Data analysis device, data analysis method, program, and recording medium

Abstract

To provide technology for evaluating the depth profile of a sample, using EDX analysis data from surfaces, nondestructively, and dispensing with assumption of initial profiles.SOLUTION: Provided is a data analyzer that analyzes the depth profile of a sample on the basis of the characteristic X ray generated from the sample upon incidence of an electron beam, said analyzer comprising: an input unit that accepts, from a measurement device that measures the characteristic X ray, a response signal that represents the measured value of intensity of the characteristic X ray; and an analysis unit that minimizes a deviation square-sum between the theoretical value of the response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample is modeled into a laminate composed of a plurality of layers, thereby analyzing the depth profile of the sample. The theoretical value of the response signal is derived on the basis of a generation function that expresses the depth distribution in the sample of a generated amount of characteristic X ray and a function that expresses the attenuation of characteristic X ray in the inside of the sample. When minimizing the deviation square-sum, the analysis unit calculates a relative concentration so as to simultaneously satisfy regularization conditions regarding the relative concentrations of chemical species of the sample.SELECTED DRAWING: Figure 5

Description

[Technical Field]

[0001] The present disclosure relates to a data analysis device, a data analysis method, a program, and a recording medium. [Background technology]

[0002] When discussing the characteristics and defects of various products, it is often extremely important to know the condition near the sample surface. In particular, evaluation of the "depth profile" of the distribution of chemical species that make up the sample in the depth direction is often required.

[0003] The most common methods for evaluating depth profiles are cross-sectional observation and cross-sectional analysis. For example, scanning transmission electron microscopy (STEM) / energy dispersive X-ray spectroscopy (EDX) analysis are used as important tools for understanding thin film multilayer structures on the nanometer order. Depth profiles can also be evaluated by combining surface analysis such as X-ray photoelectron spectroscopy (XPS) with ion sputtering. Therefore, the combination of surface analysis and ion sputtering is often used as a standard technique.

[0004] Although these methods, such as STEM / EDX analysis and XPS analysis, have been established with a high degree of reliability, they are both destructive analyses. Therefore, there is a risk that the sample's condition may change during sputtering or thinning, resulting in inaccurate information about the sample's true shape. Furthermore, these methods cannot be applied to samples that are difficult to obtain, such as defective products in the market or samples used for quality inspection on mass production lines. For this reason, there is a great need for non-destructive depth profile evaluation.

[0005] Among the various types of instrumental analysis, XPS analysis and X-ray fluorescence (XRF) analysis, which are classified as surface analysis, have long been used for non-destructive depth profile evaluation. XRF has the longest history in this field, and it is now standard practice to estimate the thickness of each layer in a multilayer thin film from the intensity ratio of fluorescent X-rays emitted from the sample using commercially available XRF equipment from various manufacturers. The thickness that can be evaluated varies greatly depending on the material, but is typically on the order of several tens of nanometers to several micrometers. These non-destructive evaluations using XRF are not profile evaluations, but rather involve specifying the layer structure of the sample in advance and estimating the thickness with only the thickness as the unknown.

[0006] In recent years, efforts to evaluate the "depth profile" itself, rather than analyzing only the thickness of a known structure, have become common in XPS analysis. Specifically, angle-resolved XPS (ARXPS) is used to vary the relative angle between the sample and detector, allowing measurements to be performed at different information depths. The depth profile is then estimated mathematically from the data. The maximum entropy method (MEM) is often used in ARXPS data analysis, and analytical laboratories offer commercial depth profile evaluation services using this method. For example, Non-Patent Document 1 ("Evaluation of Thin Films by Simultaneous Angle-Resolved Photoelectron Spectroscopy," Sumika Chemical Analysis Center, Ltd. Technical News TN413) reports the analysis of the depth profile of a self-assembled monolayer (SAM) of fluorinated alkanethiols formed on a gold (Au) substrate using simulation calculations applying MEM.

[0007] In depth profile analysis using MEM, the initial value settings have a significant impact on the final result, so an initial profile that is close to the correct answer is required. For this reason, it is difficult to apply MEM to samples with unknown depth profiles. In recent years, an alternative method, the maximum smoothness method (MSM), has been developed. By using MSM, it is possible to evaluate the depth profile nondestructively even for samples for which it is difficult to assume an initial profile (see, for example, Non-Patent Document 2 (Yutaka Hoshina, Kazuya Tokuda, and Yoshihiro Saito, "Non-destructive initial-profile-free depth profile evaluation of thin-film samples using angle-resolved X-ray photoelectron spectroscopy and profile smoothing regularization," published September 17, 2021, Jpn. J. Appl. Phys. 60 101003)). [Prior art documents] [Non-patent literature]

[0008] [Non-Patent Document 1] "Evaluation of Thin Films by Simultaneous Angle-Resolved Photoelectron Spectroscopy," Sumika Chemical Analysis Center Co., Ltd. Technical News TN413, [online], [Retrieved July 1, 2022], Internet<https: / / www.scas.co.jp / technical-informations / technical-news / pdf / tn413.pdf> [Non-patent document 2] Yutaka Hoshina, Kazuya Tokuda and Yoshihiro Saito, "Non-destructive initial-profile-free depth profile evaluation of thin-film sample using angle-resolved X-ray photoelectron spectroscopy and profile smoothing regularization" published September 17, 2021, Jpn. J. Appl. Phys. 60 101003 Summary of the Invention [Problem to be solved by the invention]

[0009] The above-mentioned Non-Patent Document 2 reports that MSM is applied to data from ARXPS analysis of thin film samples. However, from the perspective of widespread use of equipment, EDX or electron probe microanalyzer (EPMA) analysis from the surface, which is attached to scanning electron microscopy (SEM) observation and allows the acceleration voltage to be freely changed, is more widely used than XPS.

[0010] SEM / EDX analysis instruments are one of the most widely used analytical instruments, and are often owned by institutions and workplaces that do not specialize in analysis. Furthermore, because EDX uses an electron beam as its probe, it is possible to target and evaluate microscopic areas of approximately 1 μm square, which is difficult to do with XPS and XRF, which use X-ray beams as their probes. Therefore, if non-destructive depth profile evaluation using SEM / EDX analysis becomes possible, it would be extremely useful in research and industrial applications.

[0011] Therefore, an object of the present disclosure is to provide a technique for evaluating the depth profile of a sample non-destructively and without the need for an assumption of an initial profile, using data from EDX analysis from the surface. [Means for solving the problem]

[0012] The data analysis device disclosed herein is a data analysis device that analyzes the depth profile of a sample based on characteristic X-rays generated from the sample by the incidence of an electron beam, and includes an input unit that receives a response signal representing a measured value of the intensity of the characteristic X-rays from a measurement device that measures the characteristic X-rays, and an analysis unit that analyzes the depth profile of the sample by using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers and minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal, where the theoretical value of the response signal is a value derived based on a generation function that expresses the depth distribution of the generation amount of the characteristic X-rays in the sample and a function that expresses the attenuation of the characteristic X-rays inside the sample, and the analysis unit calculates the relative concentrations so as to simultaneously satisfy a regularization condition for the relative concentrations of chemical species in the sample in minimizing the sum of squares of the deviation.

[0013] The data analysis method disclosed herein includes the steps of receiving, from a measurement device, measured values ​​of the intensity of characteristic X-rays generated from a sample by the incidence of an electron beam, and analyzing a depth profile of the sample based on the measured values. The analyzing step includes a step of minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample is modeled as a stack of multiple layers. The theoretical value of the response signal is a value derived based on an emission function that expresses the depth distribution of the emission amount of characteristic X-rays in the sample and a function that expresses the attenuation of characteristic X-rays inside the sample. The minimizing the sum of squares of the deviation includes a step of calculating the relative concentrations of chemical species in the sample so as to simultaneously satisfy a regularization condition for the relative concentrations.

[0014] The program of the present disclosure causes a computer to execute the steps of receiving, from a measurement device, measured values ​​of the intensity of characteristic X-rays generated from a sample by the incidence of an electron beam, and analyzing a depth profile of the sample based on the measured values, wherein the analyzing step includes a step of minimizing the sum of squared deviations between the theoretical value of the response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample is modeled as a stack of multiple layers, where the theoretical value of the response signal is a value derived based on an generation function that expresses the depth distribution of the generation amount of characteristic X-rays in the sample and a function that expresses the attenuation of characteristic X-rays inside the sample, and the step of minimizing the sum of squared deviations includes a step of calculating the relative concentrations of chemical species in the sample so as to simultaneously satisfy a regularization condition for the relative concentrations.

[0015] The recording medium of the present disclosure is a recording medium having recorded thereon a program for causing a computer to execute the steps of receiving, from a measurement device, measured values ​​of the intensity of characteristic X-rays generated from a sample by the incidence of an electron beam, and analyzing a depth profile of the sample based on the measured values, wherein the analyzing step includes a step of minimizing the sum of squared deviations between the theoretical value of a response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample is modeled as a stack of multiple layers, wherein the theoretical value of the response signal is a value derived based on an generation function that expresses the depth distribution of the generation amount of characteristic X-rays in the sample and a function that expresses the attenuation of characteristic X-rays inside the sample, and the step of minimizing the sum of squared deviations includes a step of calculating the relative concentrations so as to simultaneously satisfy a regularization condition for the relative concentrations of chemical species in the sample. [Effects of the Invention]

[0016] According to the present disclosure, the depth profile of a sample can be evaluated non-destructively and without the need to assume an initial profile, using data from EDX analysis from the surface. [Brief explanation of the drawings]

[0017] [Figure 1] FIG. 1 is a schematic diagram showing a sample model used for EDX-MSM according to an embodiment of the present disclosure. [Figure 2] FIG. 2 is a diagram showing an example in which the model of the generation function is corrected so that the electron beam completely stops at a certain depth inside the sample. [Figure 3] FIG. 3 is a diagram illustrating an analysis system including an analysis device according to an embodiment of the present disclosure. [Figure 4] FIG. 4 is a block diagram illustrating an example of a hardware configuration of a data analysis device according to an embodiment of the present disclosure. [Figure 5] FIG. 5 is a diagram illustrating an example of functional blocks of the data analysis device 30 shown in FIG. [Figure 6] FIG. 6 is a flowchart illustrating the overall flow of a depth profile analysis method executed by the data analysis device 30 shown in FIG. [Figure 7] FIG. 7 is a flowchart illustrating a specific process flow of the depth profile analysis executed by the data analysis device 30 shown in FIG. [Figure 8] FIG. 8 shows the results of EDX-MSM analysis of the sample according to Example 1. [Figure 9] FIG. 9 shows the results of cross-sectional STEM observation of the sample of Example 1 used in the EDX-MSM analysis. [Figure 10] FIG. 10 is a view showing an SEM observation image of the surface of the sample of Example 2. [Figure 11] FIG. 11 shows the results of EDX-MSM analysis of the sample according to Example 2. [Figure 12] FIG. 12 shows the results of cross-sectional STEM observation of the sample of Example 2 used in the EDX-MSM analysis. [Figure 13] FIG. 13 shows the results of EDX-MSM analysis of the sample according to Example 3. [Figure 14] FIG. 14 shows the results of cross-sectional STEM observation of the sample at the same location as the EDX analysis. [Figure 15] FIG. 15 is a first diagram illustrating the effect of the sparsification process on the EDX-MSM analysis of the sample according to Example 2. [Figure 16]FIG. 16 is a second diagram illustrating the effect of the sparsification process on the EDX-MSM analysis of the sample according to Example 2. [Figure 17] FIG. 17 is a diagram showing the analysis results of a depth profile by another example of sparsification processing. [Figure 18] FIG. 18 is a diagram showing an example of EDX mapping data. [Figure 19] FIG. 19 is a diagram showing a location where a cross-sectional image of the distribution of chemical species in three-dimensional space is displayed. [Figure 20] FIG. 20 is a diagram showing an example of a cross-sectional image of the distribution of chemical species in three-dimensional space. [Figure 21] FIG. 21 is a diagram showing an example of a three-dimensional display of the distribution of chemical species in a three-dimensional space. DETAILED DESCRIPTION OF THE INVENTION

[0018] [Description of the embodiments of the present disclosure] First, embodiments of the present disclosure will be listed and described.

