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Abstract
The simulation of thick-mask diffraction near-field (DNF) is an indispensable process in aerial image calculation of immersion lithography. In practical lithography tools, the partially coherent illumination (PCI) is applied since it can improve the pattern fidelity. Therefore, it is necessary to precisely simulate the DNFs under PCI. In this paper, a learning-based thick-mask model proposed in our previous work is extended from the coherent illumination condition to PCI condition. The training library of DNF under oblique illumination is established based on the rigorous electromagnetic field (EMF) simulator. The simulation accuracy of the proposed model is also analyzed based on the mask patterns with different critical dimensions (CD). The proposed thick-mask model is shown to obtain high-precise DNF simulation results under PCI, and thus is suitable for 14 nm or larger technology nodes. Meanwhile, the computational efficiency of the proposed model is improved up to two orders of magnitude compared to the EMF simulator.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical lithography is a key technique in the fabrication of integrated circuit (IC) [1–3]. Figure 1(a) illustrates the basic structure of a deep ultraviolet (DUV) lithography system. The illumination systems used in practical lithography tools are always partially coherent, since it can improve the image resolution. Different from the coherent illumination, light rays emitted from the partially coherent illumination (PCI) propagate toward the mask in a range of angles, and the light rays travelling in different directions are spatially incoherent. Therefore, undesired interference pattern of incident rays can be eliminated [4]. Taking the advantage of PCI, the oblique illumination is achieved on the extended source, and the resolution limit of optical lithography system can be improved theoretically. Partial coherence factor ($\sigma = a/b$) is used to describe the oblique illumination from the light source, where a is the size of source image on entrance pupil (effective source), and b is the size of entrance pupil [5]. Figure 1(b) illustrates of the partial coherent factor and oblique illumination effect. Some resolution enhancement techniques, such as off-axis illumination [6–8] and source-mask optimization [9–12], make the use of oblique illumination. By appropriately designing the source configuration according to the characteristics of mask pattern, those resolution enhancement techniques can improve the fidelity of lithography image.
Fig. 1. The (a) basic structure of immersion DUV lithography system, and (b) the illustration of partial coherent factor and oblique illumination.
The simulation of thick-mask diffraction near-field (DNF) is an indispensable process in aerial image calculation of immersion lithography. To balance the accuracy and computational cost, many thick-mask models were proposed previously, such as the boundary-layer (BL) model [13,14], domain decomposition method (DDM) [15], filter-based methods [16–19], machine learning methods [20,21], deep-learning methods [22–24] and analytical models [25–27]. In consideration of the polarization of light waves, the DNF can be represented by four diffraction matrices denoted as $\mathbf{E}(UV)$, where U = X or Y and V = X or Y. The diffraction matrix is defined as the complex amplitude of the DNF polarized in U direction, which is generated by a unit incident electric field polarized in the V direction.
The PCI model is widely applied in lithography image simulation. Due to the spatial incoherence property of PCI, the total aerial image of lithography system can be calculated as the superposition of the aerial images contributed by every source point. Therefore, it is necessary for thick-mask model to calculate the DNFs generated by different source points independently. In [28], authors used the analytical model to discuss the impact of oblique illumination on the DNF distribution. In [20] and [29], authors proposed the thick-mask models based on non-parametric kernel regression, where the training libraries were established for different source points, so that the DNFs under oblique illuminations can be calculated. On the other hand, using the DNF under normal incident illumination to approximate the DNFs under oblique illuminations will cause error, which makes the following aerial image result unconvincing.
Moreover, the edge interference effect is a predominant factor that influences the DNF distribution. The diffraction wave generated by the boundary of thick-mask metallic layer will interferes with the incident wave, and causes the DNF fluctuation [30]. The length of fluctuation period in DNF is determined by the wavelength of incident light rays theoretically. The ratio between the period of DNF fluctuation and the pitch of mask pattern will change with the critical dimension (CD) variation. Therefore, the performance of thick-mask models needs further verification under different CD conditions.
