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

Abstract

The data analysis device according to the present disclosure analyzes a depth profile of a sample by using response signal data generated from the sample by the incidence of a probe. This data analysis device comprises: an input unit that receives response signal data obtained by a measurement device that measures a response signal while changing the probe incidence conditions so as to evaluate different depth regions of the sample; an analysis unit that, by analyzing the response signal data, estimates a relative concentration for each chemical species for each depth from the surface of the sample, and that determines average values and confidence intervals for the estimated relative concentration values; and an output unit that outputs a display of the average values and confidence intervals of the estimated relative concentration values.

Description

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

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

[0002] When discussing the characteristics and defects of various products, it is often crucial to understand 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 important.

[0003] Depth profile evaluation techniques include scanning transmission electron microscopy (STEM), energy dispersive X-ray analysis (EDX), and X-ray photoelectron spectroscopy (XPS). However, these techniques can alter the state of the sample when evaluating its depth profile. 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 at which information can be evaluated is typically on the order of a few nanometers 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 can potentially damage the sample surface.

[0004] As a method capable of evaluating the depth profile of a sample nondestructively, a method has been proposed in which a probe is incident on the sample and a response signal generated from the sample is used to estimate the depth profile of the sample. For example, Yoshiki Yonamoto, "Application of Maximum Entropy Method to Semiconductor Engineering," Entropy 2013, 15, 1663-1689; doi: 10.3390 / e15051663 (Non-Patent Document 1) discloses an analytical technique that applies the maximum entropy method (MEM) to data obtained from angle-resolved X-ray photoelectron spectroscopy (ARXPS) without sputtering. Furthermore, WO 2022 / 230112 (Patent Document 1) describes a recently developed analytical method called the Maximum Smoothness Method (MSM).

[0005] International Publication No. 2022 / 230112

[0006] “Application of Maximum Entropy Method to Semiconductor Engineering”, Yoshiki Yonamoto, Entropy 2013, 15, 1663-1689; doi:10.3390 / e15051663

[0007] A data analysis device according to one aspect of the present disclosure is a data analysis device that analyzes the depth profile of a sample using data on response signals 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 the relative concentration for each depth from the surface of the sample and for each chemical species, and calculates the average value and confidence interval of the estimated values ​​of the relative concentrations, and an output unit that outputs a display of the average value and confidence interval of the estimated values ​​of the relative concentrations.

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

[0009] [Problem to be Solved by the Present Disclosure] In MEM or MSM, the depth profile of a sample is estimated and the estimation result is output. Conventionally, analysts have not been able to know the accuracy of the estimation result. Since MEM or MSM is a "method for evaluating" a profile, it is necessary to output some degree of accuracy, as with various other analytical methods. From the viewpoint of preventing erroneous evaluation by analysts, it is desirable for analysts to be able to grasp the accuracy of the estimation result.

[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] Effect of the Present 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 Embodiments of the Present Disclosure] First, embodiments of the present disclosure will be listed and described.

[0013] (1) A data analysis device according to an embodiment of the present disclosure is a data analysis device that analyzes a depth profile of a sample using data on response signals generated from the sample by incident of a probe, and includes an input unit that accepts response signal data obtained by a measurement device that measures response signals while 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 determines the average value and confidence interval of the estimated values ​​of the relative concentrations, and an output unit that outputs a display of the average value and confidence interval of the estimated values ​​of the relative concentrations.

[0014] This configuration allows an analytical method for estimating a depth profile of a sample based on a response signal generated from the sample upon incident probe light to display the accuracy of the estimated results. The analyst (e.g., a user of a data analysis device) can grasp the accuracy of the estimated results by displaying the average value and confidence interval. Note that "outputting a display" refers to outputting the analysis results in a format that is recognizable by the user. Therefore, "outputting a display" includes displaying the analysis results, i.e., the depth profile and its accuracy, on a display screen in the form of a graph, numerical data, a table, or the like. The analysis results, i.e., the depth profile and its accuracy, may also be output on paper in the form of a graph, numerical data, a table, or the like. Alternatively, for example, a numerical string representing the analysis results may be temporarily output (saved) to a storage device or storage medium so that the analysis results can be displayed later. In the embodiments of the present disclosure, "outputting a display" also includes outputting the analysis results so that they can be displayed at a later stage.

[0015] (2) In the data analysis device described in (1), the display of the mean value and the confidence interval is in the form of a graph, and the graph includes a first axis indicating depth, a second axis indicating relative concentration, a first figure representing the change in the mean value with respect to depth, and a second figure representing the change in the confidence interval with respect to 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, and therefore 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 according to any one of (2) to (4) above, the second graphic includes an error bar, and 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 graphic.

[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 one of (2) to (5) above, the graph includes a display of at least one numerical value of the mean 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 numerically, making it easier for the analyst to understand the estimation results.

[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 numerically, so that the analyst can specifically grasp the average value and the confidence interval. The display in the form of a numerical 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 Bayesian estimation processing, and the evaluation function is expressed by the sum of a first term expressing the sum of squares of the deviation between the theoretical value of the response signal and the response signal 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.

[0028] According to this configuration, the analysis unit executes MSM based on the concept of Bayesian estimation. MSM processes an evaluation function expressed by the sum of a first term representing 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 representing the degree of continuity of the relative concentration across depth. Each point on the depth profile represented by this evaluation function essentially has concepts equivalent to the mean (μ) and variance (σ). Conventional MSMs were unable to display the mean (μ) and variance (σ). 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 concentrations 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 value.

[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 value.

[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 the posterior distribution of the parameters included in a predetermined function that represents a model of the sample with limited degrees of freedom of the depth profile according to Bayesian estimation.

[0034] This configuration limits the flexibility of the depth profile as a prerequisite for analyzing the depth profile. This allows for a user-friendly analysis method. For example, when the analysis unit calculates the mean value and confidence interval of the relative concentration from the probability distribution of parameters, the calculation time required for the calculated values ​​to reach reasonable values ​​can be shortened. Furthermore, by incorporating known elements of the profile, analysis can be performed that is specialized for only the unknown parts.

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

[0036] This configuration allows the degree of freedom of the depth profile to be limited in the estimation of the depth profile of the relative concentrations of chemical species, and also allows appropriate parameters to be selected from the above parameters depending on 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 the 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 next calculation of the forward problem evaluation unit, and the calculation of the sum of squared deviations by the forward problem evaluation unit and the recommendation of candidate values ​​of parameters by the next parameter recommendation unit are alternately repeated.

[0038] According to this configuration, the calculation of the sum of squared deviations between actual values ​​and theoretical values ​​by the forward problem evaluation unit (evaluation of the forward problem) and the recommendation of candidate parameter values ​​by the next parameter recommendation unit are alternately repeated. This makes it possible to determine the average value and confidence interval of the estimated parameter values. 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 relying on the contents of the forward problem evaluation unit. By changing the forward problem to be solved by the forward problem evaluation unit depending on the type of surface analysis, a data analysis device compatible with that surface analysis can be configured. In other words, the analysis method according to this embodiment is not particularly limited to any particular 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] With 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 for the Markov Chain Monte Carlo (MCMC) method. By using this 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 correct solution distribution 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 analyzer. Note that the above-mentioned "energy dispersive X-ray analyzer" 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 analyzer" 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 analyzer" is a device that performs "analysis in which an X-ray beam is irradiated and characteristic X-rays are observed" (X-ray fluorescence 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 mean value and variance of the estimated depth profile of the sample can be obtained from the response signal from the X-ray photoelectron spectroscopy (XPS) instrument.

