Optimization design method and manufacturing method of heating structure

Abstract

The invention discloses an optimization design method and a manufacturing method of a heating structure. The optimization design method comprises the following steps: constructing a spiral line heating wire parameter model with adjustable spacing and arc supplement; a design space is constructed, the standard deviation, the variable coefficient and the uniformity of the temperature of the temperature measuring point are used as evaluation indexes, and an optimal Latin hypercube sampling method is adopted for experimental design and simulation analysis to obtain a real response value; based on the training sample and the real response values of the three evaluation indexes, respectively constructing Gaussian process regression models, and performing model optimization by using a Bayesian optimization method; constructing a multi-objective optimization mathematical model based on design variables and objective function response values obtained by the optimized Gaussian process regression model, and solving an optimization parameter combination based on an NSGA-II algorithm; and verifying a multi-objective optimization result. According to the method, the relationship between the design parameters and the performance indexes can be more accurately captured, the optimization efficiency and accuracy are improved, and the surface temperature uniformity of the heating disc is more effectively improved.

Description

Technical Field

[0001] The present invention relates to the technical field of semiconductor processing equipment, and particularly relates to an optimized design method and manufacturing method for a heating structure. Background Art

[0002] In the semiconductor field, a hot plate is a key heat treatment equipment in the wafer processing process. Currently, the hot plate generally adopts an internal heating element design. Due to the limitations of heat conduction characteristics and layout, the surface temperature distribution of the hot plate is often uneven. This non-uniformity not only affects the consistency of the wafer bonding process but also may cause uneven expansion of the wafer during heating, thereby leading to problems such as fragmentation and bonding defects. To improve the surface temperature uniformity, the mainstream method is to adjust the heating element structure, such as adjusting the distribution of heating wires. However, this adjustment is based on experience and has certain blindness and limitations, and the optimization effect is limited. Summary of the Invention

[0003] Object of the Invention: The first object of the present invention is to provide an optimized design method that can improve the surface temperature uniformity of the heating structure, thereby improving the manufacturing quality and yield rate of semiconductor devices; the second object of the present invention is to provide a manufacturing method for the heating structure.

[0004] Technical Solution: An optimized design method for a heating structure of the present invention includes:

[0005] (1) Construct a parameter model of a spiral heating wire with adjustable spacing and arc supplement;

[0006] (2) Determine the design variables and their value ranges to construct a design space; determine the evaluation indexes, including the standard deviation T σ of the temperature at the temperature measurement points, the coefficient of variation c v and the uniformity T uni , where T σ is used to measure the absolute dispersion degree of the temperature at the temperature measurement points, c v is used to measure the relative dispersion degree of the temperature at the temperature measurement points, and T uni is used to measure the maximum deviation degree of the surface temperature of the hot plate;

[0007] Use the optimal Latin hypercube sampling method to extract training samples in the design space, and perform simulation analysis on the training samples to obtain the true response values of the three evaluation indexes;

[0008] (3) Based on the training samples and the corresponding true response values of the three evaluation indexes, construct Gaussian process regression models respectively; use the Bayesian optimization method to optimize the hyperparameter σ of the Gaussian process regression models;

[0009] (4) Based on the Gaussian process regression model optimized by Bayesian optimization, obtain the response value of the objective function according to the design variables; construct a multi-objective optimization mathematical model based on the design variables and the response value of the objective function, and perform multi-objective optimization based on the NSGA-II algorithm to solve the optimized parameter combination;

[0010] (5) Verify the multi-objective optimization results.

[0011] Furthermore, step (1) includes: constructing a parameter model of a helical heating wire with adjustable pitch, and its expression is as follows:

[0012]

[0013] where x(θ) is the x coordinate of any point on the helix; y(θ) is the y coordinate of any point on the helix; θ represents the angle from the starting point of the helix to this point, and its value range is [0, α], where α is the total angle of the helix; r(θ) is the distance of any point on the helix from the origin of the plane coordinate system, and is calculated according to the following formula:

[0014]

[0015] where r 0 is the starting radius; r 1 is the ending radius; t is a factor controlling the tightness of the helix;

[0016] Construct a parameter model of the supplementary arc segment, and its expression is as follows:

[0017]

[0018] where, is the x coordinate of any point on the supplementary arc segment; is the y coordinate of any point on the supplementary arc segment; represents the angle from the starting point of the helix to this point on the supplementary arc segment, and its value range is [α, α + β], where β is the total angle of the supplementary arc segment; R 1 is the radius of the supplementary arc segment, and its expression is as follows:

[0019]

[0020] Furthermore, in step (2), the design variables include the total angle α of the helix, the total angle β of the supplementary arc segment, the factor t controlling the tightness of the helix, and the thickness h of the heating disk, and their value ranges are as follows:

[0021]

[0022] Furthermore,

[0023]

[0024] Among them, N is the number of temperature measurement points distributed on the working surface of the heating plate; T i is the surface temperature of the heating plate measured at the i-th temperature measurement point; T μ is the average temperature of each temperature measurement point; T max is the highest temperature of the temperature measurement points; T min is the lowest temperature of the temperature measurement points.

[0025] Furthermore, step (3) includes: The basic form of the Gaussian process regression model is:

[0026] y j (x) = f(x) + ε

[0027] Among them, y j (x) is the prediction function of the j-th objective function in the Gaussian process regression model; ε is Gaussian noise with a mean of 0; f(x) is a Gaussian process, and its expression is as follows:

[0028]

[0029] Among them, m(x) is the mean function, usually a constant; k(x, x′) is the kernel function, which defines the covariance of the function values at any two points x and x′ in the input space;

[0030] Use the Bayesian optimization method to optimize the hyperparameter σ of the Gaussian process regression model, set the acquisition function to expected improvement per second, set the number of iterations, and after completing the Bayesian optimization, obtain the optimal value of the hyperparameter σ.

[0031] Furthermore, the kernel function of the Gaussian process regression model is Nonisotropic Matern 3 / 2, and the general form of the kernel function Nonisotropic Matern 3 / 2 is:

[0032]

[0033] Among them, σ 2 is the signal variance; d(x, x′) is the Euclidean distance between x and x′; l is the length parameter vector used to control the scale of each dimension.

[0034] Furthermore, in step (4), the multi-objective optimization mathematical model is expressed as:

[0035]

[0036] Among them, Obj.Minimize and d.v. are a summary description of the optimization problem, Obj.Minimize represents minimizing the objective function, and d.v. represents the design variables; Y 1, Y 2 and Y 3 represent the objective function.

[0037] Furthermore, in step (4), the multi-objective optimization based on the NSGA-II algorithm to solve the optimal parameter combination includes:

[0038] Determine the population size, iteration times, crossover probability, and mutation probability of the NSGA-II algorithm, solve the constructed multi-objective optimization mathematical model to obtain the Pareto front; screen the compromise solution of the objective function as the optimal parameter combination, and the screening criteria are as follows:

[0039]

[0040] where, T σ，0 is the standard deviation of the temperature at the initial heating plate temperature measurement points; c v，0 is the coefficient of variation of the temperature at the initial heating plate temperature measurement points; T uni，0 is the uniformity of the temperature at the initial heating plate temperature measurement points; ω 1 , ω 2 and ω 3 are the weight ratios of the objective functions Y 1 , Y 2 and Y 3 in sequence.

[0041] Furthermore, step (5) includes: performing error analysis of the multi-objective optimization results: comparing the predicted values and the simulation test calculation values of the performance indicators T σ , c v and T uni under the optimal parameter combination to perform error analysis of the multi-objective optimization results; performing performance analysis of the multi-objective optimization results: based on the three performance indicators T σ , c v and T uni , compare the temperature uniformity performance of the initial heating plate and the optimized heating plate to perform performance analysis of the multi-objective optimization results.

[0042] A manufacturing method of a heating structure according to the present invention includes: obtaining an optimized heating plate according to the optimized design method of the heating structure, exporting the geometric parameters and material parameters of the optimized heating plate, calculating the corresponding five-axis 3D printing path and determining the printing process parameters, and completing 3D printing using a five-axis 3D printer.

