Key parameter design and optimization method of inductively coupled plasma source

Abstract

The invention discloses a key parameter design and optimization method of an inductively coupled plasma source, and relates to the technical field of inductively coupled plasma sources, and the method comprises the steps: building a Plackett-Burman orthogonal matrix according to the design target of the inductively coupled plasma source; a response value coefficient matrix is calculated through a finite element analysis model, a first-order linear regression equation is established, significant variables are screened, and a Box-Behnken design matrix is calculated and established according to coefficients of main effect variables. A response value coefficient matrix is calculated through a finite element analysis model, and a high-order polynomial return equation is established; and determining the importance degree and interaction of each significant variable through variance analysis and a three-dimensional response surface. And the optimal scheme of the inductively coupled plasma source is obtained by solving the maximum value of the fitting equation. According to the scheme, a solution is provided for the problem of multi-target and multivariable plasma source design which cannot be solved by a traditional design method.

Description

Technical Field

[0001] The present invention belongs to the technical field of inductively coupled plasma sources, and particularly relates to a method for designing and optimizing key parameters of an inductively coupled plasma source. Background Art

[0002] Inductive coupled plasma sources are widely used in various fields such as microelectronic manufacturing and neutral beam injection. Among them, inductively coupled plasma sources can achieve high plasma density in a large range through radio frequency discharge, which is crucial for realizing precise etching of nanoscale devices, effective large-area plasma processing, and stable and uniform beam currents. With the improvement of process requirements, the design requirements for plasma sources are becoming increasingly stringent. Although inductively coupled plasma sources have been developed for many years, the design process still mainly relies on empirical trial and error, lacking accurate and reliable quantitative methods. Traditional empirical design often has difficulty accurately identifying key parameters when facing multi-variable complex system designs (such as the interaction of discharge pressure, radio frequency power, etc.), resulting in long experimental adjustment time and high costs. 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, so as to solve 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, including:

[0005] S101, obtaining design target parameters and constructing a response value matrix and a latent variable matrix ;

[0006] S102, constructing and simplifying a Plackett-Burman orthogonal matrix according to the latent variables , multiplying the simplified matrix by the latent variable matrix to obtain a latent variable design scheme;

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

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

[0009] S105, using analysis of variance to identify significant variables for the fitting result in S104, calculating a Box-Behnken design matrix according to the coefficients of the main effect variables, and multiplying the design matrix by the significant variable matrix to obtain a significant variable design scheme;

[0010] S106. Iteratively calculate based on the fluid model and Boltzmann model in the finite element analysis model, and sequentially solve the response value coefficient matrix according to the significant variable design scheme of S105;

[0011] S107. Fit the high-order polynomial regression equation of the response value and significant variables according to the response value coefficient matrix and design matrix of S106, and solve its goodness of fit;

[0012] S108. Perform analysis of variance on the high-order polynomial regression equation established in S107 to determine the effect of the interaction term on the response value. Solve the maximum point of the response value, and this point is the optimal design scheme.

[0013] A computing device, 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 is caused to execute the method for designing and optimizing the key parameters of 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, the computing device is caused to execute the method for designing and optimizing the key parameters of an inductively coupled plasma source.

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

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

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

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

[0019] (4) It shortens the design cycle and reduces the cost.

[0020] Therefore, this method can become a very useful tool for the design of key parameters and the optimization of operating parameters of an inductively coupled plasma source. It provides a solution to the multi-objective and multi-variable plasma source design problem that cannot be solved by traditional design methods. By effectively screening potential variables and establishing a minimum orthogonal design scheme, the design cycle is greatly shortened and the cost is reduced. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0022] Figure 2Pareto charts of the potential variables for electron density and density uniformity according to an embodiment of the present invention, where (a) is the Pareto chart of electron density and (b) is the Pareto chart of density uniformity;

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

[0024] Figure 4 Three-dimensional response surface distribution diagram of the driver length and radio frequency power on the response value of electron density according to an embodiment of the present invention. Detailed implementation manners

[0025] The object of the present invention is to provide a parameter design and optimization method for an inductively coupled plasma source, which can quickly determine and screen potential variables that have a key impact on the design target, establish a minimum orthogonal design and fit a high-order regression equation to describe the possible interaction between significant variables, and finally give the best design scheme that meets the design target parameters. It must be emphasized that the results in the orthogonal design of the present invention come from the iteration results of the fluid model and the Boltzmann model in the finite element analysis model.

