Inverse pupil wavefront optimization for immersion lithography Chunying Han, Yanqiu Li,* Lisong Dong, Xu Ma, and Xuejia Guo Key Laboratory of Photoelectronic Imaging Technology and System of Ministry of Education of China, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China *Corresponding author: [email protected] Received 30 May 2014; revised 24 August 2014; accepted 28 August 2014; posted 3 September 2014 (Doc. ID 213097); published 9 October 2014
As the critical dimension of integrated circuits is continuously shrunk, thick mask induced aberration (TMIA) cannot be ignored in the lithography image process. Recently, a set of pupil wavefront optimization (PWO) approaches has been proposed to compensate for TMIA, based on a wavefront manipulator in modern scanners. However, these prior PWO methods have two intrinsic drawbacks. First, the traditional methods fell short in building up the analytical relationship between the pupil wavefront and the cost function, and used time-consuming algorithms to solve for the PWO problem. Second, in traditional methods, only the spherical aberrations were optimized to compensate for the focus exposure matrix tilt and best focus shift induced by TMIA. Thus, the degrees of freedom were limited during the optimization procedure. To overcome these restrictions, we build the analytical relationship between the pupil wavefront and the cost function based on Abbe vector imaging theory. With this analytical model and the Fletcher–Reeves conjugate-gradient algorithm, an inverse PWO method is innovated to balance the TMIA including 37 Zernike terms. Simulation results illustrate that our approach significantly improves image fidelity within a larger process window. This demonstrates that TMIA is effectively compensated by our inverse PWO approach. © 2014 Optical Society of America OCIS codes: (110.5220) Photolithography; (110.2945) Illumination design; (220.1000) Aberration compensation. http://dx.doi.org/10.1364/AO.53.006861
1. Introduction
As optical lithography steps into the low-k1 regime [1], the critical dimension (CD) of integrated circuits becomes much smaller than the scanner exposure wavelength for 45 nm technology nodes and beyond. Thus, the wavefront phase error induced by a thick mask cannot be ignored [2,3]. To understand thick mask induced aberration (TMIA) comprehensively, Erdmann studied the impact of a thick mask on lithography performance by using rigorous electromagnetic field (EMF) simulation [4–6]. The results indicated that TMIA causes the modification of polarization and serious asymmetry of the lithographic process window. Therefore, it is necessary 1559-128X/14/296861-11$15.00/0 © 2014 Optical Society of America
to compensate TMIA for the next generation of lithography. Recently, ASML designed a freeform pupil wavefront manipulator named FlexWave in modern scanner tools [7,8]. Based on this sophisticated wavefront control equipment, Sears et al. [9–11] and Fühner et al. [12] introduced the idea of pupil wavefront optimization (PWO) and proposed PWO approaches to compensate for TMIA. It was shown that the PWO methods evidently improved the process window and counteracted thick mask induced wavefront skews. However, their PWO methods have two intrinsic drawbacks. First, an analytical relationship between the pupil wavefront and the cost function is not built in their optimization algorithms. Thus, these methods applied a nonanalytic algorithm to solve for the PWO problem, which is computationally expensive. Second, these methods only compensated 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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for the spherical aberrations in TMIA, thus limiting the degrees of optimization freedom. To overcome these limitations, this paper develops an inverse PWO approach to compensate for TMIA. The analytical framework of PWO is formulated under the Abbe vector imaging model [13–16], which is much more accurate than the scalar model for immersion lithography systems with hyper-NA (NA > 1). Here, the cost function is set to be the square of the Euler distance between the target pattern and the printed image on the focal plane with nominal dose [17,18]. And the pupil wavefront is represented as the sum of 37 Zernike polynomials. Then, the gradients of the cost function with respect to the 37 Zernike coefficients of the pupil wavefront are derived. In addition, this paper introduces a Laplacian regularization term [19,20] to smooth the distribution of the pupil wavefront. Subsequently, the Fletcher–Reeves conjugate-gradient (FR-CG) algorithm [21,22] is adopted to optimize the pupil wavefront. Compared with the traditional PWO methods, the proposed PWO algorithm optimizes 37 Zernike polynomials simultaneously, rather than only spherical aberrations. This is because the lithography performance associated with the complex thick mask and the freeform source is also sensitive to asymmetrical aberrations [9,11]. Simulation results demonstrate that the proposed method is effective and efficient in compensating for TMIA. The remainder of the paper is organized as follows. The Abbe vector imaging model used in this paper is described in Section 2. The gradient-based inverse PWO algorithm is developed in Section 3. Simulations are illustrated in Section 4. Conclusions are provided in Section 5. 2. Abbe Vector Imaging Model
In this paper, the Abbe vector imaging model is used to formulate the proposed inverse PWO framework, since the vector nature of the EMF cannot be neglected in immersion lithography [23,24]. In addition, the Hopkins approach is not accurate for imaging calculation with a thick mask [25]. A schematic of the vectorial imaging process for an optical lithography system is illustrated in Fig. 1, where the spatial coordinates on the source plane are xs ; ys , and the direction cosine of the wave propagating from the source to the mask is αs ; βs ; γ s . The spatial coordinates on the mask plane are x; y, and the direction cosine of the wave propagating from the mask to the entrance pupil is α; β; γ. The normalized spatial coordinate on the exit pupil is f ; g. The direction cosine of the wave propagating from the exit pupil to the wafer is α0 ; β0 ; γ 0 . The spatial coordinate on the wafer is xw ; yw . According to Abbe’s method [26] and Fourier optics theory [27], the aerial image on the wafer contributed by the effective source J [28] can be formulated as 6862
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Fig. 1. Vectorial imaging process of immersion lithography system.
I
1 XX J sum
×V
xs
xs ys
ys
⊙
Jxs ys
X px;y;z
‖
F −1
s ys expj2πW ⊙ Gx3D
2π ×C nw R
⊙ Exi s ys
2 ; p
‖
(1)
2
where Jxs ys is the intensity P P of the source point at xs ; ys , and J sum xs ys Jxs ys is a normalization factor. The notation ⊙ is the entry-by-entry multiplication operation, nw 1.44 is the refractive index of the media on the wafer side, R 4 is the demagnification factor, and F −1 is the inverse Fourier transform. Exi s ys is an N × N matrix with each entry equal to a 2 × 1 vector representing the electric field of incidence light in spatial coordinates. The superscript s ys xs ys denotes that Ei is the function of xs ; ys . Gx3D is the thick mask diffraction spectrum corresponding to source point xs ; ys , each entry of which is a 2 × 2 matrix. Vxs ys is an N × N matrix with each entry equal to a 3 × 2 matrix of 2 β02 α02 γ0 6 Vm; n 4
1−γ 02 α0 β 0 − 1γ 0 0
−α
0 0
αβ − 1γ 0
β02 α02 γ 0 1−γ 02 0
3 7 5;
m; n 1; 2; …; N:
−β
(2) p0 C γ∕γ is the radiometric correction factor [29]. W is the pupil wavefront distribution, which can be implemented by the FlaxWave. According to the Nijboer–Zernike theory [30–32], W can be represented as Wρ; θ
X ci Γi ρ; θ;
(3)
i
where ρ; θ are the polar coordinates on the exit pupil, ρ λ∕NAw f 2 g2 1∕2 , and θ arctanf ∕g. NAw
and λ are the numerical aperture on the wafer side and the scanner exposure wavelength, respectively. Γi ρ; θ is the ith Zernike polynomial, and ci is the corresponding Zernike coefficient. To simplify Eq. (1), we define two notations: Txps ys
F
−1
2π s ys × C × Vxs ys ⊙ Gx3D ⊙ Exi s ys nw R
(4) p
Fig. 2. Illustration of the PWO approach to compensate for TMIA.
and Θ F −1 fexpj2πWg:
(5)
Thus, the aerial image described in Eq. (1) can be reformulated as I
1 X X xs ys X xs ys 2 J ‖Tp ⊗ Θ‖2 ;
J sum
xs
ys
(6)
px;y;z
where ⊗ is the convolution operation. Then, the aerial image described in Eq. (6) enters into the photoresist and forms the printed image through development. Here, the sigmoid function is employed to approximate the constant threshold resist (CTR) model [33] due to its differentiability [34,35]. Thus, the developed resist image on the wafer can be described by a sigmoid transformation of the aerial image, that is, Z sigI
1 ; 1 exp−aI − tr
(7)
where a indicates the steepness of the sigmoid function, and tr is the process threshold. 3. Inverse Optimization Algorithm for Pupil Wavefront Optimization A.
Pupil Wavefront Optimization Algorithm
Today, a co-optimizing source and mask pattern is necessary for 45 nm technology nodes and beyond. Since the rigorous 3D mask models are complex and computationally expensive, they are almost not applied in current source and mask optimization (SMO) methods. On the other hand, the objective of this paper is to prove the capability in counteracting TMIA by the proposed PWO, not by SMO. Thus, in this paper, the vectorial SMO method described in [15] will first be used to obtain the source J and mask M. Then, we use the rigorous EMF simulator in PROLITH to calculate the thick mask diffraction specs ys trum Gx3D . Finally, the source J and the diffraction s ys are applied in the proposed PWO spectrum Gx3D algorithm. Figure 2 illustrates the PWO approach to compensate for TMIA, where the distorted wavefront W3D is introduced by TMIA and the wavefront manipulator W is the optimized pupil wavefront by the PWO method. This figure indicates that the PWO approach uses W to balance W3D , and finally makes the actual wavefront WAct significantly improve the lithography performance.
