1. Introduction

For decades, projection lithography has pursued higher resolution by various methods [1], including alternating phase shifting mask [2], off-axis illumination [3], and multiple patterning [4]. Among these methods, mask optimization (MO) [5] is one of the most promising methods to explore the limits of lithography and has been widely used in deep ultraviolet lithography (DUVL) [6–8]. As the successor of the DUVL, extreme ultraviolet lithography (EUVL) bears not solely the optical proximity effect but also the severe thick mask effect [9], which makes the MO an indispensable tool to extend the lifespan of Moore’s law [10].

The MO has been studied extensively since its first commercialization in the 2000s [11,12], which can be classified into the end-to-end and iterative MO. The former [13–16] trains neural networks with thousands of reference datasets obtained from the iterative MO, after which optimized mask patterns can be predicted directly. The end-to-end MO is fast, while it cannot guarantee obtaining a local optimal solution and usually is used to generate initial patterns for the iterative MO [17]. The iterative MO simulates the lithography process using the mask model, projection lens model and resist model to get wafer patterns, then iteratively optimizes mask patterns based on the simulation results until wafer patterns meet requirements. According to optimization algorithm, the iterative MO can be divided into the heuristic- and gradient-based MO. Although the heuristic-based MO does not need the gradient, it is typically slower than the gradient-based MO [18–20]. Therefore, the gradient-based MO is used most frequently and it is essential to calculate the gradient for reducing the runtime of the MO.

The difficulty of gradient calculations mainly depends on the thick mask model, while the derivations of other components of the lithography model like the projection lens model and resist model are well-established [21,22]. Ma et al. proposed a gradient calculation for boundary layer model [23] in the DUVL. However, the mask model is inapplicable for the EUV mask with the severe thick mask effect. Considering that the application of curvilinear patterns is inevitable with the emerging of multi-beam mask writing (MBMW) [24,25], it is also necessary for the mask model to be compatible with complex curvilinear patterns. To meet the requirements, a fast mask model based on modified Born series (MBS) [26] was proposed in our previous paper. However, there is a lack of gradient calculation methods corresponding to this mask model for the highly efficient MO.

Adjoint method is widely applied for optical inverse design [27,28]. With only two electromagnetic simulations, one can calculate the gradient of an objective function with respect to all design variables in the simulation domain [29]. In this work, the adjoint method is introduced to the gradient calculation of the EUV thick mask model for the first time. With rigorous consideration of the thick mask effect, the mask can be optimized iteratively based on the gradient. Compared to the DUVL, there are more differences between wafer patterns and mask patterns in the EUVL because of the more significant thick mask effect. Therefore, EUV masks need more optimization. To reduce the computational cost of the EUV mask optimization, a linearized gradient, which ignores resist models in the gradient calculation, is proposed. A detail of the gradient calculation is presented in Section 2. The effectiveness and superiority of the proposed method are demonstrated with two examples of the EUV mask optimization implemented in a two-step strategy in Section 3.

2. Theory and methods

2.1 Mask optimization

The MO is treated as a minimization problem, in which the mask pattern is the design variable and the normalized wafer pattern fidelity is the cost function. The mask pattern is represented by ${\mathbf {\Theta }} \in {\mathbb {R}^{M \times N}}$. The cost function is defined as the normalized square of the Euler distance between the wafer pattern and target pattern, which implies the average edge placement error along the target contour.

With the aim of high performance and manufacturability, a two-step optimization strategy is applied, which is shown in Fig. 1. In the coarse optimization step, the continuous mask is optimized with the linearized gradient, until the normalized cost function is smaller than 1 or the iteration number exceeded 60. Then, the obtained mask pattern is binarized and filtered. In the fine optimization step, the binary mask pattern is represented by the level set method [30] and optimized with the normal gradient. A fixed number of iterations are conducted to find the minimum of the cost function in this step. There is a similar strategy that uses an aerial image penalty term as the cost function in the coarse step [31]. Note this work applies different gradients instead of different cost functions between the two steps, while the substitution of the cost function might lead to ineffective optimization.

 

Fig. 1. A two-step optimization strategy. In the coarse optimization step, the continuous mask is optimized with the linearized gradient. Then, the obtained mask pattern is filtered and binarized to improve the manufacturability. In the fine optimization step, the binary mask pattern is represented by the level set method and optimized with the normal gradient.

