## Abstract

Extreme ultraviolet (EUV) lithography plays a vital role in the advanced technology nodes of integrated circuits manufacturing. Source mask optimization (SMO) is a critical resolution enhancement technique (RET) or EUV lithography. In this paper, an SMO method for EUV lithography based on the thick mask model and social learning particle swarm optimization (SL-PSO) algorithm is proposed to improve the imaging quality. The thick mask model's parameters are pre-calculated and stored, then SL-PSO is utilized to optimize the source and mask. Rigorous electromagnetic simulation is then carried out to validate the optimization results. Besides, an initialization parameter of the mask optimization (MO) stage is tuned to increase the optimization efficiency and the optimized mask's manufacturability. Optimization is carried out with three target patterns. Results show that the pattern errors (*PE*) between the print image and target pattern are reduced by 94.7%, 76.9%, 80.6%, respectively.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Lithography is a fundamental technology to drive the development of the integrated circuit. EUV lithography has become the mainstream of 5nm node and below [1]. Since the extreme ultraviolet light, whose wavelength is 13.5nm, is used as the exposure source in EUV lithography, single exposure resolution is higher than that of deep ultraviolet (DUV) lithography. Like DUV lithography, the imaging quality of EUV lithography is severely affected by optical proximity effects (OPE) when the mask pattern's critical dimensions (CD) continuously shrink. Besides, due to the oblique incidence configuration and mask thickness, three dimensions (3D) effects of the mask, including shadowing effect and focus shift effect, are distinctive in EUV lithography [2]. Thus the RET is one of the most important ways to improve the EUV lithographic imaging quality.

SMO is an effective RET widely used in the 22nm node and below. Compared with the traditional RETs such as optical proximity correction (OPC) and inverse lithography technique (ILT), SMO can significantly increase the degree of freedom by joint optimization of the source and mask [3]. Rosenbluth et al. introduced the concept of SMO in 2001 [4]. Since then, SMO techniques were intensively studied. Various representations of source and mask are utilized in different SMO techniques. Typical source representations include the parametric method [5], the pixelated method [6], and the Zernike polynomial method [7]. Typical mask representations include the pixelated method [6], the discrete cosine transform (DCT) method [8], and the compressive sensing (CS) method [9]. Pixelated representation of source and mask has higher degree of freedom. Meanwhile, the pixelated source and mask can be realized more easily in practice, which benefits from the development of freeform illumination and mask manufacture techniques [10,11]. Thus the pixelated representation becomes the most popular method in SMO researches.

According to the optimization algorithm, pixelated SMO can be mainly divided into two types, the gradient-based SMO [12–15] and the SMO using heuristic algorithms [16–18]. The gradient-based SMO techniques derive the cost function analytically by the lithographic imaging model and resist model. Then the gradients of cost function with respect to the source and mask are calculated and utilized in the gradient-based optimization algorithms, such as the steepest descent algorithm and the conjugate gradient algorithm. The efficiency of gradient-based SMO is pretty high because the variables are optimized along the descent direction of the gradients. However, the gradient-based SMO usually only obtains the optimal local solution instead of the optimal global solution. Besides, gradient-based SMO application is limited when the imaging model and resist model are too complex to be analytically expressed. In the SMO techniques using heuristic algorithms, the source and mask are encoded with various strategies. Heuristic algorithms, including the genetic algorithm (GA) and the particle swarm optimization (PSO) algorithm, are used to optimize the source and mask. Heuristic algorithms are free of gradient calculations and prior knowledge of lithography. Meanwhile, they are adaptable to various complex imaging and resist models. Nevertheless, the optimization speed and efficiency are tightly associated with the type of algorithm.

Up to now, most of the SMO techniques are developed for DUV lithography, and few previous researches have studied the SMO for EUV lithography. In 2008, Fühner et al. proposed a simulation-based SMO method for EUV [19], which optimized the source and mask using GA and the multi-object optimization framework. A fast rigorous mask model based on Waveguide (WG) method and parallelized decomposition technique were exploited in this work. However, the model speed and accuracy on complex mask patterns need further verification. In 2014, Liu et al. proposed a general EUV SMO flow for the ASML lithography tool NXE:33 × 0, which could effectively overcome the flare and 3D mask effects [20]. The thick mask model M3D+ proposed by ASML was used in the SMO flow. However, the M3D+ model needs to calculate numerous data and makes several assumptions regarding the mask shapes, such as edge angles, feature sizes, and corner sizes. This imposes limitations on the accuracy of the method when dealing with random features [21]. In 2019, Xu Ma et al. proposed a gradient-based SMO method for EUV, which expanded the SMO method for DUV they proposed earlier [22]. However, the thin mask model was used for imaging calculation, and the correction of the shadowing effect was carried out through an additional process by an empirical model.