Given a binary target pattern Z~ ∈ RN×N with all entries equal to 0 or 1, the goal of PWO is to find ˆ so that the optimal pupil wavefront denoted as W the cost function ~ F dZ; Z 1 X X xs ys X xs ys 2 ‖Tp ⊗ Θ‖2 ; Z~ d sig J J sum x y px;y;z s
s
(8) is minimized, where d·; · is the square of the Euler distance between the two arguments. According to Eq. (3), we can optimize the Zernike coefficients ci during the PWO procedure, and then the wavefront W can be reconstructed from the corresponding Zernike coefficients. Therefore, the PWO problem can be formulated as ~ cˆ i arg min dZ; Z:
(9)
ci
In this paper, the FR-CG algorithm is applied to solve for the PWO problem in Eq. (9). The efficiency and convergence property of the FR-CG algorithm in solving the linear and nonlinear problems has been widely proved in different areas [21,22]. The FR-CG algorithm requires knowledge of the gradient of the cost function with respect to the Zernike coefficients, as shown in Appendix A, which can be calculated as ∇Fci
8aπ X X xs ys X T J 1N×1 J sum x y px;y;z s s h × ReΛxps ys ⊙ fReF −1 Γi ⊙ sin2πW i ImF −1 Γi ⊙ cos2πWg 1N×1 8aπ X X xs ys X T J 1N×1 J sum xs ys px;y;z h × ImΛxps ys ⊙ fReF −1 Γi ⊙ cos2πW i (10) − ImF −1 Γi ⊙ sin2πWg 1N×1 ; −
where the superscript T is a transposition operation. 1N×1 is the one-valued vector. Re· and Im· represent the real part and the image part of the argument, respectively. Λxps ys is given by 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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∘
Λxps ys Txps ys ⊗ Z~ − Z ⊙ Z ⊙ 1 − Z ⊙ Txps ys ⊗ Θ; (11) where is the conjugate operation, and ∘ represents a rotation of the matrix in the argument by 180° in both the horizontal and vertical directions. It is noted that the derivation of Eq. (10) is based on the ideal lithography system. More works should be done in the future for the lithography system with different noises, which, however, is out of the scope of this paper. The details of the FR-CG algorithm are described by the pseudo-code in Table 1, where the variables in the kth iteration are marked by the superscript k with brackets. B.
Pupil Wavefront Regularization
In order to smooth the distribution of the optimized pupil wavefront, this paper introduces a Laplacian regularization term. The Laplacian regularization term is formulized as Z RLaplacian
Ω
2 ∂W 2 ∂W dx: ∂f ∂g
(12)
Thus, the cost function in Eq. (8) is modified as F 0 F γRLaplacian ;
(13)
where γ is the regularization weight. Since the first differential operator is calculated as [36] 2 6 6 6 D6 6 4
1
0
−1 1
−1 .. .
0
..
.
1 −1
3
7 7 7 7; 7 −1 5 1
(14)
Table 1.
∇RLaplacian ci 1TN×1 2 × DW ⊙ DΓi 2 × WDT ⊙ Γi DT 1N×1 :
(16)
4. Simulation Results
In order to demonstrate the performance of the proposed inverse PWO algorithm, two typical target patterns in Fig. 3 are used in the following simulations. The CD of both target patterns is 45 nm. Figure 3(a) is a line-space pattern represented by a 301 × 301 matrix with pixel size of 5 nm × 5 nm on wafer scale. Figure 3(b) is a complex pattern represented by a 200 × 200 matrix with pixel size of 5.625 nm × 5.625 nm on wafer scale. Consider an immersion lithography system with wavelength of λ 193 nm, and NA 1.35. We adopt Y polarization and TE polarization for the line-space pattern and the complex pattern, respectively. In Eq. (7), a 25, tr 0.1 for the line-space pattern, while tr 0.2 for the complex pattern. Some important criteria are chosen to evaluate the lithography performance, such as pattern error (PAE), CD error (CDE), placement error (PLE), depth of focus (DOF), best focus variation (ΔBF), and PW. At the focal plane with nominal exposure dose, the PAE is calculated as [15]
‖
PAE Z~ 2 1 X X xs ys X J −Ξ ‖Txps ys ⊗ Θ‖22 − tr ; J sum x y px;y;z s
s
‖
2
(17)
the matrix form of Eq. (12) can be rewritten into Eq. (15): RLaplacian ‖DW‖22 ‖WDT ‖22 :
Thus, the gradient of RLaplacian with respect to ci can be calculated as
(15)
Pseudo-Code of the FR-CG Algorithm
1. Initialize k 0, the starting Zernike coefficients c0 i 0.000001, the pupil wavefront step size sc , the maximum iteration number lpwo, and P0 −∇Fc0 i . : 2. Update the Zernike coefficients ck1 i ~ >0 while k ≤ lpwo & dZk − Z k←k 1; Calculate the gradient, i.e., ∇Fck i , using Eq. (10); k 2 ‖∇Fc ‖ i 2 Calculate βk :βk ; 2 ‖∇Fck−1 ‖2 i
where Ξf·g 1 if the argument is larger than 0; otherwise, Ξf·g 0. The CDE and PLE denote the average CDE and PLE of all critical locations on the printed image at nominal dose and focus [15]. DOF is the largest acceptable defocus range with exposure latitude (EL) of 5%. According to Refs. [4–6,10,11], TMIA can induce best focus shift. Thus, the ΔBF is taken into consideration to the advantage of our proposed algorithm in reducing the best focus offset, which is defined as the distance between the
k k−1 ; ; Update Pk :Pk −∇Fck i β P k1 k k Update ck :c c s P . c i i i
end ˆ by the 3. Calculate the optimized wavefront manipulator W optimized Zernike coefficients cˆ i using Eq. (3).
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Fig. 3. Two target patterns used in the simulations: (a) line-space pattern and (b) complex pattern. Red lines mark the positions to measure the PWs.
optimal focus position of the chosen PW and zero focus in this paper, as shown in Fig. 6. In addition, PWs are measured at the positions marked by the red lines in Fig. 3, where the linewidth measurement specification is set to be 10% and EL is 5% [37]. After that, the overlapped PWs at different measurement positions are used to evaluate the robustness of the lithography system to the defocus effect and dose variation. Besides, it is worthwhile to mention that the source and mask patterns, as shown in Figs. 4, 8, and 10, are obtained by the SMO algorithm described in [15]. The frequency spectra of the thick masks are calculated by the FDTD algorithm [5,38] in PROLITH, where the mask absorbers consist of 55 nm Cr with a refractive index of 1.48 1.76i and 18 nm CrO with a refractive index of 1.97 1.2i [5,11]. For the line-space pattern, the step size sc 0.0001 and the maximal loop lpwo 50. To prove the efficiency of the FR-CG algorithm used in the proposed PWO method, the steepest descent (SD) algorithm is compared in this paper. Because of the axial symmetry of the line-space pattern, just the even aberrations [39] need to be optimized. Figure 4 illustrates the simulation results, where Figs. 4(a) and 4(b) are the source pattern and the mask pattern, respectively. Figure 4(c) is the initial pupil wavefront. Figure 4(d) is the printed image at nominal dose and focus corresponding to the initial pupil wavefront, which is calculated by the rigorous EMF simulator in PROLITH. It clearly illustrates
Fig. 4. Simulation results of PWO based on the line-space pattern. Left to right: source pattern, mask pattern, pupil wavefront, and printed image calculated by the rigorous EMF simulator. Top to bottom: simulations for initial conditions, SD algorithm, FR-CG algorithm, and FR-CG algorithm with pupil wavefront regularization.
Fig. 5. Comparisons of the PAE convergence curves among the SD algorithm (blue solid curves), the FR-CG algorithm (green dotted curves), and the FR-CG algorithm with pupil wavefront regularization (red dash-dotted curves) for the line-space pattern.
that TMIA has a significant impact on the imaging performance. Figure 4(g) is the optimized pupil wavefront by the PWO method with the SD algorithm, and Fig. 4(h) is the corresponding printed image at nominal dose and focus. Comparison between Figs. 4(d) and 4(h) shows that the image fidelity is well improved by the PWO algorithm with the SD algorithm. The simulation results obtained by the proposed PWO method with the FR-CG algorithm are illustrated in the third row of Fig. 4. Compared with Fig. 4(h), Fig. 4(l) indicates that the FR-CG algorithm can better improve the image quality at nominal dose and focus than the SD algorithm for the line-space pattern. Figure 5 shows the convergence curves of PAE at the nominal conditions for the line-space pattern, where the blue solid and green dotted curves represent the PAE convergence of the SD algorithm and the FR-CG algorithm, respectively. By the SD algorithm, the PAE decreases from 1191 to 601 with 120 iterations. By the FR-CG algorithm, the PAE decreases from 1191 to 553 with 50 iterations. Thus, it is clearly illustrated that the FR-CG algorithm can achieve better pattern fidelity with fewer iterations than the SD algorithm for the line-space pattern. Figure 6 shows the PW simulation results of the line-space pattern, where the area enclosed by the orange curve corresponds to the focus exposure matrix (FEM) of the P1 point at the line-space pattern and the area enclosed by the purple curve corresponds to the FEM of the P2 point. The overlapped area between the FEMs of P1 and P2 is enclosed by the blue curve, and the red ellipse is the overlapped PW. Figure 6(a) shows the initial PWs of the linespace pattern. From this figure, we can clearly see that TMIA induces different FEM tilts and best focus offsets for the measured positions of P1 and P2, thus seriously reducing the PW. Figure 6(b) shows the PW of the line-space pattern after PWO by using the SD algorithm. Compared with Fig. 6(a), it illustrates that the SD algorithm reduces the ΔBF from 97 to 7 nm and increases the DOF from 22 to 97 nm for the line-space pattern. Figure 6(c) shows the PW of 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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Fig. 7. Zernike coefficients of the optimized pupil wavefront for the line-space pattern, where the blue, green, and red bars represent the Zernike coefficients of the optimized pupil wavefront by the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization, respectively.