[0043] Beneficial effects: Compared with the prior art, the present invention has the following remarkable advantages:

[0044] (1) The optimization design method of the present invention can accurately capture the relationship between design parameters and performance indicators through algorithms such as Gaussian process regression model, and achieve efficient and high-precision fitting between the design parameters of the heating plate structure and the heating plate temperature uniformity index.

[0045] (2) The present invention comprehensively considers multiple temperature uniformity evaluation indicators such as standard deviation and coefficient of variation, and realizes the comprehensive optimization of the absolute and relative dispersion degrees and the maximum deviation degree of the surface temperature of the heating plate.

[0046] In summary, the present invention has the advantages of high optimization efficiency, accurate and comprehensive optimization effect, etc., can more effectively improve the surface temperature uniformity of the heating plate, and has high practical value and promotion value. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 is a flowchart of an optimization design method for a heating structure provided by an embodiment of the present invention;

[0048] Figure 2 is a position distribution diagram of temperature measurement points on the working surface of the heating plate in an embodiment of the present invention;

[0049] Figure 3 is a schematic diagram of the fitting performance of the Gaussian process regression model in an embodiment of the present invention;

[0050] Figure 4 is an iteration diagram of three performance indicators during the optimization process in an embodiment of the present invention;

[0051] Figure 5 is a schematic diagram of the initial heating plate structure in an embodiment of the present invention;

[0052] Figure 6 is a temperature field simulation result diagram of the initial heating plate in an embodiment of the present invention;

[0053] Figure 7 is a schematic diagram of the optimized heating plate structure in an embodiment of the present invention;

[0054] Figure 8 is a temperature field simulation result diagram of the optimized heating plate in an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0055] The present invention will be further described below with reference to the accompanying drawings.

[0056] As Figure 1 shown, an embodiment of the present invention provides an optimization design method for a heating structure, and the heating structure described in this embodiment is a heating plate for wafer heating.

[0057] The optimization design method specifically includes the following steps:

[0058] (1) Construct a parametric model of a helical heating wire with adjustable pitch and arc supplement;

[0059] (1.1) Construct a parametric model of a helical heating wire with adjustable pitch, and its expression is as follows:

[0060]

[0061] where x(θ) is the x - coordinate of any point on the helix; y(θ) is the y - coordinate of any point on the helix; θ represents the angle from the starting point of the helix to this point, and its value range is [0, α], where α is the total angle of the helix; r(θ) is the distance of any point on the helix from the origin of the plane coordinate system, and is calculated according to the following formula:

[0062]

[0063] where r 0 is the starting radius; r 1 is the ending radius; t is a factor controlling the tightness of the helix;

[0064] (1.2) Construct a parametric model of the supplementary arc section, and its expression is as follows:

[0065]

[0066] where, is the x - coordinate of any point on the supplementary arc section; is the y - coordinate of any point on the supplementary arc section; represents the angle from the starting point of the helix to this point on the supplementary arc section, and its value range is [α, α + β], where β is the total angle of the supplementary arc section; R 1 is the radius of the supplementary arc section, and its expression is as follows:

[0067]

[0068] (2) Determine the design variables and their value ranges, and construct the design space; determine the evaluation indexes, including the standard deviation T σ of the temperature at the temperature - measuring points, the coefficient of variation c v and the uniformity T uni , where T σ is used to measure the absolute dispersion degree of the temperature at the temperature - measuring points, c v is used to measure the relative dispersion degree of the temperature at the temperature - measuring points, and T uni is used to measure the maximum deviation degree of the surface temperature of the heating plate; Use the optimal Latin hypercube sampling method to extract training samples in the design space, and conduct simulation analysis on the training samples to obtain the true response values of the three evaluation indexes;

[0069] (2.1) Determine the design variables and their value ranges, and construct the design space;

[0070] The design variables include the total angle α of the spiral, the total angle β of the supplementary arc, the factor t controlling the tightness of the spiral, and the thickness h of the heating plate. The value ranges are as follows:

[0071]

[0072] (2.2) Determine the evaluation index;

[0073]

[0074] Among them, N is the number of temperature measurement points distributed on the working surface of the heating plate; T i is the surface temperature of the heating plate measured at the i-th temperature measurement point; T μ is the average temperature of each temperature measurement point; T max is the highest temperature of the temperature measurement points; T min is the lowest temperature of the temperature measurement points.

