A key parameter design and optimization method for inductively coupled plasma source

Abstract

The present invention discloses a key parameter design and optimization method for an inductively coupled plasma source, which relates to the technical field of inductively coupled plasma sources, including: establishing a Plackett-Burman orthogonal matrix according to the design target of the inductively coupled plasma source. Calculating a response value coefficient matrix through a finite element analysis model, establishing a first-order linear regression equation, screening significant variables, and calculating and establishing a Box-Behnken design matrix based on the coefficients of the main effect variables. Calculating a response value coefficient matrix through a finite element analysis model, and establishing a high-order polynomial regression equation. Determining the importance and interaction of each significant variable through variance analysis and a three-dimensional response surface. Obtaining the optimal solution for the inductively coupled plasma source by solving the maximum value of the fitting equation. The solution of the present invention provides a solution to the difficult problem of multi-objective and multi-variable plasma source design that cannot be solved by traditional design methods.

Description

Technical Field

[0001] The present invention belongs to the technical field of inductively coupled plasma sources, and in particular relates to a key parameter design and optimization method of an inductively coupled plasma source. Background Art

[0002] Inductively coupled plasma (ICP) sources are widely used in a variety of fields, including microelectronics manufacturing and neutral beam injection. They can achieve high plasma density over a wide area through radio frequency (RF) discharge, making them crucial for precise etching of nanoscale devices, effective large-area plasma processing, and stable, uniform beam currents. As process requirements increase, the design requirements for plasma sources are becoming increasingly stringent. Although ICP sources have been developed for many years, the design process still relies primarily on empirical trial and error, lacking precise and reliable quantitative methods. Traditional empirical design often struggles to accurately identify key parameters when faced with complex multivariable systems (e.g., interactions between discharge pressure and RF power), resulting in time-consuming and costly experimental adjustments. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for designing and optimizing key parameters of an inductively coupled plasma source, thereby solving the problems existing in the prior art solutions.

[0004] The present invention provides a method for designing and optimizing key parameters of an inductively coupled plasma source, comprising:

[0005] S101, obtain the design target parameters and construct the response value matrix and the latent variable matrix ;

[0006] S102, construct and simplify the Plackett-Burman orthogonal matrix based on latent variables , multiply the simplified matrix by the latent variable matrix Get the latent variable design scheme;

[0007] S103, iterative calculation based on the Boltzmann module and fluid module in the finite element analysis model, and solving the response value coefficient matrix in sequence according to the potential variable design scheme in step S102 ;

[0008] S104, fitting a first-order linear regression equation of the response value and the latent variable to the response value coefficient matrix and the simplified matrix obtained in S103, and calculating the goodness of fit;

[0009] S105, using variance analysis to identify significant variables based on the fitting results of S104, calculating the Box-Behnken design matrix based on the coefficients of the main effect variables, and multiplying the design matrix with the significant variable matrix to obtain the significant variable design scheme;

[0010] S106, based on the fluid model and the Boltzmann model in the finite element analysis model, iterative calculations are performed, and the response value coefficient matrix is ​​solved in sequence according to the significant variable design scheme of S105;

[0011] S107, fitting a high-order polynomial regression equation of the response value and the significant variables according to the response value coefficient matrix and the design matrix of S106, and solving the goodness of fit;

[0012] S108, using variance analysis on the high-order polynomial regression equation established in S107 to determine the effect of the interaction term on the response value. The point of maximum response value is solved, which is the optimal design solution.

[0013] A computing device comprises: at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device executes a key parameter design and optimization method for an inductively coupled plasma source.

[0014] A readable storage medium storing program instructions, when the program instructions are read and executed by a computing device, enables the computing device to execute a key parameter design and optimization method for an inductively coupled plasma source.

[0015] The present invention has the following beneficial effects:

[0016] (1) It can quantitatively determine the optimal design of multi-objective complex plasma source systems;

[0017] (2) It can accurately identify the interaction effects between multiple variables;

[0018] (3) The designed results accurately reflect the experimental data, which can be attributed to the coupling of the Boltzmann module and the fluid module;

[0019] (4) Shortened the design cycle and reduced costs.

