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Abstract
In integrated circuit manufacturing, optical critical dimension measurement is an efficient and non-destructive metrology method. It is also a model-based metrology in which a numerical model of the target device is formed to simulate the optical spectrum. The result is then reconstructed by fitting the simulated spectrum to the experimentally measured optical spectrum. Normally, the measured optical spectrum contains a great deal of data points that consume the storage space, and increase the fitting time. Therefore, it is worth finding an appropriate approach to downsample these data points without losing much accuracy. To quickly and accurately extract critical data with high sensitivity, we propose a Laplace sensitivity operator that is widely used for feature extraction. Compared with traditional sensitivity calculation, the Laplace sensitivity operator focuses more on the correlation and coupling between multiple parameters. Thus, the sensitivity can be properly analyzed from different dimensions. To test the feasibility and correctness of the proposed method, three basic structures were used for single-parameter verification: thin film, one-dimensional grating, and two-dimensional grating, and a vertical gate-all-around device used for multi-parameter analysis. Using the Laplace sensitivity operator, the extracted data showed better results in most cases than those achieved by the traditional sensitivity calculation method. The data volume was compressed by approximately 70%, the result matching loss was not significantly increase in terms of the root mean square error, and the calculation speed was increased by a factor of 2.4. Compared to the traditional sensitivity operator, the Laplace sensitivity operator was able to reduce the RMSE by up to 50%.
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1. Introduction
In the integrated circuit (IC) industry, high-volume manufacturing requires in-line, non-destructive, high-speed inspection and metrology [1,2]. The increasing integration density of IC devices also demands high efficiency, which poses a challenge to existing metrology methods [3–5]. Optical critical dimension (OCD) measurement is a non-imaging metrology method that is considered a key technology for current and future semiconductor manufacturing. As a model-based technology, OCD combines optical detection, optical simulation, and numerical fitting. Unlike imaging techniques such as scanning electron microscopy or atomic force microscopy, OCD cannot read out the results directly from measurements. Generally, an OCD measurement has two parts, a forward model and an inverse problem solving. The forward model is a simulation model of the target, and it includes the geometry, materials and the measurement setting. To simulate the scattered optical response spectrum of the periodic structure, some reliable forward modeling techniques are applied, such as rigorous coupled wave analysis (RCWA) [6], boundary element method [7] or finite difference time domain method (FDTD) [8]. The inverse problem is to link the experimental measured optical response to the forward model. Assist by an exhaustive searching, numerical fitting or natural network, the geometric profile of the target can be reconstructed from the experimental measured optical response [9].
In inverse problems, the geometry parameters of a periodic structure, such as the line width, sidewall angle (SWA), and thickness, are determined by analyzing the reflection spectrum of the periodic structure. To solve the inverse problem, there is a matching calculation which compares the measured reflection spectrum with the simulated reflection spectrum [10]. The minimized mean square error (MSE) or root mean square error (RMSE) obtained by the point-by-point comparison is commonly used as a metric for the best matching criterion. Accordingly, various linear and nonlinear regression methods have been applied to the inverse problem solving, and the earliest method used was a library search. A library was first created with theoretical spectra that were pre-generated according to a predefined parameter range of the model [11,12]. By searching in the library, the best matching solution to the experimental spectrum was obtained. Numerical optimization methods were subsequently proposed and applied to inverse problems, such as the Levenberg-Marquardt algorithm [13,14], genetic algorithms [15], and linear regression models [16,17]. In recent years, with the development of machine learning, significant progress has been made using neural networks such as artificial neural networks and convolutional neural networks [18–20].
However, most previous studies used data from the entire measurement range to fit the measured and simulated data. With a high-resolution camera or spectrometer, one measured data point contained more than 106 data points. It is good to have high accuracy; however, it requires a large storage space and high data transfer capability, and the most critical issue is that it increases the data fitting time. Therefore, a downsampling method is required to reduce the number of data points while maintaining suitable accuracy. The use of sensitivity screening data could solve this problem. Sensitivity has diverse definitions in different metrology systems and fields [21,22]. In OCD, sensitivity can be defined as the effect of a change in the input parameter on the output spectral data. High sensitivity reflects the property of parameters of interest, and only minor adjustments of input parameters can result in significant changes in the output data [23–25]. This high sensitivity is especially helpful to achieve robustness against noise in the measurement system [26]. In addition, high sensitivity makes it easier to detect small changes in size. Research on sensitivity analysis has also been widely conducted, e.g., evaluation of the sensitivity of parameters for optimal measurement configuration [27,28] and mode simplification by feature selection [29]. In many previous studies, optimization was realized by local sensitivity analysis (LSA), an approach that focuses on one parameter of interest at a time [30–32]. This method ignores the correlation between different parameters and their non-linear coupling effects on measurement characteristics.
