Support method, recording medium, support program, support system, and substrate processing system

Abstract

This support method includes a model creation step for creating an estimation model including Equation (1). In Equation (1), D(x) represents a power law. The model creation step includes a coefficient value determination step for determining values of a first coefficient k and a second coefficient q of the power law D(x) by Bayesian estimation using a data set. The value of each explanatory variable in the data set indicates a particle size. The value of each objective variable in the data set indicates the number of particles. The coefficient value determination step includes a step in which a step for sampling the value of the first coefficient k and the value of the second coefficient q from the prior distribution of the first coefficient k and the prior distribution of the second coefficient q is repeated to acquire the posterior distribution of the first coefficient k and the posterior distribution of the second coefficient q. The prior distributions include a uniform distribution over a predetermined range.　(1): A × D(x) = A × Kx-q

Description

Support method, recording medium, support program, support system, and substrate processing system

[0001] The present invention relates to an assistance method, a recording medium, an assistance program, an assistance system, and a substrate processing system.

[0002] Optical measurement devices are known that optically measure the number of particles (particle count) adhering to a substrate such as a semiconductor wafer. The measurement results of the number of particles on the substrate are used for, for example, evaluating the performance of a substrate processing apparatus. For example, Patent Document 1 discloses a substrate processing apparatus that includes a particle measurement unit that measures particles on the wafer.

[0003] However, optical measurement devices that optically measure particles have limitations on the particle size they can measure, for example, they cannot measure particles with a particle size of less than 10 nm.

[0004] Meanwhile, research is being conducted into particle estimation technology that utilizes the power law D(x) expressed by the following formula: In the power law D(x), the variable x indicates the particle diameter: D(x) = kx-q

[0005] Specifically, particle size and particle count are measured using an optical measurement device to obtain observed value data. Then, the values ​​of the two coefficients k and q of the power law are determined by the nonlinear least squares method so that the model (graph) of the power law D(x) fits the observed value data. By using the power law D(x), it is possible to determine the number of particles with particle sizes that cannot be measured by the optical measurement device.

[0006] Japanese Patent Application Laid-Open No. 2007-88401

[0007] However, the values ​​of the coefficients k and q determined by the nonlinear least-squares method are strongly affected by noise generated in the optical measurement device. Specifically, the model (graph) of the power law D(x) may overfit the observed data, resulting in significantly large values ​​of the two coefficients k and q of the power law. Therefore, there is room for improvement in the technology for estimating the number of particles using the power law D(x).

[0008] An object of the present invention is to provide an assistance method, a recording medium, an assistance program, an assistance system, and a substrate processing system that assist in the task of estimating the number of particles on a substrate.

[0009] According to one aspect of the present invention, an assistance method is a method for assisting in an operation of estimating the number of particles on a substrate. The assistance method includes a model creation step of creating 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), constant A represents the area of ​​the substrate, D(x) represents a power law, and variable x represents the particle size. By inputting an arbitrary value into variable x of the first estimation model, the number of particles having a particle size of the value input into variable x is calculated. The model creation step includes a coefficient value determination step of determining values ​​of a first coefficient k and a second coefficient q of the power law D(x) by Bayesian estimation using a first data set, and a step of creating the first estimation model by inputting the values ​​determined in the coefficient value determination step into the first coefficient k and the second coefficient q. The first data set includes values ​​of explanatory variables and values ​​of a response variable corresponding to each value of the explanatory variables. Each value of the explanatory variable indicates a first observed value, which is an observed value of the particle diameter of the particle. Each value of the objective variable indicates a second observed value, which is an observed value of the number of particles. The coefficient value determination step includes a step of repeatedly sampling a tentative value of the first coefficient k and a tentative value of the second coefficient q from a first prior distribution, which is a prior distribution of the first coefficient k, and a second prior distribution, which is a prior distribution of the second coefficient q, to ​​obtain a posterior distribution of the first coefficient k and a posterior distribution of the second coefficient q, and a step of determining each value of the first coefficient k and the second coefficient q based on the posterior distribution of the first coefficient k and the posterior distribution of the second coefficient q. The sampling step includes a step of generating a tentative model based on first sampled values ​​sampled from the first prior distribution, second sampled values ​​sampled from the second prior distribution, and the first estimation model in which the values ​​of the first coefficient k and the second coefficient q are undetermined, and a step of evaluating a degree of fitting of the tentative model to the first data set based on the tentative model, the first data set, and a likelihood function. The second prior distribution includes a uniform distribution within a predetermined range.

[0010] In one embodiment, the model creation step further includes the steps of inputting multiple predetermined values ​​as values ​​of the explanatory variables to the first estimation model and outputting multiple values ​​of the dependent variable from the first estimation model to obtain a second dataset, adding additional information to the second dataset, and subjecting the second dataset to machine learning using a machine learning model capable of outputting a predictive distribution to create a second estimation model that outputs an expected value of the predictive distribution for the dependent variable. The second dataset includes multiple values ​​of the explanatory variables input to the first estimation model and multiple values ​​of the dependent variable output from the first estimation model. The additional information includes information of the origin (0, 0).

[0011] In one embodiment, the support method further includes a dependent variable creation step of creating a dependent variable value of a dataset used for Bayesian optimization. The dataset used for Bayesian optimization includes, as an explanatory variable, at least one parameter from among various parameters that define the operation of the substrate processing apparatus that processes the substrate. The Bayesian optimization is used to determine an optimal solution for the at least one parameter. The dependent variable creation step includes a step of acquiring an area of ​​a region equal to or greater than a minimum value set for the variable x within a region below a graph of the first estimation model for which the values ​​of the first coefficient k and the second coefficient q have been determined in the coefficient value determination step, and a step of acquiring a value of the dependent variable of the dataset used for Bayesian optimization based on the area of ​​the region equal to or greater than the minimum value and the area of ​​the substrate.

[0012] In one embodiment, the support method further includes a dependent variable generation step of generating a dependent variable value of a dataset used for Bayesian optimization. The dataset used for Bayesian optimization includes, as an explanatory variable, at least one parameter from various parameters that define the operation of a substrate processing apparatus that processes the substrate. The Bayesian optimization is used to determine an optimal solution for the at least one parameter. The dependent variable generation step includes a step of outputting an expected value of the predictive distribution from the second estimation model, a step of acquiring an area of ​​a predetermined range within a region below a graph of the expected value of the predictive distribution, and a step of acquiring a value of the dependent variable of the dataset used for Bayesian optimization based on the area of ​​the predetermined range and the area of ​​the substrate.

[0013] In one embodiment, the likelihood function comprises a Poisson distribution.

[0014] According to another aspect of the present invention, a recording medium is a computer-readable medium, and stores an assistance program that defines the above-described assistance method.

[0015] According to yet another aspect of the present invention, there is provided an assistance program executable by a computer, the assistance program defining the assistance method described above.

[0016] According to yet another aspect of the present invention, an assistance system is a system for assisting in an operation of estimating the number of particles on a substrate. The assistance system includes a processing unit and a storage unit. The processing unit creates a first estimation model for estimating the number of particles for each particle diameter. The storage 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), constant A represents the area of ​​the substrate, D(x) represents a power law, and variable x represents the particle diameter of the particles. By inputting an arbitrary value into variable x of the first estimation model, the number of particles having the particle diameter of the value input into variable x is calculated. The first data set includes values ​​of explanatory variables and values ​​of response variables corresponding to each value of the explanatory variables. Each value of the explanatory variable represents a first observed value, which is an observed value of the particle diameter of the particle. Each value of the response variable represents a second observed value, which is an observed value of the number of particles. The processing unit executes a coefficient value determination process to determine values ​​of a first coefficient k and a second coefficient q of the power law D(x) by Bayesian estimation using the first data set, and creates the first estimation model by inputting the values ​​determined by the coefficient value determination process to the first coefficient k and the second coefficient q. During the execution of the coefficient value determination process, the processing unit repeats a step of sampling a tentative value of the first coefficient k and a tentative value of the second coefficient q from a first prior distribution that is a prior distribution of the first coefficient k and a second prior distribution that is a prior distribution of the second coefficient q, thereby acquiring a posterior distribution of the first coefficient k and a posterior distribution of the second coefficient q. The processing unit determines the values ​​of the first coefficient k and the second coefficient q based on the posterior distribution of the first coefficient k and the posterior distribution of the second coefficient q. When the sampling step is performed, the processing unit creates a provisional model based on a first sampling value, which is a value sampled from the first prior distribution, a second sampling value, which is a value sampled from the second prior distribution, and the first estimation model, in which the values ​​of the first coefficient k and the second coefficient q are undetermined.The processor evaluates a degree of fitting of the tentative model to the first data set based on the tentative model, the first data set, and a likelihood function. The second prior distribution includes a uniform distribution with a predetermined range.

