Learning-based compressive sensing method for EUV lithographic source optimization

Jiaxin Lin; Lisong Dong; Taian Fan; Xu Ma; Yayi Wei; Tianchun Ye

doi:10.1364/OE.27.022563

1. Introduction

Optical lithography is a key process of integrated circuit (IC) manufacturing. Using light as writing medium, optical lithography systems transfer the IC layout patterns from the mask onto the wafer [1]. According to Rayleigh’s criterion, the lithographic resolution indicated by the minimum resolvable critical dimension (CD), is proportional to the illumination wavelength [2]. Thus, the illumination wavelength of lithography systems was reduced continuously in the past decades from 436nm to 193nm, to keep pace with the scaling of CD on the IC layouts. Recently, extreme ultraviolet (EUV) lithography with 13.5nm wavelength has been developed as the most promising technology to fabricate the semiconductor devices at 7nm technology node and beyond [3,4]. Compared to the deep ultraviolet (DUV) lithography with 193nm wavelength, the reduction of wavelength in EUV lithography brings in significant improvement of imaging resolution, and thus enables the patterning in 14nm/10nm logic nodes by single exposure [5]. To date, several leading IC manufacturers have announced to start applying EUV lithography in high volume manufacturing processes [6]. The use of EUV lithography in advanced lithography technology nodes has become a dominant trend.

EUV lithography systems use fully reflective optics because the EUV photons can be heavily absorbed by nearly all materials. A typical reflective EUV mask is constituted by 40~50 multilayers of Mo-Si stack, a capping layer, and a Ta-based absorber layer that depicts the layout pattern. As shown in Fig. 1, the EUV light rays are emitted from the laser produced plasma (LPP) source, directed by the illuminator, which then obliquely illuminate the mask with a 6° incident chief ray angle. The mask regions without the coverage of absorbers reflect the light rays, which are then transferred by the projection optics to replicate the mask pattern on the wafer. However, EUV lithography has its own adverse effects on lithography imaging performance, such as shadowing effect, non-telecentric effect, and so on [7–10]. To this end, a set of resolution enhancement techniques (RETs) have been applied to compensate these effects and further improve the imaging fidelity of EUV lithography systems.

Fig. 1 The sketch of EUV lithography system.

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Source optimization (SO) is an extensively used RET method to compensate the lithography imaging distortion by optimizing the distribution of source intensity [11–13]. In addition, SO method can be further combined with mask optimization to develop the source mask optimization (SMO) approaches [14,15]. The development of freeform illuminations in EUV lithography systems provides more flexibility for the EUV-SO methods in practice [16,17]. State-of-the-art SO methods can modify the intensity of every source pixel, thus increasing the degrees of optimization freedom to resolve more critical layout features [18]. For instance, a new illuminator module featuring flexible pixelated pupil is applied in ASML NXE:3400B EUV scanner to support freeform illumination settings [4]. Pixel-based SO/SMO methods have been realized in EUV lithography using gradient-based algorithm and particle swarm optimization (PSO) algorithm [19,20]. However, the high dimensional source variables to be optimized pose a big challenge on the computational efficiency of these methods.

Recently, compressive sensing (CS) theory was introduced to develop a set of efficient pixel-based SO methods for DUV lithography systems [21,22]. These works formulated SO as l_p-norm ( $0 < p \leq 1$ ) reconstruction problems with linear constraints. Based on the sparsity regularization, optimal source pattern can be reconstructed from a small set of sampling data on the layout, such that the dimensionality of the SO problem is reduced and the computational efficiency is significantly improved. However, these CS-SO methods were originally developed for DUV lithography systems, and fell short to consider the imaging characteristics in EUV lithography systems. In addition, the previous CS-SO methods relied on pre-defined sparse bases, such as two-dimensional discrete cosine transform (2D-DCT) basis, to represent the source patterns, and used adaptive random projection matrix to compress the dimensionality of SO problem. However, these strategies were suboptimal, since the structure of the sensing matrix in CS framework was not exploited to optimize the reconstruction performance. It has been proven that the accuracy of CS reconstruction heavily relies on two conditions, referred to as sparsity condition and incoherence condition [23]. For the sparsity condition, the optimized basis function termed as dictionary should be applied to sparsely represent the original signal, thus reducing the number of significant representation coefficients [24–26]. On the other hand, the incoherence between the rows of projection matrix and the columns of dictionary is required to attain the successful reconstruction [27–29]. Some prior works have shown the improvement of CS reconstruction performance by jointly optimizing the dictionary and projection matrix according to these conditions [29,30].

