Optimization of lithography source illumination arrays using diffraction subspaces

Xu Ma; Zhiqiang Wang; Haijun Lin; Yanqiu Li; Gonzalo R. Arce; Lu Zhang

doi:10.1364/OE.26.003738

1. Introduction

Immersion lithography is used extensively in fabricating very-large and ultra-large scale integrated circuits. As shown in Fig. 1, immersion lithography system uses deep ultraviolet illuminations to expose the mask, so as to replicate the mask pattern onto the wafer through the projection optics. The immersion medium, such as deionized water, fills up the gap between the projector and wafer to further enhance the resolution. As the critical dimension (CD) of integrated circuit continuously shrinks, the imaging distortion in lithography becomes increasingly severe, which has to be compensated by computational lithography techniques [1–4]. Illumination optimization (ILO) is an effective method used to improve lithography imaging performance by modulating the intensity distribution of the light source [5–12]. Illumination optimization is also used extensively with other computational methods, such as source mask optimization (SMO) [13–20] and source mask polarization optimization (SMPO) [21,22], so as to improve the overall imaging performance of optical lithography.

Fig. 1 Sketch of an immersion lithography system (figure revised from [12]).

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To date, the freeform light source with thousands of pixels has been realized by a programmable illumination system. Accordingly, the degrees of optimization freedom in ILO methods have been dramatically increased [23,24]. Consequently, pixelated ILO methods are time-consuming due to the high dimensionality of the optimization problem. Recently, a set of ILO algorithms based on compressive sensing (CS) has been developed [9,12]. As summarized in Appendix A, the ILO in [9,12] was formulated by the following l₁-norm reconstruction problem:

\hat{\vec{Θ}} = min_{\vec{Θ}} {‖ \vec{Θ} ‖}_{1} subject to Φ {\vec{Z}}_{s} = Φ {\vec{I}}_{s} = Φ I_{c c}^{s} \vec{J} = Φ I_{c c}^{s} Ψ \vec{Θ},

where Z⃗_s ∈ ℝ^M×1 and I⃗_s ∈ ℝ^M×1 are the vectors of target mask layout and its corresponding aerial image on some monitoring pixels, respectively.

I_{c c}^{s} \in ℝ^{M \times N_{s}^{2}}

(

M ≪ N_{s}^{2}

) is the illumination cross coefficient (ICC) matrix corresponding to the monitoring pixels. It is noted that each column of

I_{c c}^{s}

represents the coherent aerial image on the monitoring pixels generated by one source point, while each row of

I_{c c}^{s}

represents the aerial images on a wafer pixel contributed by different source points. The source pattern can be represented by the vector of J⃗ = ΨΘ⃗, where

Ψ \in ℝ^{N_{s}^{2} \times N_{s}^{2}}

is the sparse representation basis of the source pattern, and

\vec{Θ} \in ℝ^{N_{s}^{2} \times 1}

is the coefficient vector. Φ ∈ ℝ^L×M (L < M) is the projection matrix to reduce the dimension of the linear equations. Then, the linearized Bregman algorithm was applied to optimize the source pattern. However, these ILO algorithms have their own limitations. First, the ICC matrix in Eq. (1) must be calculated beforehand, which is more computationally intensive than the optimization itself. That is because each column of

I_{c c}^{s}

indicates the coherent aerial image generated by one source pixel, and the aerial image calculation is a time-consuming step. In addition, the

I_{c c}^{s}

matrix is layout dependent. If the mask pattern is changed, we need to recalculate the

I_{c c}^{s}

matrix. The specific

I_{c c}^{s}

matrix for one mask pattern cannot be reused to others. Thus, the calculation of the ICC matrix becomes the speed bottleneck of the ILO algorithm. Second, the l_p-norm (0 < p < 1) reconstruction algorithm in theory is more robust than the l₁-norm based algorithm when synthesizing sparse signals from even less CS measurements [25–31]. To this end, an l_p-norm reconstruction algorithm is used in this work to further improve the performance of the ILO algorithms.

This paper develops a method to reduce significantly the optimization complexity in ILO. It first exploits the fact that the optimization search can be restricted to a subspace of the CS framework. Secondly, it enhances the optimization framework by using the l_p-norm rather than the traditional l₁-norm used in CS. According to Abbe’s method, the aerial image under partially coherent illumination is formed by the superposition of coherent aerial images generated by all source pixels [32]. The contrast of the aerial image is contributed by the interference between different diffraction orders of the mask pattern. Thus, we first find out the source pixels that induce interference between different diffraction orders. The set of these source pixels is referred to as “source subspace”. Then, the source optimization variables are constrained within the subspace. The source pixels outside the subspace are turned off and not considered during the optimization. In this way, we only need to calculate the columns of ICC matrix supported by the source subspace. The computational complexity of ICC matrix is thus effectively reduced. Afterwards, the ILO problem is formulated as an l_p-norm (0 < p < 1) inverse reconstruction problem. An unconstrained regularized l_p-norm (URLP) algorithm is used to purse the optimal source pattern. Compared to the l₁-norm, the l_p-norm (0 < p < 1) poses more restrict constraints on the sparsity of the underlying signals. Thus, the l_p-norm based algorithm is capable of recovering the optimal source from only a few linear equations. The superiority of the proposed method is verified by the simulations at 45nm and 14nm technology nodes. The proposed method provides a significant improvement to current ILO methods in terms of the speed, robustness and imaging performance.

In the following, the estimation method of source subspace is developed in Section 2. The ILO framework based on subspace CS is formulated in Section 3. The l_p-norm based ILO algorithm is described in Section 4. Simulations and analysis are provided in Section 5. Section 6 concludes the proposed ILO method in the paper.