[0026] In order to completely and clearly describe the technical solution of the present invention, the following will further describe the present invention in detail 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 used to limit the present invention.

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

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

[0029] This step may include:

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

[0031] According to the design target, determine the potential variables that may exist in the inductively coupled plasma source to obtain a potential variable matrix. In one embodiment, the potential variables include: driver radius, driver length, discharge gas pressure, radio frequency power, number of coil turns, coil turn spacing, working temperature.

[0032] S102. Construct and simplify the Plackett - Burman orthogonal matrix according to the latent variables , and obtain the simplified matrix. Multiply the simplified matrix by the latent variable matrix to obtain the latent variable design scheme, where is the response value coefficient matrix.

[0033] This step may include:

[0034] S102 - 1. Construct K + 1 groups of Plackett - Burman orthogonal matrices according to the design target parameters and latent variables, where K refers to K latent variables.

[0035] Constructing K + 1 groups of orthogonal matrices according to the design target and latent variables may include: constructing the 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 35 orthogonal matrices. In the above - mentioned embodiment, according to the latent variables, K = 7, select 12 11 orthogonal matrices , which can satisfy up to 11 latent variables:

[0036] ,

[0037] where, represents the high level of the latent variable, represents the low level.

[0038] S102 - 2. Simplify the orthogonal matrix based on the latent variables to obtain the simplified matrix . The simplified matrix is:

[0039] ,

[0040] The latent variable matrix is:

[0041] , where to are the latent variables;

[0042] The response value matrix :

[0043] , where to are the design target parameters.

[0044] In the above - mentioned specific embodiment, the latent variable matrix is:

[0045] ,

[0046] wherein, is the radius of the driver, is the length of the driver, is the discharge gas pressure, is the RF power, is the number of turns of the coil, is the turn pitch of the coil, is the operating temperature.

[0047] Response value matrix :

[0048]

[0049] wherein, is the electron density, is the density uniformity.

[0050] S102 - 3, multiply the simplified matrix by the latent variable matrix to obtain a latent variable design scheme, also known as the Plackett - Burman orthogonal design scheme:

[0051] ,

[0052] Step S103: Based on the Boltzmann module and the fluid module in the finite element analysis model, perform iterative calculations and sequentially solve the response value coefficient matrix according to the latent variable design scheme in step S102 . In one embodiment, the response value coefficient matrix corresponds to 12 2.

[0053] This step may include: obtaining the value of the response value coefficient matrix based on the iterative results of the Boltzmann module and the fluid module. Among them, the iterative parameters output by the fluid module include: the mole fractions of metastable argon atoms and ground - state argon atoms, the ionization degree , the electron density .

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

[0055] ,

[0056] In the formula, is the mass fraction of substance j, is the mass - averaged fluid velocity vector, is the density of the mixture, is the production rate coefficient of substance j, is the diffusion flux vector of substance j, represents divergence.

[0057] ,

[0058] In the formula, is the thermal diffusion coefficient, is the working temperature, is the average molar data of the gas mixture, is the charge quantity of substance j, is the mobility of substance j, is the average diffusion coefficient of the mixture of substance j and k. is the induced electric field.

[0059] ,

[0060] Among them, is the ionization degree, is the number density of neutral particles.