Inspired by the principles mentioned above, this paper develops a fast SO method in EUV lithography using the learning-based CS framework, which is referred to as LCS-SO method for short. The contributions of this paper include three aspects. First, the CS methods are introduced to solve for the pixel-based SO problem in EUV lithography systems. The proposed LCS-SO method is derived from a rigorous imaging model that takes into account the characteristics of EUV lithography systems, such as mask shadowing effect, non-telecentric effect, and so on. Second, a circular-shaped basis function is designed to sparsely represent the source patterns, which is coined the circular exit pupil in lithography systems. Third, a learning-based method is developed to jointly optimize the source dictionary and projection matrix, thus further improving the source reconstruction accuracy and lithography imaging performance. This method is based on the concept that optimal source patterns corresponding to similar types of layouts should contain some common structure characteristics. We can optimize the dictionary based on some training source patterns. The optimized source dictionary is expected to effectively represent the common structure characteristics of the training source patterns corresponding to typical layout features. During the SO stage, the structure characteristics represented by the source dictionary will be extracted to form the optimal source patterns for the testing layouts. In particular, the current SO algorithm is applied to optimize the source patterns for some typical layout features. Then, these SO results are used as the training samples to learn the source dictionary. Based on the alternating proximal method in [26], the dictionary is iteratively updated to improve the sparsity degree of the source representation coefficients. Concatenated with the dictionary learning, the projection matrix is jointly optimized using a gradient-based algorithm to retain the incoherence condition. After that, the optimized dictionary and projection matrix are used to establish the CS reconstruction model for EUV-SO problem.

In this paper, the linearized Bregman algorithm is used to reconstruct the optimal source patterns under the CS frameworks [31]. Simulations show that the proposed method outperforms the state-of-the-art CS-SO method in terms of imaging fidelity and convergence rate. In contrast to the SO method based on PSO algorithm, the proposed method improves the computational efficiency by four orders of magnitude in each iteration. It is worth noting that the source dictionary and projection matrix need to be trained only once, and then can be repetitively used in the following simulations. Thus, except for the computational overhead of training process, the computational complexity of the proposed method is comparable to the state-of-the-art CS-SO method.

The remainder of this paper is organized as follows. The imaging model of EUV lithography systems and the CS-SO framework are described in Section 2. The joint optimization method of source dictionary and projection matrix is described in Section 3. Simulations and analysis are presented in Section 4. Finally, conclusions are presented in Section 5.

2. CS-SO framework for EUV lithography

This section establishes the CS-SO framework based on the EUV lithography imaging model. We first summarize the Abbe’s imaging model, followed by a brief discussion on the characteristics in EUV lithography systems. Then, the CS theory is exploited to formulate the EUV-SO problem as an l₁-norm inverse reconstruction problem.

2.1. Imaging model of EUV lithography systems

Let matrix $J \in ℝ^{N_{S} \times N_{S}}$ denote the lithography source pattern, and $J (x_{S}, y_{S})$ represent the light intensity of the source pixel at coordinate $(x_{S}, y_{S})$ . $M \in ℝ^{N \times N}$ denotes the mask pattern, each entry of which is equal to 1 or 0, representing the reflective region and absorber region, respectively. According to Abbe’s imaging theory, the aerial image $I \in ℝ^{N \times N}$ formed on the wafer plane can be formulated as the weighted summation of the coherent aerial images contributed by all source pixels [19,32,33]:

I = \frac{1}{J_{s u m}} \sum_{x_{s}} \sum_{y_{s}} [J (x_{s}, y_{s}) \cdot I^{x_{s} y_{s}} (M)],

where

I^{x_{s} y_{s}} (M) \in ℝ^{N \times N}

is the mask-dependent coherent aerial image contributed by the source pixel

(x_{S}, y_{S})

, and

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

is a normalization factor.

Due to the oblique incidence and three-dimensional mask topography, the imaging performance of EUV lithography is severely influenced by a set of characteristic effects, such as shadowing effect, non-telecentric effect, and so on [7–10]. It is difficult to establish an analytic model to fully characterize these effects mentioned above, instead the rigorous imaging model must be used. Fortunately, Eq. (1) shows that the source variables $J (x_{S}, y_{S})$ is separated from the coherent aerial image $I^{x_{s} y_{s}} (M)$ , which includes all of the aforementioned effects. Thus, we can use simulation software to rigorously compute $I^{x_{s} y_{s}} (M)$ corresponding to all source pixels, and then optimize the source pattern based on the rigorous imaging model.

2.2. Formulation of CS-SO problem

Let $\vec{I} \in ℝ^{N^{2} \times 1}$ and $\vec{J} \in ℝ^{N_{S}^{2} \times 1}$ be the vectorized representations of the aerial image $I$ and source pattern $J$ , respectively. $N$ and $N_{S}$ are the lateral dimensions of aerial image and source pattern, respectively. Then Eq. (1) can be rewritten into the following compact form:

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

where

I_{CC} \in ℝ^{N^{2} \times N_{S}^{2}}

is referred to as the illumination cross coefficient (ICC) matrix, indicating the intensity mapping from the source pattern to the aerial image. Figure 2 illustrates the transformation of the imaging model from Eq. (1) to Eq. (2). As shown in Fig. 2,

I_{CC} (i, j)

indicates the intensity of the

i

th aerial image pixel contributed by the

j

th source pixel [21]. It is shown that the ICC matrix is transformed from

I^{x_{s} y_{s}} (M)

in Eq. (1), which takes into account some specific imaging characteristics of EUV lithography systems.

Fig. 2 Transformation of the imaging model.

