With continuous shrinking of critical dimension (CD) and the application of immersion lithography system to technology nodes 22nm and beyond, the vector nature of electromagnetic fields propagating from mask to wafer plane cannot be ignored, rendering mask synthesis under scalar imaging model inadequate. In this paper, we develop a level-set based optimization framework for mask synthesis with a vector imaging model. The forward model of vector image formation is established, and then the photomask synthesis is addressed as an inverse imaging problem whose variational level-set reformulation is represented by a stable time-dependent model, which is solved by employing conjugate gradient methods of the cost function and readily available finite-difference schemes. Experimental results demonstrate pronounced performance in terms of pattern fidelity and edge placement error, together with notable computation acceleration and better convergence performance.
© 2017 Optical Society of America
Driven toward printing ever smaller features with increasingly large integrated circuit (IC) designs, the semiconductor industry is exploring the untapped potential of the resolution capacity of lithography systems. Along the way, resolution enhancement techniques (RETs) [1, 2] are essential in optical lithography including modified illumination schemes and optical proximity correction (OPC) [3,4], where the latter predistorts the mask patterns such that printed patterns are as close to the desired shapes as possible, among which, inverse lithography technique (ILT) is becoming a strong candidate of 22nm node and beyond and low-k1 regime, enabling the continuation of current immersion lithography.
Generally, mask synthesis in ILT involves the mathematical modeling of the imaging formation process of the lithography systems and iteratively seeking the minimization of the properly designed cost functions to improve pattern fidelity. Consequently, the performance of ILT mainly depends on the accuracy of the imaging formation model and the efficiency of the inverse imaging calculations in the optimization framework. For lithography systems with NAs less than 0.4 where the scalar imaging models provide sufficient accuracy , various model-based optimization approaches including steepest-descent methods [6, 7], active set and conjugate gradient method [8,9], augmented Lagrangian method  and level-set methods [11–15], find their applications in iterated methods for phase-shifting masks (PSMs) [8, 9, 16, 17], manufacture enhancement [18, 19], pixelated mask , illumination source optimization  and source-mask optimization [10, 22–26], double-exposure lithography  and designs with robustness to variations [28–30]. These approaches are all based on the scalar imaging models, which consider only the amplitude of the electromagnetic field ignoring its vector nature; while they make engineering sense, the current performance still falls short where the scalar imaging model cannot track the whole process of polarization state as light propagate through the optical components of the projection instruments.
With the continuous shrink of critical dimension (CD) and the NA of immersion lithography reaching as high as 1.35, the imaging process must account for the vector nature of the electromagnetic field propagating the projection system. Over the years, Yeung proposed high-NA vector imaging models incorporating thin-film model into the Hopkins formulations , whose matrix format in coherent systems developed by Flagello  with extension to partially coherent imaging systems by Pistor using Abbe method  and the vector imaging model inside resist by Adam et al. . Summarizing the above developments in high-NA projections optics, Peng et al. generalized a uniform and consistent formulation with further simplification , based on which Ma and his coworkers designed pixelated mask optimization and source-mask optimization [23, 35] and Shen proposed level-set based mask synthesis using gradient-based methods . These methods, however, are very time consuming, which limits their applicability.
This paper focuses on the development of algorithms for mask synthesis using level-set-based inverse lithography with a vector imaging model. A complete set of variational level-set formulations of the optimization problem is provided which combines an internal energy term forcing the level set function close to a signed distance function and an extra energy term that drives the zero level set toward desired mask features minimizing the pattern difference between the printed wafer and the desired pattern, thereby optimizing the mask layout without the costly re-initialization procedure. We take advantage of Fast Fourier Transform and reusable data in the computation of the evolution velocity of the level surfaces in the partial differential equation defining the time-dependent model of the level-set formulation which greatly enhances the iterative speed by easing the computation load of convolution operation; we also apply Polak-Ribière-Polyak (PRP) conjugate gradient (CG) to updating the normal velocity of the level-set function which achieves accelerated convergence. Numerical simulations demonstrate improved performance of the proposed method in both pattern fidelity and computation efficiency.