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

[0062] ,

[0063] In the formula, 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] Two approximations to the Boltzmann equation can obtain the simplified distribution function:

[0065] ,

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

[0067] ,

[0068] ,

[0069] ,

[0070] ,

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

[0072] The reduced angular frequency of the power supply in the Boltzmann model is calculated from the gas pressure P, the working temperature T, and the power supply frequency in the orthogonal design scheme of step S104 as follows:

[0073] ,

[0074] ,

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

[0076] When the difference in the degree of ionization obtained by iterating to the last calculation is less than it is regarded as convergence.

[0077] Step S104: For the response value coefficient matrix and the simplification matrix obtained in step S103, fit the first-order linear regression equation of the response value and the latent variable, and solve its goodness of fit. Due to the full rank property of the orthogonal matrix, a unique approximate solution can be obtained for the matrix.

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

[0079] .

[0080] wherein, to are the variable 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 the adjusted goodness of fit 。

[0085] ,

[0086] In the formula, is the sum of squared residuals, is the total sum of squares, is the predicted value of the fitting equation, is the average value of the response value, is the response value.

[0087] ,

[0088] In the formula, n is the number of experiments, and p is the number of latent variables.

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

[0090] Step S105: Identify significant variables using analysis of variance according to the fitting results of step S104, calculate the Box-Behnken design matrix based on the coefficients of the main effect variables, and multiply the Box-Behnken design matrix by the significant variable matrix to obtain the significant variable design scheme.

[0091] This step may include:

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

[0093] ,

[0094] ,

[0095] In the formula, 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 product of an orthogonal matrix and the transpose of an orthogonal matrix is the identity matrix, the standard error calculation is simplified to , MSE is the mean square error, equal to the sum of squared residuals divided by the degrees of freedom ( ). is the regression sum of squares, is the sum of squared residuals, and the regression sum of squares is equal to the total sum of squares minus the sum of squared residuals. The P value is determined from the F value distribution table.

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

[0097] S105-3. According to the main response value, confirm the main effect variables. Calculate the steepest ascent direction and variable range according to the coefficients of the main effect variables, calculate the step sizes of the remaining significant variables according to the coefficients of the main effect variables, and determine the Box-Behnken design center point through iterative calculations of the fluid module and the Boltzmann module:

[0098] ,

[0099] where, is the coefficient of the main effect variable, are the high-level value and low-level value of variable i respectively, is the number of step sizes, is the climbing step size of the remaining significant variable i except the main effect.

[0100] Specifically, according to the design objective, select electron density as the main response value. According to Figure 2 the T values in the Pareto chart (a) are sorted from high to low to select the main effect variables. The main effect variable can be the driver radius. According to the coefficient of the main effect variable driver radius (6.88), calculate the step sizes of the remaining significant variables. Determine the Box-Behnken design center point through iterative calculations of the fluid module and the Boltzmann module.

[0101] S105-4. According to the Box-Behnken design center point and the step sizes of the remaining significant variables, establish a design matrix, and multiply the design matrix by the latent variable matrix to obtain the corresponding design scheme. The principle is as Figure 3 shown, is the driver radius, is the driver length, is the discharge gas pressure. The Box-Behnken design is a three-level design method, and the design points are located in the center and the middle of the edge of the design space. Therefore, the points of this design are all on a spherical surface or the center of the sphere. In a specific scheme, the six-variable three-level design includes 48 + 3 groups of orthogonal design schemes. Except for the center point, each group only changes three variables and keeps the number of occurrences of each variable in the combination consistent. Multiply the design matrix by the significant variable matrix Obtain the corresponding design scheme and the design matrix which is:

[0102] .

[0103] Step S106: Based on the iterative calculation of the fluid model and the Boltzmann model in the finite element analysis model, solve the response value coefficient matrix in sequence according to the significant variable design scheme . Iterate until the difference in ionization degrees obtained from the last calculation is less than which is regarded as convergence. In one embodiment, the response value coefficient matrix is a matrix corresponding to a 51 by 2 matrix.

[0104] Step S107: Fit the high-order polynomial regression equation of the response value and the significant variables according to the response value coefficient matrix in S106, and solve its goodness of fit.

