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Abstract
Profile measurements of structures with a high aspect ratio and subwavelength features (HARSW) can be achieved using transmission electron microscopy and tilted scanning electron microscopy. Although electron microscopy can provide accurate HARSW measurements, it is laborious and destructive. In this paper, nondestructive and labor-saving methods were proposed to measure the dimensions of HARSW structures. The optical reflection spectrum, along with an artificial neural network (ANN) model, was adopted for interpolation with the simulation database to retrieve the dimensions of HARSW structures. To generate the ANN model, the experimental and simulated reflection spectra were adopted as the input and output variables for the training data, respectively. This ANN model can learn the discrepancy between simulation and experimental reflections. The finite-difference time-domain method was also adopted to calculate the simulated reflection spectra of HARSW structures with various dimensions, which can be used as a database. Once the experimental reflection of a HARSW structure with unknown dimensions was obtained, the ANN model could generate a simulation-like reflection spectrum. Linear regression was used to determine the correlation coefficients of the simulation-like reflection spectra in the database. The accurate dimensions of HARSW structures can be determined using a higher correlation coefficient. This methodology can be a prominent method for the process monitoring of HARSW structures.
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1. Introduction
Process monitoring is vital for the advancement of semiconductor fabrication. Tilted scanning electron microscopy (SEM) and transmission electron microscopy (TEM) have been used traditionally to obtain the profile information of structure with a high aspect ratio and subwavelength features (HARSW). Both methods can provide accurate profile information for HARSW structures; however, they still have considerable restrictions and limitations in process monitoring. In procedures using TEM and tilted SEM, characterizing the critical dimensions (CD) of the HARSW structure results in damage to the structure. Measurements using electron beams are complex, destructive, and time-consuming. In-line process monitoring requires more efficient measurements, and the abovementioned methods are unsuitable.
In semiconductor processes, the most appropriate process monitoring is based on optical measurement, which is a nondestructive and efficient method. However, the operating wavelength in optical measurements is constrained for HARSW structures because of strong light diffraction in subwavelength features. It is difficult to obtain an accurate CD of HARSW structures by capturing optical images directly. Optical critical dimension (OCD) metrology has been extensively adopted to monitor the CD of nanostructures [1,2]. In OCD metrology, a Mueller matrix based on ellipsometry [3] has also been adopted to monitor changes in the CD of nanostructures via the optical spectrum. The current OCD metrology can provide high measurement accuracy (< 3%), but it is not suitable for measuring the features of HARSW structures. Current OCD metrology addresses nanodevice with a small height (< 200 nm). The corresponding reflection is characterized by a monotonic signal. The nanodevices measured using OCD metrology are usually silicon-based materials. This indicates that the incident light is absorbed to a lesser extent in the OCD system. The relationship between the reflection features and the CD of the nanostructures can be established using a simple model in OCD metrology. The ellipsometry with the Muller matrix also has a more complex optical configuration, which needs to change polarization and rotate the sample. In this study, the height of HARSW structures is a few micrometers (> 1.5 µm), and the optical reflection of HARSW structures is a complicated oscillation signal because of the Fabry–Perot resonance [4,5]. The forward and backward light beams interfere with each other to form a resonance cavity in the HARSW structure. Moreover, the material of the HARSW structure is the photoresist (PR) that can absorb light. Determining the correlation between the HARSW profile and the corresponding reflection spectrum is an arduous problem.
Machine learning was studied using optical spectroscopy to extract complex physical and chemical properties. Near-field spectroscopy can resolve material properties via machine learning, and the physics-infused neural network is aware of the tip–sample interaction [6,7]. The chemical composition can be identified using a detector array and machine-learning classification [8,9]. An ellipsometer can also be empowered by neural networks for the accurate characterization of thin films [10–13]. In this paper, a machine-learning model based on an artificial neural network (ANN) and reflection spectra is proposed to retrieve the dimensions of HARSW structures. The ANN model was established by adopting the experimental reflection spectra, refractive index, and extinction coefficient as input variables. The corresponding reflection spectrum obtained using finite-difference time-domain (FDTD) simulation was the output variable in the training process of the ANN model. In principle, the experimental reflection should be equal to the simulated reflection for the same HARSW structure. However, there is a gap between the experimental and simulation reflections in practical situations. This gap is induced by the limitations of optical measurement and fabrication. At the simulation level of this study, the HARSW structure is treated as a two-dimensional (2D) periodic array with infinite length, and the reflectance is collected for all reflected diffraction orders. In this study, the HARSW structure was fabricated in a finite area using an electron beam (e-beam) writer. The optical behavior is different with an infinite periodic array because of localized effects [14,15]. Moreover, the HARSW structure was characterized using an optical microscope equipped with a spectrometer. This indicates that an objective lens was used to collect the optical reflection from the spectrometer. The objective lens has a specific numerical aperture, and its capability to collect light is restricted to the reflected diffraction light. This discrepancy in the experimental procedure caused an error between the experimental and simulation results.
