy loss function dataset, in the MonteCarlo simulation of a CD-SEM linescan profile, have a negligibleeffect on the CD measurement, even though they impact secondaryelectron emission intensity. Therefore, the CD measurement uncer-tainty by the peak/valley method is more critically related to thegeometric and beam parameters, while the accurate measurementshould be done by the MBL method.ACKNOWLEDGMENTSWe thank Professor H. M. Li and the supercomputing centerof USTC for the support of parallel computing. S. F. Mao acknowl-edges the National MCF Energy R&D Program of China (GrantNo. 2019YFE03080500) and the Collaborative Innovation Programof Hefei Science Centre, CAS (Grant No. 2022HSC-CIP010). Y. B.Zou acknowledges the Natural Science Foundation of XinjinagUygur Autonomous Region (Grant No. 2022D01A223). Y. G. Liacknowledges the National Natural Science Foundation of China(Grant No. 11975018), the National Magnetic Confinement FusionEnergy Research Project (Grant No. 2018YFE0308100), and theYouth Innovation Promotion Association of CAS (Grant No.Y202087). B. Da acknowledges the Kurata Grants from The HitachiGlobal Foundation and the Iketani Science & TechnologyFoundation. Z. J. Ding is supported by the Chinese EducationMinistry through “111 Project 2.0” (No. BP0719016). M. S. S. Khanacknowledges CAS-TWAS President’s fellowship.AUTHOR DECLARATIONSConflicts of InterestThe authors have no conflicts of interest to disclose.Author ContributionsM. S. S. Khan: Data curation (equal); Formal analysis (equal);Investigation (equal); Visualization (equal); Writing – original draft(equal); Writing – review & editing (equal). S. F. Mao:Methodology (equal); Software (equal). Y. B. Zou: Funding acqui-sition (equal); Software (equal). Y. G. Li: Conceptualization(equal); Methodology (equal); Software (equal). B. Da:Methodology (equal); Software (equal). Z. J. Ding:Conceptualization (equal); Formal analysis (equal); Funding acqui-sition (equal); Methodology (equal); Project administration(equal); Software (equal); Supervision (equal); Writing – review &editing (equal).DATA AVAILABILITYThe data that support the findings of this study are availablefrom the corresponding author upon reasonable request.REFERENCES1K. F. J. Heinrich, D. E. Newbury, and H. Yakowitz, in Use of Monte CarloCalculations in Electron Probe Microanalysis and Scanning Electron Microscopy(US Department of Commerce, National Bureau of Standards, 1976).2J. Schou, P. Kruit, and D. 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