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Abstract
With the continuous reduction of critical dimension (CD) of integrated circuits, inverse lithography technology (ILT) is widely adopted for the resolution enhancement to ensure the fidelity of photolithography, and for the process window (PW) improvement to enlarge the depth of focus (DOF) and exposure latitude (EL). In the photolithography, DOF is a critical specification which plays a vital role for the robustness of a lithographical process. DOF has been investigated to evaluate the optimization quality of ILT, but there is not a clear scenario to optimize the DOF directly. In this paper, the source and mask optimization (SMO) based on defocus generative and adversarial method (DGASMO) is proposed, which takes the source, mask and defocus as variables, and the inverse imaging framework employs the Adam algorithm to accelerate the optimization. In the optimization process, the penalty term constantly pushes the defocus outward, while the pattern fidelity pushes the defocus term inward, and the optimal source and mask are constantly searched in the confrontation process to realize the control of DOF. Compared to SMO with the Adam method (SMO-Adam), the PW and DOF (EL = 15%) in DGASMO maximally increased 29.12% and 44.09% at 85 nm technology node, and the PW and DOF (EL = 2%) at 55 nm technology node maximally increased 190.2% and 118.42%. Simulation results confirm the superiority of the proposed DGASMO approach in DOF improvement, process robustness, and process window.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As one of the manufacturing strategies of integrated circuits (IC), optical lithography systems are constantly evolving to follow the steps of Moore's Law [1]. Figure 1(a) illustrates the schematic diagram of the imaging system for deep ultraviolet (DUV) lithography with 193 nm illumination [2,3]. First, the mask carrying the integrated circuit layout is uniformly illuminated and an optical signal is generated. Second, the optical signal is then transferred to the photoresist-covered wafer through an optical projection system, and the photochemical reaction of the photoresist is induced. Finally, the wafer pattern is printed after a series of developing processes. However, the fidelity of printed wafer pattern is distorted due to optical proximity effect (OPE) and so on [4–6]. Thus, a series of ILT schemes such as mask optimization (MO) method [7–10] and source mask optimization (SMO) [11–14] were proposed and become essential to compensate for the systematic distortions of the lithographic images. As shown in Fig. 1(b), SMO improves pattern fidelity by planning the intensity distribution of the source, pre-distorting the mask pattern, and inserting sub-resolution assist features (SRAF) around the main features (MF).
Fig. 1. (a) Schematic of a DUV lithography system, in which a 193 nm annular partially coherent light source is used, the IC layout on the mask is transferred to the wafer surface by a projection exposure and development process. (b) Source and mask optimization (SMO) method based on ILT, where the pixelated light source and mask are numerically solved through an inverse imaging process to improve the pattern fidelity.
With the continuous reduction of critical dimension (CD), the optical lithography is pushed into the low-k1 regime, and the printed wafer image becomes more sensitive to process variations [15,16]. Meanwhile, in the photolithography process, due to the existence of wavefront aberrations [5], thermal aberrations [17], thermal mask effect [18], thick mask effect [19], flare effect, and unevenness of wafer surface and photoresist surface [20], the imaging of the photolithography system will inevitably be offset in the focal plane, which further presents reliability challenges and stringent requirements for ILT. Thus, various computational strategies are proposed and applied to SMO to ensure process manufacturability. Along the way, the SMO in the assigned defocus plane was studied by Peng et al [21] and further explored by Ma et al. [22], but the pattern fidelity at different defocus planes is still difficult to ensure. Recently, Jia et al. proposed the statistical multivariable-based OPC methods [23–25], which combine random defocus and a variety of evaluation indicators while searching the light source and mask, and greatly improve the pattern fidelity at different focal planes. Subsequently, based on the advent of the ASML wavefront manipulator equipment, Sears et al. proposed the pupil wavefront optimization (PWO) approaches [26–28] to compensate for thick mask induced aberration (TMIA), and a series of optimization methods based on the Zernike polynomials [29,30] are proposed and used to improve the manufacturability of the process [31,32]. Meanwhile, the differentiable expression of edge placement error (EPE) and process variation band (PVB) combined with dose and defocus was developed by Gao et al., and the MO based on process window (PW) awareness was proposed [33]. Shen et al. summarized previous approaches and combined with a reinforcement learning algorithm to optimize EPE and PVB in the vector lithography model [34], which further improved the robustness of SMO. Later, Li et al. proposed the defocus robust SMO (DRSMO) method [35], which combines defocus disturbance and mini-batch gradient descent (MBGD) algorithms, and further reduced the sensitivity of lithographic imaging to defocus. Besides, the compressive sensing focus on the manufacturability of optimized sources and masks [36,37], while also improving the PW at the same time.