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It is noted that the row dimension and column dimension of ICC matrix are equal to $N^{2}$ and $N_{S}^{2}$ , respectively. Typically, the number of image pixels is much larger than the number of source pixels, thus Eq. (2) is an overdetermined problem that cannot be solved efficiently. In order to reduce the computational complexity of the SO method, this paper adopts the method in [22] to reduce the dimensionality of the imaging model. Particularly, $M (M < N_{S}^{2})$ monitoring points are selected in the critical regions of the layout pattern using the blue noise sampling method [34–36]. Monitoring points selected by the blue noise sampling method can capture the pivotal features of the layout patterns, and thus to control the imaging fidelity of the entire layout while reducing the problem dimensionality. Let ${\vec{I}}_{S} \in ℝ^{M \times 1}$ denote the intensity of aerial image on the monitoring points. $I_{CC}^{S} \in ℝ^{M \times N_{S}^{2}}$ is a submatrix of $I_{CC}$ byextracting the $M$ rows corresponding to the monitoring points. Thus, Eq. (2) can be reduced to an underdetermined equation ${\vec{Z}}_{S} = {\vec{I}}_{S} = I_{CC}^{S} \vec{J},$ where ${\vec{Z}}_{S} \in ℝ^{M \times 1}$ represents the values of target pattern on the monitoring points. Furthermore, a projection matrix $Φ \in ℝ^{L \times M} (L < M)$ is multiplied on both sides of the imaging model to further compress the dimensionality:

Φ {\vec{Z}}_{S} = Φ {\vec{I}}_{S} = Φ I_{CC}^{S} \vec{J} .

In Eq. (3), the dimension of the problem is compressed from

N^{2}

to

L

.

Suppose the source vector $\vec{J}$ can be sparsely represented in a certain dictionary $Ψ \in ℝ^{N_{S}^{2} \times N_{A}}$ ( $N_{A} \geq N_{S}^{2}$ ), each column of which is termed as an atom. Then, the source vector $\vec{J}$ can be formulated as a linear combination of a few atoms in the dictionary: $\vec{J} = Ψ \vec{Θ}$ . $\vec{Θ} \in ℝ^{N_{A} \times 1}$ is referred to as the sparse coefficient vector, which includes only a small set of significant elements. Taking into account the sparsity prior and the linear constraint in Eq. (3), SO problem can be formulated as an l₁-norm reconstruction problem [22]:

\hat{\vec{Θ}} = \underset{\vec{Θ}}{a r g m i n} {‖ \vec{Θ} ‖}_{1} s . t . Φ {\vec{Z}}_{S} = Φ {\vec{I}}_{S} = Φ I_{CC}^{S} \vec{J} = Φ I_{CC}^{S} Ψ \vec{Θ},

where

D = Φ I_{CC}^{S} Ψ

is referred to as the sensing matrix, and

{‖ \vec{Θ} ‖}_{1}

is the sparsity prior used to reduce the solution space and guarantee the successful reconstruction of source pattern. On the other hand, the linear constraint enforces the actual aerial image close to the target pattern, and thus improves the imaging fidelity. In this paper, the linearized Bregman algorithm is applied to solve for the problem in Eq. (4) due to its computational efficiency.

3. Joint optimization of source dictionary and projection matrix

According to CS theory, the accuracy of signal reconstruction relies on the sparsity and incoherence conditions. In this section, a learning-based method is developed to jointly optimize the source dictionary and projection matrix, so as to improve the condition of sensing matrix in CS framework. It will prove that the joint optimization of dictionary and projection matrix (JODP) can effectively improve the performance of CS-SO approach. The proposed JODP algorithm includes two steps, as shown in Fig. 3. First, the source dictionary is pre-learned using the alternating proximal method. This step is computationally efficient and can effectively improve the sparsity degree of source representation. Subsequently, the dictionary and projection matrix are jointly optimized to improve both sparsity and incoherence conditions. After several iteration cycles, a pair of jointly optimal dictionary and projection matrix will be obtained. It is noted that the JODP process needs to be conducted only once. More details of the algorithm will be described in the following.

Fig. 3 Flowchart of the proposed JODP algorithm: (a) the main procedure, and (b) the procedure of “Step 2”.

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3.1. Pre-learning of source dictionary

According to Eq. (4), the source pattern is assumed to be sparse in the dictionary $Ψ$ , i.e., $\vec{J} = Ψ \vec{Θ}$ . Source dictionary can be set as a pre-defined basis matrix such as 2D-DCT basis, or learned from a set of training samples [25]. Compared to the fixed dictionaries, the dictionary learning methods can further improve the sparsity condition by constructing the redundant dictionary atoms adaptive to the structure characteristics of source patterns [24].

