Abstract

Computational lithography is a key technique to optimize the imaging performance of optical lithography systems. However, the large amount of calculation involved in computational lithography significantly increases the computational complexity. This paper proposes a model-informed deep learning (MIDL) approach to improve its computational efficiency and to enhance the image fidelity of lithography system with partially coherent illumination (PCI). Different from conventional deep learning approaches, the network structure of MIDL is derived from an approximate compact imaging model of PCI lithography system. MIDL has a dual-channel structure, which overcomes the vanishing gradient problem and improves its prediction capacity. In addition, an unsupervised training method is developed based on an accurate lithography imaging model to avoid the computational cost of labelling process. It is shown that the MIDL provides significant gains in terms of computational efficiency and imaging performance of PCI lithography system.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Moore’s law predicts that the number of transistors accommodated in an integrated circuit (IC) will double every 18-24 months [1]. Optical lithography is one of the core technologies to promote the integration of semiconductor devices on chips. Figure 1 shows the schematic diagram of a deep ultraviolet (DUV) lithography system with 193nm illumination [2,3]. The IC layout is recorded on the photomask in advance. The light source illuminates the mask uniformly, and then the layout pattern is transferred onto the wafer via the projection optics. The exposure will induce the chemical reaction of photoresist coated on the wafer. After a series of development processes of photoresist, the layout pattern is printed on the wafer.

 figure: Fig. 1.

Fig. 1. Diagram of PCI lithography system, where the commonly used partially coherent illuminations include circular, annular, dipole, quadrupole illuminations and so on.

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Due to the diffraction limit of optical lithography system, the actual print image will inevitably diverge from the target layout, as shown in Fig. 1. To overcome this problem, computational lithography technologies are used to improve the lithography resolution and image fidelity at 45nm technology node and beyond. Optical proximity correction (OPC) is a widely used computational lithography approach to optimize the lithography imaging performance by pre-warping the mask pattern and inserting sub-resolution assist features (SRAF) around the main features (MF) [47]. The pixelated model-based OPC approach using inverse optimization algorithm is also known as inverse lithography technology (ILT). In order to increase the degrees of optimization freedom, the advanced OPC methods can optimize the mask patterns at pixel scale. In the past, a variety of gradient-based OPC algorithms were proposed in the literature [59]. However, these methods are computationally intensive since the amount of optimization variables is large, and they need to repeatedly calculate the gradients of objective functions in each iteration.

Since then, machine learning and deep learning were introduced to improve the computational efficiency of OPC with various degrees of success [1012]. For example, Yang et al. proposed a mask design method using an OPC-oriented generative adversarial network [13]. Zhang et al. optimized photomasks with a variational automatic encoder [14]. Song et al. applied deep neural networks on GPU accelerated platform to improve the efficiency of mask optimization [15].

Recently, Ma et al. proposed to use the model-driven convolutional neural network (MCNN) to effectively accelerate the OPC algorithm, and improve its convergence property [16]. Different from conventional deep learning techniques, MCNN systematically derives the network architecture from the inverse lithography optimization model. Subsequently, a dual-channel feature was introduced in the MCNN framework to simultaneously optimize the MFs and SRAFs on the mask [17]. The existing MCNN methods focus on coherent illumination lithography systems. However, most of practical lithography systems use partially coherent illuminations (PCI) with finite physical dimensions [18]. Compared to the coherent illumination, PCI can improve the theoretical resolution limit of optical lithography systems [4]. As shown in Fig. 1, the commonly used partially coherent illuminations include circular, annular, dipole, and quadrupole illuminations, and so on. While the MCNN methods in [16] and [17] are effective in coherent illumination lithography systems, these algorithms will not produce adequate results when applied to a PCI lithography system.

To circumvent this problem, this paper develops a model-informed deep learning approach for PCI lithography systems, which is referred to as PCI-MIDL for short. The merits of PCI-MIDL method benefit from the explicit physical meaning of its network structure, and the remarkable ability to predict the OPC solutions under partially coherent illuminations. First of all, an inverse mask optimization framework is established based on the physical imaging model of PCI lithography system. Then, the initial structure of PCI-MIDL network is derived by unfolding and truncating the inverse optimization iterations. In order to simplify the network structure, the average coherent approximation (ACA) model is used to depict the imaging process of lithography systems [18]. In addition, a dual-channel feature is introduced in the network to predict the MFs and SRAFs simultaneously. The dual-channel structure is also beneficial to overcome the vanishing gradient problem since the residual error is propagated back through multiple paths.