[0019] (1) A data analysis device according to one embodiment of the present disclosure is a data analysis device that analyzes a depth profile of a sample based on characteristic X-rays generated from the sample by the incidence of an electron beam, and includes: an input unit that receives a response signal representing a measured value of the intensity of the characteristic X-rays from a measurement device that measures the characteristic X-rays; and an analysis unit that analyzes the depth profile of the sample by using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers and minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured response signal, where the theoretical value of the response signal is a value derived based on an generation function that represents the depth distribution of the amount of characteristic X-rays generated in the sample and a function that represents the attenuation of the characteristic X-rays inside the sample, and the analysis unit calculates the relative concentrations so as to simultaneously satisfy a regularization condition for the relative concentrations of chemical species in the sample in minimizing the sum of squares of the deviation.

[0020] This configuration allows for the non-destructive evaluation of the depth profile of a sample using EDX analysis data from the surface, without the need for an initial profile assumption. To match the depth profile to the experimental data, the sum of squared deviations between the theoretical response signal and the measured response signal is minimized. However, since there are countless potential solutions to the minimization problem, a regularization condition is imposed. By calculating the relative concentrations so as to simultaneously satisfy the regularization condition during the minimization of the sum of squared deviations, a more plausible depth profile can be obtained that is in line with common sense or prior knowledge.

[0021] (2) In the above configuration (1), the regularization condition includes a condition that the sum of squares of the difference in relative concentration between adjacent layers is minimized for all chemical species and all layers in the stack.

[0022] In this configuration, the regularization condition corresponds to a smooth variation of the relative concentrations of the chemical species in the sample among the layers of the stack, which allows for a more plausible analysis of the depth profile of the sample.

[0023] (3) In the above construction (1), the regularization conditions include a charge neutrality condition for the chemical species.

[0024] In this configuration, the regularization condition is equivalent to restricting the ratio of chemical species. This allows for more plausible analysis results of the depth profile of the sample. For example, it is possible to prevent substances that do not actually exist from appearing in the depth profile.

[0025] (4) In the configuration described in (1) above, the regularization condition includes a condition to minimize a function whose value increases when chemical species exist in a mixed state in each layer of the sample.

[0026] With this configuration, when it is known in advance that there is little mixing of chemical species in each layer of the sample, it is possible to estimate the profile so that each chemical species exists as independently as possible in each layer, thereby obtaining more plausible analysis results for the depth profile of the sample.

[0027] (5) In the above configuration (1), the regularization condition includes a condition that the sum of absolute values ​​of the differences in relative concentrations between adjacent layers for all chemical species and all layers in the stack is minimized.

[0028] This configuration allows us to obtain a depth profile in which the relative concentrations of chemical species change sharply with depth, which makes it possible to estimate the composition and thickness of layers more likely when the detailed composition of the sample is unknown but the number of layers is known.

[0029] (6) In the configuration described in any one of (1) to (5) above, the analysis unit optimizes the device constant so that the sum of squares of the deviation between the value obtained by multiplying the theoretical value of the response signal by the device constant, which is a parameter related to the measurement device, and the measured value of the response signal is minimized.

[0030] This configuration introduces an instrument constant into the minimization of the sum of squared deviations. By optimizing the instrument constant, the absolute value of the theoretical response signal can be made closer to the measured value. Therefore, more plausible analysis results can be obtained for the depth profile of the sample.

[0031] (7) In the configuration of (6) above, the analysis unit alternately repeats a first calculation that fixes the equipment constants and optimizes the relative concentration and a second calculation that fixes the relative concentration and optimizes the equipment constants, and obtains a depth profile based on the relative concentration when the results of the first calculation and the second calculation converge.

[0032] When the first and second operations are simultaneously optimized, the analysis of the depth profile exceeds the scope of a convex quadratic programming problem, requiring the assumption of initial values. With this configuration, the first operation can be placed within the scope of a convex quadratic programming problem by alternately executing the first and second operations. Meanwhile, the second operation is a simple arithmetic operation. Therefore, the analysis of the depth profile does not require the assumption of accurate initial values.

[0033] (8) In the configuration described in any one of (1) to (7) above, the analysis unit is configured to be able to combine a correction function with the generation function and the function representing the attenuation of the characteristic X-rays in deriving the theoretical value, and the correction function is a function representing that the energy of the electron beam attenuates within the sample and reaches 0 at a certain depth in the sample.

[0034] The electron beam incident on the sample gradually loses energy within the sample and completely stops at a certain depth within the sample. With this configuration, a correction function that expresses the energy attenuation of the electron beam incident on the sample is combined with the generation function and a function that expresses the attenuation of the characteristic X-rays to calculate the theoretical value of the response signal. This avoids the problem of the absolute value of the film thickness being indefinite, thereby enabling more plausible analysis results to be obtained for the depth profile of the sample.

[0035] (9) In the configuration described in any one of (1) to (8) above, the analysis unit is configured to analyze the depth profile at each point on the surface of the sample based on a response signal obtained from the measurement device when the electron beam is scanned two-dimensionally on the surface of the sample, and to be able to perform display processing to display the analysis results of the depth profile of the sample as a chemical species distribution in three-dimensional space.

[0036] This configuration enables non-destructive chemical species analysis in three-dimensional space by applying the estimated depth profile to EDX surface mapping data. Displaying the analysis results as a chemical species distribution in three-dimensional space can provide more useful information in various situations, such as product development and failure analysis.

[0037] (10) A data analysis method according to one embodiment of the present disclosure includes the steps of receiving, from a measurement device, measured values ​​of the intensity of characteristic X-rays generated from a sample by the incidence of an electron beam, and analyzing a depth profile of the sample based on the measured values, wherein the analyzing step includes the step of minimizing the sum of squares of deviations between the theoretical value of the response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample is modeled as a stack of multiple layers, wherein the theoretical value of the response signal is a value derived based on an generation function that expresses the depth distribution of the generation amount of characteristic X-rays in the sample and a function that expresses the attenuation of the characteristic X-rays inside the sample, and the step of minimizing the sum of squares of deviations includes the step of calculating the relative concentrations of chemical species in the sample so as to simultaneously satisfy a regularization condition for the relative concentrations of the chemical species in the sample.

[0038] According to this configuration, the depth profile of the sample can be evaluated non-destructively and without the need to assume an initial profile, using data from EDX analysis from the surface.

[0039] (11) In the above construction of (10), the regularization condition includes the condition that the sum of squares of the difference in relative concentration between adjacent layers is minimized for all chemical species and all layers in the stack.

[0040] This configuration makes it possible to obtain more plausible analysis results for the depth profile of the sample.

[0041] (12) In the construction of (10) above, the regularization conditions include charge neutrality conditions for the chemical species.

[0042] This configuration makes it possible to obtain more plausible analysis results for the depth profile of the sample.

[0043] (13) In the configuration described in (10) above, the regularization condition includes a condition to minimize a function whose value increases when chemical species exist in a mixed state in each layer of the sample.

[0044] With this configuration, when it is known in advance that there is little mixing of chemical species in each layer of the sample, it is possible to estimate the profile so that each chemical species exists as independently as possible in each layer, thereby obtaining more plausible analysis results for the depth profile of the sample.

[0045] (14) In the configuration described in (10) above, the regularization condition includes a condition that the sum of absolute values ​​of the differences in relative concentrations between adjacent layers for all chemical species and all layers in the stack is minimized.

[0046] This configuration allows us to obtain a depth profile in which the relative concentrations of chemical species change sharply with depth, which makes it possible to estimate the composition and thickness of layers more likely when the detailed composition of the sample is unknown but the number of layers is known.

[0047] (15) In the configuration described in any one of (10) to (14) above, the step of minimizing the sum of squared deviations includes a step of optimizing the apparatus constant so that the sum of squared deviations between the theoretical value of the response signal multiplied by the apparatus constant, which is a parameter related to the measurement apparatus, and the measured value of the response signal is minimized.

[0048] With this configuration, by optimizing the instrument constants, the absolute value of the theoretical response signal can be made closer to the measured value, thereby obtaining more plausible analysis results for the depth profile of the sample.

[0049] (16) In the configuration described in (15) above, the step of minimizing the sum of squared deviations includes a step of alternately repeating a first calculation for optimizing the relative concentration while fixing an equipment constant and a second calculation for optimizing the equipment constant while fixing the relative concentration until the results of the first calculation and the second calculation converge, and a step of obtaining a depth profile using the relative concentration when the results of the first calculation and the second calculation converge.

[0050] With this configuration, by alternately executing the first and second operations, the first operation can be contained within the scope of a convex quadratic programming problem. Meanwhile, the second operation is a simple arithmetic operation. Therefore, accurate initial values ​​are not required for analyzing the depth profile.

[0051] (17) In the configuration described in any one of (10) to (16) above, the theoretical value is derived by combining a correction function with the generation function and a function expressing the attenuation of the characteristic X-rays, and the correction function is a function expressing that the energy of the electron beam attenuates within the sample and reaches 0 at a certain depth of the sample.

[0052] This configuration calculates the theoretical value of the response signal by combining a correction function that expresses the attenuation of the energy of the electron beam incident on the sample with the generation function and the function that expresses the attenuation of the characteristic X-rays. This avoids the problem of the absolute value of the film thickness being indefinite, and allows for more plausible analysis results to be obtained for the depth profile of the sample.

[0053] (18) In the configuration described in any one of (10) to (17) above, the receiving step is a step of receiving a response signal obtained from the measuring device when the electron beam is scanned two-dimensionally on the surface of the sample, the analyzing step is a step of analyzing a depth profile at each point on the surface of the sample, and the data analysis method further includes a step of executing a display process to display the analysis results of the depth profile of the sample as a chemical species distribution in three-dimensional space.

[0054] This configuration enables non-destructive chemical species analysis in three-dimensional space by applying the estimated depth profile to EDX surface mapping data. Displaying the analysis results as a chemical species distribution in three-dimensional space can provide more useful information in various situations, such as product development and failure analysis.

[0055] (19) A program according to an embodiment of the present disclosure causes a computer to execute the steps of receiving, from a measurement device, measured values ​​of the intensity of characteristic X-rays generated from a sample by the incidence of an electron beam, and analyzing a depth profile of the sample based on the measured values, wherein the analyzing step includes a step of minimizing the sum of squares of deviations between a theoretical value of a response signal and a measured value of the response signal, using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers, wherein the theoretical value of the response signal is a value derived based on an generation function that expresses the depth distribution of the generation amount of characteristic X-rays in the sample and a function that expresses the attenuation of the characteristic X-rays inside the sample, and the step of minimizing the sum of squares of deviations includes a step of calculating relative concentrations so as to simultaneously satisfy a regularization condition for the relative concentrations of chemical species in the sample.

[0056] According to this configuration, by executing the program, the computer can evaluate the depth profile of the sample non-destructively and without the need to assume an initial profile, using data from EDX analysis from the surface.

[0057] (20) A storage medium according to an embodiment of the present disclosure is a recording medium having recorded thereon a program for causing a computer to execute the steps of receiving, from a measurement device, measured values ​​of the intensity of characteristic X-rays generated from a sample by the incidence of an electron beam, and analyzing a depth profile of the sample based on the measured values, wherein the analyzing step includes a step of minimizing the sum of squares of deviations between the theoretical value of a response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample is modeled as a stack of multiple layers, wherein the theoretical value of the response signal is a value derived based on an generation function that expresses the depth distribution of the amount of characteristic X-ray generation in the sample and a function that expresses the attenuation of the characteristic X-rays inside the sample, and the step of minimizing the sum of squares of deviations includes a step of calculating the relative concentrations so as to simultaneously satisfy a regularization condition for the relative concentrations of chemical species in the sample.

[0058] According to this configuration, a program can be provided in which a computer uses data from EDX analysis from the surface to evaluate the depth profile of a sample non-destructively and without the need to assume an initial profile.

[0059] [Details of the embodiments of the present disclosure] Hereinafter, embodiments of the present disclosure will be described with reference to the drawings. In the drawings, the same or corresponding parts are designated by the same reference numerals, and description thereof will not be repeated.

