Optimization design method and manufacturing method of heating structure

Abstract

The application discloses a kind of optimization design method and manufacturing method of heating structure, and optimization design method includes: adjustable pitch and with spiral line heating wire parameter model of arc supplementing;Design space is constructed, and standard deviation, variation coefficient and uniformity of temperature measuring point temperature are used as evaluation index, and real response value is obtained by simulation analysis using optimal Latin hypercube sampling method for test design;Based on training sample and the real response value of three evaluation indexes, Gaussian process regression model is respectively constructed, and model optimization is carried out using Bayesian optimization method;Based on design variable and the objective function response value obtained by optimized Gaussian process regression model, multi-objective optimization mathematical model is constructed, and optimization parameter combination is solved based on NSGA-Ⅱ algorithm;Multi-objective optimization result is verified.The application can more accurately capture the relationship between design parameters and performance indicators, improve optimization efficiency and accuracy, and more effectively improve the surface temperature uniformity of heating disc.

Description

Technical Field

[0001] This invention relates to the field of semiconductor processing equipment technology, and specifically to an optimized design method and manufacturing method for a heating structure. Background Technology

[0002] In the semiconductor field, heating pads are critical thermal processing equipment in wafer fabrication. Currently, heating pads generally employ an internal heating element design. Due to limitations in thermal conductivity and layout, this often results in uneven temperature distribution on the heating pad surface. This non-uniformity not only affects the consistency of wafer bonding processes but can also lead to uneven expansion of the wafer during heating, subsequently causing fragmentation and bonding defects. To improve surface temperature uniformity, the mainstream method is to adjust the heating element structure, such as adjusting the distribution of heating wires. However, such adjustments are based on experience, have certain limitations and a degree of blindness, and their optimization effect is limited. Summary of the Invention

[0003] Purpose of the invention: The first purpose of the 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 of semiconductor devices; the second purpose of the invention is to provide a manufacturing method for the heating structure.

[0004] Technical solution: The present invention provides an optimized design method for a heating structure, comprising:

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

[0006] (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 temperature measurement points. σ Coefficient of variation c v and uniformity T uni T σ Used to measure the absolute dispersion of temperature at a measuring point, c v T is used to measure the relative dispersion of temperature at a measuring point. uni Used to measure the maximum deviation of the surface temperature of the heating plate;

[0007] The optimal Latin hypercube sampling method is used to extract training samples in the design space. The training samples are then subjected to simulation analysis to obtain the true response values ​​of the three evaluation indicators.

[0008] (3) Based on the training samples and the true response values ​​of the three corresponding 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;

[0009] (4) Based on the Bayesian optimization Gaussian process regression model, the objective function response value is obtained according to the design variables; a multi-objective optimization mathematical model is constructed based on the design variables and the objective function response value; multi-objective optimization is performed based on the NSGA-II algorithm to solve the combination of optimization parameters.

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

[0011] Further, step (1) includes: constructing a parametric model of the adjustable-pitch helical heating wire, the expression of which 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 that point, with a value range of [0, α], and α is the total angle of the helix; r(θ) is the distance of any point on the helix relative to the origin of the plane coordinate system, calculated according to the following formula:

[0014]

[0015] Where r0 is the starting radius; r1 is the ending radius; and t is a factor controlling the tightness of the helix.

[0016] The parametric model of the supplementary arc segment is constructed, and its expression is as follows:

[0017]

[0018] in, It provides the x-coordinate of any point on the supplementary arc segment; It provides the y-coordinate of any point on the supplementary arc segment; R1 represents the angle from the starting point of the spiral to that point on the supplementary arc, with a value range of [α, α+β], where β is the total angle of the supplementary arc; R1 is the radius of the supplementary arc, 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 plate, with the following value ranges:

[0021]

[0022] Furthermore,

[0023]

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

[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 f(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] Where 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] The hyperparameter σ of the Gaussian process regression model was optimized using the Bayesian optimization method. The acquisition function was set to the expected improvement per second, and the number of iterations was set. After completing the Bayesian optimization, the optimal value of the hyperparameter σ was obtained.

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

[0032]

[0033] Where, σ 2 d(x,x′) is the signal variance; d(x,x′) is the Euclidean distance between x and x′; l is a 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] Here, Obj.Minimize and dv are a summary description of the optimization problem. Obj.Minimize means minimizing the objective function, and dv means design variables; Y1, Y2 and Y3 represent the objective functions.