[0045] (17) A data analysis method according to an embodiment of the present disclosure is a data analysis method for analyzing a depth profile of a sample using data on response signals generated from the sample by incident of 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 response signals while changing probe incident conditions to evaluate different depth regions of the sample; analyzing, by the data analysis device, the response signal data, to estimate relative concentrations for each depth from the surface of the sample and for each chemical species, 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 analytical 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, and an analyst (e.g., a user of the data analysis device) can understand 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 while 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.

[0048] According to this configuration, in an analytical 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, and an analyst (e.g., a user of the data analysis device) can understand 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 data on response signals obtained by a measurement device that measures response signals while changing 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, and 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 analytical 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, and an analyst (e.g., a user of the data analysis device) can understand the accuracy of the estimation result by displaying the average value and the confidence interval.

[0051] [Details of the Embodiments of the Present Disclosure] Hereinafter, embodiments of the present disclosure will be described with reference to the drawings. In the drawings, the same or corresponding parts are designated by the same reference numerals, and description thereof will not be repeated.

[0052] 1. Configuration of Analysis System and Analysis Apparatus> Fig. 1 is a diagram showing an analysis system including an analysis apparatus according to an embodiment of the present disclosure. As shown in Fig. 1, the analysis system 10 includes a measurement apparatus 20 and a data analysis apparatus 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 conditions under which the probe is irradiated onto the sample 25 so as to evaluate different depth regions of the sample 25.

[0054] The embodiments of the present disclosure are applicable to any surface analysis without limiting the type of measurement device 20. The measurement device 20 may be any device that irradiates a probe onto 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) analyzer. The measurement device 20 may be, for example, an angle-resolved photoelectron spectroscopy (e.g., ARXPS) device or an X-ray fluorescence (XRF) device.

[0055] The data analyzer 30 is realized by hardware conforming to a general-purpose computing architecture. The data analyzer 30 acquires a response signal, i.e., measurement data, from the measurement device 20. The data analyzer 30 analyzes the depth profile of the sample 25 based on the measurement data. Specifically, the data analyzer 30 estimates the relative concentration of each chemical species at each depth in the sample 25 by analyzing the measurement data. The data analyzer 30 outputs a display of the average value of the estimated relative concentration of each chemical species at each depth in the sample 25 and a confidence interval of the estimated value. This allows the data analyzer 30 to display the accuracy of the estimated depth profile of each chemical species to a user (not shown) of the data analyzer 30. The analyst (i.e., the user of the data analyzer) can understand the accuracy of the estimated result by displaying 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, the data analysis device 30 includes a processor 31, a primary storage device 32, a secondary storage device 33, an external device interface 34, an input interface 35, an output interface 36, a communication interface 37, and a bus 38.

[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 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 processor 31. 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 nonvolatile manner. In some aspects, a nonvolatile 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 to be 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 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 by the data analysis device 30 to communicate with external devices. For example, the communication interface 37 is used for communication of the data analysis device 30 via a network 44. The network 44 is, for example, a local area network (LAN). 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 include an optional 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 (e.g., an optical recording medium such as a DVD (Digital Versatile Disc)). The reading device can read 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, various programs executed by the data analysis device 30 may be downloaded from a server device (not shown) or the like via a network (e.g., 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 apparatus 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 also receives various information (e.g., information on the type of material 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. 2 .

[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. 2.

[0068] The analysis unit 52 executes the analysis program 71 to determine the depth profile and its accuracy of the sample 25. Specifically, the analysis unit 52 determines the average value of the estimated values ​​of the relative concentrations of each chemical species at each depth of the sample 25 and in accordance with the analysis method described below, and the confidence interval of the estimated values. 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 results obtained by the analysis unit 52. "Outputting a display" means outputting the analysis results in a form recognizable to the user. For example, the output unit 53 can be implemented by the output interface 36 shown in FIG. 2. For example, the analysis results, i.e., the depth profile and its accuracy, may be displayed on the screen of the display 43 (see FIG. 2) in the form of a graph, numerical data, a table, or the like. The graph, numerical data, a table, or the like representing the analysis results may be printed on paper by the printer 45 (see FIG. 2). Alternatively, the analysis results may be temporarily output (saved) in the form of a numerical string to a storage device or storage medium so that they can be displayed (e.g., displayed as a graph) by software. In the embodiments of the present disclosure, "outputting a display" also includes outputting the analysis results so that they can be displayed at a later stage.

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

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

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

[0073] φ ij (ρz ｋ ) is the mass depth ρz ｋ 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 concentrations of the chemical species of the sample is analyzed by minimizing the sum of squares of the deviations between the theoretical response signals and the measured response signals when the sample to be analyzed is modeled as a stack of multiple layers. In MSM, the relative concentrations are calculated to satisfy the maximum smoothness condition that the relative concentrations of the chemical species of the sample change smoothly among the multiple layers of the stack in the minimization of the sum of squares of the deviations.

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

[0076] In formula (1), d' ij is the theoretical measured intensity for species i at accelerating voltage j, and d ij is d' ij is the measured intensity data for chemical species i at accelerating voltage j corresponding to ik represents the relative concentration of chemical species i in the kth layer. ik is 0≦cik ≦1 and Σ i c ik = 1. I represents the total number of chemical species, J represents the total number of levels of acceleration voltage, and K represents the total number of layers.

[0077] The first term in equation (1) represents the sum of squares of the deviation between the theoretical value of the response signal and the measured value of the response signal. The second term in equation (1) represents how smoothly the profile of the relative concentration of chemical species i changes in the depth direction. The parameter λ s is a parameter that indicates how strongly the smoothness of the profile is required 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).

[0079] The vector d' on the left side of equation (2) is an (I x J) row vector that arranges the theoretical EDX measurement intensities of all chemical species and all angles in a single column. The vector c on the right side of equation (2) is an (I x 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).

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

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

[0082] In equation (5), σ ij ω i φ ij (ρz k ) is the electron beam with the acceleration voltage of level j irradiated onto the sample, and the electron beam reaches the kth layer (mass depth ρz ｋ) represents the generation function of the characteristic X-rays of chemical species i. i is the fluorescence yield of the signal of interest for the generation function of characteristic X-rays, and is an index that indicates how easily characteristic X-rays are emitted when the chemical species is ionized. The exp term in equation (5) represents the attenuation of characteristic X-rays. In equation (5), (μ / ρ) i1 , (μ / ρ) i2 , ... (μ / ρ) ik-1 represent the mass absorption coefficients of the characteristic X-rays of chemical species i in the first layer, the second layer, . . . , the (k-1)th layer, respectively.

[0083] In Equation (5), t is the mass thickness of each layer, and θ is the take-off angle of the characteristic X-ray. In EDX, the relative angle between the detector and the sample is usually fixed, so the angle θ is determined depending on the configuration of the device.

[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]

[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 an 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 allows the MSM to be applied to various situations requiring analysis of the depth profile of a sample, such as device development and failure analysis.