[0017] In one embodiment, the storage unit further stores a machine learning model capable of outputting a predictive distribution. The processing unit inputs multiple predetermined values ​​as values ​​of the explanatory variables to the first estimation model and causes the first estimation model to output multiple values ​​of the dependent variable 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 to which the additional information has been added using the machine learning model to create a second estimation model that outputs an expected value of the predictive distribution for the dependent variable. The second dataset includes multiple values ​​of the explanatory variables input to the first estimation model and multiple values ​​of the dependent variable output from the first estimation model. The additional information includes information of the origin (0, 0).

[0018] In one embodiment, the processing unit further executes a dependent variable creation process to create a dependent variable value of a dataset used for Bayesian optimization. The dataset used for the Bayesian optimization includes, as an explanatory variable, at least one parameter from among various parameters that define the operation of a substrate processing apparatus that processes the substrate. The Bayesian optimization is used to determine an optimal solution for the at least one parameter. During the dependent variable creation process, the processing unit acquires the area of ​​a region equal to or greater than a minimum value set for the variable x within a region below a graph of the first estimation model for which the values ​​of the first coefficient k and the second coefficient q have been determined by the coefficient value determination process. The processing unit acquires the value of the dependent variable of the dataset used for the Bayesian optimization based on the area of ​​the region equal to or greater than the minimum value and the area of ​​the substrate.

[0019] In one embodiment, the processing unit further executes a dependent variable creation process to create a dependent variable value of a dataset used for Bayesian optimization. The dataset used for the Bayesian optimization includes, as an explanatory variable, at least one parameter from among various parameters that define the operation of a substrate processing apparatus that processes the substrate. The Bayesian optimization is used to determine an optimal solution for the at least one parameter. When executing the dependent variable creation process, the processing unit causes the second estimation model to output an expected value of the predictive distribution. The processing unit acquires the area of ​​a predetermined range of a region below a graph of the expected value of the predictive distribution. The processing unit acquires the value of the dependent variable of the dataset used for the Bayesian optimization based on the area of ​​the predetermined range of the region and the area of ​​the substrate.

[0020] In one embodiment, the likelihood function comprises a Poisson distribution.

[0021] According to yet 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 at least one aspect of the present invention, an assistance method, a recording medium, an assistance program, an assistance system, and a substrate processing system can assist in the task of estimating the number of particles on a substrate.

[0023] FIG. 1 is a block diagram showing the configuration of an assistance system according to a first embodiment of the present invention. FIG. 2 is a flowchart showing an assistance method according to the first embodiment of the present invention. FIG. 3 is a diagram showing the flow of a coefficient value determination step. FIG. 4 is a diagram showing the flow of a step of sampling a temporary value of a first coefficient k and a temporary value of a second coefficient q. FIG. 5 is a diagram showing an example of a predicted distribution of the number of particles per substrate. FIG. 6 is a block diagram showing the configuration of an assistance system according to a second embodiment of the present invention. FIG. 7 is a schematic diagram of a substrate processing system according to a third embodiment of the present invention. FIG. 8 is a block diagram showing the configuration of an assistance system included in the substrate processing system according to the third embodiment of the present invention. FIG. 9 is a block diagram showing the configuration of an assistance system according to a fourth embodiment of the present invention. FIG. 10 is a flowchart showing an assistance method according to the fourth embodiment of the present invention. FIG. 11 is a diagram showing an example of a predicted distribution of the number of particles per substrate. FIG. 12 is a block diagram showing another example of the predicted distribution of the number of particles per substrate. FIG. 13 is a block diagram showing the configuration of an assistance system according to a fifth embodiment of the present invention. FIG. 14 is a flowchart showing an assistance method according to the fifth embodiment of the present invention. FIG. 15 is a diagram showing a dependent variable creation process. FIG. 16 is a block diagram showing the configuration of an assistance system according to a sixth embodiment of the present invention. FIG. 17 is a flowchart showing an assistance method according to the sixth embodiment of the present invention.

[0024] Hereinafter, embodiments of the support method, recording medium, support program, support system, and substrate processing system of the present invention will be described with reference to the drawings (FIGS. 1 to 17). However, the present invention is not limited to the following embodiments, and can be implemented in various forms without departing from the spirit of the present invention. Note that duplicated explanations may be omitted as appropriate. In addition, the same or equivalent parts in the drawings will be designated by the same reference numerals, and explanations will not be repeated.

[0025] In the support method, recording medium, support program, support system, and substrate processing system according to the present invention, various types of substrates can be applied to the "substrate", such as a semiconductor wafer, a glass substrate for a photomask, a glass substrate for a liquid crystal display, a glass substrate for a plasma display, a substrate for an FED (Field Emission Display), a substrate for an optical disk, a substrate for a magnetic disk, and a substrate for a magneto-optical disk.

[0026] First Embodiment First, a recording medium 200, an assistance program SP, and an assistance system 100A according to this embodiment will be described with reference to Fig. 1. Fig. 1 is a block diagram showing the configuration of the assistance system 100A according to this embodiment. The assistance system 100A is a system that assists in the task of estimating the number of particles on a substrate.

[0027] 1, the assistance system 100A of this embodiment includes a terminal device 101A. An assistance program SP is installed in the terminal device 101A from a recording medium 200. The terminal device 101A executes the assistance program SP installed from the recording medium 200 to assist in the task of estimating the number of particles. Specifically, the terminal device 101A executes the assistance program SP to create an estimation model for estimating the number of particles for each particle diameter.

[0028] The terminal device 101A is an example of an "assistance device." The terminal device 101A may be, for example, a general-purpose computer or a dedicated computer. Alternatively, the terminal device 101A may be a computer that controls a substrate processing apparatus or a computer that controls a measuring apparatus that measures particles on a substrate.

[0029] The recording medium 200 is a medium that can be read by a computer. A program (computer program) to be executed by a computer is non-temporarily recorded on the recording medium 200. An assistance program SP is recorded on the recording medium 200. In other words, the recording medium 200 stores the assistance program SP. The assistance program SP is a computer program that can be executed by a computer.

[0030] The recording medium 200 may be a medium including a semiconductor memory such as a secure digital (SD) memory card or a universal serial bus (USB) memory, or a medium including a magnetic disk such as a hard disk drive. Alternatively, the recording medium 200 may be an optical disk such as a compact disk (CD), a digital versatile disk (DVD), or a Blu-ray disk, or may be a main memory device or an auxiliary memory device installed in another computer system.

[0031] The assistance program SP includes a first estimation model EM1 whose coefficient values ​​are not yet determined and a Bayesian estimation program BE. Hereinafter, the first estimation model EM1 whose coefficient values ​​are not yet determined may be referred to as a "pre-determined model PEM."

[0032] In this embodiment, the first estimation model EM1 includes the following equation (1). In the following equation (1), A represents the area of ​​the substrate. D(x) represents a power law. The variable x represents the particle diameter. The power law D(x) includes a first coefficient k and a second coefficient q. The pre-determined model PEM represents the first estimation 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 an arbitrary value into the variable x of the first estimation model EM1 after the values ​​of the first coefficient k and the second coefficient q have been determined, the number of particles having a particle diameter equal to the value input into the variable x (the number of particles per substrate) is calculated. Specifically, by inputting an arbitrary value into the variable x, the number of particles per unit area is calculated from the power law D(x). Therefore, according to equation (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 creates an estimation model (first estimation model EM1) that estimates the number of particles per substrate.

[0034] The Bayesian inference program BE includes a computer program that determines values ​​of the first coefficient k and the second coefficient q by Bayesian inference. In this embodiment, the Bayesian inference 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 in Bayesian inference.

