A Constraint-Automated Optimization Method for Semiconductor Measurement

Abstract

This invention relates to a constrained automated optimization method for semiconductor measurement, comprising: step 3) optimizing auxiliary parameters using a training set; step 4) selecting one or more second candidate parameter combinations to form a candidate parameter combination pool, and saving the regression coefficient set corresponding to each second candidate parameter combination; step 5) using the auxiliary parameters in the optimal second candidate parameter combination as independent variables, using the corresponding regression coefficient set as the initial coefficients of the equation, and the key parameters as dependent variables to establish a second multiple linear regression equation; performing the first-stage optimization and the second-stage optimization to select the optimal predicted values ​​and optimal regression coefficients of the key parameters; and step 6) using a test set to perform the first-stage optimization and the second-stage optimization to verify the optimal candidate parameter combination and its corresponding optimal regression coefficients. The two-stage optimization strategy fundamentally solves industry pain points such as difficulty in convergence, difficulty in reproducibility, strong subjectivity, and long processing time.

Description

Technical Field

[0001] This invention relates to the field of semiconductor measurement technology, and in particular to a constraint-automated optimization method for semiconductor measurement. Background Technology

[0002] In the field of semiconductor metrology, optical critical dimension (OCD) measurement typically employs inversion analysis methods, extracting the geometric morphology parameters of the sample by fitting theoretical model spectra with measured spectra. Current mainstream methods rely on engineers' experience for multi-pass parameter tuning, and their technical limitations are mainly reflected in the following aspects:

[0003] Parameter coupling difficulties: Complex physical couplings exist between the geometric parameters of semiconductor devices (such as thin film thickness, linewidth, sidewall angles, etc.), resulting in high-dimensional and strongly nonlinear mathematical characteristics in inversion calculations. Traditional optimization methods are prone to getting trapped in local optima or failing to converge when dealing with such problems.

[0004] Limitations of manual parameter tuning: In practical engineering applications, key aspects such as setting parameter states (fixed / floating) and configuring boundary constraints still heavily rely on the individual experience of engineers. This not only makes the tuning process time-consuming and labor-intensive, but more importantly, it is highly subjective. Different technicians adopt different parameter tuning strategies, making it difficult to guarantee the reproducibility of measurement results, which is significantly different from the stringent requirements for stability and repeatability of industrial-grade equipment.

[0005] The fragmented process lacks an adaptive closed-loop mechanism: In the current technical framework, the various stages of "parameter selection - constraint setting - optimization execution" are isolated from each other, failing to form an organic whole and unable to create a closed-loop system that can self-verify and self-adjust based on optimization results. Due to the lack of a feedback mechanism for self-verification and dynamic adjustment based on intermediate optimization results, the entire process cannot adaptively correct parameter strategies, fundamentally limiting the intelligence level and robustness of the measurement system. Summary of the Invention

[0006] To address at least some of the aforementioned problems in the prior art, the present invention provides a constraint-automated optimization method for semiconductor measurement, comprising:

[0007] Step 3) Use the training set to optimize the auxiliary parameters to obtain the first optimal predicted value of the auxiliary parameters. The training set includes multiple samples, each containing a reference spectrum, reference values ​​of key parameters, and initial values ​​of multiple auxiliary parameters.

[0008] Step 4) Select k auxiliary parameters from all auxiliary parameters to obtain one or more different combinations of first candidate parameters, and use multiple linear regression analysis to screen out one or more combinations of second candidate parameters to form a pool of candidate parameter combinations, and save the set of regression coefficients corresponding to each combination of second candidate parameters.

[0009] Step 5) Using the auxiliary parameters in the optimal second candidate parameter combination as independent variables, and the corresponding regression coefficient set as the initial coefficients of the equation, and the key parameters as dependent variables, establish a second multiple linear regression equation; perform the first stage optimization, including: floating the values ​​of all auxiliary parameters, floating the values ​​of the regression coefficients, calculating the key parameters using the second multiple linear regression equation, and fitting the simulated spectrum and reference spectrum based on the key parameters and auxiliary parameters to screen out the second optimal predicted values ​​and optimal regression coefficients of all auxiliary parameters;

[0010] The second stage of optimization includes: fixing the values ​​of auxiliary parameters to the second optimal predicted values, allowing the values ​​of key parameters to float freely to fit the simulated spectrum and the reference spectrum, and selecting the optimal predicted values ​​of the key parameters.

[0011] Step 6) Use the test set to perform the first-stage optimization and the second-stage optimization to verify the optimal second candidate parameter combination and its corresponding optimal regression coefficient. If the verification is successful, output the multiple linear regression equation defined by the optimal second candidate parameter combination and its corresponding optimal regression coefficient. Otherwise, select the second-best second candidate parameter combination from the candidate parameter combination pool and execute steps 5 and 6 again.

[0012] Furthermore, step 5 also includes: calculating the coefficient of determination R by combining the reference values ​​of the key parameters with the optimal predicted values ​​of the key parameters obtained from the second-stage optimization. 2 .

[0013] Furthermore, performing the first-stage optimization and the second-stage optimization using the test set includes:

[0014] The coefficients of the second multiple linear regression equation are fixed to the optimal regression coefficients obtained in the first stage of optimization, thus obtaining the third multiple linear regression equation.