In this paper, we propose a new sensitivity operator that can filter high-sensitivity data from full data to perform matching calculations. This sensitivity operator borrows the concept of feature extraction from the Laplacian operator in the field of image processing. Compared with traditional sensitivity analysis, it can extract the data with the most significant changes more accurately and exhibits a strong response to details such as isolated points and narrow lines. This operator can also analyze the variation in scattering characteristics in two dimensions simultaneously, considering the correlation between two different parameters. Through sensitivity calculation, continuous or discrete values of the input parameters that cause significant changes in the scattering spectrum can be found. Then, these most sensitive data are used for the matching calculation, which greatly reduces the amount of data and speeds up the fitting process. For metrology systems, this is not limited by optimal measurement configuration. Besides, it saves storage space in the device, enabling accurate reconstruction of the target structure despite the use of a small amount of data.
2. Feature data algorithms
2.1 Sensitivity operator
For sensitivity analysis in scatterometry, sensitivity is usually approximated by first-order differentiation of the measured feature with respect to a single parameter:
The parameter is x and the measurement feature is the function $f(x )$. However, first-order differentiation focuses only on one parameter of interest at a time, ignoring the correlation between different parameters and non-linear coupling effects on the measured variation. Moreover, first-order differentiation is less effective in handling detail enhancement and has a weaker response to isolated points [33]. Therefore, a sensitivity-operator-based metrology method is an effective approach to solve this problem.
The Laplacian operator is an isotropic differential operator often used in digital image processing. It has a strong edge localization capability that can enhance data details by highlighting grayscale areas containing abrupt changes and weakening grayscale areas featuring smooth changes [34]. The Laplacian operator in a binary image function $f({x,y} )$ is defined as follows:
By defining second-order differentiation in terms of differences, the two-dimensional Laplace numerical implementation can be expressed as follows:
With multiple parameters, the Laplace operator can be expressed as:
However, in actual optical measurements, the change trend of the characteristic response is complex owing to the coupling influence of multi-dimensional parameters. By combining Laplace operators, we propose the following sensitivity operator:
where r is the collected reflectance data, and ${P_1}$, ${P_2}$, …, ${P_n}$ are variable parameters, such as the wavelength or angle of the incident light and critical dimension of the sample.In OCD measurements, the sensitivity operator not only enhances the response details of the discrete spectrum, which is the case of the Laplacian operator, but also analyzes the interaction between different parameters. With this operator, we can find spectral data that are strongly responsive to changes in input parameters, thereby obtaining input parameter values with high sensitivity. In this study, one of the parameters that changes is the angle or wavelength of the incident light; the remaining parameters are structural parameters of the sample.
2.2 Matching algorithm
To reduce the storage pressure and computation time, our fitting algorithm uses the aforementioned sensitivity operator to filter parameter datasets with high sensitivity from the raw spectral data for accurate matching.
Consider a variable wavelength measurement as an example. First, we need to determine the approximate size of the target structure, wavelength range of the incident light, and key size interval, which varies with a certain step. Then, we generate the corresponding theoretical spectral database of variable wavelength. Each set of spectral data ${R_i} = ({{r_1},{r_2},{r_3}, \ldots \ldots ,{r_m}} ){\; }$ in the simulation database contains information on the full measurement band, which corresponds to the simulation geometry model individually. A database consisting of multiple datasets can be regarded as a matrix R defined as $R = \{{{R_1},{R_2},{R_3}, \ldots \ldots ,{R_n}} \}$. Then, the sensitivity operator is used to calculate the sensitivity of matrix R, normalize it, and finally obtain a sensitivity matrix $S = \{{{S_1},{S_2},{S_3}, \ldots \ldots ,{S_n}} \}$ corresponding to the original theoretical spectral database. Concerning wavelengths, the data with high sensitivity indicate that the incident light at these wavelengths can produce a significant response to parameter changes of the target structure; therefore, matching based on feature data of these wavelengths can accurately find the best set of spectral data and hence the most suitable geometric model. Figure 1 illustrates the framework for applying our sensitivity operator to scatterometry.