[0088] 3. The Necessity of Accuracy Evaluation Figure 5 shows an example of the analysis results of EDX measurement using a conventional MSM. The "conventional MSM" refers to the MSM described above. The sample to be analyzed is a silicon (Si) substrate with aluminum (Al), titanium (Ti), chromium (Cr), and nickel (Ni) layered on it in order from the surface of the sample.

[0089] The analysis results shown in Figure 5 show the distribution of 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 concentrations of each chemical species and the accuracy of the analysis results. As shown in Figure 6, the analysis results of the relative concentrations of each chemical species should have a range that depends on the accuracy of the analysis. However, the display shown in Figure 5 may lead a user to believe that the error of the analysis results is zero. It is expected that such a perception will affect the interpretation of the analysis results.

[0091] Therefore, when analyzing a depth profile using 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 for 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 an MSM. To distinguish it from a conventional MSM (an MSM that does not output accuracy), an MSM to which Bayesian estimation is applied is referred to as a "Bayesian MSM" in this specification.

[0093] In conventional MSM, the relative concentration c ik Assuming that there exists a true value of , we search for one solution that is most likely from the 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 use the equation (1) to calculate the relative concentration c ikThe 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 use the equation (1) to ik Considering it as a distribution function for each c ik In the Bayesian MSM, Equation (1) is formulated as a posterior distribution in the framework of Bayesian estimation.

[0095] Bayesian estimation is generally performed in the following steps: 1. Assume a prior distribution of the parameters. 2. Assume a likelihood function of the observed data. 3. Calculate the posterior probability using Bayes' theorem. 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. Assumption of Prior Distribution of Parameters) The prior distribution for the relative concentration c of a chemical species is represented as p(c), the likelihood function when measurement data d is obtained from a certain depth profile is represented as p(d|c), and the posterior distribution of the depth profile after obtaining the measurement data d is represented 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 equation (8) below:

[0099] Each relative concentration is assumed to be a normal distribution centered on the relative concentration of the layer immediately adjacent to the surface side of that layer. The concentration distribution of chemical species at the outermost surface or the 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), σs 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 the value, the smoother the profile changes at the layer boundaries.

[0101] (2. Assumption of Likelihood Function of Observation Data) It is assumed that the measurement 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).

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

[0104] After the data is observed, 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), it is sufficient to consider only the numerator on the right-hand side of equation (10).

[0105] Here, by taking the logarithm of the numerator on the right side of equation (10), the following equation (11) is obtained.

[0106] In reality, the accuracy of each individual measurement level is often unknown. For simplicity, we will define the accuracy of each measurement level as a constant value σ d The variance between neighboring depth profiles is σ S The ratio λ S is defined according to equation (12).

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

[0108] The equation in brackets [ ] on the right side of equation (13) is the same as equation (1). In MSM, the c ik Find the set of .

[0109] Since the left side of equation (11) is a negative logarithm of the posterior distribution p(c|d), minimizing the right side of equation (11) corresponds to maximizing the posterior distribution p(c|d). In other words, conventional MSM corresponds to a MAP estimation (maximum a posteriori estimation) 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) Essentially, each point on the depth profile has concepts equivalent to a mean (μ) and a 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 information about the mean and variance 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 the prior probability distribution in Bayesian estimation.

[0112] <<4.2 Bayesian MSM by Sampling>> In sampling, the distribution function is assigned a variable, the relative concentration c ik Input a large number of pairs of values ​​and calculate the output value from the distribution function. ik When creating pairs of , if it can be said that the number of pairs is sufficient if certain rules are followed, 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. Also, the results obtained by sampling tend to be difficult to understand qualitatively.

[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 in a realistic time.

[0115] In Bayesian estimation, MCMC (Markov Chain Monte Carlo) is often used as a method for approximating a posterior distribution when it is difficult to calculate the posterior distribution. MCMC is a method that cleverly realizes the mechanism that "where a function is large (small), values ​​occur with a relatively high (low) frequency" for high-dimensional systems. In one embodiment, the Metropolis method, the most basic MCMC method, can be employed. 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 performed 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 (a vector having as many elements as the number of dimensions) c0. From the viewpoint of calculation efficiency, it is preferable to set the solution obtained by the conventional MSM as the initial value c0.

[0118] 2. Set the initial value c0 to a real value b ik In MSM, the relative concentration c ij In the Metropolis method, it is preferable to generate random numbers freely according to, for example, a normal distribution. However, the relative concentration c ij In the ik ≦1 and Σ i c ikThe restriction that ρ = 1 is imposed. 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, a real value b that takes an arbitrary range ik Consider a variable transformation using a softmax function as shown in equation (14).

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

[0121] Real value b ik From the 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 of the sample, a real value b ik to a constant A k Add.

[0122] Real value b ik Even if the relative concentration c in Equation (14) is determined according to Equation (15), ij The value of b remains unchanged. ik But the constant A k This means that there is a degree of freedom to shift the value by the amount of b. ik It is convenient to set the mean value for 0.

[0123] Therefore, the relative concentration c ij to real value b ik In the conversion to k By adding ik The average value of is adjusted to 0.

[0124] 3. Set the proposal distribution. In theory, any function can be applied to the proposal distribution. In one embodiment, a multivariate Gaussian distribution expressed by the following equation (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 between chemical species and depths).

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

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

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

[0128] 6. If the calculated value of formula (17) is 0 or greater, 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 not adopted.

[0130] 7. Updated c n+1 Use to return to step 4. Steps 4 to 6 are repeated the specified number of times.

[0131] 8. By repeating steps 4 to 6 up to the specified number of times, the relative concentration estimates (relative concentration c ik ) and calculate the average value 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 whose mean and variance are easy to grasp, and the approximated function is used to evaluate the mean and variance. Variational inference has a limited range of application compared to the MCMC method described above. However, variational inference generally has low computational cost. 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).

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

[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 equation (21) 1k is adjusted to make the approximate posterior distribution q(c) expressed by equation (21) 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, the following describes KL (Kullback-Leibler) divergence, which is often used in Bayesian estimation. Note that the method for 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 also be used.

[0138] Using 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).

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

[0140] On the other hand, the second term on the right side of equation (22) is a term corresponding 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.

[0141] Therefore, by rearranging and calculating each content of Σ in equation (24), equation (25) is obtained.

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

[0143] Although no major problems arise if equation (25) is used as is, from the viewpoint of speeding up calculations, equation (25) is approximated as follows:

[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 αIf ≠1, there is no difference from the third equation. In the case of the upper and middle equations, the error is allowed since it is about 1 / (IK) of the whole and about 1 / K of the whole, and the upper and middle equations are considered to be the same as the lower equation. Then, equation (25) can be expressed as the following equation (27).

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

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

[0147] That is, when performing variational inference using KL divergence, Equation (29) is changed to the parameter a ik It is sufficient to minimize with respect to

[0148] When no prior distribution of the profile is assumed (a uniform distribution), the MSM function is simply the deviation from the measured data, as shown in equation (30).

[0149] In this case, the KL divergence is (sum of squared deviations from the measured data) - (system entropy), which is consistent with the maximum entropy method (MEM). In other words, the Bayesian MSM based on the variational inference described above 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, encompassing them as special cases.

[0150] <<4.4 Processing Flow of Bayesian MSM by the Data Analysis Apparatus>> Figure 7 is a flowchart showing an example of the processing flow of the data analysis method (Bayesian MSM) executed by the data analysis apparatus according to the first embodiment. The analysis unit 52 shown in Figure 3 executes the analysis program stored in the storage unit 54. This executes the processing of the flowchart shown in Figure 7.

