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

Abstract

The data analysis device disclosed herein is a data analysis device that analyzes the depth profile of a sample using data on a response signal generated from the sample by the incidence of a probe, and is equipped with an input unit that accepts response signal data obtained by a measurement device that measures the response signal by changing the incidence conditions of the probe to evaluate different depth regions of the sample, an analysis unit that analyzes the response signal data to estimate the relative concentration for each depth from the surface of the sample and for each chemical species, and determines the average value and confidence interval of the estimated values ​​of the relative concentrations, and an output unit that outputs an indication of the average value and confidence interval of the estimated values ​​of the relative concentrations.

Description

[Technical field]

[0001] The present disclosure relates to a data analysis device, a data analysis method, a program, and a recording medium. This application claims priority based on Japanese Patent Application No. 2023-132504 filed on August 16, 2023. All contents of the Japanese patent application are incorporated herein by reference. [Background technology]

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

[0003] Depth profile evaluation methods include, for example, scanning transmission electron microscopy (STEM), energy dispersive X-ray analysis (EDX), and X-ray photoelectron spectroscopy (XPS). However, these methods may change the state of the sample when evaluating the depth profile of the sample. For example, when performing analysis using STEM-EDX, pretreatment is required to thin the sample prior to the analysis. On the other hand, in the case of analysis using XPS, the depth of information that can be evaluated is typically about a few nm from the surface, although this depends on the measurement conditions. Therefore, in order to evaluate deeper regions of the sample using XPS, etching of the sample surface by ion sputtering is required. However, ion sputtering may damage the surface of the sample.

[0004] As a method capable of evaluating the depth profile of a sample non-destructively, a method has been proposed in which the depth profile of the sample is estimated using a response signal generated from the sample by injecting a probe into the sample. For example, "Application of Maximum Entropy Method to Semiconductor Engineering", Yoshiki Yonamoto, Entropy 2013,15,1663-1689;doi:10.3390 / e15051663 (Non-Patent Document 1) discloses an analysis method in which the Maximum Entropy Method (MEM) is applied to data obtained from angle-resolved XPS (ARXPS) acquired without sputtering. In addition, International Publication No. 2022 / 230112 (Patent Document 1) describes the Maximum Smoothness Method (MSM), which is a recently developed analysis method. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] International Publication No. 2022 / 230112 [Non-patent literature]

[0006] [Non-Patent Document 1] “Application of Maximum Entropy Method to Semiconductor Engineering”, Yoshiki Yonamoto,Entropy 2013,15,1663-1689;doi:10.3390 / e15051663 Summary of the Invention

[0007] A data analysis device according to one aspect of the present disclosure is a data analysis device that analyzes a depth profile of a sample using data on a response signal generated from the sample by the incidence of a probe, and includes an input unit that accepts response signal data obtained by a measurement device that measures response signals by changing the incidence conditions of the probe to evaluate different depth regions of the sample, an analysis unit that analyzes the response signal data to estimate relative concentrations for each depth from the surface of the sample and for each chemical species, and determines the average value and confidence interval of the estimated values ​​of the relative concentrations, and an output unit that outputs an indication of the average value and confidence interval of the estimated values ​​of the relative concentrations. [Brief description of the drawings]

[0008] [Figure 1] FIG. 1 is a diagram illustrating an analysis system including an analysis device according to an embodiment of the present disclosure. [Diagram 2] FIG. 2 is a block diagram illustrating an example of a hardware configuration of a data analysis device according to an embodiment of the present disclosure. [Diagram 3] FIG. 3 is a diagram illustrating an example of functional blocks of the data analysis device illustrated in FIG. [Figure 4] FIG. 4 is a schematic diagram showing a response signal generated by the incidence of a probe on a sample. [Diagram 5] FIG. 5 shows an example of an analysis result of EDX measurement using a conventional MSM. [Figure 6] FIG. 6 is a schematic diagram showing the distribution of relative concentrations of each chemical species and the accuracy of the analysis results. [Figure 7] FIG. 7 is a flowchart showing an example of a processing flow of the data analysis method (Bayesian MSM) executed by the data analysis device according to the first embodiment. [Figure 8] FIG. 8 is a flowchart showing another example of the processing flow of the data analysis method (Bayesian MSM) executed by the data analysis device according to the first embodiment. [Figure 9] FIG. 9 is a flowchart showing an example of the processing flow of the Bayesian MSM to which the sampling method is applied, which is executed by the analysis unit. [Figure 10] FIG. 10 shows the structure of a model sample used to verify the analysis results by Bayesian MSM. [Figure 11] FIG. 11 is a diagram showing an example of the results of analyzing the model shown in FIG. 10 by Bayesian MSM using the sampling method (Metropolis method). [Figure 12] FIG. 12 is a diagram showing three representative points of the depth profile shown in FIG. [Figure 13] FIG. 13 shows a trace plot of relative concentration values ​​at three locations in the depth profile shown in FIG. [Figure 14] FIG. 14 is a diagram showing an example of the results of analyzing the model shown in FIG. 10 by Bayesian MSM using variational inference. [Figure 15] FIG. 15 is a model diagram of a sample to which Bayesian estimation according to the second embodiment is applied. [Figure 16] FIG. 16 is a diagram showing a schematic representation of a sigmoid function type depth profile. [Figure 17] FIG. 17 is a functional block diagram showing an example of the configuration of the analysis unit shown in FIG. 3 according to the second embodiment. [Figure 18] FIG. 18 is a block diagram showing an example of the configuration of the forward problem evaluation unit and the next parameter recommendation unit shown in FIG. [Figure 19] FIG. 19 is a diagram showing a STEM cross-sectional image of sample 1. [Figure 20] FIG. 20 shows the results of EDX surface analysis of Sample 1. [Figure 21] FIG. 21 is a diagram showing an analytical model of Sample 1 (Ni layer on an InP substrate). [Figure 22] FIG. 22 shows the results of calculations performed by the analysis unit on the thickness of the Ni layer and the In composition of InP according to the MCMC method. [Figure 23] FIG. 23 shows the analysis results of sample 1 (Ni layer on an InP substrate) by the analysis unit. [Figure 24] FIG. 24 is a diagram showing an analytical model of Sample 1 (SiN layer on an InP substrate). [Diagram 25] FIG. 25 is a diagram showing the results of calculations performed by the analysis unit according to the MCMC method on the N composition of the SiN layer, the thickness of the SiN layer, and the In composition of the InP layer. [Figure 26] FIG. 26 is a diagram showing the analysis results of sample 1 (SiN layer on an InP substrate) by the analysis unit. [Figure 27] FIG. 27 is a diagram showing a STEM cross-sectional image of sample 2. [Figure 28] FIG. 28 shows the results of a cross-sectional analysis of Sample 2 by STEM-EDX. [Figure 29] FIG. 29 is a diagram showing an analytical model of sample 2. [Diagram 30] FIG. 30 is a diagram showing the results of calculations performed by the analysis unit on the carbon composition of the PFA layer and the thickness of the PFA layer according to the MCMC method. [Diagram 31] FIG. 31 shows the results of analysis of sample 2 by the analysis unit. [Diagram 32] FIG. 32 is a diagram showing a STEM cross-sectional image of sample 3. [Diagram 33] FIG. 33 shows the results of EDX surface analysis of sample 3. [Diagram 34] FIG. 34 shows the results of calculations made by the analysis unit on the thickness of the multilayer film of sample 3 according to the MCMC method. [Diagram 35] FIG. 35 is a diagram showing the results of the profile evaluation of the sample 3 by the analysis unit 52. As shown in FIG. [Diagram 36] FIG. 36 shows the results of calculations made by the analysis unit for the thickness of the multilayer film of sample 3 according to the MCMC method when the number of EDX measurement data items is reduced. [Figure 37] FIG. 37 is a diagram showing the influence of the number of measurement data on the estimation of the film thickness of Sample 3 by the MCMC method. [Figure 38] FIG. 38 is a diagram showing a first example of a display output by the data analyzing device according to the present embodiment. [Figure 39] FIG. 39 is a diagram showing a second example of the display output by the data analyzing device according to the present embodiment. [Diagram 40] FIG. 40 is a diagram showing a third example of the display output by the data analyzing device according to the present embodiment. [Diagram 41] FIG. 41 is a diagram showing a fourth example of the display output by the data analyzing device according to the present embodiment. [Diagram 42] FIG. 42 is a diagram showing a fifth example of a display output by the data analyzing device according to the present embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0009] [Problem that this disclosure aims to solve] In MEM or MSM, the depth profile of the sample is estimated and the estimated results are output. Conventionally, analysts have not been able to know the accuracy of the estimated results. Since MEM and MSM are "evaluation methods" of the profile, they should essentially output some level of accuracy, just like other analytical methods. From the viewpoint of preventing analysts from making erroneous evaluations, it is desirable for analysts to be able to grasp the accuracy of the estimated results.

[0010] Therefore, an object of the present disclosure is to make it possible to display the accuracy of the estimation results in an analysis method that estimates the depth profile of a sample based on a response signal generated from the sample by the incidence of a probe.

[0011] [Effects of this disclosure] According to the present disclosure, in an analysis method for estimating a depth profile of a sample based on a response signal generated from the sample by the incidence of a probe, the accuracy of the estimation result can be displayed.

[0012] [Description of the embodiments of the present disclosure] First, the embodiments of the present disclosure will be listed and described.

[0013] (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 using data on a response signal generated from the sample by incidenting a probe, and includes an input unit that accepts response signal data obtained by a measurement device that measures response signals by changing probe incident conditions to evaluate different depth regions of the sample, an analysis unit that analyzes the response signal data to estimate relative concentrations for each depth from the surface of the sample and for each chemical species and to determine average values ​​and confidence intervals of the estimated values ​​of the relative concentrations, and an output unit that outputs an indication of the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations.

[0014] According to this configuration, in an analysis method for estimating a depth profile of a sample based on a response signal generated from the sample by the incidence of a probe, the accuracy of the estimation result can be displayed. The analyst (e.g., a user of a data analysis device) can grasp the accuracy of the estimation result by displaying the average value and the confidence interval. Note that "outputting a display" means outputting the analysis result in a form that can be recognized by the user. Therefore, "outputting a display" includes displaying the depth profile, which is the analysis result, and its accuracy in the form of a graph, numerical data, a table, etc. on the screen of a display. The depth profile, which is the analysis result, and its accuracy may be output on paper in the form of a graph, numerical data, a table, etc. Alternatively, for example, a numerical string representing the analysis result may be temporarily output (saved) to a storage device or storage medium so that the analysis result can be displayed later. In the embodiment of the present disclosure, "outputting a display" also includes outputting the analysis result so that it can be displayed at a later stage.

[0015] (2) In the data analysis device described in (1), the display of the mean value and confidence interval is in the form of a graph, the graph including a first axis indicating depth, a second axis indicating relative concentration, a first graph showing the change in the mean value versus depth, and a second graph showing the change in the confidence interval versus depth.

[0016] According to this configuration, the accuracy of the estimation result can be displayed in a format that is easy for the analyst to understand. The "figure" includes, for example, a point, a line, a surface, and a solid. The type of figure is not particularly limited.

[0017] (3) In the data analysis device described in (2) above, the first graphic is a curve, and the second graphic includes an area that overlaps with the curve.

[0018] According to this configuration, the width of the region corresponds to the width of the confidence interval, so that the accuracy of the estimation result can be displayed so that the analyst can easily grasp the accuracy of the estimation result.

[0019] (4) In the data analysis device described in (2) above, the second graphic includes a first curve representing an upper limit of the confidence interval and a second curve representing a lower limit of the confidence interval.

[0020] According to this configuration, the confidence interval is expressed by the first curve and the second curve, so that the accuracy of the estimation result can be displayed so that the analyst can easily grasp the accuracy of the estimation result.

[0021] (5) In the data analysis device described in any one of (2) to (4) above, the second figure includes an error bar, the length of which corresponds to the width of the confidence interval of the estimated value at a specified position on the first figure.

[0022] According to this configuration, the width of the confidence interval is expressed by the length of the error bar, so that the accuracy of the estimation result can be displayed so that the analyst can easily grasp the accuracy of the estimation result.

[0023] (6) In the data analysis device described in any of (2) to (5) above, the graph includes a display of at least one numerical value of the average value, the upper limit of the confidence interval, and the lower limit of the confidence interval at a specified position on the first figure.

[0024] According to this configuration, at least one of the average value, the upper limit value of the confidence interval, and the lower limit value of the confidence interval is displayed as a numerical value, which makes it easier for the analyst to understand the estimation result.

[0025] (7) In the data analysis device according to any one of (1) to (6) above, the output unit outputs the display of the mean value and the confidence interval in the form of a numerical table.

[0026] According to this configuration, the average value and the confidence interval are displayed as numerical values, so that the analyst can specifically grasp the average value and the confidence interval. The display in the form of a numerical value table may be output alone or together with the display of the graph.

[0027] (8) In the data analysis device described in any one of (2) to (7) above, the analysis unit calculates the average value and confidence interval of the estimated value by processing the evaluation function according to a Bayesian estimation process, and the evaluation function is expressed by the sum of a first term expressing the sum of squared deviation between the theoretical value of the response signal when the sample is modeled as a laminate consisting of multiple layers and the response signal data, and a second term expressing the degree of continuity of the relative concentration at depth.

[0028] According to this configuration, the analysis unit executes MSM based on the idea of ​​Bayesian estimation. In MSM, an evaluation function expressed by the sum of a first term expressing the sum of squared deviations between the theoretical value of the response signal and the measured data when the sample is modeled as a laminate consisting of multiple layers, and a second term expressing the degree of continuity of the relative concentration at depth is processed. Each point on the depth profile expressed by this evaluation function essentially has concepts equivalent to the mean (μ) and variance (σ). In conventional MSM, the mean (μ) and variance (σ) could not be displayed. By applying Bayesian estimation processing in minimizing the evaluation function, it is possible to display the average value and confidence interval of the estimated value.

[0029] (9) In the data analysis device described in (8) above, the Bayesian estimation process includes a sampling algorithm that uses depth and relative concentration of each chemical species as variables.

[0030] According to this configuration, it is possible to obtain the posterior probability distribution in Bayesian estimation, and therefore the mean value and variance of the estimated values ​​can be obtained.

[0031] (10) In the data analysis device described in (8) above, the Bayesian estimation process is a process according to variational inference.

[0032] According to this configuration, it is possible to obtain the posterior probability distribution in Bayesian estimation, and therefore the mean value and variance of the estimated values ​​can be obtained.

[0033] (11) In the data analysis device described in any one of (2) to (7) above, the analysis unit determines the average value and the confidence interval of the estimated value of the relative concentration by determining, according to Bayesian estimation, the posterior distribution of parameters included in a predetermined function that represents a model of a sample with limited degrees of freedom of the depth profile.

