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Abstract
The effects of thick-mask and oblique incidence in extreme ultraviolet (EUV) lithography system make the aerial image calculation a challenging task. This paper develops a fast EUV lithography aerial image model based on a new kind of deep learning framework called adjoint fully convolutional network (AFCN). The AFCN consists of two adjoint data paths to respectively recover the real part and imaginary part of the complex mask diffraction-near-field (DNF). The feature-swapping technique is introduced to exchange the information between the real and imaginary data paths. The AFCN is trained based on a pre-calculated rigorous thick-mask DNF dataset. Given the estimated thick-mask DNF, the Abbe’s method is used to calculate the aerial image of the partially coherent lithography system. Compared to the traditional non-parametric kernel regression method, the proposed model reduces the error by more than 80% and achieves 25-fold improvement in computational efficiency.
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1. Introduction
Extreme ultraviolet (EUV) lithography with 13.5nm wavelength has become the most advanced semiconductor fabrication process for the 7nm and beyond technology nodes. EUV lithography system is designed based on the fully reflective optics with oblique incident illumination [1], as shown in Fig. 1(a). The layout pattern of integrated circuit is first fabricated on the reflective mask with 4× magnification. The illuminator reflects the EUV light rays, which then obliquely irradiate the mask. The light rays are diffracted by the EUV mask and form the diffraction-near-field (DNF) close to the top mask surface [2,3]. The mask DNF can be represented by four diffraction matrices, denoted by F(XX), F(XY), F(YX), and F(YY). The diffraction matrix F(UV) represents the complex amplitude of DNF with U-polarization, generated by a unit incident electric field with V-polarization, where U and V = X or Y. Then, the image of mask is transferred to the wafer by the reflective projection optics. After the development of photoresist coated on the wafer, the layout pattern is ultimately recorded by the photoresist profile.
In general, the overall lithography image process is described by the optical model and the resist model. The output of the optical model is the aerial image, which stands for the intensity image of the mask on the wafer [4]. It is also used as the input of the resist model. The calculation of aerial image is indispensable to most of the lithography simulation tasks. For example, some important resolution enhancement techniques of lithography system, such as the source mask optimization and scattering bar insertion methods, need to rely on the simulation of aerial image to find the optimal solutions [5–8].
Aerial image calculation can be separated into two parts: the first is the mask DNF simulation, and the second is the aerial image simulation based on the DNF [9]. The second step can be implemented by the Abbe’s model [10], and the mask DNF model has significant influence on the simulation accuracy and speed. Furthermore, the characteristics of EUV lithography system lead to notable thick-mask effects, making the calculation of EUV mask DNF even more challenging [11]. Rigorous electromagnetic field (EMF) simulation methods, such as the rigorous coupled wave analysis (RCWA) method [12] and the waveguide method [13], can accurately calculate the DNF of thick-mask. However, these methods are computationally intensive, thus not suitable for the full-chip level simulation. In the past few years, attempts have been made to accelerate the calculation of thick-mask DNF by using deep learning [14–16]. However, these methods were developed based on DUV lithography and the details of the algorithms were not discussed at length in [14–16]. For EUV lithography, Tanabe, et al. proposed a convolutional neural network method to predict the far-field of the thick-mask diffraction amplitudes [17]. However, the precision is limited by the linear approximation of the incident angle dependent amplitudes.
Recently, our group developed a non-parametric kernel regression method to accelerate the DNF computation for EUV lithography [18]. This method decomposes the layout pattern into smaller patches, then uses the non-parametric kernel regression algorithm to predict the local DNF for each patch based on the training library. The entire mask DNF is then synthesized by stitching up all of the local DNFs together. However, this method has to deal with the trade-off between the computational accuracy and efficiency, since the regression process will consume more time as the volume of training library increases. Besides, the mask decomposition process introduces an inevitable overhead to the global runtime.