2. Forward vector imaging model
The projection lithographic imaging process can be divided into two function blocks, namely the aerial image formation and resist effects. A schematic of aerial image formation based on a vector imaging model is illustrated in Fig. 1. It is often found convenient to use the polarization systems when tracking the projection of light, while a spatial coordinate system (x, y, z) is preferred when computing the actual field or intensity. Consider a monochromatic wave propagating in the direction , where is the direction cosine with . Without loss of generality, the intrinsic local coordinate (e⊥, e║) which point to the TE- or s-direction and TM- or p-direction respectively, can be defined as 
For a point source (αs, βs) emanating an electric field E0 = [E⊥, E║]T, the emitting electric field in spatial coordinate is computed as Denoting the mask as the scalar matrix M(r) ∈ ℛN×N where N is the size of M(r) and the bold r denotes spatial coordinate (x, y), each entry of which represents the corresponding transmission coefficient on the mask, the mask near field in spatial coordinate can be calculated as37], where the entry in B is defined as
According to the Abbe method and assuming has identical entries, the aerial image intensity under a partially coherent illumination can be described as :
The resist effect can be approximated using a logarithmic sigmoid function
Figure 2(a) shows the annular illumination source with σin = 0.6 and σout = 0.9, Fig. 2(b) depicts the target pattern and Figs. 2(c) and 2(d) depict the resist images based on the scalar imaging model [13,14] with pattern error (PE) 2210 and edge placement error (EPE) 1251 and the vector imaging model with PE 2517 and EPE 1277, respectively, where pattern error is defined as the square of the L2 norm of the difference between the target pattern and the resist image and edge placement error is defined as the CD error at one side to convey CD information, especially in the polygon-based OPC . Severe distortion incurred by the low pass nature of the pupil function is observed in both Figs. 2(c) and 2(d), however, degrading pattern fidelity and EPE in Fig. 2(d), for example, the vanishing contacts, are noticed which is accountable by the degradation of p-polarized images compared to s-polarized images from a polarized-light emanating source .
3. Level-set based optimization framework
Given a target pattern I0 ∈ ℛN×N, the goal of mask synthesis is to find an optimal which minimizes the distance between and I0, namely
The level-set based optimization framework with vector imaging formation can be developed following the steps in [13,14,36]. We give M a level set description by introducing an unknown function ϕ which relates to M by definingEq. (9) with a least squares fit leads to minimizing
In previous level-set based methods of mask synthesis [13–15,36], re-initialization of level set function ϕ is usually not practiced, however, it has been observed that if ϕ is not smooth or much steeper on one side of the interface than the other, the zero level set of ϕ can move incorrectly from that of the original function , therefore, re-initialization has been extensively used as a numerical remedy for stable curve evolution and ensuring desirable results. The standard re-initialization method is to keep the evolving ϕ as a signed distance function during the evolution, especially in a neighborhood of the zero level set. With the signed distance function satisfying |∇ϕ| = 1, the closeness of ϕ to a signed distance function is characterized byEq. (11) with a multiplier μ gives the variational level set formulation
4. Conjugate gradient based mask optimization
The computation of v(r, t) contains three convolution operations, two explicit in Eq. (13) and one implicit in the computation of the resist image I. Although the results of in Eq. (8) can be repeatedly used in Eq. (13) therefore lightening the load of one convolution operation, the computation complexity of convolution is superlinear with respect to N which makes iteratively updating ϕ to convergence very time consuming. Applying Fast Fourier Transform (FFT) to remove the convolution operations will effectively reduce the computation intensity of Eq. (13), giving
In this paper, we update mask by employing the widely used Polak-Ribière-Polyak (PRP) conjugate gradient (CG)  which is believed most efficient, instead of the steepest descent (SD) practiced in [13,14,36] to achieve faster convergence and improved performance. It should be noted that, for convergence and stability when solving the partial differential equation Eq. (12), Courant-Friedrichs-Lewy (CFL) condition is often applied asserting that the numerical wave speed must be at least as fast as the physical wave which is described asAlgorithm (1).
5. Numerical results
The stable time-dependent model in Eq. (15) is a partial differential equation, which can be solved by readily available first-order temporal accurate and second-order accurate spatial finite-difference schemes [13, 14]. The imaging system parameters used in the numerical simulations are: λ = 193nm, NA = 1.35, resolution δx = 4nm/pixel, steepness of the sigmoid function a = 0.85, threshold tr = 0.3, annular illumination source with σin = 0.6 and σout = 0.9 which is shown in Fig. 2(a), and the target pattern is given in Fig. 2(b).
Figure 3 illustrates the OPC results using the scalar imaging formation, where the synthesized mask using the scalar imaging model [13, 14] is depicted in Fig. 3(a), the aerial image and the resist image with PE 392 of Fig. 3(a) using the scalar imaging model are given in Figs. 3(b) and 3(c), respectively. The same proposed level-based PRP conjugate gradient method is applied to the mask synthesis using the scalar imaging model to accelerate the optimization process. It is noted that while the pattern fidelity with PE 392 is greatly improved in the scalar imaging model, when the synthesized mask using the scalar imaging model in Fig. 3(a) as input enters the vector imaging model, the resist image PE reaches 1291, showing the incapability of handling large NA systems where the vector nature of the electromagnetic fields must be taken into account.