The mask model is crucial to the EUV SMO flow. The optimization results are unconvincing when the prediction result of the mask model is away from the reality. It is essential for the mask model to describe the diffraction process and the 3D mask effects effectively. Besides, models with high calculation speed is need since the calculation speed of the mask model directly affects the optimization speed of the SMO. The thick mask model for EUV based on the structure decomposition method (SDM) has been extensively studied [23–25]. It calculates the mask near-field or diffraction spectrum by describing the propagation of incident light in the absorber and the multilayer [24]. SDM model can calculate the mask spectrum fast and accurately. Moreover, the separation of the absorber and multilayer model make the SDM suitable for SMO flow. The reflection of the multilayer can be calculated in advance, and only the absorber diffraction spectrum needs to be calculated in each iteration. However, the SDM with the pixelated mask, which is usually the form of mask in SMO application, is rarely studied.

In this paper, a compact, effective and fast EUV thick mask model based on SDM is realized for pixelated masks and used for efficient forward imaging calculation in SMO application. With the thick mask model's help, the mask near-field and aerial image can be calculated with much higher accuracy than the thin mask model. Mask 3D effect which is significant in EUV lithography can be predicted effectively by the thick mask model while it is impractical by the thin mask model. By using the thick mask model, effective SMO is realized. The mask 3D effects can be mitigated in the proposed SMO flow without additional process. Besides, an SL-PSO algorithm [26] is employed in the proposed SMO. Compared with the traditional heuristic algorithm, such as PSO [17], CMA-ES [18], and JADE [27], SL-PSO shows higher optimization speed and efficiency. Besides, SL-PSO poses inherent advantages for solving large-scale problems. Thus, it is suitable for optimizing the large-area and complex mask patterns. At last, an initialization strategy at the MO stage is introduced in this paper. An initialization parameter is tuned for improving the optimization efficiency and the manufacturability of the optimized mask pattern.

## 2. Methodology

#### 2.1 EUV lithography imaging model

The EUV imaging system and the schematic of EUV SMO are presented in Fig. 1. The lights from the source illuminate the mask through the illumination system, with a chief ray angle at object (CRAO). Then the projection optics collects and combines the diffraction lights generated by the reflection on the mask, therefore transferring the pattern information from the mask to the wafer. Due to the diffraction limited property of the optics and the complex physical and chemical reactions in the resist, the print images on the wafer suffer severe distortion compared with the target pattern. It is a typical inverse problem of reducing the distortion of the print image by modifying the mask pattern and source shape. SMO provides a way to search for the optimal source and mask by iteration.

Similar to the DUV, EUV is a partially coherent imaging system, whose aerial image intensity can be calculated by Abbe model:

*I*is the intensity of the aerial image,

*S*is the source pattern,

*H*is the pupil function of the projection optics and can be regarded as a low pass filter,

*B*is the diffraction spectrum generated by the reflection on the mask.

The integration order in Eq. (1) is usually adjusted for raising the calculation speed in SMO, and the integration is numerically implemented by summation operation. At the source optimization (SO) stage, the projection optics and the mask pattern remain unchanged. Thus, the image intensity contributed by each source point can be pre-calculated and stored as illumination cross coefficient (*ICC*). Equation (1) can be rewritten with *ICC*:

At the MO stage, the projection optics and the source pattern remain unchanged. The transmission cross coefficient (*TCC*) is extensively exploited to speed up the imaging calculation, and Eq. (1) can be rewritten as:

*B** is the complex conjugation of

*B*.

