New strategy to improve the accuracy of quantitative analysis of energy dispersive spectroscopy

Elemental quantification of several single crystalline TEM samples of intermediate thickness range that cannot be quantified by the thin-film approximation method and the ZAF correction were investigated using wedge-shaped samples of known thickness fabricated by the FIB technique. ‘Thickness factor (T F )’ and ‘Thickness correction coefficient (T C )’ were proposed as error correction items of ‘thin-film approximation method’ to minimize the quantitative error that occurs when quantifying samples of intermediate thickness with TEM-EDS. As the result of T F correction, the quantification error in an Al2O3 TEM sample by TEM-EDS was reduced from about 52.15% to less than 3.28 (± 2.57) % by the correction of only one time. The self-absorption corrected line profiles and self-absorption corrected elemental mapping images in TEM samples of intermediate thickness range were also obtained by T F correction. As an example of application by T F correction, we proposed a novel method measuring thin-film thickness in normal TEM without the EELS system. The T F correction technique is a unique method to overcome the limitation of EDS quantification in the intermediate thickness range. This technique can effectively quantify characteristic x-ray lines less than 1,000 eV in the sample having intermediate thickness ranges using the conventional TEM-EDS system. It is expected to contribute significantly to understanding various characteristics or material nature related to material composition and sample thickness in nanomaterials because it provides more precise quantitative analysis information than current commercial EDS systems.


Introduction
Over the past few decades, transmission electron microscope (TEM) and scanning electron microscope (SEM) have been recognized as powerful tools for characterizing the microstructure of materials.It is important to realize that most of the energy harvesting, optical sensing, and electrical materials are composed of multiple components, and that their characteristics are determined by their composition.Energy dispersive spectroscopy (EDS) attached to TEM and SEM has been widely used for qualitative and quantitative analysis of elements because it can easily match the image and elemental analysis results at nanometer scales.Recently, an example of the ultimate improvement of the qualitative analysis technique can be found in the atomic mapping image in the unit cell by the double Cs-TEM-EDS which is one of high-performance TEM [1][2][3][4].
There is a fundamental difference in quantitative analysis methods between the TEM-EDS and SEM-EDS systems.The ZAF correction [5][6][7][8][9] related to EDS quantitative analysis is a well-known concept that includes atomic number effect (Z), absorption effect (A), and fluorescence excitation effect (F).In general, the quantitative analysis method in the SEM-EDS system [8,9] applies the 'ZAF correction method' which is considering all corrections related to atomic number, absorption, and fluorescence excitation.The effective thin-film thickness range obtained with the ZAF correction method in the SEM-EDS system normally is above about 1,000 nm.However, the quantitative analysis method in the TEM-EDS system uses the 'thin-film approximation method' based on Cliff-Lorimer ratio method [5][6][7][8][9][10][11][12][13] that mainly considers only the atomic number effect (Z) without considering the absorption effect (A) in the ZAF correction [10][11][12][13].Similarly, the Phi-Rho-Z model which is an evolution from ZAF correction is also effective for limited thickness.The effective thin film thickness range of the TEM-EDS system applying the thin film approximation method is less than tens of nanometers for most characteristic x-ray lines less than 1,000 eV emitting from light elements.This is very thin as the effective thin-film thickness range of about 1/10 than the effective thin-film thickness range of ZAF correction method in SEM-EDS systems.Here, there is a gap between the effective thin film thickness ranges of TEM-EDS and SEM-EDS, which cannot be quantified with current EDS system.In this work, the thickness range corresponding to the gap will be called the intermediate thickness range or the blank area.
As a result of these efforts, the thin-film approximation method in the TEM-EDS system is well-established.However, the thin-film approximation approach cannot quantify thick samples.Notably, the blank area is difficult to quantify using the ZAF correction and the thin-film approximation method.In terms of effective thin-film thickness, the blank area is in the ranges from about 100 to 1,000 nm.As mentioned above, this range cannot be supported by TEM-EDS and SEM-EDS.Because the thickness of most TEM samples including nanopowders, thin foils, and thin films, are in the intermediate thickness range, most TEM-EDS results obtained from these ranges are also inaccurate.A number of correction models were developed to overcome quantitative analysis problem which is light elements have a larger quantitative error than heavy elements.However, it is difficult to completely solve the inaccuracy issue of quantitative elemental analysis because there is still considerable uncertainty about the x-ray generation depth distribution of light elements.Figure S1(A) shows three regimes (1) TEM-EDS based on thin-film approximation, (2) intermediate thickness regime and (3) SEM-EDS based on ZAF correction.Here, the regime which has inaccurate EDS quantitative analysis is termed intermediate thickness.Investing the cause of inaccuracies in quantitative elemental analysis, we compared x-ray intensities obtained from Monte Carlo simulation and TEM-EDS.In figures S2(b)-(i), we present the x-ray intensity emitted from the sample, including mass absorption correction ((b), (c)), integration of generated x-ray ((d), (e)), and dead time correction ((f), (g)).These processes collectively contribute to the final quantitative elemental analysis.We can find a notable discrepancy between Monte Carlo simulations (f) and TEM-EDS results (g).The discrepancy is clear in figures S1(h) and (i), which shows elemental compositions of Monte Carlo simulations and TEM-EDS calculated from figures S1(f) and (g).It implies that EDS data correction cannot be performed with elemental composition calculated by Monte Carlo simulation.Therefore, the theoretical approach with Monte Carlo simulation to quantify the intermediate thickness range is not clear yet.Improving the accuracy of quantitative elemental analysis requires new approaches, which we have demonstrated through experimental experience.Due to these reasons, it has been taught so far to TEM operators or scientists as one of common sense that 'Please be careful when mentioning composition by TEM-EDS'.
This investigation is deal with 'How to quantify characteristic line less than 1,000 eV in the sample having intermediate thickness ranges using the EDS system.Moreover, we attempt to quantify TEM samples of intermediate thickness range with a combination of quantitative analysis results obtained with the TEM-EDS system and wedge-shaped samples with known thickness by FIB technique.From viewpoint of the theoretical approach, we examine and discuss whether it is possible to quantify thickness and absorption correction items from the analysis results of intermediate thickness samples by current TEM-EDS system.