[0060] First, a specific method for non-destructive depth profile evaluation using EDX surface analysis data will be explained.

[0061] EDX is a technique for elemental and compositional analysis by detecting characteristic X-rays emitted from a sample when the sample is irradiated with an electron beam and then dispersing these characteristic X-rays by energy. In the case of ARXPS, the detection depth of the sample is changed by changing the tilt angle of the sample relative to the analyzer. In contrast, EDX changes the detection depth of the sample by changing the acceleration energy of the electron beam incident on the sample.

[0062] ARXPS and EDX have in common the following points: a probe is injected into a sample to detect a response signal corresponding to the depth of the sample, and the depth of analytical information can be changed by controlling the measuring device. Therefore, in the embodiment described below, MSM analysis (see Non-Patent Document 2), which was developed for ARXPS, is applied to EDX analysis.

[0063] In the following, for the purpose of understanding MSM, the MSM analysis of ARXPS analytical data will be described. Therefore, the MSM analysis of ARXPS analytical data will be referred to as "ARXPS-MSM," and the MSM analysis of EDX analytical data according to the embodiment of the present disclosure will be referred to as "EDX-MSM" to distinguish between the two.

[0064] <1. Modeling of samples and physical phenomena related to EDX analysis> To perform nondestructive depth profile evaluation from EDX analysis data, it is necessary to model the physical phenomena involved in EDX analysis using functions that are as simple as possible. The more complex the modeling function, the more likely it is that the phenomenon will be reproducible by the theoretical formula, i.e., the greater the theoretical estimation accuracy. However, on the other hand, the more complex the theoretical formula, the more complex the profile estimation calculations using it become, which runs the risk of making the method less practical. In EDX-MSM, as in ARXPS-MSM, the sample is treated as a multilayer laminate film consisting of many extremely thin layers.

[0065] Figure 1 is a schematic diagram showing a sample model used in EDX-MSM according to an embodiment of the present disclosure. As shown in Figure 1, in EDX-MSM, the sample is considered to be a stack of K thin layers. The thickness of each layer is denoted as t.

[0066] Let I be the number of target chemical species, and let c be the relative concentration of chemical species i (i indicates the label of the chemical species, 1≦i≦I) in the kth layer. ik In each layer, the relative concentration of species i, c ik The sum of is 1 (Σ i c ik =1).

[0067] EDX-MSM deals with the behavior of characteristic X-rays within a sample. When dealing with the behavior of X-rays, it is more convenient to use "mass thickness" rather than simple thickness as the thickness of the sample. Hereinafter, in EDX-MSM, mass thickness will be used, and to clarify this, all thicknesses will be expressed as "ρz." Note that mass thickness alone does not correspond to the actual thickness, making it difficult to compare and discuss with other methods. Therefore, in an embodiment of the present disclosure, mass thickness is converted to simple thickness after EDX-MSM analysis is completed. A specific method for converting mass thickness to simple thickness will be described later.

[0068] As shown in Figure 1, an electron beam incident on a sample at an accelerating voltage j (j indicates the level of the accelerating voltage, where 1≦j≦J) attenuates within the sample, successively exciting characteristic X-rays corresponding to the chemical species present at that location. The degree of attenuation of the electron beam depends heavily on the accelerating voltage j of the electron beam. The characteristic X-rays generated within the sample travel toward the sample surface while attenuating within the sample, and are emitted from the sample surface at an angle θ to be detected by a detector (not shown in Figure 1). The angle θ represents the "take-off angle."

[0069] 2. Modeling the process from electron beam injection to characteristic X-ray generation EDX analysis involves irradiating a sample with an electron beam and observing the characteristic X-rays generated. EDX-MSM analysis takes into account the process by which the electron beam irradiated onto the sample excites characteristic X-rays as it attenuates within the sample, and the process by which the generated characteristic X-rays attenuate within the sample as they reach the sample surface. The function that represents the depth distribution of the amount of characteristic X-rays generated by excitation by the electron beam within the sample is generally called the "generation function."

[0070] The generation function is actually quite complex. However, various models have been proposed that approximate the generation function using simple analytical expressions, such as the Gaussian model, the product of an exponential function and the electron mean path, the quadrilateral model, the parabolic model, and the exponential model. In recent years, more detailed models incorporating the effects of fluorescence have also been proposed. In the embodiments of the present disclosure, as an example, a partially modified version of the exponential function and electron mean path product model derived by Philibert (J. Philibert, in X-Ray Optics and X-Ray Microanalysis, pp. 379-392, ed. H.H. Patee, V.E. Cosslett, and A. Engstrom, Academic Press, New York, 1963) is used as the generation function. The model derived by Philibert is not only relatively long-established, but is also easy to handle because it can be expressed by a relatively simple equation. However, there is no simple analytical model that is accurate for all materials, and the various models described above have their advantages and disadvantages depending on the situation or purpose. It should be noted that the model formula shown below is merely an example of this embodiment. Other generation function models may be used depending on various situations in the analysis scene.

[0071] According to Philibert's model, when an electron beam with an accelerating voltage of level j is irradiated onto a sample, the electron beam reaches the kth layer (mass depth ρz k The generation function of the characteristic X-rays of chemical species i generated at

[0072]

number

[0073] ω in Equation (1) i is the fluorescence yield of the target signal, which is an index of how easily the characteristic X-rays are emitted when the chemical species is ionized. R surf and R infare indices that represent the average deviation of the electron beam from the perpendicular near the sample surface and at the full diffusion depth, respectively. In the following explanation, following Philibert's paper, R surf =1, R inf = 2. Q in Equation (1) i ,σ j ,k j is expressed by the following equations (2) to (4).

[0074]

number

[0075]

number

[0076]

number

[0077] E in equation (2) Ci represents the minimum energy (unit: keV) of the electron beam required to excite the characteristic X-ray of the target. 0j represents the acceleration voltage of the electron beam (unit: kV). U0 (=E 0j / E Ci ) is called the overvoltage ratio, and E Ci With respect to the accelerating voltage E 0j In equation (4), Z represents the average atomic number of the sample, and A represents the average atomic weight of the sample.

[0078] In the embodiment of the present disclosure, the model of formula (1) is used as a base, and the following two-stage modifications are made for EDX-MSM analysis.

[0079] (i) Electron beam attenuation The first step of the correction is to deal with the attenuation of the electron beam. inf-(R inf -R surf )exp(-k j ρz k )) indicates that the generation function shows a peak at a position slightly deeper than the surface of the sample. j ρz k When considering the attenuation of an electron beam over a long distance range where exp(-σ) is sufficiently small, the value in the brackets in equation (1) can be considered to be approximately a constant. In other words, in equation (1), the attenuation of an electron beam over a sufficiently long distance is expressed by exp(-σ) j ρz k ) section.

[0080] Considering the above points, the model in equation (1) expresses that the electron beam decays exponentially from the surface of the sample, and has a non-zero intensity up to infinity. In reality, the electron beam incident on the sample gradually loses energy within the sample, and completely stops at a certain depth within the sample. However, this fact is not taken into account in the model in equation (1).

[0081] EDX-MSM analysis directly considers the compositional dependence of the sample thickness. Therefore, this is a key issue. For example, in the analysis of a layered structure consisting of chemical species A, chemical species B, and chemical species C (substrate), if the electron beam decays exponentially and maintains a nonzero intensity to infinity, there are effectively an infinite number of layer A and B thickness combinations that achieve the same EDX signal intensity ratio for chemical species A, B, and C. This is because the model assumes that the characteristic X-rays travel from an infinite depth into the sample to the detector. Even if layer A is thickened from a certain thickness, there will always be a thickness somewhere that results in the same intensity ratio if layer B is thickened accordingly. This creates a problem in MSM analysis: the absolute film thickness is indeterminate.

[0082] Therefore, in this embodiment, the fact that the electron beam completely stops at a certain depth inside the sample (the intensity of the generated characteristic X-rays becomes zero) is incorporated into equation (1).inf corresponds to the behavior of the electron beam at the full diffusion depth, so the brackets in equation (1) are mostly R inf The depth at which the electron beam can be considered as the complete diffusion depth is considered to be the complete diffusion depth. The electron beam in the sample is considered to stop completely at a depth twice the complete diffusion depth. To express this, we add F to equation (1) according to the following equation (5): CSj (ρz k ) is multiplied by the function F CSj (ρz k ) corresponds to a correction function that corrects the generation function to approximate the actual behavior of the electron beam inside the sample.

[0083]

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[0084] Equation (5) is k j ρz k This equation expresses that the depth where p is p is considered to be the complete diffusion depth, and that the value is 1 (uncorrected) in shallower regions, smoothly decays from 1 to 0 in the region from p to 2p, and becomes zero at depths of 2p or greater. The value of p in equation (5) corresponds to "considering that depth to be the complete diffusion depth," and should be a value greater than or equal to 1 in accordance with the original intent. However, it is difficult to theoretically determine the exact value of p. In the inventor's experience, it has been found that EDX-MSM analysis results are more likely to match actual conditions when p = 3 for inorganic materials and p = 2 for organic materials. For the original purpose of EDX-MSM analysis, which is to "understand the outline of the layer structure of an unknown sample," a value of p = 3 is practically sufficient.

[0085] It should be noted that equation (5) is merely an example of a simplified representation of the complete cessation of the electron beam inside the sample. Any function that expresses the electron beam smoothly decaying to zero at an appropriate depth inside the sample can be used as the correction function. Furthermore, some of the generation function models mentioned above incorporate this decay to zero, and if such a model is used, consideration of the correction term is unnecessary.

[0086] Figure 2 shows an example of correcting the model of the generation function so that the electron beam stops completely at a certain depth inside the sample. The generation function is the function expressed by equation (1), and it is assumed that a pure silicon (Si) sample is irradiated with an electron beam at an accelerating voltage of 20 kV. Figure 2(A) shows the case where the function c is calculated from equation (1). iz ω i Q i Figure 2(B) shows the change in the mass thickness of the part excluding c. Figure 2(C) shows the change in the mass thickness of the correction function according to equation (5). iz ω i Q i represents the change in mass (excluding mass) with respect to thickness.

[0087] As shown in Figure 2(A), if equation (1) is used as is, the generation function has a non-zero value up to infinity. By multiplying equation (1) by the correction function expressed by equation (5), the value of the corrected generation function becomes zero at a mass thickness of about 0.0006 [g / cm2]. The density of silicon (2.3 g / cm 2 ) considering the mass thickness 0.0006 [g / cm 2 ] corresponds to a thickness of about 2.6 μm. In other words, in the above model, at an accelerating voltage of 20 kV, characteristic X-rays of silicon are generated up to a thickness of about 2.6 μm. In reality, the characteristic X-rays generated inside the sample are attenuated as they reach the sample surface, so the information depth that actually contributes to the EDX signal is shallower than 2.6 μm.

[0088] (ii) Depth dependence of generation function The second correction is the depth dependence of the generation function. Equation (1) originally assumed a sample that is uniform in the depth direction. Therefore, it does not directly correspond to samples with composition that varies in the depth direction, as assumed by EDX-MSM analysis.

[0089] In the embodiment of the present disclosure, it is assumed that Equation (1) is strictly valid within each thin layer (see FIG. 1) treated as an analytical model of EDX-MSM. Based on this assumption, a generation function that takes into account the depth dependency can be obtained by going through the following procedure.

[0090] (Step 1) Equation (1) is applied as it is to the value of the generation function at the top surface layer of the sample. That is, φ ij (0)=c iz ω i Q i R surf is.

[0091] (Step 2) To calculate the value of the generation function for the second layer and beyond, first calculate the local k j The values ​​of the average atomic number Z and average atomic weight A in equation (4) are calculated by weighting based on the relative concentrations of the chemical species in the layer.

[0092] (Step 3) The local k obtained in Step 2 j Using the value of k j ρz k The cumulative value from the sample surface is updated.

[0093] (Step 4) k obtained in Step 3 j ρz k Using the cumulative value of k in Eq. (1), j ρz k and the value at k j (ρz k -t) and calculate the ratio.

[0094] (Step 5) Previous φ ij (ρz k-1 ) by the ratio obtained in step 4 to obtain the φ at that depth. ij (ρz k )

[0095] (Step 6) k j ρz kAccording to the cumulative value of φ obtained in step 5, ij (ρz k ) is corrected according to equation (5).