[0035] The first prior distribution PD1 may include a non-informative prior distribution. For example, the first prior distribution PD1 may be a half-normal distribution or a Cauchy distribution. In this embodiment, a half-normal distribution with an expectation value of 0 and a standard deviation of 100 is used for the first prior distribution PD1. The second prior distribution PD2 includes a uniform distribution within a predetermined range. In this embodiment, a uniform distribution within a range of 2.5 to 3.0 is used for the second prior distribution PD2. The range of the uniform distribution may be experimentally determined from actual measurements (observed values) of particle diameter and particle count. The likelihood function LF may include a probability mass function. In this embodiment, a Poisson distribution is used for the likelihood function LF.

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

[0037] The operation unit 102 includes a user interface device operated by the worker. The operation unit 102 inputs a signal corresponding to the worker's operation to the processing unit 106. The operation unit 102 may include, for example, a keyboard, a mouse, and a touch sensor. The touch sensor may be superimposed on the display surface of the display unit 103. A graphical user interface may be configured by superimposing the touch sensor on the display surface of the display unit 103. For example, the worker can operate the operation unit 102 to instruct the installation of the assistance program SP. The worker can also operate the operation unit 102 to instruct the execution of the assistance program SP.

[0038] The display unit 103 displays various screens under the control of the processing unit 106. For example, the display unit 103 displays a graph of the first estimation model EM1. The display unit 103 includes a display device such as a liquid crystal display device or an organic electroluminescence (EL) display device.

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

[0040] For example, the interface unit 104 may be electrically connected to the recording medium 200 to input and output information, data, or signals to and from the recording medium 200. Specifically, the interface unit 104 may include a slot or a USB terminal. A card-shaped information carrier such as an SD memory card may be inserted into the slot. A USB memory may be inserted into the USB terminal, or the other end of a USB cable, one end of which is electrically connected to a hard disk drive, may be inserted into the USB terminal. Alternatively, the interface unit 104 may include an optical disc drive. The optical disc drive reads information (data) from a CD, DVD, or Blu-ray disc.

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

[0042] The first data set LD1 includes values ​​of explanatory variables and values ​​of response variables corresponding to the values ​​of explanatory variables. Each value of the explanatory variables indicates a first observation value, which is an observation value of particle diameter. Each value of the response variable indicates a second observation value, which is an observation value of the number of particles. More specifically, the second observation value (response variable) indicates the number of particles per substrate. For example, the first data set LD1 indicates the number of particles per substrate for each particle diameter obtained from the measurement results of the optical measurement 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 dedicated computing device. For example, the processing unit 106 may include a field-programmable gate array (FPGA) or an application-specific integrated circuit (ASIC).

[0044] The processing unit 106 performs various processes such as numerical calculations, information processing, and device control by executing computer programs stored in the storage unit 105 based on instructions from an operator input via the operation unit 102. For example, the processing unit 106 reads out an assistance program SP from the recording medium 200 via the interface unit 104 and stores the assistance program SP in the storage unit 105.

[0045] In this embodiment, the processing unit 106 executes the assistance program SP to create an estimation model that estimates the number of particles for each particle size. Specifically, the processing unit 106 creates an estimation model (first estimation model EM1) that estimates the number of particles per substrate for each particle size. Specifically, the processing unit 106 executes the Bayesian 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 create the first estimation model EM1.

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

[0047] Step S1 is a coefficient value determination step in which the values ​​of the first coefficient k and the second coefficient q included in the pre-determined model PEM are determined by Bayesian estimation using a first data set LD1. Step S2 is a step in which the values ​​determined in the coefficient value determination step (step S1) are input to the first coefficient k and the second coefficient q to create a first estimated model EM1.

[0048] The assistance method of this embodiment is executed, for example, by the terminal device 101A included in the assistance system 100A described with reference to Fig. 1. More specifically, the assistance method shown in Fig. 2 is executed by the terminal device 101A executing the assistance program SP read from the recording medium 200. In this case, the flowchart shown in Fig. 2 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, Fig. 2 shows the processing executed by the processing unit 106 included in the assistance system 100A of this embodiment.

[0049] The process (assistance method) shown in Fig. 2 begins when the operator operates the operation unit 102 to instruct execution of the assistance program SP. When the process shown in Fig. 2 begins, the processing unit 106 executes a coefficient value determination process (step S1). Specifically, the processing unit 106 determines the values ​​of the first coefficient k and the second coefficient q by Bayesian estimation using the first data set LD1. The processing unit 106 then 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, respectively, to create a first estimated model EM1 (step S2). This completes the process shown in Fig. 2.

[0050] More specifically, the processing unit 106 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, and inputs the value of the area of ​​the substrate to the constant A of the pre-determined model PEM. The value of the area of ​​the substrate may be input by, for example, an operator operating the operation unit 102.

[0051] Next, the coefficient value determination process (step S1 in FIG. 2) will be described with reference to FIG. 3. FIG. 3 is a diagram showing the flow of the coefficient value determination process. As shown in FIG. 3, the coefficient value determination process includes steps S11 and S12.

[0052] During the coefficient value determination process, the processing unit 106 repeatedly samples a tentative value of the first coefficient k and a tentative value of 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 a posterior distribution of the first coefficient k and a posterior 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 based on the posterior distribution of the first coefficient k and the posterior distribution of the second coefficient q (step S12). This completes the process shown in FIG. 3 .

[0053] More specifically, the processing unit 106 randomly samples a tentative value of the first coefficient k and a tentative value of the second coefficient q from the first prior distribution PD1 and the second prior distribution PD2, respectively, based on the Markov chain Monte Carlo method. The algorithm of the Markov chain Monte Carlo method is not particularly limited, and may be, for example, 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 acquires a representative value from the posterior distribution of the first coefficient k as the value of the first coefficient k. Similarly, the processing unit 106 acquires a representative value from the posterior distribution of the second coefficient q as the value of the second coefficient q. The representative value may be, for example, an expected value (average value) or a median.

[0055] Next, the step of sampling the temporary value of the first coefficient k and the temporary value of the second coefficient q (step S11 in FIG. 3) will be described with reference to FIG. 4. FIG. 4 is a diagram showing the flow of the step of sampling the temporary value of the first coefficient k and the temporary value of the second coefficient q. As shown in FIG. 4, the step of sampling the temporary value of the first coefficient k and the temporary value of the second coefficient q includes steps S111 to S113.

[0056] First, the processing unit 106 inputs the first sampled value, which is a value sampled from the first prior distribution PD1, and the second sampled value, which is a value sampled from the second prior distribution PD2, into the first coefficient k and the second coefficient q included in the pre-determined model PEM, respectively, to create a provisional model (step S111).

[0057] Next, the processing unit 106 evaluates the degree of fitting of the provisional model to the first data set LD1 based on the created provisional model, the first data set LD1, and the likelihood function LF (step S112).Then, based on the evaluated degree of fitting, the processing unit 106 determines whether or not to add the currently sampled values ​​of the first coefficient k and the second coefficient q (first sampled value and second sampled value) to the posterior distribution (step S113).

[0058] Specifically, the processing unit 106 inputs the values ​​of the explanatory variables (first observation values) included in the first data set LD1 into the provisional model and acquires a provisional number of particles (provisional estimates) from the provisional model. Specifically, the processing unit 106 acquires a provisional number of particles (provisional estimates) for each value of the explanatory variables included in the first data set LD1. Then, the processing unit 106 evaluates the degree of fitting of the provisional model to the first data set LD1 based on each value of the objective variable (second observation values) included in the first data set LD1, the likelihood function LF, and the acquired provisional number of particles.

[0059] More specifically, when the likelihood function L F is a Poisson distribution, the processing unit 106 inputs the acquired tentative number of particles into the coefficient λ of the probability mass function of the Poisson distribution expressed by the following equation (2).

[0060] Next, the processing unit 106 inputs the value of the objective variable (second observation value) included in the first data set LD1 into the variable x of the probability mass function of the Poisson distribution, and calculates the probability (value of the probability mass function) that the second observation value (actual measured value of the number of particles) occurs on the Poisson distribution.

[0061] For each explanatory variable value (first observation value) included in the first data set LD1, the processing unit 106 acquires the probability (value of the probability mass function) that each corresponding value (second observation value) of the objective variable occurs on the Poisson distribution. Then, the processing unit 106 evaluates the degree of fitting of the provisional model to the first data set LD1 from a value obtained by multiplying the acquired probabilities. The value obtained by multiplying the acquired probabilities is an example of an "evaluation value." Specifically, the larger the evaluation value (the value obtained by multiplying the probabilities), the greater the degree of fitting of the provisional model to the first data set LD1. In other words, the larger the evaluation value, the better the provisional model fits to the first data set LD1. The processing unit 106 may determine whether to add a sampled value to the posterior distribution, for example, according to a Markov chain Monte Carlo algorithm.