2. Estimation of the source subspace

This section describes the estimation method of source subspace for the ILO approach based on CS framework. In 2002, Rosenbluth et al. proposed a method to find global solutions for SMO problem, where the continuous space of possible illumination directions was divided into several sub-regions, and each sub-region corresponded to the same set of diffraction orders within the collection pupil [13]. That method treated the intensity in one diffraction pupil region as one optimization variable. Thus, the source optimization method in [13] has lower degree of optimization freedom than the pixelated ILO method in this paper. The ILO method in this paper uses a CS algorithm to implement the source optimization, which is significantly more efficient than the numerical algorithm used in [13]. In addition, this paper focuses on more advanced technology nodes than those considered in [13].

The estimation method of source subspace is described as below. Given the mask pattern M ∈ ℝ^N×N, its spectrum M_f = ℱ {M} is first calculated, where ℱ{•} represents the Fourier transform. Mask layouts can be large but due to their sparse representation, fast sparse FFT algorithms reduce the complexity of the Fourier transform by orders of magnitude [33,34]. The elements in M_f that are smaller than a pre-defined threshold t_M are set to be zero, since the small spectrum components have little influence on the lithography image. The spectrum pattern after thresholding is denoted as $M_{f}^{t}$ . For instance, Fig. 2(a) is the line-space mask pattern, and Fig. 2(b) shows the thresholded spectrum pattern $M_{f}^{t}$ . It is noted that the Fourier transform of a periodic pattern with infinite dimension is a discrete function, where the diffraction orders can be represented by a set of discrete impulses. However, realistic masks may have finite dimensions and aperiodic layout patterns. Thus, the Fourier transform of a realistic mask is a continuous function. The diffraction orders are represented by the narrow-band impulses constrained within small regions. Thus, we use the peaks of impulse functions in $M_{f}^{t}$ to indicate the locations of diffraction orders. Based on this concept, we locate the positions of diffraction orders by finding out the local maximums in $M_{f}^{t}$ . These local maximums are recorded by a binary matrix W ∈ ℝ^N×N. The (m, n)th element of W is given by

W (m, n) = {\begin{array}{l} 1 & if M_{f}^{t} (m, n) > 0 and M_{f}^{t} (m, n) is the local maximum in \\ its a pixels \times a pixels neighbourhood \\ 0 & otherwise \end{array},

where m, n = 1, 2, . . ., N. Therefore, the one-valued elements in W indicate the positions of mask diffraction orders. For example, the blue points in Fig. 2(c) represent the positions of diffraction orders for the line-space mask.

Fig. 2 The examples of (a) the line-space mask pattern, (b) the thresholded mask spectrum, (c) the positions of diffraction orders, (d) the magnified central spectrum region, and (e) the corresponding source subspace.

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The lithography projection optics can be treated as a low-pass filter with the cutoff frequency of NA_m/λ, where NA_m is the numerical aperture on the mask side, and λ is the illumination wavelength. So, the spectrum components beyond the cutoff frequency cannot be collected by the projection optics. The various illumination points in the source, notably, produce different observable phenomena, which is the basis of the proposed subspace CS approach. The oblique incidence from each of the off-axis source pixels will induce different relative displacements between the mask spectrum and the low-pass filter. Thus, the projection optics may collect different diffraction orders when the mask is illuminated by different source pixels. This phenomenon is illustrated in Figs. 2(d) and 2(e). Figure 2(d) magnifies the central spectrum region including the 0th and ±1st diffraction orders. Figure 2(e) illustrates the source plane, where the blue circle represents the full-pupil boundary. Consider the on-axis source pixel and another off-axis source pixel indicated by the green plus sign and red cross in Fig. 2(e), respectively. If the mask is illuminated by the on-axis source pixel, the scope of low-pass filter is depicted by the green solid circle in Fig. 2(d). In this case, only the 0th diffraction order is collected by the projection optics. When we use the off-axis source pixel to illuminate the mask, the scope of low-pass filter will be shifted to the red dashed circle, while both 0th and +1st diffraction orders will be collected. The displacements of low-pass filter along the x and y axes due to the oblique incidence are given by

d_{x} = - \frac{{NA}_{m} x_{s}}{λ}, d_{y} = - \frac{{NA}_{m} y_{s}}{λ},

where x_s and y_s are the normalized coordinates of the source pixel, and the minus sign indicates that the source pixel and its corresponding low-pass filter always move in the opposite directions. The derivation of Eq. (3) is provided in the Appendix B. Based on Eq. (3), the source pixels that allow two or more diffraction orders falling into the scope of the low-pass filter can be identified. The collection of all such source pixels constitute the source subspace. For example, the source subspace corresponding to the line-space mask is represented by the shadow areas in Fig. 2(e).

It is noted that the aforementioned method can rapidly estimate the source subspace based on the thin-mask assumption, where the mask pattern is regarded as a thin film with infinitesimal thickness. Next, we will briefly discuss how to generalize the subspace estimation method to take into account the thick-mask effects. For a thick mask M ∈ ℝ^N×N, we first calculate its diffraction near-field contributed by every source pixel. Let M′(x_s, y_s) be the diffraction near-field contributed by the source pixel at coordinate (x_s, y_s). In order to improve the computational efficiency, some of the approximate thick-mask models could be used to calculate M′(x_s, y_s) [35–37]. Then, the far-field of the thick mask corresponding to source pixel (x_s, y_s) can be calculated as M_f (x_s, y_s) = ℱ {M′(x_s, y_s)}. After that, we can use the method mentioned above to find out all of the source pixels that induce two or more diffraction orders within the cutoff frequency. The collection of all such source pixels constitute the source subspace for the thick mask. It is noted that this paper focuses on the ILO methods based on thin-mask assumption. In the future work, the proposed ILO approach will be generalized to take into account the thick-mask effects in lithography systems.