[0034] According to this configuration, the degree of freedom of the depth profile is limited as a premise for the analysis of the depth profile. This allows for an analysis method that is easier for users to use. For example, when the analysis unit calculates the average value and the confidence interval of the relative concentration from the probability distribution of the parameters, the calculation time required for the calculated value to reach a reasonable value can be shortened. Also, by incorporating known elements of the profile, it is possible to perform an analysis specialized for only the unknown parts.

[0035] (12) In the data analysis device described in (11) above, the model of the sample is a laminate consisting of a plurality of layers, the predetermined function is a function expressing that the depth profile has rises and falls at the boundaries of each of the plurality of layers, and the parameters include a composition of the sample, a thickness of each of the plurality of layers, a gradual rise in the depth profile of each of the plurality of layers, and a gradual fall in the depth profile of each of the plurality of layers.

[0036] According to this configuration, the degree of freedom of the depth profile can be restricted in the estimation of the depth profile of the relative concentration of the chemical species, and further, appropriate parameters can be selected from the above parameters according to the problem to be solved.

[0037] (13) In the data analysis device described in (12) above, the analysis unit includes a forward problem evaluation unit that uses parameter values ​​as input to the forward problem model to calculate a sum of squared deviations between actual values ​​and theoretical values, and a next parameter recommendation unit that uses the sum of squared deviations calculated by the forward problem evaluation unit to recommend to the forward problem evaluation unit candidate values ​​of parameters to be used in the forward problem evaluation unit's next calculation, and the calculation of the sum of squared deviations by the forward problem evaluation unit and the recommendation of candidate parameter values ​​by the next parameter recommendation unit are alternately repeated.

[0038] According to this configuration, the calculation of the sum of squared deviations between the actual values ​​and the theoretical values ​​by the forward problem evaluation unit (evaluation of the forward problem) and the recommendation of the candidate values ​​of the parameters by the next parameter recommendation unit are repeated alternately. This makes it possible to obtain the average value and the confidence interval of the estimated values ​​of the parameters. Furthermore, the next parameter recommendation unit can recommend the next parameters based on the posterior distribution calculated by the forward problem evaluation unit and the previous parameters, without depending on the contents of the forward problem evaluation unit. By changing the forward problem to be solved by the forward problem evaluation unit according to the type of surface analysis, it is possible to configure a data analysis device corresponding to that surface analysis. In other words, the analysis method according to this embodiment is not particularly limited to any analysis method, and can be applied to any surface analysis.

[0039] (14) In the data analysis device described in (13) above, the next parameter recommendation unit includes an MCMC implementation unit that proposes candidate parameter values ​​using a predetermined probability distribution in accordance with the Markov chain Monte Carlo method, and an optimal solution determination unit that determines initial values ​​for the Markov chain Monte Carlo method executed by the MCMC implementation unit by executing a predetermined optimization method using the sum of squared deviations calculated by the forward problem evaluation unit.

[0040] According to this configuration, the optimal solution determination unit can determine the target value of the parameter or a value close to the target value. The value determined by the optimal solution determination unit is used as the initial value of the Markov Chain Monte Carlo (MCMC) method. By using the initial value, the MCMC implementation unit can shorten the burn-in period. Therefore, the calculation time required for the MCMC implementation unit to reach the distribution of the correct solution can be shortened.

[0041] (15) In the data analysis device according to any one of (1) to (14) above, the measurement device is an energy dispersive X-ray analyzer.

[0042] According to this configuration, the average value and variance of the estimated values ​​of the depth profile of the sample can be obtained from the response signal from the energy dispersive X-ray analysis device. The above-mentioned "energy dispersive X-ray analysis device" may be any device that analyzes characteristic X-rays generated from a sample by a probe input into the sample. One example of an "energy dispersive X-ray analysis device" is a device that performs "analysis in which an electron beam is irradiated and characteristic X-rays are observed" (so-called "EDX" analysis). Another example of an "energy dispersive X-ray analysis device" is a device that performs "analysis in which an X-ray beam is irradiated and characteristic X-rays are observed" (fluorescent X-ray analysis).

[0043] (16) In the data analysis device according to any one of (1) to (14) above, the measurement device is an X-ray photoelectron spectrometer.

[0044] According to this configuration, the average value and variance of the estimated values ​​of the depth profile of the sample can be obtained from the response signal from the X-ray photoelectron spectroscopy (XPS).

[0045] (17) A data analysis method according to one embodiment of the present disclosure is a data analysis method for analyzing a depth profile of a sample using data on a response signal generated from the sample by incidenting a probe, the data analysis method including the steps of: receiving, by the data analysis device, response signal data obtained by a measurement device that measures the response signal by changing the incident conditions of the probe to evaluate different depth regions of the sample; estimating relative concentrations for each depth from the surface of the sample and for each chemical species by analyzing the response signal data, and determining average values ​​and confidence intervals of the estimated values ​​of the relative concentrations; and outputting, by the data analysis device, a display of the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations.

[0046] According to this configuration, in an analysis method for estimating a depth profile of a sample based on a response signal generated from the sample by the incidence of a probe, the accuracy of the estimation result can be displayed. The analyst (e.g., a user of the data analysis device) can grasp the accuracy of the estimation result by displaying the average value and the confidence interval.

[0047] (18) A program according to an embodiment of the present disclosure causes a computer to execute the steps of acquiring response signal data obtained by a measurement device that measures response signals by varying probe incidence conditions to evaluate different depth regions of a sample, estimating relative concentrations for each depth from the surface of the sample and for each chemical species by analyzing the response signal data, determining average values ​​and confidence intervals of the estimated values ​​of the relative concentrations, and outputting a display of the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations.

[0048] According to this configuration, in an analysis method for estimating a depth profile of a sample based on a response signal generated from the sample by the incidence of a probe, the accuracy of the estimation result can be displayed. The analyst (e.g., a user of the data analysis device) can grasp the accuracy of the estimation result by displaying the average value and the confidence interval.

[0049] (19) A recording medium according to an embodiment of the present disclosure is a recording medium having a program recorded thereon, which causes a computer to execute the steps of acquiring response signal data obtained by a measurement device that measures response signals by changing the incident conditions of a probe to evaluate different depth regions of a sample, estimating relative concentrations for each depth from the surface of the sample and for each chemical species by analyzing the response signal data, determining average values ​​and confidence intervals of the estimated values ​​of the relative concentrations, and outputting a display of the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations.

[0050] According to this configuration, in an analysis method for estimating a depth profile of a sample based on a response signal generated from the sample by the incidence of a probe, the accuracy of the estimation result can be displayed. The analyst (e.g., a user of the data analysis device) can grasp the accuracy of the estimation result by displaying the average value and the confidence interval.

[0051] [Details of the embodiment 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 characters and their description will not be repeated.

[0052] <1. Configuration of analysis system and analysis device> 1 is a diagram showing an analysis system including an analysis device according to an embodiment of the present disclosure. As shown in FIG. 1, the analysis system 10 includes a measurement device 20 and a data analysis device 30.

[0053] In the embodiment of the present disclosure, the measurement device 20 irradiates a probe such as an X-ray or an electron beam onto the sample 25 and measures the intensity of a response signal generated from the sample 25. The measurement device 20 can change the incidence conditions of the probe onto the sample 25 so as to evaluate different depth regions of the sample 25.

[0054] The embodiment of the present disclosure is applicable to any surface analysis without limiting the type of the measurement device 20. The measurement device 20 may be any device that makes a probe incident on a sample 25 and measures the intensity of a response signal generated from the sample 25. Although not particularly limited, in one embodiment, the measurement device 20 is an energy dispersive X-ray fluorescence (EDX) analysis device. The measurement device 20 may be, for example, an angle resolved photoelectron spectroscopy (e.g., ARXPS) device or an X-ray fluorescence analysis (XRF) device.

[0055] The data analysis device 30 is realized by hardware that conforms to a general-purpose computing architecture. The data analysis device 30 acquires a response signal, i.e., measurement data, from the measurement device 20. The data analysis device 30 analyzes the depth profile of the sample 25 based on the measurement data. In detail, the data analysis device 30 estimates the relative concentration for each depth and each chemical species of the sample 25 by analyzing the measurement data. The data analysis device 30 outputs an average value of the estimated value of the relative concentration for each depth and each chemical species of the sample 25 and a display of the confidence interval of the estimated value. This allows the data analysis device 30 to display the accuracy of the estimated result of the depth profile for each chemical species to a user (not shown) of the data analysis device 30. The analyst (i.e., the user of the data analysis device) can grasp the accuracy of the estimated result by the display of the average value and the confidence interval.

[0056] 2 is a block diagram showing an example of a hardware configuration of a data analysis device according to an embodiment of the present disclosure. As shown in FIG. 2, a 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.

[0057] Elements such as the processor 31 and the primary storage device 32 exchange data, signals, and the like via a bus 38. The processor 31 processes various types of data by executing programs stored in the primary storage device 32. The primary storage device 32 stores the programs executed by the processor 31 and data referenced by the programs. In one aspect, a dynamic random access memory (DRAM) may be used as the primary storage device 32.

[0058] The secondary storage device 33 stores programs, data, etc. in a non-volatile manner. In a certain aspect, a non-volatile memory such as a hard disk drive (HDD) or a solid state drive (SSD) 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 executed by a computer.

[0059] 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.

[0060] 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 user inputs through these input devices.

[0061] The output interface 36 is used to connect an output device such as a display 43 .

[0062] The communication interface 37 is used for the data analysis device 30 to communicate with an external device. For example, the communication interface 37 is used for communication of the data analysis device 30 via a network 44. The network 44 is, for example, a LAN (Local Area Network). In FIG. 2, a printer 45 is shown as an example of an external device that can communicate with the data analysis device 30 via the network 44.

[0063] The data analysis device 30 may further have, as an option, a reading device for reading information from a recording medium. The "recording medium" here may include a recording medium that non-transiently stores a computer-readable program (for example, an optical recording medium such as a DVD (Digital Versatile Disc)). The reading device can read out the program stored in the recording medium. The program read from the recording medium may be installed in the secondary storage device 33 or the like. In addition, the various programs executed by the data analysis device 30 may be downloaded from a server device (not shown) or the like via a network (for example, the network 44) and installed in the data analysis device 30.

[0064] Fig. 3 is a diagram illustrating an example of functional blocks of the data analysis device illustrated in Fig. 2. In one aspect, each block illustrated in Fig. 3 is realized by a computer that executes a program according to an embodiment of the present disclosure.

[0065] As shown in FIG. 3, the data analysis device 30 includes an input unit 51, an analysis unit 52, an output unit 53, and a storage unit .

[0066] The input unit 51 receives measurement data output from the measurement device 20 (see FIG. 1). The input unit 51 further receives various information (e.g., information on the types of materials and chemical species) required for analyzing the depth profile of the sample 25. For example, the input unit 51 can be realized by the input interface 35 shown in FIG.

[0067] 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, etc. Furthermore, the storage unit 54 may store measurement data input to the data analysis device 30. The storage unit 54 can be realized by the primary storage device 32 and the secondary storage device 33 shown in FIG.

[0068] The analysis unit 52 executes the analysis program 71 to obtain a depth profile of the sample 25 and its accuracy. Specifically, the analysis unit 52 obtains the average value of the estimated values ​​of the relative concentrations for each depth and each chemical species of the sample 25 and the confidence interval of the estimated values ​​according to an analysis method described later. The analysis unit 52 can be realized by the processor 31 (arithmetic circuit) shown in FIG. 2.

[0069] The output unit 53 outputs a display of the analysis result by the analysis unit 52. "Outputting a display" means outputting the analysis result in a form that can be recognized by the user. For example, the output unit 53 can be realized by the output interface 36 shown in FIG. 2. For example, the analysis result, i.e., the depth profile and its accuracy, may be displayed in the form of a graph, numerical data, table, etc. on the screen of the display 43 (see FIG. 2). The graph, numerical data, table, etc. representing the analysis result may be printed on paper by the printer 45 (see FIG. 2). Alternatively, the analysis result may be temporarily output (stored) in a storage device or storage medium in the form of a numerical string so that display processing (e.g., graph display) can be performed by software. In the embodiment of the present disclosure, "outputting a display" also includes outputting the analysis result so that it can be displayed at a later stage.

[0070] The depth profile analysis method applicable to the embodiment of the present disclosure is not limited. In one embodiment, the analysis method used by the data analysis device 30 is MSM. In the following, a method for determining the estimation accuracy in estimating the depth profile by MSM will be described.

[0071] <2. Overview of MSM> Figure 4 is a schematic diagram showing the response signal generated by the incidence of a probe on a sample. An example of EDX analysis is shown in Figure 4. The probe in EDX is an electron beam, and the response signal is characteristic X-rays generated within the sample.

[0072] In MSM, the sample is considered to be a stack of K thin layers. Each layer has a thickness of t. An electron beam incident on the sample at an accelerating voltage j (j is the level of accelerating voltage, 1≦j≦J) attenuates within the sample and successively excites characteristic X-rays according to the chemical species present at that location. The degree of attenuation of the electron beam is highly dependent on the accelerating voltage j of the electron beam.

[0073] φ ij (ρz k ) is the kth layer (mass depth ρz k ) represents the generation function of characteristic X-rays of chemical species i generated at the sample. The characteristic X-rays generated within the sample travel toward the sample surface while attenuating within the sample, are emitted from the sample surface at an angle θ, and are detected by the measurement device 20 (see FIG. 1). j represents the acceleration voltage level.

[0074] In MSM, the depth profile of the relative concentration of the chemical species of the sample is analyzed by minimizing the sum of squares of the deviation between the theoretical response signal and the measured response signal, using the theoretical response signal when the sample to be analyzed is modeled as a stack of multiple layers. In minimizing the sum of squares of the deviation, MSM calculates the relative concentration so as to satisfy the maximum smoothness condition that the relative concentration of the chemical species of the sample changes smoothly between the multiple layers of the stack.

[0075] Specifically, in MSM, the function expressed by equation (1) is the object of analysis.

[0076]

number

[0077] The first term in equation (1) represents the sum of squares of the deviation between the theoretical response signal and the measured response signal. The second term in equation (1) represents how smoothly the profile of the relative concentration of chemical species i changes with depth. The parameter λ s is a parameter that indicates how strongly it is desired that the profile be smooth compared to the sum of squared deviations.