[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 in response to the incidence of the probe.

[0152] In step S12, the analysis unit 52 executes 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 the relative concentration c for each depth from the surface of the sample and for each chemical species. ik We estimate the mean value μ and the confidence interval of the estimate.

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

[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 values ​​and confidence intervals of the estimated values ​​of the relative concentrations 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 an 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, except that 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 the relative concentrations c ik The analysis unit 52 calculates the average value μ and standard deviation σ of the estimated values ​​of the parameter a 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 variational inference, and may be configured, for example, to be able to select either the sampling method or 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 determined in advance. Alternatively, in step S21, the analysis unit 52 may determine the initial value of the relative concentration by executing a conventional MSM analysis method. The conventional MSM analysis method is a method of converging the value of the relative concentration c by repeating the following steps: obtaining an optimal solution c of Equation (6), updating the matrix S using the optimal solution c, and obtaining the next optimal solution c. Relative concentration c ｉｋ If the change in is sufficiently small (for example, equal to or less than a predetermined threshold), the analysis unit 52 can determine that the solution has converged, and the relative concentration c ｉｋcan 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 preset function. For example, the multivariate Gaussian distribution expressed 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 n-th (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 A 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 formula (17) is equal to or greater than 0. If the calculated value of formula (17) is equal to or greater than 0, in step S26A, the analysis unit 52 calculates 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 determines whether the calculated value of formula (17) is equal to or greater than log r. If the calculated value of formula (17) is equal to or greater than the value of log r, in step S26A, the analysis unit 52 n+1 If the calculated value of equation (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 processing of step S26A or step S26D, in step S27, the analysis unit 52 determines whether the processing of steps S24 to S26A or S26D has been performed a specified number of times. The number of times can be set arbitrarily by the user. If the number of times the processing has been performed is less than the specified number of times, the processing returns to step S24. On the other hand, if the number of times the processing has been performed reaches the specified number of times, in step S28, the analysis unit 52 calculates the estimated value of the relative concentration for each depth and for each chemical species (i.e., the relative concentration c ik ) and calculate the average value and standard deviation. When the process of step S28 is completed, the process shown in FIG.

[0167] 4.5 Example of Analysis by Bayesian MSM Figure 10 shows the structure of a model sample used to verify the analysis results by Bayesian MSM. As shown in Figure 10, a model sample 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 used as input to the Bayesian MSM executed by the data analyzer 30. The data analyzer 30 executed the process according to the flowcharts shown in FIGS. 7 and 8 to output the average values ​​and confidence intervals of the estimated values ​​of the relative concentration for each depth and each chemical species using both the sampling method and variational inference. Based on the output from the data analyzer 30, an evaluation equivalent to the "accuracy" of the depth profile was performed. Note that the values ​​calculated by the conventional MSM were used as the initial values ​​of the relative concentration depth profile.

[0169] FIG. 11 shows an example of the results of analyzing the model shown in FIG. 10 using Bayesian MSM with the Metropolis sampling method. In FIG. 11, the dashed lines indicated by "Al(true)" and "Si(true)" represent depth profiles corresponding to those shown in FIG. 10. In FIG. 11, the solid lines indicated by "Al(MSM) mean value μ" and "Si(MSM) mean value μ" represent the mean values ​​of the estimated relative concentrations for each depth and each chemical species obtained by the Bayesian MSM with the Metropolis method. In FIG. 11, the solid lines indicated by "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 the Bayesian MSM with the Metropolis method.

[0170] In the analysis of Bayesian MSM using the Metropolis method, σ in Equation (12) d is set to 0.1, and λ in Equation (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 values ​​and confidence intervals of the estimated relative concentrations of Al and Si can be displayed. Furthermore, the depth profiles estimated by Bayesian MSM are generally consistent with the true depth profiles.

[0171] In addition, λ in equation (12) s Increasing λ corresponds to a small difference in the relative concentration c between two adjacent layers in the prior distribution in the Bayesian MSM. That is, similar to the conventional MSM, in the Bayesian MSM, the change in the mean value μ of the estimated value of the relative concentration c in the depth direction is s 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 the Bayesian MSM, the variance σ represents the "uncertainty of the prediction by the Bayesian MSM" (which can also be rephrased as "lack of confidence in the prediction"). Therefore, the narrower the width of the confidence interval (μ±3σ), the more "confident" the estimation result by the Bayesian MSM can be evaluated. On the other hand, the wider the width of the confidence interval (μ±3σ), the more "unconfident" the estimation result by the Bayesian MSM can be evaluated. In the example shown in FIG. 11, the estimation results by the Bayesian MSM can be evaluated as "confident" in each of the Al region and the Si region. On the other hand, at the interface between the Al region and the Si region, the estimation results by the 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 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, respectively. Fig. 13 shows the relative concentration c corresponding to each of the positions of "Point A", "Point B", and "Point C" in Fig. 12. ik The distribution of values ​​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 obtained reached a steady state without any problems.

[0174] In addition, the relative concentration c obtained by conventional MSM ik Even if the Metropolis method is used with an initial value completely different from the solution of (1), 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 desirable to use the solution of

[0175] FIG. 14 shows an example of the results of analyzing the model shown in FIG. 10 using Bayesian MSM with variational inference. Similar to FIG. 11 , the dashed lines in FIG. 14 , identified by the notations “Al(true)” and “Si(true),” represent depth profiles corresponding to those shown in FIG. 10 . In FIG. 14 , the solid lines identified by the notations “Al(MSM) mean value μ” and “Si(MSM) mean value μ” represent the mean values ​​of the estimated relative concentrations for each depth and each chemical species obtained by Bayesian MSM with variational inference. In FIG. 14 , 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 with variational inference.

[0176] Similar to the analysis results shown in Figure 11, the analysis results shown in Figure 14 display the mean values ​​and confidence intervals of the estimated relative concentrations for both Al and Si. However, compared to the analysis results shown in Figure 11, the change in the mean values ​​of the estimated relative concentrations for each depth and each chemical species near the boundary between the Al and Si regions is somewhat less smooth. This change in mean values ​​is thought to be due to the approximation of the posterior distribution function. However, variational inference generally has low computational cost and is easy to interpret because the results are expressed as analytical formulas. Either method can be selected depending on whether smoothness of the results is prioritized or computational cost 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 concentrations of chemical species can be obtained by executing Bayesian MSM using EDX measurement data and theoretical EDX measurement values ​​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 flexibility.

[0178] On the other hand, from a practical standpoint, estimations with a high degree of freedom may result in difficult interpretation of the results, or may lead to multiple interpretations. Furthermore, depending on the computing power of the computer, the calculation time required to obtain analysis results may be long.

[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 when estimating the depth profile of the relative concentrations of chemical species. In other words, constraints are imposed on the depth profile. This reduces the number of calculations required by the data analysis device 30 to arrive at a depth profile that is close to the correct answer. According to the second embodiment, an analysis method that is easier for users to use can be realized in that it reduces the time required to obtain analysis results. Furthermore, reducing the number of calculations required by the data analysis device 30 makes it possible to save computational resources, for example.