Fig. 1. A matching framework applied to scattering measurements. There are four modes for filtering feature data using sensitivity operators: Feature range, Discrete feature data, Hybrid, and Full range.
We explored four modes for selecting the feature data. Mode 1: Feature range. For all spectral data sets, according to the concentrated area of the data points with the highest sensitivity, a wavelength or angle range is selected and defined as the feature range. Mode 2: Discrete feature data. For each group of spectral data, a certain number of data points with the highest sensitivity are selected and defined as discrete feature data. Mode 3: Hybrid. Take all the points in the feature range and the remaining certain number of data points with the highest sensitivity to combine the feature range and discrete feature data. Mode 4: Full range. Use all the data in the measurement range, which is used to compare the results of other models. In the experiments described next, there were 1280 points of angle data and 1000 points of wavelength data. Both Modes 1 and 2 selected 300 points of data, reducing the data by 70% approximately, while Mode 3 selected 600 points, reducing the data by 50% approximately.
Each set of selected feature data is treated as a new feature spectrum $R_i^\ast{=} ({r_1^\ast ,r_2^\ast , \ldots \ldots ,r_j^\ast } )$, which is defined as a feature spectrum. These feature spectra are matched to the experimental spectrum of the target structure ${R^e} = ({r_1^e,r_2^e, \ldots \ldots ,r_m^e} )$. The metric of the best matching criterion is defined as matching error.
3. Simulated experimental results
3.1 Scatterometry measuring setups and three basic test samples
Spectral data of target structures in OCD are generally detected using scattering measurement devices. Two basic devices are shown in Fig. 2. Figure 2(a) depicts a schematic diagram of an angular scatterometry. The light source emits laser beams with different incidence angles that reach the sample surface through optical elements and are then diffracted, and the detector collects the diffracted light. The wavelength of the incident light is a parameter of the experiment and remains constant during the measurement; the spectral data are a function of the angle. Given that the 0-th order usually includes the maximum intensity and is a good signal to measure, 0-th-order diffraction data are generally used for subsequent analysis [10,35]. In spectral scatterometry (Fig. 2(b)), the fixed parameter is the angle of the incident light source whereas the variable parameter is the wavelength of the light source. Consequently, the final optical response is a function of the wavelength. In general, spectral scatterometry uses a broadband light source, followed by a monochromator to separate light within a certain wavelength range. The polarization state of the incident light is modulated using multiple polarizers. Similar to angular scatterometry, spectral scatterometry typically uses the 0-th-order diffraction method. In this study, the variable angle ranged from 0° to 64° to represent a numerical aperture of 0.9, and the variable wavelength ranged from 200 to 1200 nm.
Fig. 2. Scatterometry and test models: (a) angular scatterometry, (b) spectral scatterometry. (c) thin film; (d) 1D grating, (e) 2D grating, (f) critical dimensions of the gratings: pitch, height, sidewall angle (SWA), Top CD, Bot CD, etc.
The data discussed in this paper were obtained using Rsoft DiffractMOD, which is a design tool for diffractive optical structures, such as gratings, photonic bandgap crystals, and sub-wavelength periodic structures. DiffractMOD is capable of yielding all-vector solutions to Maxwell's equations. Its core algorithm is based on rigorous coupled wave analysis and is enhanced by the mode transmission line theory.
In order to simulate the effects of manufacturing errors and measurement noise, we add 20% random noise to the theoretical spectrum data R to generate four groups of spectral data with noise [36]. Then, we superimposed the four noisy spectra randomly to form a final experimental spectra ${R^e}$. In this method, the underlying condition is that the random weights ${w_i}$ of the four spectra amount to 1 [37]. Therefore:
with
We selected three basic structures as test samples and verified some parameters of these structures with sensitivity operators. These parameters of interest are critical dimensions that are relatively unstable in the manufacturing process, such as film thickness and Top CD of grating, and accurate dimensional measurements of them are very important in applications such as thin film optical coating and semiconductor devices [38,39]. Initially, we used a simple thin film structure (Fig. 2(c)) for verification by varying the thickness of the film. Next, we extended our conclusions to reconstruction of one-dimensional (Fig. 2(d)) and two-dimensional (Fig. 2(e)) gratings. Important parameters of the gratings are shown in Fig. 2(f). Some of the parameters of each structure as well as optical properties of the materials (refractive index n and extinction coefficient k) are listed in Table 1.