In this study, the experimental optical spectrum of a HARSW structure with undetermined dimensions can be converted into a physical-aware (simulation-like) reflection spectrum using the proposed ANN model. The FDTD method was adopted to calculate the simulated reflection spectra for various dimensions of the HARSW structure. Each simulated reflection can be labeled with a specific CD of the HARSW structure. These simulation results can act as a lookup table that provides leverage between the simulation-like reflection and FDTD results. The dimensions of the HARSW structure can be determined by finding the best correlation coefficient (R2) between the simulation-like reflection and the database of FDTD results. The term “simulation-like reflection” refers to a reflection generated not by the rigorous FDTD simulation but by the ANN model. To enhance the diversity and robustness of the ANN model, positive and negative PRs were adopted to fabricate HARSW structures using e-beam lithography. These two PRs exhibited complementary structures after exposure and development. This indicates that various reflection spectra can be obtained to train the ANN model using HARSW structures with a wide variety of geometries. This generates diverse experimental data that can be used to improve the accuracy of the ANN model.
2. Optical behavior of HARSW structures with various geometries
Figure 1(a) shows a schematic of the HARSW structure, which is a trapezoidal cylinder arranged in a 2D periodic array. The features of optical reflection vary with the top width (TW), bottom width (BW), and height (H) of the HARSW structure. The optical behavior of the HARSW structure is shown in Fig. 1(b). The incident light was polarized in the linear X-direction, and the material properties of PMMA (e-beam PR) were adopted for the PR, as shown in Fig. 1(b). The details of the simulation method (mesh and region sizes and optical properties) are listed in Supplement 1 (Figs. S1 and S2, Table S1). The solid lines of different colors represent the reflection spectra with different HARSW profiles in the lower part of Fig. 1(b). In the spectrum, the wavelengths corresponding to multiple peaks are the wavelengths of incident light that can produce constructive interference in the HARSW structure. Optical reflection revealed that HARSW structures with different profiles exhibited peak variations and shifts in the optical spectra. The change in the reflection features can be attributed to the optical modes in the different HARSW profiles. The upper part of Fig. 1(b) shows the optical mode of the HARSW structure. The optical intensity was concentrated in different vertical regions for the HARSW structures with vertical, trapezoidal, and inverted trapezoid shapes. The reflectance via the wavelength correlates with the shape of the HARSW structure. The dotted line in Fig. 1(b) represents the optical spectrum of a subwavelength structure with a small height (H: 200 nm). The dotted line is smoother than the solid lines. This reveals that the optical reflection of the HARSW structure is a complicated oscillation curve because of the induced cavity mode. In the optical spectrum shown in Fig. 1(b), it is difficult to identify the features of the change in the profile information of the HARSW structure by tracking the wavelength.
Fig. 1. (a) Schematic of a 2D periodic HARSW structure. The periodicity is 700 nm in both directions. The substrate of the HARSW structure is silicon. (b) The upper part is the distribution of optical intensity along the X-Z plane for different HARSW structures, and the wavelength of incident light is 532 nm. The color bar shown in the simulation represents the normalized intensity. The reflection spectra of various HARSW structures are shown in the lower part.