[0096] By applying the above two-stage correction to equation (1), the attenuation (generation function) of the electron beam in a sample whose composition changes in the thickness direction of the sample can be more accurately expressed than the original equation (1).

[0097] 3. Attenuation modeling of generated characteristic X-rays The characteristic X-rays generated within the sample attenuate within the sample before reaching the detector. The degree to which the characteristic X-rays generated by chemical species i in the kth layer attenuate before reaching the sample surface is expressed by the following equation (6):

[0098]

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[0099] As expressed by equation (6), the characteristic X-rays of chemical species i generated in the kth layer are attenuated in each layer (the (k-1)th layer, ..., the second layer, the first layer) located on the surface side of the kth layer. In equation (6), (μ / ρ) i1 , (μ / ρ) i2 , (μ / ρ) ik-1 are the mass absorption coefficients of chemical species i in the first, second, ..., (k-1)th layers, respectively. The mass absorption coefficients (μ / ρ) in each layer ik The method for determining is described later. In equation (6), t is the mass thickness of each layer, and θ is the take-off angle of the characteristic X-rays. In EDX, the relative angle between the detector and the sample is usually fixed, so the angle θ is determined depending on the configuration of the device.

[0100] In actual EDX analysis, the sum of signals generated from all K layers is observed. By combining equations (1), (5), and (6), the relationship between the EDX measurement data and the sample depth profile can be expressed as a matrix-vector relationship, as shown in equation (7).

[0101]

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[0102] The vector c on the right side of equation (7) is a vector with (I × K) rows, which is a line of depth profiles of all chemical species and all layers. Specifically, vector c can be expressed as equation (8).

[0103]

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[0104] The d on the left side of equation (7) is a vector with (I × J) rows and is the theoretical value of the measurement value obtained when EDX analysis is performed on a sample having the profile on the right side of equation (7). Specifically, the theoretical value d can be expressed as in equation (9).

[0105]

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[0106] Theoretical value d of component d' ij represents the measured intensity of characteristic X-rays when chemical species i and electron beam acceleration voltage j are used (see Figure 1). Equation (9) is treated as the theoretical value of EDX analysis, but in the following, it will be used as the actual measured data of EDX analysis in the same format as Equation (9). To distinguish it from the theoretical value, the actual measured data of EDX analysis will be represented as d.

[0107] The matrix S on the right side of equation (7) is a matrix with (I × J) rows and (I × K) columns, and links the depth profile and the EDX theoretical values. Specifically, matrix S is expressed as in equation (10). Note that hereinafter, (I × J) will be abbreviated as "IJ" and (I × K) will be abbreviated as "IK".

[0108]

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[0109] Coefficient r in matrix S jAlthough j does not appear in equations (1), (5), and (6), it is a parameter established to simplify the handling of relative concentrations. In ARXPS-MSM, j is an index of the take-off angle in ARXPS analysis, but in EDX-MSM, j is used as an index of the accelerating voltage. The matrix component s in equation (10) jk (i) is expressed by the following equation (11).

[0110]

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[0111] F in Equation (11) CSj is a correction term that expresses the complete cessation of the electron beam within the sample, as shown in equation (5). Equation (11) is an equation obtained by multiplying the product of the generation function in equation (1) and the function in equation (6) by the correction term. Note that equation (11) is merely a conceptual representation of the measured values ​​of characteristic X-rays. In actual calculations, rather than simply substituting each parameter into equation (11), the generation function is calculated sequentially according to the above steps (1) to (6).

[0112] <4. Application of regularization conditions> In the embodiment of the present disclosure, IK relative concentrations c ik By minimizing the sum of squared deviations with respect to the variables, it is possible to obtain a depth profile that best reproduces the measured data. However, minimizing the sum of squared deviations in this case has the problem that it is mathematically very unstable.

[0113] From the measurement data obtained by EDX analysis, the relative concentration c ik Estimating is a so-called inverse problem. According to Jacques Salomon Hadamard, a problem is generally well-posed if it satisfies all three requirements: (1) existence of a solution, (2) uniqueness of the solution, and (3) continuity or stability of the solution. A problem that does not satisfy any one of these requirements is an ill-posed problem.

[0114] When minimizing the sum of squared deviations between actual measured data and theoretical values, there are an infinite number of candidate solutions. The lack of a unique solution corresponds to an "ill-posed problem" in the Hadamard sense. To obtain a depth profile that best reproduces the measured data, constraints are required to select one solution from the infinite number of candidate solutions.

[0115] There is some reasonable thinking (or "common sense") about the system shown in Figure 1. This "common sense" is a constraint for choosing one solution from an infinite number of potential solutions, and corresponds to a regularization condition for solving ill-posed problems.

[0116] In this embodiment, the relative concentrations of the chemical species in the sample are calculated so as to simultaneously satisfy the regularization condition for the relative concentrations of the chemical species in the sample when minimizing the sum of squared deviations. In the EDX-MSM analysis, the function to be optimized (minimized) can be expressed as the following equation (12).

[0117]

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[0118] The first two terms in equation (12) are the sum of squares of the deviations between the EDX measurement data and the theoretical EDX values ​​predicted from the depth profile. When estimating a depth profile, it is always required that the theoretical values ​​match the experimental data, so these two terms are always present in equation (12). The function ψ included in equation (12) is a term that represents the normalization condition, and is a function that expresses the "common sense" or "prior knowledge" required for the depth profile, which cannot be determined solely by the sum of squares of the deviations between the EDX measurement data and the theoretical EDX values.

[0119] Below, several examples of the regularization condition in this embodiment will be described. Note that the regularization condition may be a function other than those described below. In this embodiment, various functions corresponding to "common knowledge" or "prior knowledge" can be used as the "regularization term." This allows various requirements to be imposed on the depth profile depending on the form of the function. Therefore, it is possible to obtain a depth profile that is consistent with the measurement data and also consistent with various common knowledge.

[0120] <5. Examples of regularization conditions> (5.1 Maximum smoothing condition) One of the conditions that can be used as the regularization condition is the condition that "the sum of squares of the deviation of the relative concentration between two adjacent layers is minimum." In this specification, this condition is called the "maximum smoothing condition." The maximum smoothing condition is expressed by the following equation (13). Here, Q s is a matrix with IK rows and IK columns expressed by equation (14).

[0121]

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[0122]

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[0123] A small sum of squares of the difference in relative concentration between two adjacent layers means that the change in relative concentration between layers is smooth. In other words, the maximum smoothness condition is a regularization condition that states that "for each chemical species, the depth profile is smooth." Therefore, the maximum smoothness condition can be said to be an essential part of MSM.

[0124] MSM is a method that overcomes the drawback of MEM, which is the need to input initial values. Of particular note is that MSM completely eliminates the need for an initial profile by reducing the determination of the depth profile to a convex quadratic programming problem for which a global optimal solution can be obtained.

[0125] matrix Q S corresponds to the calculation of the "unevenness of the system", and c T Q Sc A smooth profile is obtained by minimizing c T Q Sc The coefficient λ multiplied before S is a hyperparameter that determines how much importance is attached to the smoothness of the profile. S Increasing λ gives a smoother profile, while S When is made small, a profile that varies significantly in the depth direction is obtained.

[0126] (5.2 Charge neutrality condition) As the regularization condition, the "charge neutrality condition" described below may be applied.

[0127] Consider minimizing the quantity expressed by equation (15). In equation (15), e i represents a constant that constrains the abundance ratio of chemical species i.

[0128]

number

[0129] Here, Q EN is a matrix with IK rows and IK columns expressed by the following equation (16): Furthermore, E in equation (16) is a unit matrix with K rows and K columns.

[0130]

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[0131] For example, if it is reasonable from common sense that the ratio of chemical species i' to chemical species i" in a sample is 1:3, then in equation (15), e i′ =3, e i″ =-1, and other e i is set to 0. Note that e i′ and e i″The sign of (15) may be reversed from that described above. In other words, in this case, Equation (15) imposes a penalty for deviation of the concentration ratio of chemical species i′ to chemical species i″ from 1:3 in all K layers.

[0132] By applying the sum of equations (13) and (15) to the regularization term of equation (12), equation (12) can be expressed as equation (17).

[0133]

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[0134] In MSM, equation (17) is minimized with vector c as a variable. T Q CN By minimizing c, we can obtain a profile that constrains the abundance ratios of chemical species. T Q CN The coefficient λ multiplied before c CN is a hyperparameter that determines how much the ratio of chemical species is constrained. CN Increasing λ gives a profile that strictly constrains the ratio to the specified value, while CN By reducing , a profile can be obtained in which the abundance ratios of chemical species are not so restricted.

[0135] These parameters λ s and λ CN The degree of smoothness and charge neutrality of the obtained solution varies depending on . Since there is no absolute correct solution, the parameters are adjusted to obtain a plausible solution. In the examples described below, all λ S = 1. In the examples described below, when imposing charge neutrality constraints, all λ CN =1.

[0136] In the above description, the charge neutrality condition is used as a regularization condition together with the maximum smoothness condition. However, the charge neutrality condition does not necessarily have to be a condition associated with the maximum smoothness condition. The charge neutrality condition may also be used together with a regularization condition other than the maximum smoothness condition.

[0137] (5.3 Sparsification conditions) ARXPS-MSM and EDX-MSM analyses tend to output smooth depth profiles, which is reasonable from a common sense perspective. However, EDX-MSM analysis often produces problems with mixing of chemical species due to the solution being too smooth.

[0138] For example, even if the analyzed sample actually has a clear three-layer structure consisting of a layer of chemical species A, a layer of chemical species B, and a layer of chemical species C, the EDX-MSM analysis results may give a profile in which chemical species A, B, and C are uniformly mixed and distributed from the outermost surface to the innermost part. In many cases, the estimated profile is sparse, meaning that the chemical species are not evenly distributed (Q in the above-mentioned equation (17)). CN It is desirable for them to exist alone as much as possible (except for those that restrict their existence ratio).

[0139] To solve this problem, in this embodiment, a sparsification condition may be applied as a regularization condition. In order to obtain a sparse solution, "L1 norm regularization" is generally used. However, in this embodiment, the estimation parameter c ik Since there are constraints such as "the sum of the relative concentrations of chemical species must be 1," general L1 norm regularization cannot be applied. Therefore, the inventors developed the following simple sparsification process. In the simple sparsification process, the following equation (18) is minimized.

[0140]

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[0141] The first and third terms of equation (18) represent the sum of squares of the deviation between the measured data and the theoretical value, and the charge neutrality constraint, respectively, and are also included in the original EDX-MSM analysis equation (17). Therefore, the second and fourth terms of equation (18) are new elements for the simple sparsification process. The c in the second term of equation (18) ik (0) represents the depth profile output from the EDX-MSM analysis performed in advance. This second term allows us to search for an optimal (sparse) solution only in the vicinity of the original EDX-MSM analysis result.

[0142] The parameter λ of the second term L is a parameter that specifies how tightly the solution is constrained to the vicinity of the EDX-MSM result. In the examples described below, all λ L = 1. When equation (18) is expressed as a matrix, it becomes the following equation (19).

[0143]

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[0144] In equation (19), E represents an identity matrix with IK rows and IK columns. (0) is an IK row vector, which represents the profile of the EDX-MSM output result as expressed by the following equation (20).

[0145]

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[0146] The fourth term in equation (18) plays an essential role in sparsification. Therefore, from the fourth term in equation (18) and equation (19), the regularization term that indicates the essential "sparseness" in the simple sparsification process can be expressed as the following equation (21).

[0147]

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[0148] Similarly, from equation (19), the regularization term indicating "closeness to the original solution" in the simple sparsification process can be expressed as the following equation (22).

[0149]

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[0150] Target relative concentration c ik Since the sum of the relative concentrations of each chemical species is 1, the following equation (23) holds for the sum of squares of each chemical species.

[0151]

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[0152] Since the content of sigma in the second term of equation (23) is non-negative, the maximum value of the left side of equation (23) is 1. The maximum value of equation (23) is 1 when the relative concentration c ik is true when only one of the expressions is 1 and the others are zero. In other words, minimizing equation (21) is equivalent to maximizing the value of equation (23), which makes it easier to obtain a sparse solution.