Table 1. Details of the test samples
3.2 Results and discussion
3.2.1 Thin films
Regarding thin films, we verified the spectral data at various angles. The incident light was transverse electric (TE) polarized, and the fixed wavelength was set to 900 nm because of its high sensitivity. Figure 3(a) shows the theoretical data for an Si3N4 film with thickness and incident angle set as variables. This indicates that the spectral data are sensitive to angles from approximately 40° to 64°. Figure 3(b) depicts the sensitivity image calculated by the sensitivity operator, whereas Fig. 3(c) shows the image of the data after first-order differentiation of the thickness. It can be concluded by comparing Figs. 3(b) and 3(c) that first-order differentiation produces wider edges whereas the sensitivity operator can further enhance the details with a double response to changes in two-dimensional data. In addition, both plots clearly show the same range of features, with the thickness sensitivity reaching its maximum in the range 49°-64°. We performed variable wavelength simulations for the same thin-film model with a fixed incident light angle of 20° and TE polarization. The results show that the sensitivity of the feature data to thickness and wavelength ranges approximately from 200 nm to 800 nm. Processing with the sensitivity operator and first-order differentiation was then carried out. The results are shown in Figs. 3(d)-(f). The sensitivity operator and first-order differentiation exhibit the same feature region, reaching the maximum sensitivity within the wavelength range, i.e., 200-500 nm.
Fig. 3. Data of Si3N4 thin films with different thicknesses; the thickness of the films varied from 50 to 150 nm: (a) Variable angle spectral data; the incident light was TE-polarized with a fixed wavelength of 900 nm. Sensitivity data processed by both methods on these spectral data; (b) results of the sensitivity operator; (c) results of the first-order differentiation; (d) variable wavelength spectral data, the incident light was TE-polarized with a fixed angle of 20°. Sensitivity data processed by both methods on these spectral data: (e) results of the sensitivity operator; (f) results of first-order differentiation.
We then performed four mode matches for spectral profiles of multiple thicknesses and compared the effects of the sensitivity operator and first-order differentiation. Given that the feature regions obtained by the sensitivity operator and first-order differentiation were the same, we focused on the differences between the results of Modes 2 and 3. We compared the feature data obtained using the sensitivity operator and first-order differentiation. Figure 4(a) shows the variable angle data for a 60-nm-thick Si3N4 film whereas Fig. 4(b) depicts the variable wavelength data for a 120-nm-thick Si3N4 film; the blue line represents the respective spectral data of the full measurement range. It can be seen that the high-sensitivity data obtained by the sensitivity operator and first-order differentiation differ. To obtain a more significant difference, we verified several datasets and calculated the root mean square error (RMSE) of the matching results.
where N is the number of experimental spectra, $P_i^e$ is the structural parameters in each group of the experimental spectra, and $P_i^m$ is the structural parameters of the matching results. Figure 4(c) shows the matching results of the experimental data for 30 groups of variable angle films. As expected, the mode with the lowest RMSE is Mode 4 (Full range), 0.057 nm, followed by the RMSE of Mode 1 (Feature range), 0.086 nm. And the RMSE of Mode 3 (Hybrid) is less than that of Mode 2 (Discrete feature data). In Mode 2, the RMSE (0.172 nm) of the sensitivity operator is much lower than that of the first-order differential (0.246 nm). In Mode 3, the sensitivity operator RMSE (0.102 nm) is very close to the first-order differential (0.093 nm). Figure 4(d) depicts the results for 30 sets of variable wavelength data, which show the same trend. Mode 4 has the lowest RMSE of 0.037 nm, while Mode 1 has a slightly higher RMSE of 0.041 nm. In Mode 2, The RMSE of the sensitivity operator (0.048 nm) is much lower than that of the first-order differential (0.061 nm). In Mode 3, the RMSE of the two is very close, with 0.055 nm and 0.052 nm, respectively.Fig. 4. Comparison of high sensitivity data obtained by the sensitivity operator and first-order differentiation: (a) Discrete feature data mode. The full-angle spectra of the Si3N4 film with a thickness of 60 nm is represented in blue. (b) Hybrid mode. The full-band spectra of the Si3N4 film with a thickness of 120 nm is represented in blue. RMSE between experimental data and best matching results for Si3N4 films: (c) 30 sets of variable angle data, (d) 30 sets of variable wavelength data.