First, we apply the existing SO algorithm to calculate the optimized source patterns for some typical mask layouts, and then use them as the training source patterns. Let $J T = [{\vec{J}}_{T 1}, {\vec{J}}_{T2}, ..., {\vec{J}}_{T P}] \in ℝ^{N_{S}^{2} \times P}$ denote the set of training source patterns, where ${\vec{J}}_{T p}$ $(p = 1, 2, ..., P)$ is the vectorized representation of the $p$ th training source pattern. Let $Ψ \in ℝ^{N_{S}^{2} \times N_{A}}$ be the source dictionary composed of $N_{A}$ atoms. The vector ${\vec{c}}_{p}$ $(p = 1, 2, ..., P)$ denotes the representation coefficients of the training source ${\vec{J}}_{T p}$ in the dictionary $Ψ$ . $C = [{\vec{c}}_{1}, {\vec{c}}_{2}, ..., {\vec{c}}_{P}] \in ℝ^{N_{A} \times P}$ is the coefficient matrix corresponding to the entire training source set $J_{T}$ . The feasible set of the dictionary is given by $χ = {Ψ : {‖ {\vec{ψ}}_{p} ‖}_{2} = 1, 1 \leq p \leq N_{A}}$ , where every atom is constrained to have unit energy. The feasible set of the coefficient matrix is given by $C = {C : {‖ {\vec{c}}_{p} ‖}_{\infty} \leq u, 1 \leq p \leq P}$ , where ${‖ \cdot ‖}_{\infty}$ means the maximum absolute value of the elements in ${\vec{c}}_{p}$ , and $u$ is a pre-defined upper bound to avoid unexpected large sparse coefficients. Then, the source dictionary learning problem can be formulated as:

\hat{Ψ} = \underset{Ψ}{a r g m i n} (\frac{1}{2} {‖ J T - Ψ C ‖}_{2}^{2} + γ {‖ C ‖}_{0}), s . t . Ψ \in χ, C \in C,

where the quadratic term ensures the accuracy of sparse representation, the l₀-norm penalty aims at retaining the sparsity condition, and

γ

is a weight parameter. The source dictionary pre-learning step in Fig. 3 is to solve for the problem described in Eq. (5). However, Eq. (5) is a non-convex problem, the solution of which is time consuming. To address this issue, Bao et al. proposed an alternating proximal method to improve the computational efficiency and global convergence property of the dictionary learning process [26]. In this paper, we adopt this method to solve the problem in Eq. (5). First, define a quadratic function

Q (Ψ, C)

and two auxiliary variables

T_{C}^{k}

and

{\vec{s}}_{p}^{k}

as following:

Q (Ψ, C) = \frac{1}{2} {‖ J_{T} - Ψ C ‖}_{2}^{2},

T_{C}^{k} = C^{k - 1} - \frac{1}{λ^{k}} {\nabla_{C} Q |}_{C = C^{k - 1}, Ψ = Ψ^{k}},

{\vec{s}}_{p}^{k} = {\vec{ψ}}_{p}^{k} - \frac{1}{μ_{p}^{k}} {\nabla_{{\vec{ψ}}_{p}} Q |}_{C = C^{k}, Ψ = {\tilde{Ψ}}_{p}^{k}},

where the superscript

k

denotes the iteration count,

\nabla_{C} Q

denotes the gradient of

Q

with respect to

C

.

λ^{k}

and

μ_{p}^{k}

(p = 1, 2, ..., N_{A})

are the step lengths. During the learning process, the dictionary is updated atom by atom.

{\vec{s}}_{p}^{k}

(p = 1, 2, ..., N_{A})

is the auxiliary vector used to update the

p

th atom at the

k

th iteration.

{\tilde{Ψ}}_{p}^{k}

is the current dictionary where the first

(p - 1)

atoms have been updated, while the rest

(N_{A} - p + 1)

atoms remain unchanged, i.e.,

{\tilde{Ψ}}_{p}^{k} = [{\vec{ψ}}_{1}^{k + 1}, {\vec{ψ}}_{2}^{k + 1}, ..., {\vec{ψ}}_{p - 1}^{k + 1}, {\vec{ψ}}_{p}^{k}, {\vec{ψ}}_{p + 1}^{k}, ..., {\vec{ψ}}_{N_{A}}^{k}]

. The flowchart of the dictionary learning algorithm is shown in Algorithm 1, where

h > 0

is a pre-defined upper bound for

λ^{k}

and

μ_{p}^{k}

,

ρ > 1

is a weight parameter,

Γ_{a} (b)

is the ramp function defined as

Γ_{a} (b) = b

if

| b | > a

, and

Γ_{a} (b) = 0

otherwise.

Algorithm 1. Alternating proximal method for source dictionary learning

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In some previous works, 2D-DCT basis was used as the sparse basis to obtain good performance in CS-SO [21,22]. Therefore, it is natural to choose the 2D-DCT basis as the initial dictionary $Ψ^{0}$ in Algorithm 1. The atoms in the standard 2D-DCT basis are represented by squared matrices, as shown in Fig. 4(a). However, the effective source in lithography systems is constrained within a circular region by the exit pupil of the projection optics. Although an $N_{S} \times N_{S}$ squared matrix $J$ is used to represent the source pattern in Eq. (1), the source pixels outside the circular pupil region should have zero intensity and make no contribution to the aerial image. Due to the inconsistency between the standard 2D-DCT basis and circular exit pupil, more coefficients are needed to represent the circular source patterns. To further improve the sparsity condition, we design a circular-shaped DCT basis to represent the effective source pattern. In order to achieve this goal, the circular exit pupil is used to truncate the squared 2D-DCT basis functions. As shown in Fig. 4(a), the standard 2D-DCT basis functions are represented by the $N_{S} \times N_{S}$ squared matrices ( $N_{S} = 21$ ). Figure 4(b) shows the circular DCT basis functions after truncation, where each circular basis function includes 317 pixels. Afterwards, each circular DCT basis function is raster-scanned to form a 317 $\times$ 1 atom vector. Stacking all of these atom vectors together, we can obtain the initial source dictionary in Algorithm 1.