This paper develops an adjoint fully convolutional network (AFCN) model to further improve the computational efficiency and accuracy of the thick-mask DNF and aerial image in EUV lithography system. To our best knowledge, this paper is the first to propose the AFCN model. The proposed method takes advantage of the powerful fully convolutional network (FCN) to capture the diffraction characteristics of the EUV thick-mask. The AFCN method owes its name to the unique architecture, which consists of two fully convolution data paths having the same hyperparameters. With the mask pattern as the input layout to the adjoint network, these two data paths respectively recover the real part and imaginary part of the complex diffraction matrix F(UV).
Because the thick-mask diffraction changes the phase of the incident light, the optical energy will exchange between the real part and imaginary part of the DNF. Thus, we further introduce a feature-swapping structure between the two data paths to take into account the interaction between them. At the front-end of the network, the “real” path (hereafter to be shorted as R-path) and the “imaginary” path (hereafter to be shorted as I-path) are parallelly constructed, which means that they will independently extract the feature maps from the input layout. This process increases the diversity of feature maps. On the contrary, at the back-end of the network, the R-path and I-path are cross-connected. Half of the feature maps in the R-path are transferred into the I-path, while half of the feature maps in the I-path are transferred into the R-path. Finally, the real part and imaginary part together constitute the complex amplitudes of the thick-mask DNF. The feature-swapping structure emulates the energy exchange between the real part and imagery part, so as to improve the prediction accuracy. Another contribution of the proposed method is to use the dilated convolution function to simulate the optical proximity effect [19].
To build up the training library for the AFCN model, a set of representative mask clips are selected as the training samples. The waveguide method is used to calculate the rigorous DNF data of the training masks under different point illuminations. After training, the network can rapidly estimate the DNF for an arbitrary thick-mask pattern. Then, the Abbe’s model is used to calculate the aerial image of EUV lithography system with partially coherent illumination. Simulations show that the proposed method can reduce the error of aerial image prediction by more than 80% compared to the non-parametric kernel regression method in [18]. In addition, the computational efficiency is improved by more than 600 times and 25 times in contrast to the waveguide method and the non-parametric kernel regression method, respectively.
In the remainder of this paper, the fundamentals of EUV thick-mask effects and FCN framework are introduced in Section 2. The proposed AFCN model is described in Section 3. In Section 4, the simulations are provided to demonstrate the proposed method, and compare it to other thick-mask models. Finally, the conclusion is provided in Section 5.
2. Preliminaries
In this section, the fundamental of EUV thick-mask effects is reviewed, and the background of FCN is introduced.
2.1 Fundamental EUV thick-mask effects
EUV lithography system is a fully reflective optical system, and the incidence light beams have an approximately 6° chief ray angle at the mask plane, rendering the system non-telecentric at the object side [1]. Figure 2 shows the structure of EUV mask. The mask is composed of Mo/Si multilayers and Ta-based absorber. The multilayers are designed to increase the reflectivity of the EUV light. The absorber layer is deposited on the reflective multilayer, which depicts the layout pattern to be printed. The typical thickness of the absorber is 60nm [20], which is about 5 times as the EUV wavelength, making the thick-mask effects noteworthy. The most notable effect due to the non-telecentricity is the shadowing effect, which becomes more pronounced as the increment of the absorber thickness [21].
For the EUV lithography system with 0.25 numerical aperture (NA), the deviation of the Kirchhoff approximate model, i.e., the thin-mask model, is no longer negligible [20]. Rigorous methods, including the RCWA and waveguide methods, solve the Maxwell’s equations in the spatial frequency domain, making them well suited to accurately simulate the thick-mask effects [22]. However, these methods are highly time-consuming, rendering them impractical to simulate a large-scale mask layout.
2.2 FCN framework
FCN is first introduced in 2015 as a powerful tool for the semantic segmentation application [23]. It enables an end-to-end pixel-wise data transformation. In this work, transformation means the mapping from the mask pattern to the corresponding diffraction matrices.