Unfortunately, the improvement of the DOF by the above methods is always indirect, namely, the control of DOF is passive, and the corresponding optimization scheme and quantifiable optimization indicators are not clearly given. To eliminate the change and uncertainty of the pattern error (PAE) caused by the inherent defocus effectively, it is necessary to develop a method which can directly plan and control the DOF in the SMO, thereby reducing the sensitivity of the optimized source and mask to the defocus variation. Thus, reducing the sensitivity of the optimal source and optimal mask to defocusing under the condition of expanding defocusing is an effective strategy, based on which we propose a defocus generative and adversarial SMO (DGASMO) method. In the generative adversarial model, the optimized objects not only contain source and mask, but also focus on defocus interference. Specifically, the cost function in the DGASMO method includes the pattern error (PAE) at the focal plane, the PAE under defocus plane, and the defocus generation term. Here, the defocus generating term, also known as the penalty term in convex optimization, is constantly generating defocus conditions and disturbing the imaging system. On another hand, the PAE convergence at the focal plane and the defocus plane drives the optimization towards the reduction of defocus. As a result, the defocus reduction and defocus generation are constantly confronted and search for a robust solution of the source and mask. Meanwhile, we analyze the relationship between normalized image log slope (NILS) and EL and equate the PW as a Defocus-NILS window, which shows the superiority of the proposed DGASMO method in improving process manufacturability. It is worth noting that we borrow the Adam reinforcement learning algorithm to improve the optimization efficiency.
2. Forward imaging model
As depicted in Fig. 1(a), the transition from mask patterns to wafer patterns can be decomposed into two numerical models: the optical projection (aerial image formation) based on Fourier optics and Abbe method, and the resist effect (wafer image formation) described by the sigmoid function. Considering a polarized electric field emanated by a point source at $({{\alpha_s},{\beta_s}} )$, the analytic expression of aerial image ${{\bf I}_a}$ at the focal plane can be described as [5,38,39]
It should be noted that the aerial images ${{\bf I}_a}$ at defocus plane can be described as the effect of aberration on imaging [35]. And the defocus, ${\bf Def}(\delta )$, which can be thought of as the phase change on equivalent filters ${\bf H}_p^{{\alpha _s}{\beta _s}}$, it can be computed by
The photochemical reaction of photoresist is described by the differentiable sigmoid model $sig(x) = {1 / {({1 + {e^{ - a(x - {t_r})}}} )}}$, with a and ${t_r}$ are the steepness and threshold of sigmoid function respectively. Together with the optical projection, the wafer image can be uniformly calculated as
3. Defocus generative and adversarial SMO