Materials
As standard material, single crystal substrates used in this study were purchased from MTI corporation (USA) for Y 3 Al 5 O 12 , and Kyocera (Japan) for α-Al 2 O 3 .They were miller-polished wafers along (001) for Y 3 Al 5 O 12 , and miller-polished wafers along C-plane (0001) for α-Al 2 O 3 .From these single-crystal substrates, wedge-shaped TEM samples with known thickness were prepared by focused ion beam (FIB) technique [20][21][22][23][24], which can fabricate thin films below the thickness of 100 nm and measure the thin-film thickness with SEM observation.

Preparation and characterization
Thickness measurements of wedge-shaped samples were performed with an Electron Energy-Loss Spectroscopy (EELS, Quantum 965, Gatan) equipped with Cs-TEM (Titan Themis, FEI) at accelerating voltage of 300 keV.By analyzing the amount of inelastic scattering that increases with specimen thickness using EELS, we can obtain thickness information.The factors entered to calculate the thickness of the sample are as follows: The convergence semi-angle was 15 mrad, the collection semi-angle was 25 mrad, and the effective atomic number corresponding to the sample was applied.We were able to successfully compute the thickness of the wedgeshaped sample by applying the deconvolution method in the thick thickness region.
SEM-EDS analysis was performed using a FE-SEM (S-4700, Hitachi) at an accelerating voltage of 30 keV.The EDS system (Bruker, UK) in SEM used the Esprit software based on the P/B ZAF correction method.TEM-EDS analysis was performed using a FE-TEM (TECANI-F20, FEI) at an accelerating voltage of 200 keV (at a beam current of 3.71 μA).The TEM-EDS system was operated with TIA software based on the thin-film approximation method.Technically, quantitative analysis data produced by the TEM-EDS system does not include absorption effect correction (A) and fluorescence excitation effect (F) among ZAF correction.If we carry out quantitative analysis with current commercial TEM-EDS system and SEM-EDS system for the sample of intermediate thickness range, the former will obtain data no self-absorption corrected, the latter will obtain data self-absorption corrected excessively.Thus, quantitative analysis of the TEM-EDS system is valid only under conditions where self-absorption does not occur.
Normally, quantitative analysis of the TEM-EDS system is valid only under conditions where self-absorption does not occur.The quantitative data of thicker regime thin-film than effective thin-film thickness of 'thin-film approximation method' could not be accurate, and the error could be increased until the effective thin-film thickness of the ZAF correction.In this work, the quantitative data obtained from the intermediate thickness range were regarded as 'raw data' requiring correction.In order to minimize the quantitative error that occurs when quantifying intermediate thickness sample by TEM-EDS, a new error correction factor (T F ) was proposed in addition to the k factor.
The relation of standard composition, measured sample composition, and T F for single crystalline Al 2 O 3 sample can be expressed as follows.
The elemental composition in this work was dealt with this atomic percentage (at %).We ignored the fluorescence excitation effect in wedge-shaped TEM samples with an intermediate thickness range.