[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 chemical species is that it smoothly changes at the boundary between two adjacent layers. In the second embodiment, the degrees of freedom of the depth profile are further restricted in the estimation of the depth profile of the relative concentration of chemical species.

[0184] In one example, the depth profile of the relative concentration of the chemical species is subject to the condition that it rises at a first depth and falls at a second depth deeper than the first depth. The function that expresses 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 a range from 0 to 1 and outputs the converted value.

[0186] The function F shown in the following equation (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 as a sum of sigmoid functions. The first term in the parentheses on the right side of equation (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]

[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). 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 Equation (31), the depth profile of the sample is calculated using 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 random variables for Bayesian estimation. Hereinafter, the set of the above five parameters will be referred to 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 consisting 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 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 is selected from the five parameters according to the problem to be solved. This makes it possible to estimate the depth profile. 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 temporarily set to a normal distribution. In this case, the prior distribution of the parameter θ is expressed according to the following equation (32).

[0194]

[0195] In equation (32), θ miq μ denotes the parameter q among the parameters θ related to 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] 16 is a diagram showing a schematic representation of a sigmoidal 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 sigmoidal 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, the σ miq Treating is very large, in other words, treating it as "not confident in the prior distribution," is equivalent to assuming no prior distribution at all.

[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]

[0200] In formula (33), d ij represents the measurement data of 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 the accelerating voltage j for the 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 values ​​of the EDX analysis follows equations (2) to (5) above.

[0201] 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 in the sample vary smoothly among the layers of the stack in minimizing the sum of squared deviations," is not required in the second embodiment.

[0202] (3. Expression of Posterior Distribution) The posterior distribution of the parameter θ after observing 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] Therefore, if we can plot equation (34) as a function of the parameter θ, we can obtain information about the mean and variance of the parameter θ.

[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 of the posterior distribution p(θ|d) (log[p(d|θ)p(θ)]) is expressed according to the following equation (35). Note that, 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] After observing the EDX measurement data, p(d) (marginal likelihood) in Bayes' theorem can be ignored thereafter because it is a constant that does not depend on the parameter θ.

[0206] The first and second terms on the right 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 there is "no confidence at all" in the estimation of the parameter θ. "No confidence at all" 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. Below, we will deal with the most common case where there is no confidence 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] Although not explicitly stated, the right-hand side of equation (36) is a function that depends on the parameter θ. Therefore, equation (36) is a function that represents the posterior distribution of the parameter θ.

[0209] 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] Forward problem analysis is an analytical method for estimating a result (output) from a cause (input). The forward problem evaluation unit 61 receives the 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 for 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 for the parameter θ is also referred to as the “next parameter θ” ​​in this specification.

[0212] The processing of the data analysis method executed by the data analysis device according to the second embodiment is similar to the processing shown in the flowchart of Fig. 7. However, the data analysis method according to the second embodiment differs in the processing 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 number of times the evaluation of the forward problem and the recommendation of the next parameter θ is repeated reaches a predetermined number. This allows the analysis unit 52 to calculate the average value and 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 also calculate the average value and confidence interval of the estimated values ​​of the relative concentrations.

[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 theoretical measurement 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 equation (31) to determine the depth profile of the relative concentration c of each chemical species for each layer and for each chemical species.

[0216] The theoretical measurement value constructing unit 61B performs a 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 theoretical measurement 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 equation (36) to obtain a value representing the deviation between the theoretical value of the EDX measurement and the measured value. This value is 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 (z ｕ , z ｌ ) and parameters (σ ｕ , σ ｌ ) 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 determination unit 62A is not particularly limited. For example, the optimal solution determination unit 62A may determine the next optimal parameter θ using Bayesian optimization. Other optimization methods, including, but not limited to, gradient methods or covariance matrix adaptive evolutionary strategies (CMA-ES), may also be employed. When executing an optimization method such as Bayesian optimization or CMA-ES, the optimal solution determination unit 62A repeatedly calculates the value of the logarithmic posterior distribution by changing multiple variables included in the parameter θ (schematically represented as θ1 and θ2 in FIG. 18 ). If necessary, the optimal solution determination unit 62A may refer to multiple past histories. In this way, the optimal solution determination unit 62A determines the optimal parameter θ (i.e., an optimal set of parameters).

[0220] In order to execute the above optimization method 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 uses the value of the parameter θ obtained by the optimal solution determination 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 MCMC method, to propose the next parameter θ. Other methods that are more advanced versions of the Metropolis method, such as the Hamiltonian Monte Carlo method or 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, but may be, for example, a Gaussian distribution that is independent for each variable included in the parameter θ (uncorrelated with each variable). The MCMC implementation unit 62B proposes the value of the next parameter θ by randomly sampling from the proposal distribution, centered on the value of the parameter θ at a certain point in time, and taking the variance.

[0224] The MCMC implementation unit 62B calculates the logarithmic posterior distribution using the proposed next value. In the second embodiment, the logarithmic posterior distribution corresponds to the sum of squared deviations between the theoretical EDX measurement value and the measured EDX value. If the value of the logarithmic 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 logarithmic posterior distribution is worse (larger) than the previous value, the MCMC implementation unit 62B generates a uniform random number r ranging from 0 to 1.

[0225] If the absolute value of the difference between the value of the logarithmic 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 logarithmic 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, the MCMC method adopts the proposed value with a certain probability even if the value of the logarithmic posterior distribution to be evaluated is worse than the previous value. 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 initial value it starts from, if it is run over an infinite amount of time using the correct method. In the MCMC method, the period from the initial point to convergence to the target distribution is called burn-in. Generally, data in the burn-in state is discarded before various quantitative values ​​are calculated.

[0227] In practice, the time available for calculation is limited. Therefore, in the second embodiment, the optimal solution determination unit 62A determines the target value or a value close to the target value of the parameter θ, and uses this value as the initial value for the MCMC method. By using this initial value, the MCMC implementation unit 62B can shorten the burn-in period. Therefore, the MCMC implementation unit 62B can shorten the calculation time required to reach the correct distribution. Note that, although the MCMC method is generally used to evaluate the posterior distribution of parameters based on Bayesian inference, 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 determined.

[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 layer interface.

[0229] <<5.3 Analysis Examples>> The following examples show the results of calculations of layer thicknesses and compositions for several samples by the analysis unit 52. 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 of the Gaussian distribution is σ for the layer thickness. 2 =1 2 (nm ２ ) and σ for the composition 2 = 0.01 2 These 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 electrode and SiN film on InP wafer 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. Sample 1 was subjected to surface analysis by EDX. Thereafter, a cross-sectional analysis of the sample was performed by STEM-EDX to verify the validity of the analysis according to the second embodiment.

[0232] FIG. 19 shows STEM cross-sectional images of Sample 1. As shown in FIGS. 19(A) and 19(B), the STEM images indicated that the thicknesses of the Ni layer and SiN layer of Sample 1 were approximately 181 nm and 262 nm, respectively. The STEM-EDX analysis results indicated that the ratio of indium to phosphorus (In:P) in the semiconductor substrate was approximately 0.55:0:45. The STEM-EDX analysis results indicated that a small amount of phosphorus (P) 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 as being silicon-rich.