3. ILO framework based on subspace CS

Let Z be the target layout, which means the desired image on the wafer. Figures 3(a) and 3(b) show the target layouts of vertical line-space pattern and horizontal block pattern, respectively. According to [8], the critical regions of Z include the inner (green) and outer (red) margins of the target layout, as well as a part of non-pattern regions (blue). Let Z_m represent the critical regions. Then, we use the blue noise method in [12,38,39] to select M monitoring pixels from the critical regions. The pixel at (m, n) is selected as a monitoring pixel if αZ_m(m, n) + β ≥ B(m, n), where B is the blue noise pattern, B(m, n) ∈ [0, 1], and α and β are the parameters to determine the number of monitoring pixels. Compared to random noise sampling, blue noise sampling distribute the samples more uniformly, benefiting from extracting the high frequency components of the target layout, such as the edges and corners [12].

Fig. 3 Target layouts and critical regions for (a) vertical line-space pattern, and (b) horizontal block pattern (figure revised from [12]).

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The ILO problem is then formulated as an inverse l_p-norm reconstruction problem:

{\hat{\vec{Θ}}}^{sp} = min_{{\vec{Θ}}^{sp}} {‖ {\vec{Θ}}^{sp} ‖}_{p} subject to Φ {\vec{Z}}_{s} = Φ {\vec{I}}_{s} = Φ I_{c c}^{sp} {\vec{J}}^{sp} = Φ I_{c c}^{sp} Ψ^{sp} {\vec{Θ}}^{sp},

where 0 < p < 1, and ‖ • ‖_p is the l_p-norm of the argument. The definitions of Z⃗_s, I⃗_s and Φ are the same as those in Eq. (1). This paper adopts the adaptive projection matrix proposed in [12]. The adaptive projection matrix is constructed from the a-priori knowledge of the target layout and the nonlinear thresholding operation. Fixing the dimensionality of the optimization problem, the adaptive projection matrix has been proved to achieve superior ILO performance over the random projection matrix by exploiting the structure information of the underlying target layout. Particularly, the (m, n)th element of Φ is designed as

Φ (m, n) = \frac{sgn ({\vec{Z}}_{s, n} - Λ_{m n})}{\sqrt{M}},

where Z⃗_s,n is the n th element of Z⃗_s, Λ_mn is a Gaussian random threshold with zero-mean and variance of

σ_{Λ}^{2}

, and sgn(·) is the sign operator. In Eq. (4),

I_{c c}^{sp} \in ℝ^{M \times N_{sp}}

(

N_{sp} < N_{s}^{2}

) is the ICC matrix supported by the source subspace, where N_sp is the number of source pixels in the subspace. J⃗^sp ∈ ℝ^N_sp×1 represents all source pixels in the subspace, which is referred to as the source subspace vector. From Eq. (4), we have

{\vec{J}}^{sp} = Ψ^{sp} {\vec{Θ}}^{sp},

where Ψ^sp ∈ ℝ^N_sp×N_sp and Θ⃗^sp ∈ ℝ^N_sp×1 are the sparse basis and coefficients of J⃗^sp. The row dimension of Φ is referred to as the measurement number L. In general, L should be larger than or comparable to the sparsity degree of the source pattern. The sparsity degree is defined as the number of elements in Θ⃗^sp that are non-zero or much larger than zero. However, the sparsity degree of the optimized source pattern is unknown before optimization. Thus, we defined the number of L empirically. In the following, we will prove that the proposed method is more robust to the change of L over the method in [12].

Compared to the prior ILO formulation in Eq. (1), the proposed method effectively reduces the dimension of ICC matrix by constraining the optimization variables within the source subspace. As shown in Fig. 4, the matrix $I_{c c}^{sp}$ is a compressed version of $I_{c c}^{s}$ , which only consists of the matrix columns supported by the source subspace. Given that $N_{sp} < N_{s}^{2}$ , the computation time of ICC matrix is effectively shortened, and the speed of ILO algorithm is also improved by reducing the number of optimization variables. In addition, the proposed method solves the ILO problem under the l_p-norm, which is more robust than the l₁-norm framework. The l_p-norm algorithm can stably reconstruct the underlying sparse signals from only a small set of compressive measurements [27,28,31]. In the following, we will show that the proposed l_p-norm algorithm can actually optimize the source pattern using only 5 or even less linear equations in Eq. (4).

Fig. 4 Shrinkage of the ICC matrix according to the source subspace. From left to right, it shows (a) the matrix of $I_{c c}^{sp}$ , (b) the matrix of $I_{c c}^{s}$ , and (c) the source subspace, where the source subspace is represented by the white pixels.