Fig. 6. Simulation of PWs for the line-space pattern: (a) initial PW, (b) optimized PW by the SD algorithm, (c) optimized PW by the FR-CG algorithm, and (d) optimized PW by the FR-CG algorithm with pupil wavefront regularization.
the line-space pattern after PWO by using the FRCG algorithm. The corresponding ΔBF and DOF are 4 and 100 nm, respectively. Compared with the SD algorithm, the FR-CG algorithm can further enlarge the PW for the line-space pattern. To smooth the distribution of the optimized pupil wavefront, the Laplacian regularization term in Section 2.B is used in the proposed PWO method, where the regularization weight γ 0.00001. Figure 4(o) is the optimized pupil wavefront by the FR-CG algorithm with pupil wavefront regularization, and Fig. 4(p) is the corresponding printed image at nominal dose and focus. Here, the peak-to-valley (PV) and root-mean-square (RMS), which are widely applied in wavefront aberration theory [40], will be used to evaluate the smoothness of the pupil wavefront. Generally, smaller values of PV and RMS correspond to a pupil wavefront with better smoothness. Comparison between Figs. 4(k) and 4(o) shows that the PV of the optimized pupil wavefront decreases from 0.389λ to 0.364λ and the RMS of the optimized pupil wavefront decreases from 0.094λ to 0.086λ by using pupil wavefront regularization. The red dash-dotted curve shown in Fig. 5 is the PAE convergence at the nominal dose and focus for the linespace pattern by the FR-CG algorithm with pupil wavefront regularization. The corresponding PAE decreases from 1191 to 561 with 50 iterations. Figure 6(d) shows the PW of the line-space pattern after PWO by using the FR-CG algorithm with pupil wavefront regularization. The corresponding ΔBF and DOF are 8 and 105 nm, respectively. Compared with the FR-CG algorithm, the FR-CG algorithm with pupil wavefront regularization can further enlarge the PW for the line-space pattern. 6866
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The results above prove that our proposed PWO algorithm not only improves the image fidelity at nominal conditions, but also reduces the ΔBF and enlarges the PW for the line-space pattern. To further understand the proposed PWO algorithm, Fig. 7 illustrates the Zernike coefficients of the optimized pupil wavefronts shown in Fig. 4. It is noted that we use all of the even aberrations, not just the spherical aberrations, to compensate for TMIA. The results in Fig. 7 indicate that the other even aberrations, such as c5, c12 , c17 , and c32 , are also important for counteracting TMIA, beside the spherical aberrations. This phenomenon can be proved by the simulation described in Ref. [11]. Thus, our proposed PWO algorithm further improves the degree of freedom in the optimization procedure. Table 2 summarizes the values of PAE, CDE, PLE, DOF, and ΔBF corresponding to the initial case, the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization for the line-space pattern. All of these criteria of lithography performance Table 2. Comparisons of PAE, CDE, PLE, DOF, and ΔBF Among the Initial Case, the SD Algorithm, the FR-CG Algorithm, and the FR-CG Algorithm with Pupil Wavefront Regularization for the Line-Space Pattern and the Complex Pattern, Respectively
PWO Algorithm Target
Criterion
Line-Space PAE CDE (nm) PLE (nm) DOF (nm) ΔBF (nm) Runtime (s) Complex PAE CDE (nm) PLE (nm) DOF (nm) ΔBF (nm) Runtime (s)
Initial
SD
1191 14.4 2.7 22 97 — 1533 6.6 5.8 19 71 —
601 6.3 2.5 97 7 5704 1184 5.8 4.6 74 32 9508
FR-CG FR-CG+Reg 553 5.7 2.5 100 4 2519 958 4.9 4.2 116 21 3608
561 5.8 2.5 105 8 2827 978 4.9 4.2 119 12 4023
are improved by the proposed PWO algorithm. It is worthwhile to note that the simulations in this paper are carried out using MATLAB on an Intel(R) Core (TM) i5-2400 CPU, with 3.10 GHz, and 4 GB of RAM. The memory requirement of the PWO procedure is 300 MB for the line-space pattern and 260 MB for the complex pattern. The runtimes of the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization for the line-space pattern are summarized in Table 2. Comparison of the runtimes between the SD algorithm and the FR-CG algorithm further reveals that the FR-CG algorithm is more efficient than the SD algorithm for the line-space pattern. Thus, these values demonstrate that the proposed PWO method using the FR-CG algorithm with pupil wavefront regularization is effective and efficient to reduce the impact of TMIA for the line-space pattern. Figure 8 illustrates the simulations of the proposed PWO method for the line-space patterns with CDs equal to 40 and 35 nm, respectively. Figures 8(d) and 8(h) clearly show that the line-space patterns with CDs equal to 40 and 35 nm cannot be resolved at the nominal dose and focus due to TMIA. In the second row of Fig. 8, the simulation with CD 40 nm leads to PAE of 774, CDE of 7.3 nm, and PLE of 2.5 nm. In the fourth row of Fig. 8, the simulation with CD 35 nm leads to PAE of 1457, CDE of 6.4 nm, and PLE of 6.2 nm. Figure 9 shows the PWs for the line-space patterns with CDs equal to 40 and 35 nm. For the line-space pattern with CD 40 nm, the DOF increases from 11 to 22 nm.
Fig. 8. Simulations of the proposed PWO method for the line-space pattern with different CDs. Top to bottom: initial case with CD 40 nm, PWO with CD 40 nm, initial case with CD 35 nm, and PWO with CD 35 nm. Left to right: source patterns, mask patterns, pupil wavefronts, and printed images.
Fig. 9. Simulation of PWs for the line-space pattern with different CDs. Top to bottom: the CDs of the target patterns are 40 and 35 nm, respectively. Left to right: the initial case and the optimized PW by the proposed method.
For the line-space pattern with CD 35 nm, the DOF with EL of 2% increases from 4 to 10 nm. In contrast with the initial pupil wavefront for a given lithography system, the proposed PWO method can thus compensate for TMIA and improve the image fidelity for CDs equal to 40 and 35 nm. However, for the 22 nm lithography node, our PWO method should be combined with other resolution enhancement technology, such as double patterning technology (DPT) [41]. In order to prove the universality of the proposed PWO approach, another experiment based on the complex pattern is conducted with sc 0.0001, lpwo 37. Figure 10 shows the corresponding PWO results of the complex pattern in a way similar to Fig. 4. From this figure, we can see that the pattern fidelity of the printed image is evidently improved by the proposed PWO method, and the FR-CG algorithm can better improve the image quality at nominal dose and focus than the SD algorithm for the complex pattern. Figure 11 shows the convergence curves of PAE at the nominal conditions for the complex pattern. The blue solid curve in Fig. 11 reveals that the SD algorithm converges slowly and is apt to be trapped into the local optimum. The corresponding PAE decreases from 1533 to 1184 with 100 iterations. However, by using the FR-CG algorithm, the PAE decreases from 1533 to 958 with 37 iterations. Thus, it clearly illustrates that the FR-CG algorithm gets rid of the local optimum and achieves better pattern fidelity with fewer iterations than the SD algorithm. Figure 12 shows the PW simulation results of the complex pattern. Figure 12(a) illustrates that TMIA clearly degrades the lithography performance in PW and ΔBF similar to the case of the line-space 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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Fig. 11. Comparisons of the PAE convergence curves among the SD algorithm (blue solid curves), the FR-CG algorithm (green dotted curves), and the FR-CG algorithm with pupil wavefront regularization (red dash-dotted curves) for the complex pattern.
Fig. 10. Simulation results of PWO based on the complex pattern. Left to right: source pattern, mask pattern, pupil wavefront, and printed image calculated by the rigorous EMF simulator. Top to bottom: simulations for the initial conditions, the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization.
pattern. However, the degradation of this lithography performance can be well compensated by the proposed PWO method, especially by the PWO method with the FR-CG algorithm, as shown in Fig. 12(c). To smooth the distribution of the optimized pupil wavefront, the Laplacian regularization term is also used in the proposed PWO method for the complex pattern with the regularization weight γ 0.00002. Figure 10(o) is the optimized pupil wavefront by the FR-CG algorithm with pupil wavefront regularization, and Fig. 10(p) is the corresponding printed image at nominal dose and focus. Comparison between Figs. 10(k) and 10(o) shows that the PV of the optimized pupil wavefront decreases from 0.558λ to 0.327λ and the RMS of the optimized pupil wavefront decreases from 0.072λ to 0.05λ by using pupil wavefront regularization. The red dash-dotted curve shown in Fig. 11 is the PAE convergence at the nominal dose and focus for the complex pattern by the FR-CG algorithm with pupil wavefront regularization. The corresponding PAE decreases from 1533 to 978 with 37 iterations. Figure 12(d) shows the PW of the complex pattern after PWO by using the FR-CG algorithm with pupil wavefront regularization. The corresponding ΔBF and DOF are 12 and 119 nm, respectively. Compared with the FR-CG algorithm, the FR-CG algorithm with pupil wavefront regularization can also further enlarge the PW for the complex pattern. Figure 13 illustrates the Zernike coefficients of the optimized pupil wavefronts shown in Fig. 10. It is noted that some odd 6868
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aberrations [39], such as c14 , c23 , c24 , and c31 , are required to be modified to reduce the PLE induced by TMIA. Table 2 summarizes the PAE, CDE, PLE, DOF, and ΔBF for the complex pattern. The values of these criteria clearly indicate that the proposed PWO algorithm can also effectively improve the lithography performance for the complex pattern. In addition, the runtimes shown in the last row of Table 2 also prove that the FR-CG algorithm is more efficient than the SD algorithm for the complex pattern. All of the above simulation results demonstrate that our proposed inverse PWO approach performs well in compensating for TMIA and improves the lithography performance.