Fig. 4 Comparison between (a) the standard 2D-DCT basis functions, and (b) the circular DCT basis functions.

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3.2. Cooperative optimization of source dictionary and projection matrix

After the pre-learning process, the source dictionary and projection matrix will be cooperatively optimized to improve the incoherence condition between them. The dictionary and projection matrix will be updated alternatively for several iteration cycles. The details of the cooperative optimization method will be described in the following.

In Eq. (4), the sensing matrix of the CS framework is given by

D = Φ I_{CC}^{S} Ψ,

where

I_{CC}^{S}

includes the information of mask pattern and lithography system parameters, while

Φ

and

Ψ

are the projection matrix and source dictionary to be optimized. According to [29], the incoherence condition can be guaranteed if the Gram matrix of

D

is very close to the identity matrix

I_{E}

. The Gram matrix of

D

is defined as:

G = Gram (\tilde{D}) = {\tilde{D}}^{T} \tilde{D},

where

\tilde{D}

is the sensing matrix with normalized columns. In other words,

G (i, j)

indicates the correlation between the

i

th column and the

j

th column of sensing matrix. Based on the incoherence condition, we need to find out the optimal

\hat{Ψ}

and

\hat{Φ}

to minimize the distance between the Gram matrix

G

and the identity matrix

I_{E}

. Thus, the cooperative optimization problem can be formulated as:

\hat{Ψ}, \hat{Φ} = \underset{Ψ, Φ}{a r g m i n} {‖ G - I_{E} ‖}_{2}^{2} .

It should be noticed that $I_{CC}^{S}$ is the function of the mask pattern. Thus, the value of $I_{CC}^{S}$ changes for different mask layouts. In order to obtain the universal solution ${\hat{Ψ}, \hat{Φ}}$ for different mask patterns, we need to learn the matrices ${\hat{Ψ}, \hat{Φ}}$ from a set of training mask layouts. Now, consider we have $P$ different training mask layouts, whose corresponding ICC matrices are denoted by $I_{CC p}^{S}$ $(p = 1, 2, ..., P)$ . These ICC matrices can be pre-calculated using the existing software. Let $G_{p}$ represent the Gram matrix corresponding to the $p$ th training layout. Then, the cooperative optimization problem in Eq. (11) can be generalized as the following form:

\hat{Ψ}, \hat{Φ} = \underset{Ψ, Φ}{a r g m i n} \sum_{p = 1}^{P} {‖ G_{p} - I_{E} ‖}_{2}^{2} .

Taking into account both sparsity and incoherence conditions, the cost functions in Eqs. (5) and (12) are combined together, and the cooperative optimization problem is reformulated as:

\hat{Ψ}, \hat{Φ} = \underset{Ψ, Φ}{a r g m i n} (\frac{1}{2} {‖ J_{T} - Ψ C ‖}_{2}^{2} + γ {‖ C ‖}_{0} + ν \sum_{p = 1}^{P} {‖ G_{p} - I_{E} ‖}_{2}^{2}),

where

ν

is a weight coefficient to balance between the sparsity and incoherence cost terms. In Fig. 3, the step of cooperative optimization of source dictionary and projection matrix is to solve for the problem described in Eq. (13). The source dictionary

Ψ

can still be updated using Algorithm 1 mentioned before. The only difference is to replace the function

Q

in Algorithm 1 by function

\tilde{Q}

defined as:

\tilde{Q} (Ψ, C, Φ) = \frac{1}{2} {‖ J_{T} - Ψ C ‖}_{2}^{2} + ν \sum_{p = 1}^{P} {‖ G_{p} - I_{E} ‖}_{2}^{2} .

Accordingly, when referring to Eq. (8) in Algorithm 1, the gradient

\nabla_{{\vec{ψ}}_{k}} Q

is replaced by

\nabla_{{\vec{ψ}}_{k}} \tilde{Q}

. At the same time, since the second term in Eq. (13) is independent of the projection matrix

Φ

, the cost function related to

Φ

only includes two quadratic terms. Thus, we can use the gradient-based algorithm to iteratively update the projection matrix as following:

Φ^{k + 1} = Φ^{k} - s {\nabla_{Φ} \tilde{Q} |}_{Φ = Φ^{k}},

where

s

is the step length.

4. Simulation and analysis

The proposed SO method is assessed by a set of simulations in this section. Several representative layout patterns are chosen and grouped into two categories: the training data set and testing data set. The training layouts are used to implement the dictionary pre-learning and cooperative optimization of dictionary and projection matrix. On the other hand, the testing layouts are used to verify the sparsity and incoherence conditions, as well as the performance of the proposed LCS-SO method. In the following, Section 4.1 will provide the simulations of the JODP method. Section 4.2 will provide the LCS-SO results based on the optimized source dictionary and projection matrix.