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Another important issue is how to design the sparse basis Ψ^sp in Eq. (4). Some prior work utilizes the spatial basis to represent the source pattern [9,12]. However, the l_p-norm algorithm combined with the CS subspace is likely to obtain a sparse source pattern on the spatial domain. These kind of sources have very low pupil fill percentage, and may induce the aberrations due to the lens heating, or even lens-damage effect [18,40]. In order to solve this problem, we assume that a source point is represented by a two-dimensional (2D) Gaussian function, instead of a single pixel. The source pattern is formed by a set of overlapped 2D Gaussian functions. Thus, the pupil fill percentage is increased, and the singular bright source pixels are eliminated. Based on this assumption, Ψ^sp is defined as a Gaussian basis with dimension of N_sp × N_sp, the columns of which are the raster-scanned vectors of the 2D Gaussian functions with different locations. The method to generate Ψ^sp is described below. Figure 5(a) illustrates the Gaussian basis Ψ^sp ∈ ℝ^N_sp×N_sp. Figure 5(b) illustrates the entire source region, where the areas surrounded by the red dashed lines are the regions of source subspace. The 2D Gaussian function is constrained within the green square in Fig. 5(b). Assume J_s represents the areas of source subspace including the 2D Gaussian function. As mentioned above, J_s consists of N_sp pixels. Then, we raster-scan J_s to generate an N_sp × 1 vector denoted by J⃗_s. The vector J⃗_s is then placed in one column of Ψ^sp, which is shown by the green slot in Fig. 5(a). After that, we move the 2D Gaussian function in Fig. 5(b), and get another different raster-scanned vector J⃗_s, which is then placed in another column of Ψ^sp. When we go over all of the possible positions of the 2D Gaussian function in subspace, the entire sparse basis Ψ^sp will be constructed. Note that the sparse basis Ψ^sp may also be constructed by other smooth functions other than Gaussian function. However, in this paper we restrict our discussion using Gaussian basis.

Fig. 5 The (a) Gaussian basis Ψ^sp to represent source pattern, and the (b) corresponding 2D Gaussian function in the source pattern.

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4. L_p-norm based ILO algorithm

In order to solve the ILO problem in Eq. (4), we adopt the unconstrained regularized l_p-norm (URLP) algorithm in [31]. The URLP algorithm is chosen since it is computational efficient and yields superior signal reconstruction performance. Let $Φ_{a} = Φ I_{c c}^{sp} Ψ^{sp}$ be the composite sensing matrix. The solution set of Eq. (4) can be represented as

{\vec{Θ}}^{sp} = {\vec{Θ}}_{s}^{sp} + V_{r} \vec{ξ},

where

{\vec{Θ}}_{s}^{sp}

is a specific solution, the columns of V_r ∈ ℝ^{N_sp×(N_sp−L)} compose an orthonormal basis of the null space of Φ_a, and ξ⃗ ∈ ℝ^{(N_sp−L)×1} is a parameter vector [41]. The main workflow of the URLP algorithm is summarized in Table 1. The Step 4 in the workflow invokes the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, which is described in Table 2.

Table 1. The main workflow of the URLP algorithm.

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Table 2. The workflow of the BFGS algorithm used in the URLP algorithm.

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According to [31], solving problem in Eq. (7) can be converted into the unconstrained problem as

\hat{\vec{ξ}} = \underset{\vec{ξ}}{arg min} F_{p, ∊} (\vec{ξ}) \underset{\vec{ξ}}{arg min} \sum_{i = 1}^{N_{sp}} {[{(Θ_{s, i}^{sp} + {\vec{V}}_{r, i} \cdot \vec{ξ})}^{2} + ∊^{2}]}^{p / 2},

where 0 < p < 1, V⃗_r,i is the ith row of V_r,

Θ_{s, i}^{sp}

is the ith element of

{\vec{Θ}}_{s}^{sp}

, ∊ is a small positive parameter. In the URLP algorithm, the parameter ∊ plays two important roles. First, the function F_p,∊ (ξ⃗) keeps differentiable as long as ∊ is positive. Second, the convex region of function F_p,∊ (ξ⃗) can be controlled by ∊, i.e., the greater the ∊, the larger the convex region. This conclusion can be proved by solving and analyzing the Jacobian and Hessian matrices of F_p,∊ (ξ⃗) with respect to ξ⃗. Based on the above analysis, the solving process of ∊ is described in the Step 2 of Table 1.

5. Simulation and analysis

A set of simulations of the proposed ILO method at 45nm and 14nm technology nodes based on the target layouts in Fig. 3 are presented next. We also compare the proposed method to the adaptive CS (ACS) method in [12]. Figure 6 illustrates the simulations using the line-space layout with CD=45nm. We use an immersion lithography system with the illumination wavelength of λ =193nm. The NA on the wafer side is NA_w = 1.2, and the demagnification factor is R = 4. That means the NA on the mask side is NA_m = NA_w/R = 0.3. The refractive index of the immersion medium is 1.44. The parameter a in Eq. (2) is set to be 4. The variance of the 2D Gaussian basis functions is set to be 1, and the side length of the Gaussian functions is 5 pixels. The first row in Fig. 6 shows the optimized source patterns, which are normalized to have unit energy. The second row shows the corresponding print images on the wafer. The print images are calculated based on the constant threshold resist model such that Z′ = Γ{I−t_r}, where I ∈ ℝ^N×N is the aerial image, Γ{·} is a hard threshold function, and the threshold value is t_r = 0.25. In all of the following simulations, we select 300 (M = 300) monitoring pixels on the target layout.

Fig. 6 Simulations of different ILO methods based on the vertical line-space pattern.

