Support methods, recording media, support systems, and substrate processing systems

Abstract

The supporting method of the present invention includes: a model building step, which builds a presumed model including equation (1). In equation (1), D(x) represents the power law. The model building step includes: a coefficient value determination step, which uses a dataset and Bayesian estimation to determine the values ​​of the first coefficient k and the second coefficient q of the power law D(x). The values ​​of the explanatory variables of the dataset represent the particle size. The values ​​of the target variables of the dataset represent the number of particles. The coefficient value determination step includes the following steps: repeating the steps of sampling the values ​​of the first coefficient k and the second coefficient q from the prior distribution of the first coefficient k and the prior distribution of the second coefficient q to obtain the post-distribution of the first coefficient k and the post-distribution of the second coefficient q. The prior distribution includes a uniform distribution within a predetermined range. A×D(x)=A×kx-q … (1)

Description

[Technical Field]

[0001] This invention relates to a support method, recording medium, support program, support system, and substrate processing system. [Previous Technology]

[0002] Optical measuring apparatuses are known for optically measuring the number of particles (particle count) attached to substrates such as semiconductor wafers. The measurement results of the number of particles on the substrate are used for performance evaluation of substrate processing apparatuses, etc. For example, Patent Document 1 discloses a substrate processing apparatus equipped with a particle measuring unit that measures particles on a wafer.

[0003] However, there are limitations to the particle size that optical measuring devices can measure. For example, optical measuring devices cannot measure particles with a diameter of less than 10 nm.

[0004] On the other hand, a technique for estimating particle size using a power law D(x) expressed by the following mathematical formula is being studied. In the power law D(x), the variable x represents the particle size. D(x) = kx - q

[0005] Specifically, an optical measuring device is used to measure the particle size and number of particles, and observation data is obtained. Then, by fitting a model (curve) of the power law D(x) to the observation data, the values ​​of the two coefficients k and q of the power law are defined using the nonlinear least squares method. By using the power law D(x), the number of particles with particle sizes that cannot be measured by the optical measuring device can be determined. [Prior Art Documents] [Patent Documents]

[0006] [Patent Document 1] Japanese Patent Application Publication No. 2007-88401 [Summary of the Invention]

[0007] (Problem to be Solved by the Invention) However, the values ​​of the coefficients k and q defined by the nonlinear least squares method are strongly affected by noise generated in the optical measuring device. Specifically, there is a situation where the model (curve) of the power law D(x) overfits the observed data, and the values ​​of the two coefficients k and q of the power law become significantly larger. Therefore, there is still room for improvement in the technique of estimating the number of particles using the power law D(x).

[0008] One objective of this invention is to provide a method, recording medium, support program, support system, and substrate processing system for supporting the operation of estimating the number of particles on a substrate. (Technical means to solve the problem)

[0009] According to one aspect of the present invention, the support method is a method for supporting the estimation of the number of particles on a substrate. The support method includes a model building step, which establishes a first estimation model for estimating the number of particles for each particle size. The first estimation model includes the following equation (1). A×D(x)=A×kx-q … (1) In the above equation (1), the constant A represents the area of ​​the substrate, D(x) represents the power law, and the variable x represents the particle size. By inputting an arbitrary value into the variable x of the first estimation model, the number of particles having the particle size of the input variable x is calculated. The model building steps described above include the following steps: a coefficient value determination step, which uses the first data set and Bayes' estimation to determine the values ​​of the first coefficient k and the second coefficient q of the power law D(x); and a step of building the first estimation model, which inputs the values ​​determined in the coefficient value determination step into the first coefficient k and the second coefficient q. The first data set includes the values ​​of the explanatory variables and the values ​​of the target variables corresponding to the values ​​of the explanatory variables. The values ​​of the explanatory variables represent the observed values ​​of the particle size, i.e., the first observation. The values ​​of the target variables represent the observed values ​​of the number of particles, i.e., the second observation. The steps for determining the coefficient values ​​mentioned above include the following steps: repeating the steps of sampling the temporary values ​​of the first coefficient k and the second coefficient q from the prior distribution of the first coefficient k (i.e., the first prior distribution) and the prior distribution of the second coefficient q (i.e., the second prior distribution) to obtain the subsequent distribution of the first coefficient k and the subsequent distribution of the second coefficient q; and determining the values ​​of the first coefficient k and the second coefficient q respectively based on the subsequent distribution of the first coefficient k and the subsequent distribution of the second coefficient q. The sampling steps described above include the following steps: establishing a provisional model based on the values ​​sampled from the first prior distribution (i.e., the first sampled value), the values ​​sampled from the second prior distribution (i.e., the second sampled value), and the first presumed model for which the values ​​of the first coefficient k and the second coefficient q are undetermined; and evaluating the goodness of fit of the provisional model to the first data set based on the provisional model, the first data set, and the likelihood function. The second prior distribution includes a uniform distribution within a predetermined range.

[0010] In one embodiment, the above model building step further includes the following steps: inputting a plurality of predetermined values ​​as values ​​of the explanatory variables into the first presupposition model, and outputting a plurality of values ​​of the target variable from the first presupposition model to obtain a second dataset; adding supplementary information to the second dataset; and enabling a machine learning model capable of outputting a predictive distribution to perform machine learning on the second dataset with the added supplementary information, and establishing a second presupposition model that outputs the expected value of the predictive distribution relative to the target variable. The second dataset includes a plurality of values ​​of the explanatory variables input into the first presupposition model, and a plurality of values ​​of the target variable output from the first presupposition model. The supplementary information includes information about the origin (0,0).

[0011] In one embodiment, the above-described support method further includes: a target variable establishment step, which establishes the value of a target variable for a dataset used for Bayesian optimization. The dataset used for Bayesian optimization includes at least one parameter among various parameters defining the operation of a substrate processing apparatus for processing the substrate as an explanatory variable. The Bayesian optimization is used to determine the optimal solution for the at least one parameter. The target variable establishment step includes the following steps: obtaining the area above the minimum value set for the variable x in the lower region of the curve of the first estimation model in the coefficient value determination step, where the values ​​of the first coefficient k and the second coefficient q have been determined; and obtaining the value of the target variable for the dataset used for Bayesian optimization based on the area above the minimum value and the area of ​​the substrate.

[0012] In one embodiment, the above-described support method further includes: a target variable establishment step, which establishes the value of the target variable for a dataset used for Bayesian optimization. The dataset used for Bayesian optimization includes at least one parameter among various parameters defining the operation of the substrate processing apparatus that processes the substrate as an explanatory variable. The Bayesian optimization is used to determine the optimal solution for the at least one parameter. The target variable establishment step includes the following steps: outputting the expected value of the predicted distribution from the second estimation model; obtaining the area of ​​a predetermined range in the region below the curve of the expected value of the predicted distribution; and obtaining the value of the target variable for the dataset used for Bayesian optimization based on the area of ​​the predetermined range and the area of ​​the substrate.

[0013] In a certain implementation, the above likelihood function contains a Boisson distribution.

[0014] According to another aspect of the present invention, the recording medium is a computer-readable medium. The recording medium records a support program that defines the above-described support method.

[0015] According to another aspect of the present invention, the support program can be executed by a computer. The support program defines the above-described support method.

[0016] According to another aspect of the present invention, the support system is a system that supports the operation of estimating the number of particles on a substrate. The support system includes a processing unit and a memory unit. The processing unit establishes a first estimation model for estimating the number of particles for each particle size. The memory unit stores a first data set. The first estimation model includes the following equation (1). A×D(x)=A×kx-q … (1) In the above equation (1), the constant A represents the area of ​​the substrate, D(x) represents the power law, and the variable x represents the particle size. By inputting an arbitrary value into the variable x of the first estimation model, the number of particles having the particle size of the input variable x is calculated. The first data set includes each value of the explanatory variable and each value of the target variable corresponding to each value of the explanatory variable. Each value of the explanatory variable represents the observed value of the particle size, i.e., the first observation value. The values ​​of the aforementioned target variables represent the observed values ​​of the particle number, i.e., the second observed value. The processing unit performs coefficient value determination processing, using the aforementioned first dataset and Bayes' estimation to determine the values ​​of the first coefficient k and the second coefficient q of the aforementioned power law D(x). The values ​​determined by the coefficient value determination processing are input to the aforementioned first coefficient k and the aforementioned second coefficient q to establish the aforementioned first estimation model. During the coefficient value determination processing, the processing unit repeats the step of sampling the temporary values ​​of the first coefficient k and the second coefficient q from the prior distribution (i.e., the first prior distribution) and the prior distribution (i.e., the second prior distribution) of the aforementioned first coefficient k to obtain the subsequent distributions of the aforementioned first coefficient k and the aforementioned second coefficient q. The processing unit determines the values ​​of the first coefficient k and the aforementioned second coefficient q based on the subsequent distributions of the first coefficient k and the aforementioned second coefficient q. The processing unit, during the sampling step, establishes a temporary model based on the values ​​sampled from the first prior distribution (i.e., the first sampled value), the values ​​sampled from the second prior distribution (i.e., the second sampled value), and the first presumed model where the values ​​of the first coefficient k and the second coefficient q are undetermined. The processing unit then evaluates the goodness of fit of the temporary model to the first data set based on the temporary model, the first dataset, and the likelihood function. The second prior distribution includes a uniform distribution within a predetermined range.