[0075] The diameter of the heating plate is 200 mm, and 8 temperature measurement points are arranged on its working surface. The distribution of the positions of the temperature measurement points is as Figure 2 shown.

[0076] (2.3) Conduct optimal Latin hypercube sampling and perform simulation analysis to obtain the true response values

[0077] Adopt the optimal Latin hypercube sampling method to extract training samples in the constructed design space and form a training sample set x (i) (i = 1,..., n); n represents the number of training samples. In this embodiment, n = 40;

[0078] Perform simulation analysis on the training samples to obtain the true response values g j (x (i) )(j = 1, 2, 3), and form a database DB[x (i) |g j (x (i) )](i = 1,..., n, j = 1, 2, 3); where x = (α, β, t, h) is a 4-dimensional input variable.

[0079] (3) Based on the training samples and the corresponding true response values of the three evaluation indexes, construct Gaussian process regression models respectively; use the Bayesian optimization method to optimize the hyperparameter σ of the Gaussian process regression model;

[0080] (3.1) Determine the data dimension D = 4, select the basis function of the Gaussian process regression model as a constant, the kernel function as Nonisotropic Matern 3 / 2, and define the initial value of the hyperparameter σ as 0.0035;

[0081] (3.2) Based on the training sample points in the database DB and the true response values of the corresponding three evaluation indicators, Gaussian process regression models are constructed respectively. Its basic form is:

[0082] y j (x) = f(x) + ε

[0083] Among them, y j (x) is the prediction function of the jth objective function in the Gaussian process regression model; ε is Gaussian noise with a mean of 0; f(x) is a Gaussian process, and its expression is as follows:

[0084]

[0085] Among them, m(x) is the mean function, usually a constant; k(x, x′) is the kernel function, which defines the covariance of the function values at any two points x and x′ in the input space. The general form of the kernel function Nonisotropic Matern 3 / 2 is:

[0086]

[0087] Among them, σ 2 is the signal variance; d(x, x′) is the Euclidean distance between x and x′; l is the length parameter vector used to control the scale of each dimension.

[0088] (3.3) Use the Bayesian optimization method to optimize the hyperparameter σ of the Gaussian process regression model. Set the acquisition function to expected improvement per second (plus), and set the number of iterations to 100 times. After completing the Bayesian optimization, the optimal values of the hyperparameter σ are 0.00010325, 0.050005, and 1.3042 respectively, and three Gaussian process regression models are reconstructed. As Figure 3 shown, the R 2 values of the three models are all greater than 0.8, indicating high fitting accuracy and can be used for subsequent predictions.

[0089] (4) Based on the Gaussian process regression model optimized by Bayesian optimization, obtain the response value of the objective function according to the design variables; construct a multi-objective optimization mathematical model based on the design variables and the response value of the objective function, and perform multi-objective optimization based on the NSGA-II algorithm to solve the optimal parameter combination;

[0090] (4.1) Construct a multi-objective optimization mathematical model

[0091] Based on the Gaussian process regression model optimized by Bayesian optimization, obtain the response value of the objective function according to the heating plate design variables, and construct a multi-objective optimization mathematical model based on the design variables and the response value of the objective function, which is expressed as:

[0092]

[0093] Among them, Obj.Minimize and d.v. are a summary description of the optimization problem. Obj.Minimize represents minimizing the objective function, and d.v. represents design variables; Y 1 、Y 2 and Y 3 represent the objective function.

[0094] (4.2) Perform multi-objective optimization based on the NSGA-II algorithm

[0095] Determine that the population size of the NSGA-II algorithm is 200 individuals, the crossover probability is 0.8, and the mutation probability is 0.2. Solve the constructed multi-objective optimization mathematical model. After 300 iterations, the Pareto front is obtained.