[0153] The parameter λ in Eq. (21) SP is a parameter that specifies how strongly sparsity is desired. Note that the fourth term in equation (18) subtracts the quadratic coefficient of the original quadratic programming, so the parameter λ SP There is a limit to the size of the parameter λ SP If λ is made too large, any quadratic coefficient in the original quadratic programming becomes zero or negative, making the solution of the quadratic programming extremely unstable. In one example of the embodiment of the present disclosure, the parameter λ is adjusted using the following method. SP Determine.

[0154] In equation (19), the matrix S corresponding to the second-order coefficients when the sparsifying term is removed is TS+λ LE +λ CN Q CN The minimum value among the diagonal elements of is found. The minimum value is multiplied by an appropriate coefficient (but less than 1) and the resulting value is used as the parameter λ SP The value of the coefficient is, for example, 0.9, but is not limited to this. In this way, the parameter λ SP By determining the relative concentration c, we can obtain a sparse solution while maintaining the stability of the solution. ik By utilizing the constraint that the sum of the relative concentrations of each chemical species in each layer must be 1, it is possible to obtain a sparse solution within the scope of quadratic programming, as in the original EDX-MSM analysis, as described above.

[0155] It is not necessary to apply all of the above functions as regularization conditions, and any function may be selected as the regularization condition. The above functions may be used alone or in combination with other functions.

[0156] In the examples described below, simple sparsification processing is used in all cases. Therefore, unless otherwise specified, the entire process, including this simple sparsification processing, will be referred to as "EDX-MSM analysis" for convenience.

[0157] <6. A simple way to handle relative concentrations> In general, it is difficult to handle absolute values ​​of signal intensity and absolute concentrations of chemical species in EDX analysis. Therefore, it is common to handle only relative values ​​of both signal intensity and concentration of chemical species. Therefore, when comparing the theoretical value d' of EDX signal intensity with experimental data in the calculation of Equation (12), the theoretical value d' must be converted to a relative value.

[0158] From equation (7), the theoretical value d' is linear with respect to the relative concentration c. However, the value obtained by dividing each component of d' by the total component value of d' to convert the theoretical value d' to a relative value is nonlinear with respect to the relative concentration c. Therefore, the optimization problem of equation (17) goes beyond the scope of convex quadratic programming problems.

[0159] To solve this problem, in MSM, the constant r j Use the constant r j can be considered as an "instrument constant" that reflects unknown factors such as the absolute sensitivity of the instrument. In other words, the constant r j can be regarded as a parameter for virtually calculating the theoretical value d' of absolute signal intensity from the relative concentration c. Naturally, the constant r j The value of is unknown, but the constant r j In the case of ARXPS-MSM, the value of the constant r j is a separate value for each angle j. In contrast, in the case of EDX-MSM, the constant r j is a different value for each acceleration voltage j.

[0160] At some point in the analysis, the constant r j Assume that the provisional value of the constant r j A separate constant r j ' and multiply each constant r j Consider "updating" the constant r j The update policy is clear and minimizes equation (12). This means that the system constant r j This means that by optimizing the theoretical value, we can derive the value that is closest to the experimental data.

[0161] Submatrix S of matrix S expressed in equation (10) (i) is expressed by the following equation (24).

[0162]

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[0163] Each constant r j contributes only to the j component of the acceleration voltage in the experimental and theoretical values. Therefore, we can express Eq. (12) as the sum of squares of the deviations, j Assume that the result of partial differentiation with respect to ' is 0. This allows us to obtain the update equation expressed by equation (25).

[0164]

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[0165] Note that the relative concentration c and the constant r j Simultaneously optimizing (17) and (25) goes beyond the scope of convex quadratic programming and becomes a problem that requires initial value assumptions. However, in MSM, optimization of (17) and updating of (25) are alternately performed. Optimizing (17) is a convex quadratic programming problem, and updating (25) is a simple arithmetic operation. Therefore, constructing a depth profile becomes a problem that does not require initial value assumptions.

[0166] 7. Correction of mass absorption coefficient for each depth In ARXPS-MSM analysis, the value of the inelastic mean free path λ, which governs the attenuation of photoelectrons within the sample, is updated for each calculation loop. In EDX-MSM analysis, the mass absorption coefficient (= μ / ρ) plays the same role as the inelastic mean free path in ARXPS-MSM analysis. The mass absorption coefficient included in equation (6) depends on the composition in the depth direction of the sample, so the value of the mass absorption coefficient must be corrected according to the composition value. In EDX-MSM analysis, the mass absorption coefficient is updated for each layer by calculating the following equation (26) each time the depth profile is updated.

[0167]

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[0168] (μ / ρ) on the right side of equation (26) ii’ is the mass absorption coefficient of the characteristic X-rays of chemical species i in the pure substance of chemical species i' (μ / ρ). ii’ The relative mass concentration of species i' is c i’k (w) The mass absorption coefficient of the characteristic X-ray of chemical species i in the k layer can be calculated by multiplying the weighted values ​​by c and adding them up for all i'. i’k and calculated from the atomic weight of the chemical species.

[0169] <8. Calculating density at each depth> After the EDX-MSM analysis and subsequent sparsification process are completed, when outputting the final results, it is necessary to convert the mass thickness to thickness, as it is difficult to evaluate the results later. k ) to thickness (z k ) is the density at each depth ρ k is performed according to the following equation (27) using the value of

[0170]

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[0171] ρ in Eq. (27) k The value of is calculated using the depth profile obtained as a result of the MSM analysis and sparsification process described above, i.e., the relative concentration of chemical species at each depth. However, the density of a substance varies depending on the type of substance, and cannot be determined solely by the relative concentration of chemical species. Therefore, ρ is calculated using the following method. k Determine the value of

[0172] First, M types (M is plural) of reference materials with known physical properties (e.g., a group of materials for which literature values ​​for physical property parameters exist, such as semiconductors (e.g., silicon (Si)) and metals (e.g., gold (Au))) are selected in advance. The density of the reference material m is defined as ρ rm and the density ρ rm When expressed in terms of I chemical species, the relative concentration of the ith chemical species is defined as cr im The final depth profile c ik is approximated by the sum of M types of reference substances as shown in the following equation (28).

[0173]

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[0174] a in equation (28) mkis the coefficient used to approximate the relative concentration of a chemical species at that depth by the sum of the reference substances. The coefficient a mk can be determined using, for example, the same quadratic programming method as in the EDX-MSM analysis. mk The density at each depth is determined by calculating the following equation (29) using the set of

[0175]

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[0176] <9. Example of overall flow of EDX-MSM analysis> In one example of this embodiment, the steps for estimating a depth profile at a point (or an average over a wide area) in a sample from EDX surface analysis data for that point are as follows: In the following example, the maximum smoothing condition, charge neutrality condition, and sparsification condition are applied as regularization conditions.

[0177] (1) EDX surface analysis data is obtained at multiple levels of accelerating voltage. (2) For all chemical species (EDX signals) under consideration, select the mass absorption coefficient, fluorescence yield, charge-neutrality constraint in Eq. (16), and reference material to be used in Eq. (27). Known values ​​can be used for the mass absorption coefficient and fluorescence yield. In the example below, the mass absorption coefficient is given by KFJ Heinrich, Proc. ICXOM Vol. 11 (JD Brown, RH Packwood, eds.), p. 67 (1986), and the fluorescence yield is given by W. Bambynek, B. Crasemann, RW Fink, H.-U. Freund, H. Mark, CD Swift, RE Price, and PV Rao, Rev. Mod. Phys. Vol. 44, 716 (1972).

[0178] (3) Parameter λ in Eq. (17) S ,λ CN and the parameter λ in Eq. (18) L ,λ SP In the example described below, λ S ,λ CN ,λ L are all fixed to 1. SP is the matrix S that corresponds to the second-order coefficient when the sparsification term is removed from equation (19) as described above. T S++λ L E+λ CN Q CN The minimum value of the diagonal elements of is found and fixed to a value 0.9 times that minimum value.

[0179] (4) As the initial value of the desired depth profile, c ik = 1 / I. The mass absorption coefficient at each depth is the average for all chemical species.

[0180] (5) The equipment constant r in Eq. (10) j are all set to 1, and the equipment constant r j Update.

[0181] (6) The quadratic programming problem in equation (17) is solved by solving the relative concentration c ik and use the results to update the mass absorption coefficient at each depth according to equation (26). Then, the instrument constant r in equation (10) is j Update.

[0182] (7) Relative concentration c after step (6) ik If the change in is small enough, it is determined that the solution has converged and proceed to the next step (8). ik If the change in is large, step (6) is performed again.

[0183] (8) The quadratic programming problem in equation (18) is solved by solving the relative concentration c ik and use the results to update the mass absorption coefficient at each depth according to equation (26). Then, the instrument constant r in equation (10) isj Update the vector c in equation (18). (0) The components of are the relative concentrations c ik The value of is used.

[0184] (9) Relative concentration c after step (8) ik If the change in is small enough, it is determined that the solution has converged and proceed to the next step (10). ik If the change is large, execute step (8) again.

[0185] (10) The relative concentration c at the time when the solution is determined to have converged ik Using equations (27), (28), and (29) as the final result, we calculate the local density at each depth and convert the “mass thickness” to “thickness.”

[0186] (11) Calculated relative concentration c ik The pair of the value of and the depth (thickness) calculated in step (10) is output as the final solution.

[0187] The greatest feature of EDX-MSM analysis compared to existing techniques (such as MEM) is step (4) above. Conventionally, when estimating film thickness from surface EDX data, initial values ​​related to the depth distribution of chemical species, such as layer structure, have been input. This required inputting a structure that was almost "close to the correct answer." EDX-MSM analysis is characterized by starting from an unbiased state in step (4), where the initial values ​​are equal across the entire depth. This makes it possible to analyze samples with unknown layer structure without having to assume initial values ​​related to the structure.

[0188] 10. Analysis Device and Analysis Method In the embodiment of the present disclosure, a computer executes the above steps (1) to (11) to analyze a depth profile from EDX analysis data of a sample.

[0189] 3 is a diagram showing an analysis system including an analysis device according to an embodiment of the present disclosure. The analysis system 10 includes an energy dispersive X-ray spectroscopy (EDX) device 20 and a data analysis device 30.

[0190] The EDX apparatus 20 irradiates the sample 25 with an electron beam as a probe and measures the intensity of characteristic X-rays (response signals) generated from the sample 25. The EDX apparatus 20 may be combined with, for example, an SEM. The sample 25 may be, for example, a solid sample having a layer structure.

[0191] The data analysis device 30 is realized by hardware that conforms to a general-purpose computing architecture. The data analysis device 30 acquires measurement data from the EDX device 20. The data analysis device 30 analyzes the depth profile of the sample 25 from the measurement data.

[0192] 4 is a block diagram showing an example hardware configuration of a data analysis device according to an embodiment of the present disclosure. The data analysis device 30 includes a processor 31, a primary storage device 32, a secondary storage device 33, an external device interface 34, an input interface 35, an output interface 36, a communication interface 37, and a bus 38. Elements such as the processor 31 and the primary storage device 32 exchange data, signals, and the like via the bus 38.

[0193] The processor 31 processes programs and data stored in the primary storage device 32. The primary storage device 32 stores the programs executed by the processor 31 and the data referenced by the processor 31. In some aspects, a dynamic random access memory (DRAM) may be used as the primary storage device 32.

[0194] The secondary storage device 33 stores programs, data, etc. in a nonvolatile manner. In some aspects, a nonvolatile memory such as an HDD (Hard Disk Drive) or an SSD (Solid State Drive) may be used as the secondary storage device 33. Therefore, the secondary storage device 33 corresponds to a computer-readable recording medium that records a program to be executed by a computer.

[0195] The external device interface 34 is used when connecting an external device to the data analysis device 30. The external device interface 34 may be, for example, a USB (Universal Serial Bus) interface.

[0196] The input interface 35 is used to connect input devices such as a keyboard 41 and a mouse 42. The input interface 35 accepts user operations and inputs through these input devices.

[0197] The output interface 36 is used to connect an output device such as a display 43 .