3.2.2 One-dimensional and two-dimensional gratings
To explore the effect of the sensitivity operator, 1D and 2D grating models were used for further validation. Table 2 lists the details of the model parameter variations for variable angle measurements. For the 1D grating, in case of variable angle, the incident beam was set to TE polarization at a fixed wavelength of 600 nm. In case of variable wavelength, the incident beam was set to TE polarization with a fixed angle of 30°. For the 2D grating, the incident beam was polarized at 20° for the variable angle case and the fixed wavelength was 960 nm. For variable wavelength, the incident beam was polarized at 45°, with a fixed angle of 30°.
Table 2. Parameter settings for the grating models.
Figure 5 shows a schematic diagram of the 1D grating data. The Top CD of the grating varied from 450 nm to 500 nm in steps of 0.05 nm, the Bot CD maintained the same size, and the SWA changed correspondingly. Figures 5(a), (b), and (c) show the results for variable angle. According to Figs. 5(b) and (c), angles from 20° to 35° should be selected as the feature range. Figures 5(d), (e), and (f) show the results for variable wavelength. The data show that 200-500 nm is the feature range with high sensitivity for almost all variations.
Fig. 5. One-dimensional grating data for Top CD varying from 450 to 500 nm: (a) spectral data for variable angle; the incident light was TE polarized with a fixed wavelength of 600 nm. Sensitivity data were processed with two methods and these spectral data: (b) results of the sensitivity operator; (c) results of first-order differentiation; (d) variable wavelength spectral data for which the incident light was TE polarized at a fixed angle of 30°. The sensitivity data of these spectral data were processed with two methods: (e) results of the sensitivity operator; (f) results of first-order differentiation.
Figure 6 shows a schematic diagram of the 2D-grating data. The Top CD of the gratings varied from 500 nm to 540 nm in steps of 0.04 nm, the Bot CD maintained the same size, and the SWA changed correspondingly. Figures 6(a), (b), and (c) show the results for variable angle, whereas Figs. 6(d), (e), and (f) show the results for variable wavelength. It is evident from Figs. 6(b) and (c) that 25-40° is the feature range with highest sensitivity. Likewise, Figs. 6(e) and (f) indicate that 900-1200 nm is the feature range with the highest sensitivity.
Fig. 6. Two-dimensional grating data with Top CD ranging from 500 nm to 540 nm: (a) spectral data for variable angle; the incident light had a polarization angle of 20° and a fixed wavelength of 960 nm. Sensitivity data processed using two approaches and these spectral data: (b) results of the sensitivity operator; (c) results of first-order differentiation. (d) Spectral data of variable wavelength with an incident light polarization angle of 45° and an incidence angle of 30°. The sensitivity data of these spectral data were processed using two approaches: (e) results of the sensitivity operator; (f) results of first-order differentiation.
To test the feature data, we selected 30 sets of spectral data from each of the aforementioned types of gratings. Noise was added to these spectral data to generate experimental data, and the best model was searched in four modes. The sensitivity operator and first-order differentiation effects were also compared. Table 3 presents the matching performance of the four modes. The performance results show that the feature data obtained by the sensitivity operator and first-order differentiation exhibit satisfactory prediction accuracy, and the RMSE was less than 0.6 nm. For Modes 1 (Feature range) and 4 (Full range) and for the 1D and 2D gratings, the sensitivity operator and first-order differentiation selected the same range of data, and the RMSE was the same; therefore, here we only focus on Modes 2 and 3. In Mode 2 (Discrete feature data) and Mode 3 (Hybrid) for 1D-grating, the RMSE of the sensitivity operator was lower than the first-order differentiation, whether it was variable angle or variable wavelength data. For 2D-grating, the sensitivity operator was less effective for variable angle data, possibly because the sensitivity of the structure to Top CD and angle was not significant, as shown in Fig. 6(b). For variable wavelength, in Mode 3, the RMSE of the sensitivity operator was lower than that of the first-order differentiation by 0.014 nm. Therefore, it can be concluded that, in most cases, the sensitivity operator works better than the first-order differentiation on the structure reconstruction.