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The optimized patterns might contain small undesired artifacts, such as isolated voids, protrusions, and irregular edges, which can result in poor manufacturability [21,32–33]. These artifacts can be taken as noise and removed by filtering operation between the two steps [34]. During the level set evolution, an extra motion, driven by the mean curvature of the pattern contour [30], is applied, which can contribute to meeting the curvature requirements of the mask manufacturing rules checking (MRC) and prohibit the emergence of small features. To some extent, these methods can enhance the manufacturability.

2.2 Rigorous EUVL model

A lithography simulation model usually consists of a mask model, a vector projection lens model, and a resist model. Considering the complex thick mask effect, a mask model based on the MBS is adopted, which significantly outperforms the FDTD in terms of speed while maintaining comparable accuracy [26,35]. The mask model can be represented as

The feature of resists is the nonlinear response to the intensity [37], which is usually approximated by a sigmoid function. The wafer pattern can be obtained from the aerial image.

2.3 Gradient calculation

The primary challenge in gradient calculations of the rigorous EUVL model lies with the EUV mask model. To solve this problem, the adjoint method is introduced for the first time. The gradient of the cost function can be expressed as

Different from the DUV MO, there are much more modifications needed in the EUV MO due to the more prominent thick mask effect. Besides, it is found that the modification of the mask pattern for each iteration mainly occurs at the edge of previous patterns. Many iterations with heavy computational costs are needed to obtain satisfactory mask patterns, which can be attributed to the nonlinear response of the resist. At the region with the intensity away from the threshold of the resist, a small perturbation in the intensity cannot induce a discernible change on the wafer pattern. Therefore, the gradient away from the edge of the mask pattern is nearly zero. Another reason for the inefficient modification is that the definition of the gradient is based on the assumption of linearity. However, the cost function is inconsistent with the assumption because of the nonlinear response of resists.

To solve this contradiction and accelerate the optimization, the linearized gradient is proposed. The partial derivative of the cost function $F$ with respect to the intensity ${\mathbf {I}}$ is

Then, the linearized gradient can be obtained by ignoring the resist model.

The linearized gradient allows pattern modification anywhere, rather than being confined solely to the edge of previous patterns. Note that the linearized gradient is not as accurate as the normal gradient, but it is more effective than the latter. Two examples are provided in Section 3. to show the superiority of the linearized gradient.

3. Results and discussion

The proposed method can be applied to various situations. It is necessary to clarify the model and optimization parameters. The wavelength was 13.5 nm. The s-polarized light hit the mask at an incident angle of 6$^{\circ }$. The binary dark-field EUV mask consisted of the absorber and multilayer [38]. The absorber material was TaN with the refractive index $\rm {N} = 0.9385 + 0.03776 i$. The absorber thickness was 50 nm. The multilayer stack parameters from Ref. [39] were adopted. Note that the proposed method is also applicable to the attenuated phase-shift mask with standard multilayer blanks [40]. For the projection lens model, the demagnification factor and numerical aperture (NA) were 4 and 0.33, respectively. The aberration term was ignored. The steepness $a$ and threshold $t_{\rm {r}}$ in the resist model were 80 and 0.4, respectively. The resist model was used in the lithography simulations and the calculation of the normal gradient, while it would be ignored in the calculation of the linearized gradient. Equation (2) was solved by the steepest descent method iteratively. The initial step size $s$ for the first iteration was $0.75\pi /\max (\partial F/\partial {\mathbf {\Theta }})$, while $s$ for other iterations was the optimized step size of previous iterations. The actual step sizes were obtained by the Brent’s method [41] with an evaluation number of 5 and a maximum step size of $2s$. The filter used was a Gaussian filter with a standard deviation of 6. The strength factor of the curvature motion in the level set evolution is 0.008.

All the computations were performed on a virtual machine based on the Intel Xeon Gold 6230R architecture, with 8 allocated cores and 32 GiB of memory. Besides, one NVIDIA Tesla V100S with 32 GiB of memory was used. The calculations were implemented using MATLAB. The maximum GPU memory usage was approximately 7.5 GiB during the MO.

3.1 Gradient verification

Before optimizing the EUV masks, it is necessary to verify the accuracy of gradient calculations. The reference gradient was obtained from a simple first-order difference.

 

Fig. 2. An example for verification of the gradient calculation. (a) The distribution of design variable ${\mathbf {\Theta }}$. (b) The target wafer pattern. (c) Comparison between the reference gradients with perturbations of different sizes and that from the adjoint simulation at a line segment with y = 0 nm and x = [-200, 300] nm.