[0198] The communication interface 37 is used for the data analysis device 30 to communicate with external devices. For example, the communication interface 37 is used for communication of the data analysis device 30 via a network. Communication with external devices may be wireless or wired.

[0199] The data analysis device 30 may optionally include an optical drive. The optical drive reads a computer-readable program stored in a recording medium (for example, an optical recording medium such as a DVD (Digital Versatile Disc)) that non-transiently stores the program. The program read from the recording medium may be installed in a secondary storage device 33 or the like. Furthermore, various programs executed by the data analysis device 30 may be downloaded from a server device or the like on a network and installed in the data analysis device 30.

[0200] Fig. 5 is a diagram illustrating an example of functional blocks of the data analysis device 30 illustrated in Fig. 4. In one aspect, each block illustrated in Fig. 5 is realized by a computer that executes a program according to an embodiment of the present disclosure.

[0201] As shown in FIG. 5, the data analysis device 30 includes an input unit 51, an analysis unit 52, an output unit 53, and a storage unit .

[0202] The input unit 51 receives measurement data output from the EDX device 20 (see FIG. 3 ). The input unit 51 also receives various information necessary for analyzing the depth profile of the sample 25 (for example, information on the type of material and chemical species).

[0203] The storage unit 54 stores an analysis program 71 for the depth profile of the sample 25, parameters 72 required for executing the analysis program 71, etc. Furthermore, the storage unit 54 may store measurement data input to the data analysis device 30.

[0204] The analysis unit 52 uses the measurement data input to the input unit 51 and the parameters 72 to obtain a depth profile of the sample 25. The analysis unit 52 includes a parameter determination unit 61, a regularization function determination unit 62, a hyperparameter determination unit 63, and a calculation unit 64.

[0205] The parameter determination unit 61 determines the mass absorption coefficient μ / ρ, the fluorescence yield ω, and the density ρ of the reference substance included in the formula (11). rm The parameter determining unit 61 determines parameters related to chemical species contained in the sample, the value of the parameter e for the charge neutrality condition, etc. For example, the parameter determining unit 61 determines the relevant parameters from among the parameters 72 stored in the storage unit 54.

[0206] The regularization function determination unit 62 determines a function corresponding to the regularization conditions required for analyzing the depth profile. Examples of the determined regularization function are described in <5. Examples of Regularization Conditions>, so the description will not be repeated here. Note that functions that can be used as regularization functions are not limited to the functions described in <5. Examples of Regularization Conditions>.

[0207] The hyperparameter determination unit 63 determines the parameter λ included in the equation (17). S , λ CN and the parameter λ in Eq. (18). L , λ SP Determine the value of λ as described above. S ,λ CN ,λ L may all be fixed to 1. Alternatively, the user may S ,λ CN ,λ L The hyperparameter determination unit 63 determines the specified values ​​as λ S ,λ CN ,λ L The hyperparameter determination unit 63 may determine the values ​​of the matrix S corresponding to the second-order coefficients when the sparsification term is removed from the equation (19). T S++λ L E+λ CN Q CN The minimum value of the diagonal elements of is found, and the minimum value is multiplied by a predetermined coefficient (for example, 0.9) to obtain λ SP The value of may be determined.

[0208] The calculation unit 64 receives the parameter values ​​determined by the parameter determination unit 61 and the hyperparameter determination unit 63. The calculation unit 64 also receives information on the regularization function determined by the regularization function determination unit 62. The calculation unit 64 executes, for example, the processing according to steps (6) to (10) of the above-mentioned <9. Example of Overall Flow of EDX-MSM Analysis> to calculate the relative concentrations c for all chemical species and all layers. This allows the depth profiles of all chemical species to be obtained.

[0209] The output unit 53 outputs the depth profile of the sample 25 obtained by the analysis unit 52 as the analysis result. The analysis result is displayed on the display 43 (see FIG. 4), for example.

[0210] 6 is a flowchart illustrating the overall flow of the depth profile analysis method executed by the data analysis device 30 shown in FIG. 6. As shown in FIG. 6, when the process starts, in step S11, the input unit 51 acquires a response signal (i.e., EDX measurement data) representing the measured value of the intensity of characteristic X-rays from the EDX device 20. The analysis unit 52 executes the processes of steps S12 to S17 and S20. The processes of steps S12 to S17 may be executed in any order. Alternatively, the processes of a plurality of steps out of steps S12 to S17 may be executed simultaneously.

[0211] In step S12, the parameter determination unit 61 determines the values ​​of parameters related to the chemical species. The parameters related to the chemical species are, for example, the values ​​of the mass absorption coefficient μ / ρ and the fluorescence yield ω. The values ​​of the mass absorption coefficient μ / ρ and the fluorescence yield ω depend on the material constituting the sample 25. For example, the storage unit 54 may store a database that defines the values ​​of the mass absorption coefficient μ / ρ and the fluorescence yield ω for each type of material. The parameter determination unit 61 can acquire the values ​​of parameters such as the mass absorption coefficient μ / ρ and the fluorescence yield ω by referring to the database based on the information on the chemical species. Furthermore, the analysis unit 52 can acquire information on the chemical species constituting the sample from the user via the input unit 51.

[0212] The parameter determination unit 61 determines the parameters e1 to e2 included in the formula (16) based on the information on the composition of the sample 25 input to the input unit 51 by the user, for example. I The value and sign of may be determined.

[0213] In step S13, the regularization function determination unit 62 selects one or more regularization conditions and determines a function corresponding to the selected regularization conditions. For example, the regularization function determination unit 62 selects a maximum smoothing condition, a charge neutrality condition, and a sparsification condition as regularization conditions and determines functions representing the maximum smoothing condition, the charge neutrality condition, and the sparsification condition, respectively.

[0214] In step S14, the parameter determination unit 61 determines the type of reference material to be used in the calculation of equation (25). In step S15, the hyperparameter determination unit 63 determines the parameter λ S ,λ CN and the parameter λ in Eq. (18) L ,λ SP The hyperparameter determination unit 63 determines the parameter λ until the analysis of the depth profile is completed. S ,λ CN ,λ L ,λ SP Do not change the value of

[0215] In step S16, the calculation unit 64 calculates the depth profile (relative concentration c ik ) at the full depth, ik In step S17, the calculation unit 64 determines the initial value of the device constant. For example, r j = 1. The calculation unit 64 updates the device constant r j However, the initial value of the device constant can be determined randomly. The reason is that when the first calculation loop is executed according to the flow shown in FIG. 7, the process of step S23 determines r j This is because the influence of the value of r is reset.j If the initial value of is set to 100 instead of 1, in the process of step S23 described later, j is multiplied by a ratio of 1 / 100 of the original value.

[0216] After the instrument constants are updated in step S17, the calculation unit 64 analyzes the depth profile in step S20. The calculation unit 64 calculates the relative concentrations of chemical species so as to minimize the sum of squares of the deviations between the measured values ​​and the theoretical values ​​and simultaneously satisfy the regularization condition for the relative concentrations of the chemical species in the sample. When the processing of step S20 is completed, the entire processing is completed.

[0217] Fig. 7 is a flowchart illustrating a specific processing flow of the depth profile analysis executed by the data analysis device 30 shown in Fig. 5. The flow shown in Fig. 7 corresponds to the detailed processing flow of step S20 shown in Fig. 6.

[0218] As shown in FIG. 7, in step S21, the calculation unit 64 calculates the depth profile (relative concentration c ik ) is determined. The processing in step S21 is minimization of equation (17) using vector c as a variable. Therefore, the calculation unit 64 minimizes the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal, using the theoretical value of the response signal when the sample to be analyzed is modeled as a laminate consisting of multiple layers. The processing in step S21 corresponds to finding a solution to a convex quadratic programming problem. Therefore, various algorithms known as algorithms for finding a solution to a convex quadratic programming problem can be applied to the processing in step S21.

[0219] In minimizing the sum of squared deviations, the calculation unit 64 calculates the relative concentrations of the chemical species in the sample so as to simultaneously satisfy regularization conditions for the relative concentrations. For example, in the case of EDX-MSM, the maximum smoothing condition and the charge neutrality condition are applied as regularization conditions in the processes of S21 to S24.

[0220] In step S22, the calculation unit 64 updates the mass absorption coefficient μ / ρ at each depth of the sample according to equation (26). In step S23, the calculation unit 64 calculates the depth profile (relative concentration c ik Specifically, the calculation unit 64 updates the equipment constant r in accordance with the equation (25). j The process in step S23 corresponds to the optimization of the device constants.

[0221] In step S24, the calculation unit 64 determines whether the solution to the quadratic programming problem obtained by the processing in step S21 has converged. ik If the change in the relative concentration c is sufficiently small (for example, equal to or smaller than a predetermined threshold), the calculation unit 64 determines that the solution has converged, and proceeds to step S25. ik If the change in is large (exceeds a predetermined threshold), the calculation unit 64 returns the process to step S21. That is, the processes of steps S21 to S23 are repeated until the solution converges, whereby the first calculation (processing of step S21) for optimizing the relative concentration while fixing the equipment constants and the second calculation (processing of step S23) for optimizing the equipment constant while fixing the relative concentration are alternately repeated.

[0222] When the sparsification condition is set as the regularization condition, the calculation unit 64 executes a simple sparsification process in step S25. For example, the calculation unit 64 solves the quadratic programming problem of equation (18) by using the relative concentration c ik In step S26, the calculation unit 64 calculates the relative concentration c obtained by the process in step S25. ik In step S27, the calculation unit 64 updates the mass absorption coefficient at each depth according to equation (26) using the apparatus constant r j Update.

[0223] In step S28, the calculation unit 64 determines whether the solution to the quadratic programming problem obtained by the processing in step S25 has converged. ikIf the change in the relative concentration c is sufficiently small (for example, equal to or smaller than a predetermined threshold), the calculation unit 64 determines that the solution has converged, and proceeds to step S29. ik If the change in is large (exceeds the predetermined threshold), the calculation unit 64 returns the process to step S25. That is, the processes of steps S25 to S27 are repeated until the solution converges, whereby the first calculation (processing of step S25) for optimizing the relative concentration while fixing the equipment constants and the second calculation (processing of step S27) for optimizing the equipment constant while fixing the relative concentration are alternately repeated.

[0224] In step S29, the calculation unit 64 calculates the relative concentration c ik Using equations (27), (28), and (29) as the final result, the calculation unit 64 calculates the local density at each depth. In this way, the calculation unit 64 converts the "mass thickness" into "thickness." In step S30, the calculation unit 64 converts the calculated relative concentration c ik The value of and the depth (thickness) are output, thereby obtaining the depth profile of the sample.

[0225] 7, the sparsification process is performed after the solution has converged in step S24. However, looking at the overall process flow of step S20, the data analysis method according to this embodiment can be said to include a step of minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal, and a step of calculating the relative concentrations so as to simultaneously satisfy the regularization condition for the relative concentrations of the chemical species in the sample.

[0226] 11. Example of depth profile analysis Using the above analytical equipment and method, EDX-MSM analysis was performed on EDX analysis data for a single point on the sample surface. The three examples described below are for samples with different thicknesses. Cross-sectional STEM observations were performed for all three examples to compare the surface EDX data and MSM analysis results. All surface EDX data were acquired using a JEOL SEM / EDX system (SEM: JSM-IT200, JEOL; EDX: Xplore30, Oxford Instruments). Cross-sectional STEM observations for comparison with EDX-MSM analysis results were performed using a scanning transmission electron microscope (JEOL JEM-ARM300F2) after thinning the samples using a focused ion beam processing system (Helios G4 PFIB Uxe, Thermo Fisher Scientific).

[0227] (Example 1: Cr / Ni / Si laminated sample) Specimens were prepared by vapor-depositing nickel (Ni) and chromium (Cr) onto a Si wafer. Conditions were set so that the thickness of the Cr layer differed between specimens A and B. EDX analysis was performed on 100 μm square areas from the surface of specimens A and B at multiple accelerating voltages, and the intensities of the Cr L line, Ni L line, and Si K line were obtained as characteristic X-ray intensities. The accelerating voltage and parameters used for EDX-MSM analysis are shown in Table 1 below and are common to specimens A and B.