Results and discussion
Figure 1(a) represents a typical SEM image of a wedge-shaped sample with known thin-film thickness prepared by dual beam FIB system.Figure 1(b) is a schematic diagram of the wedge-shaped sample.This shape is suitable for analyzing the difference in elemental quantitative values according to the change in thin-film thickness.In addition, the 500 nm region was prepared in the lateral direction at intervals of about 2.5 μm of the sample to measure the distance in the lateral direction and check the tendency at the same thickness (figure 1(a)).With the change in the wedge angle, the thickness range of the wedge-shaped sample was controlled from tens of nanometers to several thousand nanometers.For example, wedge-shaped sample shown in figure 1(a) had a thickness ranging from about 107 to 528 nm, with a wedge angle of 2.7 degrees.
Figures 1(c) and (e) denote HAADF-STEM images of wedge-shaped samples fabricated from Al 2 O 3 and Y 3 Al 5 O 12 (YAG) substrates, respectively.In addition, the electron energy loss spectroscopy (EELS) analysis method was applied to ensure the sample thickness.The thicknesses were calculated using the wedge angle and distance and closely matched with the results measured by EELS system (shown in figures 1(d) and (f)).Both samples had linearly increasing thickness.The width of the sample was about 10 μm, and the maximum thickness was 600 nm.
We performed a quantitative analysis of EDS according to the change in thin-film thickness of sample by scanning electron microscopy (SEM) and transmission electron microscopy (TEM).To reduce the deviation of quantitative analysis, line profile analysis was performed three or more times in the lateral and vertical directions at various locations of the sample (figures  Next, we measured the elemental quantitative value according to the change in thickness using TEM-EDS.Interestingly, it was similar to the tendency of SEM-EDS, but in figures 2(c) and (d), the lateral line scan results showed that all elements had a more abrupt compositional change than the SEM results (figures S2.1(c) and (d)).Especially, in the thin region of the YAG sample, the elemental quantitative values were the same as those of the standard material, but the values changed rapidly with increased thickness of thin film.In contrast, figures 2(e) and (f) show that the elemental quantitative values did not change in the vertical direction.Both Al 2 O 3 and YAG samples exhibited a common tendency.Since the accelerating voltages of SEM and TEM had interaction volumes of several μm or more [5][6][7][8][9][10][11][12][13], x-rays could be emitted by electron excitation in all thickness regions of the fabricated samples.Nevertheless, the difference in the quantitative element values according to the change in thin-film thickness could be described by the self-absorption coefficient [14][15][16][17][18][19].Since multi-component compounds have different self-absorption coefficients depending on the elements, the quantitative values for TEM-EDS based on the thin-film approximation method showed a difference in that of standard material in the intermediate thickness range, which cannot ignore absorption effect.
We should be aware of the problem of inaccuracy in the elemental quantitative values according to sample thickness.To compensate for this, we propose a simple and new concept called the thickness factor (T F ).The T F is defined as a thickness-dependent simple correction value.New correction factor (T F ) of intermediate thickness sample is necessary to overcome elemental inaccuracy for thin-film thickness.From the viewpoint of TEM sample, the optimal thickness range of TEM sample that can be observed at an accelerating voltage of 200 keV is from a few nm to 500 nm.Almost all of them belong to narrow intermediate thickness range.Hence, the TEM-EDS results obtained from narrow intermediate thickness range could be corrected with linear function of equation (3).In this investigation, unless otherwise specified, we have evaluated the thickness factor (T F ) using a linear function.
The quantitative values of each element before T F correction differ from that of standard sample in all thickness ranges except near 100 nm as shown in figures 4(a) and (b) for Al 2 O 3 and YAG samples.Figure 4(c For YAG sample (in figure 4(b)), the quantitative values of Al and Y (cations) gradually increase as the thickness increases, while that of O (anion) decreases.Figure 4(d) shows the quantitative values of each element of YAG after T F correction and they are similar to that of standard YAG samples across all tested thickness ranges.In addition, we performed TEM-EDS line profiles at three positions in the same YAG sample (dotted arrows in figure S3(d)) to confirm T F consistency according to sample thickness in narrow intermediate thickness range.The plots of T F vs. thickness plots at three positions are almost similar, as shown in figure S3(e).Consequently, it is clear that the results of figures 4(c) and (d) are 'self-absorption corrected line profiles' that improve the accuracy of quantitative analysis of TEM-EDS by minimizing the quantitative error caused by selfabsorption.
As another example of thickness correction with T F and T C , the 'self-absorption corrected elemental mapping image' of Al 2 O 3 was investigated by applying the T F correction on each pixel of the elemental mapping images obtained from TEM-EDS.The elemental mapping images with a total pixel size of 105 (15 × 7 pixels, 1 μm/pixel) in a 15 × 7 μm area were obtained with a dwell time of 5 s per pixel.Although the resolu figure 3(e) show quantitative tion of the elemental mapping image is poor, it provides valuable information for understanding the self-absorption correction process for individual pixels.