[0017] In one embodiment, the memory unit further stores a machine learning model that can output a predictive distribution. The processing unit inputs a plurality of predetermined values ​​as values ​​of the explanatory variables into the first estimation model, and outputs a plurality of values ​​of the target variable from the first estimation model to obtain a second dataset. The processing unit adds additional information to the second dataset. The processing unit performs machine learning on the second dataset with the added additional information to establish a second estimation model that outputs the expected value of the predicted distribution of the target variable. The second dataset includes a plurality of values ​​of the explanatory variables input into the first estimation model and a plurality of values ​​of the target variable output from the first estimation model. The additional information includes information about the origin (0,0).

[0018] In one embodiment, the processing unit further performs a target variable establishment process, which establishes the value of a target variable for a dataset used for Bayesian optimization. The dataset used for Bayesian optimization includes at least one parameter among various parameters defining the operation of the substrate processing apparatus that processes the substrate as an explanatory variable. The Bayesian optimization is used to determine the optimal solution for the at least one parameter. When performing the target variable establishment process, the processing unit obtains the area above the minimum value set for the variable x in the lower region of the curve of the first estimation model, where the values ​​of the first coefficient k and the second coefficient q have been determined by the coefficient value determination process. The processing unit obtains the value of the target variable for the dataset used for Bayesian optimization based on the area of ​​the area above the minimum value and the area of ​​the substrate.

[0019] In one embodiment, the processing unit further performs a target variable establishment process, which establishes the value of the target variable for the dataset used for Bayesian optimization. The dataset used for Bayesian optimization includes at least one parameter among various parameters defining the operation of the substrate processing apparatus that processes the substrate as an explanatory variable. The Bayesian optimization is used to determine the optimal solution for the at least one parameter. When performing the target variable establishment process, the processing unit outputs the expected value of the predicted distribution from the second estimation model. The processing unit obtains the area of ​​a predetermined range in the region below the curve of the expected value of the predicted distribution. The processing unit obtains the value of the target variable for the dataset used for Bayesian optimization based on the area of ​​the predetermined range and the area of ​​the substrate.

[0020] In a certain implementation, the above likelihood function contains a Boisson distribution.

[0021] According to another aspect of the present invention, a substrate processing system is a system for processing a substrate. The substrate processing system includes a substrate processing apparatus and a control device. The substrate processing apparatus processes the substrate. The control device controls the substrate processing apparatus. The control device also serves as the support system.

[0022] According to the present invention, at least one type of support method, recording medium, support program, support system and substrate processing system can support the operation of estimating the number of particles on the substrate.

Implementation Method

[0024] Hereinafter, embodiments related to the support method, recording medium, support program, support system, and substrate processing system of the present invention will be described with reference to Figures 1 to 17. However, the present invention is not limited to the following embodiments and can be implemented in various forms without departing from its spirit. Furthermore, repeated descriptions are sometimes appropriately omitted below. Also, identical or equivalent parts are marked with the same reference numerals in the figures without repeating their descriptions.

[0025] In the support method, recording medium, support program, support system and substrate processing system of the present invention, "substrate" can be applied to various substrates such as semiconductor wafers, photomask glass substrates, liquid crystal display glass substrates, plasma display glass substrates, field emission display (FED) substrates, optical disc substrates, magnetic disc substrates and magneto-optical disc substrates.

[0026] [Embodiment 1] First, the recording medium 200, the support program SP, and the support system 100A of this embodiment will be described with reference to FIG1. ​​FIG1 is a block diagram showing the configuration of the support system 100A of this embodiment. The support system 100A is a system that supports the operation of estimating the number of particles on the substrate.

[0027] As shown in FIG1, the support system 100A of this embodiment includes a terminal device 101A. A support program SP is installed on the recording medium 200 into the terminal device 101A. The terminal device 101A executes the support program SP installed on the recording medium 200 to support the operation of estimating the number of particles. Specifically, the terminal device 101A executes the support program SP to establish an estimation model for the estimated number of particles of various sizes.

[0028] Terminal device 101A is an example of a "support device". Terminal device 101A may be a general-purpose computer or a special-purpose computer. Alternatively, terminal device 101A may be a computer that controls a substrate processing device or a computer that controls a measuring device for measuring particles on a substrate.

[0029] Recording medium 200 is a computer-readable medium. A program (computer program) that enables a computer to execute is non-temporarily recorded in recording medium 200. Support program SP is recorded in recording medium 200. In other words, recording medium 200 stores support program SP. Support program SP is a computer program that can be executed by a computer.

[0030] The recording medium 200 may be, for example, a semiconductor memory containing a secure digital (SD) memory card and a universal serial bus (USB) memory, or a magnetic disk containing a hard drive. Alternatively, the recording medium 200 may be an optical disc such as a CD (Compact Disk), DVD (Digital Versatile Disk), or Blu-ray Disc, or a main memory device or auxiliary memory device in other computer systems.

[0031] The support program SP includes the first estimated model EM1 with undetermined coefficient values ​​and the Bayesian estimation program BE. Hereinafter, the first estimated model EM1 with undetermined coefficient values ​​will sometimes be referred to as "pre-determined model PEM".

[0032] In this embodiment, the first presumed model EM1 includes the following equation (1). In the following equation (1), A represents the area of ​​the substrate. D(x) represents the power law. The variable x represents the particle size. The power law D(x) includes the first coefficient k and the second coefficient q. The pre-determined model PEM represents the first presumed model EM1 in which the values ​​of the first coefficient k and the second coefficient q are not determined. A×D(x)=A×kx-q … (1)

[0033] By inputting arbitrary values ​​into the variable x of the first estimation model EM1 after determining the values ​​of the first coefficient k and the second coefficient q, the number of particles with the particle size input to the variable x (the number of particles per substrate) can be calculated. In detail, by inputting arbitrary values ​​into the variable x, the number of particles per unit area is calculated according to the formula of the power law D(x). Therefore, according to formula (1), the number of particles per unit area is multiplied by the area of ​​the substrate to calculate the number of particles per substrate. In this embodiment, the terminal device 101A establishes an estimation model (first estimation model EM1) for estimating the number of particles per substrate.

[0034] The Bayesian estimation program BE includes a computer program that determines the values ​​of the first coefficient k and the second coefficient q by Bayesian estimation. In this embodiment, the Bayesian estimation program BE includes a first prior distribution PD1, a second prior distribution PD2, and a likelihood function LF. The first prior distribution PD1 represents the prior distribution of the first coefficient k. The second prior distribution PD2 represents the prior distribution of the second coefficient q. The first prior distribution PD1, the second prior distribution PD2, and the likelihood function LF are used for Bayesian estimation.

[0035] The first prior distribution PD1 may include a prior distribution without information. For example, the first prior distribution PD1 may be a seminormal distribution or a Cauchy distribution. In this embodiment, the first prior distribution PD1 uses a seminormal distribution with an expected value of 0 and a standard deviation of 100. The second prior distribution PD2 includes a uniform distribution within a predetermined range. In this embodiment, the second prior distribution PD2 uses a uniform distribution within a range of 2.5 to 3.0. The range of the uniform distribution can be determined experimentally based on actual measurements (observations) of the particle size and number. The likelihood function LF may include a probability mass function. In this embodiment, the likelihood function LF uses a Poisson distribution.

[0036] The terminal device 101A includes an operation unit 102, a display unit 103, an interface unit 104, a memory unit 105, and a processing unit 106.

[0037] The operation unit 102 includes a user interface device for operation by an operator. The operation unit 102 inputs signals corresponding to the operator's operation to the processing unit 106. The operation unit 102 may also include, for example, a keyboard, a mouse, and a touch sensor. The touch sensor may also be superimposed on the display surface of the display unit 103. A graphical user interface (GUI) may also be formed by superimposing the touch sensor on the display surface of the display unit 103. For example, the operator may operate the operation unit 102 to instruct the installation of the support program SP. Furthermore, the operator may operate the operation unit 102 to instruct the execution of the support program SP.

[0038] The display unit 103 is controlled by the processing unit 106 to display various screens. For example, the display unit 103 displays the curve of the first estimated model EM1. The display unit 103 may include a display device such as a liquid crystal display device or an organic electroluminescence (EL) display device.

[0039] Interface 104 exchanges information, data, or signals with recording medium 200. Specifically, interface 104 reads support program SP from recording medium 200. The support program SP read from recording medium 200 is stored in memory unit 105 by processing unit 106. As a result, support program SP is installed on terminal device 101A.

[0040] For example, the interface 104 may also be electrically connected to the recording medium 200, allowing for the input and output of information, data, or signals between the interface 104 and the recording medium 200. Specifically, the interface 104 may include a slot or a USB terminal. For example, a card-shaped information carrier such as an SD memory card can be inserted into the slot. For example, a USB memory can be inserted into the USB terminal, or the other end of a USB cable electrically connected to a hard drive can be inserted into the USB terminal. Alternatively, the interface 104 may also include an optical disc drive. The optical disc drive reads information (data) from CDs, DVDs, or Blu-ray Discs.

[0041] The memory unit 105 has a main memory device. The main memory device may include, for example, a semiconductor memory. The memory unit 105 may also have an auxiliary memory device. The auxiliary memory device may include, for example, at least one of a semiconductor memory and a hard disk. The memory unit 105 stores various computer programs and various data. Specifically, the memory unit 105 stores a first data set LD1. In addition, the memory unit 105 stores a support program SP installed from the recording medium 200. The support program SP is non-temporarily recorded in the memory unit 105. The memory unit 105 is a recording medium that non-temporarily records the support program SP.