As illustrated in Fig. 2, the forward calculation of the DGASMO framework computes the wafer images at focal plane ${\bf I}_w^{\delta \textrm{ = }0}$ and defocus plane ${\bf I}_w^\delta$. Combined with the desired target pattern and defocus generating term $\mathrm{{\cal S}}_\delta ^{gen}$, the inverse optimization updates the source, mask, and defocus parameters by minimizing the “score $({{\mathrm{{\cal S}}_L}} )$ “. In this work, the Lagrange equation based formula ${\mathrm{{\cal S}}_L}$ is defined as ${\mathrm{{\cal S}}_L} = {\lambda _{pe}}\mathrm{{\cal S}}_{pe}^{\delta = 0} + \;{\lambda _{ad}}\mathrm{{\cal S}}_\delta ^{ad} - {\lambda _{gen}}\mathrm{{\cal S}}_\delta ^{gen}$, in which, $\mathrm{{\cal S}}_{pe}^{\delta = 0}$ and $\mathrm{{\cal S}}_\delta ^{gen}$ are the PAEs at focus and defocus plane, the weights ${\lambda _{pe}} = {1 / {|{\mathrm{{\cal S}}_{pe}^{\delta = 0}} |}}$, ${\lambda _{ad}} = {1 / {|{\mathrm{{\cal S}}_\delta^{ad}} |}}$, ${\lambda _{gen}} = {1 / {|{ - \mathrm{{\cal S}}_\delta^{gen}} |}}$ are set and used to normalize the various cost functions, $| \cdot |$ means that the value has no differentiable properties. $\mathrm{{\cal S}}_\delta ^{gen}$ are derived by Karush-Kuhn-Tucker (KKT) condition, which is used to ensure a solution of the optimization problem. For brevity in the subsequent derivation, we define ${\bf Def}(\delta )\textrm{ = }{\mathrm{{\cal F}}^{\textrm{ - }1}}\{{\textrm{exp} [{j\delta \cdot {{\bf \Gamma }_{def}}} ]} \}$, details of ${{\bf \Gamma }_{def}}$ will be shown in the Appendix A. For the binary mask, the pixel values are between 0 and 1, we apply the parametric transformation such that
Fig. 2. Forward calculation and inverse optimization process for defocus generative adversarial SMO (DGASMO).
3.1 DGASMO inverse optimization framework
To detail, given a desired target pattern ${{\bf I}_0} \in {{\cal R}^{N \times N}}$, the goal of DGASMO is to search for the optimal source $\hat{{\bf J}} \in {{\cal R}^{{N_s} \times {N_s}}}$ and optimal mask $\hat{{\bf M}} \in {{\cal R}^{N \times N}}$ which minimize the mismatch between ${{\bf I}_0}$ and $\mathrm{{\cal T}}\{\{{\; \cdot \;} \}$ through inverse imaging and numerical optimization algorithms while maximizing defocus $\delta $. In this work, the measured dissimilarity is defined as “score $(\mathrm{{\cal S}} )$ “, and $\mathrm{{\cal S}}_{pe}^{\delta \ne 0}$ represents the pattern fidelity (PAE) at a defocus plane. Thus, the uniform expression of PAE at focal plane and defocus plane can be calculated as
The traditional SMO methods improve the pattern fidelity at the focal plane by minimizing $F = \mathrm{{\cal S}}_{pe}^{\delta = 0}$, but the image fidelity with an uncertain defocus plane is difficult to control, the optimized source and mask still has a relatively high sensitivity to defocus. In order to effectively eliminate the effect of defocus, it is necessary to introduce a defocus term in the penalty function. Thus, Peng et al. [21] set the defocus to 100 nm by minimize $F = {\lambda _{pe}}\mathrm{{\cal S}}_{pe}^{\delta = 0} + \;{\lambda _{ad}}\mathrm{{\cal S}}_\delta ^{ad}$, Jia et al. [23] introduced random defocus in $F = {\lambda _{pe}}\mathrm{{\cal S}}_{pe}^{\delta = 0} + \;{\lambda _{ad}}\mathrm{{\cal S}}_\delta ^{ad}$. At the same time, we pay more attention to the generative and adversarial problem of defocus, namely, the optimization problem with boundary conditions given by the Eq. (10), which leads the update of defocus $\delta $ to be considered based on Eq. (12). In Section 3.2, a defocus generation term $\mathrm{{\cal S}}_\delta ^{gen} ={-} ({{\delta_0}/\delta } )$ is designed depending on the defocus value $\delta $. The update direction vector ${{\partial \mathrm{{\cal S}}_\delta ^{gen}} / {\partial \phi }}$ is derived in the Appendix A.