Fig. 12. Simulation of PWs for the complex pattern: (a) initial PW, (b) optimized PW by the SD algorithm, (c) optimized PW by the FR-CG algorithm, and (d) optimized PW by the FR-CG algorithm with pupil wavefront regularization.
s ys where T xp;m−r;n−s is the m − r; n − sth entry of Txps ys , and Θrs is the r; sth entry of Θ. According to Eqs. (A.1) and (A.2), the gradients of the cost function with respect to ReΘrs and ImΘrs are, respectively, calculated as
N X N X ∂F 4a ⃗ − z⃗ mn · z⃗ mn ∂ReΘrs m1 n1 J sum z~ mn XX · 1 − z⃗ mn Jxs ys xs
·
ys
(
X
N X N X
Re
px;y;z
Fig. 13. Zernike coefficients of the optimized pupil wavefront for the complex pattern, where the blue, green, and red bars represent the Zernike coefficients of the optimized pupil wavefront by the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization, respectively.
5. Conclusion
In conclusion, this paper proposed an inverse PWO approach to compensate for TMIA based on the analytic vector imaging theory. The FR-CG algorithm was used to solve for the PWO problem, and the Laplacian regularization term was introduced to smooth the distribution of the pupil wavefront. Besides, all of the Zernike terms of the pupil wavefront are simultaneously optimized to compensate for the impact of TMIA in our PWO algorithm, thus improving the degree of freedom in the optimization process. To prove the effectiveness and efficiency of the approach, two typical target patterns were used to test the proposed algorithm. The simulation results indicated that the inverse PWO algorithm can improve the pattern fidelity, reduce the best focus shift, and enlarge the PW for the immersion lithography system. Hence, our proposed inverse PWO approach is demonstrated to be effective and efficient to compensate for TMIA and improve lithography performance.
·
)
s ys T xp;m−r;n−s
(A3)
N X N X ∂F 4a ⃗ − z⃗ mn · z⃗ mn ∂ImΘrs m1 n1 J sum z~ mn XX · 1 − z⃗ mn J xs ys xs
·
(
X
Im
px;y;z
ys N X N X
! s ys T xp;m−r;n−s
· Θrs
r1 s1
)
s ys ·T xp;m−r;n−s
:
(A4)
According to Eqs. (3) and (5), Θ F −1 fexpj2πWg
N X N X 1 X expj2π ci Γi;gh N 2 g1 h1 i gr hs · exp j2π ; N N
(A5)
where Γi;gh is the g; hth entry of Γi. Thus, ∂ReΘrs ReF −1 fsin2πW · 2πΓi g ∂ci
The cost function described in Eq. (8) can be reformulated as F
· Θrs
r1 s1
and
Appendix A: Derivation of Gradients of the Cost Function
N X N X
! s ys T xp;m−r;n−s
ImF −1 fcos2πW · 2πΓi g
(A6)
and 2
⃗z~ mn − z⃗ mn ;
(A1)
∂ ImΘrs −ImF −1 fsin2πW · 2πΓi g ∂ci
m1 n1
~ and z⃗ mn is the where z⃗ ~ mn is the m; nth entry of Z, m; nth entry of Z; that is,
z⃗ mn
1 exp
n
−a J sum
PP xs ys
J
xs ys
ReF −1 fcos2πW · 2πΓi g:
1 P PN PN p
r1
xs ys s1 T p;m−r;n−s Θrs
2
atr
o;
10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
(A7)
(A2)
6869
According to the chain rule, the gradient of the cost function with respect to ci can be calculated as N X N ∂F X ∂F ∂ ReΘrs ∂F ∂ ImΘrs · ⊙ ∂ci ∂ReΘrs ∂ci ∂ImΘrs ∂ci r1 s1 h i X X X 8aπ Jxs ys 1TN×1 ReΛxps ys ⊙ fReF −1 Γi ⊙ sin2πW ImF −1 Γi ⊙ cos2πWg 1N×1 J sum x y px;y;z s s h i X X X 8aπ − Jxs ys 1TN×1 ImΛxps ys ⊙ fReF −1 Γi ⊙ cos2πW − ImF −1 Γi ⊙ sin2πWg 1N×1 : J sum x y px;y;z s
We thank the financial support by the Key Program of the National Natural Science Foundation of China under Grant No. 60938003, the National Science and Technology Major Project, the National Natural Science Foundation of China (Grant No. 61204113), the Program for New Century Excellent Talents in University (NCET, Grant No. NCET-100042), the Basic Research Foundation of Beijing Institute of Technology (Grant No. 20120442001), and the Technology Foundation for Selected Overseas Chinese Scholar. We also thank the KLA-Tencor Corporation for providing academic use of PROLITH. References 1. M. Rothschild, “A roadmap for optical lithography,” Opt. Photon. News 21(6), 26–31 (2010). 2. J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003). 3. M. Bai, L. S. Melvin III, Q. Yan, J. P. Shiely, B. J. Falch, C. Fu, and R. Wang, “Approximation of three dimensional mask effects with two dimensional features,” Proc. SPIE 5751, 446–454 (2005). 4. A. Erdmann, “Mask modeling in the low k1 and ultrahigh NA regime: phase and polarization effects,” Proc. SPIE 5835, 69–81 (2005). 5. A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. De Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: an assessment for typical hyper NA imaging problems,” Proc. SPIE 6283, 628319 (2006). 6. A. Erdmann and P. Evanschitzky, “Rigorous electromagnetic field mask modeling and related lithographic effects in the low k1 and ultrahigh numerical aperture regime,” J. Microlithogr., Microfabr., Microsyst. 6, 031002 (2007). 7. J. Finders, M. Dusa, J. Mulkens, Y. Cao, and M. Escalante, “Solutions for 22 nm node patterning using ArFi technology,” Proc. SPIE 7973, 79730U (2011). 8. F. Staals, A. Andryzhyieuskaya, H. Bakker, M. Beems, J. Finders, T. Hollink, J. Mulkens, A. Nachtwein, R. Willekers, and P. Engblom, “Advanced wavefront engineering for improved imaging and overlay applications on a 1.35 NA immersion scanner,” Proc. SPIE 7973, 79731G (2011). 9. M. K. Sears, G. Fenger, J. Mailfert, and B. W. Smith, “Extending SMO into the lens pupil domain,” Proc. SPIE 7973, 79731B (2011). 10. M. K. Sears, J. Bekaert, and B. W. Smith, “Pupil wavefront manipulation for optical nanolithography,” Proc. SPIE 8326, 832611 (2012). 11. M. K. Sears, J. Bekaert, and B. W. Smith, “Lens wave front compensation for 3D photomask effects in subwavelength optical lithography,” Appl. Opt. 52, 314–322 (2013). 12. T. Fühner, P. Evanschitzky, and A. Erdmann, “Mutual source, mask and projector pupil optimization,” Proc. SPIE 8326, 83260l (2012). 6870
(A8)
s
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13. X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29, 1300–1312 (2012). 14. X. Ma, Y. Li, X. Guo, L. Dong, and G. R. Arce, “Vectorial mask optimization methods for robust optical lithography,” J. Microlithogr., Microfabr., Microsyst. 11, 043008 (2012). 15. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30, 112–123 (2013). 16. X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52, 4200–4211 (2013). 17. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17, 5783–5793 (2009). 18. X. Ma and G. R. Arce, Computational Lithography, 1st ed., Wiley Series in Pure and Applied Optics (Wiley, 2010). 19. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17, 23690–23701 (2009). 20. Z. Geng, Z. Shi, X. Yan, and K. Luo, “Regularized levelset-based inverse lithography algorithm for IC mask synthesis,” J. Zhejiang University Sci. C 14, 799–807 (2013). 21. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964). 22. X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19, 2165–2180 (2011). 23. D. G. Flagello and A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992). 24. G. M. Gallatin, “High-numerical-aperture scalar imaging,” Appl. Opt. 40, 4958–4964 (2001). 25. P. Evanschitzky, A. Erdmann, and T. Fühner, “Extended Abbe approach for fast and accurate lithography imaging simulations,” Proc. SPIE 7470, 747007 (2009). 26. R. Schlief, A. Liebchen, and J. F. Chen, “Hopkins vs. Abbe, a lithography simulation matching study,” Proc. SPIE 4691, 1106–1117 (2002). 27. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGrawHill Science, 1996). 28. A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001). 29. M. S. Yeung, D. Lee, R. Lee, and A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” Proc. SPIE 1927, 452–463 (1993). 30. B. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (Groningen University, 1942). 31. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). 32. P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer-Zernike theory,” Proc. SPIE 5754, 262–273 (2005).
33. W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE 5377, 1536–1543 (2001). 34. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007). 35. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15066–15079 (2007). 36. J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18, 23331–23342 (2010). 37. KLA-Tencor, “ProDATA advanced CD analysis software— lithography modeling,” http://www.kla‑tencor.com/lithography‑ modeling/pro‑data.html.
38. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). 39. W. Liu, S. Liu, T. Shi, and Z. Tang, “Generalized formulations for aerial image based lens aberration metrology in lithographic tools with arbitrarily shaped illumination sources,” Opt. Express 18, 20096–20104 (2010). 40. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 36–38 (1992). 41. R. C. Peng, I. H. Huang, H. H. Liu, H. J. Lee, J. Lin, A. Lin, A. Chang, B. S. Lin, and I. Lalovic, “Process requirements for pitch splitting LELE double patterning at advanced logic technology node,” Proc. SPIE 8326, 83260X (2012).