Table 3. RMSE of 1D and 2D gratings
3.3 Multi-parameter analysis
To verify the effect of the sensitivity operator for multiple parameters, we chose a state of art vertical gate-all-around (GAA) device for the analysis, which was introduced by SUNY Polytechnic Institute, College of Nanoscale Sciences group for optical measurement of feature dimensions and shapes research purpose [40]. Figure 7 shows the nanodiffract model of the vertical GAA structure at the dummy a-Si etch back step. The simulated structure has a pitch of 40 nm, nanowire height of 120 nm, SiO2 hardmask height of 10 nm, and Si3N4 thickness of 30 nm. In fabrication, the parameters of interest are the bottom and top critical dimensions of the nanowire and the thickness of the amorphous silicon (a-Si) after etching, so they were set as variables. The critical dimension (CD) of the nanowires ranges from 10 to 20 nm with 0.5 nm steps, and the thickness of the a-Si ranges from 50 to 70 nm with 1 nm steps.
Fig. 7. NanoDiffract model of the vertical GAA structure at the dummy a-Si etch back step. (Adapted from Ref. [40], Fig. 13)
Next, the vertical GAA was measured at variable angle and variable wavelength. At variable angle, the incident light polarization was 0° and the fixed wavelength was 500 nm. And at variable wavelength, the incident light polarization was 0° and the fixed angle was 60°. The spectral data were processed with the sensitivity operator and the first-order differentiation, respectively. In order to display the data better, high-sensitivity data are screened by threshold and shown in Fig. 8. Figures 8(a), (b), and (c) show the results for variable angle, and 0-15° is the feature range with high sensitivity. Figures 8(d), (e), and (f) show the results of variable wavelength. Different from the previous structure, the data of sensitivity operator and first-order differentiation of a-Si thickness show a feature range of 200-500 nm, and the nanowire CD shows a feature range of 400-700 nm.
Fig. 8. Vertical GAA sensitivity data filtered by threshold, nanowire CD ranging from 10 nm to 20 nm and a-Si thickness ranging from 50 nm to 70 nm. Variable angle with polarization angle of 0° and fixed wavelength of 500 nm: (a) results of the sensitivity operator; (b) the first-order differentiation results of nanowire CD; (c) the first-order differentiation results of a-Si thickness; Variable wavelength with polarization angle of 0° and fixed angle of 60°: (d) results of the sensitivity operator; (e) the first-order differentiation results of nanowire CD; (f) the first-order differentiation resuts of a-Si thickness.
We randomly selected 30 sets of spectral data with variable angle and wavelength, and performed matching calculations in four modes based on sensitivity data of sensitivity operator and first-order differentiation. Table 4 shows the RMSE of the matching results. At variable angle, for the nanowire CD, in Mode 2 (Discrete feature data), the sensitivity operator is lower than the first-order differentiation of nanowire CD and first-order differentiation of a-Si thickness by 0.129 nm respectively. For the a-Si thickness, the RMSE of Mode 2 (Discrete feature data) and Mode 3 (Hybrid) for the sensitivity operator (0.983 nm and 1.017 nm, respectively) were lower than that of first-order differentiation of nanowire CD (1.643 nm and 1.633 nm, respectively), and also lower than that of the first-order differentiation of a-Si thickness (1.975 nm and 1.592 nm, respectively). Compared with the first-order differentiation, the sensitivity operator reduced the RMSE by 50% at most. For the variable wavelength, the RMSE between the sensitivity operator and the first-order differential is not significantly different. From that, we can conclude that for the multi-parameter sensitivity analysis of the structure, the sensitivity operator is better than the first-order differentiation.
Table 4. RMSE of vertical GAA
4. Summary
We propose a method for screening feature spectral data based on a Laplace sensitivity operator. This method is used to determine the influence of input parameters on the output optical spectrum and extract a small number of high-sensitivity data points from a large number of measured spectrum data points. To verify the feasibility and correctness of the proposed method, three basic devices were selected for single-parameter analysis: thin-film, 1D and 2D gratings, and vertical GAA was used for multi-parameter analysis. It was found that in the process of matching with feature range, despite a data compression of 70%, the RMSE didn't increase significantly, and the computational speed was increased by a factor of 2.4. For the single parameter analysis of thin film and grating structure, the RMSE of sensitivity operator is lower than the first order differential in most cases. For multi-parameter analysis of vertical GAA structures, RMSE of sensitivity operators is significantly lower than that of first-order differentiation, up to 50% reduction. It is efficient to use high-sensitivity operators for model reconstruction.
Funding
National Key Research and Development Program of China (2022YFB3206000); Key Research and Development Program of Hubei (2021BAA173).
Acknowledgments
The authors acknowledge support from National Key R&D Program of China and Key R&D program of Hubei.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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