3.2 Regularization of defocus generation term
As mentioned previously, $\mathrm{{\cal S}}_\delta ^{gen}$ generates and plans defocus $\delta $ during SMO to ensure the robustness of optimized source $\hat{{\bf J}}$ and mask $\hat{{\bf M}}$. Focusing on the defocus generative and adversarial problem, the Euler-Lagrange equation corresponding to $\delta $ is computed as:
Thus, ${{\partial \mathrm{{\cal S}}_\delta ^{gen}} / {\partial \delta }}$ is designed to be a conservative function with a lower bound. In cooperation with defocus generative problem, which is employed to generate defocus near the focal plane and converge on the infimum when the defocus is too large, a monotonic lower bound equation ${{\partial \mathrm{{\cal S}}_\delta ^{gen}} / {\partial \delta }}\textrm{ = }{1 / {{\delta ^2}}}$ is devised to guarantee that the formula has a solution (other functions can be readily applied), with the defocus generation term can be expressed as:
3.3 Adam optimizer
Although SGD provides a feasible scheme for minimizing ${\mathrm{{\cal S}}_L}$, however, the optimization suffers from the challenge of the sensitivity on the step-size $\eta $, which is extremely serious in multi-task optimization (optimization with constraints) as we studied before. Thus the Adam optimizer [43] is employed in this paper, with the updated rules at time-step t+1 is
4. Defocus-NILS window
The lithography process robustness is usually described by the focus-exposure process window (PW), which includes all pairs of DOF and EL that satisfy the CD error condition: $\varDelta \textrm{CD} \le 10{\%}$. Without loss of generality, Fig. 3(a) gives the cross-sections of the aerial image exposed by nominal dose and dose’, with the CD of wafer image corresponding to CD1 and CD2 after photoresist effect. Figure 3(b) shows the slope of the aerial image on the image contour, which is defined as NILS:
Fig. 3. Relationship between exposure latitude (EL) and the NILS. (a) The cross sections of aerial images with different dose and (b) the cross sections based on NILS.
5. Numerical results
Numerical simulations are performed on a DUV lithographic imaging system with 85 nm and 55 nm technology node, and the parameters used for simulations are: $\lambda = 193\textrm{nm}$, $\textrm{NA} = 1.35$, resolution $\Delta x = \Delta y = 4\textrm{nm/pixel}$, ${N_S} = 29$, ${N_M} = 256$, transverse magnification R = 4, refractive index ${n_w} = 1.44$, the steepness and threshold of sigmoid function being $a = 80$ and $tr = 0.2$. The system is illuminated by a partially coherent annular source ${\bf J}$ with an inner radius ${\delta _{inner}} = 0.6$ and the outer radius ${\delta _{outer}} = 0.9$. ${\bf J}$ and two desired target pattern ${{\bf I}_{01}}$ (CD = 85 nm) and ${{\bf I}_{02}}$ (CD = 55 nm) are given in Fig. 4.
Fig. 4. (a) The annular source ${\bf J}$. (b) The desired target pattern ${{\bf I}_{01}}$ and (c) The desired target pattern ${{\bf I}_{02}}$. The averaged-NILS and PW are computed based on the critical locations marked by the yellow lines.