4.1. Simulations of JODP method

In this paper, fifteen representative layout patterns are chosen for the simulations. These layouts contain typical 1D and 2D features such as the line-space pattern, T-shape pattern, L-shape pattern, contact holes, and so on. Twelve of them are used as the training layouts, while the other three are used as the testing layouts. The training layouts are shown in the first row and the third row of Fig. 5.

Fig. 5 Twelve training layouts and the corresponding optimized source patterns obtained by the ACS-SO method.

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The ICC matrices of the training layouts are calculated using the S-Litho software [37], which involves the rigorous EUV lithography imaging model to take into account the shadowing effect, non-telecentric effect, etc. These simulations are based on an EUV lithography system with 13.5nm illumination wavelength and 0.33 numerical aperture. The source grid is set as 21 $\times$ 21 ( $N_{S} =21$ ). The dimension of the layout patterns is 153nm $\times$ 153nm, which is gridded into 153 $\times$ 153 pixels ( $N =153$ ). The CD on the layouts is equal to 30nm. The adaptive projection CS-SO (ACS-SO) method proposed in [22] is used to calculate the optimized source patterns for all training layouts, and the results are shown in the second row and the fourth row in Fig. 5. In the following simulations, fourfold symmetry is imposed on the source patterns, since the source with increased degree of symmetry is beneficial to improve the imaging contrast [38] and mitigate the pattern shift effect [39]. However, the proposed method can also be used to optimize asymmetric source patterns.

Based on the training source patterns in Fig. 5, the source dictionary and projection matrix can be optimized using the proposed JODP algorithm as described in Section 3. The initial source dictionary $Ψ^{0}$ is set as the circular DCT basis with dimension of 317 $\times$ 441, and the initial projective matrix $Φ^{0}$ is set as a 25 $\times$ 300 Bernoulli random matrix. First, the source dictionary is pre-learned for 80 iterations according to Algorithm 1. Then, the projection matrix and dictionary are cooperatively optimized according to the flowchart in Fig. 3(b). In each iteration cycle, the projection matrix is first updated for 5 times, then the dictionary is updated for 5 times. This kind of iteration cycle is repeated for 3 times to obtain the final optimized ${\hat{Ψ}, \hat{Φ}}$ . Figures 6 and 7 provide illustrative comparisons between ${Ψ^{0}, Φ^{0}}$ and ${\hat{Ψ}, \hat{Φ}}$ . In Fig. 6, the first row and second row present four atoms in the initial dictionary $Ψ^{0}$ and optimized dictionary $\hat{Ψ}$ , respectively. The third row shows the difference between the atoms in $Ψ^{0}$ and $\hat{Ψ}$ . Figure 7, from top to bottom, illustrates the initial projection matrix $Φ^{0}$ , the optimized projection matrix $\hat{Φ}$ , and the absolute value of difference between them. It is observed that the optimized dictionary atoms include some of the structure characteristics of the training source patterns, which benefits to improve the sparsity degree in source representation.

Fig. 6 Comparison between the initial and optimized dictionaries: four atoms in (a) $Ψ^{0}$ and (b) $\hat{Ψ}$ , as well as (c) the difference between them.

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Fig. 7 Comparison of the initial and optimized projection matrices: (a) the initial Bernoulli random projection matrix $Φ^{0}$ , (b) the optimized projection matrix $\hat{Φ}$ , and (c) the absolute value of difference between them.

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Next, we will verify the improvement of sparsity and incoherence conditions by the JODP method based on the testing layouts. Figure 8 shows the three testing layouts and the corresponding optimized source patterns obtained by the ACS-SO method proposed in [22]. Then, we use both $\hat{Ψ}$ and $Ψ^{0}$ to represent these three testing source patterns. Figure 9 illustrates the absolute values of the representation coefficients, where the coefficients are sorted in the descending order. It is observed that the coefficients on $\hat{Ψ}$ descend more quickly than the coefficients on $Ψ^{0}$ . That means the sparsity degree of the testing source patterns can be improved by optimizing the dictionary. The improvement of sparsity condition can be illustrated in another way. By truncating the smaller coefficients, we can use some significant coefficients to approximately represent the source patterns. The representation error (RE) induced by the truncation of coefficients is defined as:

RE = \frac{||approximate source - original {source||}_{2}}{||original {source||}_{2}} .

Figure 10 plots the curves of RE versus the number of significant coefficients used to represent the testing source patterns. Compared to the initial dictionary, the optimized dictionary can use less coefficients to achieve a given RE level. Thus, the dictionary learning is shown to improve the sparsity condition in the source representation.

Fig. 8 Three testing layouts and the corresponding optimized source patterns obtained by the ACS-SO method.

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Fig. 9 The absolute values of coefficients to represent the testing source patterns on the initial and optimized dictionaries.

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Fig. 10 The curves of REs versus the number of significant coefficients used to represent the testing source patterns.