[0105] In one embodiment, the goodness of fit of the quadratic linear regression equation is the highest. Therefore, establish the quadratic linear regression equation and solve it. The solution is:

[0106] ,

[0107] ,

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

[0109] This step may include:

[0110] S108-1, Calculate the F value and the P value, and determine the statistical significance degree of the potential variable to the response value according to the P value. If the P value is less than 0.05, it is considered that the variable is statistically significant to its response value, and confirm the importance of each variable according to the F value in descending order.

[0111] S108-2, Judge the interaction of two significant variables on the response value Figure 4 shows the interaction of the driver radius and the radio frequency power on the electron density.

[0112] S108-3, Draw a three-dimensional response surface according to the high-order polynomial regression equation established in step S107, judge the selection range of the maximum response value with the help of the three-dimensional response surface, and solve the maximum value point within the selection range of the maximum response value, that is, solve the maximum point of the response value. This point is the optimal design scheme.

[0113] According to an embodiment of the present invention, there is also provided a computing device, including: at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device is caused to execute a method for designing and optimizing key parameters of an inductively coupled plasma source.

[0114] According to an embodiment of the present invention, there is also provided a readable storage medium storing program instructions, when the program instructions are read and executed by a computing device, the computing device is caused to execute a method for designing and optimizing key parameters of an inductively coupled plasma source.

[0115] In the specification provided herein, a large number of specific details are set forth. However, it can be understood that embodiments of the present invention may be practiced without these specific details. In some instances, well-known methods, structures and techniques have not been shown in detail so as not to obscure the understanding of this specification.

[0116] Although the present invention has been described based on a limited number of embodiments, those skilled in the art in this technical field will understand, upon the above description, that other embodiments can be conceived within the scope of the present invention thus described. In addition, it should be noted that the language used in this specification has been mainly selected for readability and teaching purposes rather than 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 Including: S101, Obtain design target parameters and construct a response value matrix and a latent variable matrix ; S102, construct and simplify the Plackett-Burman orthogonal matrix according to the latent variables , multiply the simplified matrix by the latent variable matrix to obtain the latent variable design scheme; S103, perform iterative calculations based on the Boltzmann module and the fluid module in the finite element analysis model, and sequentially solve the response value coefficient matrix according to the latent variable design scheme in step S102 ; S104: For the response value coefficient matrix and the simplification matrix obtained in S103, fit the first-order linear regression equation of the response value and the latent variable, and solve its goodness of fit; S105: Use analysis of variance on the fitting result of S104 to identify significant variables, calculate the Box-Behnken design matrix according to the coefficients of the main effect variables, and multiply the design matrix by the significant variable matrix to obtain the significant variable design scheme; S106: Based on the iterative calculation of the fluid model and the Boltzmann model in the finite element analysis model, solve the response value coefficient matrix in turn according to the significant variable design scheme of S105; S107: According to the response value coefficient matrix and the design matrix in S106, fit the high-order polynomial regression equation of the response value and the significant variables, and solve its goodness of fit; S108: Use analysis of variance on the high-order polynomial regression equation established in S107 to determine the effect of the interaction term on the response value, solve the maximum point of the response value, and this point is the optimal design scheme.

2. The key parameter design and optimization method of the inductively coupled plasma source according to claim 1, characterized in that S101 includes: obtaining the design target parameters of the inductively coupled plasma source, and constructing a response value matrix according to the design target parameters ; determining the latent variables of the inductively coupled plasma source according to the design target parameters, and obtaining a latent variable matrix .

3. 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 groups of Plackett-Burman orthogonal matrices according to the design target parameters and latent variables, where K refers to K latent variables; , where K refers to K latent variables; S102-2, simplify the Plackett-Burman orthogonal matrix based on latent variables to obtain a simplified matrix ; S102-3, multiply the simplified matrix by the latent variable matrix Obtain the latent variable design scheme, also known as the Plackett-Burman orthogonal design scheme.