[0078] The relationship between the profile and the theoretical value of the EDX measurement can be expressed according to the following equation (2).

number

[0079] The vector d' on the left side of equation (2) is an (I×J) row vector that arranges the theoretical EDX measured intensity values ​​for all chemical species and all angles in a single column. The vector c on the right side of equation (2) is an (I×K) row vector that arranges the depth profiles of all chemical species and all layers in a single column. Specifically, vector c can be expressed as in equation (3).

number

[0080] S on the right hand side of equation (2) is a matrix with (I×J) rows and (I×K) columns, and is expressed according to equation (4).

number

[0081] S in equation (4)(1) ~S (I) is a submatrix with J rows and K columns. j is a parameter established to conveniently handle relative concentrations. (i) jk is expressed according to equation (5).

number

[0082] In formula (5), σ ij ω i φ ij (ρz k ) is the mass depth ρz k ) represents the emission function of the characteristic X-rays of chemical species i. i is the fluorescence yield of the signal of the characteristic X-ray generation function, and is an index of how easily the characteristic X-ray is emitted when the chemical species is ionized. The exp term in equation (5) represents the attenuation of the characteristic X-ray. In equation (5), (μ / ρ) i1 , (μ / ρ) i2 , (μ / ρ) ik-1 denote the mass absorption coefficients of the characteristic X-rays of chemical species i in the first, second, …, (k-1)th layers, respectively.

[0083] In equation (5), 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 according to the configuration of the device.

[0084] Therefore, the function to be minimized in the MSM, equation (1), can be expressed by the matrices shown in equations (6) and (7).

[0085]

number

number

[0086] Strictly speaking, the matrix S also depends on the relative concentration profile c (because the X-ray absorption coefficient depends on the relative concentration). However, as a simple approximate solution, S is treated as a constant. Then, the above equation (6) becomes a convex quadratic programming problem, so a self-consistent solution method can be adopted in which the optimal solution c is obtained, the matrix S is updated using that optimal solution c, and the next optimal solution c is obtained.

[0087] Therefore, MSM does not require an accurate initial value, but the relative concentration c ik This makes it possible to apply MSM to a variety of situations that require analysis of the depth profile of a sample, such as device development and failure analysis.

[0088] <3. The need for accuracy evaluation> Figure 5 shows an example of the analysis results of EDX measurement using a conventional MSM. The "conventional MSM" is the MSM described above. The sample to be analyzed is a silicon (Si) substrate on which aluminum (Al), titanium (Ti), chromium (Cr), and nickel (Ni) are layered in order from closest to the sample surface.

[0089] The analysis results shown in Figure 5 show the distribution of the relative concentrations of each chemical species along the depth direction from the sample surface, but do not provide any information regarding the accuracy of the analysis results.

[0090] Figure 6 is a schematic diagram showing the distribution of the relative concentration of each chemical species and the accuracy of the analysis results. As shown in Figure 6, the analysis results of the relative concentration of each chemical species should have a range according to the accuracy of the analysis. However, in the display shown in Figure 5, the user may perceive the error of the analysis results as zero. It is expected that such a perception will affect the interpretation of the analysis results.

[0091] Therefore, when analyzing a depth profile by applying a non-destructive analysis method such as MSM, it is desirable to present not only the depth profile obtained as the analysis result but also information regarding the accuracy of the analysis result to the user. In the following first embodiment, a method of outputting accuracy in MSM will be described.

[0092] <4. First embodiment> <<4.1 Accuracy evaluation using Bayesian estimation>> Bayesian estimation is known as a method for estimating the reliability of some measurement value based on measurement data. In the first embodiment, Bayesian estimation is applied to MSM. In order to distinguish from conventional MSM (MSM that does not output accuracy), MSM to which Bayesian estimation is applied is referred to as "Bayesian MSM" in this specification.

[0093] In conventional MSM, the relative concentration c ik Assuming that the true value of exists, we search for one solution that is most likely in the given situation. This corresponds to a convex quadratic programming problem that can obtain a globally optimal solution (a solution that is guaranteed to be the best in the entire feasible region). In conventional MSM, we express equation (1) as the relative concentration c ik The relative concentration c ik In the conventional MSM, equation (1) is formulated (derived) as a data term and a regularization term.

[0094] On the other hand, in Bayesian MSM, the relative concentration c ik As a random variable, we search for a plausible distribution. In other words, in Bayesian MSM, we rewrite equation (1) as ik Considering it as a distribution function for each c ik In Bayesian MSM, equation (1) is formulated as a posterior distribution in the Bayesian estimation framework.

[0095] Bayesian estimation is generally performed in the following procedure. 1. Assume a prior distribution for the parameters. 2. Assume a likelihood function for the observed data. 3. Use Bayes' theorem to calculate posterior probabilities. 4. Use the predictive distribution to find the unknown probability distribution.

[0096] Each of the above steps in Bayesian MSM will be specifically explained below.

[0097] (1. Assumptions on the prior distribution of parameters) The prior distribution for the relative concentration c of a chemical species is denoted as p(c), the likelihood function when measurement data d is obtained from a certain depth profile is denoted as p(d|c), and the posterior distribution of the depth profile after obtaining the measurement data d is denoted as p(c|d).

[0098] For the depth profile prior, we only assume that the distribution of relative concentrations is smooth. The prior distribution p(c) can be expressed according to the following equation (8):

number

[0099] Each relative concentration is assumed to be a normal distribution centered on the relative concentration of the layer adjacent to the layer on the surface side. The concentration distribution of chemical species at the top surface or deepest part is assumed to be a non-informative prior distribution, and is a Dirichlet distribution with a parameter α of 1. In equation (8), the factorial of (I-1) (i.e., (I-1)!) is the normalization term of the Dirichlet distribution.

[0100] In equation (8), σ s is a hyperparameter that represents the degree of change in the relative concentration of chemical species at the layer boundary. s The larger the value of σ, the larger the change in the profile at the layer boundary. s The smaller is the smoother the profile changes at the layer boundaries.

[0101] (2. Assumptions about the likelihood function of observed data) Assuming that the measured data d varies within an appropriate range around the theoretical value d' of the response signal, the likelihood function p(d|c) can be expressed according to the following equation (9).

number

[0102] In equation (9), σ ij is a hyperparameter that represents the reliability of the measurement results obtained by the measurement device under the conditions of chemical species i and the penetration condition j of the probe into the sample (the accelerating voltage condition in the case of EDX). ij The larger σ is, the more unreliable the measurement data is. On the other hand, the hyperparameter σ ij The smaller the , the more reliable the measurement data is.

[0103] (3. Calculation of posterior probability) According to Bayes' theorem, the posterior distribution p(c|d) is expressed by the following equation (10).

number

[0104] After observing the data, the denominator on the right-hand side of equation (10) becomes a constant independent of the relative concentration c. Therefore, for the posterior distribution p(c|d), we need to consider only the numerator on the right-hand side of equation (10).

[0105] Now, if we take the logarithm of the numerator on the right-hand side of equation (10), we obtain the following equation (11).

number

[0106] In reality, the accuracy of each measurement level is often unknown. For simplicity, we set the accuracy of each measurement level as a constant value σ d The variance between neighbors in the depth profile is σ S Ratio of λ S is defined according to equation (12).

number

[0107] By removing the constant part from equation (11) and applying equation (12) to equation (11), we obtain the following equation (13).

number

[0108] The equation in [ ] on the right-hand side of equation (13) is the same as equation (1). In MSM, the solution to minimize this equation is c ik Find the pair of .

[0109] Since the left side of equation (11) is the logarithm of the posterior distribution p(c|d) with a negative sign, minimizing the right side of equation (11) corresponds to maximizing the posterior distribution p(c|d). In other words, conventional MSM corresponds to a maximum a posteriori estimation (MAP) procedure that maximizes the posterior distribution when the measurement data generation process is modeled as in equations (8) and (9).

[0110] (4. Finding the unknown probability distribution) In reality, each point on the depth profile has concepts equivalent to the mean (μ) and variance (σ). However, the function expressed by equation (1) is a complex function expressed by many variables (I × K variables) related to the relative concentration c. Therefore, it is difficult to immediately grasp the mean and variance information of the relative concentration c of each chemical species from equation (1).

[0111] In this embodiment, the prior probability distribution can be obtained using each of two types of techniques, sampling and variational inference, which are typical techniques for obtaining a prior probability distribution in Bayesian estimation.

[0112] <<4.2 Bayesian MSM by Sampling>> In sampling, the distribution function has the relative concentration c as a variable ikInput a large number of pairs of and obtain the output value from the distribution function. ik If it can be said that the number of pairs is sufficient if certain rules are followed when creating pairs of , the created distribution can be considered to be the true distribution.

[0113] Sampling is a method that can handle any distribution function. However, sampling can be computationally expensive. In addition, it tends to be difficult to qualitatively understand the results obtained by sampling.

[0114] In MSM, the dimension of the relative concentration c is (I × K). Although it depends on the type of sample to be analyzed, the (I × K) dimension is generally high, and in one example, it is about 5000 dimensions. Therefore, c ik Substituting all combinations of pairs of these into the predictive distribution function to calculate the function value would take a long time to calculate, which could result in a problem that the analysis cannot be performed within a realistic time frame.

[0115] In Bayesian estimation, MCMC (Markov Chain Monte Carlo) is often used as a method for approximating the posterior distribution when it is difficult to obtain the posterior distribution by calculation. The MCMC method is a method for skillfully realizing the mechanism that "where a function is large (small), values ​​occur relatively frequently (low frequency)" for a high-dimensional system. In one embodiment, the Metropolis method, which is the most basic method of MCMC, can be adopted. Note that other methods that are more advanced than the Metropolis method, such as the Hamiltonian Monte Carlo method and the replica exchange Monte Carlo method, may also be used.

[0116] In this embodiment, MSM by the Metropolis method is carried out as follows to obtain a relative concentration c ik Calculate the mean and standard deviation for

[0117] 1. Set the initial value of the variable, c0 (a vector with as many elements as the number of dimensions). From the viewpoint of calculation efficiency, it is preferable to use the solution obtained by the conventional MSM as the initial value c0.

[0118] 2. Set the initial value c0 to real value b ik Convert to. In MSM, the relative concentration c ij is treated as the analysis object. In the Metropolis method, it is preferable to generate random numbers freely according to, for example, normal distribution. However, the relative concentration c ij For 0≦c ik ≦1 and Σ i c ik = 1. Such a restriction makes it difficult to generate random numbers. Therefore, in the first embodiment, the relative concentration c ij This problem is solved by dealing with

[0119] First, let b be a real number in any range. ik Consider a variable transformation using a softmax function as shown in equation (14).

number

[0120] Equation (14) is a real-valued ik The relative concentration c ranges from 0 to 1. ij Note that the softmax function is a function that converts an input value so that the sum of multiple output values ​​is 1, and outputs the converted value.

[0121] Real value b ik from relative concentration c ij However, the conversion to the relative concentration c ij is a ratio, so the relative concentration c ij to real value b ik Therefore, as shown in the following equation (15), for each depth k in the sample, a real value b ik to constant A k Add.

number

[0122] Real value bik Even if the relative concentration c in Eq. (14) is determined according to Eq. (15), ij This means that the real value b ik But the constant A k This means that we have the freedom to shift the value by the amount of b ik It is convenient to set the mean value for to zero.

[0123] Therefore, the relative concentration c ij to real value b ik In the conversion to, a constant A is added for each depth k. k By adding ik Adjust so that the average value is 0.

[0124] 3. Set the proposal distribution. In theory, any function can be applied to the proposed distribution. In one embodiment, a multivariate Gaussian distribution expressed by the following formula (16) is used. That is, the next b' is generated from a "range" of about Σ centered on the current b. Σ is a diagonal matrix with all components 0.1 (no correlation for each chemical species and depth).

number

[0125] 4. nth (n is an integer equal to or greater than 0) b n Then, using q(b',b) expressed by equation (16), one b n+1 Generate. That b n+1 According to equation (15), n+1 Convert to.

[0126] 5. Calculate the logarithmic difference of the true posterior distribution, as shown in equation (17) below.

number

[0127] G and h in equation (17) are expressed according to the following equations (18) and (19), respectively.

number

number

[0128] 6. If the calculated value of formula (17) is 0 or more, c n+1 will be adopted.

[0129] If the calculated value of equation (17) is less than 0, a uniform random number r is generated that takes a value between 0 and 1. If the calculated value of equation (17) is equal to or greater than the value of log(r) (which is a negative value), the calculated value is adopted, and if it is less than the value of log(r), the calculated value of equation (17) is rejected.

[0130] 7. Updated c n+1 Return to step 4 using the command. Repeat steps 4 to 6 the specified number of times.

[0131] 8. Repeat steps 4 to 6 up to the specified number of times to obtain the estimated relative concentration for each depth and each chemical species (relative concentration c ik ) and calculate the average and standard deviation.

[0132] <<4.3 Bayesian MSM using variational inference>> In Bayesian MSM using variational inference, the posterior distribution function is approximated to a simple function that is easy to grasp the mean and variance, and the mean and variance are evaluated using the approximated function. Variational inference has a limited range of applications compared to the MCMC method mentioned above. However, variational inference generally has low computational costs. Furthermore, variational inference allows for a qualitative understanding of the results obtained by the approximation formula.

[0133] In Bayesian MSM, the true posterior distribution function can be expressed by equation (20).

number

[0134] The posterior distribution function that approximates equation (20) is the Dirichlet distribution expressed by the following equation (21). Note that equation (21) is a function of c ik ∈(0,1) and Σ i=1 K c ik = 1. ik In equation (21), C D (a 1k ,···,a IK ) is a regularization term, and c ik aik-1 The term represents the distribution.

number

[0135] As shown in equation (21), each relative concentration c ik The mean (expected value) and variance of ik Therefore, by obtaining equation (21), the relative concentration c ik It is possible to grasp the distribution of.

[0136] In this embodiment, the parameter a in the formula (21) 1k By adjusting the distribution, the approximate posterior distribution q(c) expressed by equation (21) is made as close as possible to the true posterior distribution p(c). Various criteria can be applied to measure the closeness of the distributions.

[0137] As an example, KL (Kullback-Leibler) divergence, which is often used in Bayesian estimation, is shown below. Note that the method of evaluating the closeness between the approximate posterior distribution q(c) and the true posterior distribution p(c) is not limited to the method using KL divergence. For example, other indices equivalent to the similarity between functions, such as "the sum of squared differences between the values ​​of both functions in the variable range in which the functions are defined," may be used.

[0138] Using the KL divergence, the “closeness” between the approximate posterior distribution q(c) and the true posterior distribution p(c) can be expressed as the following equation (22).

number

[0139] The first term on the right-hand side of equation (22) corresponds to the entropy of the approximate posterior distribution q(c). Due to the nature of the Dirichlet distribution, this term can be expressed as the following equation (23).

number

[0140] On the other hand, the second term on the right-hand side of equation (22) is a term that corresponds to the entropy of the true posterior distribution q(c). Using a general expression for the posterior distribution p(c|d), the following equation (24) holds.

number

[0141] Therefore, by rearranging the contents of Σ in equation (24) and calculating it, we obtain equation (25).

number

[0142] c ik The integrals with respect to c ik Since it is the expected value of the first or second power of , we can obtain the expression of equation (25) by using the properties of the Dirichlet distribution.