[0062] Next, the assistance method, recording medium 200, assistance program SP, and assistance system 100A of this embodiment will be described with reference to FIGS. 1 and 5. FIG. 5 is a diagram showing an example of a predicted distribution of the number of particles per substrate. In FIG. 5, the horizontal axis represents particle diameter, and the vertical axis represents the number of particles per substrate. Furthermore, black circles represent observed values ​​included in the first data set LD1, and the solid line represents a graph of the first estimation model EM1. The graph of the first estimation model EM1 shows estimated values ​​of the number of particles (number of particles per substrate) versus particle diameter.

[0063] As shown in Fig. 5, the processing unit 106 executes the assistance program SP to display a graph of the first estimation model EM1 (estimated particle count) on the display unit 103. In this embodiment, the processing unit 106 further executes the assistance program SP to display a predicted distribution of the particle count per substrate on the display unit 103. The predicted distribution is, for example, a Poisson distribution. As shown in Fig. 5, the processing unit 106 may display, for example, a range of 5% to 95% of the Poisson distribution (predictive distribution) on the display unit 103.

[0064] Specifically, the processing unit 106 constructs a predictive distribution of the first coefficient k based on the posterior distribution of the first coefficient k. Similarly, the processing unit 106 constructs a predictive distribution of the second coefficient q based on the posterior distribution of the second coefficient q. Then, the processing unit 106 causes the display unit 103 to display the predictive distribution of the number of particles per substrate based on the predictive distribution of the first coefficient k and the predictive distribution of the second coefficient q.

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

[0066] The first embodiment of the present invention has been described above with reference to FIGS. 1 to 5 . According to the first embodiment, it is possible to assist in the task of estimating the number of particles on a substrate. More specifically, according to the first embodiment, the values ​​of the first coefficient k and the second coefficient q of the power law D(x) are determined by Bayesian estimation. As a result, overfitting of the model (graph) of the power law D(x) to the observed value data is mitigated. Therefore, the operator can use the first estimation model EM1 to more accurately estimate the number of particles of a given particle size. For example, the operator can input a given particle size into the first estimation model EM1 by operating the operation unit 102, thereby obtaining an estimate of the number of particles having the input particle size.

[0067] As will be described later, 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 particle diameter greater than 7 nm. However, optical measurement devices cannot measure particles with a particle diameter less than 10 nm. In contrast, according to the first embodiment, the number of particles with a particle diameter greater than 7 nm and less than 10 nm can be more accurately estimated, allowing the operator to obtain an estimate of the number of particles with a particle diameter that cannot be measured by an optical measurement device.

[0068] Furthermore, according to embodiment 1, a uniform distribution within a predetermined range is used as the prior distribution of the second coefficient q, so the model (graph) of the first estimation model EM1 is less likely to overfit the observed value data compared to when a half-normal distribution is used as the prior distribution of the second coefficient q.

[0069] Furthermore, according to the first embodiment, since the Markov Chain Monte Carlo method is used to sample the first coefficient k and the second coefficient q, the predictive distribution of the particle count can be constructed from the posterior distribution of the first coefficient k and the second coefficient q. Therefore, in addition to the estimated value of the particle count, the predictive distribution of the particle count can be visualized. This further supports the worker in estimating the particle count.

[0070] Furthermore, slight variations occur in each measurement with optical measurement devices, and particles having the same particle size may be measured as particles of different particle sizes. This phenomenon becomes more pronounced as the particle count increases. In contrast, according to the first embodiment, the predicted particle count distribution can be displayed on the display unit 103, thereby visualizing this phenomenon. This further supports the operator's work of estimating the particle count.

[0071] In the embodiment described with reference to FIGS. 1 to 5, the assistance system 100A (terminal device 101A) acquires the assistance program SP from the recording medium 200. However, the assistance system 100A (terminal device 101A) may acquire the assistance program SP from another computer system. For example, the terminal device 101A may be communicatively connected to the other computer system via a cable and acquire the assistance program SP from the other computer system. Alternatively, the terminal device 101A may be communicatively connected to the other computer system via a line network such as the Internet and acquire the assistance program SP from the other computer system. The other computer system may be a general-purpose computer or a dedicated computer. The other computer system may also be a server.

[0072] 1 to 5, the assistance program SP is installed in the terminal device 101A, but the assistance program SP does not have to be installed in the terminal device 101A. The terminal device 101A may execute the assistance program SP stored in the recording medium 200.

[0073] [Embodiment 2] Next, a second embodiment of the present invention will be described with reference to Figures 2 to 6. However, differences from the first embodiment will be described, and a description of the same aspects as in the first embodiment will be omitted. The second embodiment differs from the first embodiment in that the server 300 creates a first estimation model EM1 based on the assistance program SP. More specifically, in the second embodiment, the server 300 determines the values ​​of the first coefficient k and the second coefficient q of the first estimation model EM1.

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

[0075] In the second embodiment, the terminal device 101B includes an operation unit 102, a display unit 103, a storage 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 communication device. The communication unit 107 is, for example, a network interface controller.

[0077] The communication unit 107 is controlled by the processing unit 106 to exchange information, data, or signals with the server 300. For example, the communication unit 107 transmits 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 storage unit 302 , and a processing unit 303 .

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

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

[0081] The processing unit 303 includes a processor. The processing unit 303 may include, for example, a CPU, a GPU, an NPU, or a quantum computer. Alternatively, the processing unit 303 may include a general-purpose computing device or a dedicated computing device. For example, the processing unit 303 may include an FPGA or an ASIC. The processing unit 303 executes the assistance program SP based on an instruction from the terminal device 101B.

[0082] Next, the support method, support program SP, and support system 100B of the second embodiment will be described with reference to Figures 2 to 6. The processing unit 303 starts the processes shown in Figures 2 to 4 based on a command from the terminal device 101B.

[0083] 2 starts, the processing unit 303 executes a coefficient value determination process (step S1). Specifically, the processing unit 303 executes the process described with reference to FIGS. 3 and 4 to determine the values ​​of the first coefficient k and the second coefficient q. The processing unit 303 then inputs the values ​​determined by the coefficient value determination process into the first coefficient k and the second coefficient q of the pre-establishment model PEM, respectively, to create a first estimation model EM1 (step S2). As a result, the processing shown in FIG. 2 ends.

[0084] More specifically, the processing unit 303 transmits the created first estimation model EM1 to the terminal device 101B. As a result, a worker performing work using the terminal device 101B can estimate the number of particles on the substrate using the first estimation model EM1. For example, the processing unit 106 of the terminal device 101B may display a graph of the first estimation model EM1 (estimated value of the number of particles) 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 the predicted distribution of the number of particles described with reference to Fig. 5. Then, the processing unit 303 transmits information indicating the predicted distribution of the number of particles to the terminal device 101B. The processing unit 106 of the terminal device 101B causes the display unit 103 to display the predicted distribution of the number of particles received from the server 300.

[0086] The second embodiment of the present invention has been described above with reference to Figures 2 to 6. According to the second embodiment, similar to the first embodiment, it is possible to assist in the task of estimating the number of particles on a substrate.

[0087] 7 and 8, a third embodiment of the present invention will be described. However, differences from the first and second embodiments will be described, and a description of the same aspects as the first and second embodiments will be omitted. The third embodiment differs from the first and second embodiments in that the control device 10 included in the substrate processing system 1000 also functions as a support system 100C.

[0088] Fig. 7 is a schematic diagram of the substrate processing system 1000 of this embodiment. In detail, Fig. 7 is a schematic plan view of the substrate processing apparatus 400 included in the substrate processing system 1000 of this embodiment.

[0089] 7, the substrate processing system 1000 includes a substrate processing apparatus 400 and a control device 10. The substrate processing apparatus 400 processes a substrate W. The control device 10 controls the substrate processing apparatus 400. In the third embodiment, the substrate W is a disk-shaped semiconductor wafer. The substrate processing apparatus 400 is a single-wafer processing apparatus, and processes the substrates W one by one.

[0090] More specifically, the substrate processing apparatus 400 includes a plurality of substrate processing units 2, a fluid cabinet 401, a plurality of fluid boxes 402, a plurality of load ports LP, an indexer robot IR, and a center robot CR.