[0015] All auxiliary parameters are floated, and then the key parameters are calculated based on the third multiple linear regression equation. The key parameters and auxiliary parameters are used to fit the simulated spectrum and the reference spectrum, and the third best predicted value of all auxiliary parameters is selected.

[0016] All auxiliary parameters are fixed at the third best predicted value, and the key parameters are allowed to float freely to fit the simulated spectrum and the reference spectrum in order to select the best validation predicted value of the key parameters.

[0017] Calculate the coefficient of determination between the optimal validation prediction value and the reference value of the key parameter. If the coefficient of determination is greater than the second threshold, the validation is successful; otherwise, the validation is unsuccessful.

[0018] Furthermore, auxiliary parameter optimization using the training set to obtain the first optimal predicted value for the auxiliary parameters includes:

[0019] The values ​​of the key parameters are fixed as reference values, while the values ​​of the auxiliary parameters are floated. The simulated spectrum and the reference spectrum are fitted to obtain the first optimal predicted value of the auxiliary parameters, forming an N×M parameter matrix, where N is the number of training set samples and M is the sum of the number of key parameters and the number of auxiliary parameters.

[0020] Furthermore, if all second candidate parameter combinations under the current k value fail the verification, then modify the k value and repeat steps 4 to 6.

[0021] Further, in step 3, using the key parameter as the dependent variable and k first candidate parameters as independent variables, a first multiple linear regression equation is established, and multiple linear regression analysis is performed. Multiple combinations of first candidate parameters are iterated through, and one or more second candidate parameter combinations with a determination coefficient greater than a first threshold are selected to form a candidate parameter combination pool. The regression coefficient set corresponding to each candidate parameter combination is then saved, where:

[0022] The coefficient of determination is calculated from the key parameter values ​​derived from the first candidate parameter and the key parameter reference values.

[0023] The first multiple linear regression equation is: The set of regression coefficients includes ,in For the intercept term, Represent the independent variable The coefficients, 1≤k≤M-1.

[0024] Furthermore, after selecting multiple second candidate parameter combinations, they are sorted based on the magnitude of the coefficient of determination.

[0025] Furthermore, in the first stage of optimization in step 5, a grid search method is used, with the regression coefficients obtained in step 4 as the center, to search within the neighborhood of the center point, including:

[0026] Define the search range for each coefficient dimension, and divide the search range evenly into G points;

[0027] Traverse all (k+1) dimensional grids At each point in the grid, the first stage of optimization is performed: all auxiliary parameters are floated, and key parameters are calculated. Then, based on the key parameters and auxiliary parameters, the simulated spectrum and reference spectrum are fitted to screen out the second best predicted value and the best regression coefficient of all auxiliary parameters.

[0028] Furthermore, it also includes:

[0029] Step 1) Provide a training dataset consisting of multiple samples, each sample containing a reference spectrum, reference values ​​for key parameters, and initial values ​​for multiple auxiliary parameters;

[0030] Step 2) Split the training dataset into a training set and a test set.

[0031] The present invention has at least the following beneficial effects:

[0032] This invention proposes a constraint-automated optimization method for semiconductor measurement. Its core lies in automatically establishing the constraint relationship between key parameters and auxiliary parameters through a data-driven approach. It adopts a two-stage optimization strategy of "first constraining key parameters and optimizing auxiliary parameters, and then refining key parameters" to achieve efficient parameter inversion, thereby improving the goodness of fit between the calculated predicted values ​​of key parameters and the reference values ​​of key parameters.

[0033] The two-stage optimization strategy effectively decomposes the problem of inversion of high-dimensional and strongly coupled parameters in semiconductor measurement from a mathematical perspective, transforming the complex global joint optimization into two more easily convergent continuous optimization stages, fundamentally solving the industry pain points of traditional methods being prone to getting trapped in local optima and unstable convergence.

[0034] The first optimal predicted value of the auxiliary parameters is obtained to form a parameter matrix. Multiple linear regression analysis is used to automatically screen multiple second candidate parameter combinations that are highly correlated with the key parameters. Then, two-stage optimization is used to select the final parameter combination. At the same time, grid search is used to fine-tune the regression coefficients to select the optimal regression coefficients. The selected parameter combinations and corresponding optimal regression coefficients are verified by performing the first and second stage optimizations using a test set. If the verification fails, the parameter combinations and regression coefficients are re-selected, realizing an adaptive feedback closed loop, realizing the self-verification and intelligent fault tolerance of the algorithm, and ensuring the robustness of the system.

[0035] The constraint-automated optimization method of this invention automates the entire process, transforming the traditional trial-and-error and parameter tuning process that relies on human experience into an efficient and automated parallel computing process. This not only frees up manpower but also reduces the optimization time from several days to several hours, achieving a key leap from algorithm innovation to industrial applicability. Attached Figure Description

[0036] Figure 1The flowchart of a constraint-automated optimization method for semiconductor measurement according to an embodiment of the present invention is shown. Detailed Implementation

[0037] It should be noted that the components in the accompanying drawings may be shown exaggerated for illustrative purposes and may not be to scale.

[0038] In this invention, the various embodiments are merely intended to illustrate the solutions of the invention and should not be construed as limiting.

[0039] In this invention, unless otherwise specified, the quantifiers “a” and “one” do not exclude scenarios involving multiple elements.

[0040] It should also be noted that, in the embodiments of the present invention, only a portion of the parts or components may be shown for clarity and simplicity. However, those skilled in the art will understand that, under the teachings of the present invention, the required parts or components can be added as needed for specific scenarios.