[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 electron beam acceleration voltage levels in the EDX surface analysis were 10 kV, 15 kV, 20 kV, 25 kV, and 30 kV. Fig. 20(A) shows the results of EDX surface analysis on the Ni electrode surface, and Fig. 20(B) shows the results of 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 of the InP substrate according to the MCMC method. The analysis results by the analysis unit 52 will be explained 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 consists of two layers: a Ni layer and an InP layer. The thickness d of the Ni layer NiThe 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. 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 calculated the thickness d of the Ni layer according to the Bayesian optimization method. Ni The initial values ​​of the composition of InP were determined.

[0237] 22A and 22B show the results of calculations performed by the analysis unit 52 on the thickness of the Ni layer and the In composition of InP using the MCMC method. Fig. 22A shows the variation in the calculation results for the thickness of the Ni layer versus the number of repetitions of the MCMC method. Fig. 22B 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 Figure 22A, the calculated thickness of the Ni layer fluctuates around 172 nm, but reaches a steady state as the MCMC method loop is repeated. The Ni layer thickness obtained by STEM-EDX analysis is approximately 181 nm. Therefore, according to the second embodiment, a value close to that obtained by STEM-EDX analysis can be obtained for the Ni layer thickness by non-destructive analysis.

[0239] Similarly, as shown in Figure 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 loop is repeated. The analysis according to the second embodiment shows that the In composition of the InP layer exceeds 0.5. That is, according to the second embodiment, the results obtained show that phosphorus is being removed from the InP layer, similar to the results of the STEM-EDX analysis.

[0240] 23 shows the analysis results of sample 1 (Ni layer on an InP substrate) by the 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 chemical species. The MCMC calculation was repeated 10,000 times.

[0241] The analysis unit 52 discarded the calculation results of the first 1,000 MCMC loops and calculated the mean value (μ) and variance (σ) of the thickness of the Ni layer and the mean value (μ) and confidence interval (σ) of the In content of the InP layer from the remaining calculation results. The mean value (μ) and variance (σ) of the thickness of the Ni layer were 172 nm and 0.99 nm, respectively. The mean value (μ) and confidence interval (σ) of the In content of the InP layer were 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 MCMC loop from 1000th to 10000th, 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 approximately 180 nm to 183 nm. The width of the depth range corresponds to the range of ±3σ of the average thickness of the Ni layer. The σ obtained by the MCMC method can be regarded as the Bayesian confidence interval, i.e., the accuracy of the analysis results in the second embodiment. Therefore, the curves shown in Figure 23 represent the confidence interval of the estimated values ​​of the relative concentrations of 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 composed 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. 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 of the SiN layer according to the Bayesian optimization method. SiN The initial values ​​of the composition of InP were determined.

[0246] 25A and 25B show 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 using the MCMC method. Fig. 25A shows the variation in the calculation results for the N composition of the SiN layer versus the number of repetitions of the MCMC method. Fig. 25B shows the variation in the calculation results for the thickness of the SiN layer versus the number of repetitions of the MCMC method. Fig. 25C shows the variation in the calculation results for the In composition of the InP layer versus the number of repetitions of the MCMC method.

[0247] As shown in Figure 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 Figure 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 Figure 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 Figure 25 demonstrate that the MCMC method can be used to estimate the thickness of the SiN layer even when the compositions of the two layers are simultaneously varied. The thickness of the SiN layer is close to the thickness (approximately 262 nm) determined from the cross-sectional STEM image.

[0249] 26 shows the analysis results of sample 1 (SiN layer on an InP substrate) by the 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 MCMC calculation was repeated 10,000 times.

[0250] The analysis unit 52 discarded the calculation results of the first 1,000 MCMC loops and calculated the mean (μ) and variance (σ) of the 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 from the remaining calculation results. The μ and σ of the 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 MCMC loop from 1000th to 10000th, 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 revealed that the SiN layer was Si-rich (approximately 0.56) and the InP layer was In-rich (approximately 0.55). The analysis results shown in FIG. 26 indicate that the SiN layer was Si-rich and the InP layer was In-rich. The analysis of Sample 1 according to the second embodiment was non-destructive and showed 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, to a copper (Cu) foil and then baking it. As in Example 1, Sample 2 was subjected to surface analysis 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, the thickness of the PDF layer was evaluated to be about 1.1 µm from the STEM image.

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

[0256] 29 is a diagram showing an analytical model of sample 2. In the analytical model shown in FIG. 29, it is assumed that the sample is composed 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 ,σ l is fixed at 50 nm. 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 ​​that were significantly different from the optimum 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 and thickness of the PFA layer according to the MCMC method. FIG. 30(a) shows the variation in the calculation results for the C composition of the PFA layer versus the number of repetitions of the MCMC method. FIG. 30(b) shows the variation in the calculation results 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 calculation repetitions increases. In the example shown in FIG. 30, the calculated value can be considered to have reached a steady state after 6,000 calculations.

[0259] As is clear from FIG. 30 , with the MCMC method, even if a value that deviates from the optimal value is used as the initial value, the calculated value will reach the correct value by repeating the calculation loop a sufficiently large number of times. Therefore, determining an appropriate initial value for the MCMC method using an optimization technique is not an essential requirement for executing the MCMC method. However, as can be seen from a comparison between FIG. 22 and FIG. 30 , using the parameter θ determined by the optimization technique as the initial value for the MCMC loop can shorten the burn-in period. Therefore, in the second embodiment, the optimal solution determination unit 62A provides the MCMC implementation unit 62B with a value close to the optimal value as the initial value for the MCMC method. This reduces the number of calculations required by the data analysis device 30 to arrive at a depth profile close to the correct solution, thereby shortening the calculation time required for analyzing the depth profile.

[0260] Figure 31 shows the analysis results of sample 2 by the analysis unit 52. In Figure 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 MCMC calculation was repeated 10,000 times. However, as shown in Figure 30, the results of the first 6,000 calculations were not stable. For this reason, the analysis unit 52 discarded the results of the first 6,000 calculations and calculated the mean (μ) and variance (σ) of the PFA layer film thickness, as well as the μ and σ of the C composition in the PFA layer from the remaining calculation results. The μ and σ of the PFA layer film thickness were 1292 nm and 10.2 nm, respectively. The μ and σ of the C composition in the PFA layer were 0.38 and 0.0021, respectively.

[0261] The analysis unit 52 picked up the calculation results of the depth profile of the relative concentration every 500 times from 6000 to 10000 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 are a diagram showing the calculation results picked up and displayed every 500 times from 6000 to 10000 of the MCMC loops. 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 at which the curve showing the relative concentration of F intersects with the curve showing the relative concentration of Cu, and the depth at which 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 obtained by the MCMC method are close to the thickness of the PFA layer evaluated by STEM (approximately 1100 nm) and the composition evaluated by GCIB-XPS (C:F ~ approximately 1:2). Therefore, according to the second embodiment, it is clear that the sample can be correctly evaluated non-destructively in the analysis of Example 2.

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

[0265] Fig. 32 is a diagram showing an STEM cross-sectional image of Sample 3. As shown in Fig. 32, a Ni layer, a Cr layer, a Ti layer, and an Al layer are stacked on the surface of the Si substrate in 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. EDX surface analysis was performed at accelerating voltages of 10 kV to 30 kV in 1 kV increments (21 levels in total). The MCMC method was performed based on the data obtained from the EDX surface analysis. Note that the structure of the sample was assumed to be known, an Al / Ti / Cr / Ni / Si structure, and the only variables were the thicknesses of the four layers on the Si substrate.