In order to avoid the time-consuming labelling process in supervised learning, this paper applies an unsupervised training method to the proposed PCI-MIDL network. Particularly, the network is regarded as an encoder that translates the training layouts to their OPC solutions. As the counterpart, a physical imaging model of PCI lithography system is used as the auto-decoder to optimize the network parameters. In order to improve the accuracy and stability of training process, a more rigorous imaging model, namely Fourier series expansion (FSE) model, is used to build up the auto-decoder [19]. This unsupervised training method does not need to label the training samples, thus effectively reduces the computational cost of the training stage. After the training process, the PCI-MIDL can implement a very fast prediction of the approximate OPC solutions. Then, we can run the gradient-based algorithm for a few iterations to refine the predicted mask patterns to achieve promising lithography image fidelity.

This paper provides a set of simulations to assess the proposed PCI-MIDL method. Compared to the traditional gradient-based OPC algorithm, the PCI-MIDL method can improve the computational efficiency by up to 200-fold, and obtain higher imaging performance of the PCI lithography systems.

The remainder of this paper is organized as follows. The two kinds of physical imaging models of PCI lithography systems are described in Section 2. The network structure of PCI-MIDL is derived in Section 3. The unsupervised training method is described in Section 4. Simulations and analysis are presented in Section 5. Section 6 provides the conclusions and discusses the future work.

2. Physical imaging models for PCI lithography systems

As mentioned above, the OPC mask is composed of a set of MFs and SRAFs. The MFs refer to the major mask geometries that are distorted from the target layout. On the other hand, the SRAFs refer to the minor assistant features distributed around the MFs. Although SRAFs themselves do not print on the wafer, they can effectively compensate the image distortion caused by the optical proximity effect. The imaging process of PCI lithography system is described in Fig. 2. Let ${{\textbf M}_b} \in {{\mathbb R}^{N \times N}}$ represent the binary mask patt ${{\textbf M}_b}$ ern with N being the lateral dimension of mask. The values of elements in are either 0 or 1, which represent the opaque (grey area) and transparent (white area) pixels. Let ${{\textbf M}_m} \in {{\mathbb R}^{N \times N}}$ and ${{\textbf M}_s} \in {{\mathbb R}^{N \times N}}$ represent the MF and SRAF patterns, respectively. Thus, the entire mask pattern can be formulated as [17]

$${{\textbf M}_b} = \Gamma \textrm{\{ }{{\textbf M}_g} - {t_m}\textrm{\} = }\Gamma \textrm{\{ (}{{\textbf M}_m} + {{\textbf M}_s}\textrm{)} - {t_m}\textrm{\} }, $$
where ${{\textbf M}_g} = {{\textbf M}_m} + {{\textbf M}_s}$ is the superposition of MFs and SRAFs, and $\Gamma \{{\cdot} \}$ is the hard threshold function with ${t_m} = 0.5$. Since ${{\textbf M}_m}$ and ${{\textbf M}_s}$ are allowed to be partially overlap together, ${{\textbf M}_g}$ is referred to as the grey-scale mask pattern. After the hard thresholding operation, we can obtain the binary mask pattern ${{\textbf M}_b}$. In order to make the imaging model differentiable, the hard threshold function can be replaced by the sigmoid function ${\textrm{sig} _m}({\cdot} , \cdot )$, and Eq. (1) is modified as:
$${{\textbf M}_b} = {\textrm{sig} _m}({{\textbf M}_g},{t_m}) = \frac{1}{{1 + \exp [ - {a_m}({{\textbf M}_m} + {{\textbf M}_s} - {t_m})]}}, $$
where ${a_m}$ is the steepness index of the sigmoid function.

 figure: Fig. 2.

Fig. 2. The imaging process of PCI lithography system.