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The first and second columns in Fig. 6 illustrate the results of the ACS method in [12], where the initial source before optimization is a circular illumination with partial coherence factor of σ = 1. In the first column, the dimension of projection matrix Φ in Eq. (4) is 25 × 300. That means the dimensionality of linear equations is compressed from 300 to 25 (L = 25). In the second column, the dimension of Φ is 5 × 300, so the dimensionality of linear equations is further compressed down to 5 (L = 5). The third and fourth columns illustrate the results of the proposed ILO method with L = 25 and L = 5, respectively. The initial source fills up the entire subspace region with unit energy. Figure 6 also provides the values of pattern errors (PE), edge placement errors (EPE), normalized image log slopes (NILS), and contrasts for different methods. The pattern error is defined as $PE = {‖ Z - Z^{'} ‖}_{2}^{2}$ , where Z and Z′ represent the target layout and print image, respectively. The EPE indicates the average deviation of the actual printed edges from the target layout including the corners. The contrast is calculated along the red dashed lines on the aerial images. The NILS measures the slope of aerial image normalized by CD along the contour of target layout. It is observed that the proposed method achieves superior imaging performance over the ACS method for the line-space layout in Fig. 6. Additionally, for the ACS method the decrease of L will deteriorate the imaging performance due to the ill-posedness of the linear equations in Eq. (4). On the other hand, the proposed method is robust to the reduction of L. As shown in Fig. 6(h), the proposed method leads to good imaging performance using only 5 linear equations. Based on the CS theory, the l_p-norm (0 < p < 1) is more close to the l₀-norm, and poses restrict constraint on the sparsity of the underlying signals. Thus, only a small set of compressive measurements are sufficient to recover the original signals under the l_p-norm [27, 31, 42]. In the following, we will briefly explain this idea. It is known that the l_p-norm of a signal X⃗ = [x₁, x₂, . . ., x_N]^T is defined as

{‖ \vec{X} ‖}_{p} = {(\sum_{n = 1}^{N} {| x_{n} |}^{p})}^{1 / p},

where 0 < p < 1. For simplicity, let’s only consider one element in X⃗. Figure 7 shows the curves for the exponential function of y = |x_n|^p for p =0, 0.3, 0.7 and 1, respectively. It is shown that the function is more and more close to the step function as p approaches 0. If ‖X⃗‖_p is used as the cost function, larger p will pose heavier penalty on the large magnitudes of |x_n|, while the smaller p is apt to constrain the number of non-zero elements in X⃗. Thus, the behavior of l₀-norm is more like the l_p-norm (0<p <1) than the l₁-norm. Compared to the l₁-norm, the l_p-norm (0<p <1) will pose more restrict on the sparsity of the underlying signal.

Fig. 7 The curves of exponential functions of y = |x_n|^p for p =0, 0.3, 0.7 and 1, respectively.

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Figures 8(a) and 8(b) are the convergence curves of the PE and contrast for different ILO methods. It is noticed that the initial PE and contrast of the proposed method are much better than the ACS method. That is because the source subspace provides a good starting point for the ILO method, which is close to the optimal solution. In addition, we observe a few kinks in the convergence curves of the ACS method due to the oscillation of l₁-norm algorithm. On the other hand, the proposed method based on l_p-norm algorithm leads to much smoother convergence curves. Benefiting from the good initial point and the promising convergence properties, the proposed method with no more than 30 iterations will achieve superior imaging performance over the ACS method with 60 iterations.

Fig. 8 The convergence of the PE and contrast for different ILO methods, where (a) and (b) are the convergence curves for the vertical line-space pattern, while (c) and (d) are the convergence curves for the horizontal block pattern.

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Figure 9(a) compares the overlapped process windows (PW) obtained by different ILO methods, where the x and y axes represent the depth of focus (DOF), and exposure latitude (EL), respectively. As shown in Fig. 6, we first calculate the PWs at the intersections between the red dashed lines and the first and third bars, then their overlapped PWs are illustrated in Fig. 9(a). It is observed that decreasing L will reduce the PW of ACS method. The reason is the same as the deterioration of pattern error. On the other hand, the proposed method leads to comparable PWs with different L.

Fig. 9 The overlapped PWs obtained by different ILO methods for the (a) vertical line-space pattern and (b) horizontal block pattern.

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The 2nd to 4th columns of Table 3 present the average runtimes of the above simulations. All programming are implemented by Matlab and the computations are carried out on a server with an Intel(R) Xeon(R) E7-4830 CPU(4 cores), 2.13GHz, 128GB of RAM. We carry out each of the simulations for 100 times to calculate the average runtimes. The overall runtime includes the computation times of subspace, ICC matrix, and optimization procedure. It is noted that the ACS method doesn’t need to estimate the source subspace. We found that the proposed method can significantly speed up the ICC matrix calculation and the optimization procedure. The improvement of speed is mainly attributed to the reduction of the number of optimization variables by using source subspace. The extra runtime induced by the subspace estimation can be ignored compared to the overall runtime reduction.

Table 3. The average runtimes of different ILO methods. The computation times of the subspace, ICC matrix and optimization procedure are abbreviated by “Sub.”, “ICC” and “Opt.”, respectively.

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Next, we study the impacts of compression ratio on the imaging performance and runtimes. The compression ratio is defined as R_c = M/L, where L and M are the numbers of rows and columns of the projection matrix Φ. In the following simulations, we fix M = 300 and increase L from 2 to 25. Table 4 presents the average PEs, EPEs, NILSs, contrasts and runtimes corresponding to different compression ratios. We observe that the imaging performance of ACS method will be generally degraded by increasing R_c, since less linear equations are used to recover the source pattern. At the same time, the speed of ACS method is slightly improved, because the dimensionality of ILO problem is reduced by increasing R_c. On the other hand, the proposed method achieves better imaging performance than the ACS method regardless of the value of R_c. In addition, the proposed method results in comparable imaging performance for different compression ratios. That means the proposed method is more robust to the compression ratio in contrast to the ACS method. It is also interesting to find that the smaller R_c may even leads to shorter runtime for the proposed method. The reason is that decreasing R_c will accelerate the convergence of BFGS algorithm in Table 2, thus benefits to reduce the overall runtime.