[0143] Although there is no particular problem if equation (25) is used as is, from the viewpoint of speeding up calculations, etc., equation (25) is approximated as follows.

number

[0144] The upper part of equation (26) is a α ,a xα If ≫1, there is no difference with the third equation. The middle equation of equation (26) is a α If it is >>1, there will be no difference with the third equation. In the case of the upper equation and the middle equation, the error is about 1 / (IK) of the whole and about 1 / K of the whole, so the upper equation and the middle equation can be considered to be the same as the lower equation, allowing for some error. Then, equation (25) can be expressed as the following equation (27).

number

[0145] In equation (27), vector a is expressed as in equation (28) below. Note that equation (28) represents the expected value of the relative concentration c.

number

[0146] In other words, by using the Dirichlet distribution as an approximate distribution and performing the above approximation, the expected value of the true log-posterior distribution is expressed in the same way as the true log-posterior distribution. From the above, the KL divergence is expressed as the following equation (29).

number

[0147] In other words, when using KL divergence to perform variational inference, we use the formula (29) with the parameter a ik It is sufficient to minimize with respect to

[0148] In addition, when no prior distribution of the profile is assumed (uniform distribution), the MSM function is only the deviation from the measured data, as shown in equation (30).

number

[0149] In this case, the KL divergence is (sum of squared deviations from the measured data) - (entropy of the system), which is identical to the maximum entropy method (MEM). In other words, the Bayesian MSM based on the above variational inference is equivalent to the MEM when no prior distribution or normalization other than entropy is assumed. On the other hand, when the EDX measurement accuracy is high (or is considered to be high), the Bayesian MSM is equivalent to the conventional MSM. Therefore, the Bayesian MSM based on variational inference using the Dirichlet distribution can be said to be equivalent to a superordinate concept of both the conventional MSM and the MEM, incorporating them as special cases.

[0150] <<4.4 Processing flow of Bayesian MSM by data analysis device>> Fig. 7 is a flowchart showing an example of a process flow of the data analysis method (Bayesian MSM) executed by the data analysis device according to the first embodiment. The analysis unit 52 shown in Fig. 3 executes the analysis program stored in the storage unit 54. This causes the process of the flowchart shown in Fig. 7 to be executed.

[0151] 7, when the process starts, in step S11, the analysis unit 52 acquires measurement data (e.g., EDX measurement data) output from the measurement device 20 via the input unit 51. This measurement data is data of a response signal generated from the sample by the incidence of the probe.

[0152] In step S12, the analysis unit 52 executes a Bayesian MSM analysis using the sampling method using the measurement data acquired in step S11, and calculates each relative concentration c ik The analysis unit 52 calculates the average value μ and standard deviation σ of the estimated values ​​of each relative concentration c ik That is, in step S12, the analysis unit 52 calculates the average value μ of the estimated value of each chemical species and each depth from the surface of the sample, and calculates the confidence interval of the relative concentration c ik We estimate the mean μ and analyze the confidence interval of the estimate.

[0153] In this embodiment, the confidence interval can be arbitrarily set by the user, and may be set to μ±3σ, for example.

[0154] In step S13, the analysis unit 52 outputs the results of the analysis performed in step S12 to the output unit 53. The output unit 53 performs processing for displaying the analysis results, and outputs a display of the analysis results. Therefore, in step S13, the output unit 53 outputs a display of the average value and the confidence interval of the estimated value of the relative concentration for each depth and each chemical species. An example of the display output will be described later.

[0155] Fig. 8 is a flowchart showing another example of the processing flow of the data analysis method (Bayesian MSM) executed by the data analysis device according to the first embodiment. The processing of the flowchart shown in Fig. 8 may be performed by the analysis unit 52 shown in Fig. 3 executing the analysis program stored in the storage unit 54. The processing of the flowchart shown in Fig. 8 is basically the same as the processing of the flowchart shown in Fig. 7, and the processing of step S12 is replaced with the processing of step S12A.

[0156] In step S12A, the analysis unit 52 executes a Bayesian MSM analysis using variational inference to calculate each relative concentration c ik The analysis unit 52 calculates the average value μ and standard deviation σ of the estimated values ​​of ik By minimizing with respect to ik Calculate the mean value μ and confidence interval of the estimated value of .

[0157] The analysis unit 52 is not limited to performing only one of the sampling method and the variational inference. For example, the analysis unit 52 may be configured to select either the sampling method or the variational inference according to a user's selection.

[0158] 9 is a flowchart showing an example of the processing flow of the Bayesian MSM to which the sampling method is applied, which is executed by the analysis unit. In FIG. 9, processing according to the Metropolis method is shown as a specific example of the processing of step S12 in FIG.

[0159] In step S21, the analysis unit 52 sets an initial value of the relative concentration. The initial value of the relative concentration may be obtained in advance. Alternatively, in step S21, the analysis unit 52 may obtain the initial value of the relative concentration by executing a conventional analysis method using MSM. The conventional analysis method using MSM is a method in which the value of the relative concentration c is converged by repeating a process of obtaining an optimal solution c of equation (6), updating the matrix S using the optimal solution c, and obtaining the next optimal solution c. The relative concentration c ik If the change in is sufficiently small (for example, equal to or smaller than a predetermined threshold), the analysis unit 52 can determine that the solution has converged, and the relative concentration c ik can be used as the initial value.

[0160] In step S22, the analysis unit 52 uses the formula (15) to set the initial value to the dummy variable b ik Convert to.

[0161] In step S23, the analysis unit 52 sets a proposal distribution. The proposal distribution may be a function set in advance. For example, the multivariate Gaussian distribution represented by equation (16) is set as the proposal distribution. Alternatively, in step S23, the user may set a function representing the proposal distribution to the analysis unit 52.

[0162] In step S24, the analysis unit 52 calculates the nth (n is an integer equal to or greater than 1) b n From b n+1 Furthermore, the analysis unit 52 generates one b n+1 c n+1 Convert to.

[0163] In step S25, the analysis unit 52 calculates the difference in logarithms of the true posterior distributions using equation (17).

[0164] In step S26, the analysis unit 52 determines whether the calculated value of the formula (17) is equal to or greater than 0. If the calculated value of the formula (17) is equal to or greater than 0, in step S26A, the analysis unit 52 calculates c n+1 If the calculated value of equation (17) is less than 0, the analysis unit 52 generates a uniform random number r that takes a value between 0 and 1 in step S26B.

[0165] In step S26C, the analysis unit 52 judges whether the calculated value of the formula (17) is equal to or greater than log r. If the calculated value of the formula (17) is equal to or greater than the value of log r, in step S26A, the analysis unit 52 calculates c n+1 If the calculated value of the formula (17) is less than the value of log r, in step S26D, the analysis unit 52 uses c n+1 will be rejected.

[0166] After the process of step S26A or step S26D, in step S27, the analysis unit 52 determines whether the process of steps S24 to S26A or S26D has been executed a designated number of times. The number of times can be set arbitrarily by the user. If the process has been executed less than the designated number of times, the process returns to step S24. On the other hand, if the process has been executed the designated number of times, in step S28, the analysis unit 52 calculates the estimated value of the relative concentration for each depth and each chemical species (i.e., the relative concentration c ik ) and calculate the average value and standard deviation. When the process of step S28 ends, the process shown in FIG.

[0167] <<4.5 Example of Bayesian MSM analysis>> Fig. 10 shows the structure of the model specimen used to verify the analysis results by Bayesian MSM. As shown in Fig. 10, a model specimen having a two-layer structure of aluminum (Al) and silicon (Si) was used for the analysis by Bayesian MSM.

[0168] The theoretical values ​​of the intensity of the response signal (characteristic X-rays) when EDX analysis was performed on a model sample having the depth profile shown in FIG. 10 were calculated. The obtained theoretical values ​​were input to the Bayesian MSM executed by the data analysis device 30. The data analysis device 30 executed the process according to the flowcharts shown in FIG. 7 and FIG. 8 to output the average value and the confidence interval of the estimated value of the relative concentration for each depth and each chemical species by both the sampling method and the variational inference. Based on the output of the data analysis device 30, an evaluation equivalent to the "accuracy" of the depth profile was performed. Note that the value calculated by the conventional MSM was used as the initial value of the depth profile of the relative concentration.

[0169] FIG. 11 is a diagram showing an example of the results of analyzing the model shown in FIG. 10 by Bayesian MSM using a sampling method (Metropolis method). In FIG. 11, the dashed lines identified by the notations "Al(true)" and "Si(true)" represent the depth profiles corresponding to the depth profiles shown in FIG. 10. In FIG. 11, the solid lines identified by the notations "Al(MSM) average value μ" and "Si(MSM) average value μ" represent the average values ​​of the estimated relative concentrations for each depth and each chemical species obtained by Bayesian MSM using the Metropolis method. In FIG. 11, the solid lines identified by the notations "Al(MSM) μ±3σ" and "Si(MSM) μ±3σ" represent the confidence intervals of the estimated relative concentrations for each depth and each chemical species obtained by Bayesian MSM using the Metropolis method.

[0170] In the analysis of Bayesian MSM using the Metropolis method, σ in Eq. (12) d is set to 0.1, and λ in Eq. (12) s was set to 0.1. The calculation was repeated 10,000 times. The analysis results shown in Figure 11 show that the average and confidence intervals of the estimated relative concentrations of both Al and Si can be displayed. Furthermore, the depth profile estimated by Bayesian MSM is generally consistent with the true depth profile.

[0171] In addition, λ in equation (12) sIncreasing λ corresponds to a small difference in the relative concentration c between two adjacent layers in the prior distribution in the Bayesian MSM. In other words, in the Bayesian MSM, as in the conventional MSM, the change in the average value of the estimated relative concentration c, μ, in the depth direction is s The larger the value, the smoother the image becomes. On the other hand, σ generally tends to be at its maximum when the relative density is around 0.5, and λ s If is large, "σ" will be large overall.

[0172] In Bayesian MSM, the variance σ represents the "uncertainty of the prediction by Bayesian MSM" (or, alternatively, the "lack of confidence in the prediction"). Therefore, the narrower the confidence interval (μ±3σ), the more the estimation result by Bayesian MSM can be evaluated as "confident." On the other hand, the wider the confidence interval (μ±3σ), the more the estimation result by Bayesian MSM can be evaluated as "unconfident." In the example shown in Figure 11, the estimation result by Bayesian MSM can be evaluated as "confident" in each of the Al region and Si region. On the other hand, at the interface between the Al region and the Si region, the estimation result by Bayesian MSM can be evaluated as "unconfident" compared to the estimation results in each of the Al region and the Si region.

[0173] FIG. 12 is a diagram showing three representative points in the depth profile shown in FIG. 11. FIG. 13 is a diagram showing trace plots of relative concentration values ​​at three points in the depth profile shown in FIG. 12. "Point A," "Point B," and "Point C" in the depth profile shown in FIG. 12 respectively represent the position of the top surface of the sample, the boundary between the top surface layer of the sample and the next layer, and a position in the layer next to the top surface layer of the sample. FIG. 13 shows the relative concentration c ik The distribution of the values ​​of is shown. At points A, B, and C, there is no tendency for the distribution to change as the Metropolis method loop is repeated, and it can be determined that the results reached a steady state without any problems.

[0174] In addition, the relative concentration c obtained by conventional MSM ikEven if the Metropolis method is used with an initial value completely different from the solution of, it is practically difficult to obtain the results shown in Figure 11. Therefore, in the case of the Bayesian MSM to which the Metropolis method is applied, the relative concentration c obtained by the conventional MSM is ik It is preferable to use the solution of

[0175] FIG. 14 is a diagram showing an example of the results of analyzing the model shown in FIG. 10 by Bayesian MSM using variational inference. As in FIG. 11, the dashed lines in FIG. 14, identified by the notations "Al(true)" and "Si(true)", are depth profiles corresponding to the depth profiles shown in FIG. 10. In FIG. 14, the solid lines identified by the notations "Al(MSM) average value μ" and "Si(MSM) average value μ" indicate the average values ​​of the estimated relative concentrations for each depth and each chemical species obtained by Bayesian MSM using variational inference. In FIG. 14, the solid lines identified by the notations "Al(MSM) μ±3σ" and "Si(MSM) μ±3σ" indicate the confidence intervals of the estimated relative concentrations for each depth and each chemical species obtained by Bayesian MSM using variational inference.

[0176] As with the analysis results shown in Fig. 11, the analysis results shown in Fig. 14 are able to display the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations for both Al and Si. However, compared to the analysis results shown in Fig. 11, the changes in the average values ​​of the estimated values ​​of the relative concentrations for each depth and each chemical species near the boundary between the Al and Si regions are somewhat less smooth. This change in the average value is thought to be due to the effect of approximation of the posterior distribution function. However, variational inference generally has low computational costs, and the results are expressed as analytical expressions, so the results are highly interpretable. Either method can be selected depending on whether smoothness of the results or computational costs and interpretability of the results are prioritized.

[0177] <5. Second embodiment> According to the first embodiment, information about the sample to be analyzed (e.g., the number of layers, the type of layers, etc.) does not need to be obtained in advance. Even for a completely unknown sample, information about the depth profile of the relative concentration of chemical species can be obtained by executing Bayesian MSM using EDX measurement data and theoretical values ​​of EDX measurement on a computer. Therefore, the analysis method (Bayesian MSM) according to the first embodiment can be said to be an estimation method with a high degree of freedom.

[0178] On the other hand, from a practical standpoint, estimation with a high degree of freedom may result in the results being difficult to interpret or may lead to multiple interpretations. Furthermore, depending on the computing power of the computer, it may take a long time to obtain analytical results.

[0179] Therefore, the second embodiment relates to a depth profile analysis method that is easier for users to use. Note that, like the first embodiment, the analysis method according to the second embodiment uses Bayesian estimation.

[0180] In the second embodiment, if some information about the sample is available in advance, that information is used as prior knowledge for estimating the depth profile of the relative concentrations of chemical species, such as the number of layers that make up the sample or the composition of each layer.

[0181] Furthermore, in the second embodiment, the degree of freedom regarding the depth profile is limited in estimating the depth profile of the relative concentration of chemical species. That is, a constraint is imposed on the depth profile. This makes it possible to reduce the number of calculations of the data analysis device 30 until a depth profile close to the correct answer is reached. According to the second embodiment, it is possible to realize an analysis method that is easier for users to use in that it is possible to shorten the time required to obtain an analysis result. Furthermore, by reducing the number of calculations of the data analysis device 30, it is possible to save, for example, computational resources.