[0020] Therefore, this method can be a very useful tool for designing key parameters and optimizing operating parameters of inductively coupled plasma sources. It provides a solution to the multi-objective, multi-variable plasma source design challenges that traditional design methods struggle to address. By effectively screening potential variables and establishing a minimum orthogonal design solution, the design cycle is significantly shortened and costs are reduced. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 Flowchart of a method for designing and optimizing key parameters of an inductively coupled plasma source according to an embodiment of the present invention;

[0022] Figure 21. Pareto diagram of potential variables for electron density and density uniformity according to an embodiment of the present invention, wherein (a) is the electron density Pareto diagram, and (b) is the density uniformity Pareto diagram;

[0023] Figure 3 : a distribution diagram of Box-Behnken design points according to a three-variable embodiment of the present invention;

[0024] Figure 4 3D response surface distribution diagram of driver length and RF power versus response value electron density according to an embodiment of the present invention. DETAILED DESCRIPTION

[0025] The present invention aims to provide a parameter design and optimization method for an inductively coupled plasma source. This method can rapidly identify and screen potential variables that have a critical impact on the design objectives, establish a minimum orthogonal design, and fit a high-order regression equation to describe possible interactions between significant variables. Ultimately, it provides an optimal design solution that meets the design objectives. It is important to emphasize that the results of the orthogonal design of the present invention are derived from the iterative results of the fluid model and the Boltzmann model in the finite element analysis model.

[0026] In order to fully and completely describe the technical solution of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings in the embodiments of the invention. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0027] Figure 1 Flowchart of the key parameter design and optimization method of the inductively coupled plasma source according to an embodiment of the present invention. Figure 1 As shown, the method includes:

[0028] Step S101: Obtain design target parameters and construct a response value matrix and the latent variable matrix .

[0029] This step may include:

[0030] Obtain the design target parameters of the inductively coupled plasma source, including electron density and density uniformity, and construct a response value matrix based on the design target parameters .

[0031] Based on the design objectives, potential variables of the inductively coupled plasma source are determined to obtain a potential variable matrix. In one embodiment, the potential variables include: actuator radius, actuator length, discharge gas pressure, RF power, coil turns, coil turn spacing, and operating temperature.

[0032] S102, construct and simplify the Plackett-Burman orthogonal matrix based on latent variables , get the simplified matrix, and multiply the simplified matrix with the latent variable matrix Get the latent variable design scheme, where is the response coefficient matrix.

[0033] This step may include:

[0034] S102-1, construct K+1 sets of Plackett-Burman orthogonal matrices based on the design target parameters and latent variables, where K refers to K latent variables.

[0035] Constructing K+1 sets of orthogonal matrices based on the design objectives and potential variables can include: constructing an orthogonal matrix based on the Plackett-Burman design, where K+1 is a multiple of 4, usually 12 11, 20 19, 24 23 and 36 In the above embodiment, according to the latent variables, K=7, 12 11 Orthogonal Matrix , which can satisfy up to 11 latent variables:

[0036] ,

[0037] in, represents a high level of the latent variable, Represents low level.

[0038] S102-2, simplify the orthogonal matrix based on the latent variables to obtain a simplified matrix . Simplify the matrix for:

[0039] ,

[0040] Latent variable matrix for:

[0041] ,in to is a latent variable;

[0042] Response value matrix :

[0043] ,in to is the design target parameter.

[0044] In the above specific implementation, the latent variable matrix for:

[0045] ,

[0046] in, is the actuator radius, is the drive length, is the discharge gas pressure, is the RF power, is the number of coil turns, is the coil turn spacing, For the operating temperature.

[0047] Response value matrix :

[0048]

[0049] where is the electron density, Density uniformity.