Table 4. The average PEs, EPEs, NILSs, contrasts and runtimes corresponding to different compression ratios, where the vertical line-space pattern is used.

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In the simulations mentioned above, the parameters of the 2D Gaussian basis function are selected empirically. In order to verify the robustness of the proposed ILO method to these parameters, we repeat the simulations in Fig. 6 using different parameters of Gaussian basis functions. In particular, the variance of the 2D Gaussian basis functions is set to be 2, and the side length of the Gaussian functions is 7 pixels. Figure 10 illustrates the optimized sources and print images based on the proposed algorithm for L = 25 and L = 5. Comparing Fig. 10 to Fig. 6, it is observed that the increment of variance and side length of Gaussian basis functions will lead to more uniform energy distributions on source patterns. In addition, the two pairs of parameters for Gaussian basis functions result in comparable imaging performance.

Fig. 10 Simulations of the proposed ILO method using different parameters of Gaussian basis.

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In order to prove the validity of the proposed method on different layout patterns, Fig. 11 illustrates the simulations based on another horizontal block pattern with CD=45nm. Figures 8(c) and 8(d) provide the convergences of PE and contrast, respectively. The contrast is calculated along the red dashed lines on the aerial images in Fig. 11. Figure 9(b) compares the overlapped PWs obtained by different ILO methods. In particular, we first calculate the PWs at the intersections between the red dashed lines and the first and third blocks. Then, the overlap of these two PWs is presented in Fig. 9(b). The 5th to 7th columns of Table 3 provide the average runtimes of different ILO methods by implementing the simulations for 100 times. The simulations also prove the superiority of the proposed method over the ACS method in both of imaging performance and computational efficiency. In the future, we will study the proposed method based on other representative layout patterns to demonstrate the imaging performance improvement over the ACS method for general cases.

Fig. 11 Simulations of different ILO methods based on the horizontal block layout pattern.

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Figure 12 illustrates another simulation to co-optimize the imaging fidelity of the vertical line-space layout and horizontal block layout. From top to bottom, Fig. 12 shows the optimized source patterns, the print images of vertical line-space layout, and the print images of horizontal block layout. In the proposed method, the source subspace is defined by the union of the two subspaces estimated from the two layouts. It is shown that the optimized sources are quadrapole-like illuminations, which benefit in improving the imaging performance of the layout features oriented in orthogonal directions. According to the simulations in Fig. 12, the proposed method is proved to achieve superior imaging performance over the ACS method. In the future, we will study the subspace CS approaches to co-optimize multiple mask clips together.

Fig. 12 Simulations of different ILO methods to co-optimize two layout patterns.

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At the end of this section, we present the simulations of ILO and SMO methods based on line-space layout pattern at 14nm technology node. According to the International Technology Roadmap for Semiconductors (ITRS), the half-pitch at 14nm technology node is 28nm [43]. If we use the triple patterning technique to fabricate the dense line-space layout, the single mask for one patterning process should have the duty ratio of 1:5. The parameter a in Eq. (2) is set to be 3. The first row of Fig. 13 shows the simulations of the proposed ILO method. From left to right, it shows the optimized source, mask, and the corresponding print image, where the mask is the same as the target layout. The second row illustrates the results of SMO method, which jointly optimizes the source and mask. In the SMO simulation, we first optimize the source using the proposed ILO method for 20 iterations, then optimize the mask using the optical proximity correction (OPC) method in [44] for 500 iterations. After that, we optimize the source again for 20 iterations. It is obvious that the SMO method achieves much lower PE and EPE than the ILO method due to the higher degrees of optimization freedom. However, the NILS and contrast of the SMO method are slightly inferior than the ILO method. That is because the OPC method in [44] falls short to take into account the NILS and contrast in its objective function. In our future work, we will study on the SMO method to further enhance the NILS and contrast. Figure 13 provides simple examples to prove that the proposed ILO method is potential to be applied for the individual source optimization and SMO algorithms at 14nm technology node. In our future work, we will study the optimal SMO workflow to co-optimize the source and mask, and consider some of the realistic constraints on the physically manufacturable masks. In addition, we will make comprehensive comparison between the ILO and SMO algorithms at 14nm technology node.

Fig. 13 Simulations of the ILO method and SMO method using the line-space pattern at 14nm technology node.

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6. Conclusion

A novel efficient and robust ILO method is proposed based on the subspace CS and l_p-norm (0 < p < 1) reconstruction algorithm. We first identify the source subspace consisting of all source pixels that induce the interference between different diffraction orders. The illumination optimization is then formulated as an l_p-norm inverse problem, where the optimization variables are constrained within the source subspace. Subsequently, an unconstrained regularized l_p-norm algorithm is used to solve for the ILO problem. The subspace CS mechanism effectively reduces the dimensionality of the ILO problem, thus significantly improving the speed of the ILO algorithm. In addition, the l_p-norm based ILO algorithm is more robust than the l₁-norm based ILO algorithm. The simulations also prove that the proposed ILO method can achieve superior imaging performance over the ACS method.