[0182] The analysis method according to the second embodiment will be described in detail below.

[0183] <<5.1 Estimation method>> In the Bayesian MSM according to the first embodiment, the only condition imposed on the depth profile c of the relative concentration of the chemical species is that it changes smoothly at the boundary between two adjacent layers. In the second embodiment, the degree of freedom of the depth profile is further restricted in the estimation of the depth profile of the relative concentration of the chemical species.

[0184] In one example, the depth profile of the relative concentration of the chemical species is subject to a condition that the profile rises at a first depth and falls at a second depth deeper than the first depth. The function expressing the depth profile of the relative concentration of the chemical species is not particularly limited as long as it satisfies the above condition. A function that can express a natural depth profile is more preferable.

[0185] For example, a depth profile can be expressed using a sigmoid function, which converts any input value into an output value in the range 0 to 1.

[0186] The function F shown in the following formula (31) is an example of a function that expresses the depth profile of the relative concentration c of a chemical species i, and is expressed by the sum of sigmoid functions. The first term in the parentheses on the right side of formula (31) is a sigmoid function that expresses the rising edge of the depth profile, and the second term is a sigmoid function that expresses the falling edge of the depth profile.

[0187]

number

[0188] In equation (31), z represents the depth from the sample surface. A represents the composition of the sample (relative concentration of chemical species) and is a value between 0 and 1. u ,z l represents the position of the layer boundary (depth from the sample surface). Therefore, z u,z l is a variable related to the layer thickness. σ u ,σ l represent the gradual rise and fall of the depth profile, respectively.

[0189] As shown in Eq. (31), the depth profile of the sample is calculated by five parameters related to the composition and thickness of the layer (A, z u ,z l ,σ u ,σ l ) In the second embodiment, the set of these five parameters is treated as a random variable for Bayesian estimation. Hereinafter, the set of the above five parameters is expressed as "parameter θ".

[0190] FIG. 15 is a model diagram of a sample to which Bayesian estimation according to the second embodiment is applied. As shown in FIG. 15, the sample is modeled as a stack of M (M≧1) layers. In each layer of the stack, parameters are set for each chemical species. Since the total number of chemical species constituting the sample is I, the depth profile of the relative concentration of the chemical species in each layer is expressed by applying (5×I) parameters θ to Equation (31). Of the five parameters, parameter z u ,z l ,σ u ,σ l is assumed to be common to all chemical species in the same layer. In the second embodiment, the same chemical species is allowed to exist in multiple layers.

[0191] By setting the parameter θ as described above, an appropriate parameter according to the problem to be solved is selected from the five parameters. This makes it possible to estimate the depth profile. The Bayesian estimation according to the second embodiment will be described below.

[0192] (1. Prior distribution of parameter θ) The depth profile of the relative concentration of chemical species in the sample model shown in Fig. 15 is defined by (5 x I x M) parameters. Therefore, a prior distribution of the (5 x I x M) parameters is set.

[0193] As an example, the prior distribution is provisionally set to a normal distribution. In this case, the prior distribution of the parameter θ is expressed according to the following equation (32).

[0194]

number

[0195] In equation (32), θ miq means parameter q of the parameters θ for the chemical species i in the m-th layer. miq and σ miq are the parameters θ miq means the mean and variance of

[0196] Fig. 16 is a diagram showing a schematic representation of a sigmoid function type depth profile. In Fig. 16, the horizontal axis of the graph represents the depth of the sample, and the vertical axis of the graph represents the relative concentration of the chemical species. The graph shows a schematic representation of a sigmoid function type depth profile for each layer from layer 1 to layer M.

[0197] In the second embodiment, a normal distribution is provisionally set as the prior distribution. However, in the end, σ miq Treating is very large, in other words, treating it as "we have no confidence in the prior distribution," is equivalent to not assuming any prior distribution.

[0198] (2. Likelihood function) In this embodiment, the likelihood function is expressed as p(d|θ). That is, the likelihood function is expressed as a function of the parameter θ when EDX measurement data (response signal d from EDX) is obtained from the parameter θ. Specifically, in this embodiment, the likelihood function is expressed according to equation (33).

[0199]

number

[0200] In equation (33), d ij represents the measurement data of the EDX analysis at accelerating voltage j for chemical species i, and d' ij represents the theoretical value (theoretical value of the response signal) of the EDX analysis at accelerating voltage j for chemical species i. EDX_ij represents the variability of the EDX analysis at accelerating voltage j for chemical species i. The relationship between the depth profile and the theoretical value of the EDX analysis follows the above-mentioned formulas (2) to (5).

[0201] Note that in the second embodiment, the shape of the depth profile is constrained to follow a sigmoid function, so the maximum smoothness condition imposed on the MSM, i.e., "the relative concentrations of the chemical species of the sample vary smoothly among the layers of the stack in the minimization of the sum of squared deviations," is not necessary in the second embodiment.

[0202] (3. Representation of the Posterior Distribution) The posterior distribution of the parameter θ after observing the EDX data can be expressed according to the following equation (34) using Bayes' theorem (see equation (10)). Note that the likelihood function p(d|θ) included in equation (34) is expressed by equation (33).

[0203]

number

[0204] As in the transformation from equation (10) to equation (11), the logarithm of the posterior distribution p(θ|d) is taken. By taking the logarithm of the right-hand side of equation (34), the logarithm (log[p(d|θ)p(θ)]) of the posterior distribution p(θ|d) is expressed according to the following equation (35). For convenience of notation, the sign of log[p(d|θ)p(θ)] on the left-hand side in equation (35) is made negative.

[0205]

number

[0206] The first and second terms on the right-hand side of equation (35) are terms resulting from the prior distribution of the parameter θ. Since the prior distribution of the parameter θ is uncertain, we assume that we are "not at all confident" in estimating the parameter θ. "Not at all confident" means that the variance σ miq is sufficiently large. Therefore, the second term on the right-hand side of equation (35) can be ignored. Note that if some information about the parameters is obtained in advance, for example, through another analysis or feedback from the manufacturing process, a prior distribution may be used proactively. In the following, we will deal with the most common case where we are not confident in the prior distribution.

[0207] From the above, the logarithm of the posterior distribution p(θ|d) of the parameter θ is expressed according to the following equation (36) using the first term on the right-hand side of equation (35).

[0208]

number

[0209] <<5.2 Analysis section configuration>> Fig. 17 is a functional block diagram showing an example of the configuration of the analysis unit 52 shown in Fig. 3 according to the second embodiment. As shown in Fig. 17, the analysis unit 52 includes a forward problem evaluation unit 61 and a next parameter recommendation unit 62.

[0210] The forward problem is an analysis method for estimating a result (output) from a cause (input). The forward problem evaluation unit 61 receives a value of a parameter θ from a next parameter recommendation unit 62. The forward problem evaluation unit 61 uses the value of the parameter θ as an input to the forward problem model, and calculates the sum of squared deviations between the theoretical measurement value and the actual measurement value when the parameter is used. The forward problem evaluation unit 61 outputs the calculation result of the sum of squared deviations as an output y.

[0211] The next parameter recommendation unit 62 receives the output y from the forward problem evaluation unit 61. Using the output y, the next parameter recommendation unit 62 recommends a candidate value of the parameter θ to be used in the next calculation of the sum of squared deviations by the forward problem evaluation unit 61. This candidate value of the parameter θ is also referred to as the "next parameter θ" in this specification.

[0212] The process of the data analysis method executed by the data analysis device according to the second embodiment is similar to the process shown in the flowchart in Fig. 7. However, the data analysis method according to the second embodiment differs in the process corresponding to step S12 in Fig. 7. Specifically, evaluation of the forward problem by the forward problem evaluation unit 61 (calculation of the posterior distribution of the parameter θ) and recommendation of the next parameter θ by the next parameter recommendation unit 62 are alternately repeated. This makes it possible to obtain the mean value and confidence interval of the variable selected from the parameter θ according to the problem to be solved.

[0213] The analysis unit 52 can output the value of the parameter θ each time the evaluation of the forward problem and the recommendation of the next parameter θ are repeated a predetermined number of times. This allows the analysis unit 52 to obtain the average value and the confidence interval of each variable included in the parameter θ for each layer of the sample model. Therefore, the analysis unit 52 can estimate the relative concentration for each depth from the surface of the sample and for each chemical species, and can obtain the average value and the confidence interval of the estimated value of the relative concentration.

[0214] Fig. 18 is a block diagram showing an example of the configuration of the forward problem evaluation unit 61 and the next parameter recommendation unit 62 shown in Fig. 17. As shown in Fig. 18, the forward problem evaluation unit 61 includes a profile construction unit 61A, a measurement theoretical value construction unit 61B, and a deviation sum of squares evaluation unit 61C.

[0215] The profile constructing unit 61A substitutes the parameter θ output from the next parameter recommending unit 62 into the equation (31) to determine, for each layer and for each chemical species, a depth profile of the relative concentration c of that chemical species.

[0216] The measurement theoretical value constructing unit 61B performs conversion according to equation (2) on the relative concentration c according to the depth profile determined by the profile constructing unit 61A. In this way, the measurement theoretical value constructing unit 61B converts the relative concentration c into a theoretical value d' of the EDX measurement (theoretical value of the response signal).

[0217] The deviation sum of squares evaluation unit 61C applies the EDX theoretical measurement value d' and the EDX measurement data d obtained by the measurement theoretical value construction unit 61B to the formula (36) to obtain a value that indicates the deviation between the theoretical value and the measured value of the EDX measurement. This value is a value included in the posterior distribution of the parameter θ, and corresponds to the output y of the forward problem evaluation unit 61. Note that, among the parameters θ, the parameter related to the layer thickness d (z u ,z l ) and the parameters (σ u ,σ l The value of ) is given to the forward problem evaluation unit 61 when the forward problem evaluation unit 61 performs its first calculation.

[0218] The next parameter recommendation unit 62 includes an optimal solution determination unit 62A and an MCMC implementation unit 62B. The optimal solution determination unit 62A determines (estimates) the optimal value of the next parameter θ based on all past history of the parameter θ. The optimal value of the parameter θ estimated by the optimal solution determination unit 62A becomes the initial value of the MCMC method executed by the MCMC implementation unit 62B.

[0219] The optimization method executed by the optimal solution judgment unit 62A is not particularly limited. For example, the optimal solution judgment unit 62A may use Bayesian optimization to determine the next optimal parameter θ. As other optimization methods, it is possible to adopt, for example, a gradient method or a covariance matrix adaptive evolution strategy (CMA-ES), without being particularly limited. When executing an optimization method such as Bayesian optimization or CMA-ES, the optimal solution judgment unit 62A repeatedly calculates the value of the logarithmic posterior distribution by changing multiple variables (schematically represented as θ1, θ2 in FIG. 18) included in the parameter θ. If necessary, the optimal solution judgment unit 62A may refer to multiple past histories. In this way, the optimal solution judgment unit 62A finds the optimal parameter θ (i.e., one optimal set of parameters).

[0220] In addition, since the above optimization method is executed by a computer, a known algorithm can be adopted, and the computer program language for writing the algorithm is not particularly limited.

[0221] The MCMC implementation unit 62B sets the value of the parameter θ obtained by the optimal solution judgment unit 62A as the initial value, and proposes the next parameter θ using a predetermined probability distribution according to the MCMC method. The MCMC implementation unit 62B outputs the parameter θ to the forward problem evaluation unit 61.

[0222] In one embodiment, the MCMC implementation unit 62B executes the Metropolis method, which is the most basic method of MCMC, to propose the following parameter θ. Other methods that are more advanced versions of the Metropolis method, such as the Hamiltonian Monte Carlo method and the replica exchange Monte Carlo method, may also be used. The MCMC implementation unit 62B sets the value of the parameter θ input from the optimal solution determination unit 62A to the initial value θ (t=0) of the Metropolis method.

[0223] Next, the MCMC implementation unit 62B randomly samples a candidate value a for the parameter θ(t+1) following the parameter θ(t) according to a certain proposal distribution. The proposal distribution is not particularly limited, and may be, for example, a Gaussian distribution that is independent for each variable included in the parameter θ (without correlation for each variable). The MCMC implementation unit 62B proposes the value of the next parameter θ by taking the variance by random sampling from the proposal distribution, centered on the value of the parameter θ at a certain point in time.

[0224] The MCMC implementation unit 62B uses the proposed next value to calculate the log posterior distribution. In the second embodiment, the log posterior distribution corresponds to the sum of squared deviations between the EDX measurement theoretical value and the EDX measurement value. If the value of the log posterior distribution is improved (smaller) than the previous value, the MCMC implementation unit 62B adopts the proposed value of the next parameter θ. On the other hand, if the value of the log posterior distribution is worsened (larger) than the previous value, the MCMC implementation unit 62B generates a uniform random number r that takes a value between 0 and 1.

[0225] If the absolute value of the difference between the value of the log posterior distribution and the previous value is less than the absolute value of log(r), the MCMC implementation unit 62B adopts the proposed next value of the parameter θ. On the other hand, if the absolute value of the difference between the value of the log posterior distribution and the previous value is equal to or greater than the absolute value of log(r), the MCMC implementation unit 62B discards the proposed value of the parameter θ and adopts the previous value of the parameter θ. As in the first embodiment, in the MCMC method, even if the value of the log posterior distribution to be evaluated is worse than the previous value, the proposed value is adopted with a certain probability. By appropriately determining this adoption probability, the posterior distribution of the parameter θ can be obtained.

[0226] In theory, the MCMC method can obtain the correct distribution no matter what the initial value is, if it is run over an infinite amount of time in the correct way. In the MCMC method, the period from the initial point to convergence to the desired distribution is called burn-in. Generally, the data in the burn-in state is discarded to calculate various quantitative values.

[0227] In practice, the time available for calculation is limited. Therefore, in the second embodiment, the optimal solution determination unit 62A obtains a target value or a value close to the target value of the parameter θ, and uses the value as an initial value for the MCMC method. The MCMC implementation unit 62B uses the initial value, thereby shortening the burn-in period. Therefore, the calculation time required for the MCMC implementation unit 62B to reach the distribution of the correct solution can be shortened. In addition, in the evaluation of the posterior distribution of parameters based on Bayesian inference, the MCMC method is generally used, but a method based on variational inference as in the first embodiment may also be used. For example, a method is also possible in which the posterior distribution is assumed to be a parametric distribution such as a Gaussian distribution or a Gamma distribution, and the parameters are obtained.