4. A method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 1, characterized in that, S103 includes: obtaining a response value coefficient matrix according to the iteration results of the Boltzmann module and the fluid module in the finite element analysis model values, where the iteration parameters output by the fluid module include: the mole fractions of metastable atoms and ground state atoms, and the degree of ionization , and electron density , where the mole fractions of metastable atoms and ground state atoms are calculated by the Maxwell-Stefan equation in the heavy ion model: ， wherein, is the mass fraction of substance j, is the mass-averaged fluid velocity vector, is the mixture density, is the production rate coefficient of substance j, is the diffusion flux vector of substance j, represents divergence; ， In the formula, is the thermal diffusion coefficient, is the working temperature, is the average molar data of the gas mixture, is the electric charge of substance , is the mobility of substance , is the average diffusion coefficient of substance and the mixture, is the induced electric field; , where 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 mobility coefficient and reduced energy diffusion coefficient , and these parameters are calculated by solving the Boltzmann equation: ， Wherein, 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; Perform two-term approximation on the Boltzmann equation to obtain a simplified distribution function: ， In the formula, is the flow velocity, is the diffusion coefficient, is the scattering coefficient, is the electron energy; ， ， ， ， wherein, is a coefficient, is the average electron energy, is the effective momentum transfer cross section; Power reduction angular frequency in the Boltzmann module Given by the gas pressure P, working temperature T, and power supply frequency in the orthogonal design scheme of step S104 Calculated as follows: ， ， Among them, is the power supply angular frequency; The difference in the degree of ionization obtained by iterating to the last calculation is less than It is regarded as convergent.

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

6. A method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 1, characterized in that, 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 the adjusted goodness of fit : ， In the formula, is the sum of squared residuals, is the total sum of squares, is the predicted value of the fitting equation, is the average value of the response, is the response value, is the goodness of fit; ， where n is the number of experiments, p is the number of latent variables, is the adjusted goodness of fit.

7. A method for designing and optimizing key parameters of an inductively coupled plasma source according to claim 1, characterized in that, S105 includes: S105-1: Calculate the T value, F value, and P value: ， ， In the formula, is the coefficient of variable i in the fitted first-order linear regression equation, The standard error is calculated from the mean square error, and the calculation is simplified to , where MSE is the mean square error, equal to the sum of squared residuals divided by the degrees of freedom , is the regression sum of squares, is the sum of squared residuals. The regression sum of squares is equal to the total sum of squares minus the sum of squared residuals, and the P-value is determined from the F-value distribution table; S105-2: According to the variance calculation result of step S108, if the P value is less than 0.05, consider this variable statistically significant and it is a significant variable; S105-3: According to the main response value, confirm the main effect variables, calculate the steepest ascent direction and variable range according to the coefficients of the main effect variables, calculate the step sizes of the other significant variables according to the coefficients of the main effect variables, and determine the Box-Behnken design center point through iterative calculation of the fluid module and the Boltzmann module; S105-4: According to the Box-Behnken design center point and the step sizes of the other significant variables, establish a design matrix, and multiply the design matrix by the latent variable matrix to obtain the corresponding design scheme.

8. The key parameter design and optimization method of an inductively coupled plasma source according to claim 6, characterized in that S108 includes: S108-1: Calculate the F value and P value, and determine the statistical significance degree of the significant variables on the response value according to the P value; S108-2: Judge the interaction effect of two variables on the response value; S108-3: Draw a three-dimensional response surface according to the high-order polynomial regression equation established in step S107, judge the selection range of the maximum response value with the help of the three-dimensional response surface, and solve the maximum value point within the selection range of the maximum response value, that is, solve the maximum point of the response value, and this point is the optimal design scheme.

9. A computing device, characterized in that, Including: At least one processor and a 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 as described in any one of claims 1-8.

10. A readable storage medium storing program instructions, characterized in that, When the program instructions are read and executed by the computing device, the computing device is caused to execute the key parameter design and optimization method of the inductively coupled plasma source as described in any one of claims 1-8.

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