*TCC*can be approximated by singular value decomposition (SVD):

*K*is the truncate order of the decomposition. With the approximate representation of

*TCC*, Eq. (3) is transformed to:

An accurate and calibrated resist model is vital to calculate the resist image and print image from the aerial image. For simplification, a constant threshold resist model is employed in this paper:

where ${I_p}$ is the print image, ${I_n}$ is the normalization intensity, ${t_r}$ is the resist threshold. $\Gamma ({\cdot} )$ represents the hard threshold function, which means that $\Gamma (x )= 1$ if $x \ge 0$ and $\Gamma (x )= 0$ if $x < 0$.#### 2.2 EUV thick mask model

In DUV lithography, the diffraction spectrum $B({\hat{f}^{\prime},\hat{g}^{\prime}} )$ is usually calculated by Kirchhoff thin mask model, which is the Fourier transform of the mask pattern $M({{{\hat{x}}_o},{{\hat{y}}_o}} )$. The $({{{\hat{x}}_o},{{\hat{y}}_o}} )$ represents the normalized spatial coordinates on the object plane.

Due to the oblique incidence configuration and significant mask 3D effects, the traditional Kirchhoff thin mask model is no longer sufficient for the EUV mask. Thus, a series of fast and accurate thick mask model for EUV are proposed. A fast and accurate EUV thick mask model based on SDM, which is applicable in pixelated mask simulation, is illustrated in this section. Figure 2(a) presents the structure of the EUV mask, where $\theta $ is the incident angle of light and $\varphi $ is the azimuth angle of light. Typical EUV mask is mainly composed of TaN absorber and Mo/Si multilayer. Figure 2(b) shows the schematic of the exposure ring slit in the EUV system. Along the ring slit, the CRAO remains 6°, and the azimuth varies from -30° to 30° [28].

In numerical simulations, the diffraction spectrum *B* is denoted by a ${N_O} \times {N_O}$ matrix, where ${N_O}$ is the number of diffraction orders in the *x* or *y* direction. The mask pattern *M* is denoted by a ${N_M} \times {N_M}$ matrix, of which each element represents a mask pixel. The element is 0 for background pixel and 1 for pattern pixel. Generally, ${N_O}$ is smaller than ${N_M}$. The reason is that the projection lens is a low-pass filter, and only a few diffraction lights participate in the imaging formation. Besides, the value of ${N_O}$ has a great influence on the calculation speed of the RCWA. The choice of ${N_O}$ is a trade-off between the calculation accuracy and the speed.

After the diffraction on the absorber and reflection on the multilayer, the angular spectrum can be calculated by:

where ${\odot} $ represents the element-wise multiply of the matrix. ${\phi _a} = exp \left( { - j\frac{{2\pi }}{\lambda }\cos \theta \times \frac{1}{2}{d_{abs}}} \right)$ is the phase propagation factor from the upper surface to the virtual middle plane of the absorber, where $\lambda $ is the wavelength of the incident light, and ${d_{abs}}$ is the absorber thickness. ${B_a}$ is the diffraction matrix at the middle plane of the absorber. ${\phi _b} = exp \left( { - j\frac{{2\pi }}{\lambda }\cos {\theta_m} \times \frac{1}{2}{d_{abs}}} \right)$ is the phase propagation matrix from the middle plane to the bottom surface of the absorber, where ${\theta _m}$ represents the angles between the diffraction lights and the*z*axis. ${R_m}$ is the reflection matrix of the multilayer, which can be calculated by the equivalent multilayer method (ELM) [29]. ${B_a}$,${\phi _b}$ and ${R_m}$ are all denoted by ${N_O} \times {N_O}$ matrixes.

The mask spectrum is the superposition of the secondary absorber diffraction of lights from different directions generated by the multilayer reflection. Rigorously, the diffraction matrix ${B_a}$ is different for each reflection light. However, the differences are not considered in the fast thick mask model. Lights from different directions have the same ${B_a}$, with only the frequency coordinates changing [24]. The element of the diffraction spectrum can be calculated by:

*B*is upgraded by:

The transmission matrix at the middle plane of the absorber ${B_a}$ can be represented by:

where $\Lambda ({\cdot} )$ represents the edge detection function, which can set the edge pixels to 1 and other pixels to 0. ${t_a}$ is the transmission of the background area and ${t_b}$ is the transmission of the pattern area on the absorber. ${\delta _e}$ is the point pulses factor. These three parameters are complex values and can be calibrated by the rigorous electromagnetic simulation approach, such as the rigorous coupled wave analysis (RCWA) method. Then ${B_a}$ can be calculated by:Most researches on the SDM apply the point pulses to the edge diffraction via the analytical method. In this paper, the pixelated mask representation is exploited so that the point pulses are added to the edge pixels. The accuracy will be reduced when the pixel size is too large at the object plane. Thus, the up-sampling process is carried out on the mask pattern *M* in this paper when calculating the $\tilde{M}$. Then Eq. (12) is implemented by the fast Fourier transform (FFT), and a ${N_O} \times {N_O}$ part is truncated to represent ${B_a}$ from the center of the FFT result. At the MO stage,${\phi _a}$,${\phi _b}$,${R_m}$,${\phi _d}$,${t_a}$,${t_b}$ and ${\delta _e}$ all remain unchanged. These thick mask model's parameters are pre-calculated and stored to speed up the imaging calculation and decrease the runtime of the optimization.

#### 2.3 SMO flow

The source and mask need to be encoded in SMO. The encoding strategies of source and mask are illustrated in Fig. 3.

As shown in Fig. 3(a), the source *S* is denoted by a ${N_S} \times {N_S}$ matrix, of which each element corresponds to the intensity of a source point ranging in $[{0,1} ]$. The inner and outer partial coherent factors are represented by ${\sigma _{in}}$ and ${\sigma _{out}}$. Only the elements located in the first quadrant are selected for encoding to enforce the source's symmetry. These elements, whose corresponding indices in *S* are denoted by ${P_S}$, are stacked as a vector ${J_S}$ representing the coded source. With the information of ${P_S}$ and ${J_S}$, a ${N_S} \times {N_S}$ matrix $\tilde{S}$ is generated according to the source symmetry when decoding the source. A blur function is then employed to roughly account for the finite resolution of the illumination optics [30]. The decoded source is thus calculated by:

The mask is similarly encoded, as shown in Fig. 3(b). Elements located in different quadrants are selected according to the symmetry of the mask pattern. These elements, whose corresponding indices in *M* are denoted by ${P_M}$, are stacked as a vector ${T_M}$ representing the coded mask. With the information of ${P_M}$ and ${T_M}$, a ${N_M} \times {N_M}$ matrix $\tilde{M}$ is generated according to the pattern symmetry when decoding the mask. The hard threshold function is then employed to create a binary mask pattern to denote the decoded mask:

As mentioned in section 2.2, mask defocus is necessary to eliminate the pattern shift at the wafer plane. The shift of the pattern position can be represented by the edge placement error (*EPE*). There are two kinds of *EPE* calculation:

*C*is the contour of the target pattern with the total length ${L_C}$, $dl$ is the segment length between the two metrology points.

*EPE*

^{1}approaches 0 when the pattern shift at the wafer plane is eliminated, and

*EPE*

^{2}embodies the absolute difference between the contours of the print image and the target pattern. In this paper, the optimized mask defocus $\varDelta {z^ \ast }$ is obtained by minimizing the

*EPE*

^{1}using the linear searching method.

As exhibited in Eq. (6), the contour of the print image transformed from the aerial image is mainly determined by the resist threshold ${t_r}$. The overall differences between the print image and the target pattern, which can be described by *PE*, are different with various threshold values. Thus, the optimized threshold $t_r^ \ast $ is obtained by minimizing the *PE* using the linear searching method before optimizing the source and mask. After optimization, the $t_r^ \ast $ remains unchanged during the following SMO flow. The *PE* can be calculated by:

*L*

_{2}-norm.

With the optimized $\varDelta {z^ \ast }$ and $t_r^ \ast $, the source and mask are optimized via SL-PSO based on the thick mask model and imaging model described in section 2.1 and section 2.2. SL-PSO algorithm is inspired by learning mechanisms in social learning of animals. It has a higher computational efficiency in comparison with a few representative PSO variants [26]. Compared to most modern meta-heuristics for optimization, SL-PSO is easy to implement and computationally efficient. Besides, SL-PSO performs much better for solving large-scale problems, and is suitable for optimizing the large-area and complex mask pattern.