3. Methodology of measuring HARSW profile
Figure 2 shows the schematic of the strategy adopted in this study. The reflectance between the experiment and simulation shows an intrinsic discrepancy in how the spectra shift and change with wavelength. As shown in Fig. 2, an ANN model was adopted to compensate for the error between the experimental and simulated reflections. In this ANN model, the input layer Xn and output layer Yn can be described by the following equations:
where λn is the wavelength of the incident light, Ren is the experimental reflectance of the HARSW structure, Nn is the refraction index of the PR, Kn is the extinction coefficient of the PR, and Rsn is the simulation reflectance of the HARSW structure. In Eqs. (1) and (2), Ren, Nn, Kn, and Rsn are functions of λn. To build an ANN model with a perceptron in the optical spectrum, each layer was interconnected with artificial neurons. The weights and biases were iteratively updated during model training using a backpropagation algorithm [16,17]. In the training procedure of the ANN model, the optical properties (Nn and Kn) of the PR were used to enable the ANN model to recognize the type of PR (negative- or positive-tone resists). These optical properties were obtained by the ellipsometer (Fig. S2 in Supplement 1). Four hidden layers were used. The numbers of neurons were 1024, 512, 256, and 128 in the first, second, third, and fourth hidden layers, respectively. A rectified linear unit was adopted as the activation function in the ANN model.Fig. 2. Strategy of retrieving HARSW profile. The color bar shown in the simulation database represents the reflectance.
The measured optical reflection was converted into a simulation-like reflection using the ANN model. The simulation database shown in Fig. 2 can be interpolated with the simulation-like reflection to measure the profile of the HARSW structure with linear regression. The simulation database shown in Fig. 2 is a three-dimensional dataset, which is constructed using the TW, BW, and wavelength. Each optical spectrum in the simulation database can be associated with a specific TW and BW for the HARSW structure. Once the experimental reflection was obtained, it could be converted into a corresponding simulation-like reflection using the ANN model. Some correlation coefficients can be defined by simulation-like reflections and each reflection of the simulation database is based on the wavelength. The TW and BW of the HARSW structure can be determined by finding the best correlation coefficient through the simulation database and simulation-like reflection. This methodology can provide another route via the use of the experimental optical reflection, ANN model, and simulation database to obtain HARSW profile information.
4. Experimental results and discussion
An e-beam writer (ELS-7500EX, Elionix) was used for the fabrication of HARSW structures. Positive (950 PMMA A9, MicroChem) and negative (maN-2410, Microresist) PRs were selected as the HARSW materials. The detailed procedure for the e-beam lithography is described in Supplement 1 (Table S2). The regions of the remaining PR in the positive and negative PRs were opposite under electron-beam irradiation. Complementary PR structures that can provide optical reflection with a greater variety of experimental data can be obtained. During the training of an ANN model, diverse data can make the model more robust. SEM images (FEI, Helios G3 CX) of the positive and negative PRs are shown in Fig. 3. The profiles of the positive and negative PRs are shown in Fig. 3. Trapezoidal and inverted trapezoidal shapes were obtained using negative and positive PRs, respectively. In this study, the thickness of the PR and the pitch of the HARSW structures were maintained constant at 1800nm and 700 nm for both PRs, respectively. The dosage of the e-beam writer and the designed geometry were altered to obtain HARSW structures with various TW and BW values. Sixteen types (PMMA: 9, maN: 7) of HARSW structures were fabricated using the e-beam writer. Each HARSW structure had a different TW and BW. Nine types of reflection spectra (PMMA: 5, maN: 4) were adopted as training data for the ANN model. Each reflection spectrum had 126 datasets, in which the reflectance corresponded to wavelengths from 450 to 800 nm. The reflection spectra of the remaining seven types (PMMA: 4 and maN: 3) were used to validate the ANN model. Table S3 of Supplement 1 shows the HARSW structure used to train and validate the ANN model. In the training data, there are 9 × 126 × 4 data points (nine types of spectral data and two types of PR material in refraction index and extinction coefficient). In this study, the top and bottom CDs are considered as the averaged top and bottom CDs across the sample. Five positions in the measurement area are taken to obtain the average data for the top and bottom CDs to eliminate the effect of CD variation in the training and verification data.
Fig. 3. SEM images (viewing angle is 52°) of HARSW structures with negative (maN) and positive (PMMA) PRs. The scale bar is 1 µm in both SEM images. For the maN-based HARSW structure, the corresponding TW and BW are 410 nm and 450 nm, respectively. For the PMMA-based HARSW structure, the corresponding TW and BW are 480 nm and 350 nm, respectively.