Generally, the FCN consists of a downsampling path and an upsampling path, as shown in Fig. 3. The downsampling path extracts and interprets the input data as multiple feature maps. Then, the upsampling path recovers the pixel-wise information based on these feature maps. Along the downsampling path, the number of feature maps is gradually increased and the size of feature maps is accordingly reduced, while along the upsampling path is the opposite. In the context of the thick-mask DNF calculation, these two paths first capture the geometric features of the mask pattern (input), then assign different weights to those features located at different spots, and finally recover the diffraction matrices (output).
The structure of FCN is based on the locally-connected layers, including the convolution layer, activation layer, pooling layer, and unpooling layer. The activation layer usually locates after the convolution layer to introduce nonlinearity, so as to improve the prediction capacity of the model. The pooling layer summarizes the features and reduces the dimension of feature map, which can reduce the computational complexity of the feedforward network. The unpooling layer, on the contrary, increases the dimension of feature map via interpolation, zero-paddling, or replication operation. In order to control the dimension of the output data, the pooling layers and unpooling layers should be configured intentionally.
3. Fast aerial image calculation based on the AFCN model
In this section, the AFCN method is developed to calculate the EUV aerial image efficiently and accurately. We first briefly describe the process to construct the training dataset in Section 3.1. The dataset includes the pre-calculated rigorous DNFs of the representative mask samples. In Section 3.2, we demonstrate the proposed adjoint network with the feature-swapping structure. The input and output of the network are the mask image and the diffraction matrices, respectively. Once the network is trained, the DNF of an arbitrary thick-mask can be rapidly estimated. Thereafter, the aerial image of the EUV lithography system is calculated based on the estimated DNF, which is described in Section 3.3.
3.1 Training dataset construction
Figure 4 shows the sketch of the training dataset in this work, where Fig. 4(a) is the pixelated source pattern. The diffraction behavior of the thick-mask is associated with the incident angle. Thus, we assign a set of equidistant reference points on a sparse mesh to cover the entire source pattern. The blue dots in Fig. 4(a) represent the sampling points. We also select a set of representative mask segments to cover the diverse geometries and environments of the layout features. For each pair of reference source point and mask segment, the thick-mask DNF is rigorously calculated using the waveguide method in Sentaurus Lithography software [24], and then stored in the training library.
Fig. 4. Sketch of the training dataset: (a) the reference points on the source pattern, (b) the representative mask segments with different features, and (c) the thick-mask DNFs associated with different pairs of reference source points and mask segments.
3.2 Architecture of the AFCN model
Figure 5 illustrates the architecture of the proposed AFCN model. The detailed structural parameters in each layer are provided in Table 5 of Appendix A. The input of the network is the mask image denoted by a binary matrix, where each entry is equal to 0.1 or 1, representing the reflective region and absorber region, respectively. The network consists of two adjoint paths, i.e., the R-path and I-path. The mask image is simultaneously inputted to both of the R-path and I-path, which respectively predict the real part and imaginary part of the diffraction matrix. The total number of layers in each path is 22, excluding the input layer and output layer. The Layer 1 to Layer 12 conduct the downsampling, while the Layer 13 to Layer 22 conduct the pling. Feature-swapping is implemented twice, as shown by the red and green arrows. The first one is in between Layer 18 and Layer 19, and the second one is in between Layer 21 and Layer 22. Next, we will firstly explain some notable layers in a single path. Then, we will further describe the feature-swapping mechanism between the two adjoint paths.
The “D. Conv” in Fig. 5 is short for the term of dilated convolution. Dilated convolution introduces an extra parameter called dilation rate to control the gap between adjacent pixels of the convolution kernel. Figure 6 shows three 3×3 convolution kernels with the dilation rate = 1, 2, and 3, respectively. The dilated convolution is used, since it conforms with the optical proximity effect (OPE), which is a prominent problem in optical lithography [15]. OPE implies that the diffraction spectrum and aerial image of a main pattern can be strongly impacted by its surrounding environment. Thus, proposed network should be able to process the neighboring pixels. The dilation rate of the kernel helps to enlarge and control the size of receptive field [25]. Compared to the regular convolution, the dilated convolution can obtain a larger receptive filed without increasing the number of trainable variables.