The aim of the optimization can be symbolized by:

Figure 4 provides the whole flow of the proposed EUV SMO method and illustrates the SL-PSO sub-flow. At the beginning of the SL-PSO algorithm, several hyper-parameters are set, including the maximum number of iteration ${N_{iter}}$, the maximum number of calling of fitness function (*CoF*)${N_{CoF}}$, the size of particle swarm ${M_{pop}}$, and the particle dimension ${D_{pop}}$. The social influence factor is then defined as:

The particle swarm is then initialized. The ${i_{th}}$ particle’s position is denoted by a vector ${p_i}$ whose length is ${D_{pop}}$, and the positions of all particles are denoted by a ${M_{pop}} \times {D_{pop}}$ matrix ${P_{pop}}$. The ${i_{th}}$ particle’s velocity is denoted by a vector ${v_i}$ whose length is ${D_{pop}}$, and the velocities of all particles are denoted by a ${M_{pop}} \times {D_{pop}}$ matrix ${V_{pop}}$.

In this paper, the particles’ velocities are all initialized to 0, but the initialization strategies of the particle positions are different for SO and MO. At the SO stage, ${D_{pop}}$ is the length of ${J_S}$. The first particle's position is set to ${J_{init}}$, which is generated by encoding the initial source pattern. The positions of other particles are randomly initialized in the range of $[{0,1} ]$. At the MO stage, ${D_{pop}}$ is the length of ${T_S}$. The first particle's position is set to ${T_{init}}$, which is generated by encoding the initial mask pattern. For the position of the ${s_{th}}$ particle except for the first particle, the ${t_{th}}$ dimension is initialized to:

*r*is defined as the mask initialization probability factor, which ranges in $[{0,0.5} ]$.

Each dimension of the particle position corresponds to a mask pixel. According to Eq. (19), for the dimensions corresponding to the background pixels of the initial mask, the values randomly range in $[{0,1 - r} ]$. For the dimensions corresponding to the pattern pixels of the initial mask, the values randomly range in $[{r,1} ]$. As a result of Eq. (14), when the initialized particle is decoded to mask pattern, the mask pixel corresponding to a certain particle dimension will have a probability of $\frac{{0.5}}{{1 - r}}$ to remain the background or pattern area pixel, as it is in the initial mask pattern.

For example, with $r = 0.2$ and $T_{init}^t = 0$, the ${t_{th}}$ dimension of ${T_{init}}$ corresponds to a background pixel of the initial mask pattern. Then the ${t_{th}}$ dimension of ${p_s}$ will be initialized to a random value in range of (0, 0.8). The value has a probability of 62.5% to be smaller than 0.5, and 37.5% to be larger than 0.5. This dimension will be decoded to a background pixel if the value is smaller than 0.5. So the ${t_{th}}$ dimension of ${p_s}$ has a probability of 62.5% to remain a background pixel after decoding.

When $r = 0$, the positions of particle swarm are generated randomly ranging in $[{0,1} ]$. When $r = 0.5$, all the positions of particles are the same as ${T_{init}}$. Thus the positions of the initial particles in the solution space are partly restricted by *r*. Details about the impact of *r* on SMO are simulated and discussed in section 3.3.

Iteration starts after the particle swarm initialization. Taking ${F_{SMO}}$ as the fitness function, the fitness values of each particle are calculated. The better particle has lower fitness values. The swarm is then sorted according to the decreasing order of the particles’ fitness values. The learning probability ${P^L}$ is calculated for each particle in the sorted swarm:

Behavior learning occurs when the particles are updated. Except for the best particle, each particle (imitator) will correct its behaviors by learning from those particles (demonstrators) with lower fitness values. The velocity of the imitator is updated firstly by:

The ${N_{iter}}$ or ${N_{CoF}}$ is selected to be the termination condition of the SL-PSO sub-flow. The SO and MO processes are repeated ${N_{SMO}}$ times, and the optimized source ${S^ \ast }$ and mask ${M^ \ast }$ are then obtained by decoding the best particle in the swarm at the last iteration.

