Local thickness and composition measurements from scanning convergent-beam electron diffraction of a binary non-crystalline material obtained by a pixelated detector.

We measured the local composition and thickness of SiO2-based glass material from diffraction. By using four dimensional scanning transmission electron microscopy (4D-STEM), we obtained diffraction at each scanning point. Comparing the obtained diffraction with simulated diffraction patterns, we try to measure the local composition and thickness. Although this method requires some constraints, this method measured local composition and thickness with 1/10 or less electron dose of EELS.


Introduction
Thickness and composition of a specimen is crucial information to analyze image, diffraction, and spectrum observed by transmission electron microscopy (TEM). The thickness and composition are mainly measured by spectroscopic methods, such as electron energy loss spectroscopy (EELS) or energy dispersive X-ray spectroscopy (EDS). In EELS, the relative thickness is measured by the log-ratio method, and the composition can be determined by the ratios between core-loss intensities of each atom [1]. In EDS, the thickness and composition of the sample are simultaneously determined by the ζ factor method [2]. Since these spectroscopic methods require a large electron dose, they cannot be applied to electronsensitive materials.
In crystalline materials, in addition to spectroscopy, imaging and diffraction are used to measure the thickness and composition. High-angle annular dark-field (HAADF) imaging enables the thickness and column composition to be determined at the single-atom level [3][4][5][6]. The convergent-beam electron diffraction (CBED) method enables the determination of the thickness at a sub-nanometer accuracy by analyzing the intensity distribution of diffraction disks [7,8]. The difference in the thickness affects the number of scattering events, and the difference in the composition affects the scattering cross section. These differences eventually appear in the CBED patterns. By matching the CBED patterns with the library of simulated diffraction patterns, the local thickness and composition can be determined. The development of the high-performance electron pixelated detector technique has enabled the acquisition of CBED patterns easily at each scanning point with high speed. This method, which acquires diffraction patterns (reciprocal 2D) at all scanning points (real 2D), is called 4D-scanning transmission electron microscopy (STEM) (reciprocal 2D × real 2D = 4D) [9], and enables a mapping of the thickness or composition of crystalline materials [10].
As stated above, there are alternatives to spectroscopic techniques for measuring the thickness and composition of crystalline materials. However, there are few alternatives to spectroscopic techniques for measuring the thickness and composition of non-crystalline materials [11]. In particular, the thickness and composition of non-crystalline materials have not been measured by diffraction. This is because noncrystalline materials do not have long-range order and do not make regular diffraction patterns like crystals. However, the intensity of the high angle region of diffraction increases as the thickness and the atomic number of the sample increase.
Since non-crystalline materials are macroscopically isotropic, the diffractions of non-crystalline materials mainly depend on the thickness and composition of the samples. Therefore, there is a possibility that information regarding the thickness and composition can be extracted from the diffraction simultaneously.
In this article, we applied the 4D-STEM method to a BaO-SiO2 glass system. By comparing the diffraction obtained by the experiment with the simulated diffraction under some constraints, we obtained both the thickness and composition maps. To verify the accuracy of these mappings, these were compared with the maps obtained by EELS and HAADF. As a result, through some constraints as this method requires, we succeed in the determination of both thickness and composition mappings with one-tenth of the electron irradiation of the EELS method.

Procedure to determine thickness and composition
The schematic procedure used to determine the thickness and composition is shown in Fig. 1. The diffraction pattern obtained by 4D-STEM was compared with those obtained by simulation. For this comparison, using the property that diffractions of glasses have no azimuthal dependency, only the azimuthally-averaged intensity (AAI) of the diffraction pattern was compared. An AAI is obtained by dividing a diffraction pattern into regions every 10 mrad from the center and averaging each region (the top row of Fig. 1).
There are three reasons why we used discrete expressions.
First, in this experiment, we only can acquire discrete diffraction pattern by pixelated detector. Second, discrete expressions are easier to handle in computer. Third ,by averaging in 10 mrad increments, the effect of noise is reduced. The obtained AAIs ( ) were compared with the simulated AAI ( sim ) (the second and third row of Fig.   1). We measured the mean squared error (mse) according to respectively. An azimuthally-averaged intensity (AAI) is obtained by dividing a diffraction pattern into regions every 10 mrad from the center and averaging in each region. The obtained AAIs are compared with the simulated AAI and calculate mean square error (mse). We adopt the thickness and composition that produced the AAI that has the lowest mse as the composition and thickness at the scanning point. the following formula, where is the number of divided regions of the AAI, AAI(r) is the intensity of the AAI at radius r (mrad) in the diffraction pattern. The ratio is calculated to adjust the total electron dose. Low mse means a high similarity and high mse means a low similarity between the two AAIs. We adopted the thickness and composition that produced the AAI that had the lowest mse as the composition and thickness at the scanning point (the bottom row of Fig. 1).
We performed the above comparison for all scanning points and obtained the thickness and composition maps.