Recently, we introduced a thick-mask model based on decomposition machine learning method [30], which rendered fast and high-precise DNF simulation results with intense edge interference effect. Comparing to the conventional filter-based model, the proposed thick-mask model can reduce the RMS error by about 70%, meanwhile maintain high computation speed. However, this previous work focuses on the coherent lithography system, and thus the thick-mask model is inadequate to simulate the aerial image in partially coherent lithography systems. Furthermore, in our previous work the CDs of mask patterns were limited within a specific range. In order to certify the effectiveness and applicability of the proposed model in different layout circumstances, more investigation and analysis should be performed under different simulation conditions.
In this paper, we further expand the decomposition-learning-based thick-mask model to the partially coherent lithography systems. The DNF simulations with different mask CDs and different incident angles are studied and discussed. Training libraries of thick-mask DNFs are established respectively under different conditions based on the rigorous EMF simulator. It demonstrates that the proposed model can obtain high-precise results with pattern CDs above 28 nm on the wafer scale, which is corresponding to the 14 nm technology node. It also shows that the proposed model is stable under oblique illuminations, which is applicable to the partially coherent lithography systems.
The rest of this paper is organized as follows. Section 2 describes Abbe’s model to calculate the aerial image of partially coherent lithography system. Section 3 provides the details of the proposed thick-mask model and the process to establish the training libraries. Simulations and discussions are presented in Section 4. Conclusions are given in Section 5.
2. Abbe’s model for partially coherent lithography system
In this section, Abbe’s model to calculate the aerial image of partially coherent lithography system is described. Based on the Abbe’s model, the necessity to extend the thick-mask model to PCI condition is explained. We first show an instance of DNF simulation under oblique illumination. Then, Abbe’s imaging model for PCI is summarized. Finally, the simulation results of the accurate Abbe’s model and an approximate Abbe’s model are compared.
The DNF of thick-mask depends on the impending direction of light rays. Figure 2 illustrates an instance of DNF simulation. In this case, the amplitude of diffraction matrix E(XX) calculated by EMF simulator is studied. Figure 2(a) shows the source configuration in this simulation. There are 13 reference source points selected to cover the entire source pattern, which will be discussed in the next section. Three reference points are selected to study the oblique illumination effect, and their spatial coordinates on the pupil plane are (0,0.5), (0.5,0.5), and (0.5,0), respectively. The green crosses in Fig. 2(a) denote the selected reference points. On the other hand, the yellow cross in Fig. 2(a) represents the central source point of normal incidence, whose spatial coordinate is (0,0) on the pupil plane. Figure 2(b) displays the simulation results of DNFs. From left to right, it shows the results under normal incidence illumination and three oblique illuminations, respectively. The top row shows the amplitude of E(XX), and the bottom row shows the absolute value of difference between the results of normal incidence illumination and oblique illuminations. It can be observed that, changing the incident angle will cause the shift and variation of DNF distribution. Therefore, the thick-mask DNFs of different source points are required to be calculated individually.
Fig. 2. The simulations to show the incidence-dependency of thick-mask DNF: (a) the reference points on the source pattern, and (b) the thick-mask DNFs calculated under difference illumination conditions.
The imaging model of partially coherent lithography system requires the calculation of thick-mask DNFs under different incident angles. Based on the spatially incoherence of PCI, the aerial image on the wafer can be calculated as the summation of the aerial images generated by all source points. Abbe’s model is widely used to calculate the aerial image of partially coherent lithography system, which can be expressed as [4]:
Using the DNF under coherent illumination to approximate the DNFs under PCI will cause error in the simulation result. Figure 3 shows an instance of aerial image calculation. The simulations use the 193 nm immersion lithography system, where the refractive factor of the immersion medium is 1.44. The numerical aperture of the projection optics is 1.35, and the demagnification factor is 4. Figure 3(a) illustrates the source configuration, where the annular illumination is used, and the inner and outer partial coherence factors are 0.4 and 0.6. In this case, four reference source points are used to represent the annular illumination, and their spatial coordinates on the pupil plane are (0.5,0), (0,0.5), (-0.5,0), and (0, -0.5), respectively. The red crosses in Fig. 3(a) illustrate the 4 selected reference source points. Figure 3(b) shows the aerial image calculated by Abbe’s model in Eq. (1), which is referred to as the accurate Abbe’s model.