[0228] [Table 1]

[0229] FIG. 8 shows the results of EDX-MSM analysis of samples according to Example 1. FIG. 8(A) shows the results of EDX-MSM analysis of sample A, and FIG. 8(B) shows the results of EDX-MSM analysis of sample B. For both sample A and sample B, the analysis results indicated a structure of Cr / Ni / Si. The thickness of the Cr layer was estimated to be about 1 nm for sample A and about 10 nm for sample B. The thickness of the Ni layer was estimated to be about 5 nm for both sample A and sample B.

[0230] Figure 9 shows the results of cross-sectional STEM observation of the sample of Example 1 used in the EDX-MSM analysis. Figure 9(A) shows a cross-sectional STEM image of sample A, and Figure 9(B) shows a cross-sectional STEM image of sample B. The magnification of both Figure 9(A) and Figure 9(B) is 5,000,000 times.

[0231] As shown in Figure 9, the thickness of the Cr layer was approximately 5 nm for sample A and approximately 10 nm for sample B. The thickness of the Ni layer was approximately 10 nm for both sample A and sample B. In the EDX-MSM analysis results, the absolute thicknesses of the Cr and Ni layers were smaller than those in the cross-sectional STEM results. However, the EDX-MSM analysis correctly estimated that both sample A and sample B had a layered structure of Cr / Ni / Si and that the thickness of the Cr layer in sample A was approximately half that of the Cr layer in sample B. It should be emphasized again that the results shown in Figure 8 were obtained without assuming any layered structure and starting from an initial value of a 1:1:1 relative concentration ratio of Cr, Ni, and Si throughout the entire sample, from the outermost surface to the deepest part.

[0232] (Example 2: Ni electrode pattern sample on InP) A sample was prepared by forming a Ni electrode and a silicon nitride (SiN) protective film on an indium phosphide (InP) wafer. Figure 10 shows a surface SEM image of the sample of Example 2. The surface SEM image shown in Figure 10 was obtained under conditions of 5 kV and 100x magnification. Regions A and B in the surface SEM image are the regions to be analyzed. The composition in the depth direction in region A is SiN / InP, and the composition in the depth direction in region B is Ni / InP.

[0233] EDX-MSM analysis and cross-sectional STEM observation were performed on Region A and Region B. The EDX-MSM analysis parameters are shown in Table 2 below. In the sample of Example 2, Si and N, and In and P are not thought to exist independently, so the Si:N and In:P ratios were constrained to approximately 1:1 as a condition for the charge neutrality constraint of Equation (19).

[0234] [Table 2]

[0235] Figure 11 shows the results of EDX-MSM analysis of the sample according to Example 2. As shown in Figure 11(A), in region A, it is evaluated that a SiN layer with a thickness of about 260 nm is formed on the InP wafer. As shown in Figure 11(B), in region B, it is evaluated that a Ni layer with a thickness of about 280 nm is formed on the InP wafer.

[0236] FIG. 12 shows the results of cross-sectional STEM observation of the sample of Example 2 used in EDX-MSM analysis. FIG. 11(A) shows a cross-sectional STEM image of region A, and FIG. 11(B) shows a cross-sectional STEM image of region B. As shown in FIG. 11, the thickness of the SiN layer in region A is approximately 260 nm, and the thickness of the Ni layer in region B is approximately 180 nm. According to the EDX-MSM analysis results, the thickness of the SiN layer is equivalent to the cross-sectional STEM observation results, while the thickness of the Ni layer is estimated to be approximately 1.5 times thicker than the cross-sectional STEM observation results. However, according to the EDX-MSM analysis, the thicknesses of both the SiN layer and the Ni layer are correct. Furthermore, the EDX-MSM analysis correctly estimated the layered structure of the sample.

[0237] (Example 3: PFA sample on Cu foil) A sample was prepared by applying PFA (perfluoroalkoxy alkane), a type of fluororesin, to copper (Cu) foil and then baking it. A 100 μm square area of ​​this sample was subjected to surface EDX analysis, as in Example 1. The measurement conditions and EDX-MSM analysis conditions are shown in Table 3 below. Note that, under the charge neutrality constraint of Equation (19), the composition ratio of the chemical species C (carbon) to the chemical species F (fluorine) was set to approximately 1:2, based on the chemical formula of PFA.

[0238] [Table 3]

[0239] 13 shows the results of EDX-MSM analysis of the sample according to Example 3. From the distribution of C and F, the thickness of the PFA was estimated to be about 1150 nm.

[0240] FIG. 14 shows the results of cross-sectional STEM observation of the sample at the same location as the EDX analysis. Note that in the cross-sectional STEM image shown in FIG. 14, a Pt layer is formed on top of the PFA. The Pt layer is a coating layer provided to make the edge of the PFA easier to see in the cross-sectional STEM image of the sample. According to the cross-sectional STEM image, the thickness of the PFA is approximately 1.12 μm (1120 nm). Therefore, it can be seen that the results of the EDX-MSM analysis and the cross-sectional STEM image are in good agreement.

[0241] In all three examples, the absolute values ​​of sample thickness estimated by EDX-MSM analysis were on the same order of magnitude when compared with the results of cross-sectional STEM observation. While EDX-MSM analysis provides lower accuracy in determining film thickness than destructive analyses such as cross-sectional STEM observation, it can analyze the layer structure from an unknown state over a wide range of three orders of magnitude, from 10 nm to 1 μm. For this reason, EDX-MSMS analysis can provide valuable data for the initial evaluation of unknown samples before proceeding to other analyses such as destructive analysis.

[0242] (Effects of sparsification processing) The above-mentioned simple sparsification process was performed in all of the EDX-MSM analyses in Examples 1 to 3. The effect of the sparsification process will be explained using a specific example. Note that the regularization functions expressed by the above-mentioned formulas (21) and (22) were used in the sparsification process.

[0243] FIG. 15 is a first diagram illustrating the effect of sparsification treatment in EDX-MSM analysis of a sample according to Example 2. FIG. 16 is a second diagram illustrating the effect of sparsification treatment in EDX-MSM analysis of a sample according to Example 2. The four graphs shown in FIG. 15 show the results of EDX-MSM analysis of region A (SiN / InP) of the sample of Example 2, where the sparsification treatment conditions are varied. The four graphs shown in FIG. 16 show the results of EDX-MSM analysis of region B (Ni / InP) of the sample of Example 2, where the sparsification treatment conditions are varied.

[0244] The "coefficients" shown in Figures 15 and 16 are the parameters λ SP Specifically, the coefficients used to calculate the matrix S in Equation (21) are T S+λ LE +λ CN Q CN is a coefficient by which the smallest value of the diagonal elements of is multiplied. The magnitude of the coefficient corresponds to the degree of sparsification. "Coefficient = 0" (Fig. 15(A) and Fig. 16(A)) means that no sparsification is performed. "Coefficient = 0.9" (Fig. 15(D) and Fig. 16(D)) means that sparsification is performed in the EDX-MSM analysis of Example 2 above. "Coefficient = 0.2" (Fig. 15(B) and Fig. 16(B)) and "Coefficient = 0.5" (Fig. 15(C) and Fig. 16(C)) mean that the degree of sparsification is small.

[0245] As shown in Figures 15 and 16, when there is no sparsification process or when the degree of sparsification is small, the depth profile of each chemical species becomes smooth overall. In other words, when there is no sparsification process or when the degree of sparsification is small, a depth profile is obtained in which each chemical species is mixed from the outermost surface to the deepest part of the sample. By increasing the coefficient (i.e., the parameter λ SP It can be seen that the larger the θ is, the less the mixing of chemical species in the depth direction becomes, and the clearer the layer structure becomes.

[0246] Furthermore, the regularization function applied to the sparsification process is not limited to the functions expressed by equations (21) and (22). For example, the function expressed by the following equation (30) may be applied to the sparsification process.

[0247]

number

[0248] In equation (17), the term λ representing the maximum smoothing condition S / 2C T Q S c is expressed as in equation (31)

[0249]

number

[0250] Equation (30) corresponds to equation (31) where the sum of squared differences is replaced with the absolute value of the difference. Optimization problems involving this type of regularization are called "fused lasso" problems, and are used for image noise removal, etc.

[0251] When the regularization function expressed by Equation (31) is used, it is required that the difference in relative concentration be small overall, but it is not required that the difference be zero. Therefore, the analysis results tend to produce a smooth depth profile overall. On the other hand, when the regularization function expressed by Equation (30) is used, it is required that the number of points where the difference in relative concentration is zero in the sample model shown in Figure 1 be increased as much as possible. As a result, a depth profile with uniform concentration can be produced except for a few transition regions. An example of this will be explained with reference to Figure 17.

[0252] FIG. 17 shows the results of depth profile analysis using another example of sparsification processing. FIG. 17 shows an example of depth profile analysis based on EDX measurement data of the PFA sample on Cu foil shown in Example 3 above. In the analysis of the depth profile, the parameter λ in the following equation (32) is FL λ FL = 0.01. Minimizing equation (32) goes beyond the scope of convex quadratic programming, but it can be solved using, for example, the alternating determination method of multipliers (ADMM) method. A sharp profile, not a smooth one, was obtained, which cannot be obtained by MSM analysis without sparsification processing.

[0253]

number

[0254] The depth profile shown in Figure 13 was obtained by MSM and simple sparsification processing. However, even without these two-stage processing steps, a sharp depth profile can be obtained simply by optimizing Equation (32), as shown in Figure 17. However, because no processing equivalent to sparsification processing is performed, the Cu concentration in the region less than 1000 nm deep is not zero.

[0255] By using Equation (30) as a regularization function, a depth profile in which the change in the relative concentration of chemical species along the depth direction is steeper can be obtained. FL The larger the parameter λ, the more it is required to reduce the number of layers as much as possible. FL If is set too large, the entire sample will be represented as a single layer, while the parameter λ FL As σ decreases, a number of layers with step-like changes in relative concentration along the depth direction are expressed. Therefore, by using Eq. (30) as a function for the sparsification process, it is possible to estimate the layer composition and thickness more plausibly, for example, when the detailed composition of the sample is unknown but the number of layers is known in advance. Depending on the situation or purpose of the problem, it is necessary to appropriately combine various regularization functions, such as the maximum smoothness condition shown in Eq. (17), the sparsity condition shown in Eq. (18), and the layer number minimization condition shown in Eq. (30).

[0256] 12. MSM Analysis of EDX Mapping Data The depth profile analysis described above is based on EDX measurement data at a single point on the sample surface. In EDX analysis, mapping analysis is easily possible by scanning the sample surface with an electron beam. Since mapping analysis data is a collection of EDX point analysis data within the XY plane, the depth profile at each point within the XY plane can be estimated by repeating the above process within the XY plane. In other words, by combining EDX mapping analysis with depth profile analysis based on point analysis data, the distribution of chemical species in three-dimensional space can be estimated nondestructively. Below, we will explain nondestructive three-dimensional chemical species distribution evaluation using the sample of Example 2 above (Ni circuit on an InP wafer) as an example.

[0257] The accelerating voltage and EDX signal mapping data shown in Table 2 are acquired for the entire region of the SEM image shown in Figure 10. The EDX intensity data acquired for chemical species i at the analysis point (x, y) in the plane at the accelerating voltage j of the electron beam are hereinafter referred to as d xyij The mapping data described in this specification is data in which EDX signal intensities are arranged every 256 pixels horizontally and 192 pixels vertically.

[0258] FIG. 18 shows an example of EDX mapping data. All of FIG. 18(A) to FIG. 12(D) show measurement results at an accelerating voltage of 15 kV. If there is a lot of noise in the data for each pixel, EDX-MSM analysis of all pixel data as is may result in unstable results. In this case, smoothing can be performed on each piece of data. For example, data smoothing may be performed by performing a conversion according to the following equation (33):

[0259]

number

[0260] Equation (33) means that the data of a pixel is replaced with the sum of the values ​​of the surrounding two pixels, totaling 25 points. At the edges of the data, the data is treated as if it continues for two extra pixels. This conversion reduces the effects of measurement noise.