Optimization results with desired target pattern ${{\bf I}_{01}}$ are given in Fig. 5. In the columns of Fig. 5, from left to right are illumination source, mask, wafer images at $\delta \;$= 0 (focal plane), 100 and 200 nm. In row (a) of Fig. 5, optimized source and mask are derived using the traditional SMO, with the improved PAEs are 290, 723, and 6268 at $\delta \;\textrm{ = }$ 0, 100 and 200 nm, and the NILS from 1.207 decreases to 0.929 and 0.264 with the defocus increases from 0 to 100 and 200 nm, row (b) shows the imaging results with SMO-Adam method, and the improved PAEs = 230, 584, 4271 and NILS = 1.407,1.136, 0.449 at $\delta \;\textrm{ = }$ 0, 100 and 200 nm, respectively. Row (c) gives the DGASMO imaging results using defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$, with the PAEs are 215464, and 3621 at $\delta \;\textrm{ = }$ 0, 100 and 200 nm, where the proposed method achieves better performance of PAEs. Similarly, compared with the traditional SMO algorithm, NILS at $\delta \;\textrm{ = }$ 0, 100 and 200 nm increased to 1.428, 1.176 and 0.511, which has the similar performance to the method proposed by Jia [23]. Row (d) compares the imaging results with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / {{\delta ^4}}}$, with the improved PAE is 227 at $\delta \;\textrm{ = }$ 0, and the better performance of PAEs are 357 and 2561 at $\delta \;\textrm{ = }$ 100 and 200 nm, with the NILS improved to 1.389, 1.186 and 0.638. Row (e) is optimized using defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / {{\delta ^9}}}$, which has the best imaging results at defocus plane, the improved PAEs are 244, 281, and 1378 at $\delta \;\textrm{ = }$ 0, 100 and 200 nm, and the NILS from 1.399 decreases to 1.2267 and 0.873 with the defocus increases from 0 to 100 and 200 nm. In particular, it can be observed that the assistant features of the mask (the area in the red box) become more and more obvious as the $\mathrm{{\cal S}}_\delta ^{gen}$ is set more aggressively, because the aggressive $\mathrm{{\cal S}}_\delta ^{gen}$ generates a higher defocus value to be included in SMO, while the defocus adversarial term $\mathrm{{\cal S}}_\delta ^{ad}$ continuously reduces the influence of defocusing on SMO, and two terms form a dynamic confrontation. Thus, the negative impact of defocus on imaging can be effectively compensated, and the optimization process does not lose robustness due to the modulation by KKT conditions. Although the generative and adversarial problem introduces interference to the optimization of source and mask, DGASMO achieves better global performance with the same iterations, which is realized based on the adaptive adjustment ability of the Adam enhanced algorithm.
Fig. 5. Simulation results with ${{\bf I}_{01}}$ as target pattern. Columns from left to right: the optimized source pattern $\hat{{\bf J}}$, the optimized mask pattern $\hat{{\bf M}}$, and the printed wafer image ${\bf I}_w^\delta $ at 0 nm, 100 nm and 200 nm defocus. Rows (a-e): the traditional SMO method, the SMO-Adam method, the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$, $\textrm{ - }{{1 / \delta }^4}$ and $\textrm{ - }{{1 / \delta }^9}$, respectively.
Figure 6(a) depicts the optimized defocus–PAE of desired target pattern ${{\bf I}_{01}}$, with the defocus range from −200 nm to 200 nm. Where the black dotted, black, green, blue, and red curve corresponds to the imaging results of rows (a-e) in Fig. 5, respectively. Where PAEs greatly decreased from traditional SMO (black dotted curve) to SMO-Adam method (black curve), while the PAEs of each curve gradually decreased in the order of DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Compared with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve)and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve), the most aggressive defocus generation term $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve) greatly reduces the sensitivity of imaging to defocusing, and the printed wafer image ${\bf I}_w^\delta $ within the defocusing range of ${\pm} 100$nm is almost uniform, which provides a better defocus tolerance for lithography, and the process robustness benefit can be realized by setting the degree of aggressiveness of the defocus generation term. Figure 6(b) gives the convergence performance of defocus value $\delta$ for DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). At the initial stage of the confrontation process (Iterations < 50), the properties of the source and mask under the low $\delta$ are also optimized. Compared with Peng's method [21], we do not directly set the defocusing value $\delta$, and $\delta$ is continuously adjusted by the generation terms and the adversarial terms to achieve a steady state. At the middle and late stages of the confrontation, the defocusing converges to ${\pm} 37$ (green curve), ${\pm} 67$ (blue curve), and ${\pm} 103$nm (red curve), respectively, and the robustness of the optimized source and mask is continuously improved. Thus, the PAE in Fig. 6(a) shows a uniform error magnitude within the defocusing range of 50 (green curve), 70 (blue curve) and 90 nm (red curve) respectively. Compared with Li’s method [35], the proposed method clearly gives the maximum defocusing value $\delta$ that SMO can tolerate, and can determine the improvement of DOF without the help of tedious calculation operations.