[0050] S102-3, multiply the simplified matrix by the latent variable matrix This results in a latent variable design, also known as a Plackett-Burman orthogonal design:

[0051] ,

[0052] Step S103: Iterative calculation based on the Boltzmann module and fluid module in the finite element analysis model, and solving the response value coefficient matrix in sequence according to the potential variable design scheme of step S102 In one embodiment, the response coefficient matrix Corresponding to 12 2.

[0053] This step may include: obtaining a response value coefficient matrix according to the results of the iteration of the Boltzmann module and the fluid module The iterative parameters output by the fluid module include: the mole fraction of metastable argon atoms and ground state argon atoms, the ionization degree , electron density .

[0054] The mole fractions of metastable argon atoms and ground-state argon atoms are calculated using the Maxwell-Stefan equation in the heavy ion model:

[0055] ,

[0056] Where, is the mass fraction of substance j, is the mass-averaged fluid velocity vector, is the density of the mixture, is the generation rate coefficient of species j, is the diffusion flux vector of species j, Represents divergence.

[0057] ,

[0058] Where, is the thermal diffusivity, is the operating temperature, is the average molar data of the gas mixture, is the charge of substance j, is the mobility of species j, is the average diffusion coefficient of the mixture of species j and k. is the induced electric field.

[0059] ,

[0060] in, is the degree of ionization, is the neutral particle number density.

[0061] The parameters of the Boltzmann module input fluid module include: electron energy distribution function f, reduced electron mobility coefficient , reduced electron diffusion coefficient , reduced energy transfer coefficient and reduced energy diffusion coefficient These parameters are calculated by solving the Boltzmann equation:

[0062] ,

[0063] Where, is the electron density, is the velocity coordinate, is the velocity gradient operator, is the collision change rate. is the electron energy distribution function, is the electron charge.

[0064] The two-term approximation to the Boltzmann equation yields a simplified distribution function:

[0065] ,

[0066] In the formula is the flow rate, is the diffusion coefficient, is the scattering coefficient, is the electron energy.

[0067] ,

[0068] ,

[0069] ,

[0070] ,

[0071] Where, is the coefficient, is the average electron energy, is the effective momentum transfer cross section.

[0072] Reduced Angular Frequency of Power Source in the Boltzmann Model The gas pressure P, operating temperature T and power frequency in the orthogonal design scheme of step S104 are Calculation yields:

[0073] ,

[0074] ,

[0075] in, is the power supply angular frequency.

[0076] The difference in ionization degree from the last iteration to the last calculation is less than Considered convergent.

[0077] Step S104: Calculate the response coefficient matrix obtained in step S103 Simplify the matrix, fit the first-order linear regression equation between the response value and the latent variable, and solve its goodness of fit. Because the orthogonal matrix is ​​full rank, a unique approximate solution can be found for the matrix.

[0078] The first-order linear regression equation for fitting the response value and the latent variable includes:

[0079] .

[0080] in, to are the coefficients of the fitted first-order linear regression equation.

[0081] In one embodiment, the solution obtained is:

[0082] ,

[0083] .

[0084] Solving its goodness of fit includes: performing residual analysis based on the first-order linear regression equation of the response value and the latent variable, and calculating its goodness of fit and adjusted goodness of fit .

[0085] ,

[0086] Where, is the residual sum of squares, is the total sum of squares, is the predicted value of the fitting equation, is the mean of the response values, is the response value.

[0087] ,

[0088] Where n is the number of experiments and p is the number of latent variables.

[0089] Specifically, in a specific application example, the electron density R 2 =0.9538, Adjust R 2 =0.8730, R of density uniformity 2 =0.9872, Adjust R 2 =0.9647, indicating that the first-order linear regression has a high goodness of fit.

[0090] Step S105: According to the fitting results of step S104, variance analysis is used to identify significant variables, and the Box-Behnken design matrix is ​​calculated according to the coefficients of the main effect variables. The Box-Behnken design matrix is ​​multiplied by the significant variable matrix to obtain the significant variable design scheme.