[0041] It should also be noted that within the scope of this invention, the terms "same", "equal", and "equal to" do not mean that the two values ​​are absolutely equal, but allow for a certain reasonable error. In other words, the terms also cover "substantially the same", "substantially equal", and "substantially equal to".

[0042] It should also be noted that in the description of this invention, the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not explicitly or implicitly suggest that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0043] Furthermore, the embodiments of the present invention describe the process steps in a specific order. However, this is only for the convenience of distinguishing each step, and is not a limitation on the order of each step. In different embodiments of the present invention, the order of each step can be adjusted according to the process.

[0044] Figure 1 The flowchart of a constraint-automated optimization method for semiconductor measurement according to an embodiment of the present invention is shown.

[0045] like Figure 1 As shown, a constraint-automated optimization method for semiconductor measurement includes:

[0046] Step 1) Provide a training dataset consisting of multiple samples, each sample containing a reference spectrum, reference values ​​for key parameters, and initial values ​​for multiple auxiliary parameters;

[0047] Step 2) Split the training dataset into a training set and a test set;

[0048] Step 3) Use the training set to optimize the auxiliary parameters to obtain the first optimal predicted value of the auxiliary parameters;

[0049] Step 4) Select k auxiliary parameters from all auxiliary parameters to obtain one or more different combinations of first candidate parameters, and use multiple linear regression analysis to screen out one or more combinations of second candidate parameters to form a pool of candidate parameter combinations, and save the set of regression coefficients corresponding to each combination of second candidate parameters.

[0050] Step 5) Using the auxiliary parameters in the optimal second candidate parameter combination as independent variables, and the corresponding regression coefficient set as the initial coefficients of the equation, and the key parameters as dependent variables, establish a second multiple linear regression equation; perform the first stage optimization, including: floating the values ​​of all auxiliary parameters, floating the values ​​of the regression coefficients, calculating the key parameters using the second multiple linear regression equation, and fitting the simulated spectrum and reference spectrum based on the key parameters and auxiliary parameters to screen out the second optimal predicted values ​​and optimal regression coefficients of all auxiliary parameters; perform the second stage optimization, including: fixing the values ​​of the auxiliary parameters to the second optimal predicted values, freely floating the values ​​of the key parameters to fit the simulated spectrum and reference spectrum, and screening out the optimal predicted values ​​of the key parameters;

[0051] Step 6) Use the test set to perform the first-stage optimization and the second-stage optimization to verify the optimal second candidate parameter combination and its corresponding optimal regression coefficient. If the verification is successful, output the multiple linear regression equation defined by the optimal second candidate parameter combination and its corresponding optimal regression coefficient. Otherwise, select the second-best second candidate parameter combination from the candidate parameter combination pool and execute steps 5) and 6) again.

[0052] Steps 1 through 6 are described in detail below.

[0053] Step 1) Provide a training dataset, which includes: multiple reference spectra and reference values ​​of corresponding key parameters, as well as initial values ​​of multiple auxiliary parameters.

[0054] Multiple sets of reference spectra (such as Mueller matrix spectra containing multiple wavelength points) of the semiconductor sample under test are acquired, and a series of key parameter reference values ​​are simultaneously acquired through reliable measurement methods (such as transmission electron microscopy, TEM). The key parameter reference values ​​correspond one-to-one with the measured reference spectra.

[0055] In the field of semiconductor metrology, the reference value obtained through TEM technology is generally considered the standard value. Whether through manual adjustment or the automated method proposed in this invention, the goal is to maximize the correlation between the calculated result and the reference value. This correlation is represented by the statistical index, the coefficient of determination (R²). 2 express.

[0056] In addition, several auxiliary parameters are provided to assist in the calculation of simulated spectra.

[0057] Simultaneously, it receives the physical meaning, initial values, and physical boundaries (maximum and minimum values) of all user-defined parameters to be optimized (critical and auxiliary parameters). These parameters characterize the geometric features of the semiconductor structure (such as depth, thickness, angle, offset, etc.), and their physical boundaries are usually determined based on physical possibilities and process knowledge.

[0058] The following example illustrates step 1:

[0059] The semiconductor sample under test is a product of a certain SDB (single diffusion break) process. The measurement system receives three sets of spectra from different wafers in different process slots. Each set of spectra contains three measured spectra, for a total of nine spectra (nSpectra=9). The corresponding physical model has 11 parameters to be optimized (nDOF=11), including: trench depth (Tr_Depth), trench bottom critical dimension offset (Tr_BCD_off), hard mask nitride layer thickness (HM_NIT), trench middle critical dimension offset (Tr_MCD_off), fin sidewall angle (Fin_SWA), polysilicon gate height (Poly_HT), polysilicon gate top critical dimension (Poly_TCD), recess depth (Recess_HT), sidewall thickness (SP_CESL), fin height (Fin_HT), and fin bottom oxide loss (FinBotOxLoss).

[0060] The trench depth was set as the critical parameter, and nine TEM measurement data were used as a set of reference values ​​for the critical parameter (CP), which corresponded strictly to the nine spectra mentioned above.

[0061] In addition, the measurement system also receives the initial settings for each parameter, including the initial value, upper limit, and lower limit. For example, the initial value for the trench depth is 125 nm, the upper limit is 95 nm, and the lower limit is 170 nm. The initial value for the fin sidewall angle is 87°, the upper limit is 80°, and the lower limit is 93°.