[0091] A cassette CA is placed on each load port LP. The cassette CA accommodates a stack of substrates W. The cassette CA is, for example, a front opening unified pod (FOUP), a standard mechanical interface (SMIF) pod, or an open cassette (OC).

[0092] The indexer robot IR transports substrates W between the cassettes CA and the center robot CR. The center robot CR transports substrates W between the indexer robot IR and a plurality of substrate processing units 2. Note that a placement stage (path) on which substrates W are temporarily placed may be provided between the indexer robot IR and the center robot CR, and the device may be configured so that substrates W are indirectly transferred between the indexer robot IR and the center robot CR via the placement stage.

[0093] The substrate processing units 2 form a plurality of towers TW (four towers TW in FIG. 7 ). The towers TW are arranged to surround the center robot CR in a plan view. Each tower TW includes a plurality of substrate processing units 2 (three substrate processing units 2 in FIG. 7 ) stacked one above the other.

[0094] The fluid cabinet 401 contains a fluid. Specifically, the fluid cabinet 401 contains a processing liquid. Alternatively, the fluid cabinet 401 may contain a processing liquid and a gas.

[0095] The processing liquid is not particularly limited as long as it is a liquid that comes into contact with the substrate W. Examples of the processing liquid include dilute hydrofluoric acid (DHF), hydrofluoric acid (HF), hydronitric 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 water, hydrogen peroxide water, organic acids (e.g., citric acid, oxalic acid), organic alkalis (e.g., TMAH: tetramethylammonium hydroxide), sulfuric acid / hydrogen peroxide water mixture (SPM), ammonia / hydrogen peroxide water mixture (SC1), hydrochloric acid / hydrogen peroxide water mixture (SC2), isopropyl alcohol (IPA), surfactants, corrosion inhibitors, pure water (e.g., deionized water), carbonated water, electrolytic ionized water, hydrogen water, ozone water, and diluted hydrochloric acid water (e.g., about 0.001 wt % to about 0.01 wt %). The gas includes, for example, an inert gas, such as nitrogen gas.

[0096] Each of the fluid boxes 402 corresponds to one of the plurality of towers TW. The fluid in the fluid cabinet 401 is supplied to all of the substrate processing units 2 included in the corresponding tower TW via one of the fluid boxes 402.

[0097] Each of the substrate processing units 2 processes one substrate W at a time. Specifically, each of the substrate processing units 2 processes the substrate W by supplying a processing liquid to the substrate W. For example, each of the substrate processing units 2 performs a cleaning process or an etching process on the 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 box 402, the load port LP, the indexer 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 unit of the substrate processing apparatus 400 based on various information stored in the storage unit 12. The control unit 11 includes a processor. The control unit 11 may include, for example, a CPU, a GPU, an NPU, or a quantum computer. Alternatively, the control unit 11 may include a general-purpose computing device or a dedicated computing device. For example, the control unit 11 may include an FPGA or an ASIC.

[0100] The memory unit 12 stores various types of information for controlling the operation of the substrate processing apparatus 400. For example, the memory unit 12 stores various types of data and various computer programs. The various types of data include recipe data. The recipe data indicates a recipe that defines the processing content, processing conditions, and processing procedure for the substrate W. In the recipe, various setting values ​​(recipe parameter values) are set as processing conditions.

[0101] The storage unit 12 includes a main storage device. The main storage device includes, for example, a semiconductor memory. The storage unit 12 may further include an auxiliary storage device. The auxiliary storage device includes, for example, at least one of a semiconductor memory and a hard disk drive.

[0102] 8 is a block diagram showing the configuration of a support system 100C included in the substrate processing system 1000 of this embodiment. In detail, FIG. 8 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 functions as an assistance system 100C. More specifically, as shown in Figure 8, an assistance program SP is installed in the control device 10 from a recording medium 200. Similar to the terminal device 101A described with reference to Figures 1 to 5, the control device 10 executes the assistance 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 estimation model EM1. The control device 10 is an example of an "assistance device."

[0104] Specifically, as shown in Fig. 8, the control device 10 further includes an operation unit 13, a display unit 14, and an interface unit 15. The configurations of the operation unit 13, the display unit 14, and the interface unit 15 are similar to those of the operation unit 102, the display unit 103, and the interface unit 104 described with reference to Fig. 1, and therefore a description thereof will be omitted.

[0105] The storage unit 12 stores the first data set LD1 and the assistance program SP read from the recording medium 200 by the interface unit 15, similar to the storage unit 105 described with reference to FIG.

[0106] The control unit 11 executes the assistance program SP in the same manner as the processing unit 106 described with reference to FIGS. 1 to 5, determines the values ​​of the first coefficient k and the second coefficient q, and creates the first estimation model EM1.

[0107] The third embodiment of the present invention has been described above with reference to Figures 7 and 8. According to the third embodiment, similarly to the first and second embodiments, it is possible to assist in the task of estimating the number of particles on a substrate.

[0108] In the third embodiment, the substrate processing apparatus 400 performs a cleaning process or an etching process on the substrate W. However, the substrate processing apparatus 400 is not particularly limited as long as it is an apparatus for processing a substrate. For example, the substrate processing apparatus 400 may be a coating apparatus, a developing apparatus, an exposure apparatus, a baking apparatus, or a film forming apparatus.

[0109] Furthermore, in the third embodiment, the substrate processing apparatus 400 is of a single wafer type, but the substrate processing apparatus 400 may be of a batch type.

[0110] 2 to 4, 9, 10, 11A, and 11B, a fourth embodiment of the present invention will be described. However, only differences from the first to third embodiments will be described, and a description of the same aspects as the first to third embodiments will be omitted. The fourth embodiment differs from the first to third embodiments in that the terminal device 101A executes the assistance program SP to create a second estimation model EM2.

[0111] Fig. 9 is a block diagram showing the configuration of a support system 100A according to embodiment 4. As shown in Fig. 9, the support program SP further includes a pre-learning machine learning model GP and a second dataset creation program CP in addition to a pre-determined model PEM and a Bayesian inference program BE.

[0112] The machine learning model GP includes a model capable of outputting a predictive distribution. The machine learning model GP performs machine learning on the second dataset LD2 and outputs an expected value of the predictive distribution for the number of particles per substrate (objective variable). The algorithm of the machine learning model GP is not particularly limited as long as it includes an algorithm capable of outputting a predictive distribution. The machine learning model GP may include, for example, a Gaussian process regression model. For example, the machine learning model GP may be a Gaussian process regression model including a Poisson distribution as a likelihood function.

[0113] The second dataset creation program CP is a computer program that creates a second dataset LD2 from the first estimation model EM1. Specifically, by executing the second dataset creation program CP, the processing unit 106 inputs multiple predetermined values ​​as explanatory variable (particle size) values ​​to the first estimation model EM1 and causes the first estimation model EM1 to output multiple values ​​of the objective variable (number of particles) to acquire the second dataset LD2. For example, by executing the second dataset creation program CP, the processing unit 106 may input 500 particle sizes x (i = 1, 2, ..., 500) equally spaced in the range of 7 nm to 60 nm to the first estimation model EM1 to acquire 500 particle counts.

[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 a model creation process for creating a second estimation model EM2.

[0115] The model creation process shown in Fig. 10 may be started after step S2 shown in Fig. 2. That is, the model creation process shown in Fig. 10 may be started after the first estimation model EM1 is created. As shown in Fig. 10, the model creation process of the fourth embodiment further includes steps S3 to S5.

[0116] Step S3 represents a step of inputting multiple predetermined values ​​as values ​​of the explanatory variable (particle size) into the first estimation model EM1 and outputting multiple values ​​of the objective variable (number of particles) from the first estimation model EM1 to obtain a second data set LD2. Step S4 represents a step of adding additional information to the second data set LD2. Step S5 represents a step of using the machine learning model GP to machine-learn the second data set LD2 to which the additional information has been added, to create an estimation model (second estimation model EM2) that outputs an expected value of the predictive distribution for the objective variable (number of particles). The second estimation model EM2 is the machine learning model GP after machine learning of the second data set LD2 to which the additional information has been added.

[0117] The support method of the fourth embodiment is executed, for example, by the terminal device 101A included in the support system 100A described with reference to Fig. 9. More specifically, the support method shown in Fig. 10 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 Fig. 10 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, Fig. 10 shows the processing executed by the processing unit 106 included in the support system 100A of the fourth embodiment.