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As mentioned before, the incoherence condition is improved if the Gram matrix is close to the identity matrix. To evaluate the incoherence condition, the Gram matrices of the testing layouts before and after JODP are calculated using Eqs. (9) and (10). Figure 11 shows the histograms of the absolute values of off-diagonal elements in the Gram matrices corresponding to the three testing layouts. The blue solid histograms and red shadowing histograms depict the distributions of off-diagonal elements before and after JODP, respectively. The JODP method is shown to reduce the number of larger off-diagonal elements (absolute value >0.6), and increase the number of smaller off-diagonal elements (absolute value <0.2). This reveals that the proposed JODP method can effectively improve the incoherence condition.

Fig. 11 Histograms of the absolute values of off-diagonal elements in the Gram matrices corresponding to the testing layouts.

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4.2. Simulations of LCS-SO using optimized dictionary and projection matrix

This section presents the LCS-SO results for the testing layouts using the optimized dictionary and projection matrix. The optimized source patterns can be obtained by solving the l₁-norm CS reconstruction problem in Eq. (4). Many algorithms were developed to seek for the l₁-norm minimization solution under CS framework [40–42]. In this paper, we choose the linearized Bregman algorithm to solve the LCS-SO problem, due to its high computational efficiency and the ability to enhance imaging contrast [31,43]. The initial source pattern before optimization is set to be a full pupil illumination with normalized dose. Figures 12–14 provide the simulation results for the three testing layouts, respectively. In each figure, we compare the results obtained by the proposed LCS-SO method (1st column), the traditional ACS-SO method (2nd column), the SO method based on conjugate gradient (CG) algorithm (3rd column), and the SO method based on PSO algorithm (4th column). Hereafter, the last two SO methods are referred to as the CG-SO and PSO-SO, respectively. From top to bottom, Figs. 12–14 illustrate the optimized source patterns, the corresponding aerial images, and print images on the wafer. The photoresist effect is represented by a constant threshold resist model, and the print image on the wafer can be calculated as [44,45]:

P = Γ_{t r} (I),

where

I

is the aerial image described in Eq. (1), and

Γ_{t r} (\cdot)

is a hard threshold function with threshold value of

t r = 0.25

. Pattern error (PE) is used as the metric to assess the lithography imaging fidelity, where the PE is defined as the square of Euclidean distance between the print image and the target layout. The proposed LCS-SO method is shown to achieve superior imaging fidelity over other SO methods for Testing Layouts #2 and #3. For Testing Layout #1, the proposed LCS-SO achieves the second-best imaging fidelity among the four types of SO methods. Although the PSO-SO algorithm leads to the lowest PE for Testing Layout #1, it is much more computationally intensive than the LCS-SO method as described shortly.

Fig. 12 Simulation results obtained by the proposed LCS-SO method, ACS-SO method, CG-SO method, and PSO-SO method based on Testing Layout #1.

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Fig. 13 Simulation results obtained by the proposed LCS-SO method, ACS-SO method, CG-SO method, and PSO-SO method based on Testing Layout #2.

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Fig. 14 Simulation results obtained by the proposed LCS-SO method, ACS-SO method, CG-SO method, and PSO-SO method based on Testing Layout #3.

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Table 1 shows the average PEs of print images corresponding to the initial source patterns, and the optimized source patterns obtained by different types of SO methods, as well as the runtime per iteration for these methods. It is noted that the ACS-SO method uses the adaptive random projection matrix to reduce the dimensionality of SO problem [22]. The randomness of the projection matrix may lead to a slightly different result in each implementation. To make a fair comparison, we repeat the ACS-SO algorithm for 50 times, and use the average PE to evaluate the imaging fidelity. On the other hand, the projection matrix of the proposed LCS-SO method is fixed when the JODP is completed. Thus, the average PE of LCS-SO method is the same as that obtained by one implementation. From this point of view, the LCS-SO method is more robust than the ACS-SO method. It is shown that the LCS-SO method leads to lower average PEs than the ACS-SO and CG-SO methods for all testing layouts. In each iteration, the runtime of LCS-SO method is comparable to the ACS-SO method, but less than the CG-SO method. Furthermore, the LCS-SO method can improve the computational efficiency by four orders of magnitude in each iteration compared to the PSO-SO algorithm. Therefore, the LCS-SO method results in good performance in both computational efficiency and imaging fidelity among different SO methods.

Table 1. The PEs and runtimes of the proposed LCS-SO and other state-of-the-art SO methods.

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Figure 15 illustrates the convergence curves of PEs for the three kinds of fast SO methods, i.e., the LCS-SO, ACS-SO, and CG-SO methods. The proposed LCS-SO method is shown to improve the convergence rate compared to the other methods. According to the simulations mentioned above, the LCS-SO method can further improve the lithography imaging fidelity and convergence characteristics in contrast to the ACS-SO method. The performance gain is mainly attributed to the improvement of sparsity and incoherence conditions benefited from the proposed JODP method.

Fig. 15 Convergence curves of PEs for the proposed and traditional SO methods based on the testing layouts.