Fig. 6. (a) The optimized defocus-pattern error (PAE) of desired target pattern ${{\bf I}_{01}}$ for traditional SMO (black dotted curve), the SMO-Adam method (black curve), the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). (b) The confrontation process of defocus value using desired target pattern ${{\bf I}_{01}}$ for DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve).
To compare the similarity between Defocus-NILS window and PW, the averaged NILS corresponding to different defocus value $\delta$ is given in Fig. 7(a), and Fig. 7(b) shows the rectangle- based PW. Figure 7(a) shows the optimized Defocus-NILS window of desired target pattern ${{\bf I}_{01}}$ for traditional SMO (black dotted curve), the SMO-Adam method (black curve), and the DGASMO methods with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve), with the defocus range from −200 nm to 200 nm. It is illustrated that the NILS of the proposed DGASMO method with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) is larger than the traditional SMO method (black curve) while defocus value $\delta \ge 70$nm, and the NILS optimized by the most aggressive defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ has a better performance within the defocusing range of ${\pm} 200$nm. Figure 7(b) shows the PWs of desired target pattern ${{\bf I}_{01}}$ for traditional SMO method (black dotted curve), the SMO-Adam method (black curve), and the DGASMO methods with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Due to the optimization ability of the Adam method, the increased PW and DOF (EL = 15%) of the SMO-Adam method are 33.63% and 38.81%, respectively. Compare with the SMO-Adam method, the increased PW being 11.15%, 21.4% and 29.12%, and the increased DOFs (EL = 15%) are 19.35%, 32.26% and 44.09% by the proposed DGASMO method. The PW optimized by DGASMO with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve) have a best performance in DOF and EL. It should be noted that PWs and Defocus-NILS window has similar trends in DOF and EL.
Fig. 7. Comparison of Defocus-NILS window and PW for desired target pattern ${{\bf I}_{01}}$ (a) The Defocus-NILS window for traditional SMO (black dotted curve), the SMO-Adam method (black curve), the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). (b) The PWs for traditional SMO (black dotted curve), the SMO-Adam method (black curve), the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Details of PWs Fig. S1 of the supplementary material.
Optimization results with desired target pattern ${{\bf I}_{0\textrm{2}}}$ are given in Fig. 8. The columns from left to right lies the illumination source, mask, wafer images at $\delta \;$= 0 (focal plane), 100, and 200 nm. Rows (a-e) of Fig. 10 are imaging results optimized by the traditional SMO, SMO-Adam, and DGASMO with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$, $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$. Where the PAE = 347, 710, 5145 and NILS = 0.905, 0.806, 0.530 optimized by SMO are given in row (a), with the defocus of optical system are $\delta \;\textrm{ = }$ 0, 100 and 200 nm. The huge improvement of PAE = 277, 575, 4005 and NILS = 1.030, 0.916, 0.599 are given in row (b), which are optimized by the SMO-Adam method. Row (c) gives the DGASMO imaging results using defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$, with the improved PAE = 264, 470, 3166 and NILS = 1.042, 0.936, 0.639 at $\delta \;\textrm{ = }$ 0, 100 and 200 nm. Row (d) shows the DGASMO imaging results using defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^\textrm{4}}$, with the improved PAE = 270, 395, 2680 and NILS = 1.045, 0.948, 0.673, while $\delta \;\textrm{ = }$ 0, 100 and 200 nm. Row (e) shows the DGASMO imaging results using defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^\textrm{9}}$, with the better imaging performance under defocusing $\delta \;\textrm{ = }$ 100 and 200 nm, in which, PAE = 278, 341, 2316 and NILS = 1.064, 0.968, 0.698. And the changes of assistant features in mask (the area in the red box) become more and more important to compensate for wafer image distortion.