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Given the mask models in Eqs. (1) and (2), we can calculate the aerial image of PCI lithography system, which indicates the distribution of light intensity projected on the photoresist surface. According to the Hopkins diffraction model, the aerial image can be formulated as [20]

$${\textbf I}({\textbf r^{\prime}}) = \int {\int {{{\textbf M}_b}({\textbf r}){{\textbf M}_b}(\tilde{{\textbf r}}){\mathbf \gamma }({\textbf r},\tilde{{\textbf r}}){{\textbf h}^ \ast }(\tilde{{\textbf r}},{\textbf r^{\prime}}){\textbf h}({\textbf r},{\textbf r^{\prime}})\textrm{d}{\textbf r}\textrm{d}\tilde{{\textbf r}}} }, $$
where ${\textbf r} = (x,y)$ and $\tilde{{\textbf r}} = (\tilde{x},\tilde{y})$ are the coordinates on the object plane located on the mask. ${\textbf r^{\prime}} = (x^{\prime},y^{\prime})$ is the coordinate on the image plane located on the wafer. ${{\textbf M}_b}$ is given by Eq. (2), ${\mathbf \gamma }({\textbf r},\tilde{{\textbf r}})$ is the complex degree of coherence, and ${\textbf h}$ is the point spread function of PCI lithography system. In this paper, only binary mask is considered, but the proposed method can be easily generalized to the optimization of phase-shifting masks [21,22].

Given the aerial image, the print image on the wafer can be calculated based on the photoresist model. In the following, we introduced two kinds of optical imaging models i.e., the FSE model and ACA model [18,19]. Both of these models are derived from the Hopkins imaging model. The FSE model represents the PCI imaging system as the sum of several coherent imaging systems [23]. The accuracy of FSE model is comparable to the discrete Hopkins diffraction model, but it has to increase the computational complexity to achieve higher simulation accuracy [23]. On the other hand, ACA model is an approximate compact imaging model, where the PCI imaging system is approximately represented by the superposition of a coherent imaging system and an incoherent imaging system. Compared to the FSE model, ACA model can significantly improve computational efficiency at the cost of reduced accuracy.

Although the FSE model is more accurate than the ACA model, it will greatly increase the computational complexity as shown in [23]. If the FSE model is used to construct the network, the network structure will become much more complicated than that obtained from the ACA model. This will multiply the computational complexity in both of the training process and inference process. Thus, the ACA model is used to derive the PCI-MIDL network to reduce the complexity of network structure, and to improve the speed of the forward prediction process. On the other hand, the more rigorous model, i.e. FSE model, serves as the decoder to train the network in an unsupervised manner, which is beneficial to improve the accuracy of training process. Next, the two models mentioned above are briefly described.

2.1. Fourier series expansion model

Assume the mask is constrained in a squared area defined by $x,y \in [{{{ - D} / 2},{D / 2}} ]$. Therefore, we can expand ${\mathbf \gamma }({\textbf r})$ as a two-dimensional Fourier series in both x and y directions [19]:

$${\mathbf \gamma }({\textbf r}) = \sum\limits_\textrm{m} {{{\mathbf \Gamma }_\textrm{m}}\exp (j{\omega _0}{\textbf m} \cdot {\textbf r})} ,\,{{\mathbf \Gamma }_\textrm{m}} = \frac{1}{{{D^2}}}\int_{{A_\gamma }} {{\mathbf \gamma }({\textbf r})\exp (j{\omega _0}{\textbf m} \cdot {\textbf r})\textrm{d}{\textbf r}}, $$
where ${\omega _0}\textrm{ = }{\pi / D}$, ${\textbf m} = ({m_x},{m_y})$ represents the integer grids on the source plane, and “${\cdot}$” is the inner-product operation. Substituting Eq. (4) into Eq. (3), the aerial image can be expressed as [19]
$${\textbf I} = {\sum\limits_{\textbf m} {{{\mathbf \Gamma }_{\textbf m}}|{{{\textbf M}_b} \otimes {{\textbf h}^{\textbf m}}} |} ^2}, $$
where ${\otimes} $ is the convolution operator, and ${{\textbf M}_b}$ is given by Eq. (2). The convolution kernel in Eq. (5) can be calculated as
$${{\textbf h}^{\textbf m}} = {\textbf h} \cdot \exp (j{\omega _0}{\textbf m} \cdot {\textbf r}), $$
where ${\textbf h}$ is the point spread function given by
$${\textbf h} = \frac{{{J_1}(2\pi rNA/\lambda )}}{{2\pi rNA/\lambda }}. $$

In the above equation, ${J_1}({\cdot} )$ is the first kind of Bessel function, NA is the numerical aperture of projector, $r = \sqrt {{x^2} + {y^2}}$, $\lambda$ is the illumination wavelength. In this paper, the photoresist effect is approximately represented by the hard threshold model, so the print image can be calculated as [2,5]

$${\textbf Z} = \Gamma \{ {\textbf I} - {t_r}\} , $$
where ${t_r}$ is the photoresist threshold. Similar to Eq. (2), we use the sigmoid function to replace the hard threshold function, and Eq. (8) is adjusted as follows
$${\textbf Z} \approx {\textrm{sig} _r}({\textbf I},{t_r}) = \frac{1}{{1 + \exp [ - {a_r}({\textbf I} - {t_r})]}}, $$
where ${a_r}$ is the steepness index.