Fig. 3. Aerial image calculation of partially coherent lithography system: (a) the reference points on source pattern; (b) aerial image calculated by the accurate Abbe’s model; (c) the aerial image calculated by the approximate Abbe’s model; (d) the absolute value of difference between (b) and (c); and (e) the intensity cross sections in (b) and (c). The positions of sections are illustrated by the grey arrows in (b) and (c), and the black arrow in (e) denotes the target CD.
In order to show the impact of oblique illumination on the imaging simulation, another approximate Abbe’s model is used as the comparison. In the approximate model, the DNF under normal incidence illumination is used to approximate the DNFs of all source points. Other parameters are the same as the accurate Abbe’s model. The yellow cross in Fig. 3(a) represents the source point of normal incidence illumination, and Fig. 3(c) displays the aerial image calculated by the approximate Abbe’s model. Figure 3(d) shows the absolute value of difference between Figs. 3(b) and 3(c). The RMS error of the result calculated by the approximate Abbe’s model is $1.1602 \times {10^{\textrm{ - 5}}}$ (The expression of RMS error is provided in Sec. 4). Figure 3(e) shows the cross sections of intensities calculated by the two models. The positions are illustrated by the gray arrows in Fig. 3(b) and Fig. 3(c). The constant threshold function is used to represent the effect of photoresist, where the pattern is printed on the wafer if the intensity of its aerial image is higher than 0.3. The length of original layout section is 54 nm, and the section lengths of the printed patterns on wafer that calculated by the accurate Abbe’s model and the approximate Abbe’s model are 27 nm and 38 nm, respectively. The error of approximate Abbe’s model is larger than 20% of the layout CD, which indicates the model is not accurate enough for aerial image simulation under PCI. It can be observed that the normal incidence approximation will induce pronounced error in the aerial image result. Therefore, it is essential to extend the thick-mask model to the PCI condition.
3. Decomposition-learning-based thick-mask model
In this section, the thick-mask model based on decomposition-learning method is first introduced. The decomposition process is also discussed in detail. Then, we describe the process to build the training libraries for partially coherent lithography system. Finally, we briefly describe the process to build the training libraries for mask patterns with different CDs.
3.1 Proposed thick-mask model
In our previous work, we proposed a fast learning-based thick-mask model. The proposed model obtained high-precision in DNF simulation with intense edge interference effect. The diffraction transfer matrix (DTM), indicating the mapping from the mask pattern to its DNF amplitude, was introduced in this model. Notice that the ${N_d}$ and ${N_m}$ are lateral dimensions of the predicted DNF segment and the mask segment, respectively. Suppose $\vec{M} \in {\mathbf{{\mathbb R}}^{{N_m}^2 \times 1}}$ represents the vectorized representation of a thick-mask segment denoted by M, and $\vec{A} \in {\mathbf{{\mathbb R}}^{{N_d}^2 \times 1}}$ represents the vectorized representation of the corresponding DNF segment denoted by A. Then, the vectorized DNF can be expressed as:
where $\mathbf{T} \in {\mathbf{{\mathbb R}}^{{N_d}^2 \times {N_m}^2}}$ is called DTM. Figure 4 illustrates the relationship among $\vec{A}$, $\vec{M}$ and T.Fig. 4. The relationship among the vectorized DNF segment, vectorized mask segment, and DTM, where the DTM indicating the intensity mapping from the mask pattern to the corresponding DNF.
The diffraction behaviors of different geometric patterns are dissimilar. Therefore, the proposed model decomposes the mask pattern into four categories of segments based on its geometric features, including the convex corners, concave corners, horizontal edges and vertical edges. A corresponding DTM is used for each category of features, and four DTMs are calibrated for different categories independently. Hence, the relationship between the DNF and mask pattern in Eq. (2) can be further extended as following:
The flow of the algorithm is summarized below. First, training libraries of thick-mask DNFs are established based on some training mask patterns. Then, the training mask patterns and their corresponding DNFs are decomposed into segments. The methods to establish libraries and decompose masks are discussed in the following. After the decomposition process, mask patterns in different sizes can be divided into different amounts of segments with similar sizes. So, the DNF segments can be further calculated with the same DTM. Therefore, the generalizability of the algorithm is improved.