[0118] The processing unit 106 may start the processing shown in Fig. 10 after executing the processing described with reference to Fig. 2 to Fig. 4. In other words, the processing unit 106 may start the processing shown in Fig. 10 after creating the first estimation model EM1.

[0119] 10 starts, the processing unit 106 executes the second dataset creation program CP to acquire the second dataset LD2 (step S3). Specifically, the processing unit 106 inputs a plurality of predetermined values ​​as values ​​of the explanatory variable (particle size) to the first estimation model EM1, and causes the first estimation model EM1 to output a plurality of values ​​of the objective variable (number of particles), thereby acquiring the second dataset LD2. The second dataset LD2 includes the plurality of values ​​of the explanatory variable input to the first estimation model EM1 and the plurality of values ​​of the objective variable output from the first estimation model EM1.

[0120] For example, as already explained, the processing unit 106 may input 500 particle diameters xi (i = 1, 2, ..., 500) into the first estimation model EM1 to obtain 500 particle count values. In detail, the processing unit 106 executes processing based on the following formula (3) on the value of the objective variable output from the first estimation model EM1 to obtain the second data set LD2. In formula (3), round() indicates processing to round off the value in parentheses (in formula (3), the value of the objective variable output from the first estimation model EM1). yi = round(A × D(xi)) ... (3)

[0121] Next, the processing unit 106 adds additional information to the second data set LD2 (step S4). The additional information includes information about the origin (0, 0). In other words, the processing unit 106 adds the information about the origin (0, 0) to the second data set LD2. The information about the origin (0, 0) may be included in the assistance program SP in advance.

[0122] The processing unit 106 may add, as additional information, information on the number of particles and particle diameters per substrate obtained from a measurement device other than an optical measurement device to the second data set LD2. Specifically, as additional information, information indicating the number of particles with particle diameters less than 10 nm may be added to the second data set LD2. According to this embodiment, information on particles with particle diameters less than 10 nm that cannot be measured by an optical measurement device can be added to the second data set LD2.

[0123] Next, the processing unit 106 performs machine learning on the second data set LD2 to which the additional information has been added using the machine learning model GP to generate a second estimation model EM2, thereby completing the processing shown in FIG.

[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 a predicted distribution of the number of particles per substrate. In Figures 11A and 11B, the horizontal axis represents particle diameter, and the vertical axis represents the number of particles per substrate.

[0125] 11A and 11B, the processing unit 106 executes the assistance program SP to display 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 indicates an estimated value of the particle number (number of particles per substrate) relative to the particle diameter. The variance of the predicted distribution indicates an estimated value of the variation in the particle number (number of particles per substrate) relative to the particle diameter.

[0126] 11A shows a predicted distribution output from a machine learning model GP (second estimation model EM2) that has been machine-learned on a second data set LD2 to which only information about the origin (0,0) has been added as additional information. FIG. 11B shows a predicted distribution output from a machine learning model GP (second estimation model EM2) that has been machine-learned on a second data set LD2 to which, together with information about the origin (0,0), particle information indicating a particle size (explanatory variable) of 5 nm and a particle count (objective variable) of 5 has been added as additional information. As shown in FIGS. 11A and 11B , by adding information about particles with particle sizes less than 10 nm that cannot be measured by an optical measurement device to the second data set LD2 as additional information, the added particle information can be reflected in the outline of the predicted distribution.

[0127] Furthermore, research using atomic force microscopy (AFM) and cryo-electron microscopy (cryo-EM) suggests that the number of particles with a particle size of 7 nm or less may not follow the power law D(x). For ease of understanding, the graphs of the first estimated model EM1 are shown by dashed lines in Figures 11A and 11B.

[0128] 11A and 11B, the estimated value output from the first estimation model EM1 increases exponentially as the value of the explanatory variable (particle size) approaches 0 nm. However, the number of particles with a particle size of 0 nm is expected to be 0. Furthermore, experimental examples using atomic force microscopes and cryo-electron microscopes suggest that the number of particles may converge to 0 before the particle size reaches 0 nm.

[0129] In contrast, according to the fourth embodiment, by subjecting the machine learning model GP to machine learning of a dataset (second dataset LD2) obtained by inputting values ​​of particle diameters of 7 nm or more into the first estimation model EM1, the expected values ​​of the predictive distribution output from the second estimation model EM2 in the particle diameter range of greater than 7 nm can be made to conform to the graph of the first estimation model EM1, as shown in Figures 11A and 11B. Furthermore, by adding information about the origin (0, 0) to the second dataset LD2, the expected values ​​of the predictive distribution output from the second estimation model EM2 in the particle diameter range of 7 nm or less can be made to converge toward the origin (0, 0), as shown in Figures 11A and 11B.

[0130] Therefore, according to the fourth embodiment, the operator can use the second estimation model EM2 to more accurately estimate (predict) the number of particles of an arbitrary particle size. For example, the operator can operate the operation unit 102 to input an arbitrary particle size into the second estimation model EM2, thereby obtaining an estimated value (predicted value) of the number of particles having the input particle size. In particular, according to the fourth embodiment, it is possible to more accurately estimate (predict) the number of particles having a particle size that cannot be measured by an optical measurement device. Furthermore, according to the fourth embodiment, it may be possible to more accurately estimate (predict) the number of particles having a particle size of 7 nm or less that does not follow the power law D(x).

[0131] 2 to 4, 9, 10, 11A, and 11B, the fourth embodiment of the present invention has been described. According to the fourth embodiment, similar to the first to third embodiments, it is possible to assist in the task of estimating the number of particles on a substrate.

[0132] 12 to 14, a fifth embodiment of the present invention will be described. However, differences from the first to fourth embodiments will be described, and a description of the same aspects as the first to fourth embodiments will be omitted. The fifth embodiment differs from the first to fourth embodiments in that a first estimation model EM1 is used to generate values ​​of the objective variables of the dataset used in Bayesian optimization.

[0133] 12 is a block diagram showing the configuration of a support system 100A according to the fifth embodiment. As shown in FIG. 12, the support program SP includes a first objective variable generation program VC1 in addition to a pre-determination model PEM and a Bayesian estimation program BE. The first objective variable generation program VC1 is a computer program that uses a first estimation model EM1 to generate objective variable values ​​for a dataset used in Bayesian optimization.

[0134] Here, the objective variable of the data set used for Bayesian optimization includes particle density, and the explanatory variables of the data set used for Bayesian optimization include at least one parameter among various parameters that define the operation of the substrate processing apparatus, and Bayesian optimization is used to determine an optimal solution for the explanatory variables (at least one parameter).

[0135] The explanatory variables used in Bayesian optimization may include at least one of various setting values ​​(recipe parameter values) defined by a recipe and various setting values ​​(equipment parameter values) set for a substrate processing apparatus. The substrate processing apparatus may be any apparatus that processes substrates. Examples of the substrate processing apparatus include 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 assistance method, recording medium 200, assistance program SP, and assistance system 100A of embodiment 5 will be described with reference to FIGS. 12 to 14. FIG. 13 is a flowchart showing the assistance method of embodiment 5. FIG. 14 is a diagram showing a dependent variable creation process. In embodiment 5, the assistance method includes a dependent variable creation step of creating dependent variable values ​​of a dataset used in Bayesian optimization. FIG. 13 shows the flow of the dependent variable creation step. As shown in FIG. 13, the dependent variable creation step includes steps S21 and S22.

[0137] Step S21 is a coefficient value determination step in which the values ​​of the first coefficient k and the second coefficient q of the power law D(x) are determined by Bayesian estimation using the first data set LD1. Step S21 is similar to step S1 described with reference to Figures 2 to 4, and therefore a detailed description thereof will be omitted.

[0138] Step S22 indicates a step of acquiring an area D1 of a region below the graph of the first estimation model EM1 for which the values ​​of the first coefficient k and the second coefficient q have been determined in the coefficient value determination step (step S21), the region being equal to or greater than the minimum value set for the variable x. Hereinafter, the minimum value set for the variable x may be referred to as the "minimum particle size Xmin." The minimum particle size Xmin may be preset to any value greater than, for example, 7 nm. This is because, as already explained, the number of particles with a particle size of 7 nm or less may not follow the power law D(x).