[0042] Data set LD1 contains the values ​​of the explanatory variables and the values ​​of the target variables corresponding to the values ​​of the explanatory variables. The values ​​of the explanatory variables represent the observed values ​​of the particle size, i.e., the first observations. The values ​​of the target variables represent the observed values ​​of the number of particles, i.e., the second observations. In detail, the second observations (target variables) represent the number of particles per substrate. For example, data set LD1 represents the number of particles per substrate for each particle size obtained from the measurement results of the optical measuring device.

[0043] The processing unit 106 includes a processor. The processing unit 106 may include, for example, a central processing unit (CPU), a graphics processing unit (GPU), a neural network processing unit (NPU), or a quantum computer. Alternatively, the processing unit 106 may include a general-purpose computing device or a special-purpose computing device. For example, the processing unit 106 may also include a field-programmable gate array (FPGA) or an application-specific integrated circuit (ASIC).

[0044] The processing unit 106 executes the computer program stored in the memory unit 105 according to the instructions input by the operator via the operation unit 102, thereby performing various processes such as numerical calculation, information processing, and machine control. For example, the processing unit 106 reads the support program SP from the recording medium 200 via the interface 104 and stores the support program SP in the memory unit 105.

[0045] In this embodiment, the processing unit 106 executes the support program SP to establish an estimation model for the estimated number of particles of each particle size. Specifically, the processing unit 106 establishes an estimation model (first estimation model EM1) for the estimated number of particles of each particle size on each substrate. Specifically, the processing unit 106 executes the Bausch estimation program BE to determine the values ​​of the first coefficient k and the second coefficient q, and inputs the determined values ​​of the first coefficient k and the second coefficient q into the pre-determination model PEM to establish the first estimation model EM1.

[0046] Next, the support method, recording medium 200, support program SP, and support system 100A of this embodiment will be described with reference to FIGS. 1 to 4. FIG. 2 is a flowchart showing the support method of this embodiment. Here, the support method refers to a method for supporting the operation of estimating the number of particles on a substrate. In this embodiment, the support method includes a model building step of building a first estimating model EM1. FIG. 2 shows the flowchart of the model building step. As shown in FIG. 2, the model building step includes step S1 and step S2.

[0047] Step S1 represents the coefficient value determination step, which uses the first data set LD1 and Bayesian estimation to determine the values ​​of the first coefficient k and the second coefficient q included in the pre-determined model PEM. Step S2 represents the step of inputting the values ​​determined in the coefficient value determination step (step S1) into the first coefficient k and the second coefficient q to establish the first estimated model EM1.

[0048] The support method of this embodiment is executed, for example, by the terminal device 101A included in the support system 100A described with reference to FIG1. ​​More specifically, the support method shown in FIG2 is executed by the terminal device 101A executing the support program SP read from the recording medium 200. In this case, the flowchart shown in FIG2 corresponds to the processing flow executed by the processing unit 106 included in the terminal device 101A. In other words, FIG2 shows the processing executed by the processing unit 106 included in the support system 100A of this embodiment.

[0049] The processing (support method) shown in Figure 2 is initiated by an operator using the operation unit 102 to instruct the execution of the support program SP. When the processing unit 106 begins the processing shown in Figure 2, it executes the coefficient value determination process (step S1). Specifically, the processing unit 106 uses the first dataset LD1 and determines the values ​​of the first coefficient k and the second coefficient q using Bayesian estimation. Then, the processing unit 106 inputs the values ​​determined by the coefficient value determination process into the first coefficient k and the second coefficient q of the pre-determined model PEM to establish the first estimated model EM1 (step S2). As a result, the processing shown in Figure 2 ends.

[0050] More specifically, the processing unit 106 inputs the values ​​determined by the coefficient value determination process into the first coefficient k and the second coefficient q of the pre-determination model PEM, and inputs the value of the substrate area into the constant A of the pre-determination model PEM. The value of the substrate area can be input, for example, by an operator operating the operation unit 102.

[0051] Next, the step of determining the coefficient value (step S1 in Figure 2) will be explained with reference to Figure 3. Figure 3 is a flowchart showing the step of determining the coefficient value. As shown in Figure 3, the step of determining the coefficient value includes steps S11 and S12.

[0052] When performing the coefficient value determination process, the processing unit 106 repeats the step of sampling the temporary values ​​of the first coefficient k and the second coefficient q from the prior distribution of the first coefficient k (first prior distribution PD1) and the prior distribution of the second coefficient q (second prior distribution PD2) respectively, to obtain the post-distribution of the first coefficient k and the post-distribution of the second coefficient q (step S11). Then, the processing unit 106 determines the values ​​of the first coefficient k and the second coefficient q respectively based on the post-distribution of the first coefficient k and the post-distribution of the second coefficient q (step S12). As a result, the process shown in FIG3 ends.

[0053] More specifically, the processing unit 106 randomly samples temporary values ​​of the first coefficient k and the second coefficient q from the first prior distribution PD1 and the second prior distribution PD2, respectively, according to the Markov chain Monte Carlo method. The algorithm of the Markov chain Monte Carlo method is not particularly limited, for example, it can be a No-U-Turn-Sampler (NUTS) or Gibbs sampling.

[0054] Furthermore, when determining the values ​​of the first coefficient k and the second coefficient q, the processing unit 106 obtains a representative value from the post-hoc distribution of the first coefficient k as the value of the first coefficient k. Similarly, the processing unit 106 obtains a representative value from the post-hoc distribution of the second coefficient q as the value of the second coefficient q. The representative value may be, for example, the expected value (average) or the median value.

[0055] Next, the steps for sampling the temporary values ​​of the first coefficient k and the second coefficient q will be described with reference to FIG4 (step S11 in FIG3). FIG4 is a flowchart showing the steps for sampling the temporary values ​​of the first coefficient k and the second coefficient q. As shown in FIG4, the steps for sampling the temporary values ​​of the first coefficient k and the second coefficient q include steps S111 to S113.

[0056] First, the processing unit 106 inputs the values ​​sampled from the first prior distribution PD1 (i.e., the first sampled value) and the values ​​sampled from the second prior distribution PD2 (i.e., the second sampled value) into the first coefficient k and the second coefficient q contained in the determined pre-model PEM to establish a temporary model (step S111).

[0057] Next, the processing unit 106 evaluates the fit of the temporary model to the first data set LD1 based on the established temporary model, the first data set LD1, and the likelihood function LF (step S112). Then, based on the evaluated fit, the processing unit 106 decides whether to add the values ​​of the first coefficient k and the second coefficient q (the first sample value and the second sample value) of this sample to the post-distribution (step S113).

[0058] In detail, the processing unit 106 inputs the values ​​of the explanatory variables (first observations) contained in the first dataset LD1 into the temporary model, and obtains the number of temporary particles (temporary estimates) from the temporary model. Specifically, the processing unit 106 obtains the number of temporary particles (temporary estimates) for each value of the explanatory variable contained in the first dataset LD1. Then, the processing unit 106 evaluates the fit of the temporary model to the first dataset LD1 based on the values ​​of the target variables contained in the first dataset LD1 (second observations), the likelihood function LF, and the obtained number of temporary particles.

[0059] More specifically, when the likelihood function LF is a Boisson distribution, the processing unit 106 inputs the obtained temporary particle number to the coefficient λ of the probability mass function of the Boisson distribution expressed by the following equation (2).

[0060] Next, the processing unit 106 inputs the value of the target variable (the second observation value) contained in the first data set LD1 to the variable x of the probability mass function of the Boisson distribution, so as to calculate the probability (value of the probability mass function) of the second observation value (the actual measurement value of the number of particles) generated on the Boisson distribution.

[0061] For each value of the explanatory variable (first observation) contained in the first dataset LD1, the processing unit 106 obtains the probability (value of the probability mass function) of the corresponding value of the target variable (second observation) on the Boisson distribution. Then, it evaluates the fit of the temporary model to the first dataset LD1 based on the value obtained by multiplying by the obtained probability. The value obtained by multiplying by the obtained probability is an example of an "evaluation value". Specifically, the larger the evaluation value (the value obtained by multiplying by the probability), the better the temporary model fits to the first dataset LD1. In other words, the larger the evaluation value, the better the temporary model fits to the first dataset LD1. The processing unit 106 may also decide whether to add the sampled values ​​to the post-distribution based on, for example, the algorithm of the Markov chain Monte Carlo method.

[0062] Next, the support method, recording medium 200, support program SP, and support system 100A of this embodiment will be described with reference to Figures 1 and 5. Figure 5 is a graph showing an example of the predicted distribution of the number of particles per substrate. In Figure 5, the horizontal axis represents the particle size, and the vertical axis represents the number of particles per substrate. Furthermore, the black dots represent the observations included in the first dataset LD1, and the solid line represents the curve of the first estimation model EM1. The curve of the first estimation model EM1 represents the estimated value of the number of particles (number of particles per substrate) for the particle size.