[0228] In both the first and second embodiments, the confidence interval (σ) of the estimated value of the relative concentration means a “Bayesian confidence interval.” However, while “σ” in the first embodiment means an error including the certainty of the layer structure, in the second embodiment in which the layer structure is assumed in advance, “σ” means an error related to the thickness and composition of each layer and the steepness of the profile at the interface of the layers.

[0229] <<5.3 Analysis example>> The following examples show the results of the analysis unit 52 calculating the layer thickness and composition for several samples. Equation (31) was used to express the depth profile. A Gaussian distribution was set as the proposed distribution for the Metropolis method. A Gaussian distribution was set for each of the five parameters (variables) included in the parameter θ. In the proposed distribution, it was assumed that there was no correlation between the variables.

[0230] The standard deviation σ in the Gaussian distribution is σ for the layer thickness. 2 =1 2 (nm 2 ) and σ 2 =0.01 2These σ values ​​correspond to a change in the layer thickness of about 1 nm and a change in the layer composition of about 0.01 for each loop of the MCMC calculation.

[0231] (Example 1: Analysis of Ni electrodes and SiN films on InP wafers) A sample (sample 1) was prepared by forming a nickel (Ni) electrode and a silicon nitride (SiN) protective film on an indium phosphide (InP) semiconductor wafer by patterning. A surface analysis was performed on sample 1 using EDX. Then, in order to verify the validity of the analysis according to the second embodiment, a cross-sectional analysis of the sample was performed using STEM-EDX.

[0232] FIG. 19 shows a STEM cross-sectional image of sample 1. As shown in FIG. 19(A) and FIG. 19(B), the STEM images show that the thicknesses of the Ni layer and the SiN layer of sample 1 were approximately 181 nm and 262 nm, respectively. The STEM-EDX analysis results showed that the ratio of indium to phosphorus (In:P) in the semiconductor substrate was approximately 0.55:0:45. The STEM-EDX analysis results showed that a small amount of phosphorus (P) element was missing from the InP substrate in sample 1. The ratio of silicon to nitrogen (Si:N) in the SiN film was approximately 0.56:0.44. Therefore, this SiN film was evaluated to be a silicon-rich film.

[0233] FIG. 20 shows the results of EDX surface analysis of sample 1. EDX surface analysis was performed on both the Ni electrode surface and the SiN film surface. The accelerating voltage levels of the electron beam in the EDX surface analysis were 10 kV, 15 kV, 20 kV, 25 kV, and 30 kV. FIG. 20(A) shows the results of the EDX surface analysis on the Ni electrode surface, and FIG. 20(B) shows the results of the EDX surface analysis on the SiN film surface.

[0234] The EDX data at the above five accelerating voltage levels was input to the analysis unit 52 shown in Fig. 17. The analysis unit 52 analyzed the thickness of the Ni layer and the In composition in the InP substrate according to the MCMC method. The analysis results by the analysis unit 52 will be described in detail below.

[0235] FIG. 21 shows an analytical model of sample 1 (Ni layer on an InP substrate). In the analytical model shown in FIG. 21, it is assumed that the sample is made up of two layers, a Ni layer and an InP layer. The thickness d Ni The MCMC method was performed using the In composition of the InP layer as a variable.

[0236] In the model shown in FIG. 21, among the five parameters of layer 1 (Ni layer) and layer 2 (InP layer), the parameter σ u ,σ l is fixed at 50 nm. The parameters θ, composition A and film thickness z u ,z l The depth profile analysis was performed using the variables. Regarding the initial values ​​used in the MCMC method, the analysis unit 52 uses the thickness d Ni And the initial composition of InP was determined.

[0237] Fig. 22 shows the results of calculations performed by the analysis unit 52 on the thickness of the Ni layer and the In composition of InP according to the MCMC method. Fig. 22(A) shows the variation in the calculation results for the thickness of the Ni layer versus the number of repetitions of the MCMC method. Fig. 22(B) shows the variation in the calculation results for the In composition of the InP layer versus the number of repetitions of the MCMC method.

[0238] As shown in Fig. 22(A), the calculated thickness of the Ni layer fluctuates around 172 nm, but reaches a steady state as the MCMC method loop is repeated. The thickness of the Ni layer obtained by STEM-EDX analysis is about 181 nm. Therefore, according to the second embodiment, a value close to that of the STEM-EDX analysis can be obtained by non-destructive analysis of the thickness of the Ni layer.

[0239] Similarly, as shown in Fig. 22(B), the calculated value of the In composition of the InP layer fluctuates around 0.616, but reaches a steady state as the MCMC method loop is repeated. In the analysis according to the second embodiment, the In composition of the InP layer exceeds 0.5. That is, according to the second embodiment, the result is obtained that indicates that phosphorus is removed from the InP layer, similar to the result of the STEM-EDX analysis.

[0240] Fig. 23 is a diagram showing the analysis results of sample 1 (Ni layer on an InP substrate) by analysis unit 52. In Fig. 23, the horizontal axis of the graph indicates the depth from the sample surface, and the vertical axis of the graph indicates the relative concentration of the chemical species. The calculation of the MCMC method was repeated 10,000 times.

[0241] The analysis unit 52 discarded the first 1000 calculation results of the MCMC loop, and from the remaining calculation results, determined the average value (μ) and variance (σ) of the thickness of the Ni layer, and the average value (μ) and confidence interval (σ) of the In composition in the InP layer. The average value (μ) and variance (σ) of the thickness of the Ni layer are 172 nm and 0.99 nm, respectively. The average value (μ) and confidence interval (σ) of the In composition in the InP layer are 0.616 and 0.0014, respectively.

[0242] The analysis unit 52 picked up the calculation results of the depth profile of the relative concentration for every 1000th loop from 1000th to 10000th of the MCMC loops, and the output unit 53 (see FIG. 3) output a display of the picked-up calculation results. The curves in the graph correspond to the display of the depth profile of the relative concentration of each of Ni, In, and P output from the output unit 53.

[0243] The depth from the surface corresponding to the position where the curve showing the relative concentration of Ni and the curve showing the relative concentration of In intersect is about 180 nm to 183 nm. The width of the depth range corresponds to the range of the average thickness of the Ni layer ±3σ. The σ obtained by the MCMC method can be regarded as the Bayesian confidence interval, that is, the accuracy of the analysis result in the second embodiment. Therefore, the curves shown in FIG. 23 show the confidence interval of the estimated value of the relative concentration of the chemical species.

[0244] FIG. 24 shows an analytical model of sample 1 (SiN layer on an InP substrate). In the analytical model shown in FIG. 24, it is assumed that the sample is made up of two layers, a SiN layer and an InP layer. The thickness d of the SiN layer is SiN The MCMC method was performed using the In composition of the InP layer as a variable.

[0245] In the model shown in FIG. 24, among the five parameters of layer 1 (SiN layer) and layer 2 (InP layer), the parameter σ u ,σ l is fixed at 50 nm. The parameters θ, composition A and film thickness z u ,z l The depth profile analysis was performed for each of Si, N, and In using the variables. Regarding the initial values ​​used in the MCMC method, the analysis unit 52 calculated the thickness d SiN And the initial composition of InP was determined.

[0246] Fig. 25 shows the results of calculations performed by the analysis unit 52 on the N composition of the SiN layer, the thickness of the SiN layer, and the In composition of the InP layer according to the MCMC method. Fig. 25(A) shows the variation in the calculation results with respect to the number of repetitions of the MCMC method for the N composition of the SiN layer. Fig. 25(B) shows the variation in the calculation results with respect to the number of repetitions of the MCMC method for the thickness of the SiN layer. Fig. 25(C) shows the variation in the calculation results with respect to the number of repetitions of the MCMC method for the In composition of the InP layer.

[0247] As shown in Fig. 25(A), the calculated value of the N composition of the SiN layer fluctuates around 0.363, but reaches a steady state as the MCMC loop is repeated. As shown in Fig. 25(B), the calculated value of the thickness of the SiN layer fluctuates around 253 nm, but reaches a steady state as the MCMC loop is repeated. As shown in Fig. 25(C), the calculated value of the In composition of the InP layer fluctuates around 0.624, but reaches a steady state as the MCMC loop is repeated.

[0248] The analysis results shown in Fig. 25 indicate that the MCMC method can be used to estimate the thickness of the SiN layer even when the compositions of the two layers are changed simultaneously. The thickness of the SiN layer is close to the thickness (approximately 262 nm) obtained from the cross-sectional STEM image.

[0249] Fig. 26 is a diagram showing the analysis results of sample 1 (SiN layer on an InP substrate) by analysis unit 52. In Fig. 26, the horizontal axis of the graph indicates the depth from the sample surface, and the vertical axis of the graph indicates the relative concentration of the chemical species. The number of iterations of the MCMC method calculation is 10,000.

[0250] The analysis unit 52 discarded the calculation results of the first 1000 times of the MCMC loop, and from the remaining calculation results, determined the average (μ) and variance (σ) of the film thickness of the SiN layer, the μ and σ of the N composition in the SiN layer, and the μ and σ of the In composition in the InP layer. The μ and σ of the film thickness of the SiN layer are 252 nm and 1.83 nm, respectively. The μ and σ of the N composition in the SiN layer are 0.364 and 0.0018, respectively. The μ and σ of the In composition in the InP layer are 0.624 and 0.0010, respectively.

[0251] The analysis unit 52 picked up the calculation results of the depth profile of the relative concentration for every 1000th loop from 1000th to 10000th of the MCMC loops, and the output unit 53 (see FIG. 3) output a display of the picked-up calculation results. The curves in the graph correspond to the display of the depth profile of the relative concentration of each of Si, N, In, and P output from the output unit 53.

[0252] Analysis of sample 1 by STEM-EDX showed that the SiN layer was Si-rich (about 0.56) and the InP layer was In-rich (about 0.55). The analysis results shown in Fig. 26 show that the SiN layer is Si-rich and the InP layer is In-rich. The analysis of sample 1 according to the second embodiment is non-destructive and shows results consistent with the results of the STEM-EDX analysis.

[0253] (Example 2: Analysis of PFA film on Cu foil) A sample (sample 2) was prepared by applying PFA (perfluoroalkoxy alkane), a type of fluororesin, onto a copper (Cu) foil and then baking the coated film. As in Example 1, a surface analysis was performed on sample 2 using EDX. Thereafter, in order to verify the validity of the analysis according to the second embodiment, a cross-sectional analysis of the sample was performed using STEM-EDX.

[0254] Fig. 27 is a diagram showing a STEM cross-sectional image of Sample 2. As shown in Fig. 27, from the STEM image, the thickness of the PDF layer was evaluated to be about 1.1 µm.

[0255] Figure 28 shows the results of cross-sectional analysis of sample 2 by STEM-EDX. As shown in Figure 28, EDX analysis was performed at seven levels of accelerating voltage. Since the PFA film is easily damaged by ions, it is difficult to evaluate its composition by STEM-EDX. For this reason, GCIB (Gas Cluster Ion Beam)-XPS depth analysis was performed, and the composition ratio (C:F) of carbon (C) and fluorine (F) in the PFA film was evaluated to be approximately 1:2.

[0256] Fig. 29 is a diagram showing an analytical model of the sample 2. In the analytical model shown in Fig. 29, it is assumed that the sample is made up of two layers, a PFA layer and a Cu layer. The analysis unit 52 calculates the thickness d PFA The MCMC method is performed using the composition of the PFA layer as a variable. In the model shown in FIG. 29, among the five parameters θ of layer 1 (PFA layer) and layer 2 (Cu layer), the parameter σ u ,σl is fixed at 50 nm. The parameters θ, composition A and film thickness z u ,z l The depth profile analysis was carried out using as a variable.

[0257] In Example 2, the initial values ​​used in the MCMC method were intentionally set to values ​​far removed from the optimal values ​​in order to confirm the stability of the method.

[0258] FIG. 30 shows the results of the analysis unit 52 calculating the carbon composition of the PFA layer and the thickness of the PFA layer according to the MCMC method. FIG. 30(a) shows the variation of the calculation result for the C composition of the PFA layer versus the number of repetitions of the MCMC method. FIG. 30(b) shows the variation of the calculation result for the thickness of the PFA layer versus the number of repetitions of the MCMC method. As shown in FIG. 30, even if the initial value of the MCMC method is significantly different from the optimal value, the calculated value reaches a steady state as the number of repetitions of the calculation increases. In the example shown in FIG. 30, the calculated value can be considered to have reached a steady state after 6000 calculations.

[0259] As is clear from FIG. 30, according to the MCMC method, even if a value other than the optimal value is used as the initial value, the calculation value reaches the correct value by repeating the calculation loop a sufficiently large number of times. Therefore, it is not an essential requirement for executing the MCMC method to determine an appropriate initial value of the MCMC method by the optimization method. However, as can be seen from a comparison between FIG. 22 and FIG. 30, the burn-in state can be shortened by using the parameter θ obtained by the optimization method as the initial value of the MCMC loop. Therefore, in the second embodiment, the optimal solution determination unit 62A gives a value close to the optimal value to the MCMC implementation unit 62B as the initial value of the MCMC method. This can reduce the number of calculations of the data analysis device 30 until a depth profile close to the correct solution is reached, thereby making it possible to shorten the calculation time required for analyzing the depth profile.

[0260] FIG. 31 is a diagram showing the analysis results of the sample 2 by the analysis unit 52. In FIG. 31, the horizontal axis of the graph indicates the depth from the sample surface, and the vertical axis of the graph indicates the relative concentration of the chemical species. The number of repetitions of the MCMC calculation is 10,000. However, as shown in FIG. 30, the results of the first 6,000 calculations are not stable. For this reason, the analysis unit 52 discards the results of the first 6,000 calculations, and from the remaining calculation results, the average value (μ) and variance (σ) of the thickness of the PFA layer, as well as the μ and σ of the composition of C in the PFA layer are obtained. The μ and σ of the thickness of the PFA layer are 1292 nm and 10.2 nm, respectively. The μ and σ of the composition of C in the PFA layer are 0.38 and 0.0021, respectively.

[0261] The analysis unit 52 picked up the calculation results of the depth profile of the relative concentration for every 500 times from 6000 to 10000 in the MCMC loop, and the output unit 53 (see FIG. 3) output a display of the picked up calculation results. The curves in the graph are a diagram in which the calculation results for every 500 times from 6000 to 10000 in the MCMC loop are picked up and displayed. The curves in the graph correspond to the display of the depth profile of the relative concentrations of F, C, and Cu output from the output unit 53.

[0262] The thickness of the PFA layer is estimated to be about 1200 nm from the depth where the curve showing the relative concentration of F intersects with the curve showing the relative concentration of Cu, and the depth where the curve showing the relative concentration of C intersects with the curve showing the relative concentration of Cu. Furthermore, the ratio of C to F in the PFA layer is estimated to be about 1:2.