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5. Conclusion

This paper proposed an efficient LCS-SO method to improve the imaging performance of EUV lithography systems. The LCS-SO problem was formulated as an l₁-norm inverse reconstruction problem based on the rigorous EUV lithography imaging model. The learning-based JODP method was developed to jointly optimize the source dictionary and projection matrix based on a set of representative training layouts. In order to improve the sparsity degree, the circular-shaped basis function was designed to sparsely represent the source patterns. The proposed JODP method was proved to effectively ameliorate the sparsity and incoherence conditions in CS theory, thus improving the performance of source reconstruction algorithm. Benefiting from the learning-based CS strategy, the proposed LCS-SO method can effectively improve the lithography imaging fidelity and convergence rate in contrast to the state-of-the-art CS-SO method. The merits of the JODP and LCS-SO approaches were verified by a set of simulations using different testing layouts. For the real lithography systems, the proposed LCS-SO method can be combined with mask optimization to develop the SMO workflow. The training layouts can be generated according to the industrial design rules of real mask patterns. In our future work, more metrics can be taken into account in the optimization process, such as the normalized image log slope, pattern placement error, and so on.

Funding

National Science and Technology Major Project of China (2017ZX02315001-003); National Natural Science Foundation of China (NSFC) (61804174, 61675021).

Acknowledgments

We would like to acknowledge the support from EUV lithography simulation joint lab of Synopsys and IMECAS.

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Input:	Training source matrix $J T$
Output:	Source dictionary $Ψ$
Initialize:	$Ψ^{0}$ , $h$ , $ρ$ , $u$ , $γ$ .
Main procedure:	loop for $k = 0, 1, 2, ..., \max k$ :
	compute: $λ^{k} = max{ρ {‖ Ψ^{k}^{T} Ψ^{k} ‖}_{2}, h}$
	compute $T_{C}^{k}$ using Eq. (7)
	compute $C^{k}$ using $C^{k}_{i, j} = \min {Γ_{\sqrt{2 γ / λ^{k}}} (T_{C i, j}^{k}), u}$
	loop for $p = 1, 2, ..., N_{A}$ , compute:
	$μ_{p}^{k} = \max {ρ {(C^{k} C^{k}^{T})}_{p, p}, h}$
	compute ${\vec{s}}_{p}^{k}$ using Eq. (8)
	${\vec{ψ}}_{p}^{k + 1} = {{\vec{s}}_{p}^{k} / ‖ {\vec{s}}_{p}^{k} ‖}_{2}$
	end loop
	$Ψ^{k + 1} = [{\vec{ψ}}_{1}^{k + 1}, {\vec{ψ}}_{2}^{k + 1}, ..., {\vec{ψ}}_{N_{A}}^{k + 1}]$
	$k = k + 1$ end loop

Source Pattern	PE			Runtime (s) per Iteration
Source Pattern	Testing Layout #1	Testing Layout #2	Testing Layout #3	Runtime (s) per Iteration
Initial Source	2737	2209	2844	-
Proposed LCS-SO	2179	1705	2200	1.47E-5
ACS-SO	2287	1721	2289	1.66E-5
CG-SO	2321	1959	2568	1.19E-4
PSO-SO	2077	1734	2242	0.61

Input:	Training source matrix $J T$
Output:	Source dictionary $Ψ$
Initialize:	$Ψ^{0}$ , $h$ , $ρ$ , $u$ , $γ$ .
Main procedure:	loop for $k = 0, 1, 2, ..., \max k$ :
	compute: $λ^{k} = max{ρ {‖ Ψ^{k}^{T} Ψ^{k} ‖}_{2}, h}$
	compute $T_{C}^{k}$ using Eq. (7)
	compute $C^{k}$ using $C^{k}_{i, j} = \min {Γ_{\sqrt{2 γ / λ^{k}}} (T_{C i, j}^{k}), u}$
	loop for $p = 1, 2, ..., N_{A}$ , compute:
	$μ_{p}^{k} = \max {ρ {(C^{k} C^{k}^{T})}_{p, p}, h}$
	compute ${\vec{s}}_{p}^{k}$ using Eq. (8)
	${\vec{ψ}}_{p}^{k + 1} = {{\vec{s}}_{p}^{k} / ‖ {\vec{s}}_{p}^{k} ‖}_{2}$
	end loop
	$Ψ^{k + 1} = [{\vec{ψ}}_{1}^{k + 1}, {\vec{ψ}}_{2}^{k + 1}, ..., {\vec{ψ}}_{N_{A}}^{k + 1}]$
	$k = k + 1$ end loop

Source Pattern	PE			Runtime (s) per Iteration
Source Pattern	Testing Layout #1	Testing Layout #2	Testing Layout #3	Runtime (s) per Iteration
Initial Source	2737	2209	2844	-
Proposed LCS-SO	2179	1705	2200	1.47E-5
ACS-SO	2287	1721	2289	1.66E-5
CG-SO	2321	1959	2568	1.19E-4
PSO-SO	2077	1734	2242	0.61

Learning-based compressive sensing method for EUV lithographic source optimization

Abstract

1. Introduction

2. CS-SO framework for EUV lithography

2.1. Imaging model of EUV lithography systems

2.2. Formulation of CS-SO problem

3. Joint optimization of source dictionary and projection matrix

3.1. Pre-learning of source dictionary

3.2. Cooperative optimization of source dictionary and projection matrix

4. Simulation and analysis

4.1. Simulations of JODP method

4.2. Simulations of LCS-SO using optimized dictionary and projection matrix

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (15)

Tables (2)

Equations (17)

Optics Express