Figure 7(a) shows the detailed structure of Layer 1. The input image is convoluted using five parallel branches of dilated kernels with different dilation rates (dr). Each branch extracts the features of input image based on a different local view. After the convolution, the outputs of the five branches are concatenated together, and inputted to the next layer. Figure 7(b) shows the equivalent kernel considering all of the five branches, that is the superposition of the five dilated kernels in Fig. 7(a). It is observed that the equivalent kernel pays more attention to the central region and less to the far-end, which is in accordance with the fact that the influence of OPE decreases with distance.
Fig. 7. The sketches of (a) the Layer 1 in the network, and (b) the equivalent kernel considering all of the five parallel branches.
Layer 2 uses the rectified linear unit (ReLU) as the activation function to introduce nonlinearity [26]. The ReLU function is defined as:
The ReLU layer is commonly used after a convolution layer. Layer 3 has the same structure as Layer 1, which was mentioned above.
Layer 5 is a max pooling layer. The pooling size is 2×2 and the stride size is 2. Thus, after Layer 5 the size of the feature maps is reduced by 2 times compared to the input image. Thereafter, three sequential “Convolution-ReLU” combinations are used to extract features. In Layer 12, the pooling size is 3×3 and the stride size is 3. After Layer 12, the size of the feature maps is reduced by 6 times compared to the network input.
From Layer 13 to Layer 16, there are two “Convolution-ReLU” combinations. Layer 17 is an unpooling layer, where every pixel is replicated to fill in a 3×3 patch. Thus, the size of the feature maps increases by 3 times in Layer 17. Another layer that changes the feature size is Layer 20, where the unpooling size is 2×2. After Layer 20, the size of the feature maps is the same as the network input.
Since the optical energy will exchange between the real part and imaginary part of the thick-mask DNF, thus the interaction between them should be considered to improve the simulation accuracy. In the proposed network, a feature-swapping structure is implemented in between Layer 18 and Layer 19 for the first time, and in between Layer 21 and Layer 22 for the second time. Figure 8 demonstrates the process of feature-swapping. In the R-path and the I-path, the output of previous layer is convoluted by two groups of kernels independently, so as to produce two sets of feature maps. Then, one of the feature map sets is transferred to the adjoint path, while the other set remains in the original path. Thus, each path will extract the feature maps not only from its own dataflow, but also from the adjoint dataflow. Finally, the two sets of feature maps are concatenated together and transferred to the next layer.
3.3 Aerial image calculation
The thick-mask DNF obtained by the AFCN model can be used to calculate the aerial image. According to Abbe’s imaging model, the aerial image of the EUV lithography system can be formulated as the superposition of the aerial images contributed by all of the source points [10,27]:
4. Simulations
This section provides a set of simulations to assess the proposed AFCN method. The performance of the proposed method is compared to the non-parametric kernel regression method in [18] (to be shorted as regression method in this section) and the regular FCN-based method.
4.1 Data preparation and network training
In this work, we select 120 mask segments in total to represent the typical environments and geometries of the layout features. Among these mask segments, 108 of them are used for training, and 12 of them are used for testing. The dimension of each mask segment is 300 pixels × 300 pixels, where each pixel stands for 1 nm on the wafer scale. The minimal critical dimension (CD) of these features is 27 nm on the wafer scale. The waveguide method is used to prepare the rigorous thick-mask DNF data in the training library. The illumination wavelength is 13.5nm and the NA of lithography system is 0.33.
We consider 17 different illuminating source points as the reference points to approximately represent the diffraction spectrum generated by a range of incident angles. The coordinates of the reference points are listed in Table 1.
Table 1. The coordinates of the 17 reference source points.