[0063] As shown in FIG. 5, the processing unit 106 executes the support program SP and displays the curve (estimated value of particle count) of the first estimated model EM1 on the display unit 103. In this embodiment, the processing unit 106 further executes the support program SP and displays the predicted distribution of particle count for each substrate on the display unit 103. The predicted distribution is, for example, a Boisson distribution. As shown in FIG. 5, the processing unit 106 can also display, for example, a range of 5% to 95% of the Boisson distribution (predicted distribution) on the display unit 103.

[0064] In detail, the processing unit 106 constructs a predicted distribution of the first coefficient k based on the subsequent distribution of the first coefficient k. Similarly, the processing unit 106 constructs a predicted distribution of the second coefficient q based on the subsequent distribution of the second coefficient q. Then, the processing unit 106 displays the predicted distribution of the number of particles in each substrate on the display unit 103 based on the predicted distribution of the first coefficient k and the predicted distribution of the second coefficient q.

[0065] For example, the processing unit 106 may also use the range of 5% to 95% of the Boisson distribution (predicted distribution) of the first coefficient k and the range of 5% to 95% of the Boisson distribution (predicted distribution) of the second coefficient q to display the Boisson distribution (predicted distribution) of the particle count on the display unit 103. In this case, the range of 5% to 95% of the Boisson distribution (predicted distribution) of the particle count is displayed on the display unit 103.

[0066] The present invention embodiment 1 has been described above with reference to FIGS. 1 to 5. Embodiment 1 supports the estimation of the number of particles on a substrate. More specifically, according to Embodiment 1, the values ​​of the first coefficient k and the second coefficient q of the power law D(x) are determined by Bayes estimation. As a result, it mitigates the overfitting of the model (curve) of the power law D(x) to the observed data. Therefore, operators can more accurately estimate the number of particles of any size using the first estimation model EM1. For example, by operating the operation unit 102 to input an arbitrary particle size into the first estimation model EM1, operators can obtain an estimated value of the number of particles with the input particle size.

[0067] Furthermore, as described below, the number of particles that can be accurately estimated using the first estimation model EM1 may be limited to the number of particles with a diameter greater than 7 nm. However, optical measuring devices cannot measure particles with a diameter less than 10 nm. In contrast, according to Embodiment 1, the number of particles with a diameter greater than 7 nm and less than 10 nm can be estimated more accurately. Therefore, operators can obtain an estimated value for the number of particles with a diameter that cannot be measured by optical measuring devices.

[0068] Furthermore, according to implementation form 1, a uniform distribution within a predetermined range is used as the prior distribution of the second coefficient q. Therefore, compared with the case where a semi-normal distribution is used for the prior distribution of the second coefficient q, the model (curve) of the first presupposition model EM1 is less likely to overfit the observed data.

[0069] Furthermore, according to Embodiment 1, the Markov chain Monte Carlo method is used to sample the first coefficient k and the second coefficient q. Therefore, a predicted distribution of the particle number can be constructed based on the ex-post distribution of the first coefficient k and the second coefficient q. Thus, in addition to the estimated value of the particle number, the predicted distribution of the particle number can also be visualized. This further supports operators in estimating the particle number.

[0070] Furthermore, the optical measuring device produces slight variations in each measurement, and particles with the same diameter may be measured as particles of different diameters. Moreover, the larger the number of particles, the more pronounced this phenomenon becomes. In contrast, according to Embodiment 1, by displaying the predicted distribution of the particle number on the display unit 103, this phenomenon can be visualized. Therefore, it can further support operators in estimating the particle number.

[0071] Furthermore, in the embodiments described with reference to Figures 1 to 5, the support system 100A (terminal device 101A) obtains the support program SP from the recording medium 200, but the support system 100A (terminal device 101A) can also obtain the support program SP from other computer systems. For example, the terminal device 101A can also be connected to other computer systems via a cable-based communication method to obtain the support program SP from other computer systems. Alternatively, the terminal device 101A can also be connected to other computer systems via a network such as the Internet to obtain the support program SP from other computer systems. The other computer system can be a general-purpose computer or a special-purpose computer. The other computer system can also be a server.

[0072] Furthermore, in the embodiments described with reference to Figures 1 to 5, the support program SP is installed on the terminal device 101A, but the support program SP may not be installed on the terminal device 101A. The terminal device 101A may also execute the support program SP stored in the recording medium 200.

[0073] [Embodiment 2] Next, Embodiment 2 of the present invention will be described with reference to FIGS. 2 to 6. However, only the differences from Embodiment 1 will be described, and the descriptions of the same aspects as Embodiment 1 will be omitted. The difference between Embodiment 2 and Embodiment 1 is that the server 300 establishes a first presumed model EM1 based on the support program SP. More specifically, in Embodiment 2, the server 300 determines the values ​​of the first coefficient k and the second coefficient q of the first presumed model EM1.

[0074] Figure 6 is a block diagram showing the configuration of the support system 100B in Embodiment 2. As shown in Figure 6, the support system 100B includes a terminal device 101B and a server 300.

[0075] In embodiment 2, the terminal device 101B includes an operation unit 102, a display unit 103, a memory unit 105, a communication unit 107, and a processing unit 106.

[0076] The communication unit 107 is connected to a network and performs communication with the server 300. The network includes, for example, the Internet, a local area network (LAN), a public telephone network, and a short-range wireless network. The communication unit 107 includes a communicator. The communication unit 107 is, for example, a network interface controller.

[0077] The communication unit 107 is controlled by the processing unit 106 and exchanges information, data, or signals with the server 300. For example, the communication unit 107 sends the first data set LD1 and information indicating the area of ​​the substrate to the server 300.

[0078] The server 300 includes a communication unit 301, a memory unit 302, and a processing unit 303.

[0079] The communication unit 301 is connected to a network and performs communication with the terminal device 101B. The communication unit 301 includes a communicator. The communication unit 301 is, for example, a network interface controller. The communication unit 301 is controlled by the processing unit 303 to exchange information, data, or signals with the terminal device 101B. For example, the communication unit 301 receives a first data set LD1 and information indicating the area of ​​the substrate from the terminal device 101B.

[0080] The memory unit 302 has a main memory device and an auxiliary memory device. The main memory device includes, for example, a semiconductor memory. The auxiliary memory device includes, for example, a hard disk. The memory unit 302 stores the support program SP. Furthermore, the memory unit 302 stores the first data set LD1 received from the terminal device 101B and information indicating the area of ​​the substrate.

[0081] The processing unit 303 includes a processor. The processing unit 303 may include, for example, a CPU, GPU, NPU, or quantum computer. Alternatively, the processing unit 303 may include a general-purpose computing device or a special-purpose computing device. For example, the processing unit 303 may also include an FPGA or ASIC. The processing unit 303 executes the support program SP according to commands from the terminal device 101B.

[0082] Next, the support method, support program SP, and support system 100B of Embodiment 2 will be described with reference to FIGS. 2 to 6. The processing unit 303 starts the processing shown in FIGS. 2 to 4 according to the command from the terminal device 101B.

[0083] When the processing unit 303 starts the processing shown in FIG2, it performs the coefficient value determination process (step S1). Specifically, the processing unit 303 performs the processing described with reference to FIG3 and FIG4 to determine the values ​​of the first coefficient k and the second coefficient q. Then, the processing unit 303 inputs the values ​​determined by the coefficient value determination process to the first coefficient k and the second coefficient q of the pre-determined model PEM to establish the first estimated model EM1 (step S2). As a result, the processing shown in FIG2 ends.

[0084] More specifically, the processing unit 303 sends the established first estimation model EM1 to the terminal device 101B. As a result, the operator using the terminal device 101B can use the first estimation model EM1 to estimate the number of particles on the substrate. For example, the processing unit 106 of the terminal device 101B can also display the curve (estimated value of the number of particles) of the first estimation model EM1 on the display unit 103 based on the first estimation model EM1 received from the server 300.

[0085] In this embodiment, the processing unit 303 further constructs a predicted distribution of the number of particles as described with reference to FIG5. Then, the processing unit 303 sends information representing the predicted distribution of the number of particles to the terminal device 101B. The processing unit 106 of the terminal device 101B displays the predicted distribution of the number of particles received from the server 300 on the display unit 103.

[0086] The present invention embodiment 2 has been described above with reference to FIGS. 2 to 6. According to embodiment 2, similar to embodiment 1, the operation of estimating the number of particles on the substrate can be supported.

[0087] [Embodiment 3] Next, Embodiment 3 of the present invention will be described with reference to FIGS. 7 and 8. However, only the differences from Embodiments 1 and 2 will be described, while the differences from Embodiments 1 and 2 will be omitted. The difference between Embodiment 3 and Embodiments 1 and 2 is that the control device 10 included in the substrate processing system 1000 is also used as the support system 100C.

[0088] FIG7 is a schematic diagram of the substrate processing system 1000 of this embodiment. More specifically, FIG7 is a schematic top view of the substrate processing apparatus 400 included in the substrate processing system 1000 of this embodiment.

[0089] As shown in FIG. 7, the substrate processing system 1000 includes a substrate processing apparatus 400 and a control device 10. The substrate processing apparatus 400 processes the substrate W. The control device 10 controls the substrate processing apparatus 400. In Embodiment 3, the substrate W is a disk-shaped semiconductor wafer. Furthermore, the substrate processing apparatus 400 is a monolithic device that processes the substrate W one wafer at a time.