[0062] This step provides the sole data input and physical constraint basis for the entire automated optimization process.

[0063] Step 2) Split the training dataset into a training set and a test set.

[0064] To ensure that the established parameter constraints have good generalization ability and avoid being effective only for specific data and causing overfitting, this invention adopts a training set-test set splitting strategy. The training set is specifically used to perform subsequent parameter coupling relationship (optimal candidate parameter combination) search and constraint coefficient (regression coefficient) tuning; the test set serves as simulated unknown production samples, used to finally and independently evaluate the prediction accuracy and robustness of the established constraint relationships.

[0065] The 9 spectra obtained in step A can be divided according to a common empirical ratio in machine learning (e.g., 2 / 3 for training, 1 / 3 for testing). For example, if the 9 spectra are indices 0-8, they can be divided into 3... The spectrum is arranged in three combinations, where 0, 1, 2 form the first group of spectra, 3, 4, 5 form the second group, and 6, 7, 8 form the third group. One spectrum from each group is selected as the test set spectrum, and the rest are used as the training set spectra. For example, spectra with indices 0, 1, 3, 4, 6, 7, and their corresponding key and auxiliary parameters are assigned to the training set, while spectra with indices 2, 5, 8, and their corresponding key and auxiliary parameters are assigned to the test set.

[0066] The above are just examples. The division method is not unique. The division process can be achieved through a preset index list or random sampling, but it is necessary to ensure that all data (spectrum, key parameters and auxiliary parameters) of the same spectral sample are completely divided into the same subset.

[0067] This step, by introducing a hold-out verification mechanism during the algorithm development phase, simulates the real performance of the constrained optimization method when faced with new data, providing a crucial pre-evaluation step for generalization performance in the core optimization process.

[0068] Step 3) Optimize auxiliary parameters using the training set to obtain the first optimal predicted value of the auxiliary parameters. Specifically, fix the values ​​of the key parameters to reference values ​​and adjust the values ​​of the auxiliary parameters. Fit the simulated spectrum and the reference spectrum to obtain the first optimal predicted value of the auxiliary parameters, forming an N×M parameter matrix, where N is the number of training set samples and M is the sum of the number of key parameters and the number of auxiliary parameters.

[0069] By minimizing the error between the simulated spectrum and the reference spectrum through nonlinear optimization, the first optimal predicted values ​​of a set of auxiliary parameters are obtained. The optimization results of all samples are combined to form a parameter matrix, where each row corresponds to all physical parameters of a sample, and each column corresponds to the same physical parameter of all samples.

[0070] Step 3 aims to obtain the most probable combinations of values ​​for other parameters while ensuring the absolute accuracy of the key parameters, providing a clean data foundation for discovering the intrinsic relationships between parameters. The parameter matrix obtained in this step eliminates the interference caused by the uncertainty of the key parameters, and the numerical relationships between its columns (between different parameter columns) more purely reflect the physical intrinsic coupling and covariance between different structural parameters, making it an ideal input for subsequent statistical modeling.

[0071] Step 3 is explained in detail below:

[0072] For each sample (index i) in the training set, a constant constraint is constructed to strictly fix the key parameter (such as the trench depth) to its corresponding TEM reference value TEM[i]. This constraint ensures that the key parameter remains unchanged in subsequent regression calculations. Taking the measured spectrum of sample i as the objective, and starting from the initial values ​​of all auxiliary parameters in step 1, the remaining 10 auxiliary parameters are floated. A nonlinear optimizer (such as the Levenberg-Marquardt algorithm or LM algorithm) is called to perform regression calculations. The optimization objective is to minimize the difference between the calculated simulated spectrum and the measured spectrum (such as the chi-square value or MSE). During the regression process, it is strictly ensured that all auxiliary parameter values ​​are within the upper and lower boundaries set in step 1. After the calculation of each sample is completed, the key parameter values ​​and the first optimal predicted values ​​of the auxiliary parameters are saved.

[0073] After all samples have been processed, an N×M parameter matrix ParaMatrix is ​​obtained, where M is the total number of parameters (in this embodiment, M=nDOF=11, which includes 1 key parameter and 10 auxiliary parameters). ParaMatrix[i][j] represents the value of the j-th parameter calculated by regression optimization for the i-th sample.

[0074] It is worth noting that since the regression calculations for each sample are independent, parallel programming techniques can be used to process all training set samples in multiple threads using tools such as OpenMP, thereby greatly reducing computation time. This technique can also be applied to several subsequent steps.

[0075] Step 4) Select k auxiliary parameters from all auxiliary parameters to obtain one or more different combinations of first candidate parameters, and use multiple linear regression analysis to screen out one or more combinations of second candidate parameters to form a pool of candidate parameter combinations, and save the set of regression coefficients corresponding to each combination of second candidate parameters.

[0076] In one embodiment, a first multiple linear regression equation is established with the key parameter as the dependent variable and the first candidate parameter as the independent variable. Multiple linear regression analysis is then performed, iterating through multiple combinations of the first candidate parameters. One or more second candidate parameter combinations with a coefficient of determination greater than a first threshold are selected, forming a pool of candidate parameter combinations. The regression coefficient set corresponding to each candidate parameter combination is then saved. Here, the coefficient of determination is calculated from the key parameter value derived from the first candidate parameter and the key parameter reference value. 1 ≤ k ≤ M-1. The value of k can be set manually.