The synthesized masks acquired using level-set methods based on steepest descent (SD) and proposed Polak-Ribière-Polyak conjugate gradient (PRP-CG) are given in Figs. 4(a) and 4(e), with corresponding aerial images in Figs. 4(b) and 4(f), and resist images in Figs. 4(c) and 4(f), respectively. Several trials of using different μ = 0.01, 0.1, 0.5, 1 in Eq. (15) show no significant distinction in terms of the synthesized mask, hence, μ = 0.1 is applied in the following simulations. Bearing almost the same pattern error around 400 and edge placement error around 150, the level-set based ILT with SD and proposed method both demonstrate great improvement in terms of pattern fidelity comparing with Fig. 2(d). Apart from a few scattered little blocks observed in Fig. 4(a), the similarity of the synthesized masks in Figs. 4(a) and 4(e) suggests that the SD and proposed techniques reach very close local minimum which is justified by the underlying gradient-based nature of both methods.
In Fig. 5, the computation time of the simulations with the SD method and the proposed method is depicted, illustrating computation linearity for both methods and much faster speed of the proposed method than that of the SD method. A more straight-forward comparison of the computation time of the two methods is given in Table 1 with the observation that it took over 230 hours for the SD method to complete 200 mask updates amounting to 1.16 hours each iteration in contrast to only 5.60 hours total time and 0.028 hours each iteration using the proposed method. While the strenuous SD method taxes the computation load in every iteration rendering the mask synthesis intolerably slow, the proposed method achieves over 40 times of acceleration in terms of computation speed owing to the application of Eq. (16) and the repeated usage of to the calculation of the evolution velocity. The computation time of mask synthesis using the scalar imaging model of up to 50 iterations is also given in Table 1, amounting to 0.0095 hours each iteration which is about of that using the vector imaging model. This observation conforms to the involving of the equivalent filters of the x, y, z components in the computation of the resist image in Eq. (8) and the velocity function in Eq. (13) using the vector imaging model instead of only one filter in the scalar imaging model, hence tripling the computation.
The convergence drift of the pattern error and edge placement error which is popularly used in RETs to convey CD information, are illustrated in Figs. 6(a) and 6(b) respectively. It is duely noted from Fig. 6 that the edge placement error (EPE) has a positive relationship with the pattern error (PE) in general aside from small local fluctuations indicating the equivalency of minimizing PE and EPE in mask synthesis, however, explicit incorporation of EPE into the cost function is often too complicated, therefore the minimization of PE is often applied in ILT instead of EPE. As shown in Fig. 6(a), both the SD method and the proposed method do not need 200 iterations to reach the local minimum, although it takes the proposed method around 20 iterations compared to the 40 iterations by the SD method. Naturally, the PRP-CG method applied in the proposed method contributes to the 50% convergency performance improvement.
We develop a complete variational level-set based optimization framework for inverse lithography with a vector imaging model. Detailed level-set formulation represented by a time-dependent model is extensively investigated by minimizing the sum of the mismatches between the printed image and the desired one over all locations. Moreover, a distance metric preserving the closeness of the evolution function ϕ to a signed distance function is incorporated into the cost function which eliminates the necessity of re-initialization in every iteration which helps stable curve evolution and ensure desirable results. Calculation of convolution in the frequency domain with Fast Fourier Transform (FFT) and repeated usage of in the computation of the evolution velocity V (r, t) expedite the iterative mask synthesis procedure. We also employ the Polak-Ribière-Polyak (PRP) conjugate gradient (CG) in the proposed level-set based ILT. Numerical results demonstrate a acceleration of the mask updating by over 40 times compared to the conventional steepest descent (SD) method and around 50% improved convergency performance. The theoretic and numerical analysis of the proposed ILT brings algorithmic insights into the exploitation of vector imaging model for mask optimization problems in next generation lithography with 22nm technology node and beyond.
Natural Science Foundation of Guangdong Province, China (2016A030313709, 2015A030310290). Guangzhou Science and Technology Project, China (201607010180).
The work in this paper is partially supported by Natural Science Foundation of Guangdong Province, China (2016A030313709, 2015A030310290) and Guangzhou Science and Technology Project, China (201607010180). The author greatly appreciates the suggestions from Professor Edmund Lam (Imaging System Lab, the University of Hong Kong).
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