[0090] In detail, the substrate processing apparatus 400 includes: a plurality of substrate processing units 2, a fluid cabinet 401, a plurality of fluid tanks 402, a plurality of loading ports LP, an indexer robot IR, and a center robot CR.

[0091] The cassette CA is placed in each loading port LP. The cassette CA stacks and houses multiple substrates W. The cassette CA is, for example, a front-opening unified pod (FOUP), a standard mechanical interface (SMIF) cassette, or an open cassette (OC).

[0092] The indexing robot IR transports the substrate W between the box CA and the central robot CR. The central robot CR transports the substrate W between the indexing robot IR and a plurality of substrate processing units 2. Alternatively, it can be configured such that a platform (path) for temporarily placing the substrate W is provided between the indexing robot IR and the central robot CR, and the substrate W is indirectly transferred between the indexing robot IR and the central robot CR via the platform.

[0093] A plurality of substrate processing units 2 form a plurality of towers TW (4 towers TW in FIG7). The plurality of towers TW are configured to surround the central robot CR from a top view. Each tower TW contains a plurality of substrate processing units 2 stacked on top of each other (3 substrate processing units 2 in FIG7).

[0094] Fluid cabinet 401 contains fluid. Specifically, fluid cabinet 401 is used to contain processing liquid. Alternatively, fluid cabinet 401 may also contain processing liquid and gas.

[0095] The processing solution is not particularly limited as long as it is a liquid that comes into contact with the substrate W. Examples of processing solutions include dilute hydrofluoric acid (DHF), hydrofluoric acid (HF), nitric acid (a mixture of hydrofluoric acid and nitric acid (HNO3)), buffered hydrofluoric acid (BHF), ammonium fluoride, HFEG (a mixture of hydrofluoric acid and ethylene glycol), phosphoric acid (H3PO4), sulfuric acid, acetic acid, nitric acid, hydrochloric acid, ammonia, hydrogen peroxide solution, organic acids (e.g., citric acid, oxalic acid), organic bases (e.g., TMAH: tetramethylammonium hydroxide), sulfuric acid-hydrogen peroxide solution (SPM), ammonia-hydrogen peroxide solution (SC1), hydrochloric acid-hydrogen peroxide solution (SC2), isopropanol (IPA), surfactants, preservatives, pure water (e.g., deionized water), carbonated water, electrolyzed ionized water, hydrogen water, ozone water, or hydrochloric acid solution diluted to a concentration of (e.g., approximately 0.001% by weight to approximately 0.01% by weight). Gases may include, for example, inert gases. Inert gases, such as nitrogen, can be used.

[0096] The fluid tank 402 corresponds to one of the plurality of towers TW. The fluid in the fluid tank 401 is supplied to all the substrate processing units 2 contained in the corresponding tower TW via any one of the fluid tanks 402.

[0097] Each substrate processing unit 2 processes substrate W one by one. Specifically, each substrate processing unit 2 supplies processing liquid to substrate W to process substrate W. For example, each substrate processing unit 2 performs cleaning or etching processing on substrate W.

[0098] The control device 10 controls the operation of each part of the substrate processing apparatus 400. For example, the control device 10 controls the substrate processing unit 2, the fluid cabinet 401, the fluid tank 402, the loading port LP, the indexing robot IR, and the center robot CR. The control device 10 includes a control unit 11 and a memory unit 12.

[0099] The control unit 11 controls the operation of each part of the substrate processing device 400 based on various information stored in the memory unit 12. The control unit 11 includes a processor. The control unit 11 may include, for example, a CPU, GPU, NPU, or quantum computer. Alternatively, the control unit 11 may include a general-purpose computing device or a special-purpose computing device. For example, the control unit 11 may also include an FPGA or ASIC.

[0100] The memory unit 12 stores various information used to control the operation of the substrate processing device 400. For example, the memory unit 12 stores various data and various computer programs. The various data includes process recipe data. The process recipe data represents a process recipe that defines the processing content, processing conditions, and processing procedures of the substrate W. Various setting values ​​(process recipe parameter values) are set in the process recipe as processing conditions.

[0101] The memory unit 12 has a main memory device. The main memory device includes, for example, a semiconductor memory. The memory unit 12 may also have an auxiliary memory device. The auxiliary memory device includes, for example, at least one of a semiconductor memory and a hard disk.

[0102] FIG8 is a block diagram showing the configuration of the support system 100C included in the substrate processing system 1000 of this embodiment. More specifically, FIG8 is a block diagram showing the configuration of the control device 10.

[0103] As shown in Figures 7 and 8, the control device 10 also serves as the support system 100C. Specifically, as shown in Figure 8, the support program SP is installed from the recording medium 200 to the control device 10. Similar to the terminal device 101A described with reference to Figures 1 to 5, the control device 10 executes the support program SP installed from the recording medium 200 to determine the values ​​of the first coefficient k and the second coefficient q of the first estimated model EM1. The control device 10 is an example of a "support device".

[0104] Specifically, as shown in FIG8, the control device 10 further includes an operation unit 13, a display unit 14, and an interface unit 15. The configuration of the operation unit 13, the display unit 14, and the interface unit 15 is the same as that of the operation unit 102, the display unit 103, and the interface unit 104 described with reference to FIG1, therefore, their description is omitted.

[0105] The memory unit 12, like the memory unit 105 described with reference to FIG1, stores the first data set LD1 and the support program SP that is read from the recording medium 200 through the interface 15.

[0106] The control unit 11, similar to the processing unit 106 described with reference to Figures 1 to 5, executes the support program SP to determine the values ​​of the first coefficient k and the second coefficient q, and establishes the first estimation model EM1.

[0107] The present invention embodiment 3 has been described above with reference to Figures 7 and 8. According to embodiment 3, similar to embodiments 1 and 2, it can support the operation of estimating the number of particles on the substrate.

[0108] Furthermore, in Embodiment 3, the substrate processing apparatus 400 performs cleaning or etching processing on the substrate W. However, the substrate processing apparatus 400 is only required to process the substrate and is not particularly limited. For example, the substrate processing apparatus 400 may also be a coating apparatus, a developing apparatus, an exposure apparatus, a baking apparatus, or a film forming apparatus.

[0109] Furthermore, in Embodiment 3, the substrate processing apparatus 400 is a single-piece type, but the substrate processing apparatus 400 may also be a batch type.

[0110] [Embodiment 4] Next, Embodiment 4 of the present invention will be described with reference to FIGS. 2 to 4, 9, 10, 11A and 11B. However, the following description focuses on the differences from Embodiments 1 to 3, while omitting the description of the same aspects as Embodiments 1 to 3. The difference between Embodiment 4 and Embodiments 1 to 3 is that the terminal device 101A executes the support program SP to establish the second presumed model EM2.

[0111] Figure 9 is a block diagram showing the structure of the support system 100A in implementation form 4. As shown in Figure 9, the support program SP includes the pre-determined model PEM and the Bayesian estimation program BE, as well as the pre-learning machine learning model GP and the second dataset creation program CP.

[0112] The machine learning model GP contains a model that can output a predictive distribution. The machine learning model GP performs machine learning on the second dataset LD2 and outputs the expected value of the predicted distribution of the number of particles (target variable) for each substrate. The algorithm of the machine learning model GP only needs to contain an algorithm that can output a predictive distribution, and there are no particular limitations. For example, the machine learning model GP can contain a Gaussian process regression model. For example, the machine learning model GP can also be a Gaussian process regression model that uses the Poisson distribution as the likelihood function.

[0113] The second dataset creation program CP is a computer program that creates the second dataset LD2 from the first presumed model EM1. Specifically, the processing unit 106 executes the second dataset creation program CP to input a plurality of predetermined values ​​as explanatory variables (particle size) into the first presumed model EM1, and outputs a plurality of values ​​of the target variable (particle number) from the first presumed model EM1 to obtain the second dataset LD2. For example, the processing unit 106 can also execute the second dataset creation program CP to input 500 particle sizes xi (i = 1, 2, ..., 500) arranged at equal intervals in the range of 7 nm to 60 nm into the first presumed model EM1 to obtain the particle number of 500 points.

[0114] Next, the support method, recording medium 200, support program SP, and support system 100A of Embodiment 4 will be described with reference to Figures 2 to 4, 9, and 10. Figure 10 is a flowchart showing the support method of Embodiment 4. In detail, Figure 10 shows the flow of the model building steps for building the second presumed model EM2.

[0115] The model building steps shown in Figure 10 can also begin after step S2 shown in Figure 2. That is, the model building steps shown in Figure 10 can also begin after the first presumed model EM1 is built. As shown in Figure 10, the model building steps of embodiment 4 further include steps S3 to S5.

[0116] Step S3 represents the following steps: inputting a plurality of predetermined values ​​as explanatory variables (particle size) into the first presupposition model EM1, and outputting a plurality of values ​​of the target variable (particle number) from the first presupposition model EM1 to obtain the second dataset LD2. Step S4 represents the step of adding additional information to the second dataset LD2. Step S5 represents the following steps: performing machine learning on the second dataset LD2 with the added additional information to establish a presupposition model (second presupposition model EM2) that outputs the expected value of the predicted distribution of the target variable (particle number). The second presupposition model EM2 is the machine learning model GP after performing machine learning on the second dataset LD2 with the added additional information.