[0077] This step, based on the N×M parameter matrix ParaMatrix, automatically and quantitatively selects the subset of parameters that have the strongest statistical correlation with the key parameter (TEM reference value) and are most suitable for building a linear prediction model from all auxiliary parameters, and establishes a preliminary linear relationship for each subset.

[0078] The process is as follows: First, define the search parameter space, with M-1 auxiliary parameters in the parameter matrix ParaMatrix. Select k auxiliary parameters from this space as combinations of independent variables. The number of all possible combinations is determined by the combination number formula C(M-1, k). After confirming the search parameter space, use an exhaustive method to perform multiple linear regression (LINEST) calculations on all combinations sequentially. Since the multiple linear regression function LINEST is relatively fast, even if the number of parameter combinations increases exponentially, the time to traverse all combinations is still acceptable. To avoid missing better combinations, a global traversal method should be used.

[0079] For each candidate parameter combination, using the k columns of data corresponding to that combination in the parameter matrix ParaMatrix as the independent variable X, and the reference values ​​of the key parameters as the dependent variable Y, the multiple linear regression function LINEST is called to calculate the model. The regression coefficient set {β} and the determination coefficient R² of the model. The regression coefficient set {β} contains ,in For the intercept term, Represent the independent variable The coefficient.

[0080] After the traversal calculation is completed, all auxiliary parameter combinations are sorted from high to low according to their corresponding coefficient of determination R². The closer R² is to 1, the higher the correlation between the auxiliary parameter combination and the reference value of the key parameter.

[0081] Next, highly correlated combinations are selected, and a first threshold (e.g., 0.85) is set. All auxiliary parameter combinations with a determination coefficient greater than the first threshold are selected as candidate parameter combinations, forming a candidate parameter combination pool, and the regression coefficient set corresponding to the candidate parameter combinations is saved. At the same time, the optimal candidate parameter combination corresponding to the largest determination coefficient is recorded as the initial optimal combination.

[0082] The following example illustrates step 4.

[0083] Selecting k=3 columns of auxiliary parameters from the parameter matrix as combinations of independent variables yields C(10,3)=120 combinations. Using the key parameter as the dependent variable and the first candidate parameter as the independent variable, establish and calculate the first multiple linear regression equation, outputting the set of regression coefficients and the coefficient of determination R. 2 .

[0084] The coefficient of determination R is obtained by using multiple linear regression to calculate all combinations of k=3. 2 The range is approximately 0.55-0.95. All combinations are processed according to R... 2 After sorting the values ​​from highest to lowest, the highest-ranking parameter combination (the optimal second candidate parameter combination) is COMB_1 = {hard mask nitride layer thickness, polysilicon gate height, trench bottom critical dimension offset}, with the corresponding regression coefficient set being BETA_1 = {179.2655, 0.1826, -0.6508, -0.7263}. In other words, for combination 1, the linear constraint model (the first multiple linear regression equation) is trench depth = 179.2655 + 0.1826. Hard mask nitride layer thickness - 0.6508 Polysilicon gate height - 0.7263 Offset of critical dimensions at the bottom of the trench.

[0085] Following the optimal second candidate parameter combination are COMB_2 = {hard mask nitride layer thickness, trench center critical dimension offset, polysilicon gate height} and its corresponding regression coefficient set BETA_2, COMB_3 = {hard mask nitride layer thickness, fin sidewall angle, polysilicon gate height} and its corresponding regression coefficient set BETA_3, and so on, up to COMB_120, BETA_120. Using a first threshold of 0.85 as a standard, combinations below the standard are discarded, and only combinations above the standard are retained to fill the candidate parameter combination pool.

[0086] It is worth noting that the value of k can vary. If all combinations of a certain k value fail to filter out R... 2 For parameter combinations with sufficiently high values, the system will adjust the k value within the range of [1, M-1] and search again.

[0087] Step 5) Using the auxiliary parameters in the optimal second candidate parameter combination as independent variables, and the corresponding regression coefficient set as the initial coefficients of the equation, and the key parameters as dependent variables, establish a second multiple linear regression equation; perform the first stage optimization, including: floating the values ​​of all auxiliary parameters, floating the values ​​of the regression coefficients, calculating the key parameters using the second multiple linear regression equation, and fitting the simulated spectrum and reference spectrum based on the key parameters and auxiliary parameters to screen out the second optimal predicted values ​​and optimal regression coefficients of all auxiliary parameters;

[0088] The second stage of optimization involves fixing the values ​​of auxiliary parameters to the second optimal predicted values, allowing the values ​​of key parameters to float freely to fit the simulated spectrum and the reference spectrum, and then selecting the optimal predicted values ​​for the key parameters.

[0089] The final multiple linear regression equation can be determined based on the optimal combination of candidate parameters and the optimal regression coefficients.

[0090] In one embodiment, during the first-stage optimization, the initial values ​​of the regression coefficients are the values ​​obtained in step 3, and the initial values ​​of the auxiliary parameters are the values ​​of the training set.

[0091] In the second stage of optimization, the initial values ​​of the key parameters are reference values.