Figure 5 shows a series of procedures to obtain a 'self-absorption corrected elemental mapping image'.Figure 5(a) represents a STEM image of wedge-shaped TEM sample with a thickness range of about 300 to 2,100 nm, with a wedge angle of 6.5 degrees.The rectangle of the blue dotted box in the STEM image is an elemental mapping area by TEM-EDS.5(c)) was maximum in the thin area of the sample and decreased as thin-film thickness increased, as shown in the contour map of green color.However, the quantitative value Al (in figure 5(d)) was minimum in thin area of the sample and increased with an increase in the thin-film thickness.This gradient of composition distribution along the wedge direction must be an artifact arising from the limitations of quantification method of TEM-EDS, which could not support the intermediate thickness range, as shown in the results of line profiles (in figures 2 and 3).
In reference, we could not infer the composition of each pixel in the mapping image of TEM-EDS because the general mapping image of TEM-EDS expressed only the degree of abundance of any element in the elemental area without a composition guide through the map legend.Here, we attempted T F correction in EDS mapping images by extending the T F correction technique of line analysis.Figure S5 shows the numerical analysis results for individual pixels in the elemental mapping images (in figures 5(c) and (d)).Figure Consequently, it is clear that contour maps in figures 5(g) and (h) show self-absorption corrected elemental mapping images.Since the T F correction technique performs x-ray self-absorption correction in individual points or pixels, it is also possible to obtain self-absorption corrected point analysis, line profile, and elemental map.The elemental mapping analysis by the general TEM-EDS system is valid only for a thin region that satisfies thin-film approximation.The elemental mapping analysis of TEM samples with an intermediate thickness range could be achieved by redrawing contour maps using T F corrected compositions.The wedge-shaped TEM samples with intermediate thickness which could not be quantified by TEM-EDS and SEM-EDS were successfully prepared by dual-beam FIB system.'Thickness factor (T F )' and 'Thickness correction coefficients (T C )' were proposed as error correction items of TEM-EDS.The T F value increased as thinfilm thickness increased within intermediate thickness range, implying that an increase of thin-film thickness caused an increase in the error from standard composition.As the result of T F correction, the quantification error in an Al 2 O 3 TEM sample reduced from about 54.8% to less than 1.86 (±1.6) % by thickness factor correction of only one time.Our new concept of T F and T C could provide not only quantitative analysis methods of intermediate thickness by TEM-EDS but also sample thickness measurement methods by TEM-EDS.Finally, our work provides a new strategy for the quantitative analysis of TEM samples within the intermediate thickness range.Technically, new approach can be extended to enable composition analysis by combination of electron energy loss spectroscopy (EELS) and EDS system.If the capability for thickness analysis through EELS and composition analysis through EDS is in place, T F can be determined for all tested materials.Subsequently, composition with the sample thickness can be acquired via the procedure in figure S1.It is expected to contribute significantly to understanding various properties related to material composition in intermediate thickness range.
Figure1(a) represents a typical SEM image of a wedge-shaped sample with known thin-film thickness prepared by dual beam FIB system.Figure1(b) is a schematic diagram of the wedge-shaped sample.This shape is suitable for analyzing the difference in elemental quantitative values according to the change in thin-film thickness.In addition, the 500 nm region was prepared in the lateral direction at intervals of about 2.5 μm of the sample to measure the distance in the lateral direction and check the tendency at the same thickness (figure1(a)).With the change in the wedge angle, the thickness range of the wedge-shaped sample was controlled from tens of nanometers to several thousand nanometers.For example, wedge-shaped sample shown in figure1(a) had a thickness ranging from about 107 to 528 nm, with a wedge angle of 2.7 degrees.Figures1(c) and (e) denote HAADF-STEM images of wedge-shaped samples fabricated from Al 2 O 3 and Y 3 Al 5 O 12 (YAG) substrates, respectively.In addition, the electron energy loss spectroscopy (EELS) analysis method was applied to ensure the sample thickness.The thicknesses were calculated using the wedge angle and distance and closely matched with the results measured by EELS system (shown in figures 1(d) and (f)).Both samples had linearly increasing thickness.The width of the sample was about 10 μm, and the maximum thickness was 600 nm.We performed a quantitative analysis of EDS according to the change in thin-film thickness of sample by scanning electron microscopy (SEM) and transmission electron microscopy (TEM).To reduce the deviation of quantitative analysis, line profile analysis was performed three or more times in the lateral and vertical directions at various locations of the sample (figures 2(a), (b) and S2.1).The thickness of the sample precisely fabricated with FIB varied along the lateral direction but remained the same along the vertical direction.The characteristics of these samples can provide different elemental quantification values with thickness change.Figures S2.1(c) and