[0117] The support method of Embodiment 4 is executed, for example, by the terminal device 101A included in the support system 100A described with reference to FIG9. More specifically, the support method shown in FIG10 is executed by the terminal device 101A executing the support program SP read from the recording medium 200. In this case, the flowchart shown in FIG10 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, FIG10 shows the processing executed by the processing unit 106 included in the support system 100A of Embodiment 4.

[0118] The processing unit 106 may also start the processing shown in FIG10 after performing the processing described with reference to FIGS. 2 to 4. In other words, the processing unit 106 may also start the processing shown in FIG10 after establishing the first presumed model EM1.

[0119] When the processing unit 106 starts the processing shown in FIG10, it executes the second data set creation program CP to obtain the second data set LD2 (step S3). Specifically, the processing unit 106 inputs a plurality of predetermined values ​​as explanatory variables (particle size) to the first presumed model EM1, and outputs a plurality of values ​​of the target variable (particle number) from the first presumed model EM1 to obtain the second data set LD2. The second data set LD2 includes a plurality of values ​​of the explanatory variables input to the first presumed model EM1, and a plurality of values ​​of the target variable output from the first presumed model EM1.

[0120] For example, as explained above, the processing unit 106 can also input the particle size xi (i = 1, 2, ..., 500) of 500 points into the first presupposition model EM1 to obtain the particle number value of 500 points. In detail, the processing unit 106 performs processing on the target variable value output from the first presupposition model EM1 according to the following formula (3) to obtain the second dataset LD2. Furthermore, in formula (3), round() indicates that the value within the parentheses (in formula (3), is the value of the target variable output from the first presupposition model EM1) is rounded. yi = round(A × D(xi)) … (3)

[0121] Next, the processing unit 106 adds the additional information to the second data set LD2 (step S4). The additional information includes the information of the origin (0,0). In other words, the processing unit 106 adds the origin (0,0) information to the second data set LD2. The origin (0,0) information can also be pre-included in the support program SP.

[0122] Furthermore, the processing unit 106 can also add information on the number of particles and particle size of each substrate obtained from a measuring device other than an optical measuring device to the second data set LD2 as supplementary information. Specifically, information on the number of particles of each particle size less than 10 nm can also be added to the second data set LD2 as supplementary information. According to this embodiment, information on particles with a particle size less than 10 nm that cannot be measured by an optical measuring device can be added to the second data set LD2.

[0123] Next, the processing unit 106 enables the machine learning model GP to perform machine learning on the second dataset LD2 with the added information to build a second inference model EM2. As a result, the processing shown in FIG10 ends.

[0124] Next, the support method, recording medium 200, support program SP, and support system 100A of this embodiment will be described with reference to Figures 9, 11A, and 11B. Figures 11A and 11B are diagrams showing an example of the predicted distribution of the number of particles in each substrate. In Figures 11A and 11B, the horizontal axis represents the particle size, and the vertical axis represents the number of particles in each substrate.

[0125] As shown in Figures 11A and 11B, the processing unit 106 executes the support program SP and displays the predicted distribution output from the second estimation model EM2 (machine learning model GP) on the display unit 103. The expected value of the predicted distribution represents the estimated value of the number of particles with the particle size (the number of particles per substrate). The dispersion of the predicted distribution represents the estimated value of the deviation from the number of particles with the particle size (the number of particles per substrate).

[0126] In detail, Figure 11A shows the predicted distribution output from the following machine learning model GP (Second Prediction Model EM2), which performs machine learning on the second dataset LD2 with only the origin (0,0) information added as supplementary information. Figure 11B shows the predicted distribution output from the following machine learning model GP (Second Prediction Model EM2), which performs machine learning on the second dataset LD2 with the origin (0,0) information added and particle information representing a particle size (explanatory variable) of 5 nm and a particle number (target variable) of 5 as supplementary information. As shown in Figures 11A and 11B, by adding particle information with a particle size of less than 10 nm, which cannot be measured by optical measuring devices, as supplementary information to the second dataset LD2, the supplementary particle information can be reflected in the approximate shape of the predicted distribution.

[0127] Furthermore, inspired by studies using atomic force microscopy (AFM) or cryo-electron microscopy (cryo-EM), the number of particles with a diameter of 7 nm or less may not follow the power law D(x). In Figures 11A and 11B, for ease of understanding, the curve of the first hypothetical model EM1 is represented by a dashed line.

[0128] In detail, as shown in Figures 11A and 11B, the closer the value of the explanatory variable (particle size) is to 0 nm, the more the estimated value output from the first estimation model EM1 increases exponentially. However, the number of particles with a particle size of 0 nm is expected to be 0. Furthermore, inspired by experimental examples using atomic force microscopy or cryo-electron microscopy, the number of particles may converge to 0 before the particle size reaches 0 nm.

[0129] In contrast, according to Embodiment 4, by having the machine learning model GP perform machine learning on the dataset (second dataset LD2) obtained by inputting values ​​with a particle size of 7 nm or larger into the first estimation model EM1, as shown in Figures 11A and 11B, the expected value of the predicted distribution output from the second estimation model EM2 can conform to the curve of the first estimation model EM1 within the range of particle sizes greater than 7 nm. Furthermore, by adding the origin (0,0) information to the second dataset LD2, as shown in Figures 11A and 11B, the expected value of the predicted distribution output from the second estimation model EM2 can converge toward the origin (0,0) within the range of particle sizes less than 7 nm.

[0130] Therefore, according to Embodiment 4, the operator can more accurately estimate (predict) the number of particles of any size using the second estimation model EM2. For example, the operator can input any particle size into the second estimation model EM2 by operating the operation unit 102, and obtain an estimated value (predicted value) of the number of particles with the input particle size. In particular, according to Embodiment 4, the number of particles with particle sizes that cannot be measured by optical measuring devices can be estimated (predicted) more accurately. Furthermore, according to Embodiment 4, it is possible to more accurately estimate (predict) the number of particles with a particle size of 7 nm or less that do not follow the power law D(x).

[0131] The present invention embodiment 4 has been described above with reference to FIGS. 2 to 4, 9, 10, 11A and 11B. According to embodiment 4, similar to embodiments 1 to 3, the operation of estimating the number of particles on the substrate can be supported.

[0132] [Embodiment 5] Next, Embodiment 5 of the present invention will be described with reference to FIGS. 12 to 14. However, only the differences from Embodiments 1 to 4 will be described, and the descriptions of the same aspects as Embodiments 1 to 4 will be omitted. The difference between Embodiment 5 and Embodiments 1 to 4 is that the values ​​of the target variables of the dataset used for Bayes optimization are established using the first presupposition model EM1.

[0133] Figure 12 is a block diagram showing the structure of the support system 100A in implementation form 5. As shown in Figure 12, the support program SP includes the pre-determined model PEM and the Bayesian estimation program BE, and further includes the first target variable establishment program VC1. The first target variable establishment program VC1 is a computer program that uses the first pre-determined model EM1 to establish the values ​​of the target variables of the dataset used for Bayesian optimization.

[0134] Here, the target variable in the dataset used for Bayes optimization includes particle density. Furthermore, the explanatory variables in the dataset used for Bayes optimization include at least one parameter among various parameters defining the operation of the substrate processing apparatus. Bayes optimization is used to determine the optimal solution for the explanatory variables (at least one parameter).

[0135] The explanatory variables used for Bausch optimization may also include at least one of the various set values ​​defined by the process recipe (process recipe parameter values) and the various set values ​​set for the substrate processing apparatus (apparatus parameter values). The substrate processing apparatus may be any apparatus that processes a substrate. The substrate processing apparatus may include, for example, a cleaning apparatus, an etching apparatus, a coating apparatus, a developing apparatus, an exposure apparatus, a baking apparatus, and a film forming apparatus.

[0136] Next, the support method, recording medium 200, support program SP, and support system 100A of Embodiment 5 will be described with reference to Figures 12 to 14. Figure 13 is a flowchart of the support method of Embodiment 5. Figure 14 is a diagram of the target variable establishment process. In Embodiment 5, the support method includes a target variable establishment step, which establishes the values ​​of target variables for the dataset used in Bayesian optimization. Figure 13 shows the flowchart of the target variable establishment step. As shown in Figure 13, the target variable establishment step includes steps S21 and S22.

[0137] Step S21 represents the coefficient value determination step, which uses the first data set LD1 and Bayes' estimation to determine the values ​​of the first coefficient k and the second coefficient q of the power law D(x). Step S21 is the same as step S1 described with reference to Figures 2 to 4, therefore, its detailed description is omitted.

[0138] Step S22 means obtaining the area D1 of the region above the minimum value set for variable x in the lower region of the curve of the first estimated model EM1, in which the values ​​of the first coefficient k and the second coefficient q have been determined in the coefficient value determination step (step S21). Hereinafter, the minimum value set for variable x is sometimes referred to as "minimum particle size Xmin". The minimum particle size Xmin can also be preset to any value, for example, greater than 7 nm. The reason is that, as explained above, the number of particles with a particle size of less than 7 nm may not follow the power law D(x).

[0139] Step S23 means the following step, namely, obtaining the value of the target variable (particle density) of the dataset used for Bausch optimization based on the area D1 of the region above the minimum particle size Xmin and the area of ​​the substrate.