The proposed AFCN model is set up on a Linux server with 20-core Intel Xeon Gold 6230 CPU @2.1GHz and NVIDIA GTX 2080TI GPU. The deep learning project is deployed in PyCharm, configuring Keras 2.3.1. Independent network will be trained for each reference point, so as to capture the DNF characteristics from a certain incident angle. During the training process, the batch size is assigned as 32, and the number of epochs is assigned as 50.
4.2 Performance evaluation
After training, the proposed method is used to calculate the DNF and aerial image of the testing masks. We select five on-axis reference source points as representatives to show the average mean square error (MSE) values of the predicted diffraction matrices over the 12 testing masks in Table 2. From left to right, Table 2 shows the results obtained by the regression method, regular FCN method, and the proposed AFCN method. The rigorous DNF data calculated by the waveguide method is regarded as the ground truth. For a fair comparison, the hyperparameters in the regular FCN are set to the same as a single path in the AFCN model. In addition, two regular FCNs are independently trained to predict the real part and the imaginary part, respectively.
Table 2. The average MSE values of the diffraction matrices calculated by different methods.
In Table 2, “F(UV) - Real” and “F(UV) - Imag” indicate the real part and imaginary part of F(UV), respectively. Here, we only assess the prediction results of F(XX) and F(YY), because the amplitudes of F(XY) and F(YX) are three orders of magnitude smaller than those of F(XX) and F(YY), thus they have little impact on the aerial image. As shown in Table 2, the proposed method achieves more than two-fold improvement on the prediction accuracy compared to the regression method. Although the AFCN model induces larger errors in the real part than the regular FCN, it alleviates the imbalance of accuracy between the real part and the imaginary part. This is beneficial in improving the accuracy of the aerial image calculation, which will be discussed shortly.
Next, we provide the simulation results of aerial images based on two kinds of commonly-used source shapes, i.e., the conventional source and quasar source. When calculating the coherent image generated by source point located in (σx, σy), we use the DNF of the reference point closest to (σx, σy) to approximately represent that of (σx, σy). Thus, the reference source points #1∼#5 will be used to simulate for the conventional source with partially coherence factor σ = 0.2. As shown in Fig. 9(a), distinctive colors are assigned to the five areas, within which the mask DNF is approximately represented by the DNF generated by the closest reference point. For the quasar source, the inner and outer partially coherence factors are set as σin = 0.5 and σout = 0.7, the opening angle is 45°, and the rotation angle is 0°. The reference source points #6∼#17 will be used to simulate for the quasar source, as shown in Fig. 9(b). In this paper, we consider the aerial image calculated based on Fig. 9 as ground truth if the mask DNF is obtained by the waveguide method.
Fig. 9. The reference points used to simulate the aerial image: (a) the five-source-points approximation for the conventional source, and (b) the twelve-source-points approximation for the quasar source.
Table 3 provides the average MSE values of aerial images over the 12 testing masks. It shows that compared to the regression method, the AFCN model under the conventional and quasar illuminations can reduce the errors by 90.95% and 88.38%, respectively. Compared to the regular FCN, the proposed method reduces the errors by 37.88% and 15.14% for the conventional source and quasar source, respectively.
Table 3. The average MSE values of the aerial images calculated by different methods.
Using the DNFs calculated by different methods, Fig. 10 and Fig. 11 illustrate the simulated aerial images of two testing masks. The conventional source and quasar source are used in the simulations of Fig. 10 and Fig. 11, respectively. Figures 12 and 13 show the error distribution maps of the aerial images using the conventional source and quasar source, respectively, where the MSE values are provided in the figures.
Fig. 10. The aerial images of testing masks produced by the conventional source, which are calculated using the rigorous method (the third column), the regression method (the fourth column), the regular-FCN method (the fifth column), and the proposed AFCN method (the sixth column).
Fig. 11. The aerial images of testing masks produced by the quasar source, which are calculated using the rigorous method (the third column), the regression method (the fourth column), the regular-FCN method (the fifth column), and the proposed AFCN method (the sixth column).