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

[0038] The population size, number of iterations, crossover probability, and mutation probability of the NSGA-II algorithm are determined. The constructed multi-objective optimization mathematical model is solved to obtain the Pareto front. The compromise solutions of the objective function are selected as the combination of optimization parameters, and the selection criteria are as follows:

[0039]

[0040] Among them, T σ，0 c is the standard deviation of the initial heating plate temperature measurement point. v，0 T is the coefficient of variation of the temperature at the initial heating plate measuring point; uni，0 ω1 represents the uniformity of temperature at the initial heating plate temperature measurement point; ω2 and ω3 are the weight proportions of the objective function Y1, Y2 and Y3, respectively.

[0041] Furthermore, step (5) includes: performing multi-objective optimization result error analysis: comparing the performance index T under the optimized parameter combination. σ c v and T uni Error analysis of multi-objective optimization results was conducted by comparing predicted values ​​with simulation experimental calculations; performance analysis of multi-objective optimization results was also performed based on three performance indicators T. σ c v and T uni By comparing the temperature uniformity performance of the initial heating plate and the optimized heating plate, a performance analysis of the multi-objective optimization results is conducted.

[0042] The present invention discloses a method for manufacturing a heating structure, comprising: obtaining an optimized heating plate according to the optimization design method of the heating structure, deriving 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 significant 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 realize efficient and high-precision fitting between heating plate structure design parameters and heating plate temperature uniformity index.

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

[0046] In summary, this invention has the advantages of high optimization efficiency and precise and comprehensive optimization effect, which can more effectively improve the surface temperature uniformity of the heating plate and has high practical value and promotion value. Attached Figure Description

[0047] Figure 1 This is a flowchart illustrating an optimized design method for a heating structure provided in an embodiment of the present invention.

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

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

[0050] Figure 4 This is an iterative graph of the three performance indicators in the optimization process in this embodiment of the invention;

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

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

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

[0054] Figure 8 This is a simulation result of the temperature field of the optimized heating plate in an embodiment of the present invention. Detailed Implementation

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

[0056] like Figure 1 As shown, this embodiment of the invention provides an optimized design method for a heating structure, wherein 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 parameter model of a spiral heating wire with adjustable spacing and circular arc supplementation;

[0059] (1.1) Construct a parameter model for the adjustable pitch helical heating wire, the expression of which 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 that point, with a value range of [0, α], and α is the total angle of the helix; r(θ) is the distance of any point on the helix relative to the origin of the plane coordinate system, calculated according to the following formula:

[0062]

[0063] Where r0 is the starting radius; r1 is the ending radius; and t is a factor controlling the tightness of the helix.

[0064] (1.2) Construct the parametric model of the supplementary arc segment, the expression of which is as follows:

[0065]

[0066] in, It provides the x-coordinate of any point on the supplementary arc segment; It provides the y-coordinate of any point on the supplementary arc segment; R1 represents the angle from the starting point of the spiral to that point on the supplementary arc, with a value range of [α, α+β], where β is the total angle of the supplementary arc; R1 is the radius of the supplementary arc, 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 indicators, including the standard deviation T of the temperature at the temperature measurement points. σ Coefficient of variation c v and uniformity T uni T σ Used to measure the absolute dispersion of temperature at a measuring point, c v T is used to measure the relative dispersion of temperature at a measuring point. uni This is 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.

[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 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 plate, with the following value ranges:

[0071]

[0072] (2.2) Determine the evaluation indicators;

[0073]

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

[0075] The heating plate has a diameter of 200mm, and eight temperature measuring points are arranged on its working surface. The positions of the temperature measuring points are as follows: Figure 2 As shown.

[0076] (2.3) Perform optimal Latin hypercube sampling and obtain the true response value through simulation analysis.

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

[0078] Simulation analysis was performed on the training samples to obtain the true response values ​​g of the three evaluation metrics. j (x (i) (j = 1, 2, 3), forming the 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 true response values ​​of the three corresponding 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;

[0080] (3.1) Determine the data dimension D=4, select the basis function of the Gaussian process regression model as 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. Their basic form is as follows:

[0082] y j (x)=f(x)+ε

[0083] Among them, y jf(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:

[0084]

[0085] Where 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] Where, σ 2 d(x,x′) is the signal variance; d(x,x′) is the Euclidean distance between x and x′; l is a length parameter vector used to control the scale of each dimension.