4D-STEM, EELS, HAADF
The silicate-based glass 27.0 BaO-73.0 SiO2 (mol%) was selected for the experiment. This composition is known to separate into Si-pure phases and Ba-rich phases [12][13][14]. We performed 4D-STEM, high-angle annular dark-field (HAADF), and EELS. We set the camera length at 25 mm to secure a large collection angle when performing 4D-STEM.
The collection angle for 4D-STEM was in the range of 0-130 mrad. The 4D-STEM measurement was performed by 4D-canvas (JEOL. Ltd.). Dwell times and pixel sizes were set to 1 ms and 1.6 nm × 1.6 nm for 4D-STEM, 10 ms and 7.7 nm × 7.7 nm for HAADF and the thickness measurement by EELS, and 2 s and 20 nm ×20 nm for the measurement of the composition by EELS.

Simulation of diffraction patterns
The diffraction patterns were simulated by the multi-slice method using the Dr. Probe package [16] with form factors of Weickenmeier & Kohl [17], which incorporates the effect of inelastic scattering. The composition of the atomic structure was varied from 0 BaO-100 SiO2 to 100 BaO-0 SiO2 in 1 mol% increments, and the thickness was varied from 0 nm to 1000 nm in increments of 1 nm. The 0 BaO-100 SiO2 glass structures were created by molecular dynamics simulation using LAMMPS code [18]. The Badoped glass structures were made by randomly replacing Si atoms with Ba atoms and erasing O atoms. Though this is a simple structure construction, the densities of these structures were almost equal to that from a previous experiment [15]. We simulated the AAIs of the electron diffraction patterns under the condition that the acceleration voltage was 200 kV, defocus was 0 nm, convergence semiangle was 20.0 mrad, AAI range was from 0 to 130 mrad, and step size of AAI was 10 mrad.    composition under certain constraints.

Experiment
First, we attempted to determine the thickness of the sample with a known composition by 4D-STEM and simulation. We In this section, we attempted to detect the thickness and composition of a sample with an unknown thickness and composition. We analyzed the annealed sample whose composition was not uniform because of phase separation.
, , = , , , ,   around the pixel of interest at 27 mol%(w dependence is shown in Fig S2). The determination algorithm of the thickness and composition is schematically shown in Fig. 7.  Fig. 8