Fig. 8. Simulation results with ${{\bf I}_{02}}$ as target pattern. Columns from left to right: the optimized source pattern $\hat{{\bf J}}$, the optimized mask pattern $\hat{{\bf M}}$, and the printed wafer image ${\bf I}_w^\delta $ at 0 nm, 100 nm and 200 nm defocus. Rows (a-e): the traditional SMO method, the SMO-Adam method, the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$, $\textrm{ - }{{1 / \delta }^4}$ and $\textrm{ - }{{1 / \delta }^9}$, respectively.
Fig. 9. (a) The optimized defocus-pattern error (PAE) of desired target pattern ${{\bf I}_{02}}$ for traditional SMO (black dotted curve), the SMO-Adam method (black curve), the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). (b) The confrontation process of defocus value using desired target pattern ${{\bf I}_{02}}$ for DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve).
Fig. 10. Comparison of Defocus-NILS window and PW for desired target pattern ${{\bf I}_{02}}$. (a) The Defocus-NILS window for traditional SMO (black dotted curve), the SMO-Adam method (black curve), the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). (b) The PWs for traditional SMO (black dotted curve), the SMO-Adam (black curve), the DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Details of PWs are given in Fig. S2 of the supplementary material.
Figure 9(a) depicts the optimized defocus–PAE using ${{\bf I}_{02}}$ as desired target pattern, with the defocus range from −200 nm to 200 nm. Where the black dotted, black, green, blue and red curve corresponds to the imaging results of rows (a-e) in Fig. 8. And the global PAE of each curve decreased in the order of traditional SMO (black dotted curve), SMO-Adam, DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Similar to the results in Fig. 6(a) (using ${{\bf I}_{01}}$ as target pattern), the sensitivity of imaging to defocusing reduced by the most aggressive defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve) greatly. Figure 9(b) shows the confrontation performance of defocus value $\delta$ for DGASMO with defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). And the defocusing value $\delta$ rises rapidly at the beginning of the confrontation, and fluctuates at ${\pm} 44$, ${\pm} 70$ and ${\pm} 91$nm at the later stage of iterations. However, the convergence speed of defocus value $\delta$ becomes slower with the aggressiveness of defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}$, so it is necessary to select an appropriate $\mathrm{{\cal S}}_\delta ^{gen}$ to ensure the robustness of optimization.
Figure 10(a) shows the Defocus-NILS window of desired target pattern ${{\bf I}_{02}}$ for traditional SMO (black dotted curve), the SMO-Adam (black curve), and the DGASMO method with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Where the NILS of SMO-Adam and proposed the DGASMO methods are larger than traditional SMO method (black dotted curve) within ${\pm} 200$nm defocusing. Figure 10(b) shows the PWs of desired target pattern ${{\bf I}_{02}}$ for traditional SMO method (black dotted curve), the SMO-Adam method (black curve), and the DGASMO with the defocus generation terms $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{1 / \delta }$ (green curve), $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^4}$ (blue curve) and $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ (red curve). Compared with the traditional SMO method, the PW optimized by SMO-Adam method is greatly improved. Compared with the SMO-Adam method, the increased PW being 54.9%, 126.8% and 190.2%, and the increased DOF (EL = 2%) are 50%, 86.84% and 118.42% by the proposed DGASMO method.