[0088] (3.3) The hyperparameter σ of the Gaussian process regression model was optimized using the Bayesian optimization method. The acquisition function was set to expected improvement per second (plus), and the number of iterations was set to 100. After Bayesian optimization, the optimal values ​​of the hyperparameter σ were obtained as 0.00010325, 0.050005, and 1.3042, respectively. The three Gaussian process regression models were then reconstructed. Figure 3 As shown, the R values ​​of the three models 2 All values ​​are greater than 0.8, indicating high fitting accuracy, which can be used for subsequent prediction.

[0089] (4) Based on the Bayesian optimization Gaussian process regression model, the objective function response value is obtained according to the design variables; a multi-objective optimization mathematical model is constructed based on the design variables and the objective function response value; multi-objective optimization is performed based on the NSGA-II algorithm to solve the combination of optimization parameters.

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

[0091] Based on the Gaussian process regression model optimized by Bayes, the objective function response value is obtained according to the design variables of the heating plate. A multi-objective optimization mathematical model is then constructed based on the design variables and the objective function response value, as follows:

[0092]

[0093] Here, Obj.Minimize and dv are a summary description of the optimization problem. Obj.Minimize means minimizing the objective function, and dv means design variables; Y1, Y2 and Y3 represent the objective functions.

[0094] (4.2) Multi-objective optimization based on NSGA-II algorithm

[0095] The population size for the NSGA-II algorithm was determined to be 200 individuals, with a crossover probability of 0.8 and a mutation probability of 0.2. The constructed multi-objective optimization mathematical model was solved, and the Pareto front was obtained after 300 iterations.

[0096] (4.3) Selecting a compromise solution for the objective function as the optimal parameter combination

[0097] The selection criteria are as follows:

[0098]

[0099] Among them, T σ，0 The standard deviation of the initial heating plate temperature measurement point is 2.033; c v，0 The coefficient of variation for the initial heating plate temperature measurement point is 1.209%; T uni，0 The initial uniformity of the temperature at the heating plate measurement point is 4.719%; ω1, ω2, and ω3 are the weight percentages of the objective function Y1, Y2, and Y3, respectively, which are 0.3, 0.3, and 0.4.

[0100] Three performance indices T corresponding to optimized parameter combinations σ c v and T uni The iterative process of the corresponding predicted value in multi-objective optimization is as follows: Figure 4 As shown in Table 1, the optimized parameter combinations and the corresponding prediction results for the three performance indices are presented.

[0101] Table 1

[0102]

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

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

[0105] Three performance indices T corresponding to optimized parameter combinations σ c v and T uni The comparison results between the predicted values ​​and the simulation test calculations are shown in Table 2.

[0106] Table 2

[0107]

[0108] As shown in Table 2, the prediction errors for all response types are less than 5%, indicating that the constructed Gaussian process regression model has high accuracy.

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

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

[0111] Table 3

[0112]

[0113] As shown in Table 3, the three optimized performance indices T σ c v and T uni All of these values ​​decreased significantly, resulting in a substantial optimization of the heating plate temperature uniformity.

[0114] This invention also provides a method for manufacturing a heating structure, comprising:

[0115] According to the heating structure optimization design method described in the embodiment of the present invention, an optimized heating plate is obtained, the geometric parameters and material parameters of the optimized heating plate are exported, the corresponding five-axis 3D printing path is calculated and the printing process parameters are determined, and 3D printing is completed using a five-axis 3D printer.

Claims

1. A method of optimizing the design of a heating structure, characterized in that, Comprise: (1) Constructing a helical heating wire parameter model with adjustable spacing and arc supplement; (2) Determining the design variables and their value range, and constructing the design space; determining evaluation indexes, including standard deviation of temperature of measuring points , coefficient of variation and uniformity , wherein for measuring the absolute dispersion degree of the temperature of the measuring points, for measuring the relative dispersion degree of the temperature of the measuring points, for measuring the maximum deviation degree of the surface temperature of the heating disc; Design variables include the total angle of the spiral , the total angle of the supplemental segment circular arc , a factor controlling the tightness of the spiral , and the thickness of the heating disk ; Using the optimal Latin hypercube sampling method to extract training samples in the design space, and performing simulation analysis on the training samples to obtain the true response values of the three evaluation indexes; (3) Based on the training samples and the true response values of the three evaluation indexes, Gaussian process regression models are constructed respectively; Using a bayesian optimization method for hyperparameters of a gaussian process regression model optimization is performed; (4) Based on the Gaussian process regression model of Bayesian optimization, the response value of the objective function is obtained according to the design variables; Based on the design variables and the response value of the objective function, a multi-objective optimization mathematical model is constructed, and NSGA-II algorithm is used for multi-objective optimization to solve the optimal parameter combination; (5) Verify the multi-objective optimization result.