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As the critical dimension of integrated circuits is continuously shrunk, thick mask induced aberration (TMIA) cannot be ignored in the lithography image process. Recently, a set of pupil wavefront optimization (PWO) approaches has been proposed to compensate for TMIA, based on a wavefront manipulator in modern scanners. However, these prior PWO methods have two intrinsic drawbacks. First, the traditional methods fell short in building up the analytical relationship between the pupil wavefront and the cost function, and used time-consuming algorithms to solve for the PWO problem. Second, in traditional methods, only the spherical aberrations were optimized to compensate for the focus exposure matrix tilt and best focus shift induced by TMIA. Thus, the degrees of freedom were limited during the optimization procedure. To overcome these restrictions, we build the analytical relationship between the pupil wavefront and the cost function based on Abbe vector imaging theory. With this analytical model and the Fletcher–Reeves conjugate-gradient algorithm, an inverse PWO method is innovated to balance the TMIA including 37 Zernike terms. Simulation results illustrate that our approach significantly improves image fidelity within a larger process window. This demonstrates that TMIA is effectively compensated by our inverse PWO approach. © 2014 Optical Society of America OCIS codes: (110.5220) Photolithography; (110.2945) Illumination design; (220.1000) Aberration compensation. http://dx.doi.org/10.1364/AO.53.006861
1. Introduction
As optical lithography steps into the low-k1 regime [1], the critical dimension (CD) of integrated circuits becomes much smaller than the scanner exposure wavelength for 45 nm technology nodes and beyond. Thus, the wavefront phase error induced by a thick mask cannot be ignored [2,3]. To understand thick mask induced aberration (TMIA) comprehensively, Erdmann studied the impact of a thick mask on lithography performance by using rigorous electromagnetic field (EMF) simulation [4–6]. The results indicated that TMIA causes the modification of polarization and serious asymmetry of the lithographic process window. Therefore, it is necessary 1559-128X/14/296861-11$15.00/0 © 2014 Optical Society of America
to compensate TMIA for the next generation of lithography. Recently, ASML designed a freeform pupil wavefront manipulator named FlexWave in modern scanner tools [7,8]. Based on this sophisticated wavefront control equipment, Sears et al. [9–11] and Fühner et al. [12] introduced the idea of pupil wavefront optimization (PWO) and proposed PWO approaches to compensate for TMIA. It was shown that the PWO methods evidently improved the process window and counteracted thick mask induced wavefront skews. However, their PWO methods have two intrinsic drawbacks. First, an analytical relationship between the pupil wavefront and the cost function is not built in their optimization algorithms. Thus, these methods applied a nonanalytic algorithm to solve for the PWO problem, which is computationally expensive. Second, these methods only compensated 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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for the spherical aberrations in TMIA, thus limiting the degrees of optimization freedom. To overcome these limitations, this paper develops an inverse PWO approach to compensate for TMIA. The analytical framework of PWO is formulated under the Abbe vector imaging model [13–16], which is much more accurate than the scalar model for immersion lithography systems with hyper-NA (NA > 1). Here, the cost function is set to be the square of the Euler distance between the target pattern and the printed image on the focal plane with nominal dose [17,18]. And the pupil wavefront is represented as the sum of 37 Zernike polynomials. Then, the gradients of the cost function with respect to the 37 Zernike coefficients of the pupil wavefront are derived. In addition, this paper introduces a Laplacian regularization term [19,20] to smooth the distribution of the pupil wavefront. Subsequently, the Fletcher–Reeves conjugate-gradient (FR-CG) algorithm [21,22] is adopted to optimize the pupil wavefront. Compared with the traditional PWO methods, the proposed PWO algorithm optimizes 37 Zernike polynomials simultaneously, rather than only spherical aberrations. This is because the lithography performance associated with the complex thick mask and the freeform source is also sensitive to asymmetrical aberrations [9,11]. Simulation results demonstrate that the proposed method is effective and efficient in compensating for TMIA. The remainder of the paper is organized as follows. The Abbe vector imaging model used in this paper is described in Section 2. The gradient-based inverse PWO algorithm is developed in Section 3. Simulations are illustrated in Section 4. Conclusions are provided in Section 5. 2. Abbe Vector Imaging Model
In this paper, the Abbe vector imaging model is used to formulate the proposed inverse PWO framework, since the vector nature of the EMF cannot be neglected in immersion lithography [23,24]. In addition, the Hopkins approach is not accurate for imaging calculation with a thick mask [25]. A schematic of the vectorial imaging process for an optical lithography system is illustrated in Fig. 1, where the spatial coordinates on the source plane are xs ; ys , and the direction cosine of the wave propagating from the source to the mask is αs ; βs ; γ s . The spatial coordinates on the mask plane are x; y, and the direction cosine of the wave propagating from the mask to the entrance pupil is α; β; γ. The normalized spatial coordinate on the exit pupil is f ; g. The direction cosine of the wave propagating from the exit pupil to the wafer is α0 ; β0 ; γ 0 . The spatial coordinate on the wafer is xw ; yw . According to Abbe’s method [26] and Fourier optics theory [27], the aerial image on the wafer contributed by the effective source J [28] can be formulated as 6862
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
Fig. 1. Vectorial imaging process of immersion lithography system.
I
1 XX J sum
×V
xs
xs ys
ys
⊙
Jxs ys
X px;y;z
‖
F −1
s ys expj2πW ⊙ Gx3D
2π ×C nw R
⊙ Exi s ys
2 ; p
‖
(1)
2
where Jxs ys is the intensity P P of the source point at xs ; ys , and J sum xs ys Jxs ys is a normalization factor. The notation ⊙ is the entry-by-entry multiplication operation, nw 1.44 is the refractive index of the media on the wafer side, R 4 is the demagnification factor, and F −1 is the inverse Fourier transform. Exi s ys is an N × N matrix with each entry equal to a 2 × 1 vector representing the electric field of incidence light in spatial coordinates. The superscript s ys xs ys denotes that Ei is the function of xs ; ys . Gx3D is the thick mask diffraction spectrum corresponding to source point xs ; ys , each entry of which is a 2 × 2 matrix. Vxs ys is an N × N matrix with each entry equal to a 3 × 2 matrix of 2 β02 α02 γ0 6 Vm; n 4
1−γ 02 α0 β 0 − 1γ 0 0
−α
0 0
αβ − 1γ 0
β02 α02 γ 0 1−γ 02 0
3 7 5;
m; n 1; 2; …; N:
−β
(2) p0 C γ∕γ is the radiometric correction factor [29]. W is the pupil wavefront distribution, which can be implemented by the FlaxWave. According to the Nijboer–Zernike theory [30–32], W can be represented as Wρ; θ
X ci Γi ρ; θ;
(3)
i
where ρ; θ are the polar coordinates on the exit pupil, ρ λ∕NAw f 2 g2 1∕2 , and θ arctanf ∕g. NAw
and λ are the numerical aperture on the wafer side and the scanner exposure wavelength, respectively. Γi ρ; θ is the ith Zernike polynomial, and ci is the corresponding Zernike coefficient. To simplify Eq. (1), we define two notations: Txps ys
F
−1
2π s ys × C × Vxs ys ⊙ Gx3D ⊙ Exi s ys nw R
(4) p
Fig. 2. Illustration of the PWO approach to compensate for TMIA.
and Θ F −1 fexpj2πWg:
(5)
Thus, the aerial image described in Eq. (1) can be reformulated as I
1 X X xs ys X xs ys 2 J ‖Tp ⊗ Θ‖2 ;
J sum
xs
ys
(6)
px;y;z
where ⊗ is the convolution operation. Then, the aerial image described in Eq. (6) enters into the photoresist and forms the printed image through development. Here, the sigmoid function is employed to approximate the constant threshold resist (CTR) model [33] due to its differentiability [34,35]. Thus, the developed resist image on the wafer can be described by a sigmoid transformation of the aerial image, that is, Z sigI
1 ; 1 exp−aI − tr
(7)
where a indicates the steepness of the sigmoid function, and tr is the process threshold. 3. Inverse Optimization Algorithm for Pupil Wavefront Optimization A.
Pupil Wavefront Optimization Algorithm
Today, a co-optimizing source and mask pattern is necessary for 45 nm technology nodes and beyond. Since the rigorous 3D mask models are complex and computationally expensive, they are almost not applied in current source and mask optimization (SMO) methods. On the other hand, the objective of this paper is to prove the capability in counteracting TMIA by the proposed PWO, not by SMO. Thus, in this paper, the vectorial SMO method described in [15] will first be used to obtain the source J and mask M. Then, we use the rigorous EMF simulator in PROLITH to calculate the thick mask diffraction specs ys trum Gx3D . Finally, the source J and the diffraction s ys are applied in the proposed PWO spectrum Gx3D algorithm. Figure 2 illustrates the PWO approach to compensate for TMIA, where the distorted wavefront W3D is introduced by TMIA and the wavefront manipulator W is the optimized pupil wavefront by the PWO method. This figure indicates that the PWO approach uses W to balance W3D , and finally makes the actual wavefront WAct significantly improve the lithography performance.