[0140] The support method of Embodiment 5 is executed, for example, by the terminal device 101A included in the support system 100A described with reference to FIG12. More specifically, the support method shown in FIG13 is executed by the terminal device 101A executing the support program SP read from the recording medium 200. In this case, the flowchart shown in FIG13 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, FIG13 shows the processing executed by the processing unit 106 included in the support system 100A of Embodiment 5.

[0141] The processing (support method) shown in Figure 13 can also be performed by an operator on the operation unit 102 to instruct the start of establishing the value of the target variable for Bayes optimization. When the processing unit 106 starts the processing shown in Figure 13, the target variable establishment process is executed.

[0142] In detail, the processing unit 106 first performs coefficient value determination processing, using the first data set LD1 and by Bayes' estimation to determine the values ​​of the first coefficient k and the second coefficient q contained in the power law D(x) (step S21).

[0143] Next, the processing unit 106 executes the first target variable establishment program VC1. Specifically, as shown in FIG14, the processing unit 106 obtains the area D1 of the region with minimum particle size Xmin and above in the lower region of the curve of the first estimated model EM1, whose values ​​of the first coefficient k and the second coefficient q have been determined by the coefficient value determination process (step S21) (step S22). The processing unit 106 may also calculate the area D1 according to the following formula (4). Then, the processing unit 106 obtains the value of the target variable (particle density) of the dataset used for Bausch optimization based on the area D1 of the region with minimum particle size Xmin and above and the area of ​​the substrate (step S23). As a result, the processing shown in FIG13 ends. In detail, the processing unit 106 divides the area D1 of the region with minimum particle size Xmin and above by the area of ​​the substrate to obtain the value of particle density.

[0144] The present invention embodiment 5 has been described above with reference to FIGS. 12 to 14. According to embodiment 5, similar to embodiments 1 to 4, it can support the estimation of the number of particles on a substrate. Furthermore, according to embodiment 5, it uses a power law D(x) to obtain the particle density; therefore, it can reflect the number of particles with particle sizes that cannot be measured by an optical measuring device in the value of the target variable (particle density) of the dataset used for Bayes optimization. Moreover, according to embodiment 5, the particle density can be obtained using a power law D(x) that mitigates overfitting to the observed data. Therefore, a more accurate particle density can be obtained.

[0145] [Embodiment 6] Next, Embodiment 6 of the present invention will be described with reference to FIGS. 15 to 17. However, only the differences from Embodiments 1 to 5 will be described, and the descriptions of the same aspects as Embodiments 1 to 5 will be omitted. The difference between Embodiment 6 and Embodiments 1 to 5 is that the values ​​of the target variables of the dataset used for Bayes optimization are established using the second estimation model EM2.

[0146] Figure 15 is a block diagram showing the structure of the support system 100A in implementation form 6. As shown in Figure 15, the support program SP includes the pre-determined model PEM, the Bayesian estimation program BE, the pre-learning machine learning model GP, and the second data set creation program CP, and further includes the second target variable creation program VC2.

[0147] The second target variable establishment program VC2 is a computer program that uses the second estimation model EM2 to establish the values ​​of the target variables in the dataset used for Bayes optimization. As explained above, the target variables in the dataset used for Bayes optimization include particle density. Furthermore, the explanatory variables in the dataset used for Bayes optimization include at least one parameter among various parameters that define the operation of the substrate processing device.

[0148] Next, the support method, recording medium 200, support program SP, and support system 100A of Embodiment 6 will be described with reference to Figures 15 to 17. Figure 16 is a flowchart of the support method of Embodiment 6. Figure 17 is a diagram of the target variable establishment process. In Embodiment 6, the support method includes a target variable establishment step, which establishes the values ​​of target variables for the dataset used for Bayes optimization. Figure 16 shows the flowchart of the target variable establishment step. The target variable establishment step shown in Figure 16 can also begin after the establishment of the second presupposition model EM2. As shown in Figure 16, the target variable establishment step includes steps S51 to S53.

[0149] Step S51 represents the step of outputting the expected value of the predicted distribution of the number of particles for each substrate from the second estimation model EM2. Step S52 represents the step of obtaining the area D2 of a predetermined range in the region below the curve of the expected value of the predicted distribution. Step S53 represents the step of obtaining the value of the target variable (particle density) of the dataset used for Bayes optimization based on the area D2 of the predetermined range and the area of ​​the substrate. Here, any value below 7 nm can be set for the lower limit of the predetermined range. The reason is that, as explained above, the predicted distribution output from the second estimation model EM2 is expected to more accurately reflect the number of particles with a diameter of 7 nm or less that do not follow the power law D(x).

[0150] The support method of Embodiment 6 is executed, for example, by the terminal device 101A included in the support system 100A described with reference to FIG15. More specifically, the support method shown in FIG16 is executed by the terminal device 101A executing the support program SP read from the recording medium 200. In this case, the flowchart shown in FIG16 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, FIG16 shows the processing executed by the processing unit 106 included in the support system 100A of Embodiment 6.

[0151] The processing (support method) shown in Figure 16 can also be performed by an operator on the operation unit 102 to instruct the start of establishing the value of the target variable for Bayes optimization. When the processing unit 106 starts the processing shown in Figure 16, the target variable establishment process is executed.

[0152] In detail, the processing unit 106 first outputs the expected value of the predicted distribution of the number of particles in each substrate from the second estimation model EM2 (step S51). Furthermore, the processing unit 106 may also establish the second estimation model EM2 corresponding to the value of the target variable instructed to be established for Bayes optimization. Alternatively, the second estimation model EM2 may also be established before performing the target variable establishment process. The process of establishing the second estimation model EM2 has been described with reference to Figures 2 to 4, 9, 10, 11A and 11B, therefore, its description is omitted here.

[0153] Next, the processing unit 106 executes the second target variable establishment program VC2. Specifically, as shown in FIG17, the processing unit 106 obtains the area D2 of a predetermined range in the region below the curve of the expected value of the predicted distribution (step S52). Then, the processing unit 106 obtains the value of the target variable (particle density) of the dataset used for Bausch optimization based on the area D2 of the predetermined range and the area of ​​the substrate (step S53). As a result, the processing shown in FIG16 ends. In detail, the processing unit 106 divides the area D2 of the predetermined range by the area of ​​the substrate to obtain the particle density value.

[0154] Furthermore, the predetermined range may be, for example, 5 nm or more and 100 nm or less. Also, the processing unit 106 may perform numerical integration to calculate the area D2. The numerical integration may be, for example, any one of trapezoidal integration, Simpson integration, or Romberg integration.

[0155] The present invention embodiment 6 has been described above with reference to FIGS. 15 to 17. According to embodiment 6, similarly to embodiments 1 to 5, it can support the estimation of the number of particles on a substrate. Furthermore, according to embodiment 6, the particle density is obtained using the expected value of the predicted distribution output from the second estimation model EM2; therefore, it can reflect the number of particles with diameters that cannot be measured by an optical measuring device in the value of the target variable (particle density) of the dataset used for Bayes optimization. Moreover, according to embodiment 6, the number of particles with diameters of 7 nm or less that may not be accurately estimated using the power law D(x) can be reflected in the value of the target variable (particle density) of the dataset used for Bayes optimization.

[0156] The embodiments of the present invention have been described above with reference to Figures 1 to 17. However, the present invention is not limited to the above embodiments and can be implemented in various forms without departing from the spirit of the present invention. Furthermore, the plurality of constituent elements disclosed in the above embodiments can be appropriately modified. For example, a constituent element from all the constituent elements shown in one embodiment may be added to the constituent elements of another embodiment, or several constituent elements from all the constituent elements shown in one embodiment may be deleted from the embodiment.

[0157] To facilitate understanding of the invention, the drawings mainly illustrate the constituent elements schematically. Furthermore, for ease of drawing production, the thickness, length, number, and spacing of the constituent elements shown in the drawings may differ from the actual components. Also, the configuration of the constituent elements shown in the above embodiments is merely an example and is not particularly limited. Of course, various modifications can be made without substantially departing from the effects of the invention.

[0158] For example, in embodiments 4 to 6, the support program SP is executed by the terminal device 101A, similar to embodiment 1. However, the support program SP can also be executed by the server 300, similar to embodiment 2, or by the control device 10 of the substrate processing system 1000, similar to embodiment 3.

[0159] The present invention can help in estimating the number of particles on a substrate.

[0160] This application claims priority to Japanese Patent Application No. 2024-047111, filed on March 22, 2024, the contents of which are incorporated herein by reference. [Simplified Explanation of the Diagram]

[0023] Figure 1 is a block diagram showing the configuration of the support system according to Embodiment 1 of the present invention. Figure 2 is a flowchart showing the support method according to Embodiment 1 of the present invention. Figure 3 is a flowchart showing the coefficient value determination step. Figure 4 is a flowchart showing the sampling step of the temporary value of the first coefficient k and the temporary value of the second coefficient q. Figure 5 is a diagram showing an example of the predicted distribution of the number of particles in each substrate. Figure 6 is a block diagram showing the configuration of the support system according to Embodiment 2 of the present invention. Figure 7 is a schematic diagram of the substrate processing system according to Embodiment 3 of the present invention. Figure 8 is a block diagram showing the configuration of the support system included in the substrate processing system according to Embodiment 3 of the present invention. Figure 9 is a block diagram showing the configuration of the support system according to Embodiment 4 of the present invention. Figure 10 is a flowchart showing the support method according to Embodiment 4 of the present invention. Figure 11A is a diagram showing an example of the predicted distribution of the number of particles in each substrate. Figure 11B is a diagram showing another example of the predicted distribution of the number of particles in each substrate. Figure 12 is a block diagram showing the configuration of the support system according to Embodiment 5 of the present invention. Figure 13 is a flowchart illustrating the support method of Embodiment 5 of the present invention. Figure 14 is a diagram illustrating the target variable creation process. Figure 15 is a block diagram illustrating the structure of the support system of Embodiment 6 of the present invention. Figure 16 is a flowchart illustrating the support method of Embodiment 6 of the present invention. Figure 17 is a diagram illustrating the target variable creation process.