The spectra of the experimental, simulated, and simulation-like reflections are shown in Fig. 4. The experimental reflection was captured using an inverted microscope (Olympus, IX73) with an objective lens of magnification 50× (NA: 0.8). The optical microscopy system was equipped with a spectrometer (ANDOR, Kymera 328i) and a charge-coupled device (ANDOR, DU416A). The correlation coefficients between the experimental and simulated reflections were relatively low, as shown in Fig. 4 (R2 < 0.5). The discrepancy between the experiment and simulation is apparent because of restrictions on fabrication and measurement. The simulation-like reflections (dotted black lines) converted by the ANN model adopted the experimental reflections as the input, as shown in the left part of Fig. 4. In the right part of Fig. 4, the figures depict the similarities in simulation reflection to simulation-like or experimental reflections. Each point on the X and Y axes represents the simulation reflectance and simulation-like or experimental reflectance at the same wavelength in the right part of Fig. 4. In Figs. 4(a) and 4(b), the simulation-like reflections exhibit a trend similar to that of the simulation reflections. The spectral data of the HARSW structures in Figs. 4(a) and 4(b) were used to train and validate the ANN model, respectively. Hence, R2 is larger in Fig. 4(a) between the simulation-like and simulated reflections. This indicates that simulation-like reflections can be adopted to find similar simulation reflections in the simulation database using regression analysis. Each simulation reflection in the database can be linked to a specific TW and BW for the HARSW structures. The measured reflection spectra were used to identify the TW and BW of the corresponding HARSW structures using the proposed methodology.
Fig. 4. Spectra (left) and regression analysis (right) for the different HARSW structures. The dotted black line in the optical spectrum is the simulation-like reflection converted by the ANN model. The simulation and experimental reflections are the training and validation data of the ANN model in (a) and (b), respectively.
Table 1 presents the comparisons between the actual and measured HARSW profiles. The actual profile of the HARSW structures was measured using tilted SEM. The predicted profiles were obtained using the ANN model and simulation database. In the simulation database, the range of TW and BW was 300 to 600 nm, and the interval was 5 nm for splitting the TW and BW. The HARSW profiles also show trapezoid (TW is less than BW) and inverted trapezoid (BW is less than TW) shapes for maN and PMMA, respectively. Each HARSW structure shown in Table 1 was first used to measure the optical reflection. Once the experimental spectrum was obtained, it was converted into a simulation-like spectrum using the ANN model. Subsequently, the simulation database was interpolated by a simulation-like reflection to obtain the corresponding TW and BW of the HARSW structure using regression analysis. The correlation coefficient R2 between the simulation-like and simulated reflections determined the TW and BW of the HARSW structure. As shown in Table 1, the predicted TW and BW values were close to the actual TW and BW values. The difference between the predicted and actual TW and BW values was less than 10%. Hence, this method can balance the accuracy and efficiency in the measurement. There is still room for improvement in accuracy in this study. The deep neural network with more precise experimental data could enhance the prediction accuracy. The accuracy of the ANN model was affected by the measured HARSW profile. In this study, tilted SEM was used to measure the dimensions of the HARSW structures. The measured TW and BW determined the output, which was a simulation reflection of the training data of the ANN model. The measurement error caused variations in the simulation reflection for the HARSW structures. This influenced the accuracy of the training process. More precise measurements in the training data can improve the accuracy of the ANN model.
Table 1. Actual and predicted TW and BW for the HARSW structures
5. Conclusion
The proposed methodology of building an ANN model and a simulation database can assist in obtaining the profile information of HARSW structures. Optical imaging makes it difficult to resolve the geometry of subwavelength structures owing to the limitations of optical diffraction. An ANN model was developed to learn the discrepancy between the experimental and simulation reflections. The simulation-like reflection converted using the ANN model was interpolated with the simulation database, and the corresponding TW and BW of the HARSW structures were obtained. Hence, the use of optical reflection can reveal the geometry of HARSW structures using spectral fingerprints. The measured accuracy of the HARSW profile obtained using the proposed method was lower than that of e-beam metrology. E-beam metrology is a powerful method for the characterization of precise geometries, but it is destructive and time-consuming. Therefore, it may not be an efficient method for monitoring mass fabrication processes. Optical measurement is an efficient and nondestructive method that is suitable for monitoring high-volume manufacturing processes. For a subwavelength structure with a high aspect ratio, the corresponding optical reflection, along with the ANN model, can be a potential method to obtain profile information accurately and in a timely manner.