## 3. Simulations and results

#### 3.1 Simulation settings

In this paper, the annular source, of which the inner and outer partial coherent factors are 0.65 and 0.95, is exploited as the initial source before SMO. The polarization mode is TE. The CRAO of the source is $\theta = {6^ \circ }$. The azimuth is $\varphi = {0^ \circ }$ unless being specified, which means the mask pattern is located at the center of the exposure ring slit. A 3×3 matrix representing the Gaussian convolution kernel is used to calculate the source blur, and ${\sigma _k}$ is 0.7 pixel. The material of the absorber is TaN, whose complex index is 0.926-0.0436*j* and thickness is 70nm. The multilayer is composed of 40 bilayers of Mo/Si whose thicknesses are 2.78/4.17nm. The complex indexes of Mo and Si are 0.9238-0.0064*j* and 0.999-0.0018*j*, respectively. The material of mask substrate is SiO_{2}, whose complex index is 0.978-0.0108*j*. The up-sampling factor is 4 when calculating $\tilde{M}$. The numerical aperture and reduction of the projection optics are 0.33 and 4×. The pitches and sizes of mask patterns are on the wafer scale. Three target patterns are used in simulations, as shown in Fig. 5(a)∼(c), and the masks are all dark-field mask. The profile of the mask structure is shown in Fig. 5(d). The ${N_{SMO}}$ is set to 1 in this paper because that the optimization results is already relatively good after one loop of SO and MO. The method is also effective with other values of ${N_{SMO}}$. The simulation hardware is a desktop with an 8-core 3.6GHz CPU and 16GB memory.

#### 3.2 SMO results and analysis

Pattern 1, which is presented in Fig. 5(a), is employed in this section for the target pattern and the initial mask pattern. The periods in the *x* and *y* directions are both 130nm. ${N_M}$ is 65 and the pixel size is 2nm. ${N_O}$ is 31 for calculating the mask diffraction spectrum. The three bars are the same, whose width is 18nm in *x* direction and length is 90nm in the *y* direction. For SL-PSO, ${M_{pop}}$ is 100, $\kappa $ is 0.01, and *r* is 0.3.

Figure 6 compares the initial and optimized source and mask. The total runtime of the SMO flow is 286.2s. The optimized mask defocus $\varDelta {z^ \ast }$ is 160.2nm, and the optimized resist threshold $t_r^ \ast $ is 0.516. In order to validate the imaging quality of the optimized source and mask, comparisons of the print image and the target pattern are exhibited in Fig. 7. The *PE* values are 605, 270, and 32 for three statuses. It is apparent that the proposed EUV SMO method significantly reduces the *PE* and improves imaging fidelity.

Rigorous electromagnetic simulation is then carried out to validate the proposed method. The print image is calculated with the optimized source and mask by the RCWA and imaging model. The imaging results of RCWA and the thick mask model used in SMO are compared in Fig. 8. Rigorous simulation shows that the *PE* after SMO is 39, which is only 7 different from the result calculated by the thick mask model. It reveals that the proposed EUV SMO method is accurate and convincing.

Together with *PE*, *EPE*^{1} and *EPE*^{2} of different statuses are presented in Table 1. As shown in Table 1, the *EPE*^{1} is reduced nearly to 0nm after setting the mask defocus, which indicates that the pattern shift is corrected. The data in Table 1 prove the effectiveness of the proposed EUV SMO. Besides, the depth of focus (DOF) is calculated before and after SMO. The reference cutline is determined by two points, of which the coordinates are (18nm, 0nm) and (54nm, 0nm). The DOF values before and after SMO are 170.6nm and 184.3nm, respectively. Although the different imaging focus planes are not considered in the proposed SMO method, the DOF is increased after optimization.

To demonstrate the necessity of using a thick mask model, the SDM model is replaced by a binary thin mask model in the proposed method, with other processes remaining unchanged. The optimization results with the thin mask model are presented in Fig. 9. The optimized source and mask are much different from the results shown in Fig. 6.

The thin mask model ignores the incident light's direction, which results in no pattern shift at the wafer scale. Thus the mask defocus optimization is omitted during the SMO. Comparison of the print image calculated via the thin mask model and the target pattern is shown in Fig. 10(a). The *PE* is 31, which indicates that the SMO seems effective with the thin mask model. However, when the rigorous simulation is carried out with the optimized source and mask, the actual print image is much different from the target pattern, as shown in Fig. 10(b). The *PE* reaches 514, which results from the inability of thin mask model to accurately predict the 3D mask effects, such as pattern shift. Thus, the source and mask optimized with the thin mask model are not convincing. Based on the rigorous results, mask defocus is optimized to mitigate the pattern shift, as shown in Fig. 10(c). However, the *PE* reaches 103, which is still much larger than that in Fig. 8(b). The simulation results demonstrate that the SMO accuracy with the thin mask model is low, and the thick mask model is necessary.