Claims

1. A support method for estimating the number of particles on a substrate; comprising: a model building step for building a first estimation model for estimating the number of particles for each particle diameter; the first estimation model comprising the following equation (1), A×D(x)=A×kx-q … (1) In the above equation (1), the constant A represents the area of ​​the substrate, D(x) represents the power law, the variable x represents the particle diameter, and the number of particles having the particle diameter input to the variable x of the first estimation model is calculated by inputting an arbitrary value to the variable x; the model building step comprising the following steps: a coefficient value determination step for determining the values ​​of the first coefficient k and the second coefficient q of the power law D(x) using a first dataset and by Bayes estimation; and a step for building the first estimation model for inputting the values ​​determined in the coefficient value determination step to the first coefficient k and the second coefficient q. The aforementioned first dataset contains the values ​​of the explanatory variables and the corresponding values ​​of the target variables. The values ​​of the explanatory variables represent the observed particle size (i.e., the first observed value), and the values ​​of the target variables represent the observed particle number (i.e., the second observed value). The coefficient value determination step includes the following steps: sampling the temporary values ​​of the first coefficient k and the second coefficient q from the prior distribution (i.e., the first prior distribution) and the prior distribution of the second coefficient q (i.e., the second prior distribution) to obtain the post-hoc distributions of the first coefficient k and the second coefficient q; and determining the values ​​of the first coefficient k and the second coefficient q based on the post-hoc distributions of the first coefficient k and the second coefficient q. The sampling step includes the following steps: The steps are as follows: establishing a provisional model based on the values ​​sampled from the first prior distribution (i.e., the first sampled value), the values ​​sampled from the second prior distribution (i.e., the second sampled value), and the first presumed model for which the values ​​of the first coefficient k and the second coefficient q are undetermined; and evaluating the goodness of fit of the provisional model to the first data set based on the provisional model, the first data set, and the likelihood function; wherein the second prior distribution comprises a uniform distribution within a predetermined range.

2. As in the support method of request item 1, wherein, The above model building steps further include the following steps: inputting a plurality of predetermined values ​​as the values ​​of the explanatory variables into the first inference model, and outputting a plurality of values ​​of the target variable from the first inference model to obtain a second dataset; adding supplementary information to the second dataset; and enabling a machine learning model capable of outputting a predictive distribution to perform machine learning on the second dataset with the added supplementary information, and establishing a second inference model that outputs the expected value of the predicted distribution relative to the target variable; the second dataset includes a plurality of values ​​of the explanatory variables input into the first inference model, and a plurality of values ​​of the target variable output from the first inference model, and the supplementary information includes information about the origin (0,0).

3. The method supported by claim 1 further includes: a target variable establishment step, which establishes the value of the target variable in the dataset used for Bayes optimization; the dataset used for Bayes optimization includes at least one parameter among various parameters defining the operation of the substrate processing apparatus that processes the substrate as an explanatory variable, and the Bayes optimization is used to determine the optimal solution of the at least one parameter; the target variable establishment step includes the following steps: obtaining the area above the minimum value set for the variable x in the lower region of the curve of the first estimation model in which the values ​​of the first coefficient k and the second coefficient q have been determined in the coefficient value determination step; and obtaining the value of the target variable in the dataset used for Bayes optimization based on the area of ​​the area above the minimum value and the area of ​​the substrate.

4. The method supported by claim 2 further includes: a target variable establishment step, which establishes the value of the target variable for the dataset used for Bayes optimization; the dataset used for the Bayes optimization includes at least one parameter among various parameters defining the operation of the substrate processing apparatus for processing the substrate as an explanatory variable, and the Bayes optimization is used to determine the optimal solution of the at least one parameter; the target variable establishment step includes the following steps: a step of outputting the expected value of the predicted distribution from the second estimation model; a step of obtaining the area of ​​a predetermined range in the region below the curve of the expected value of the predicted distribution; and a step of obtaining the value of the target variable for the dataset used for the Bayes optimization based on the area of ​​the predetermined range and the area of ​​the substrate.

5. The support method for any of the requests in items 1 to 4, wherein, The likelihood function described above includes the Boisson distribution.

6. A recording medium readable by a computer, which records a support program defining a support method for any one of the request items 1 to 4.

7. A support system for estimating the number of particles on a substrate; comprising: a processing unit that establishes a first estimation model for estimating the number of particles for each particle size; and a memory unit that stores a first data set; wherein the first estimation model comprises the following equation (1), A×D(x)=A×kx-q … (1) In the above equation (1), the constant A represents the area of ​​the substrate, D(x) represents the power law, the variable x represents the particle size, and by inputting an arbitrary value into the variable x of the first estimation model, the number of particles having the particle size of the input variable x is calculated; wherein the first data set comprises each value of an explanatory variable and each value of a target variable corresponding to each value of the explanatory variable, and each value of the explanatory variable represents an observed value of the particle size, i.e., a first observed value, and each value of the target variable represents an observed value of the number of particles, i.e., a second observed value. The aforementioned processing unit performs coefficient value determination processing, using the first dataset and Bayesian estimation to determine the values ​​of the first coefficient k and the second coefficient q of the power law D(x). The values ​​determined by this coefficient value determination processing are input into the first coefficient k and the second coefficient q to establish the first estimation model. During the coefficient value determination processing, the processing unit repeats the step of sampling the temporary values ​​of the first coefficient k and the second coefficient q from the prior distribution (first prior distribution) and the prior distribution (second prior distribution) of the first coefficient k to obtain the subsequent distributions of the first coefficient k and the second coefficient q. Based on these subsequent distributions, the values ​​of the first coefficient k and the second coefficient q are determined. During the sampling step, the processing unit... A temporary model is established based on the values ​​sampled from the first prior distribution (i.e., the first sampled value), the values ​​sampled from the second prior distribution (i.e., the second sampled value), and the first presupposition model for which the values ​​of the first coefficient k and the second coefficient q are undetermined. Based on the temporary model, the first dataset, and the likelihood function, the goodness of fit of the temporary model with respect to the first dataset is evaluated. The second prior distribution includes a uniform distribution within a predetermined range.

8. As in the support system requested in item 7, wherein, The aforementioned memory unit further stores a machine learning model that can output a predictive distribution. The aforementioned processing unit inputs a plurality of predetermined values ​​as the values ​​of the aforementioned explanatory variables into the aforementioned first estimation model, and outputs a plurality of values ​​of the aforementioned target variable from the aforementioned first estimation model to obtain a second data set. The aforementioned processing unit adds additional information to the aforementioned second data set. The aforementioned processing unit enables the aforementioned machine learning model to perform machine learning on the aforementioned second data set with the added additional information, and establishes a second estimation model that outputs the expected value of the predicted distribution of the aforementioned target variable. The aforementioned second data set includes a plurality of values ​​of the aforementioned explanatory variables input into the aforementioned first estimation model, and a plurality of values ​​of the aforementioned target variable output from the aforementioned first estimation model. The aforementioned additional information includes information about the origin (0,0).

9. As in the support system requested in item 7, wherein, The aforementioned processing unit then performs a target variable establishment process, which establishes the value of the target variable in the dataset used for Bayes optimization. The dataset used for Bayes optimization includes at least one parameter among various parameters that define the operation of the substrate processing apparatus that processes the substrate as an explanatory variable. Bayes optimization is used to determine the optimal solution for the at least one parameter. When performing the target variable establishment process, the processing unit obtains the area above the minimum value set for the variable x in the lower region of the curve of the first estimation model, which has already determined the values ​​of the first coefficient k and the second coefficient q by the coefficient value determination process. Based on the area of ​​the area above the minimum value and the area of ​​the substrate, the processing unit obtains the value of the target variable in the dataset used for Bayes optimization.

10. As in the support system requested in item 8, wherein, The aforementioned processing unit then performs a target variable establishment process, which establishes the value of the target variable for the dataset used for Bayes optimization. The dataset used for Bayes optimization includes at least one parameter among various parameters defining the operation of the substrate processing apparatus that processes the substrate as an explanatory variable. Bayes optimization is used to determine the optimal solution for the at least one parameter. When performing the target variable establishment process, the processing unit outputs the expected value of the predicted distribution from the second estimation model, obtains the area of ​​a predetermined range in the region below the curve of the expected value of the predicted distribution, and obtains the value of the target variable for the dataset used for Bayes optimization based on the area of ​​the predetermined range and the area of ​​the substrate.

11. A support system as described in any of requests 7 to 10, wherein, The likelihood function described above includes the Boisson distribution.

12. A substrate processing system for processing a substrate; comprising: a substrate processing apparatus for processing the substrate; and a control device for controlling the substrate processing apparatus; wherein the control device also serves as a support system for any one of claims 7 to 10.

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