[0096] (4.3) Screen the compromise solutions of the objective functions as the optimized parameter combinations

[0097] The screening criteria are as follows:

[0098]

[0099] Among them, T σ，0 is the standard deviation of the temperature at the initial heating plate temperature measurement points, which is 2.033; c v，0 is the coefficient of variation of the temperature at the initial heating plate temperature measurement points, which is 1.209%; T uni，0 is the uniformity of the temperature at the initial heating plate temperature measurement points, which is 4.719%; ω 1 、ω 2 and ω 3 are the weight ratios of the objective functions Y 1 、Y 2 and Y 3 in turn, which are 0.3, 0.3, and 0.4 in turn.

[0100] The predicted values of the three performance indicators T σ 、c v and T uni corresponding to the optimized parameter combinations during the iteration process in multi-objective optimization are as Figure 4 shown. The optimized parameter combinations and the predicted results of the corresponding three performance indicators are shown in Table 1.

[0101] Table 1

[0102]

[0103] (5) Verify the multi-objective optimization results

[0104] (5.1) Conduct error analysis of the multi-objective optimization results

[0105] The three performance indicators T corresponding to the optimized parameter combination σ 、c v and T uni The comparison results between the predicted values and the simulation test calculation values are shown in Table 2.

[0106] Table 2

[0107]

[0108] It can be seen from Table 2 that the prediction errors of various response types are all less than 5%, indicating that the constructed Gaussian process regression model has high accuracy.

[0109] (5.2) Conduct performance analysis of the multi-objective optimization results

[0110] The initial heating plate structure is as Figure 5 shown. Perform simulation analysis on the initial heating plate, and the corresponding temperature field simulation results are as Figure 6 shown. The optimized heating plate structure is as Figure 7 shown. Perform simulation analysis on the optimized heating plate, and the corresponding temperature field simulation results of the heating plate are as Figure 8 shown. The comparison results of the three performance indicators T σ 、c v and T uni before and after optimization are shown in Table 3.

[0111] Table 3

[0112]

[0113] As shown in Table 3, the three optimized performance indicators T σ 、c v and T uni all decrease significantly, achieving a significant optimization of the temperature uniformity of the heating plate.

[0114] The embodiment of the present invention also provides a manufacturing method of a heating structure, including:

[0115] Obtain the optimized heating plate according to the optimized design method of the heating structure described in the embodiment of the present invention, export the geometric parameters and material parameters of the optimized heating plate, calculate the corresponding five-axis 3D printing path and determine the printing process parameters, and complete 3D printing using a five-axis 3D printer.

Claims

1. A method for optimizing the design of a heating structure, characterized in that: include: (1) Construct a parameter model of a helical heating wire with adjustable spacing and arc supplement; (2) Determine the design variables and their value ranges and construct the design space; Determine the evaluation indicators, including the standard deviation T of the temperature at the measuring point σ , coefficient of variation c v and uniformity T uni , where T σ It is used to measure the absolute discreteness of the temperature at the measuring point, c v It is used to measure the relative dispersion of the temperature at the measuring point, T uni Used to measure the maximum deviation of the surface temperature of the heating plate; The optimal Latin hypercube sampling method is used to extract training samples in the design space, and the training samples are simulated and analyzed to obtain the true response values ​​of the three evaluation indicators; (3) Based on the training samples and the corresponding true response values ​​of the three evaluation indicators, Gaussian process regression models are constructed respectively; the hyperparameter σ of the Gaussian process regression model is optimized using the Bayesian optimization method; (4) Based on the Gaussian process regression model of Bayesian optimization, the objective function response value is obtained according to the design variables; Construct a multi-objective optimization mathematical model based on design variables and objective function response values, perform multi-objective optimization based on NSGA-Ⅱ algorithm, and solve the optimization parameter combination; (5) Verify the multi-objective optimization results.

2. The optimization design method of the heating structure according to claim 1, characterized in that: Step (1) includes: constructing a parameter model of a spiral heating wire with adjustable spacing, the expression of which is as follows: Where x(θ) is the x-coordinate of any point on the spiral; y(θ) is the y-coordinate of any point on the spiral; θ represents the angle from the starting point of the spiral to the point, and its value range is [0, α], where α is the total angle of the spiral; r(θ) is the distance of any point on the spiral relative to the origin of the plane coordinate system, calculated according to the following formula: Among them, r0 is the starting radius; r1 is the ending radius; t is the factor that controls the tightness of the spiral; Construct a parametric model of the supplementary segment arc, and its expression is as follows: in, is the x-coordinate of any point on the supplementary segment arc; is the y coordinate of any point on the supplementary segment arc; It represents the angle from the starting point of the spiral to the point on the supplementary arc. The value range is [α, α+β], where β is the total angle of the supplementary arc. R1 is the radius of the supplementary arc. Its expression is as follows: R1＝r1 1 / t 。 3. The optimization design method of the heating structure according to claim 2, characterized in that: In step (2), the design variables include the total angle α of the helix, the total angle β of the supplementary arc, the factor t for controlling the tightness of the helix, and the thickness h of the heating plate, and the value ranges are as follows:

4. The optimization design method of the heating structure according to claim 3, characterized in that: Where N is the number of temperature measurement points distributed on the working surface of the heating plate; T i is the surface temperature of the heating plate measured at the i-th temperature measurement point; T μ is the average temperature of each temperature measuring point; T max is the highest temperature of the temperature measuring point; T min is the lowest temperature of the measuring point.

5. The optimization design method of the heating structure according to claim 4, characterized in that: Step (3) includes: The basic form of the Gaussian process regression model is: y j (x)=f(x)+ε Among them, y j (x) is the prediction function of the jth objective function in the Gaussian process regression model; ε is Gaussian noise with a mean of 0; f(x) is a Gaussian process, and its expression is as follows: Where m(x) is the mean function, usually a constant; k(x,x′) is the kernel function, which defines the covariance of the function value at any two points x and x′ in the input space; Use the Bayesian optimization method to optimize the hyperparameter σ of the Gaussian process regression model, set the acquisition function to the expected improvement per second, set the number of iterations, and after completing the Bayesian optimization, obtain the optimal value of the hyperparameter σ.

6. The optimization design method of the heating structure according to claim 5, characterized in that: The kernel function of the Gaussian process regression model is Nonisotropic Matern 3 / 2. The general form of the kernel function Nonisotropic Matern 3 / 2 is: Among them, σ 2 is the signal variance; d(x,x′) is the Euclidean distance between x and x′; l is the length parameter vector used to control the scale of each dimension.

7. The optimization design method of the heating structure according to claim 6, characterized in that: In step (4), the multi-objective optimization mathematical model is expressed as: Among them, Obj.Minimize and dv are a summary description of the optimization problem, Obj.Minimize means minimization of the objective function, dv means the design variable; Y1, Y2 and Y3 mean the objective function.

8. The optimization design method of the heating structure according to claim 7, characterized in that: In step (4), the multi-objective optimization is performed based on the NSGA-II algorithm to solve the optimization parameter combination, including: Determine the population size, number of iterations, crossover probability and mutation probability of the NSGA-Ⅱ algorithm, solve the constructed multi-objective optimization mathematical model, and obtain the Pareto frontier; select the compromise solution of the objective function as the optimization parameter combination, and the screening criteria are as follows: Among them, T σ，0 is the standard deviation of the temperature at the initial heating plate measurement point; c v，0 is the coefficient of variation of the temperature at the initial heating plate temperature measurement point; T uni，0 is the temperature uniformity of the initial heating plate temperature measurement point; ω1, ω2 and ω3 are the weight proportions of the objective functions Y1, Y2 and Y3 respectively.

9. The optimization design method of the heating structure according to claim 8, characterized in that: Step (5) includes: performing error analysis of multi-objective optimization results: comparing the performance indicators T under the optimization parameter combination σ 、c v and T uni The predicted value and the calculated value of the simulation test are used to perform error analysis of the multi-objective optimization results; the performance analysis of the multi-objective optimization results is performed: based on three performance indicators T σ 、c v and T uni , the temperature uniformity performance of the initial heating plate and the optimized heating plate is compared, and the performance analysis of the multi-objective optimization results is carried out.

10. A method for manufacturing a heating structure, characterized in that: include: According to the optimization design method of the heating structure described in any one of claims 1 to 9, an optimized heating plate is obtained, geometric parameters and material parameters of the optimized heating plate are derived, a corresponding five-axis 3D printing path is calculated and printing process parameters are determined, and 3D printing is completed using a five-axis 3D printer.

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