Afterwards, based on the least square method, the DTM associated to each kind of features is calibrated based on the training libraries, which can be written as:
where the subscript x is the category index of segments (concave corner, convex corner, horizontal edge, or vertical edge), and the superscript T is the transpose symbol. The calibrated DTMs can be repetitively used in the following simulations of any test mask patterns.Given a test mask pattern, we first decompose it into four groups of feature segments, similar as the training mask patterns. Then, each segment is vectorized and multiplicated with the corresponding DTM as shown in Eq. (3), so that the local DNF of the mask segment is calculated. Finally, the entire DNF distribution is synthesized by stitching up all of the DNF segment. The DNF of overlapped area is calculated as the weighted average of all overlapped DNFs, which is discussed in the following.
It is worth noting that, the algorithm discussed above is not limited to calculating the amplitude of DNF. The complex amplitude of the diffraction matrices can also be calculated using the proposed model.
3.2 Mask decomposition and DNF stitching
Figure 5 shows the decomposition process of an L-shape mask pattern as an instance. First, a set of sampling points are anchored on the boundaries and corners of the mask pattern with a fixed interspace, L0. Colorful points in Figs. 5(a) and 5(b) denote the sampling points. The square area surrounding each sampling point is assigned for mask segment and DNF segment, their lateral dimensions are ${N_m}$ and ${N_d}$ respectively. Therefore, the training mask patterns and their DNF images are decomposed into four kinds of features: the concave corner segments, convex corner segments, vertical edge segments and horizontal edge segments. As shown in Fig. 5(c), all of the sampling points are then extended simultaneously to cover their adjacent regions. The extension of a segment terminates as the radius of the segment reaches the preset upper limit. Figure 5(d) is a concave corner segment on the mask, and the corresponding sampling point is placed at the corner. The lateral dimension of the mask segment is ${N_m}$. The simulated DNF amplitude is displayed in Fig. 5(e), and the lateral dimension of the DNF segment is ${N_d}$.
Fig. 5. Illustration of the decomposition process: (a) the sampling points anchored at the corners; (b) the additional sampling points anchored on edges; (c) the segment generation; (d) the concave corner segment on the mask; (e) the calculated DNF amplitude in the segment; and (f) the instance for calculating the DNF in the overlapped area.
Notice that DNF segments can be overlapped with each other, and the total DNF cannot be calculated by stitching up all DNF segments simply. To overcome this issue, the DNF in the overlapping area is calculated as the weighted average of all the overlapped DNFs. The weight coefficients are defined as the inverse of distance between the calculated pixel and the corresponding sampling point. The expression can be written as:
Figure 5(f) illustrates an instance of calculating the DNF in the overlapping area. Pixels denoted by the blue point and orange point represent two sampling points, and the DNF segments corresponding to those two points are shown as the semi-transparent area. The pixel encircled by the red dot line is the overlapped area to be calculated, and its spatial coordinates is denoted by $(x,y)$. The distance between the location of $(x,y)$ and the two sampling points are $\sqrt 2 $ and $\sqrt 5 $, respectively. Therefore, the weight coefficients of the two overlapped DNFs are set as $\frac{1}{{\sqrt 2 }}$ and $\frac{1}{{\sqrt 5 }}$, and the DNF in the overlapped area can be calculated as the weight average of the two overlapped DNFs.
3.3 Training libraries for PCI
As mentioned above, the proposed thick-mask model relies on the pre-calculated training libraries. In order to extend the thick-mask model to the partially coherent lithography system, we need to calculate the DNFs of training mask patterns under all source points in principle, since the DNF distribution depends on the incidence angle of light rays. Figure 6(a) illustrates a pixelated source pattern. Nevertheless, building training libraries based on a large number of source points will introduce huge computational burden. To compromise, we select 13 equidistance reference points on a sparse mesh to cover the entire source pattern. The blue dots in Fig. 6(a) represent the reference source points, and their coordinates and included angles are listed in the Appendix.
Fig. 6. The illustrations of (a) the pixelated source pattern with 13 reference points, and (b) the training libraries.
The training libraries are established for every reference source point individually. Figure 6(b) shows the structure of training libraries. In each library, a set of representative training mask patterns are selected, and the four diffraction matrixes of the corresponding source point are pre-calculated using the rigorous EMF simulator.