Given a binary target pattern Z~ ∈ RN×N with all entries equal to 0 or 1, the goal of PWO is to find ˆ so that the optimal pupil wavefront denoted as W the cost function ~ F dZ; Z 1 X X xs ys X xs ys 2 ‖Tp ⊗ Θ‖2 ; Z~ d sig J J sum x y px;y;z s
s
(8) is minimized, where d·; · is the square of the Euler distance between the two arguments. According to Eq. (3), we can optimize the Zernike coefficients ci during the PWO procedure, and then the wavefront W can be reconstructed from the corresponding Zernike coefficients. Therefore, the PWO problem can be formulated as ~ cˆ i arg min dZ; Z:
(9)
ci
In this paper, the FR-CG algorithm is applied to solve for the PWO problem in Eq. (9). The efficiency and convergence property of the FR-CG algorithm in solving the linear and nonlinear problems has been widely proved in different areas [21,22]. The FR-CG algorithm requires knowledge of the gradient of the cost function with respect to the Zernike coefficients, as shown in Appendix A, which can be calculated as ∇Fci
8aπ X X xs ys X T J 1N×1 J sum x y px;y;z s s h × ReΛxps ys ⊙ fReF −1 Γi ⊙ sin2πW i ImF −1 Γi ⊙ cos2πWg 1N×1 8aπ X X xs ys X T J 1N×1 J sum xs ys px;y;z h × ImΛxps ys ⊙ fReF −1 Γi ⊙ cos2πW i (10) − ImF −1 Γi ⊙ sin2πWg 1N×1 ; −
where the superscript T is a transposition operation. 1N×1 is the one-valued vector. Re· and Im· represent the real part and the image part of the argument, respectively. Λxps ys is given by 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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∘
Λxps ys Txps ys ⊗ Z~ − Z ⊙ Z ⊙ 1 − Z ⊙ Txps ys ⊗ Θ; (11) where is the conjugate operation, and ∘ represents a rotation of the matrix in the argument by 180° in both the horizontal and vertical directions. It is noted that the derivation of Eq. (10) is based on the ideal lithography system. More works should be done in the future for the lithography system with different noises, which, however, is out of the scope of this paper. The details of the FR-CG algorithm are described by the pseudo-code in Table 1, where the variables in the kth iteration are marked by the superscript k with brackets. B.
Pupil Wavefront Regularization
In order to smooth the distribution of the optimized pupil wavefront, this paper introduces a Laplacian regularization term. The Laplacian regularization term is formulized as Z RLaplacian
Ω
2 ∂W 2 ∂W dx: ∂f ∂g
(12)
Thus, the cost function in Eq. (8) is modified as F 0 F γRLaplacian ;
(13)
where γ is the regularization weight. Since the first differential operator is calculated as [36] 2 6 6 6 D6 6 4
1
0
−1 1
−1 .. .
0
..
.
1 −1
3
7 7 7 7; 7 −1 5 1
(14)
Table 1.
∇RLaplacian ci 1TN×1 2 × DW ⊙ DΓi 2 × WDT ⊙ Γi DT 1N×1 :
(16)
4. Simulation Results
In order to demonstrate the performance of the proposed inverse PWO algorithm, two typical target patterns in Fig. 3 are used in the following simulations. The CD of both target patterns is 45 nm. Figure 3(a) is a line-space pattern represented by a 301 × 301 matrix with pixel size of 5 nm × 5 nm on wafer scale. Figure 3(b) is a complex pattern represented by a 200 × 200 matrix with pixel size of 5.625 nm × 5.625 nm on wafer scale. Consider an immersion lithography system with wavelength of λ 193 nm, and NA 1.35. We adopt Y polarization and TE polarization for the line-space pattern and the complex pattern, respectively. In Eq. (7), a 25, tr 0.1 for the line-space pattern, while tr 0.2 for the complex pattern. Some important criteria are chosen to evaluate the lithography performance, such as pattern error (PAE), CD error (CDE), placement error (PLE), depth of focus (DOF), best focus variation (ΔBF), and PW. At the focal plane with nominal exposure dose, the PAE is calculated as [15]
‖
PAE Z~ 2 1 X X xs ys X J −Ξ ‖Txps ys ⊗ Θ‖22 − tr ; J sum x y px;y;z s
s
‖
2
(17)
the matrix form of Eq. (12) can be rewritten into Eq. (15): RLaplacian ‖DW‖22 ‖WDT ‖22 :
Thus, the gradient of RLaplacian with respect to ci can be calculated as
(15)
Pseudo-Code of the FR-CG Algorithm
1. Initialize k 0, the starting Zernike coefficients c0 i 0.000001, the pupil wavefront step size sc , the maximum iteration number lpwo, and P0 −∇Fc0 i . : 2. Update the Zernike coefficients ck1 i ~ >0 while k ≤ lpwo & dZk − Z k←k 1; Calculate the gradient, i.e., ∇Fck i , using Eq. (10); k 2 ‖∇Fc ‖ i 2 Calculate βk :βk ; 2 ‖∇Fck−1 ‖2 i
where Ξf·g 1 if the argument is larger than 0; otherwise, Ξf·g 0. The CDE and PLE denote the average CDE and PLE of all critical locations on the printed image at nominal dose and focus [15]. DOF is the largest acceptable defocus range with exposure latitude (EL) of 5%. According to Refs. [4–6,10,11], TMIA can induce best focus shift. Thus, the ΔBF is taken into consideration to the advantage of our proposed algorithm in reducing the best focus offset, which is defined as the distance between the
k k−1 ; ; Update Pk :Pk −∇Fck i β P k1 k k Update ck :c c s P . c i i i
end ˆ by the 3. Calculate the optimized wavefront manipulator W optimized Zernike coefficients cˆ i using Eq. (3).
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Fig. 3. Two target patterns used in the simulations: (a) line-space pattern and (b) complex pattern. Red lines mark the positions to measure the PWs.
optimal focus position of the chosen PW and zero focus in this paper, as shown in Fig. 6. In addition, PWs are measured at the positions marked by the red lines in Fig. 3, where the linewidth measurement specification is set to be 10% and EL is 5% [37]. After that, the overlapped PWs at different measurement positions are used to evaluate the robustness of the lithography system to the defocus effect and dose variation. Besides, it is worthwhile to mention that the source and mask patterns, as shown in Figs. 4, 8, and 10, are obtained by the SMO algorithm described in [15]. The frequency spectra of the thick masks are calculated by the FDTD algorithm [5,38] in PROLITH, where the mask absorbers consist of 55 nm Cr with a refractive index of 1.48 1.76i and 18 nm CrO with a refractive index of 1.97 1.2i [5,11]. For the line-space pattern, the step size sc 0.0001 and the maximal loop lpwo 50. To prove the efficiency of the FR-CG algorithm used in the proposed PWO method, the steepest descent (SD) algorithm is compared in this paper. Because of the axial symmetry of the line-space pattern, just the even aberrations [39] need to be optimized. Figure 4 illustrates the simulation results, where Figs. 4(a) and 4(b) are the source pattern and the mask pattern, respectively. Figure 4(c) is the initial pupil wavefront. Figure 4(d) is the printed image at nominal dose and focus corresponding to the initial pupil wavefront, which is calculated by the rigorous EMF simulator in PROLITH. It clearly illustrates
Fig. 4. Simulation results of PWO based on the line-space pattern. Left to right: source pattern, mask pattern, pupil wavefront, and printed image calculated by the rigorous EMF simulator. Top to bottom: simulations for initial conditions, SD algorithm, FR-CG algorithm, and FR-CG algorithm with pupil wavefront regularization.
Fig. 5. Comparisons of the PAE convergence curves among the SD algorithm (blue solid curves), the FR-CG algorithm (green dotted curves), and the FR-CG algorithm with pupil wavefront regularization (red dash-dotted curves) for the line-space pattern.
that TMIA has a significant impact on the imaging performance. Figure 4(g) is the optimized pupil wavefront by the PWO method with the SD algorithm, and Fig. 4(h) is the corresponding printed image at nominal dose and focus. Comparison between Figs. 4(d) and 4(h) shows that the image fidelity is well improved by the PWO algorithm with the SD algorithm. The simulation results obtained by the proposed PWO method with the FR-CG algorithm are illustrated in the third row of Fig. 4. Compared with Fig. 4(h), Fig. 4(l) indicates that the FR-CG algorithm can better improve the image quality at nominal dose and focus than the SD algorithm for the line-space pattern. Figure 5 shows the convergence curves of PAE at the nominal conditions for the line-space pattern, where the blue solid and green dotted curves represent the PAE convergence of the SD algorithm and the FR-CG algorithm, respectively. By the SD algorithm, the PAE decreases from 1191 to 601 with 120 iterations. By the FR-CG algorithm, the PAE decreases from 1191 to 553 with 50 iterations. Thus, it is clearly illustrated that the FR-CG algorithm can achieve better pattern fidelity with fewer iterations than the SD algorithm for the line-space pattern. Figure 6 shows the PW simulation results of the line-space pattern, where the area enclosed by the orange curve corresponds to the focus exposure matrix (FEM) of the P1 point at the line-space pattern and the area enclosed by the purple curve corresponds to the FEM of the P2 point. The overlapped area between the FEMs of P1 and P2 is enclosed by the blue curve, and the red ellipse is the overlapped PW. Figure 6(a) shows the initial PWs of the linespace pattern. From this figure, we can clearly see that TMIA induces different FEM tilts and best focus offsets for the measured positions of P1 and P2, thus seriously reducing the PW. Figure 6(b) shows the PW of the line-space pattern after PWO by using the SD algorithm. Compared with Fig. 6(a), it illustrates that the SD algorithm reduces the ΔBF from 97 to 7 nm and increases the DOF from 22 to 97 nm for the line-space pattern. Figure 6(c) shows the PW of 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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Fig. 7. Zernike coefficients of the optimized pupil wavefront for the line-space pattern, where the blue, green, and red bars represent the Zernike coefficients of the optimized pupil wavefront by the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization, respectively.