Funding
National Science and Technology Council (112-2221-E-006 -119 -MY2).
Acknowledgments
The authors would like to thank the technical services (e-beam writer) provided by the “Core Facility Center of National Cheng”.
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 results.
Supplemental document
See Supplement 1 for supporting content.
References
1. H. J. Patrick, B. M. Barnes, T. A. Germer, et al., “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7(1), 013012 (2008). [CrossRef]
2. A. Vaid, C. Osorio, J. Tsai, et al., “Hybrid metrology co-optimization of critical dimension scanning electron microscope and optical critical dimension,” J. Micro/Nanolith. MEMS MOEMS 13(4), 041413 (2014). [CrossRef]
3. C. Guo, Y. Shi, H. Wu, et al., “Integrated optical critical dimension metrology with Mueller matrix ellipsometry,” Thin Solid Films 768, 139695 (2023). [CrossRef]
4. M. Menard and A. G. Kirk, “Design of Fabry-Perot filters in planar waveguides with deep-etched features for spatial switching,” Opt. Express 17(20), 17614–17629 (2009). [CrossRef]
5. C. P. Huang, X. G. Yin, Y. Zhang, et al., “Deep subwavelength Fabry-Perot-like resonances in a sandwiched reflection grating,” Phys. Rev. B 85(23), 235410 (2012). [CrossRef]
6. X. Chen, R. Ren, and M. Liu, “Validity of Machine Learning in the Quantitative Analysis of Complex Scanning Near-Field Optical Microscopy Signals Using Simulated Data,” Phys. Rev. Applied 15(1), 014001 (2021). [CrossRef]
7. X. Chen, Z. Yao, S. Xu, et al., “Hybrid Machine Learning for Scanning Near-Field Optical Spectroscopy,” ACS Photonics 8(10), 2987–2996 (2021). [CrossRef]
8. J. Meng, J. J. Cadusch, and K. B. Crozier, “Plasmonic Mid-Infrared Filter Array-Detector Array Chemical Classifier Based on Machine Learning,” ACS Photonics 8(2), 648–657 (2021). [CrossRef]
9. H. Tan, J. J. Cadusch, J. Meng, et al., “Genetic optimization of mid-infrared filters for a machine learning chemical classifier,” Opt. Express 30(11), 18330–18347 (2022). [CrossRef]
10. I. Kim, S. Gwak, Y. Bae, et al., “Optical spectrum augmentation for machine learning powered spectroscopic ellipsometry,” Opt. Express 30(10), 16909–16920 (2022). [CrossRef]
11. J. Liu, D. Zhang, D. Yu, et al., “Machine learning powered ellipsometry,” Light: Sci. Appl. 10(1), 55 (2021). [CrossRef]
12. P. Zhu, D. Zhang, X. Niu, et al., “A Lightweight Neural Network for Spectroscopic Ellipsometry Analysis,” Adv. Opt. Mater. 12(4), 2301381 (2024). [CrossRef]
13. Q. Kim, S. Lee, A. Ma, et al., “A simulation physics-guided neural network for predicting semiconductor structure with few experimental data,” Solid-State Electron. 201, 108568 (2023). [CrossRef]
14. L. Zundel and A. Manjavacas, “Finite-size effects on periodic arrays of nanostructures,” J. Phys. Photonics 1(1), 015004 (2018). [CrossRef]
15. F. Przybilla, A. Degiron, C. Genet, et al., “Efficiency and finite size effects in enhanced transmission through subwavelength apertures,” Opt. Express 16(13), 9571–9579 (2008). [CrossRef]
16. L. G. Wright, T. Onodera, M. M. Stein, et al., “Deep physical neural networks trained with backpropagation,” Nature 601(7894), 549–555 (2022). [CrossRef]
17. T. P. Lillicrap, A. Santoro, L. Marris, et al., “Backpropagation and the brain,” Nat. Rev. Neurosci 21(6), 335–346 (2020). [CrossRef]