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

[0268] FIG. 35 shows the results of the profile evaluation of sample 3 by the analysis unit 52. In FIG. 35, the profile evaluated by the analysis unit 52 using 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. In other words, according to the second embodiment, it is possible to obtain results similar to those obtained by destructive evaluation, despite the non-destructive evaluation of the sample.

[0269] According to FIG. 35 , the intersection of the curve showing the relative concentration of Al and the curve showing the relative concentration of Ti corresponds to the interface between the Al layer and the Ti layer. Similarly, the intersection of the curve showing the relative concentration of Ni and the curve showing the relative concentration of Si corresponds to the interface between the Ni layer and the Si layer. In the evaluation according to the second embodiment, the estimation range for the position of the interface between the Ni layer and the Si layer is larger than the estimation range for the position of the Al / Ti interface. This indicates that the MCMC estimation results are less reliable at the inner side of the sample (closer to the Si substrate) than at the outermost surface. The EDX analysis measurements, which are the basis of MCMC analysis, have a greater influence at the outermost surface and a weaker influence at deeper positions. Therefore, 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 measurement data and estimation results>> Figure 36 shows the results of calculations made by the analysis unit 52 for the thickness of the multilayer film of sample 3 according to the MCMC method when the number of EDX measurement data (in other words, the number of acceleration voltage levels) was 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. Figures 36(A) to 36(D) are figures to be compared with Figures 34(A) to 34(D), respectively.

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

[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 (mean 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 (mean value μ) of each layer tends to be small, while the confidence interval of thickness (variance σ) of each layer tends to be large.

[0273] In this way, by applying Bayesian estimation to film thickness estimation, the relationship between the number of data and uncertainty can be evaluated. Therefore, the number of measurement data required for evaluation can be determined depending on 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 can be determined depending on the accuracy required for the evaluation.

[0274] 6. Examples of Display Output By executing the Bayesian MSM described above, the data analysis device 30 can determine the average value of the estimated relative concentration for each depth and each chemical species in the sample, as well as the confidence interval for that estimate. 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, for example, displayed on a display screen.

[0275] 38 is a diagram showing a first example of a display output by the data analysis device according to this embodiment. In this 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 format 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 depth, and a region (second figure) representing the confidence interval of the relative concentration with respect to depth.

[0276] In the example of Figure 38, the sample to be analyzed is a sample in which aluminum (Al), titanium (Ti), chromium (Cr), and nickel (Ni) are layered on a silicon (Si) substrate in descending order of proximity to the sample surface. In Figure 15, curve 101 and region 102 represent the mean and confidence interval of the estimated relative concentration of aluminum (Al) at each depth. The vertical width of region 102 corresponds to the width of the confidence interval. Similarly, for other chemical species, a curve (first figure) representing the change in mean value μ with respect to depth and a region (second figure) representing the change in confidence interval with respect to depth are displayed. For a given chemical species, an analyst (e.g., a user of the data analysis device) can easily grasp the accuracy of the relative concentration estimation (whether the estimation result obtained by Bayesian MSM can be considered "confident") based on the width of the region representing the confidence interval. Therefore, according to the example of Figure 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 Figure 38, the line type of the curve representing the average value of the estimated relative concentration is different for each chemical species. However, the display of the curve is not limited to this. The curves and regions may be colored differently for each chemical species. This makes it possible to display the curves and regions corresponding to each chemical species in a way that makes it easier for the user to distinguish between them.

[0278] 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 intervals of the estimated values ​​of the relative concentrations for each depth and each chemical species are expressed by curves. 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 relative concentration at each depth when the chemical species is aluminum (Al). Curves 103A and 103B represent the upper and lower limits, respectively, of the confidence interval of the estimated relative concentration at 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 that 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 relative concentration estimation (whether the estimation result by Bayesian MSM can be said to be "confident") for a certain chemical species from the width of the area representing the confidence interval.

[0280] As in the display example shown in Fig. 38, the curves representing the average value of the estimated relative concentration 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 relative concentration and the upper and lower limits of the confidence interval may be different for each chemical species.

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

[0282] The length of the 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 the cursor 104. According to the example shown in Fig. 40, 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. Note that, similarly to the display example shown in Fig. 39, when the cursor 104 is placed on the curve representing the average value of the estimates, an error bar 105 indicating the width of the confidence interval of the estimate may be displayed at the position on the curve pointed to by the cursor 104.

[0283] Figure 41 is a diagram showing a fourth example of display output by the data analysis device according to this embodiment. In the display example shown in Figure 41, a text box 106 is displayed in addition to the display shown in Figure 40. 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; at least one of (3) to (6) may be displayed in the text box.

[0284] (1) Chemical species corresponding to the curve indicated by the cursor 104 (2) Depth corresponding to the position on the curve indicated by the cursor 104 (3) Mean value μ of the estimated values ​​of relative concentration corresponding to the position on the curve indicated by the cursor 104 (4) Variance σ of the estimated values ​​of relative concentration corresponding to the position on the curve indicated by the cursor 104 (5) Upper limit value of the confidence interval (e.g. μ±3σ) corresponding to the error bar 105 (6) Lower limit value of the confidence interval (e.g. μ±3σ) corresponding to the error bar 105 Note that the display of information using text boxes is not limited to being applied only to the display example shown in FIG. 41 . The display of information using text boxes can also be applied to any of the display examples shown in FIGS. 38 to 40 . Displaying numerical values ​​makes it easier for analysts to understand the estimation results.

[0285] Alternatively, as shown in FIGS. 23, 26, 31, and 35, the width of the confidence interval of the estimated value of the relative concentration may be expressed by overlapping 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 used to represent 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 relative concentration estimate and the confidence interval of the relative concentration estimate may be displayed in a manner other than a graph.

[0288] 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 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 confidence interval are displayed numerically, allowing the analyst to specifically grasp the mean value and 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 in a format that can be displayed on a display screen (for example, CSV format data).

[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 application 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 equation (1) is ij is expressed by the following equation (37).

[0291] I in formula (37) ki (θ j ) represents the photoelectron signal for species i. j represents the take-off angle of the photoelectron signal from the sample. Take-off angle θj The level of the photoelectron signal I is J. ki (θ j ) is expressed by the following equation (38).

[0292] In formula (38), c ik represents the relative concentration of chemical species i in the kth layer. The sum of the relative concentrations of chemical species i is 1 (Σ i c ik = 1). λ li represents the inelastic mean free path of photoelectrons originating from chemical species i in the lth layer. i represents the relative ionization cross section of photoelectrons of chemical species i against X-rays. For the signal when k = 1, that is, the signal of the outermost surface layer, the total product Π is considered to be 1. Note that the layer thickness t is λ li is extremely small compared to t<<λ li ) is assumed. Therefore, in equation (38), the attenuation of photoelectrons in the layer where the photoelectrons are generated is approximated by a linear function (e -x ~1-x).

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

[0294] Similarly, the Bayesian estimation of depth profiles according to the second embodiment is not limited to applications 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 the other analyses. More specifically, Equation (2) calculated by the measurement theoretical value construction unit 61B can be transformed into a form suitable for the 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. Supplementary Notes The present disclosure includes the following embodiments.