[0092] In one embodiment, in the first stage of optimization in step 5, a grid search method is used, with the regression coefficients obtained in step 4 as the center, to search within the neighborhood of the center point, including:

[0093] Define the search range for each coefficient dimension, and divide the search range evenly into G points;

[0094] Traverse all (k+1) dimensional grids At each point in the grid, the first stage of optimization is performed: all auxiliary parameters are floated, and key parameters are calculated. Then, based on the key parameters and auxiliary parameters, the simulated spectrum and reference spectrum are fitted to screen out the second best predicted value and the best regression coefficient of all auxiliary parameters.

[0095] Step 5 is a crucial bridge connecting the statistical model with the actual inversion task. The regression coefficient set BETA obtained in Step 4 is the "statistically optimal solution" that minimizes the linear fitting error. However, this invention aims to use this linear relationship to guide nonlinear inversion optimization. Therefore, it is necessary to find an optimal coefficient set BETA_plus such that when applied as a constraint to the inversion, it yields a result closest to the reference value (TEM).

[0096] Implementation process:

[0097] Define the objective function: For a given set of second candidate parameter combinations COMB and its corresponding regression coefficient set BETA (BETA provided in step 4), define its performance score S(BETA) as follows:

[0098] S(BETA) = The coefficient of determination R² between the predicted values ​​and reference values ​​of key parameters, calculated after performing a "two-stage automated constraint optimization sub-process" on all samples in the training set using the linear constraints (second multiple linear regression equation) formed by the regression coefficient set BETA. In other words, the coefficient of determination R² is used as an evaluation criterion to select the optimal regression coefficients.

[0099] The function S(BETA) directly quantifies the effectiveness of a set of coefficients in the actual inversion task and is the sole objective of this optimization.

[0100] The optimization algorithm is executed as follows: A grid search method is used, with the regression coefficient BETA obtained in step 4 as the center, searching within the neighborhood of the center point. For each coefficient dimension, a certain percentage (e.g., ±50%) is defined for the search range, and this range is evenly divided into G points (e.g., G=5). The algorithm will traverse all points on this (k+1)-dimensional grid (including the intercept term). For each point in the grid (i.e., a coefficient vector), the "two-stage automated constraint optimization process" is invoked to calculate its S, and the coefficient vector (coefficient value) that maximizes S is selected as the optimal solution BETA_plus.

[0101] When there are many parameters in the combination, the grid search method may encounter dimensionality explosion, and the computation time may become unacceptable. In this case, common parameter tuning methods such as coordinate descent can be used. Professionals in this field should understand this and will not elaborate further.

[0102] In the first stage, the coefficients are adjusted to select the optimal regression coefficients. In the second stage, the optimal regression coefficients are fixed, and the optimal predicted values ​​of the key parameters are calculated.

[0103] Output optimal coefficients: After the tuning process is complete, the optimal coefficient set BETA_plus that maximizes S(BETA) is obtained. This determines the customized linear constraint equation under the current optimal parameter combination. This equation will be passed as the final result of the training set to the next step for verification on the test set.

[0104] The two-stage automated constraint optimization process includes a first-stage optimization and a second-stage optimization.

[0105] For any sample spectrum, the second candidate parameter combination COMB, and the corresponding regression coefficient set BETA, the automated two-stage constrained optimization process is as follows:

[0106] Phase Optimization:

[0107] By combining the regression coefficient set BETA and its corresponding second candidate parameter combination COMB, a linear equality constraint (the second multiple linear regression equation) is constructed. Where CP is the key parameter, and β is the coefficient in the regression coefficient set BETA. This is the i-th auxiliary parameter in the second candidate parameter combination. Represents the intercept value. Let i represent the coefficient of the independent variable, where 1 ≤ i ≤ k.

[0108] The values ​​of all M-1 auxiliary parameters are floated, the values ​​of the regression coefficients are floated, the key parameter values ​​are calculated by the second multiple linear regression equation, and then a regression optimization algorithm (such as the LM algorithm) is run to fit the simulated spectrum and the reference spectrum to find the auxiliary parameter values ​​that minimize the matching error between the simulated spectrum and the reference spectrum.

[0109] During the optimization process, the values ​​of key parameters are automatically calculated by the constraint equations and do not participate in the regression calculations. The first stage of optimization aims to find the auxiliary parameter values ​​and regression coefficient values ​​that minimize the current spectral matching error.

[0110] Second phase optimization:

[0111] The second stage automatically inherits the results from the first stage as the initial values ​​for optimization. The values ​​of the M-1 auxiliary parameters are fixed as the optimal predicted values ​​(second optimal predicted values) obtained from the first stage optimization. The key parameters are no longer linearly constrained and are no longer calculated by the second multiple linear regression equation. Instead, they float freely within the upper and lower bounds to fit the simulated spectrum and the reference spectrum, seeking the optimal predicted values ​​of the key parameters that minimize the matching error between the simulated spectrum and the reference spectrum.

[0112] The second stage of optimization aims to eliminate any residual errors that may exist in the linear model through further fine-tuning.

[0113] Post-processing stage: Computational performance metric S(BETA):

[0114] By combining the reference values ​​of the key parameters with the optimal predicted values ​​of the key parameters obtained from the second-stage optimization, the coefficient of determination R between the two sets of data is calculated. 2 This indicator directly reflects the prediction effect of the second candidate parameter combination and the optimal coefficient set of the first stage optimization.