[0139] Step S23 is a step of acquiring the value of the objective variable (particle density) of the data set used for Bayesian optimization based on the area D1 of the region equal to or larger than the minimum particle size Xmin and the area of ​​the substrate.

[0140] The support method of the fifth embodiment is executed, for example, by the terminal device 101A included in the support system 100A described with reference to Fig. 12. More specifically, the support method shown in Fig. 13 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 Fig. 13 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, Fig. 13 shows the processing executed by the processing unit 106 included in the support system 100A of the fifth embodiment.

[0141] 13 may be started by, for example, an operator operating the operation unit 102 to instruct the creation of values ​​of objective variables to be used in Bayesian optimization. When the processing unit 106 starts the processing shown in FIG. 13 , it executes an objective variable creation process.

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

[0143] Next, the processing unit 106 executes the first objective variable generation program VC1. Specifically, as shown in FIG. 14 , the processing unit 106 acquires the area D1 of the region equal to or greater than the minimum particle diameter Xmin within the region below the graph of the first estimation model EM1 for which the 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 calculate the area D1, for example, based on the following equation (4). Then, the processing unit 106 acquires the value of the objective variable (particle density) of the dataset used for Bayesian optimization based on the area D1 of the region equal to or greater than the minimum particle diameter Xmin and the area of ​​the substrate (step S23). As a result, the processing shown in FIG. 13 ends. Specifically, the processing unit 106 acquires the value of the particle density by dividing the area D1 of the region equal to or greater than the minimum particle diameter Xmin by the area of ​​the substrate.

[0144] The fifth embodiment of the present invention has been described above with reference to FIGS. 12 to 14 . According to the fifth embodiment, similarly to the first to fourth embodiments, the task of estimating the number of particles on a substrate can be supported. Furthermore, according to the fifth embodiment, the particle density is acquired using the power law D(x), so that the number of particles with particle sizes that cannot be measured by an optical measurement device can be reflected in the value of the objective variable (particle density) of the data set used for Bayesian optimization. Furthermore, according to the fifth embodiment, the particle density can be acquired using the power law D(x) in which overfitting to observed value data is alleviated. Therefore, a more accurate particle density can be acquired.

[0145] 15 to 17, a sixth embodiment of the present invention will be described. However, differences from the first to fifth embodiments will be described, and a description of the same aspects as the first to fifth embodiments will be omitted. The sixth embodiment differs from the first to fifth embodiments in that a second estimation model EM2 is used to generate values ​​of the objective variable of the dataset used in Bayesian optimization.

[0146] Fig. 15 is a block diagram showing the configuration of a support system 100A according to embodiment 6. As shown in Fig. 15, the support program SP includes a pre-determined model PEM, a Bayesian inference program BE, a pre-learning machine learning model GP, a second dataset creation program CP, and a second objective variable creation program VC2.

[0147] The second objective variable generation program VC2 is a computer program that generates objective variable values ​​of a dataset used for Bayesian optimization using the second estimation model EM2. As already described, the objective variable of the dataset used for Bayesian optimization includes particle density. Furthermore, the explanatory variables of the dataset used for Bayesian optimization include at least one parameter from among various parameters that define the operation of the substrate processing apparatus.

[0148] Next, the assistance method, recording medium 200, assistance program SP, and assistance system 100A of embodiment 6 will be described with reference to FIGS. 15 to 17. FIG. 16 is a flowchart showing the assistance method of embodiment 6. FIG. 17 is a diagram showing a dependent variable creation process. In embodiment 6, the assistance method includes a dependent variable creation step of creating dependent variable values ​​of a dataset used in Bayesian optimization. FIG. 16 shows the flow of the dependent variable creation step. The dependent variable creation step shown in FIG. 16 may be started after the second estimation model EM2 is created. As shown in FIG. 16, the dependent variable creation step includes steps S51 to S53.

[0149] Step S51 represents a step of outputting an expected value of the predicted distribution for the number of particles per substrate from the second estimation model EM2. Step S52 represents a step of acquiring an area D2 of a predetermined range within the region below the graph of the expected value of the predicted distribution. Step S53 represents a step of acquiring a value of the objective variable (particle density) of the data set used for Bayesian optimization based on the area D2 of the predetermined range and the area of ​​the substrate. Here, the lower limit of the predetermined range may be set to any value equal to or less than 7 nm. As already explained, this is because the predicted distribution output from the second estimation model EM2 is expected to more accurately reflect the number of particles with a particle diameter of 7 nm or less, which do not follow the power law D(x).

[0150] The support method of the sixth embodiment is executed, for example, by the terminal device 101A included in the support system 100A described with reference to Fig. 15. More specifically, the support method shown in Fig. 16 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 Fig. 16 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, Fig. 16 shows the processing executed by the processing unit 106 included in the support system 100A of the sixth embodiment.

[0151] 16 may be started by, for example, an operator operating the operation unit 102 to instruct the creation of values ​​of objective variables to be used in Bayesian optimization. When the processing unit 106 starts the processing shown in FIG. 16, it executes an objective variable creation process.

[0152] In more detail, the processing unit 106 first outputs an expected value of the predicted distribution for the number of particles per substrate from the second estimation model EM2 (step S51). The processing unit 106 may create the second estimation model EM2 in response to an instruction to create a value of the objective variable to be used in Bayesian optimization. Alternatively, the second estimation model EM2 may be created before the execution of the objective variable creation process. The process of creating the second estimation model EM2 has already been described with reference to FIGS. 2 to 4, 9, 10, 11A, and 11B, and therefore will not be described here.

[0153] Next, the processing unit 106 executes the second objective variable generation program VC2. Specifically, as shown in FIG. 17 , the processing unit 106 acquires the area D2 of a predetermined range of the region below the graph of the expected value of the predicted distribution (step S52). Then, the processing unit 106 acquires the value of the objective variable (particle density) of the data set used for Bayesian optimization based on the area D2 of the predetermined range of the region and the area of ​​the substrate (step S53). As a result, the processing shown in FIG. 16 ends. More specifically, the processing unit 106 acquires the value of the particle density by dividing the area D2 of the predetermined range of the region by the area of ​​the substrate.

[0154] The predetermined range may be, for example, 5 nm to 100 nm. The processing unit 106 may also calculate the area D2 by performing a numerical integration. The numerical integration may be, for example, a trapezoidal integral, a Simpson integral, or a Romberg integral.

[0155] The sixth embodiment of the present invention has been described above with reference to FIGS. 15 to 17 . According to the sixth embodiment, similarly to the first to fifth embodiments, the task of estimating the number of particles on a substrate can be supported. Furthermore, according to the sixth embodiment, the particle density is acquired using the expected value of the predicted distribution output from the second estimation model EM2. Therefore, the number of particles with particle diameters that cannot be measured by an optical measurement device can be reflected in the value of the objective variable (particle density) of the data set used for Bayesian optimization. Furthermore, according to the sixth embodiment, the number of particles with particle diameters of 7 nm or less, which may not be accurately estimated using the power law D(x), can be reflected in the value of the objective variable (particle density) of the data set used for Bayesian optimization.

[0156] The embodiments of the present invention have been described above with reference to the drawings (FIGS. 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 components disclosed in the above embodiments can be modified as appropriate. For example, some of the components shown in one embodiment may be added to the components of another embodiment, or some of the components shown in one embodiment may be deleted from the embodiment.

[0157] The drawings mainly show each component in a schematic manner to facilitate understanding of the invention, and the thickness, length, number, spacing, etc. of each component shown in the drawings may differ from the actual ones due to the convenience of creating the drawings. Furthermore, the configuration of each component shown in the above embodiment is merely an example and is not particularly limited, and it goes without saying that various modifications are possible within a range that does not substantially deviate from the effects of the present invention.

[0158] For example, in embodiments 4 to 6, the terminal device 101A executed the support program SP as in embodiment 1, but the server 300 may execute the support program SP as in embodiment 2, or the control device 10 of the substrate processing system 1000 may execute the support program SP as in embodiment 3.

[0159] The present invention is useful for estimating the number of particles on a substrate.