The expansion trend between defocus value $\delta$ and DOF revealed by desired target patterns ${{\bf I}_{01}}$ and ${{\bf I}_{02}}$ are summarized in Table 1. Where the optimized DOF of desired target pattern ${{\bf I}_{01}}$ by proposed DGASMO increased from 111 nm to 123 nm to 134 nm, while the defocus value $\delta$ increased from 37 nm to 67 nm to 103 nm. And the optimized DOF of desired target pattern ${{\bf I}_{02}}$ by proposed DGASMO increased from 57 nm to 71 nm to 83 nm, while the defocus value $\delta$ increased from 44 nm to 70 nm to 91 nm. Thus, the improvement of defocus value $\delta$ have the similar trend of DOF. Integrated consider the convergence performance of PAEs, $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ is a reasonable term for generating defocus value.
Table 1. The expansion trend between defocus value $\delta$ and DOF revealed by desired target patterns ${{\bf I}_{01}}$ and ${{\bf I}_{02}}$
Compared with the desired target pattern ${{\bf I}_{01}}$ (CD = 85 nm), the DOF and EL of the desired target pattern ${{\bf I}_{02}}$ (CD = 55 nm) are decreased, this is because the acyclic pattern is more sensitive to the defocus of optical system, but the proposed DGASMO can effectively perceive such drastic changes, so as to compensate the tolerance of the optimized source and mask to defocus. On the other hand, the slight difference between PWs and Defocus-NILS window is due to the average operation of NILS.
6. Conclusions
This paper proposed an efficient DGASMO method to improve the imaging performance for lithographic defocus, which was formulated as a SMO problem under defocusing interference. The inverse optimization framework using defocus generation term generates a ruler for planning defocus and against with the adversary term, thus, a more robust lithographic source and mask with better defocusing tolerance were searched. Benefit from the constraint of the KKT condition and the acceleration of the Adam algorithm, the defocus value $\delta$ changes and converges continuously, the optimization of DOF is more direct and effective. Compared to the SMO-Adam method, the DGASMO with $\mathrm{{\cal S}}_\delta ^{gen}\;\textrm{ = }\;\textrm{ - }{{1 / \delta }^9}$ resulting in the improvement of PW and DOF (EL = 15%) are 29.12% and 44.09% with CD = 85 nm, and the better performance of PW and DOF (EL = 2%) at CD = 55 nm are 190.2% and 118.42%. In addition, the similarities between Defocus-NILS window and PW is compared, and the method of directly optimizing PW will be considered in our subsequent work.
Appendix A
For brevity, Eq. (5) can be simplified as:
According to Eq. (11) and (12), we define the time-dependent scheme of the Euler-Lagrange term ${{{\partial _t}{\mathrm{{\cal S}}_L}} / {{\partial _t}\phi }}$, where $\phi $ can be $\theta ,\omega \;or\;\delta $. According to the derivation in [34], the gradients of $\mathrm{{\cal S}}_{pe}^\delta$ with respect to $\theta $ can be computed as:
similarly, for the $\omega $
Similarly, the update direction vector of defocus $\delta $ to be computed as:
For defocus generation term $\mathrm{{\cal S}}_\delta ^{gen} ={-} ({{\delta_0}/\delta } )$ with defocus scale ${\delta _0} = 1nm$, it’s obviously that ${{\partial \mathrm{{\cal S}}_\delta ^{gen}} / {\partial \theta }} = {{\partial \mathrm{{\cal S}}_\delta ^{gen}} / {\partial \omega }} = 0$, and ${{\partial \mathrm{{\cal S}}_\delta ^{gen}} / {\partial \delta }} = {1 / \delta }$.
Funding
Key R&D program of Hubei (2021BAA173).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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
Supplemental document
See Supplement 1 for supporting content.
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