[0091] This step may include:

[0092] S105-1, calculate T value, F value and P value:

[0093] ,

[0094] ,

[0095] Where, is the coefficient of variable i of the fitted first-order linear regression equation, The standard error is calculated from the mean square error. Since the orthogonal matrix and the orthogonal matrix transpose are multiplied to form the identity matrix, the standard error calculation is simplified to , MSE is the mean square error, which is equal to the residual sum of squares divided by the degrees of freedom ( ). is the regression sum of squares, is the residual sum of squares. The regression sum of squares equals the total sum of squares minus the residual sum of squares. The P value is determined from the F-value distribution table.

[0096] S105-2, according to the variance calculation result of step S108, if the P value is less than 0.05, the variable is considered statistically significant and is a significant variable. Figure 2 The standardized effect value (T value calculated in S108) and cumulative proportion in the Pareto chart ((a) is electron density, (b) is density uniformity) screen significant variables: actuator radius, RF power, discharge pressure, actuator length, and number of coil turns. The dotted line in the figure indicates the significance level. The corresponding T value in the T distribution table.

[0097] S105-3, based on the main response value, confirm the main effect variable, calculate the steepest climbing square and variable range based on the coefficient of the main effect variable, calculate the step size of the remaining significant variables based on the coefficient of the main effect variable, and determine the Box-Behnken design center point through iterative calculation of the fluid module and the Boltzmann module:

[0098] ,

[0099] in, is the coefficient of the main effect variable, are the high and low level values ​​of variable i, respectively. is the number of steps, is the climbing step length of other significant variables i except the main effect.

[0100] Specifically, according to the design goal, electron density is selected as the main response value. Figure 2 In (a), sort the T values ​​in the Pareto chart from highest to lowest and select the main effect variable. The main effect variable could be the actuator radius. Based on the actuator radius coefficient (6.88), calculate the step sizes of the remaining significant variables. Determine the Box-Behnken design center point using iterative calculations in the Fluid and Boltzmann modules.

[0101] S105-4, based on the Box-Behnken design center point and the remaining significant variable steps, establish a design matrix, and multiply the design matrix with the latent variable matrix to obtain the corresponding design scheme. Figure 3 As shown, is the actuator radius, is the drive length, is the discharge gas pressure. Box-Behnken design is a 3-level design method. The design points are located in the center and the middle of the edge of the design space. Therefore, the design points are all on a sphere or the center of a sphere. In a specific scheme, the 6-variable 3-level design includes 48+3 groups of orthogonal design schemes. Except for the center point, only 3 variables are changed in each group, and the number of occurrences of each variable in the combination is kept consistent. Multiply the design matrix with the significant variable matrix Get the corresponding design scheme, design matrix for:

[0102] .

[0103] Step S106: Based on the iterative calculation of the fluid model and the Boltzmann model in the finite element analysis model, the response value coefficient matrix is ​​solved in sequence according to the significant variable design scheme. The difference in ionization degree obtained by the last iteration is less than In one embodiment, the response coefficient matrix is ​​corresponding to 51 2's matrix of matrices.

[0104] Step S107: Fitting a high-order polynomial regression equation of the response value and significant variables according to the response value coefficient matrix and the Box-Behnken design matrix of S106, and solving its goodness of fit.

[0105] In one embodiment, the quadratic linear regression equation has the highest goodness of fit, so a quadratic linear regression equation is established and solved, and the solution is:

[0106] ,

[0107] ,

[0108] Step S108: Using variance analysis on the high-order polynomial regression equation established in step S107, determine the effect of the interaction term on the response value, and solve the point of maximum response value, which is the optimal design solution.

[0109] This step may include:

[0110] S108-1: Calculate the F-value and P-value. Use the P-value to determine the statistical significance of the potential variable to the response value. A P-value less than 0.05 is considered statistically significant for the variable. Use the F-value to determine the importance of each variable in descending order.

[0111] S108-2, determine the interaction effect of two significant variables on the response value, Figure 4 The interactive effect of the actuator radius and RF power on the electron density is demonstrated.