3.4 Training libraries under different CD conditions
In order to study the influence of CD conditions on the performance of thick-mask model, we need to build training libraries under different CD conditions. First, the selected training mask patterns are scaled down to obtain the training samples with different CDs. Figure 7 takes a training mask pattern as an instance to illustrate the process of scaling down. Each pixel of the mask pattern stands for a 1nm × 1 nm square on the wafer scale. The CD of the initial mask pattern is 54 nm, which corresponds to the CD of metal contact pattern at 32 nm technology node. Then, the CD of mask pattern is gradually scaled down to 45 nm, 32 nm, 28 nm, 22 nm, and 18 nm respectively, and the lateral dimension of mask pattern is also reduced accordingly. The corresponding technology nodes are presented in Fig. 7 [33]. For DUV lithography, the multiple patterning is applied beyond 16 nm technology node, and the extreme ultraviolet (EUV) lithography is applied beyond 7 nm technology node. After scaling down the mask patterns, the DNFs of all thick-masks are calculated using the rigorous EMF simulator, so that the training libraries are established for different CD conditions. It is noted that, the main purpose of this section is to study the relationship between CD conditions and the fidelity of the proposed model. Therefore, the DNF under of normal incidence illumination (reference point #7) is used for establishing training libraries under different CD conditions.
On the other hand, the lateral dimensions of the calculated DNF segment (${N_d}$) and the mask segment (${N_m}$) are two key parameters in the proposed model. The optimal value of ${N_m}$ is determined by the environmental conditions, like the illumination wavelength, the mask material property, and so on. ${N_d}$ is determined by the size of the calculated DNF segments. In the pattern dividing process, the sampling points are anchored on the edges and corners of the pattern. Therefore, the optimal value of ${N_d}$ should be large enough to cover the main features of mask pattern. Moreover, the DTM calculation depends on the least square method. Setting ${N_d}$ and ${N_m}$ too big will induce large computational burden. We have tested the model with different values of ${N_d}$ and ${N_m}$, and these two parameters influence the results trivially in an appropriate range. In this work, the values of ${N_d}$ and ${N_m}$ corresponding to different pattern CDs are listed in Table 1.
Table 1. The values of ${N_d}$ and ${N_m}$ corresponding to different pattern CDs.
4. Simulation and discussion
In this section, the thick-mask DNFs calculated by the rigorous EMF simulator and the proposed model are compared under different simulation conditions. As shown in Fig. 8(a), 7 patterns are used in the simulation. One of seven patterns is selected as the testing set, and other six patterns will be attributed to the training set. Figure 8(b) illustrates an instance that the pattern 1 is selected as the testing set and patterns 2 to 7 are used for DTM training. For each simulation, all 7 patterns are tested, and errors discussed below are the average simulation error of 7 patterns.
Fig. 8. The illustration of (a) seven mask patterns, and (b) an instance of patterns attribution in testing set and training set.
The waveguide method, a rigorous EMF simulator, is used to calculate the accurate DNF data of the training masks. The 193 nm argon fluoride (ArF) immersion lithography system is used as the lithography tool in the simulation. The refractive factor of the immersion medium is 1.44, and the demagnification factor of the lithography system is 4. For each mask pattern, four diffraction matrices E(XX), E(XY), E(YX) and E(YY) are calculated. All of the simulation codes are implemented by MATLAB, and the computations are carried out on a computer with Intel Core i7-10870 H CPU, 2.20 GHz, 32.0 GB of RAM.