[0115] The following example illustrates step 5:

[0116] Step 4 yielded the optimal second candidate parameter combination COMB_1 = {hard mask nitride layer thickness, polysilicon gate height, trench bottom critical dimension offset}, with the corresponding regression coefficient set BETA_1 = {179.2655, 0.1826, -0.6508, -0.7263}. The regression coefficient set BETA_1 has four dimensions, with each dimension having 10 uniformly distributed grid points. The grid boundaries are ±50% of the original values. The ranges of the four coefficients are [89.6328, 268.8983], [0.0913, 0.2739], [-0.9762, -0.3254], [-1.0895, -0.3632], respectively. =10,000 grid points need to be automatically optimized; each grid point optimization only uses the spectra of the 6 training set samples (0, 1, 3, 4, 6, 7) segmented in step 2.

[0117] During the process of traversing all grid points, R is continuously compared and updated to a higher value. 2 The optimal set of regression coefficients and their corresponding values ​​are searched until all 10,000 grid points have completed the two-stage optimization. Through the above steps, for the same parameter combination COMB_1, the optimal coefficient set is found to be BETA_plus = {89.6328, 0.1598, -0.6237, -0.9079}, and the optimal R-value is... 2 The value also increased from about 0.95 in step 4 to about 0.99, showing a significant improvement.

[0118] The two-stage automatic optimization between different samples is independent of each other and can be computed in parallel to achieve acceleration.

[0119] Step 6) Use the test set to perform the first-stage optimization and the second-stage optimization to verify the optimal second candidate parameter combination and its corresponding optimal regression coefficient. If the verification is successful, output the multiple linear regression equation defined by the optimal second candidate parameter combination and its corresponding optimal regression coefficient. Otherwise, select the second-best second candidate parameter combination from the candidate parameter combination pool and execute steps 5) and 6) again.

[0120] Specifically, performing the first and second phases of optimization using the test set includes:

[0121] The coefficients of the second multiple linear regression equation are fixed to the optimal regression coefficients obtained in the first stage of optimization, resulting in the third multiple linear regression equation. All auxiliary parameters are then floated, and key parameters are calculated based on the third multiple linear regression equation. These key parameters and auxiliary parameters are used to fit the simulated spectrum and the reference spectrum, and auxiliary parameters are optimized to select the third optimal predicted value for all auxiliary parameters. During the first stage of optimization, the coefficients of the third multiple linear regression equation do not float.

[0122] All auxiliary parameters are fixed at the third best predicted value, while the key parameters are allowed to float freely to fit the simulated spectrum and the reference spectrum in order to select the best validation predicted value for the key parameters.

[0123] Calculate the coefficient of determination between the optimal validation prediction value and the reference value of the key parameter. If the coefficient of determination is greater than the second threshold, the validation is successful; otherwise, the validation is unsuccessful.

[0124] If R 2 If the value is lower than the preset threshold, it indicates that the current constraint relationship is unfavorable, meaning that the parameter combination and regression coefficient combination selected in the current two-stage optimization are not good. We can then return to the second-best candidate parameter combination obtained in step 4, and repeat steps 5 and 6, thus forming a feedback optimization loop until the requirements are met. If R... 2 The value is already high enough (generally considered R to be high enough). 2 If the value is greater than 0.95, the second candidate parameter combination and the optimal regression coefficient that meet the conditions will be output as the measurement results for subsequent data analysis, visualization and process monitoring.

[0125] If none of the parameter combinations under the current k value (e.g., k=3) can satisfy R 2 If the value is greater than the second threshold, the system will automatically modify the value of k, for example, to k=4, and repeat steps 4 to 6.

[0126] The following example illustrates step 6:

[0127] Step 5 yields the parameter combination {hard mask nitride layer thickness, polysilicon gate height, trench bottom critical dimension offset} and the optimal regression coefficient set {89.6328, 0.1598, -0.6237, -0.9079}, whose performance on the training set is R. 2 =0.99. The linear constraints formed by the above parameter combination and regression coefficient set are applied to the test set (spectrums 2, 5, and 8) segmented in step 2. Two-stage automatic optimization is also performed on each sample spectrum in the test set. After completion, the reference values ​​of the key parameters for the test set and the R-values ​​of the key parameter verification predicted values ​​calculated by the third multiple linear regression equation formed by the above parameter combination and coefficient set are calculated. 2The value is set. A second acceptable threshold is also set, for example, 0.95. Calculation results show that the current scheme performs well on the test set with R... 2 =0.98, and the R of the training set 2 If the values ​​are not significantly different and are higher than the second threshold, the current parameter combination and regression coefficients are considered as output results.

[0128] If a low R occurs 2 If the value is not met, then step 5 needs to be returned. The second-best candidate parameter combination, COMB_2 = {hard mask nitride layer thickness, trench center critical dimension offset, polysilicon gate height}, and its corresponding coefficient set BETA_2, should be used to repeat steps 5 and 6. If none of the combinations with k=3 can meet the threshold, the system will automatically change to k=4 and repeat steps 4 to 6.

[0129] While some embodiments of the present invention have been described in this application, those skilled in the art will understand that these embodiments are merely illustrative. Numerous variations, alternatives, and improvements will arise in those skilled in the art under the teachings of this invention without departing from its scope. The appended claims are intended to define the scope of the invention and thereby cover methods and structures within the scope of the claims themselves and their equivalents.