A. Appendix: derivation of Eq. (1)

According to Abbe’s method based on the thin-mask assumption, the overall aerial image of the lithography system can be calculated as

I = \frac{1}{J_{sum}} \sum_{x_{s}} \sum_{y_{s}} (J (x_{s}, y_{s}) \sum_{p = x, y, z} {| H_{p}^{x_{s} y_{s}} \otimes (B^{x_{s} y_{s}} ⊙ M) |}^{2}),

where

H_{p}^{x_{s} y_{s}} \in ℝ^{N \times N}

is referred to as the equivalent filter along the p-axis (p = x, y, z), B^{x_sy_s ∈ ℝN×N} represents the oblique incidence effect of the source point (x_s, y_s), J ∈ ℝ^N×N is the source pattern, and

J_{sum} = \sum_{x_{s}} \sum_{y_{s}} J (x_{s}, y_{s})

is a normalization factor. Assume

G^{x_{s} y_{s}} = \frac{1}{J_{sum}} \sum_{p = x, y, z} {| H_{p}^{x_{s} y_{s}} \otimes (B^{x_{s} y_{s}} ⊙ M) |}^{2}

, Eq. (10) is transformed into

\vec{I} = I_{c c} \vec{J},

where

I_{c c} \in ℝ^{N^{2} \times N_{s}^{2}}

is the ICC matrix. The ith column of I_cc is the raster-scanned vector of G^x_sy_s corresponding to the source point J⃗_i. I⃗ ∈ ℝ^N²×1 and

\vec{J} \in ℝ^{N_{s}^{2} \times 1}

are the raster-scanned aerial image and source pattern, respectively. It is noted that each column of I_cc represents the coherent aerial image generated by one source point, while each row of I_cc represents the aerial images on a wafer pixel contributed by different source points.

When $N_{s}^{2} < N^{2}$ , Eq. (11) is an over-determined problem where the number of equations is more than the number of variables. In order to reduce the dimensionality of Eq. (11), we select M ( $M ≪ N_{s}^{2}$ ) monitoring pixels in the critical regions of the layout pattern. Subsequently, we choose the M rows of I_cc corresponding to the monitoring pixels to generate a compressed ICC matrix $I_{c c}^{s} \in ℝ^{M \times N_{s}^{2}}$ . Suppose that the aerial image is equal to the target layout on the monitoring pixels, Eq. (11) is reduced to an underdetermined problem as following

{\vec{Z}}_{s} = {\vec{I}}_{s} = I_{c c}^{s} \vec{J},

where Z⃗_s, I⃗_s ∈ ℝ^M×1 are the vectors of target layout and aerial image on the monitoring pixels, respectively. Suppose the source pattern can be sparsely represented on the basis

Ψ \in ℝ^{N_{s}^{2} \times N_{s}^{2}}

, that is J⃗ = ΨΘ⃗ and Θ⃗ represents the sparse coefficients of the source pattern. Then, according to the CS theory, the ILO can be formulated as an l₁-norm reconstruction problem [12]:

\hat{\vec{Θ}} = min_{\vec{Θ}} {‖ \vec{Θ} ‖}_{1} subject to Φ {\vec{Z}}_{s} = Φ {\vec{I}}_{s} = Φ I_{c c}^{s} \vec{J} = Φ I_{c c}^{s} Ψ \vec{Θ},

where Φ ∈ ℝ^L×M (L < M) is the projection matrix to reduce the dimension of the linear equations.

B. Appendix: derivation of Eq. (3)

Let (x_s, y_s) be the normalized coordinates on the source plane, where x_s, y_s ∈ [−1/2, 1/2]. The diffraction near-field of the mask generated by the source pixel at (x_s, y_s) can be formulated as [44]

\tilde{M} = B ⊙ M,

where M ∈ ℝ^N×N is the transmittance function of mask, and B ∈ ℝ^N×N is the matrix representing the oblique incidence effect. According to the constant scattering coefficient assumption, each element of B can be calculated as [45]:

B (x_{m}, y_{m}) = exp (\frac{j 2 π α_{s} x_{m}}{λ}) exp (\frac{j 2 π β_{s} y_{m}}{λ}),

where α_s = NA_mx_s, β_s = NA_my_s, and (x_m, y_m) are the coordinates on the mask pattern. It is noted that the constant scattering coefficient assumption is the same as the linear shift-invariant condition of the diffraction orders. Taking the Fourier transfer on both sides of Eq. (14), we have

{\tilde{M}}_{f} (u, v) = M_{f} (u - \frac{α_{s}}{λ}, v - \frac{β_{s}}{λ}) = M_{f} (u - \frac{{NA}_{m} x_{s}}{λ}, v - \frac{{NA}_{m} y_{s}}{λ}),

where (u, v) are the coordinates in the frequency domain, and M̃_f and M_f are the Fourier transforms of M̃ and M, respectively. Thus, the mask spectrum is shifted by NA_mx_s/λ and NA_my_s/λ in the x and y directions, respectively. If we fix the mask spectrum, it is equal to move the low-pass filter by d_x and d_y along the x and y axes, where

d_{x} = - \frac{{NA}_{m} x_{s}}{λ}, d_{y} = - \frac{{NA}_{m} y_{s}}{λ} .

Funding

National Natural Science Foundation of China (Grant No. 61675021 and Grant No. 61675026); Beijing Natural Science Foundation (4173078); and National Science and Technology Major Project.

Acknowledgments

We thank Mentor Graphics Corporation for providing academic use of Calibre.