In this work, the results of the EMF simulator are set as the benchmark. The root-mean-square (RMS) error and the signal-to-noise ratio (SNR) of the simulation results are used to evaluate the accuracy of different thick-mask models. The RMS error is defined as:
On the other hand, the standard definition of SNR is the ratio of signal variance to noise variance in statistical theory [31]. In this work, the diffraction matrix calculated by the EMF simulator is considered as the signal, and the difference between the results obtained by the proposed model and EMF simulator is considered as the noise. Therefore, the SNR of the result is defined as:
The performance of the proposed model under coherence illumination has been discussed detailly in our previous work [30]. Comparing to the conventional filter-based model, the proposed thick-mask model can reduce the RMS error by about 70%, meanwhile maintain high computation speed. Figure 9 provides the simulation results under oblique illuminations. Results of pattern 1 are displayed as an instance. In this simulation, three reference source points are selected to study the DNF distributions under PCI. The spatial coordinates of the three reference source points on the pupil plane are (0.5, 0), (0, 0.5), and (0.5, 0.5), as shown by the green crosses in Fig. 2(a). The lateral dimension and CD of the test pattern is 1200 nm and 54 nm, respectively. As mentioned above, changing the incident angle of illumination will cause shift and variation in DNF distribution. Table 2 provides the average RMS errors and the average SNRs of seven patterns calculated by the proposed model under oblique illuminations. The results of the proposed model show high fidelity compared with the corresponding benchmarks, and the calculation errors under different source points are similar. As a contrast, the RMS errors of E(XX) and E(YY) under normal incidence illumination are about 0.036, and the SNRs of E(XX) and E(YY) are about 70. The calculation errors under the oblique illuminations increase trivially compared to the results under the normal incidence illumination, which confirms the applicability and stability of the proposed model for PCI.
Fig. 9. The amplitudes of diffraction matrices under oblique illuminations, where the DNFs of E(XX) and E(YY) are obtained by the EMF simulator and proposed model. Pattern 1 is shown as an instance.
Table 2. The average RMS errors and average SNRs of the simulated DNFs under oblique illuminations, where the diffraction matrixes E(XX) and E(YY) are selected.
DNF simulations for mask pattern with different CDs are also studied. Notice that all the CDs mentioned in this section refer to the wafer scale. In our previous work, the simulation results of mask patterns with 54 nm CD were discussed. Next, the mask patterns with smaller CDs are used to test the proposed model, and results of mask patterns with 28 nm CD and 18 nm CD are discussed as instances. Figure 10 provides the simulation results with 28 nm CD, which is corresponding to the 14 nm technology node. The results obtained by the EMF simulator are shown in the top row, and results calculated by the proposed model are shown in the bottom row. From left to right, it shows the amplitudes of diffraction matrices E(XX), E(XY), E(YY), E(YX), respectively.
Fig. 10. The amplitudes of diffraction matrices obtained by the EMF simulator and the proposed model. The CD of mask pattern is 28 nm, which is corresponding to the 14 nm technology node. Pattern 1 is shown as an instance.
Table 3 shows the average RMS errors and average SNRs of the proposed model using the test pattern with 28 nm CD. Runtimes of the EMF simulator and the proposed model are also provided. It is shown that the proposed model can improve the computational efficiency up to two orders of magnitude compared to the EMF simulator. In addition, the proposed model can obtain relatively high fidelity. However, the noise at boundaries of DNF segments becomes more pronounced, and the SNR of the proposed model is degraded in this case.
Table 3. The average RMS errors, average SNRs and runtimes of the EMF simulator and the proposed model, where the CD of mask pattern is 28 nm.
Figure 11 provides the simulation results of the mask pattern 1 with 18 nm CD, which is corresponding to the 7 nm technology node. Four diffraction matrixes are listed in Fig. 11, from left to right. The results of EMF simulator show that the amplitude of diffraction matrixes decreases tremendously due to the small mask opening area and the edge interference effect of the thick mask. Notice that the 18 nm CD on the test pattern is far beyond the resolution limit of the single exposure using 193 nm DUV lithography system.
Fig. 11. The amplitudes of diffraction matrices obtained by the EMF simulator and the proposed model. The CD of mask pattern is 18 nm, which is corresponding to the 7 nm technology node. Pattern 1 is shown as an instance.
Table 4 shows the runtimes, average RMS errors, and average SNRs of the EMF simulator and the proposed model for the mask pattern with 18 nm CD. The efficiency of the proposed model is about 50 times higher compared to the rigorous EMF simulator. The RMS errors in this case change trivially compared with the results of mask patterns with larger CD. However, since the amplitude of diffraction matrixes is lower, the resulting noise of the proposed model is pronounced. Thus, the SNRs of the proposed model further decrease, and the fidelity of the results are unacceptable.
Table 4. The average RMS errors, average SNRs and runtimes of the EMF simulator and the proposed model, where the CD of mask pattern is 18 nm.