EELS. A HAADF image is also shown in
where HAADF is the intensity of HAADF, total is the intensity of the incident electron beam, is the Rutherford scattering cross-section of element , is the number of elements in unit volume, and t is the thickness of the sample. Rearranging the above equation, we obtain the following equation: where is almost proportional to the square of the atomic number [22,23]. The left side of the equation is the composition weight by scattering cross-sections. Thus, the right side is proportional to the composition. The compositional map created by the above equation is shown in Fig. 8(d). This map contains all the dark circular contrast, as indicated by red arrows.
The composition map created by 4D-STEM showed good method assessed a thicknesses that was 1.10-times higher than that determined by the EELS method(The detailed thickness ratio of thickness measured by 4D-STEM divided by thickness measured by EELS is shown in Fig. S1.).
Although this method required two constraints, this method can qualitatively measure the thickness and composition from AAIs.
Here, to assess the precision of detecting thickness, we measured the thickness of a SiO2 particle by the present 4D-STEM method. BF and HAADF images are shown in Fig.   9(a) and (b), respectively. Since, this sample is commercially available and made of pure SiO2 with a sphere-shape, we imposed the constraint that the composition is 0.0 BaO-100.0 SiO2. Figure 9(c) shows the thickness of the SiO2 sphere measured by the 4D-STEM method. The diameter of the sphere measured along in-plane direction is 1084 nm. We calculated the thickness ratio by dividing the thickness of a sphere with a diameter of 1084 nm by the thickness measured by 4D-STEM. The result is shown in Fig. 9(d).
The line profiles of Fig. 9(c) and (d) are shown Fig. S6. In at the edge of the sphere (whose thickness is less than 800 nm) and 1.2 near the center of the sphere (whose thickness less than 800 nm). In the thicker area, the present method tends to underestimate the thickness. The reason of We also measured thickness of the same SiO2 particle by EELS. The mean free path of SiO2 glass is 183 nm [24]. Further, this method has an additional benefit, an improvement of the image quality. A HAADF image obtained by extracting a 120 to 130 mrad region from the raw 4D-STEM data is shown in Fig. 11(a). The HAADF method only uses highly scattered electrons, which results in a low intensity, statistical fluctuations, and a noisy image.
After determining the composition and thickness, the AAI AAI. Raw AAI is extracted from the green rectangle in Fig.   9(a) and the reproduced AAI is extracted from the blue rectangle in Fig. 9 was specified from the AAI group created by simulation using the determined composition and thickness values at each pixel, and the 120 to 130 mrad region was extracted from the AAI to reconstruct a HAADF image. The reconstructed HAADF image is shown in Fig. 10(b).
Although the same electron dose was used, the reconstructed HAADF image contained less noise than the HAADF image created from the raw 4D-STEM data. When determining the composition and thickness, we compared the simulated AAI groups with the experimental AAI and chose the composition and thickness which had the most similar AAI as the answer. This operation had the same effect as fitting. This operation not only denoised but also restored the intensity of the saturated area (0-40 mrad region). Fig. 11(d) shows the bright field (BF) images obtained by extraction of the 0-10 mrad region from the reconstructed AAIs. The BF generated here corresponded to the region where the CCD was saturated and could not be obtained in the experiment.
As the contrast of the reconstructed BF was the inverse of the contrast of the HAADF image, the reconstruction of the BF was at least qualitatively correct.

Conclusions
We acquired AAIs of the electron diffraction patterns of each scanning point of 27.0BaO-73.0SiO2 glass by 4D-STEM. We also simulated diffraction patterns with several thicknesses and compositions of the BaO-SiO2 system.
Comparing the simulated AAIs with the rest of the simulated AAIs, we confirmed a one-to-one correspondence between the composition and thickness that reproduced a certain AAI.
By comparing the AAIs measured by 4D-STEM with the simulated AAIs, we determined the thickness of the sample with a known composition. Furthermore, we simultaneously determined both the composition and thickness of a sample with 27.0BaO-73.0SiO2 under two constraints; 1) thickness does not abruptly change and 2) average composition in the observed area is identical to that of the specimen. Although this method is limited compared with EELS and EDS, in that constraints are required, this method can determine both the thickness and composition with a lower electron dose and a higher accuracy than EELS or EDS. This method has an additional benefit, in an improvement of the quality of the image. It is expected that more various analyses are possible by changing how to apply the constraints.
In this experiment, we have estimated the relative accuracy of the 4D-STEM method by comparing with the EELS and HAADF methods. In future work, we will conduct the 4D-STEM method to investigate the absolute accuracy of this method by measuring a sample with a known thickness and composition. Figure S1. Ratios of thicknesses measured by 4D-STEM and EELS (4D-STEM/EELS). Top image shows the same region as Fig. 5. Bottom image shows the same region as Fig. 6.

Supplementary Information
The 4D-STEM method assesses a thickness that is thinner than the EELS method in the thin area, and thicker than the EELS method in the thick area.     The line profiles are acquired along the arrow in Fig. S6(b).
The black broken line in Fig. S6(c) shows the thickness assuming that the sphere is a perfect sphere with a diameter of 1084 nm. The black broken line in Fig. S6(d) shows the value of thickness ratio is 1.0.