Fig. 6. Simulation of PWs for the line-space pattern: (a) initial PW, (b) optimized PW by the SD algorithm, (c) optimized PW by the FR-CG algorithm, and (d) optimized PW by the FR-CG algorithm with pupil wavefront regularization.
the line-space pattern after PWO by using the FRCG algorithm. The corresponding ΔBF and DOF are 4 and 100 nm, respectively. Compared with the SD algorithm, the FR-CG algorithm can further enlarge the PW for the line-space pattern. To smooth the distribution of the optimized pupil wavefront, the Laplacian regularization term in Section 2.B is used in the proposed PWO method, where the regularization weight γ 0.00001. Figure 4(o) is the optimized pupil wavefront by the FR-CG algorithm with pupil wavefront regularization, and Fig. 4(p) is the corresponding printed image at nominal dose and focus. Here, the peak-to-valley (PV) and root-mean-square (RMS), which are widely applied in wavefront aberration theory [40], will be used to evaluate the smoothness of the pupil wavefront. Generally, smaller values of PV and RMS correspond to a pupil wavefront with better smoothness. Comparison between Figs. 4(k) and 4(o) shows that the PV of the optimized pupil wavefront decreases from 0.389λ to 0.364λ and the RMS of the optimized pupil wavefront decreases from 0.094λ to 0.086λ by using pupil wavefront regularization. The red dash-dotted curve shown in Fig. 5 is the PAE convergence at the nominal dose and focus for the linespace pattern by the FR-CG algorithm with pupil wavefront regularization. The corresponding PAE decreases from 1191 to 561 with 50 iterations. Figure 6(d) shows the PW of the line-space pattern after PWO by using the FR-CG algorithm with pupil wavefront regularization. The corresponding ΔBF and DOF are 8 and 105 nm, respectively. Compared with the FR-CG algorithm, the FR-CG algorithm with pupil wavefront regularization can further enlarge the PW for the line-space pattern. 6866
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The results above prove that our proposed PWO algorithm not only improves the image fidelity at nominal conditions, but also reduces the ΔBF and enlarges the PW for the line-space pattern. To further understand the proposed PWO algorithm, Fig. 7 illustrates the Zernike coefficients of the optimized pupil wavefronts shown in Fig. 4. It is noted that we use all of the even aberrations, not just the spherical aberrations, to compensate for TMIA. The results in Fig. 7 indicate that the other even aberrations, such as c5, c12 , c17 , and c32 , are also important for counteracting TMIA, beside the spherical aberrations. This phenomenon can be proved by the simulation described in Ref. [11]. Thus, our proposed PWO algorithm further improves the degree of freedom in the optimization procedure. Table 2 summarizes the values of PAE, CDE, PLE, DOF, and ΔBF corresponding to the initial case, the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization for the line-space pattern. All of these criteria of lithography performance Table 2. Comparisons of PAE, CDE, PLE, DOF, and ΔBF Among the Initial Case, the SD Algorithm, the FR-CG Algorithm, and the FR-CG Algorithm with Pupil Wavefront Regularization for the Line-Space Pattern and the Complex Pattern, Respectively
PWO Algorithm Target
Criterion
Line-Space PAE CDE (nm) PLE (nm) DOF (nm) ΔBF (nm) Runtime (s) Complex PAE CDE (nm) PLE (nm) DOF (nm) ΔBF (nm) Runtime (s)
Initial
SD
1191 14.4 2.7 22 97 — 1533 6.6 5.8 19 71 —
601 6.3 2.5 97 7 5704 1184 5.8 4.6 74 32 9508
FR-CG FR-CG+Reg 553 5.7 2.5 100 4 2519 958 4.9 4.2 116 21 3608
561 5.8 2.5 105 8 2827 978 4.9 4.2 119 12 4023
are improved by the proposed PWO algorithm. It is worthwhile to note that the simulations in this paper are carried out using MATLAB on an Intel(R) Core (TM) i5-2400 CPU, with 3.10 GHz, and 4 GB of RAM. The memory requirement of the PWO procedure is 300 MB for the line-space pattern and 260 MB for the complex pattern. The runtimes of the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization for the line-space pattern are summarized in Table 2. Comparison of the runtimes between the SD algorithm and the FR-CG algorithm further reveals that the FR-CG algorithm is more efficient than the SD algorithm for the line-space pattern. Thus, these values demonstrate that the proposed PWO method using the FR-CG algorithm with pupil wavefront regularization is effective and efficient to reduce the impact of TMIA for the line-space pattern. Figure 8 illustrates the simulations of the proposed PWO method for the line-space patterns with CDs equal to 40 and 35 nm, respectively. Figures 8(d) and 8(h) clearly show that the line-space patterns with CDs equal to 40 and 35 nm cannot be resolved at the nominal dose and focus due to TMIA. In the second row of Fig. 8, the simulation with CD 40 nm leads to PAE of 774, CDE of 7.3 nm, and PLE of 2.5 nm. In the fourth row of Fig. 8, the simulation with CD 35 nm leads to PAE of 1457, CDE of 6.4 nm, and PLE of 6.2 nm. Figure 9 shows the PWs for the line-space patterns with CDs equal to 40 and 35 nm. For the line-space pattern with CD 40 nm, the DOF increases from 11 to 22 nm.
Fig. 8. Simulations of the proposed PWO method for the line-space pattern with different CDs. Top to bottom: initial case with CD 40 nm, PWO with CD 40 nm, initial case with CD 35 nm, and PWO with CD 35 nm. Left to right: source patterns, mask patterns, pupil wavefronts, and printed images.
Fig. 9. Simulation of PWs for the line-space pattern with different CDs. Top to bottom: the CDs of the target patterns are 40 and 35 nm, respectively. Left to right: the initial case and the optimized PW by the proposed method.
For the line-space pattern with CD 35 nm, the DOF with EL of 2% increases from 4 to 10 nm. In contrast with the initial pupil wavefront for a given lithography system, the proposed PWO method can thus compensate for TMIA and improve the image fidelity for CDs equal to 40 and 35 nm. However, for the 22 nm lithography node, our PWO method should be combined with other resolution enhancement technology, such as double patterning technology (DPT) [41]. In order to prove the universality of the proposed PWO approach, another experiment based on the complex pattern is conducted with sc 0.0001, lpwo 37. Figure 10 shows the corresponding PWO results of the complex pattern in a way similar to Fig. 4. From this figure, we can see that the pattern fidelity of the printed image is evidently improved by the proposed PWO method, and the FR-CG algorithm can better improve the image quality at nominal dose and focus than the SD algorithm for the complex pattern. Figure 11 shows the convergence curves of PAE at the nominal conditions for the complex pattern. The blue solid curve in Fig. 11 reveals that the SD algorithm converges slowly and is apt to be trapped into the local optimum. The corresponding PAE decreases from 1533 to 1184 with 100 iterations. However, by using the FR-CG algorithm, the PAE decreases from 1533 to 958 with 37 iterations. Thus, it clearly illustrates that the FR-CG algorithm gets rid of the local optimum and achieves better pattern fidelity with fewer iterations than the SD algorithm. Figure 12 shows the PW simulation results of the complex pattern. Figure 12(a) illustrates that TMIA clearly degrades the lithography performance in PW and ΔBF similar to the case of the line-space 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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Fig. 11. Comparisons of the PAE convergence curves among the SD algorithm (blue solid curves), the FR-CG algorithm (green dotted curves), and the FR-CG algorithm with pupil wavefront regularization (red dash-dotted curves) for the complex pattern.
Fig. 10. Simulation results of PWO based on the complex pattern. Left to right: source pattern, mask pattern, pupil wavefront, and printed image calculated by the rigorous EMF simulator. Top to bottom: simulations for the initial conditions, the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization.
pattern. However, the degradation of this lithography performance can be well compensated by the proposed PWO method, especially by the PWO method with the FR-CG algorithm, as shown in Fig. 12(c). To smooth the distribution of the optimized pupil wavefront, the Laplacian regularization term is also used in the proposed PWO method for the complex pattern with the regularization weight γ 0.00002. Figure 10(o) is the optimized pupil wavefront by the FR-CG algorithm with pupil wavefront regularization, and Fig. 10(p) is the corresponding printed image at nominal dose and focus. Comparison between Figs. 10(k) and 10(o) shows that the PV of the optimized pupil wavefront decreases from 0.558λ to 0.327λ and the RMS of the optimized pupil wavefront decreases from 0.072λ to 0.05λ by using pupil wavefront regularization. The red dash-dotted curve shown in Fig. 11 is the PAE convergence at the nominal dose and focus for the complex pattern by the FR-CG algorithm with pupil wavefront regularization. The corresponding PAE decreases from 1533 to 978 with 37 iterations. Figure 12(d) shows the PW of the complex pattern after PWO by using the FR-CG algorithm with pupil wavefront regularization. The corresponding ΔBF and DOF are 12 and 119 nm, respectively. Compared with the FR-CG algorithm, the FR-CG algorithm with pupil wavefront regularization can also further enlarge the PW for the complex pattern. Figure 13 illustrates the Zernike coefficients of the optimized pupil wavefronts shown in Fig. 10. It is noted that some odd 6868
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aberrations [39], such as c14 , c23 , c24 , and c31 , are required to be modified to reduce the PLE induced by TMIA. Table 2 summarizes the PAE, CDE, PLE, DOF, and ΔBF for the complex pattern. The values of these criteria clearly indicate that the proposed PWO algorithm can also effectively improve the lithography performance for the complex pattern. In addition, the runtimes shown in the last row of Table 2 also prove that the FR-CG algorithm is more efficient than the SD algorithm for the complex pattern. All of the above simulation results demonstrate that our proposed inverse PWO approach performs well in compensating for TMIA and improves the lithography performance.