[0263] The analysis results by the MCMC method show that the thickness of the PFA layer evaluated by STEM (about 1100 nm) and the composition evaluated by GCIB-XPS (C:F about 1:2) are close to each other. Therefore, according to the second embodiment, the analysis of Example 2 can be performed correctly and non-destructively.

[0264] (Example 3): Analysis of multilayer film on Si substrate A multilayer film was formed on a silicon (Si) substrate by electron beam (EB) deposition (sample 3). A surface analysis was performed on sample 3 by EDX. Then, a cross-sectional analysis of the sample was performed by STEM-EDX to obtain information for verifying the validity of the analysis according to the second embodiment.

[0265] Figure 32 is a diagram showing a STEM cross-sectional image of sample 3. As shown in Figure 32, a Ni layer, a Cr layer, a Ti layer, and an Al layer are laminated on the surface of the Si substrate in that order from the surface of the Si substrate. From the STEM image, the thicknesses of the Ni layer, the Cr layer, the Ti layer, and the Al layer were evaluated to be approximately 302 nm, 286 nm, 314 nm, and 258 nm, respectively.

[0266] Figure 33 shows the results of EDX surface analysis of sample 3. The EDX surface analysis was performed at accelerating voltages of 10 kV to 30 kV (a total of 21 levels). The MCMC method was performed based on the data obtained by the EDX surface analysis. Note that the structure of the sample was assumed to be known, an Al / Ti / Cr / Ni / Si structure, and only the thicknesses of the four layers on the Si substrate were used as variables.

[0267] FIG. 34 shows the results of the calculation of the thickness of the multilayer film of the sample 3 by the analysis unit 52 according to the MCMC method. FIG. 34(A), FIG. 34(B), FIG. 34(C), and FIG. 34(D) show the variation of the calculation results with respect to the number of repetitions of the MCMC method for the thickness of the Al layer, the thickness of the Ti layer, the thickness of the Cr layer, and the thickness of the Ni layer, respectively. From all of FIG. 34(A) to FIG. 34(D), it can be evaluated that the calculated values ​​reach a steady state after the first several hundred cycles. Each graph shows the average thickness (μ) and variance (σ) calculated using the calculated values ​​after 1000 cycles for each film.

[0268] Fig. 35 is a diagram showing the results of the profile evaluation of the sample 3 by the analysis unit 52. In Fig. 35, the profile evaluated by the analysis unit 52 by executing the MCMC method is shown by a solid line, and the profile evaluated by STEM-EDX is shown by a dashed line. As shown in Fig. 35, the profile evaluated by the MCMC method closely approximates the profile evaluated by STEM-EDX. That is, according to the second embodiment, it is possible to obtain results close to those obtained by destructive evaluation, despite the non-destructive evaluation of the sample.

[0269] According to FIG. 35, the position where the curve showing the relative concentration of Al and the curve showing the relative concentration of Ti intersect corresponds to the interface between the Al layer and the Ti layer. Similarly, the position where the curve showing the relative concentration of Ni and the curve showing the relative concentration of Si intersect corresponds to the interface between the Ni layer and the Si layer. In the evaluation according to the second embodiment, the estimation range of the position of the interface between the Ni layer and the Si layer is larger than the estimation range of the position of the Al / Ti interface. This indicates that the estimation results by MCMC are less reliable at the inner side of the sample (the side closer to the Si substrate) than at the outermost surface of the sample. The measurement value of the EDX analysis, which is the basis of the MCMC analysis, has a greater influence at the outermost surface and a weaker influence at the deeper position, so the analysis result shown in FIG. 35 that the estimation results are less reliable at the inner side is reasonable.

[0270] <<5.4 Relationship between the number of measured data and the estimated results>> Fig. 36 shows the results of calculations of the thickness of the multilayer film of sample 3 by analysis unit 52 according to the MCMC method when the number of EDX measurement data (in other words, the number of acceleration voltage levels) is reduced. Specifically, the MCMC method was performed using measurement data at three levels (10 kV, 20 kV, and 30 kV) of the EDX measurement data. Fig. 36(A) to Fig. 36(D) are figures to be compared with Fig. 34(A) to Fig. 34(D), respectively.

[0271] Comparing the mean value and variance of each stratum, the mean value μ does not change significantly even if the number of levels is reduced. On the other hand, the variance σ is uniformly large in the evaluation results of all stratums. Figures 34 and 36 quantitatively show the fact that in Bayesian estimation, "the more measurement data used in the evaluation, the more confident the estimation results are."

[0272] Fig. 37 shows the effect of the number of measurement data on the estimation of the film thickness of sample 3 by the MCMC method. Fig. 37(A) shows the expected value of thickness (average value μ) versus the number of acceleration voltage levels. Fig. 37(B) shows the confidence interval of thickness (variance σ) versus the number of acceleration voltage levels. The greater the number of acceleration voltage levels, the greater the number of EDX measurement data. When the number of measurement data used to evaluate the film thickness is reduced (when the number of acceleration voltage levels is small), the change in the expected value of thickness (average value μ) of each layer is small, while the confidence interval of thickness of each layer (variance σ) tends to become larger.

[0273] In this way, by applying Bayesian estimation to the estimation of the film thickness, the relationship between the number of data and the uncertainty can be evaluated. Therefore, the number of measurement data required for the evaluation can be determined according to the lack of confidence in the estimation result, i.e., the degree of uncertainty of the estimation result. The degree of uncertainty of the estimation result is, so to speak, the accuracy of the evaluation. That is, according to the second embodiment, the number of measurement data required may be determined according to the accuracy required for the evaluation.

[0274] <6. Example of display output> By executing the above-mentioned Bayesian MSM, the data analysis device 30 can obtain the average value of the estimated value of the relative concentration for each depth of the sample and for each chemical species, and the confidence interval of the estimated value. The data analysis device 30 can output the display of the average value and the confidence interval in various forms. The display of the average value and the confidence interval is displayed, for example, on the screen of a display.

[0275] FIG. 38 is a diagram showing a first example of a display output by the data analysis device according to the present embodiment. In the first example, the average value and confidence interval of the estimated value of the relative concentration are displayed in the form of a graph. Therefore, the accuracy of the estimation result can be displayed in a form that is easy for the analyst to understand. The graph includes a horizontal axis (first axis) representing the depth from the surface of the sample, a vertical axis (second axis) representing the relative concentration of the chemical species, a curve (first figure) representing the average value of the estimated value of the relative concentration with respect to the depth, and an area (second figure) representing the confidence interval of the relative concentration with respect to the depth.

[0276] In the example of FIG. 38, the sample to be analyzed is a sample in which aluminum (Al), titanium (Ti), chromium (Cr), and nickel (Ni) are laminated on a silicon (Si) substrate in the order of proximity to the sample surface. In FIG. 15, a curve 101 and an area 102 represent the average value and confidence interval of the estimated value of the relative concentration of aluminum (Al) at each depth. The width of the vertical axis of the area 102 corresponds to the width of the confidence interval. Similarly, for other chemical species, a curve (first figure) representing the change in the average value μ with respect to the depth and an area (second figure) representing the change in the confidence interval with respect to the depth are displayed. An analyst (e.g., a user of the data analysis device) can easily grasp the accuracy of the estimation of the relative concentration of a certain chemical species (whether the estimation result by Bayesian MSM can be said to be "confident") from the width of the area representing the confidence interval. Therefore, according to the example of FIG. 38, the accuracy of the estimation result can be displayed so that the analyst can easily grasp the accuracy of the estimation result.

[0277] In the example of Fig. 38, the line type of the curve representing the average value of the estimated values ​​of the relative concentrations is different for each chemical species. However, the display of the curve is not limited to this. The curves and regions may be colored in different colors for each chemical species. This makes it possible to display the curves and regions so that the user can easily distinguish the curves and regions corresponding to each chemical species.

[0278] Fig. 39 is a diagram showing a second example of the display output by the data analysis device according to this embodiment. In the example shown in Fig. 39, the confidence interval of the estimated value of the relative concentration for each depth and each chemical species is expressed by a curve. In this respect, the display example shown in Fig. 39 differs from the example shown in Fig. 38.

[0279] For example, curve 101 represents the average value of the estimated value of the relative concentration for each depth when the chemical species is aluminum (Al). Curve 103A and curve 103B represent the upper and lower limits, respectively, of the confidence interval of the estimated value of the relative concentration for each depth when the chemical species is aluminum (Al). The area between the two curves corresponds to the confidence interval, and the vertical width of the area corresponds to the width of the confidence interval. This allows an analyst (e.g., a user of the data analysis device) to easily grasp the accuracy of the estimation of the relative concentration for a certain chemical species (whether the estimation result by Bayesian MSM can be said to be "confident") from the width of the area representing the confidence interval.

[0280] Similar to the display example shown in Fig. 38, the curves representing the average value of the estimated values ​​of the relative concentrations and the upper and lower limits of the confidence interval may be colored differently for each chemical species. Also, as shown in Fig. 39, the line types of the curves representing the average value of the estimated values ​​of the relative concentrations and the upper and lower limits of the confidence interval may be different for each chemical species.

[0281] Fig. 40 is a diagram showing a third example of the display output by the data analysis device according to this embodiment. The display example shown in Fig. 40 is basically the same as the display example shown in Fig. 38. The user places cursor 104 of mouse 42 (see Fig. 2) over a curve (e.g., curve 101) of a graph on the screen of display 43 (see Fig. 2). This causes error bar 105 to be displayed at the position on the curve pointed to by cursor 104. In this respect, the display example shown in Fig. 40 differs from the display example shown in Fig. 38.

[0282] The length of error bar 105 corresponds to the width of the confidence interval of the estimated value of the relative concentration corresponding to the position on the curve pointed to by cursor 104. According to the example shown in Fig. 40, since the width of the confidence interval is expressed by the length of the error bar, it is possible to display the accuracy of the estimated result so that the analyst can easily grasp the accuracy of the estimated result. Similarly, in the display example shown in Fig. 39, when cursor 104 is placed on the curve representing the average value of the estimated values, error bar 105 indicating the width of the confidence interval of the estimated value may be displayed at the position on the curve pointed to by cursor 104.

[0283] Fig. 41 is a diagram showing a fourth example of the display output by the data analysis device according to this embodiment. In the display example shown in Fig. 41, in addition to the display shown in Fig. 40, a text box 106 is displayed. The information displayed in the text box 106 includes, for example, the following information (1) to (5). Note that it is not necessary for all of the information (3) to (6) to be displayed in the text box, and at least one of (3) to (6) may be displayed in the text box.

[0284] (1) Chemical species corresponding to the curve pointed to by cursor 104 (2) The depth corresponding to the position on the curve pointed to by the cursor 104 (3) The average value μ of the estimated relative concentration corresponding to the position on the curve pointed to by the cursor 104 (4) Variance σ of the estimated relative concentration corresponding to the position on the curve pointed to by cursor 104 (5) The upper limit of the confidence interval (e.g. μ±3σ) corresponding to the error bar 105 (6) The lower limit of the confidence interval (e.g. μ±3σ) corresponding to the error bar 105 It should be noted that the display of information using text boxes is not limited to the display example shown in Fig. 41. The display of information using text boxes can be applied to any of the display examples shown in Fig. 38 to Fig. 40. The display of numerical values ​​makes it easier for analysts to understand the estimation results.

[0285] Alternatively, as shown in Figures 23, 26, 31 and 35, the width of the confidence interval for the estimated relative concentration may be represented by the overlap of multiple curves.

[0286] In the above display example, the average value of the estimated relative concentration is represented by a line, and the confidence interval of the estimated relative concentration is represented by a surface or a line. However, the figures for representing the average value of the estimated relative concentration and the confidence interval of the estimated relative concentration on the graph are not limited to lines or surfaces, and may be, for example, points or a solid.

[0287] Furthermore, the display of the mean value of the estimate of the relative concentration and the confidence interval of the estimate of the relative concentration may be in a form other than a graph.

[0288] Fig. 42 is a diagram showing a fifth example of the display output by the data analysis device according to this embodiment. In the example shown in Fig. 42, the mean value μ, variance σ, upper limit of the confidence interval (e.g. μ+3σ), and lower limit of the confidence interval (e.g. μ-3σ) of the estimated value of the relative concentration for each depth of the sample and each chemical species are output in the form of a numerical table. According to this example, the mean value and the confidence interval are displayed numerically, so that the analyst can specifically grasp the mean value and the confidence interval.

[0289] The table shown in Fig. 42 may be displayed on a display screen or output from a printer. The display in the form of a numerical table shown in Fig. 42 may be output alone or together with the graphs exemplified in Figs. 38 to 41. Furthermore, the data included in the table shown in Fig. 42 may be output to a recording medium as data in a format that can be displayed on a display screen (for example, CSV format).

[0290] As an example of an embodiment of the present disclosure, an example of analysis by EDX has been shown. However, the embodiment of the present disclosure is not limited to being applied to analysis by EDX. For example, the embodiment of the present disclosure can also be applied to analysis by XPS. In the case of XPS, the theoretical value d' of the response signal in formula (1) is ij is expressed by the following equation (37).

[0291]

number

[0292]

number

[0293] As described above, the theoretical value d' of the response signal in equation (1) ij By setting according to equations (37) and (38), the Bayesian MSM method described above can be applied to analysis by XPS. Furthermore, the theoretical value d' of the response signal in equation (1) can be calculated by ij By appropriately expressing , the Bayesian MSM according to the embodiments of the present disclosure is applicable to any kind of surface analysis, not limited to EDX and XPS.

[0294] Similarly, the Bayesian estimation of the depth profile according to the second embodiment is not limited to being applied to the analysis of EDX analysis data, but can also be applied to other analyses. The configuration of the forward problem evaluation unit 61 can be changed to a configuration appropriate for other analyses. More specifically, the formula (2) calculated by the measurement theoretical value construction unit 61B can be transformed into a form suitable for other analyses.

[0295] 17, the next parameter recommendation unit 62 recommends the next parameter θ based on a combination of the previously recommended parameter θ and the output y from the forward problem evaluation unit 61. Although the next parameter recommendation unit 62 treats the forward problem as a black box, the next parameter recommendation unit 62 can recommend the next parameter θ. Therefore, the depth profile estimation according to the second embodiment is not limited to EDX analysis and XPS analysis, but is applicable to all kinds of surface analysis.

[0296] <7. Notes> The present disclosure includes the following embodiments.