Figure 1 .
Figure 1.(a) The cross-sectional SEM image of wedge-shaped Al 2 O 3 sample.(b) The schematic diagram of wedge-shaped sample.The HAADF-STEM images of wedge-shaped Al 2 O 3 (c) and Y 3 Al 5 O 12 sample (e), respectively.Comparison of the thickness measurement results by EELS system and FIB system for Al 2 O 3 (d) and Y 3 Al 5 O 12 sample (f).

Figure 2 .
Figure 2. The HAADF-STEM images of wedge-shaped Al 2 O 3 (a) and Y 3 Al 5 O 12 sample (b), respectively.The EDS line profiles were obtained along the dotted arrows in the HAADF STEM images.The results of EDS line profile analysis carried out with various directions of Al 2 O 3 (c, e) and Y 3 Al 5 O 12 sample (d, f).Here, dotted lines is guide line showing the atomic percentage of each elements in standard materials.

2 F 3
Here, γ and δ are the thickness correction coefficients (T C ) of power function in a plot of T F -thickness valid for the sample with wide intermediate thickness ranges (from 100 to 2,000 nm for Al 2 O 3 sample) such as figure3(a).Symbol (d) in the figure 3(c) denotes the plot of T F calculated using equation (2) vs. thin-film thickness in individual analysis points in figure 3(b) and could be fitted by the power law (γ is 0.107 and δ is 0.465), as shown with a red line.Considering the narrow intermediate thickness ranges from 100 to 600 nm as shown in figure 3(c), T F increases linearly with increasing thin-film thickness; therefore, the relation can also be expressed as a linear function as Here, α and β are the thickness correction coefficients (T C ) of linear function, and is valid for the narrow intermediate thickness range with the linear function, and t denotes the sample thickness.Figure 3(e) show quantitative values of oxygen and aluminum after T F correction using power law fit for wedge-shaped Al 2 O 3 sample.Most of the quantitative values for O and Al (except values less than 300 nm) are also very close to the standard Al 2 O 3 sample.
) shows the composition of Al 2 O 3 sample after T F correction, and it is broadly similar to that of standard composition of Al 2 O 3 across all tested thickness ranges.Despite T F correction being applied only once with thickness correction coefficients, the quantitative values of each component of Al 2 O 3 in figure 4(c) are very close to that of standard sample.
Figure 5(b) shows the change in the composition of oxygen and aluminum according to thin-film thickness along the black dotted arrow in figure 5(a).All points tested differ to the composition of standard sample and were in intermediate thickness range.Figures 5(c) and (d) denote general elemental mapping images of O and Al obtained with TEM-EDS, respectively.Although the wedge-shaped TEM sample was a single crystalline Al 2 O 3 which showed no composition variation in the short-range and long-range order, elemental distributions of two elements in figures 5(c) and (d) showed a gradient of composition distribution along the wedge direction.The quantitative value of O (in figure

Figure 3 .
Figure 3. (a) The HAADF-STEM images of wedge-shaped Al 2 O 3 sample.(b) The EDS line profiles were obtained from wedge-shaped TEM sample.(c) The T F calculated from figure 3(b) and its power law fitted plots.(d) Power law fitted and linear fitted plots for T F values extracted from the gray area.(e) The quantitative value plots after T F correction.

Figure 4 .
Figure 4.The EDS line profiles were obtained from Al 2 O 3 (a) and Y 3 Al 5 O 12 (b).Quantitative values of Al 2 O 3 (c) and Y 3 Al 5 O 12 (d) after T F correction with linear fitting function.

Figure 5 .
Figure 5. (a) A HAADF-STEM image of wedge-shaped Al 2 O 3 .The EDS line profile (b) and mapping images ((c), (d)) were obtained from the black dotted arrow and blue dotted rectangular of figure 5(a), respectively.The mapping images (figures 5(c), (d)) converted to contour maps(((c), (d)) having composition information for individual pixels.Aluminum (g) and oxygen (h) contour maps after T F correction.

Table 1 .
The fitting results and quantitative values of the Al 2 O 3 sample after T F correction with power law and linear fitting function.