Fig. 12. Simulation of PWs for the complex pattern: (a) initial PW, (b) optimized PW by the SD algorithm, (c) optimized PW by the FR-CG algorithm, and (d) optimized PW by the FR-CG algorithm with pupil wavefront regularization.
s ys where T xp;m−r;n−s is the m − r; n − sth entry of Txps ys , and Θrs is the r; sth entry of Θ. According to Eqs. (A.1) and (A.2), the gradients of the cost function with respect to ReΘrs and ImΘrs are, respectively, calculated as
N X N X ∂F 4a ⃗ − z⃗ mn · z⃗ mn ∂ReΘrs m1 n1 J sum z~ mn XX · 1 − z⃗ mn Jxs ys xs
·
ys
(
X
N X N X
Re
px;y;z
Fig. 13. Zernike coefficients of the optimized pupil wavefront for the complex pattern, where the blue, green, and red bars represent the Zernike coefficients of the optimized pupil wavefront by the SD algorithm, the FR-CG algorithm, and the FR-CG algorithm with pupil wavefront regularization, respectively.
5. Conclusion
In conclusion, this paper proposed an inverse PWO approach to compensate for TMIA based on the analytic vector imaging theory. The FR-CG algorithm was used to solve for the PWO problem, and the Laplacian regularization term was introduced to smooth the distribution of the pupil wavefront. Besides, all of the Zernike terms of the pupil wavefront are simultaneously optimized to compensate for the impact of TMIA in our PWO algorithm, thus improving the degree of freedom in the optimization process. To prove the effectiveness and efficiency of the approach, two typical target patterns were used to test the proposed algorithm. The simulation results indicated that the inverse PWO algorithm can improve the pattern fidelity, reduce the best focus shift, and enlarge the PW for the immersion lithography system. Hence, our proposed inverse PWO approach is demonstrated to be effective and efficient to compensate for TMIA and improve lithography performance.
·
)
s ys T xp;m−r;n−s
(A3)
N X N X ∂F 4a ⃗ − z⃗ mn · z⃗ mn ∂ImΘrs m1 n1 J sum z~ mn XX · 1 − z⃗ mn J xs ys xs
·
(
X
Im
px;y;z
ys N X N X
! s ys T xp;m−r;n−s
· Θrs
r1 s1
)
s ys ·T xp;m−r;n−s
:
(A4)
According to Eqs. (3) and (5), Θ F −1 fexpj2πWg
N X N X 1 X expj2π ci Γi;gh N 2 g1 h1 i gr hs · exp j2π ; N N
(A5)
where Γi;gh is the g; hth entry of Γi. Thus, ∂ReΘrs ReF −1 fsin2πW · 2πΓi g ∂ci
The cost function described in Eq. (8) can be reformulated as F
· Θrs
r1 s1
and
Appendix A: Derivation of Gradients of the Cost Function
N X N X
! s ys T xp;m−r;n−s
ImF −1 fcos2πW · 2πΓi g
(A6)
and 2
⃗z~ mn − z⃗ mn ;
(A1)
∂ ImΘrs −ImF −1 fsin2πW · 2πΓi g ∂ci
m1 n1
~ and z⃗ mn is the where z⃗ ~ mn is the m; nth entry of Z, m; nth entry of Z; that is,
z⃗ mn
1 exp
n
−a J sum
PP xs ys
J
xs ys
ReF −1 fcos2πW · 2πΓi g:
1 P PN PN p
r1
xs ys s1 T p;m−r;n−s Θrs
2
atr
o;
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(A7)
(A2)
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According to the chain rule, the gradient of the cost function with respect to ci can be calculated as N X N ∂F X ∂F ∂ ReΘrs ∂F ∂ ImΘrs · ⊙ ∂ci ∂ReΘrs ∂ci ∂ImΘrs ∂ci r1 s1 h i X X X 8aπ Jxs ys 1TN×1 ReΛxps ys ⊙ fReF −1 Γi ⊙ sin2πW ImF −1 Γi ⊙ cos2πWg 1N×1 J sum x y px;y;z s s h i X X X 8aπ − Jxs ys 1TN×1 ImΛxps ys ⊙ fReF −1 Γi ⊙ cos2πW − ImF −1 Γi ⊙ sin2πWg 1N×1 : J sum x y px;y;z s
We thank the financial support by the Key Program of the National Natural Science Foundation of China under Grant No. 60938003, the National Science and Technology Major Project, the National Natural Science Foundation of China (Grant No. 61204113), the Program for New Century Excellent Talents in University (NCET, Grant No. NCET-100042), the Basic Research Foundation of Beijing Institute of Technology (Grant No. 20120442001), and the Technology Foundation for Selected Overseas Chinese Scholar. We also thank the KLA-Tencor Corporation for providing academic use of PROLITH. References 1. M. Rothschild, “A roadmap for optical lithography,” Opt. Photon. News 21(6), 26–31 (2010). 2. J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003). 3. M. Bai, L. S. Melvin III, Q. Yan, J. P. Shiely, B. J. Falch, C. Fu, and R. Wang, “Approximation of three dimensional mask effects with two dimensional features,” Proc. SPIE 5751, 446–454 (2005). 4. A. Erdmann, “Mask modeling in the low k1 and ultrahigh NA regime: phase and polarization effects,” Proc. SPIE 5835, 69–81 (2005). 5. A. Erdmann, P. Evanschitzky, G. Citarella, T. Fühner, and P. De Bisschop, “Rigorous mask modeling using waveguide and FDTD methods: an assessment for typical hyper NA imaging problems,” Proc. SPIE 6283, 628319 (2006). 6. A. Erdmann and P. Evanschitzky, “Rigorous electromagnetic field mask modeling and related lithographic effects in the low k1 and ultrahigh numerical aperture regime,” J. Microlithogr., Microfabr., Microsyst. 6, 031002 (2007). 7. J. Finders, M. Dusa, J. Mulkens, Y. Cao, and M. Escalante, “Solutions for 22 nm node patterning using ArFi technology,” Proc. SPIE 7973, 79730U (2011). 8. F. Staals, A. Andryzhyieuskaya, H. Bakker, M. Beems, J. Finders, T. Hollink, J. Mulkens, A. Nachtwein, R. Willekers, and P. Engblom, “Advanced wavefront engineering for improved imaging and overlay applications on a 1.35 NA immersion scanner,” Proc. SPIE 7973, 79731G (2011). 9. M. K. Sears, G. Fenger, J. Mailfert, and B. W. Smith, “Extending SMO into the lens pupil domain,” Proc. SPIE 7973, 79731B (2011). 10. M. K. Sears, J. Bekaert, and B. W. Smith, “Pupil wavefront manipulation for optical nanolithography,” Proc. SPIE 8326, 832611 (2012). 11. M. K. Sears, J. Bekaert, and B. W. Smith, “Lens wave front compensation for 3D photomask effects in subwavelength optical lithography,” Appl. Opt. 52, 314–322 (2013). 12. T. Fühner, P. Evanschitzky, and A. Erdmann, “Mutual source, mask and projector pupil optimization,” Proc. SPIE 8326, 83260l (2012). 6870
(A8)
s
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
13. X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29, 1300–1312 (2012). 14. X. Ma, Y. Li, X. Guo, L. Dong, and G. R. Arce, “Vectorial mask optimization methods for robust optical lithography,” J. Microlithogr., Microfabr., Microsyst. 11, 043008 (2012). 15. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30, 112–123 (2013). 16. X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52, 4200–4211 (2013). 17. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17, 5783–5793 (2009). 18. X. Ma and G. R. Arce, Computational Lithography, 1st ed., Wiley Series in Pure and Applied Optics (Wiley, 2010). 19. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17, 23690–23701 (2009). 20. Z. Geng, Z. Shi, X. Yan, and K. Luo, “Regularized levelset-based inverse lithography algorithm for IC mask synthesis,” J. Zhejiang University Sci. C 14, 799–807 (2013). 21. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964). 22. X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19, 2165–2180 (2011). 23. D. G. Flagello and A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992). 24. G. M. Gallatin, “High-numerical-aperture scalar imaging,” Appl. Opt. 40, 4958–4964 (2001). 25. P. Evanschitzky, A. Erdmann, and T. Fühner, “Extended Abbe approach for fast and accurate lithography imaging simulations,” Proc. SPIE 7470, 747007 (2009). 26. R. Schlief, A. Liebchen, and J. F. Chen, “Hopkins vs. Abbe, a lithography simulation matching study,” Proc. SPIE 4691, 1106–1117 (2002). 27. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGrawHill Science, 1996). 28. A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001). 29. M. S. Yeung, D. Lee, R. Lee, and A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” Proc. SPIE 1927, 452–463 (1993). 30. B. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (Groningen University, 1942). 31. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). 32. P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer-Zernike theory,” Proc. SPIE 5754, 262–273 (2005).
33. W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE 5377, 1536–1543 (2001). 34. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007). 35. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15066–15079 (2007). 36. J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18, 23331–23342 (2010). 37. KLA-Tencor, “ProDATA advanced CD analysis software— lithography modeling,” http://www.kla‑tencor.com/lithography‑ modeling/pro‑data.html.
38. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). 39. W. Liu, S. Liu, T. Shi, and Z. Tang, “Generalized formulations for aerial image based lens aberration metrology in lithographic tools with arbitrarily shaped illumination sources,” Opt. Express 18, 20096–20104 (2010). 40. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 36–38 (1992). 41. R. C. Peng, I. H. Huang, H. H. Liu, H. J. Lee, J. Lin, A. Lin, A. Chang, B. S. Lin, and I. Lalovic, “Process requirements for pitch splitting LELE double patterning at advanced logic technology node,” Proc. SPIE 8326, 83260X (2012).
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