[0112] S108-3, plotting a three-dimensional response surface based on the high-order polynomial regression equation established in step S107, determining the maximum response value selection range with the help of the three-dimensional response surface, and solving for the maximum point within the maximum response value selection range, i.e., solving for the maximum response value point, which is the optimal design solution.

[0113] According to an embodiment of the present invention, a computing device is also provided, comprising: at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device executes a key parameter design and optimization method for an inductively coupled plasma source.

[0114] According to an embodiment of the present invention, a readable storage medium storing program instructions is also provided. When the program instructions are read and executed by a computing device, the computing device executes a key parameter design and optimization method for an inductively coupled plasma source.

[0115] In the description provided herein, numerous specific details are described. However, it is understood that embodiments of the present invention may be practiced without these specific details. In some instances, well-known methods, structures, and techniques are not shown in detail so as not to obscure the understanding of this description.

[0116] Although the present invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of the foregoing description, will appreciate that other embodiments are contemplated within the scope of the invention thus described. Furthermore, it should be noted that the language used in this specification has been selected primarily for readability and instructional purposes, and not for the purpose of explaining or limiting the subject matter of the present invention.

Claims

1. A method for designing and optimizing key parameters of an inductively coupled plasma source, characterized in that: include: S101, obtain the design target parameters and construct the response value matrix and the latent variable matrix , including: obtaining the design target parameters of the inductively coupled plasma source, and constructing a response value matrix based on the design target parameters ; According to the design target parameters, determine the potential variables of the inductively coupled plasma source and obtain the potential variable matrix ; S102, construct and simplify the Plackett-Burman orthogonal matrix based on latent variables , multiply the simplified matrix by the latent variable matrix Get the latent variable design scheme; S103, iterative calculation based on the Boltzmann module and fluid module in the finite element analysis model, and solving the response value coefficient matrix in sequence according to the potential variable design scheme in step S102 ; S104, fitting a first-order linear regression equation of the response value and the latent variable to the response value coefficient matrix and the simplified matrix obtained in S103, and calculating the goodness of fit; S105, using variance analysis to identify significant variables based on the fitting results of S104, calculating the Box-Behnken design matrix based on the coefficients of the main effect variables, and multiplying the design matrix with the significant variable matrix to obtain the significant variable design scheme; S106, based on the fluid model and the Boltzmann model in the finite element analysis model, iterative calculations are performed, and the response value coefficient matrix is ​​solved in sequence according to the significant variable design scheme of S105; S107, fitting a high-order polynomial regression equation of the response value and the significant variables according to the response value coefficient matrix and the design matrix of S106, and solving the goodness of fit; S108, using variance analysis on the high-order polynomial regression equation established in S107 to determine the effect of the interaction term on the response value, and solving the point of maximum response value, which is the optimal design solution.

2. The key parameter design and optimization method of the inductively coupled plasma source according to claim 1, characterized in that: S102 includes: S102-1, construct K+1 sets of Plackett-Burman orthogonal matrices based on design target parameters and potential variables , K refers to K latent variables; S102-2, based on the latent variables, the Plackett-Burman orthogonal matrix is ​​simplified to obtain the simplified matrix ; S102-3, multiply the simplified matrix by the latent variable matrix The latent variable design is obtained, also known as the Plackett-Burman orthogonal design.