The mask pattern is located at the center of the ring slit in the simulations above. However, the proposed SMO flow based on the thick mask model is applicable to patterns at random positions of the slit. Considering the real exposure process, the mask defocus and exposure dose are the same for patterns in the different positions of the ring slit. Thus, the optimized mask defocus and resist threshold are calculated when $\varphi = {0^ \circ }$, and are implemented in the optimization when $\varphi = {15^ \circ }$ and $\varphi = {30^ \circ }$. As shown in Fig. 11, the optimized sources are almost the same for patterns at different positions. The optimized masks differ slightly from each other, which results from the different shadowing effects varying with the different azimuths or different positions of the slit. For these two positions, *PEs* after SMO are 32 and 35, whose rigorous simulation results are 41 and 43, respectively. The results indicate that the optimization capability of the proposed method are high at random positions of the slit. The imaging fidelity is improved with the same resist threshold for the patterns, which means that the shadowing effects are also alleviated.

#### 3.3 Performance analysis

Compared to individual learning, social learning has the advantage of allowing individuals to learn behaviors from others. The SL-PSO algorithm exploited in the SMO flow introduces social learning mechanisms into PSO. Unlike the classical PSO where the particles are updated according to historical information, each particle in the SL-PSO learns from any better particles. It can significantly improve the optimization speed and efficiency.

The performance of the proposed SMO method based on SL-PSO is compared with three other heuristic algorithms, including PSO, CMA-ES, and JADE. Comparisons of the optimization results are exhibited in Table 2 and Fig. 12.

The data in Table 2 demonstrates that the algorithm efficiency of PSO is the worst among the four algorithms. The *PE* and *EPE*^{2} values are still large after optimized with PSO. JADE algorithm has good performance for SO and lousy performance for MO. The reason is that the optimization dimension of MO is much higher than that of SO, and the optimization capability of JADE for high dimension problems does not satisfy the MO stage in the SMO. The CMA-ES algorithm performances are excellent for SO and MO. The CMA-ES results are only slightly worse than the SL-PSO results. However, the CMA-ES speed is the slowest due to the eigenvalue decomposition (EVD) operation in the algorithm. Besides, the speed of CMA-ES will be slower due to the EVD when the optimization dimension increases.

It can be seen from Fig. 12 that the difference between the print image and target pattern is smallest when using SL-PSO. Although the imaging quality is pretty good for CMA-ES, the mask complexity of the optimized mask obtained by CMA-ES is much higher than that by other algorithms, which results in low manufacturability. Further comparison of the algorithms’ performance is shown in Fig. 13 via convergence curves. The red dotted line in the Fig. 13 represents the separation of SO and MO stages. The SL-PSO starts to converge when the fitness function is called 43000 times, while the CMA-ES starts to converge when the fitness function is called more than 50000 times.

Overall, these results from Table 2, Fig. 12 and Fig. 13 demonstrate that SL-PSO has better performance than the other three algorithms. With the help of SL-PSO, the proposed SMO method has high optimization efficiency.

Furthermore, in order to validate the performance of the proposed SMO method when optimization dimension increases, the ${N_M}$ is changed to 95 while the other parameters unchanged. The performance comparisons of different algorithms are listed in Table 3. Table 3 shows that the optimization speed and efficiency of SL-PSO remains the best when optimization dimension increases. It should be noted that the *PE* values after SMO is pretty large because that the total CoF number remains unchanged. Increasing of CoF number with larger ${N_M}$ will be helpful to get better optimization results.

Next, the impact of mask initialization probability factor *r* on the SMO is analyzed. According to the discussions in section 2.3, the particles in the initial swarm are all the same as ${T_{init}}$ when $r = 0.5$. Thus the optimization will fail as the velocities of particles are initialized to 0. In this part, SMO is implemented with setting *r* to 0, 0.1, 0.2, 0.3, and 0.4. The optimized masks with different *r* are presented in Fig. 14. The mask patterns in Fig. 14 are final optimization results after SMO.