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

Claims

1. A method for assisting in an operation of estimating the number of particles on a substrate, comprising: a model creation step of creating a first estimation model for estimating the number of particles for each particle size; the first estimation model comprises the following formula (1): A×D(x)=A×kx-q ... (1) In the above formula (1), constant A represents the area of ​​the substrate, D(x) represents a power law, and variable x represents the particle size of the particle; by inputting an arbitrary value into variable x of the first estimation model, the number of particles having a particle size of the value input into variable x is calculated; the model creation step comprises: a coefficient value determination step of determining the values ​​of a first coefficient k and a second coefficient q of the power law D(x) by Bayesian estimation using a first data set; and a step of creating the first estimation model by inputting the values ​​determined in the coefficient value determination step into the first coefficient k and the second coefficient q, the first data set includes values ​​of explanatory variables and values ​​of response variables corresponding to the values ​​of the explanatory variables, each value of the explanatory variables indicating a first observed value which is an observed value of the particle diameter of the particle, and each value of the response variable indicating a second observed value which is an observed value of the number of particles; the coefficient value determining step includes: a step of repeating a step of sampling a tentative value of the first coefficient k and a tentative value of the second coefficient q from a first prior distribution which is a prior distribution of the first coefficient k and a second prior distribution which is a prior distribution of the second coefficient q, to ​​obtain a posterior distribution of the first coefficient k and a posterior distribution of the second coefficient q; and a step of determining each value of the first coefficient k and the second coefficient q based on the posterior distribution of the first coefficient k and the posterior distribution of the second coefficient q, and the sampling step includes: a step of creating a provisional model based on first sampled values ​​which are values ​​sampled from the first prior distribution, second sampled values ​​which are values ​​sampled from the second prior distribution, and the first estimation model in which the values ​​of the first coefficient k and the second coefficient q are undetermined; and a step of evaluating a degree of fitting of the provisional model to the first data set based on the provisional model, the first data set, and a likelihood function, wherein the second prior distribution includes a uniform distribution within a predetermined range.

2. The support method according to claim 1, wherein the model creation step further includes the steps of: inputting a plurality of predetermined values ​​as values ​​of the explanatory variables into the first estimation model, and outputting a plurality of values ​​of the dependent variable from the first estimation model to obtain a second dataset; adding additional information to the second dataset; and subjecting the second dataset to which the additional information has been added to a machine learning model capable of outputting a predictive distribution, to create a second estimation model that outputs an expected value of the predictive distribution for the dependent variable; wherein 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 dependent variable output from the first estimation model; and the additional information includes information of the origin (0, 0).

3. The support method according to claim 1, further comprising a dependent variable creation step of creating a value of a dependent variable of a dataset used in Bayesian optimization, wherein the dataset used in the Bayesian optimization includes, as an explanatory variable, at least one parameter from among various parameters that define the operation of a substrate processing apparatus that processes the substrate, and the Bayesian optimization is used to determine an optimal solution for the at least one parameter, and the dependent variable creation step comprises: a step of acquiring the area of ​​a region equal to or greater than a minimum value set for the variable x within the region below the graph 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 a step of acquiring the value of the dependent variable of the dataset used in the Bayesian optimization based on the area of ​​the region equal to or greater than the minimum value and the area of ​​the substrate.

4. The support method according to claim 2, further comprising a dependent variable creation step of creating a value of a dependent variable of a dataset used in Bayesian optimization, wherein the dataset used in the Bayesian optimization includes, as an explanatory variable, at least one parameter from among various parameters that define the operation of a substrate processing apparatus that processes the substrate, and the Bayesian optimization is used to determine an optimal solution for the at least one parameter, and the dependent variable creation step comprises: a step of outputting an expected value of the predictive distribution from the second estimation model; a step of acquiring the area of ​​a predetermined range of an area below a graph of the expected value of the predictive distribution; and a step of acquiring the value of the dependent variable of the dataset used in the Bayesian optimization based on the area of ​​the predetermined range of the area and the area of ​​the substrate.

5. The method of any one of claims 1 to 4, wherein the likelihood function includes a Poisson distribution.

6. A computer-readable recording medium having a support program that defines the support method according to any one of claims 1 to 5 recorded thereon.

7. A support program that can be executed by a computer and that defines the support method according to any one of claims 1 to 5.

8. A support system for supporting a task of estimating the number of particles on a substrate, comprising: a processing unit that creates a first estimation model that estimates the number of particles for each particle size; and a storage unit that stores a first dataset, wherein the first estimation model includes the following formula (1): A×D(x)=A×kx-q ... (1) In the above formula (1), constant A represents the area of ​​the substrate, D(x) represents a power law, variable x represents the particle size of the particle, and by inputting an arbitrary value into variable x of the first estimation model, the number of particles having a particle size of the value input into variable x is calculated, and the first dataset includes values ​​of explanatory variables and values ​​of objective variables corresponding to each value of the explanatory variables, and each value of the explanatory variable represents a first observed value that is an observed value of the particle size of the particle, and each value of the objective variable represents a second observed value that is an observed value of the number of particles, the processing unit executes a coefficient value determination process to determine the values ​​of each of a first coefficient k and a second coefficient q of the power law D(x) by Bayesian estimation using the first data set, and creates the first estimation model by inputting the values ​​determined by the coefficient value determination process into each of the first coefficient k and the second coefficient q; during the execution of the coefficient value determination process, the processing unit repeats a step of sampling a tentative value of the first coefficient k and a tentative value of the second coefficient q from a first prior distribution that is a prior distribution of the first coefficient k and a second prior distribution that is a prior distribution of the second coefficient q, to ​​obtain a posterior distribution of the first coefficient k and a posterior distribution of the second coefficient q; and determines the values ​​of each of the first coefficient k and the second coefficient q based on the posterior distribution of the first coefficient k and the posterior distribution of the second coefficient q; during the execution of the sampling step, the processing unit creating a provisional model based on first sampled values ​​sampled from the first prior distribution, second sampled values ​​sampled from the second prior distribution, and the first estimation model in which the values ​​of the first coefficient k and the second coefficient q are undetermined; evaluating a degree of fitting of the provisional model to the first data set based on the provisional model, the first data set, and a likelihood function;The second prior distribution comprises a uniform distribution over a predetermined range.

9. The assistance system described in claim 8, wherein the storage unit further stores a machine learning model capable of outputting a predictive distribution, and the processing unit: inputs a plurality of predetermined values ​​as values ​​of the explanatory variables to the first estimation model and causes the first estimation model to output a plurality of values ​​of the dependent variable to obtain a second dataset; adds additional information to the second dataset; and causes the machine learning model to machine-learn the second dataset with the added additional information to create a second estimation model that outputs an expected value of the predictive distribution for the dependent variable, wherein the second dataset includes a plurality of values ​​of the explanatory variables input to the first estimation model and a plurality of values ​​of the dependent variable output from the first estimation model, and the additional information includes information of the origin (0, 0).

10. The assistance system described in claim 8, wherein the processing unit further executes a dependent variable creation process to create a value of a dependent variable of a dataset to be used for Bayesian optimization, the dataset to be used for Bayesian optimization including, as an explanatory variable, at least one parameter among various parameters that define the operation of a substrate processing apparatus that processes the substrate, and the Bayesian optimization is used to determine an optimal solution for the at least one parameter, and the processing unit, when executing the dependent variable creation process, obtains the area of ​​a region below the graph of the first estimation model in which the values ​​of the first coefficient k and the second coefficient q have been determined by the coefficient value determination process, that is equal to or greater than a minimum value set for the variable x, and obtains the value of the dependent variable of the dataset to be used for the Bayesian optimization based on the area of ​​the region equal to or greater than the minimum value and the area of ​​the substrate.

11. The assistance system described in claim 9, wherein the processing unit further executes a dependent variable creation process to create a value of a dependent variable of a dataset to be used for Bayesian optimization, the dataset to be used for Bayesian optimization includes, as an explanatory variable, at least one parameter from among various parameters that define the operation of a substrate processing apparatus that processes the substrate, and the Bayesian optimization is used to determine an optimal solution for the at least one parameter, and when executing the dependent variable creation process, the processing unit causes the second estimation model to output an expected value of the predictive distribution, obtains the area of ​​a predetermined range of the area below a graph of the expected value of the predictive distribution, and obtains the value of the dependent variable of the dataset to be used for the Bayesian optimization based on the area of ​​the predetermined range of the area and the area of ​​the substrate.

12. The assistance system according to any one of claims 8 to 11, wherein the likelihood function includes a Poisson distribution.

13. A substrate processing system for processing substrates, comprising: a substrate processing apparatus for processing the substrates; and a control device for controlling the substrate processing apparatus, wherein the control device also serves as a support system according to any one of claims 8 to 12.

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