3. The key parameter design and optimization method of an inductively coupled plasma source according to claim 1, characterized in that: S103 includes: obtaining a response value coefficient matrix according to the iterative results of the Boltzmann module and the fluid module in the finite element analysis model The values ​​of , where the iterative parameters output by the fluid module include: the mole fraction of metastable atoms and ground state atoms, the degree of ionization , electron density , where the mole fractions of metastable atoms and ground-state atoms are calculated using the Maxwell-Stefan equation in the heavy-ion model: ， Where, is the mass fraction of substance j, is the mass-averaged fluid velocity vector, is the density of the mixture, is the generation rate coefficient of species j, is the diffusion flux vector of species j, represents divergence; ， Where, is the thermal diffusivity, is the operating temperature, is the average molar data of the gas mixture, For material The amount of charge, For material The migration rate, For material and The average diffusion coefficient of the mixture, is the induced electric field; ,in, is the neutral particle number density; The parameters of the Boltzmann module input fluid module include: electron energy distribution function f, reduced electron mobility coefficient , reduced electron diffusion coefficient , reduced energy transfer coefficient and reduced energy diffusion coefficient , these parameters are calculated by solving the Boltzmann equation: ， Where, is the electron density, is the velocity coordinate, is the velocity gradient operator, is the collision change rate, is the electron energy distribution function, is the electron charge; The two-term approximation to the Boltzmann equation gives the simplified distribution function: ， Where, is the flow rate, is the diffusion coefficient, is the scattering coefficient, is the electron energy; ， ， ， ， Where, is the coefficient, is the average electron energy, is the effective momentum transfer cross section; Power reduction angular frequency in the Boltzmann module The gas pressure P, operating temperature T and power frequency in the orthogonal design scheme of step S104 are The calculation gives: ， ， in, is the power supply angular frequency; The difference in ionization degree obtained by the last iteration is less than Considered convergent.

4. The method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 1, wherein: In S104, the first-order linear regression equation for fitting the response value and the latent variable includes: ， in, to are the coefficients of the fitted first-order linear regression equation.

5. The method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 1, wherein: In S104, solving the goodness of fit includes: performing residual analysis based on the first-order linear regression equation of the response value and the latent variable, and calculating its goodness of fit and adjusted goodness of fit : ， Where, is the residual sum of squares, is the total sum of squares, is the predicted value of the fitting equation, is the mean of the response values, is the response value, is the goodness of fit; ， Where n is the number of experiments, p is the number of potential variables, is the adjusted goodness of fit.

6. The method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 5, wherein: S105 includes: S105-1, calculate T value, F value and P value: ， ， Where, is the coefficient of variable i of the fitted first-order linear regression equation, The standard error is calculated from the mean square error, and the calculation is simplified to , MSE is the mean square error, which is equal to the residual sum of squares divided by the degrees of freedom , is the regression sum of squares, is the residual sum of squares. The regression sum of squares is equal to the total sum of squares minus the residual sum of squares. The P value is determined by the F value distribution table. S105-2, based on the variance calculation result of step S108, if the P value is less than 0.05, the variable is considered statistically significant and is a significant variable; S105-3: Based on the main response value, identify the main effect variable, calculate the steepest climbing square and variable range based on the coefficient of the main effect variable, calculate the step size of the remaining significant variables based on the coefficient of the main effect variable, and determine the Box-Behnken design center point through iterative calculation of the fluid module and the Boltzmann module; S105-4: Based on the Box-Behnken design center point and the remaining significant variable steps, a design matrix is ​​established, and the design matrix is ​​multiplied by the latent variable matrix to obtain the corresponding design scheme.

7. The method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 5, wherein: S108 includes: S108-1, calculate the F value and P value, and determine the statistical significance of the significant variable to the response value based on the P value; S108-2, determine the interaction effect of two variables on the response value; S108-3, plotting a three-dimensional response surface based on the high-order polynomial regression equation established in step S107, determining the maximum response value selection range with the help of the three-dimensional response surface, and solving for the maximum point within the maximum response value selection range, i.e., solving for the maximum response value point, which is the optimal design solution.

8. A computing device, characterized in that include: at least one processor and memory storing program instructions; When the program instructions are read and executed by the processor, the computing device is caused to execute the key parameter design and optimization method of the inductively coupled plasma source according to any one of claims 1 to 7.

9. A readable storage medium storing program instructions, characterized in that: When the program instructions are read and executed by a computing device, the computing device is caused to execute the key parameter design and optimization method for an inductively coupled plasma source according to any one of claims 1 to 7.

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