Only optimized masks are presented since *r* has no impact on the SO stage. From the visual point of view, the mask complexity decreases with the increase of *r*. Total variation (TV) is usually used to represent the mask complexity quantitatively [15]. The lower TV is, the higher manufacturability the mask will have. The *PE* and TV values with different *r* are listed in Table 4.

For a particle's specific dimension in the initial swarm, the corresponding pixel after decoding will have a higher probability of maintaining the background pixel or pattern pixel when *r* increases. The particles are restricted by *r* to the neighboring area of the first particle, which corresponds to the initial mask pattern. Thus, the optimized particle will have a higher probability of being found near the particle corresponding to the initial mask pattern when *r* gets larger.

The *PE* values are larger with *r*=0 and *r*=0.4. When *r* is too small, the distribution of the initial swarm is more stochastic and broader. It results in difficulties to efficiently find the best particle in a few iterations. When *r* is too large, the searching space of the initial swarm is so small that the algorithms will be less likely able to find the optimized particle globally. The results indicate that *r* can also influence the imaging quality after SMO besides the mask manufacturability. So *r*=0.3 in other simulations is the trade-off between the algorithm efficiency and mask complexity.

Further comparison of different *r* values is presented in Fig. 15 via convergence curves. As mentioned above, the SO stage is not influenced by *r*. At the MO stage, the *PE* will decrease more early with larger *r*, which indicates that it is easier to find a better particle when the initial searching space is small. In conclusion, although the heuristic algorithms such as SL-PSO benefit from global optimization capability, the optimized mask's manufacturability may be lousy if the searching space is too large. The restriction of the searching space in MO, which uses the proposed initialization strategy, can improve the optimization efficiency and mask manufacturability. Compared to traditional methods to improve the mask manufacturability, such as the penalty function method and mask filter method, the approach in this paper introduces no extra calculation and has little impact on the SMO's speed.

#### 3.4 SMO results of complex patterns

Pattern 1 exploited in section 3.2 and 3.3 is a simple and XY-symmetric pattern. Validation about the generality of the proposed SMO method is carried out with pattern 2 and pattern 3 in this section. Pattern 2 is a complex and XY-symmetric pattern. The periods in the *x* and *y* directions are both 405nm. ${N_M}$ is 81 and the pixel size is 5nm. ${N_O}$ is 41 for the calculation of mask diffraction spectrum. CD of pattern 2 is 15nm. Pattern3 is an asymmetric pattern. The periods in the *x* and *y* directions are both 159nm. ${N_M}$ is 53 and the pixel size is 3nm. ${N_O}$ is 31 for the calculation of mask diffraction spectrum. CD of pattern 3 is 15nm. Besides, the capping layer of Ru with 2nm thickness above the multilayer structure was considered in the simulations of pattern 3. The complex index of Ru is 0.9032-0.0127*j*.

The optimization results of these two target patterns are presented in Fig. 16, and the merit function values are listed in Table 5.

Simulations results show that the proposed EUV SMO method is applicable to different target patterns and has the capability to improve imaging quality.

## 4. Conclusion

An SMO method for EUV lithography based on the thick mask model and SL-PSO algorithm has been proposed in this paper. A series of simulations show that pattern error is significantly reduced, and the imaging fidelity is improved by the proposed method. Simulations with patterns located at different positions of the ring slit show that the proposed method is able to not only improve the imaging fidelity, but also alleviate the shadowing effect for patterns at random positions of the slit. In comparisons between different heuristic algorithms, SL-PSO shows better performance in efficiency and speed. The impact of the mask initialization probability factor *r* on the optimization is then analyzed. By tuning *r*, the optimization efficiency of MO and the manufacturability of optimized mask are both improved. Simulation with two complex patterns verified the generality of the proposed SMO method. Further research will focus on improving optimization efficiency when the number of mask pixels increases significantly. Researches on eliminating the pattern shift at the wafer defocus plane via SMO will be carried out.

## Funding

National Major Science and Technology Projects of China (2017ZX02101004-002); Natural Science Foundation of Shanghai (17ZR1434100).

## Disclosures

The authors declare no conflicts of interest.

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