Fig. 12. The error distribution maps of testing masks produced by the conventional source, which are calculated using the regression method (the first column), the regular-FCN method (the second column), and the proposed AFCN method (the third column).
Fig. 13. The error distribution maps of testing masks produced by the quasar source, which are calculated using the regression method (the first column), the regular-FCN method (the second column), and the proposed AFCN method (the third column).
Because all of the simulations mentioned above use the same aerial image model as described in Section 3.3, the computational efficiency of different methods mainly depends on the calculation process of thick-mask DNF. The third row of Table 4 lists the average runtimes to calculate the DNF of one testing mask. The fourth row of Table 4 provides the standard deviation of runtimes over the 12 testing masks. The simulations mentioned above are based on the illumination source point (0, 0). It is noted that the proposed AFCN method achieves more than 600× speedup and 25× speedup on average compared to the rigorous method and the regression method, respectively. In addition, the proposed method obtains the lowest standard deviation of runtimes among all of the four methods. Since the computations based on rigorous method and regression method rely heavily on the mask feature geometries, the variation of runtimes over different masks is more notable. In contrast, the runtimes of FCN-based methods mainly depend on the mask dimension rather than the feature geometries. Thus, the regular FCN and AFCN methods achieve much smaller standard deviations. Thus, the proposed AFCN method is most robust in the sense of computational efficiency.
Table 4. The average runtimes and the runtime standard deviations of different DNF calculation methods based on the source point (0, 0).
It is noted that the number of used reference points could affect the accuracy of aerial image calculation. As a comparison to Fig. 9, Fig. 14(a) shows a more simplified way to simulate the aerial image for the conventional source with only the central reference source point #1 being used. Figure 14(b) shows the simplified four-source-points approximation for the quasar source, where only reference point #7, #8, #13, and #16 are used. Based on Fig. 14, we re-calculate the aerial image for the two testing masks using the AFCN method and then obtain the aerial image error distribution maps. Figure 15 shows the error distribution maps, where the MSE values of the aerial images labeled in each subfigure. It is shown that for the conventional source, the accuracy of using the one-point-approximation is at the same level as using the five-points-approximation. The one-point-approximation even has a slightly smaller MSE for Mask #1 in our simulation, which can be attributed to the error of the estimated diffraction spectrums on the other four reference points. In contrast, using the simplified four-points-approximation for quasar source result in up to 4 times the MSE of using 12 reference source points. Therefore, it is recommended to use more source points to simulate the aerial image, especially for those illumination shapes with larger incident angles. Since the proposed AFCN method can estimate the diffraction spectrums with considerably high efficiency, using more source points can benefit from it.
Fig. 14. The reference points used to simulate the aerial image: (a) the one-source-point approximation for the conventional source, and (b) the four-source-points approximation for the quasar source.
Fig. 15. The error distribution maps of testing masks based on the AFCN method, which are calculated using: (a) the five-source-points approximation and the one-source-point approximation for the conventional source, and (b) the twelve-source-points approximation and the four-source-points approximation for the quasar source.
5. Conclusion
This paper proposed an AFCN modelling method to rapidly calculate the thick-mask DNF and aerial image of the EUV lithography system. The network takes the mask image as the input, and simultaneously estimates the real part and imagery part of the diffraction matrices based on the adjoint fully convolution data paths. The aerial image is then calculated using the Abbe’s method based on the estimated DNF. The proposed method improves the computational efficiency of the thick-mask DNF up to 600 times compared to the waveguide method, and 25 times compared to the non-parametric kernel regression method. Additionally, the proposed method reduces the simulation errors of aerial images by more than 80% in contrast to the non-parametric kernel regression method.
Appendix A. Detailed structural parameters of the proposed network
Table 5. Detailed structural parameters of each path in the proposed AFCN network.
Funding
National Natural Science Foundation of China (61804174); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021115).
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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