[0297] [Supplementary Note 1] (1) A data analysis device that analyzes a depth profile of a sample using data on a response signal generated from the sample by incident of a probe, the data analysis device comprising: an input unit that accepts data on the response signal obtained by a measurement device that measures the response signal while changing incident conditions of the probe so as to evaluate different depth regions of the sample; and 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 on the response signal, and calculates an average value and a confidence interval of the estimated values ​​of the relative concentrations, wherein the analysis unit calculates the average value and the confidence interval of the estimated values ​​by processing, according to Bayesian estimation, an evaluation function expressed by the sum of a first term that expresses the sum of squares of deviation between a theoretical value of the response signal when the sample is modeled as a stack of multiple layers and the data of the response signal, and a second term that expresses the degree of continuity of the relative concentration at the depth.

[0298] (2) The data analysis device according to (1), wherein the Bayesian estimation process includes a sampling algorithm that uses 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 variational inference.

[0301] [Supplementary Note 2] (1) A data analysis method for analyzing a depth profile of a sample using data on response signals generated from the sample by incident of a probe, the data analysis method comprising the steps of: a data analysis device receiving data on the response signals obtained by a measurement device that measures the response signals while changing incident conditions of the probe to evaluate different depth regions of the sample; a data analysis device analyzing the data on the response signals to estimate relative concentrations for each depth from the surface of the sample and for each chemical species, and determining average values ​​and confidence intervals for the estimated relative concentrations; and a data analysis device outputting a display of the average values ​​and confidence intervals for the estimated relative concentrations, the display of the average values ​​and the confidence intervals being in the form of a graph, the graph including a first axis indicating the depth, a second axis indicating the relative concentration, a first figure representing the change in the average value with respect to the depth, and a second figure representing the change in the confidence interval with respect to the depth.

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

[0303] (3) The data analysis method according to claim (1), 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.

[0304] (4) The data analysis method according to any one of (1) to (3), wherein the second graphic includes an error bar, and 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 graphic.

[0305] (5) The data analysis method according to any one 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 outputs the display of the mean value and the confidence interval in the form of a numerical table.

[0307] (7) The data analysis method according to any one of (1) to (6), wherein 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, and the evaluation function is expressed by the sum of a first term expressing the sum of squares of the 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.

[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] [Supplementary Note 3] (1) A data analysis method for analyzing a depth profile of a sample using data on a response signal generated from the sample by incident of a probe, the data analysis method comprising: a step of receiving, by a data analysis device, data on the response signal obtained by a measurement device that measures the response signal while changing incident conditions of the probe so as to evaluate different depth regions of the sample; and 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 signal, and determining an average value and a confidence interval of the estimated values ​​of the relative concentration, wherein 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 values ​​by processing an evaluation function according to Bayesian estimation processing, and wherein the evaluation function is expressed by the sum of a first term that expresses the sum of squares of deviation between a theoretical value of the response signal and the data of the response signal when the sample is modeled as a stack of multiple layers, and a second term that expresses 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] [Supplementary Note 4] (1) A data analysis method for analyzing a depth profile of a sample using data on response signals generated from the sample by incident of a probe, comprising: a step of receiving, by a data analysis device, data on the response signals obtained by a measurement device that measures the response signals while changing incident conditions of the probe so as to evaluate different depth regions of the sample; and a step of estimating relative concentrations for each depth from the surface of the sample and for each chemical species by analyzing the data of the response signals, and determining an average value and a confidence interval of the estimated values ​​of the relative concentrations, wherein the step of determining the average value and the confidence interval includes 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 limited degrees of freedom of the depth profile.

[0317] (2) The data analysis method according to (1), wherein 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 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.

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

[0319] (4) The data analysis method according to (3), wherein the step of recommending the candidate values ​​of the parameters includes: a step of proposing the candidate values ​​of the parameters using a predetermined probability distribution according to a Markov Chain Monte Carlo (MCMC) method; and a step of determining optimal initial values ​​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 limiting. The scope of the present invention is defined by the claims, not by the above-described embodiments, and is intended to include meanings equivalent to the claims and all modifications within the scope thereof.

[0321] 10 Analysis system, 20 Measuring 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 Deviation square sum 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 box, S11 to S13, S12A, S21 to S28, S26A to S26D steps.

Claims

1. 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, comprising: an input unit that accepts data on the response signal 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 data of the response signal 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 concentration; and an output unit that outputs an indication of the average value and the confidence interval of the estimated values ​​of the relative concentration.

2. The data analysis device of claim 1, wherein the display of the mean and the confidence interval is in the form of a graph, the graph including a first axis indicating the depth, a second axis indicating the relative concentration, a first graph representing the variation of the mean with respect to the depth, and a second graph representing the variation of the confidence interval with respect to the depth.

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

4. The data analysis device 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. A data analysis device as described in any one of claims 2 to 4, wherein the second figure includes an error bar, the length of the error bar corresponding to the width of the confidence interval of the estimated value at a specified position on the first figure.

6. A data analysis device as claimed in any one of claims 2 to 5, 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.

7. A data analysis device according to any one of claims 1 to 6, wherein the output section outputs the representation of the mean and the confidence interval in the form of a numerical table.

8. A data analysis device as described in any one of claims 2 to 7, wherein the analysis unit calculates the average value and the confidence interval of the estimated value by processing an 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 data of the response signal, and a second term expressing the 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 that uses 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 variational inference.

11. A data analysis device as claimed in any one of claims 2 to 7, wherein 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 the sample with restricted degrees of freedom of the depth profile.

12. The data analysis device of claim 11, wherein the model of the sample is a stack of multiple layers, the predetermined function is a function that expresses that the depth profile has rises and falls at the boundaries of each of the multiple layers, and the parameters include: a composition of the sample, a thickness of each of the multiple layers, a gradual rise in the depth profile of each of the multiple layers, and a gradual fall in the depth profile of each of the multiple layers.

13. The data analysis device according to claim 12, wherein the analysis unit includes: a forward problem evaluation unit that uses values ​​of the parameters as input to a 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 the parameters to be used in a next calculation by the forward problem evaluation unit, wherein the calculation of the sum of squared deviations by the forward problem evaluation unit and the recommendation of the candidate values ​​of the parameters by the next parameter recommendation unit are alternately repeated.

14. The data analysis device described in claim 13, wherein the next parameter recommendation unit includes: an MCMC implementation unit that proposes candidate values ​​for the parameters using a predetermined probability distribution in accordance with a 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.

15. A data analysis device according to any one of claims 1 to 14, wherein the measurement device is an energy dispersive X-ray analysis device.

16. The data analysis device according to any one of claims 1 to 14, 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 on a response signal generated from the sample by the incidence of a probe, comprising the steps of: a data analysis device receiving data on the response signal 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; a data analysis device analyzing the response signal data to estimate relative concentrations for each depth from the surface of the sample and for each chemical species, and determining average values ​​and confidence intervals of the estimated values ​​of the relative concentrations; and a data analysis device outputting an indication of the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations.

18. A program causing a computer to execute the steps of: acquiring response signal data obtained by a measuring 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, and determining average values ​​and confidence intervals of the estimated values ​​of the relative concentrations; and outputting an indication of the average values ​​and confidence intervals of the estimated values ​​of the relative concentrations.

19. A recording medium having recorded thereon a program for causing a computer to execute the steps of: acquiring data on response signals obtained by a measuring 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 data on the response signals, and 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.

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