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Step 1	Initialize p (0 < p < 1) and the maximum iteration number N. Set the current iteration number as n = 1. We also initialize ξ⃗⁰ as a zero-valued vector. In addition, calculate the specific solution in Eq. (7) as ${\vec{Θ}}_{s}^{sp} = Ψ^{s p^{- 1}} {\vec{J}}_{ini}^{sp}$ , where ${\vec{J}}_{ini}^{sp}$ is the initial source subspace vector.
Step 2	Generate a parameter sequence ∊⃗ = [∊₁, ∊₂, . . ., ∊_N]^T. The initial value ∊₁ should satisfy $∊_{1} \geq \sqrt{1 - p} \cdot max {\| {\vec{Θ}}_{si}^{sp} \|}$ , where $max {\| {\vec{Θ}}_{si}^{sp} \|}$ represents the maximum absolute value of the elements in ${\vec{Θ}}_{s}^{sp}$ . Also, the target value ∊_N should be given. The other (N − 2) values of ∊⃗ are calculated as ∊_n = e^−βn (n = 2, 3, . . ., N − 1), where β = log (∊₁/∊_N) /(N − 1).
Step 3	Calculate V_r by applying the QR decomposition on $Φ_{a}^{T}$ , such that $Φ_{a}^{T} = Q {[R^{T} 0^{T}]}^{T}$ . Then, V_r is obtained by removing the leftmost L columns of the matrix Q.
Step 4	In the nth iteration, input ∊_n and update ξ⃗ⁿ by using the BFGS algorithm, which will be described in Table 2. Then, update n = n + 1.
Step 5	If n < N, then return Step 4. Otherwise, denote the current ξ⃗^N as $\hat{\vec{ξ}}$ , then goto Step 6.
Step 6	Calculate the optimal coefficient vector as ${\hat{\vec{Θ}}}^{sp} = {\vec{Θ}}_{s}^{sp} + V_{r} \hat{\vec{ξ}}$ . According to Eq. (6), the optimal source subspace vector is calculated as ${\hat{\vec{J}}}^{sp} = Ψ^{sp} {\hat{\vec{Θ}}}^{sp}$ , and the optimal source pattern can be acquired from ${\hat{\vec{J}}}^{sp}$ .

Step 1	For the n th iteration, define a dummy vector Θ⃗′ ∈ ℝ^N_sp×1. Set the termination condition parameter δ_t. In our simulations, we set δ_t = 1. Set α₁ = 0, l = 1, and δ_a = δ_t + 1.
Step 2	Iteratively update the step length α as following: Calculate the ith element of Θ⃗′ as ${\vec{Θ}}_{i}^{'} = {\vec{Θ}}_{si}^{sp} + V_{r i}^{T} {\vec{ξ}}^{n - 1}$ , where ${\vec{Θ}}_{si}^{sp}$ is the ith element of ${\vec{Θ}}_{s}^{sp}$ , and $V_{r i}^{T}$ is the ith row of V_r. The ξ⃗ⁿ⁻¹ is the vector in the last iteration. Then, we calculate a parameter ν_i as $ν_{i} = V_{r i}^{T} \cdot {\vec{d}}_{r}$ . In our simulations, d⃗_r is set to be a one-valued vector. Then, we calculate $γ_{i} = {({\vec{Θ}}_{i}^{'} + α_{l} ν_{i})}^{2} + ∊_{n}^{2}, and α_{l + 1} = - \frac{\sum_{i = 1}^{N_{sp}} {\vec{Θ}}_{i}^{'} \cdot ν_{i} \cdot γ_{i}^{p / 2 - 1}}{\sum_{i = 1}^{N_{sp}} ν_{i}^{2} \cdot γ_{i}^{p / 2 - 1}} .$ Then, we update δ_a = α_l+1 − α_l, and l = l + 1. Repeat Step 2 until δ_a ≤ δ_t. After that, we get the optimal step length α̂.
Step 3	Update ξ⃗ⁿ = ξ⃗ⁿ⁻¹ + α̂d⃗_r and output it, then terminate the BFGS algorithm.

Runtime(s) of	Vertical Line-Space			Horizontal Block
Runtime(s) of	Sub.	ICC	Opt.	Sub.	ICC	Opt.
ACS(L=25)	–	201.636	0.090	–	357.711	0.089
ACS(L=5)	–	201.636	0.086	–	357.711	0.086
Proposed(L=25)	0.567	17.806	0.026	0.569	19.272	0.026
Proposed(L=5)	0.567	17.806	0.033	0.569	19.272	0.033

ACS method
L	2	3	5	10	25
R_c	150:1	100:1	60:1	30:1	12:1
PE	2241	2027	1643	1449	1339
EPE(nm)	12.451	11.261	9.129	8.052	7.376
NILS	1.137	1.285	1.655	1.971	2.217
Contrast	0.461	0.531	0.693	0.809	0.871
Runtime(s)	0.083	0.087	0.086	0.088	0.090

Proposed method
L	2	3	5	10	25
R_c	150:1	100:1	60:1	30:1	12:1
PE	1242	1264	1291	1299	1298
EPE(nm)	6.900	7.024	7.171	7.216	7.208
NILS	2.423	2.393	2.340	2.303	2.295
Contrast	0.926	0.920	0.906	0.897	0.894
Runtime(s)	0.034	0.033	0.033	0.032	0.026

Optimization of lithography source illumination arrays using diffraction subspaces

Abstract

1. Introduction

2. Estimation of the source subspace

3. ILO framework based on subspace CS

4. L_p-norm based ILO algorithm

5. Simulation and analysis

6. Conclusion

A. Appendix: derivation of Eq. (1)

B. Appendix: derivation of Eq. (3)

Funding

Acknowledgments

References and links

Cited By

Figures (13)

Tables (4)

Equations (18)

Optics Express

Abstract

1. Introduction

2. Estimation of the source subspace

3. ILO framework based on subspace CS

4. Lp-norm based ILO algorithm

5. Simulation and analysis

6. Conclusion

A. Appendix: derivation of Eq. (1)

B. Appendix: derivation of Eq. (3)

Funding

Acknowledgments

References and links

Cited By

Figures (13)

Tables (4)

Equations (18)

Optics Express

4. L_p-norm based ILO algorithm