[0261] FIG. 18(A) shows mapping data of Ni characteristic X-rays (Ni Lα) before smoothing. FIG. 18(B) shows mapping data of Ni characteristic X-rays (Ni Lα) after smoothing. Similarly, FIG. 18(C) shows mapping data of Si characteristic X-rays (Si Kα) before smoothing. FIG. 18(D) shows mapping data of Si characteristic X-rays (Si Lα) after smoothing. Noise can be reduced by smoothing.

[0262] Note that equation (33) shows the simplest example of smoothing processing. Depending on the purpose of analysis and the state of the data, an appropriate smoothing processing or noise removal processing can be adopted.

[0263] The results of performing MSM analysis on all points of the EDX surface mapping data are data indicating the distribution of chemical species in the three-dimensional space (x, y, z). In this embodiment, the output unit 53 shown in FIG. 5 outputs the data indicating the distribution of chemical species in the three-dimensional space obtained by the analysis unit 52 as the analysis result. The display 43 shown in FIG. 4 displays the analysis result. In one example, the analysis unit 52 generates data for displaying a cross-sectional slice image of the sample.

[0264] Figure 19 shows the location where a cross-sectional image of the distribution of chemical species in three-dimensional space is displayed. Figure 20 shows an example of a cross-sectional image of the distribution of chemical species in three-dimensional space. In this example, EDX-MSM analysis was performed on smoothed EDX data of 256 points in the X direction and 192 points in the Y direction, for a total of 49,152 points. The analysis conditions were the same as those shown in Table 2.

[0265] The cross-sectional images shown in Figures 20(A), 20(B), and 20(C) represent the distribution of chemical species (Ni electrode, SiN protective film, and InP wafer) at the positions of lines A, B, and C in Figure 19, respectively. Although the term "cross-section" is used for convenience, the shape of the cross-sectional image shown in Figure 20 differs slightly from the shape of a cross-section obtained by actually destroying the sample. For example, the right edge of the cross-section at position C in Figure 19 is the area where the InP wafer is exposed. The height of this area should be lower than the surfaces of the Ni electrode and SiN protective film, but in Figure 20, they appear to be at the same height. This display is due to the fact that EDX-MSM analysis recognizes the outermost surface of the sample as z = 0 regardless of surface irregularities. As shown in Figure 20, EDX-MSM analysis of mapping data is extremely useful because it allows nondestructive estimation of the depth distribution of chemical species at any cross-section of the sample.

[0266] In Figure 20, the brightness of the map differs between Ni and other chemical species, which indicates the relative concentration values ​​of the chemical species (Ni is approximately 1, and the others are approximately 0.5).

[0267] Displaying a slice image in a virtual cross section, such as that shown in Figure 20, is not easy to achieve using conventional analysis methods that assume a layered structure and use film thickness as the only variable. When attempting to achieve something similar using conventional methods, it is necessary to assume an initial structure at each point in the mapping data. However, the appropriate initial structure to assume varies greatly from point to point. For example, in the example above, regions such as Ni / InP, SiN / InP, InP, and intermediate states are mixed in a complex manner within the plane.

[0268] As shown in step (4) of Section 9. Example of the Overall Flow of EDX-MSM Analysis, in EDX-MSM analysis, the extremely simple setting of "1 / I at all depths" can be used as the initial value of the relative concentration of chemical species at all points in the X and Y directions. This makes it easy to evaluate the depth profile at all points and display the virtual cross-sectional slice image shown in Figure 20.

[0269] FIG. 21 is a diagram showing an example of a three-dimensional display of the distribution of chemical species in a three-dimensional space. EDX-MSM analysis was performed on smoothed EDX data of 256 points in the X direction and 192 points in the Y direction, for a total of 49,152 points. The analysis conditions were the same as those shown in Table 2. As shown in FIG. 21, the distribution of each of the chemical species In, N, Ni, P, and Si in three-dimensional space is displayed as a 3D graph. According to this embodiment, data indicating the distribution of chemical species in three-dimensional space obtained by the analysis unit 52 can also be displayed as a 3D graph.

[0270] In the above example, evaluation of a sample having a relatively simple structure was shown in order to explain the depth profile evaluation according to this embodiment. However, according to this embodiment, by not assuming the structure in advance, it is possible to discover an unexpected profile region (for example, a foreign substance embedded locally) existing within the surface.

[0271] As described above, according to the embodiments of the present disclosure, it is possible to estimate the depth profile of a sample from measurement data obtained by surface EDX without destroying the sample. According to the embodiments of the present disclosure, it is possible to estimate the depth profile non-destructively without assuming the layer structure of the sample in advance.

[0272] Furthermore, according to the embodiments of the present disclosure, applying the estimated depth profile to EDX surface mapping data enables non-destructive chemical species analysis in three-dimensional space. Furthermore, the analysis results can be displayed as a chemical species distribution in three-dimensional space. Therefore, more useful information can be provided in various situations, such as product development and failure analysis.

[0273] The embodiments disclosed herein are illustrative in all respects and should not be considered limiting. The scope of the present invention is defined by the claims rather than the above-described embodiments, and it is intended to include any modifications within the scope of the claims and meanings equivalent to the claims. [Explanation of symbols]

[0274] 10 Analysis System 20 Energy Dispersive X-ray Spectroscopy (EDX) Equipment 25 samples 30 Data analysis equipment 31 processors 32 Primary storage 33 Secondary storage device 34 External Device Interface 35 Input Interfaces 36 Output Interfaces 37 Communication Interface 38 Bus 41 keyboard 42 Mouse 43 Display 51 Input section 52 Analysis Department 53 Output section 54 Memory section 61 Parameter determination unit 62 Regularization function determination unit 63 Hyperparameter determination unit 64 Arithmetic section 71 Analysis Program 72 parameters S11~S17, S21~S30 steps

Claims

1. 1. A data analysis device that analyzes a depth profile of a sample based on characteristic X-rays generated from the sample by incident electron beams, comprising: an input unit that receives a response signal representing a measured value of the intensity of the characteristic X-ray from a measurement device that measures the characteristic X-ray; an analysis unit that analyzes the depth profile of the sample by using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers and minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal, the theoretical value of the response signal is a value derived based on a generation function that expresses a depth distribution of the generation amount of the characteristic X-rays in the sample and a function that expresses attenuation of the characteristic X-rays inside the sample, The analysis unit calculates the relative concentrations of the chemical species in the sample so as to simultaneously satisfy a regularization condition for the relative concentrations of the chemical species in the sample in minimizing the sum of squared deviations.

2. The data analysis device according to claim 1 , wherein the regularization condition includes a condition that the sum of squares of the differences in relative concentrations between adjacent layers is minimized for all chemical species and all layers in the stack.

3. The data analysis device according to claim 1 , wherein the regularization condition includes a condition that restricts the abundance ratio of the chemical species in the sample to a predetermined ratio.

4. The data analysis device according to claim 1 , wherein the regularization condition includes a condition that minimizes a function whose value increases when the chemical species are present in a mixed state in each layer of the sample.

5. The data analysis device according to claim 1 , wherein the regularization condition includes a condition that the sum of absolute values ​​of the differences in relative concentrations between adjacent layers for all chemical species and all layers in the stack is minimized.

6. 2. The data analysis device according to claim 1, wherein the analysis unit optimizes the apparatus constant, which is a parameter related to the measurement device, so that the sum of squares of the deviation between a value obtained by multiplying the theoretical value of the response signal by the apparatus constant and the measured value of the response signal is minimized.

7. 7. The data analysis device according to claim 6, wherein the analysis unit alternately repeats a first calculation that fixes the apparatus constant and optimizes the relative concentration and a second calculation that fixes the relative concentration and optimizes the apparatus constant, and determines the depth profile based on the relative concentration when the results of the first calculation and the second calculation converge.

8. 2. The data analysis device according to claim 1, wherein the analysis unit is configured to be able to combine a correction function with the generation function and the function expressing the attenuation of the characteristic X-rays in deriving the theoretical value, and the correction function is a function expressing that the energy of the electron beam attenuates within the sample and reaches 0 at a certain depth of the sample.

9. 9. The data analysis device according to claim 1, wherein the analysis unit is configured to analyze the depth profile at each point on the surface of the sample based on the response signal obtained from the measurement device when the electron beam is scanned two-dimensionally on the surface of the sample, and to perform a display process to display the analysis results of the depth profile of the sample as a chemical species distribution in three-dimensional space.

10. receiving, from a measurement device, a measured value of the intensity of characteristic X-rays generated from the sample by the incidence of the electron beam; and analyzing a depth profile of the sample based on the measurements, the analyzing step comprising: minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers; the theoretical value of the response signal is a value derived based on a generation function that expresses a depth distribution of the generation amount of the characteristic X-rays in the sample and a function that expresses attenuation of the characteristic X-rays inside the sample, A data analysis method, wherein the step of minimizing the sum of squared deviations includes the step of calculating the relative concentrations of chemical species in the sample such that the relative concentrations simultaneously satisfy a regularization condition for the relative concentrations.

11. The data analysis method according to claim 10 , wherein the regularization condition includes a condition that the sum of squares of the differences in relative concentrations between adjacent layers for all chemical species and all layers in the stack is minimized.

12. The data analysis method according to claim 10 , wherein the regularization condition includes a condition that restricts the abundance ratio of the chemical species in the sample to a predetermined ratio.

13. The data analysis method according to claim 10 , wherein the regularization condition includes a condition that minimizes a function whose value increases when the chemical species exist in a mixed state in each layer of the sample.

14. The data analysis method according to claim 10 , wherein the regularization condition includes a condition that the sum of absolute values ​​of the differences in relative concentrations between adjacent layers for all chemical species and all layers in the stack is minimized.

15. The step of minimizing the sum of squared deviations comprises:

11. The data analysis method according to claim 10, further comprising a step of optimizing the apparatus constant, which is a parameter related to the measurement apparatus, so that the sum of squares of the deviation between a value obtained by multiplying the theoretical value of the response signal by the apparatus constant and the measured value of the response signal is minimized.

16. The step of minimizing the sum of squared deviations comprises: a step of alternately repeating a first calculation for optimizing the relative concentration while fixing the equipment constant and a second calculation for optimizing the equipment constant while fixing the relative concentration until the results of the first calculation and the second calculation converge; and determining the depth profile using the relative concentrations when the results of the first calculation and the second calculation have converged.

17. 11. The data analysis method according to claim 10, wherein the theoretical value is derived by combining a correction function with the generation function and the function expressing the attenuation of the characteristic X-rays, and the correction function is a function expressing that the energy of the electron beam attenuates within the sample and reaches 0 at a certain depth of the sample.

18. the receiving step is a step of receiving the response signal obtained from the measurement device when the electron beam is two-dimensionally scanned on the surface of the sample, the analyzing step includes analyzing the depth profile at each point on the surface of the sample; The data analysis method includes:

18. The data analysis method according to claim 10, further comprising the step of executing a display process for displaying the analysis results of the depth profile of the sample as a chemical species distribution in three-dimensional space.

19. On the computer, receiving, from a measurement device, a measured value of the intensity of characteristic X-rays generated from the sample by the incidence of the electron beam; and analyzing a depth profile of the sample based on the measurements, the analyzing step comprising: minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers; the theoretical value of the response signal is a value derived based on a generation function that expresses a depth distribution of the generation amount of the characteristic X-rays in the sample and a function that expresses attenuation of the characteristic X-rays inside the sample, The program, wherein the step of minimizing the sum of squared deviations includes a step of calculating the relative concentrations of chemical species in the sample such that a regularization condition for the relative concentrations is simultaneously satisfied.

20. On the computer, receiving, from a measurement device, a measured value of the intensity of characteristic X-rays generated from the sample by the incidence of the electron beam; and analyzing a depth profile of the sample based on the measurements, the analyzing step comprising: minimizing the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal using a theoretical value of the response signal when the sample is modeled as a stack of multiple layers; the theoretical value of the response signal is a value derived based on a generation function that expresses a depth distribution of the generation amount of the characteristic X-rays in the sample and a function that expresses attenuation of the characteristic X-rays inside the sample, A recording medium having a program recorded thereon, wherein the step of minimizing the sum of squared deviations includes a step of calculating the relative concentrations of chemical species in the sample so as to simultaneously satisfy a regularization condition for the relative concentrations.

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