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3.2 Mask optimization examples

Two examples of the EUV thick mask optimization are presented in this section. The target patterns, including a logic pattern and a BigMaC [43] pattern, are discretized into $664 \times 672$ and $700 \times 632$ pixels, respectively. The pixel size is 3.375 nm. The results are shown in Fig. 3 and Fig. 4. The minimum dimensions of both the patterns are 20 nm on wafer scale. The upper-left figure shows the original and optimized patterns in green and blue, respectively. The corresponding wafer patterns, compared with the target pattern in red, are shown in the right figure. The figures below are the corresponding zoomed-in views.

 

Fig. 3. The mask optimization for the logic pattern with the minimum dimension of 20 nm on wafer scale. (a) The original and optimized mask pattern. (b) The corresponding wafer pattern. The figures below are the corresponding zoomed-in views.

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Fig. 4. The mask optimization for the BigMaC pattern with the minimum dimension of 20 nm on wafer scale. (a) The original and optimized mask pattern. (b) The corresponding wafer pattern. The figures below are the corresponding zoomed-in views.

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Due to the mask shadow effect [9], the original wafer patterns shrink compared to the target patterns. To compensate for the shrinkage, the openings of the mask patterns are enlarged, which provides excellent contour fidelity in both examples. The aerial and resist images are provided in Supplement 1. Compared to the logic pattern, the smaller interval between the features in the BigMaC pattern would lead to stronger interaction. Therefore, it is more challenging to optimize the BigMaC pattern. For the logic and BigMaC pattern, the cost functions of the optimized patterns are 90% and 86% smaller than those of the original patterns, respectively.

The convergence of the cost function with the linearized and normal gradient is shown in Fig. 5. For the case of the normal gradient, the normal gradient was used in both two steps. There are some turning points at the normalized cost of 1, at which the mask patterns are binarized and filtered to reduce the complexity. With the linearized gradient, the numbers of iterations required to reach the turning points are decreased by about 44% and 57% for the logic and BigMaC pattern, respectively, which demonstrates the superiority of the linearized gradient.

 

Fig. 5. Convergence of normalized cost function with linearized and normal gradient for (a) the logic and (b) the BigMaC pattern. The optimization adopts the two steps strategy. In the coarse step, the iteration stops when the normalized cost is smaller than 1. Then, the obtained masks are binarized and filtered, which leads to the turning points of the cost function. The filtered masks are optimized for 36 iterations in the fine step to get the optimized masks.

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The runtime of the mask optimizations is shown in Table 1. The forward simulations of EUV masks are accelerated by the initial condition from the field of the mask with the last pattern (PatternACC) [26]. In other words, the greater the similarity between the patterns, the less time needed. Given the linearized gradient tends to introduce more modifications than the normal one, the reduction in time, which is 42% and 47% respectively, is slightly less pronounced than the reduction in iteration number. Due to the step size search, the forward simulation has been executed five times more frequently than the adjoint simulation. Then, the estimated average effect of the PatternACC with the linearized gradient is about 48%. The average runtime with the linearized gradient is about 400 seconds.

Table 1. Runtime of the mask model and MO. The data are presented in seconds.

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The mask optimizations in this work are conducted under a single source point. The proposed method can be extended to the partially coherent illumination with the Abbe integration [44]. However, the PatternACC may be unsuitable for the partially coherent illumination due to the difficulty of handling vast amounts of data. Assuming that the runtime of the MO is linear to the number of source points $N_{\rm {p}}$, the runtime is estimated as $400 \times 2 \times N_{\rm {p}}$ seconds. For example, the runtime of the MO with 108 source points is 1 day. Further investigation about the selection of the source points and speed improvement, is required for a reliable and pratical MO under the partially coherent illumination. Given that the current NA is 0.33, the runtime can be reduced by replacing the vector model with a scalar one, while additional research about the impact on accuracy is required. The vector model used in this work can be applied to situations with high NA.

4. Conclusion

In summary, a linearized EUV mask optimization based on the adjoint method is proposed. The adjoint method is first introduced into gradient calculations of the EUV thick mask model, which makes the gradient-based rigorous EUV MO possible. Moreover, the linearized gradient is proposed to reduce the iterations of the optimization. It was shown that the linearized gradient was more efficient than the normal one in the coarse optimization step, and half of the iterations can be saved. This work contributes a solid foundation for the further development of EUV computational lithography, and future studies can explore its application on partially coherent illumination with acceptable runtime.