2. The method of optimizing the design of a heating structure according to claim 1, characterized in that, Step (1) includes: constructing a helical heating wire parameter model with adjustable spacing, the expression is as follows: wherein is the coordinate of an arbitrary point on the helix line ; is the coordinate of an arbitrary point on the helix line ; denotes the angle from the starting point of the helix line to the point, which has a value range of , is the total angle of the helix line; is the distance of an arbitrary point on the helix line relative to the origin of the plane coordinate system, which is calculated according to the following formula: wherein, is the start radius; is the end radius; is a factor that controls how tight the spiral is. Construct the parameter model of the supplementary arc, the expression is as follows: wherein, is the coordinate of any point on the supplementary segment circular arc ; is the coordinate of any point on the supplementary segment circular arc ; denotes the angle from the starting point of the spiral to the point on the supplementary segment circular arc, and the value range is , is the total angle of the supplementary segment circular arc; is the radius of the supplementary segment circular arc, and the expression is as follows: 。 3. The method of optimizing the design of a heating structure according to claim 2, characterized in that, In step (2), the value range of each design variable is as follows: 。 4. The optimization design method of the heating structure according to claim 3, characterized in that, wherein, is the number of temperature measurement points distributed over the working surface of the heating plate; is the surface temperature of the heating plate measured at the th temperature measurement point; is the average temperature of the temperature measurement points; is the maximum temperature of the temperature measurement points; is the minimum temperature of the temperature measurement points.

5. The method of optimizing the design of a heating structure according to claim 4, characterized in that, Step (3) includes: the basic form of the Gaussian process regression model is: wherein, is the th objective function in the predictive function of the Gaussian process regression model; is Gaussian noise with mean 0; is a Gaussian process with expression as follows: where is a mean function, and is a kernel function that defines the covariance of the function values at any two points and in the input space. Using Bayesian optimization method to optimize hyperparameters of Gaussian process regression model Optimization is performed, the acquisition function selection is set to expected improvement per second, the number of iterations is set, after the Bayesian optimization is completed, the optimal value of the hyperparameters is obtained .

6. The method of optimizing the design of a heating structure according to claim 5, characterized in that, 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: wherein, is the signal variance; is and the Euclidean distance between is a length parameter vector to control the scale of each dimension.

7. The method of optimizing the design of a heating structure according to claim 6, characterized in that, In step (4), the multi-objective optimization mathematical model is represented as: where, and is a summary description of an optimization problem, denotes minimization of the objective function, denotes design variables; , and denotes the objective function.

8. The method of optimizing the design of a heating structure according to claim 7, characterized in that, In step (4), the multi-objective optimization based on NSGA-II algorithm is used to solve the optimal parameter combination, which includes: Determine the population size, iteration number, crossover probability and mutation probability of 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 optimal parameter combination, and the selection criteria are as follows: wherein, is a standard deviation of the temperature of the temperature measurement points of the initial heating plate; is a coefficient of variation of the temperature of the temperature measurement points of the initial heating plate; is a uniformity of the temperature of the temperature measurement points of the initial heating plate; , and are weight proportions of the objective functions , and in turn.

9. The method of optimizing the design of a heating structure according to claim 8, characterized in that, Step (5) includes: performing multi-objective optimization result error analysis; comparing performance indicators under different combinations of optimization parameters. , and Error analysis of multi-objective optimization results was conducted by comparing predicted values ​​with simulation experimental calculations; performance analysis of multi-objective optimization results was also performed based on three performance indicators. , and By comparing the temperature uniformity performance of the initial heating plate and the optimized heating plate, a performance analysis of the multi-objective optimization results is conducted.

10. A method of manufacturing a heating structure, characterized by, Comprise: The optimization design method of the heating structure according to any one of claims 1 to 9 obtains the optimized heating disc, derives the geometric parameters and material parameters of the optimized heating disc, calculates the corresponding five-axis 3D printing path and determines the printing process parameters, and completes 3D printing by using a five-axis 3D printer.

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