[0297] [Appendix 1] (1) A data analysis device for analyzing a depth profile of a sample using data of a response signal generated from the sample by the incidence of a probe, comprising: an input unit that receives data of the response signal obtained by a measurement device that measures the response signal while changing an incidence condition of the probe so as to evaluate different depth regions of the sample; an analysis unit that estimates a relative concentration for each depth from the surface of the sample and for each chemical species by analyzing the data of the response signal, and determines an average value and a confidence interval of the estimated values ​​of the relative concentrations; The analysis unit calculates the average value and the confidence interval of the estimated value by processing an evaluation function expressed by the sum of a first term expressing the sum of squared deviation between the theoretical value of the response signal when the sample is modeled as a laminate consisting of a plurality of layers and the data of the response signal, and a second term expressing the degree of continuity of the relative concentration at the depth, according to a Bayesian estimation process.

[0298] (2) The data analysis device according to (1), wherein the Bayesian estimation process includes a sampling algorithm using the relative concentrations for each depth and each chemical species as variables.

[0299] (3) The data analysis device according to (2), wherein the sampling algorithm is an algorithm based on the Metropolis method.

[0300] (4) The data analysis device according to claim (1), wherein the Bayesian estimation process is a process according to a variational inference method.

[0301] [Appendix 2] (1) A data analysis method for analyzing a depth profile of a sample using data of a response signal generated from the sample by incidence of a probe, comprising the steps of: a step of receiving, by a data analysis device, data of the response signal obtained by a measurement device measuring the response signal while changing an incidence condition of the probe so as to evaluate different depth regions of the sample; a step of estimating a relative concentration for each depth from the surface of the sample and for each chemical species by analyzing the data of the response signals by the data analysis device, and determining an average value and a confidence interval of the estimated values ​​of the relative concentrations; and the data analysis device outputting a representation of the mean and the confidence interval of the estimate of the relative concentration; the representation of the mean and the confidence interval is in the form of a graph; A data analysis method, wherein the graph includes a first axis indicating the depth, a second axis indicating the relative concentration, a first graph representing the change in the average value with respect to the depth, and a second graph representing the change in the confidence interval with respect to the depth.

[0302] (2) The first figure is a curve, The data analysis method according to (1), wherein the second figure includes an area that overlaps with the curve.

[0303] (3) The data analysis method according to claim (1), wherein the second figure includes a first curve representing an upper limit of the confidence interval and a second curve representing a lower limit of the confidence interval.

[0304] (4) the second graphic includes error bars; The data analysis method according to any one of (1) to (3), wherein the length of the error bar corresponds to the width of the confidence interval of the estimated value at the specified position on the first figure.

[0305] (5) A data analysis method described in any of (1) to (4), wherein the graph includes a display of at least one numerical value of the average value, the upper limit of the confidence interval, and the lower limit of the confidence interval at a specified position on the first figure.

[0306] (6) The data analysis method according to any one of (1) to (5), wherein the step of outputting the display includes outputting the display of the mean value and the confidence interval in the form of a numerical table.

[0307] (7) The step of determining the average value and the confidence interval includes a step of calculating the average value and the confidence interval of the estimated value by processing an evaluation function according to a Bayesian estimation process; The data analysis method according to any one of (1) to (6), wherein the evaluation function is expressed by the sum of a first term expressing a sum of squared deviation between a theoretical value of the response signal when the sample is modeled as a laminate consisting of a plurality of layers and the data of the response signal, and a second term expressing a degree of continuity of the relative concentration at the depth.

[0308] (8) The data analysis method according to (7), wherein the Bayesian estimation process includes a sampling algorithm using the relative concentrations for each depth and each chemical species as variables.

[0309] (9) The data analysis method according to (7), wherein the Bayesian estimation process is a process according to variational inference.

[0310] (10) The data analysis method according to any one of (1) to (9), wherein the measurement device is an energy dispersive X-ray analyzer.

[0311] (11) The data analysis method according to any one of (1) to (9), wherein the measurement device is an X-ray photoelectron spectrometer.

[0312] [Appendix 3] (1) A data analysis method for analyzing a depth profile of a sample using data of a response signal generated from the sample by incidence of a probe, comprising the steps of: a step of receiving, by a data analysis device, data of the response signal obtained by a measurement device measuring the response signal while changing an incidence condition of the probe so as to evaluate different depth regions of the sample; the data analysis device estimates a relative concentration for each depth from the surface of the sample and for each chemical species by analyzing the data of the response signals, and determines an average value and a confidence interval of the estimated values ​​of the relative concentrations; the step of determining the mean value and the confidence interval includes a step of calculating the mean value and the confidence interval of the estimated value by processing an evaluation function according to a Bayesian estimation process; a data analysis method in which the evaluation function is expressed by the sum of a first term expressing a sum of squared deviation between a theoretical value of the response signal when the sample is modeled as a stack consisting of a plurality of layers and the data of the response signal, and a second term expressing a degree of continuity of the relative concentration at the depth.

[0313] (2) The data analysis method according to (1), wherein the Bayesian estimation process includes a sampling algorithm using the relative concentrations for each depth and each chemical species as variables.

[0314] (3) The data analysis method according to (2), wherein the sampling algorithm is an algorithm based on the Metropolis method.

[0315] (4) The data analysis method according to claim (1), wherein the Bayesian estimation process is a process according to variational inference.

[0316] [Appendix 4] (1) A data analysis method for analyzing a depth profile of a sample using data of a response signal generated from the sample by incidence of a probe, comprising the steps of: a step of receiving, by a data analysis device, data of the response signal obtained by a measurement device measuring the response signal while changing an incidence condition of the probe so as to evaluate different depth regions of the sample; the data analysis device estimates a relative concentration for each depth from the surface of the sample and for each chemical species by analyzing the data of the response signals, and determines an average value and a confidence interval of the estimated values ​​of the relative concentrations; The step of determining the mean value and the confidence interval comprises: A data analysis method comprising a step of determining, according to Bayesian estimation, a posterior distribution of parameters included in a predetermined function that expresses a model of the sample with restricted degrees of freedom of the depth profile.

[0317] (2) the model of the sample is a laminate consisting of a plurality of layers, the predetermined function is a function that expresses that the depth profile has a rise and a fall at the boundary of each of the plurality of layers; The parameters are: A composition of the sample; a thickness of each of the plurality of layers; and the gradual rise of the depth profile of each of the plurality of layers; The data analysis method according to (1), further comprising determining whether the depth profile of each of the plurality of layers is gradually declining.

[0318] (3) the step of determining the posterior distribution of the parameters according to the Bayesian estimation includes a step of calculating a sum of squared deviations between actual measured values ​​and theoretical values ​​by using the values ​​of the parameters in a forward problem model; the step of calculating a probability distribution of the parameter includes a step of recommending a candidate value of the parameter to be used in calculating a posterior distribution of the parameter by using the calculated sum of squared deviations; The data analysis method according to (1) or (2), wherein the calculation of the posterior distribution and the recommendation of the candidate values ​​of the parameters are repeated alternately.

[0319] (4) the step of recommending candidate values ​​for the parameters includes: proposing the candidate values ​​for the parameters using a predefined probability distribution according to a Markov Chain Monte Carlo (MCMC) method; and determining an optimal initial value for the Markov chain Monte Carlo method using the calculated sum of squared deviations.

[0320] The embodiments disclosed herein are illustrative in all respects and should not be considered as limiting. The scope of the present invention is defined by the claims, not by the embodiments described above, and is intended to include the equivalent meanings of the claims and all modifications within the scope of the claims. [Explanation of symbols]

[0321] 10 Analysis system, 20 Measurement device, 25 Sample, 30 Data analysis device, 31 Processor, 32 Primary storage device, 33 Secondary storage device, 34 External device interface, 35 Input interface, 36 Output interface, 37 Communication interface, 38 Bus, 41 Keyboard, 42 Mouse, 43 Display, 44 Network, 45 Printer, 51 Input section, 52 Analysis section, 53 Output section, 54 Memory section, 61 Forward problem evaluation section, 61A Profile construction section, 61B Measurement theoretical value construction section, 61C Sum of squared deviation evaluation section, 62 Next parameter recommendation section, 62A Optimal solution judgment section, 62B MCMC implementation section, 71 Analysis program, 72 Parameters, 101, 103A, 103B Curve, 102 Area, 104 Cursor, 105 Error bar, 106 Text boxes, S11~S13, S12A, S21~S28, S26A~S26D steps.

Claims

1. 1. A data analysis device for analyzing a depth profile of a sample using data of a response signal generated from the sample by incidence of a probe, comprising: an input unit that receives data of the response signal obtained by a measurement device that measures the response signal while changing an incidence condition of the probe so as to evaluate different depth regions of the sample; an analysis unit that determines, by analyzing the data of the response signals, expected values ​​of estimated values ​​of relative concentrations for each depth from the surface of the sample and for each chemical species, and a confidence interval of the estimated values ​​by Bayesian estimation based on a likelihood function that treats the relative concentrations as random variables and assumes that the data of the response signals varies around a theoretical value of the response signals; and an output section that outputs a representation of the expected value and the confidence interval of the estimate of the relative concentration.

2. the representation of the expected values ​​and the confidence intervals is in the form of a graph; 2. The data analysis device according to claim 1, wherein the graph includes a first axis indicating the depth, a second axis indicating the relative concentration, a first graph representing the change in the expected value with respect to the depth, and a second graph representing the change in the confidence interval with respect to the depth.

3. the first figure is a curve, The data analysis device according to claim 2 , wherein the second graphic includes an area overlapping with the curve.

4. 3. The data analysis apparatus according to claim 2, wherein the second graphic includes a first curve representing an upper limit of the confidence interval and a second curve representing a lower limit of the confidence interval.

5. the second graphic includes an error bar; The data analysis apparatus of claim 2 , wherein the length of the error bar corresponds to a width of the confidence interval of the estimate at a specified position on the first graph.

6. The data analysis apparatus according to claim 2 , wherein the graph includes a display of at least one numerical value of the expected value, an upper limit value of the confidence interval, and a lower limit value of the confidence interval at a specified position on the first graphic.

7. The data analysis apparatus according to claim 1 , wherein the output unit outputs the display of the expected value and the confidence interval in the form of a numerical table.

8. the analysis unit processes the evaluation function according to a Bayesian estimation process to calculate the expected value and the confidence interval of the estimated value; 3. The data analysis device according to claim 2, wherein the evaluation function is expressed by the sum of a first term expressing a sum of squared deviation between a theoretical value of the response signal when the sample is modeled as a laminate consisting of a plurality of layers and the data of the response signal, and a second term expressing a degree of continuity of the relative concentration at the depth.

9. The data analysis device according to claim 8 , wherein the Bayesian estimation process includes a sampling algorithm using the relative concentrations for each depth and each chemical species as variables.

10. The data analysis device according to claim 8 , wherein the Bayesian estimation process is a process according to a variational inference method.

11. 3. The data analysis device according to claim 2, wherein the analysis unit determines the expected value and the confidence interval of the estimated value of the relative concentration by determining, according to Bayesian estimation, a posterior distribution of parameters included in a predetermined function that represents a model of the sample in which the degrees of freedom of the depth profile are restricted.

12. the model of the sample is a laminate of multiple layers, the predetermined function is a function that expresses that the depth profile has a rise and a fall at the boundary of each of the plurality of layers; The parameters are: A composition of the sample; a thickness of each of the plurality of layers; and the gradual rise of the depth profile of each of the plurality of layers; The data analysis apparatus of claim 11 , further comprising a gradual decline in the depth profile of each of the plurality of layers.

13. The analysis unit is a forward problem evaluation unit that calculates a sum of squared deviations between actual measured values ​​and theoretical values ​​by using the parameter values ​​as an input of a forward problem model; a next parameter recommendation unit that uses the sum of squared deviations calculated by the forward problem evaluation unit to recommend to the forward problem evaluation unit candidate values ​​of the parameters to be used in a next calculation by the forward problem evaluation unit, 13. The data analysis device according to claim 12, wherein the calculation of the sum of squared deviations by the forward problem evaluation section and the recommendation of the candidate value of the parameter by the next parameter recommendation section are repeated alternately.

14. The next parameter recommendation unit: an MCMC implementation that proposes candidate values ​​for the parameters using a predetermined probability distribution according to a Markov Chain Monte Carlo method; and an optimal solution determination unit that determines an initial value for the Markov Chain Monte Carlo method executed by the MCMC implementation unit by executing a predetermined optimization method using the sum of squared deviations calculated by the forward problem evaluation unit.

15. The data analysis apparatus according to claim 1 , wherein the measurement device is an energy dispersive X-ray analyzer.

16. The data analysis device according to claim 1 , wherein the measurement device is an X-ray photoelectron spectroscopy device.

17. A data analysis method for analyzing a depth profile of a sample using data of a response signal generated from the sample by incidence of a probe, the method comprising: a step of receiving, by a data analysis device, data of the response signal obtained by a measurement device measuring the response signal while changing an incidence condition of the probe so as to evaluate different depth regions of the sample; a step of the data analysis device analyzing the data of the response signals to determine expected values ​​of estimated values ​​of relative concentrations for each depth from the surface of the sample and for each chemical species and a confidence interval of the estimated values ​​by Bayesian estimation based on a likelihood function assuming that the relative concentrations are treated as random variables and that the data of the response signals varies around a theoretical value of the response signals; and wherein the data analysis apparatus outputs an indication of the expected value and the confidence interval of the estimate of the relative concentration.

18. On the computer, acquiring response signal data obtained by a measurement device that measures response signals while varying the incidence conditions of the probe to evaluate different depth regions of the sample; determining an expected value of an estimated value of a relative concentration for each depth from the surface of the sample and for each chemical species and a confidence interval of the estimated value by Bayesian estimation based on a likelihood function assuming that the relative concentration is treated as a random variable and that the data of the response signal varies around a theoretical value of the response signal by analyzing the data of the response signal; and outputting an indication of the expected value and the confidence interval of the estimate of the relative concentration.

19. On the computer, acquiring response signal data obtained by a measurement device that measures response signals while varying the incidence conditions of the probe to evaluate different depth regions of the sample; determining an expected value of an estimated value of a relative concentration for each depth from the surface of the sample and for each chemical species and a confidence interval of the estimated value by Bayesian estimation based on a likelihood function assuming that the relative concentration is treated as a random variable and that the data of the response signal varies around a theoretical value of the response signal by analyzing the data of the response signal; and outputting a display of the expected value and the confidence interval of the estimated value of the relative concentration.

20. The likelihood function is The sample is modeled as a stack of layers having different relative concentrations; The probe incident on the sample is attenuated within the sample, the response signal generated from the plurality of layers by the incidence of the probe reaches the surface of the sample while being attenuated within the sample; a measured value of the response signal varies with respect to the theoretical value of the response signal obtained by adding up contributions from all of the plurality of layers, depending on the reliability of the measurement result obtained by the measurement device; 2. The data analysis apparatus according to claim 1, wherein the expression is a representation of a phenomenon model as follows:

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