Figure 12 displays the variation of simulation SNR with respect to different CDs. The amplitudes of E(XX) and E(YY) are greater than E(XY) and E(YX) remarkably, and thus the fidelity of E(XX) and E(YY) has much larger influence on the aerial image simulation. Therefore, the SNRs of E(XX) and E(YY) are used to test the applicability of the proposed model. The technology nodes corresponding to the mask CDs are also marked in the figure [32]. It can be observed that, the simulation SNR degrades with the reduction of the mask CD.
Fig. 12. The simulation SNRs of the proposed model with different mask CDs, where the corresponding technology nodes are also provided in the figure.
To assess the quality of the simulation results, we assume that the simulation result is acceptable if the SNR is greater than 20. Therefore, the fidelity of the proposed model is acceptable if mask CD is above 28 nm. Notice that the 28 nm CD corresponds to the 14 nm technology node. In conclusion, the proposed model is applicable for 14 nm technology node and above, which almost covers all of the available technology nodes supported by the 193 nm DUV lithography system. It is worth mentioning that, the standard used above is empirical, and other standards can also be used based on specific demands.
The complex amplitude of DNF can also be calculated by the proposed method. Based on the complex DNFs under different source points, the lithography aerial image can be further calculated, and the results are shown in Fig. 13. Figure 13(a) shows the aerial image calculated with the accurate Abbe’s model, which is discussed in Sec. 2. The DNFs are provided by the EMF simulator, therefore the result shown in Fig. 13(a) is used as the benchmark. On the other hand, the aerial image in Fig. 13(b) is calculated based on the complex DTMs obtained by the proposed model. Figure 13(c) displays the difference between the aerial images in Figs. 13(a) and 13(b), and the RMS error of aerial image in Fig. 13(b) is $\textrm{4}\textrm{.4236} \times {10^{\textrm{ - }6}}$. Comparing Fig. 3(d) and Fig. 13(c), it can be observed that the proposed DTM model can significantly increase the accuracy of aerial image calculation. Figure 13(d) shows the cross sections of aerial images, and their positions are illustrated by the grey arrows in Figs. 13(a) and 13(b). The threshold function is used to simulate the effect of photoresist, as discussed in Sec. 2. It can be observed that, the section length of exposed pattern calculated by the proposed model is 31 nm, and the length of the benchmark is 27 nm. The CD of the pattern is 54 nm, and the CD error of the proposed model is less than 10%, which further confirms the applicability of the proposed model.
Fig. 13. Aerial image calculation of partially coherent lithography system: (a) aerial image calculated by the accurate Abbe’s model; (b) the aerial image calculated with the DNFs that simulated by the proposed model; (c) the absolute value of difference between (a) and (b); and (d) the intensity cross sections in (a) and (b). The positions of sections are illustrated by gray arrows in (a) and (b), and the black arrow in (e) denotes the target CD.
5. Conclusion
In this paper, a thick-mask model based on decomposition-learning method is extended for partially coherent lithography systems with different mask CDs. The necessity to extend the thick-mask model for PCI is first discussed. Then, the proposed model is tested under different simulation settings, and the simulation results confirm the generalizability of the proposed model. Simulation results based on different oblique incident angles illustrate the robustness of the model, which indicates its applicability to the partially coherent lithography system. The proposed model can also obtain high-precise DNF simulation results for the 14 nm technology node and above, which covers most of the available technology nodes using 193 nm DUV lithography system. Meanwhile, the proposed model can significantly improve the computational efficiency compared to the rigorous EMF simulator.
A. Appendix
The coordinates and included angles of the 13 reference source points are listed as follows:
Funding
National Natural Science Foundation of China (61804174, 62274181); Guangdong Province Research and Development Program in Key Fields (2021B0101280002); Guangzhou City Research and Development Program in Key Fields (202103020001); Opening Project of Key Laboratory of Microelectronic Devices & Integrated Technology, Institute of Microelectronics, Chinese Academy of Sciences (E1YS05X001).
Acknowledgments
This work is supported by University of Chinese Academy of Sciences (Grant#118900M032); China Fundamental Research Funds for the Central Universities (Grant# E2ET3801); Youth Innovation Promotion Association Chinese Academy of Sciences (No. 2021115); Beijing Institute of Electronics, Beijing Association for Science and Technology.
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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