Claims

1. A constrained automation optimization method for semiconductor metrology, comprising: include: Step 3) Use the training set to optimize the auxiliary parameters to obtain the first optimal predicted value of the auxiliary parameters. The training set includes multiple samples, each containing a reference spectrum, reference values ​​of key parameters, and initial values ​​of multiple auxiliary parameters. Step 4) Select k auxiliary parameters from all auxiliary parameters to obtain one or more different combinations of first candidate parameters, and use multiple linear regression analysis to screen out one or more combinations of second candidate parameters to form a pool of candidate parameter combinations, and save the set of regression coefficients corresponding to each second candidate parameter combination; after screening out multiple combinations of second candidate parameters, sort them based on the size of the coefficient of determination. Step 5) Using the auxiliary parameters in the optimal second candidate parameter combination as independent variables, the corresponding regression coefficient set as the initial coefficients of the equation, and the key parameters as dependent variables, establish the second multiple linear regression equation. The first stage of optimization is performed, which includes: floating the values ​​of all auxiliary parameters, floating the values ​​of regression coefficients, calculating key parameters using the second multiple linear regression equation, and fitting simulated and reference spectra based on key and auxiliary parameters to screen out the second best predicted values ​​and best regression coefficients for all auxiliary parameters. The second stage of optimization includes: fixing the values ​​of auxiliary parameters to the second optimal predicted values, allowing the values ​​of key parameters to float freely to fit the simulated spectrum and the reference spectrum, and selecting the optimal predicted values ​​of the key parameters; Step 6) Use the test set to perform the first-stage optimization and the second-stage optimization to verify the optimal second candidate parameter combination and its corresponding optimal regression coefficient. If the verification is successful, output the multiple linear regression equation defined by the optimal second candidate parameter combination and its corresponding optimal regression coefficient. Otherwise, select the second-best second candidate parameter combination from the candidate parameter combination pool and execute steps 5 and 6 again. In the first stage of optimization in step 5, a grid search method is used, with the regression coefficients obtained in step 4 as the center, to search within the neighborhood of the center point, including: Define the search range for each coefficient dimension, and divide the search range evenly into G points; Traverse all of the k+1 dimensional grids At each point in the grid, the first stage of optimization is performed: all auxiliary parameters are floated, and key parameters are calculated. Then, based on the key parameters and auxiliary parameters, the simulated spectrum and reference spectrum are fitted to screen out the second best predicted value and the best regression coefficient of all auxiliary parameters.

2. The constraint-automated optimization method for semiconductor measurement according to claim 1, characterized in that, Step 5 further comprises calculating a determination coefficient R combining the reference value of the key parameter and the optimal predicted value of the key parameter obtained from the second stage optimization 2 .

3. The constraint-automated optimization method for semiconductor measurement according to claim 1, characterized in that, Performing first-stage and second-stage optimizations using the test set includes: The coefficients of the second multiple linear regression equation are fixed to the optimal regression coefficients obtained in the first stage of optimization, thus obtaining the third multiple linear regression equation. All auxiliary parameters are floated, and then the key parameters are calculated based on the third multiple linear regression equation. The key parameters and auxiliary parameters are used to fit the simulated spectrum and the reference spectrum, and the third best predicted value of all auxiliary parameters is selected. All auxiliary parameters are fixed at the third best predicted value, and the key parameters are allowed to float freely to fit the simulated spectrum and the reference spectrum in order to select the best validation predicted value of the key parameters. Calculate the coefficient of determination between the optimal validation prediction value and the reference value of the key parameter. If the coefficient of determination is greater than the second threshold, the validation is successful; otherwise, the validation is unsuccessful.

4. The constraint-automated optimization method for semiconductor measurement according to claim 1, characterized in that, Using the training set to optimize auxiliary parameters to obtain the first optimal predicted values ​​for the auxiliary parameters includes: The values ​​of the key parameters are fixed as reference values, while the values ​​of the auxiliary parameters are floated. The simulated spectrum and the reference spectrum are fitted to obtain the first optimal predicted value of the auxiliary parameters, forming an N×M parameter matrix, where N is the number of training set samples and M is the sum of the number of key parameters and the number of auxiliary parameters.

5. The constraint-automated optimization method for semiconductor measurement according to claim 3, characterized in that, If all second candidate parameter combinations under the current k value fail the verification, then modify the k value and repeat steps 4 to 6.

6. The constraint-automated optimization method for semiconductor measurement according to claim 1, characterized in that, In step 3, using the key parameter as the dependent variable and k first candidate parameters as independent variables, a first multiple linear regression equation is established. Multiple linear regression analysis is then performed, iterating through multiple combinations of first candidate parameters. One or more second candidate parameter combinations with a determination coefficient greater than a first threshold are selected, forming a pool of candidate parameter combinations. The regression coefficient set corresponding to each candidate parameter combination is then saved, where: The coefficient of determination is calculated from the key parameter values ​​derived from the first candidate parameter and the key parameter reference values. The first multiple linear regression equation is: The set of regression coefficients includes ,in For the intercept term, Represent the independent variable The coefficients, 1≤k≤M-1.

7. The constraint-automated optimization method for semiconductor measurement according to claim 1, characterized in that, Also includes: Step 1) Provide a training dataset consisting of multiple samples, each sample containing a reference spectrum, reference values ​​for key parameters, and initial values ​​for multiple auxiliary parameters; Step 2) Split the training dataset into a training set and a test set.

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