2.2. Average coherent approximation model

In ACA model, the aerial image of PCI lithography system can be expressed as [18,20]:

$${\textbf I} = {|{{{\textbf M}_b} \otimes {{\textbf h}_C}} |^2}\textrm{ + }|{{\textbf M}_b^2 \otimes {\textbf h}_I^2} |, $$
where ${{\textbf h}_C}$ and ${{\textbf h}_I}$ are the equivalent amplitude impulse responses of the coherent component and incoherent component, respectively. Furthermore,
$${{\textbf h}_C}({\textbf r^{\prime}},{\textbf r}) = {\textbf f}{({\textbf r^{\prime}},{\textbf r})^{1/2}} \cdot {\textbf h}({\textbf r^{\prime}},{\textbf r})\textrm{ and }{{\textbf h}_I}({\textbf r^{\prime},r}) = {[{{\textbf 1} - {\textbf f}({\textbf r^{\prime}},{\textbf r})} ]^{1/2}} \cdot {\textbf h}({\textbf r^{\prime}},{\textbf r}). $$

In Eq. (11),${\textbf h}$ is given in Eq. (7), ${\textbf f} \in [0,1]$ represents the fraction of the coherent incident power:

$${\textbf f}({\textbf r^{\prime}},{\textbf r})\textrm{ = }\frac{{\int {{{|{{\textbf h}({\textbf r^{\prime}},\hat{{\textbf r}})} |}^2}{\boldsymbol{\mathrm{\mu}} }({\textbf r},\hat{{\textbf r}})\textrm{d}\hat{{\textbf r}}} }}{{\int {{{|{{\textbf h}({\textbf r^{\prime}},\tilde{{\textbf r}})} |}^2}\textrm{d}\tilde{{\textbf r}}} }}\,\textrm{and}\,{\boldsymbol{\mathrm{\mu}} }({\textbf r},\hat{{\textbf r}})\textrm{ = }\frac{{{\mathbf \gamma }({\textbf r},\hat{{\textbf r}})}}{{{{[{{\mathbf \gamma }({\textbf r},{\textbf r}){\mathbf \gamma }(\hat{{\textbf r}},\hat{{\textbf r}})} ]}^{{1 / 2}}}}}, $$
where $\hat{{\textbf r}}$ and $\tilde{{\textbf r}}$ are the dummy variables. Given the aerial image, the print image on the wafer can be calculated based on Eqs. (8) and (9).

3. Structure of the PCI-MIDL network

As mentioned above, the initial structure of PCI-MIDL network is derived from the inverse mask optimization model. Thus, before deriving the structure of PCI-MIDL network, we need to first establish the inverse mask optimization framework based on the ACA imaging model. The goal of OPC algorithm is to find the optimal mask that minimizes the difference between the target layout $\tilde{{\textbf Z}}$ and the actual print image ${\textbf Z}$. Thus, the objective function of OPC can be formulated as

$$F = ||{\tilde{{\textbf Z}} - {\textbf Z}} ||_2^2 \approx ||{\tilde{{\textbf Z}} - {{\textrm{sig} }_r}({\textbf I},{t_r})} ||_2^2, $$
where $\textrm{||}\cdot \textrm{|}{\textrm{|}_2}$ represents the l2-norm, and ${\textbf Z}$ is calculated based on the ACA imaging model and photoresist model. Therefore, the inverse mask optimization problem is modelled as
$$\{ {\hat{{\textbf M}}_m},{\hat{{\textbf M}}_s}\} = \arg \mathop {\min }\limits_{{{\textbf M}_m},{{\textbf M}_s}} F, $$
where ${\hat{{\textbf M}}_\textrm{m}}$ and ${\hat{{\textbf M}}_s}$ represent the optimal MF pattern and SARF pattern, respectively.

The problem in Eq. (14) can be solved by the gradient-based algorithm, where the MF pattern and SARF pattern are iteratively updated as follows [2,9]:

$${\textbf M}_m^{k + 1} = {\textbf M}_m^k - ste{p_{\textbf M}} \cdot {\nabla F} |{{\textbf M}_m} \textrm{and} {\textbf M}_s^{k + 1} = {\textbf M}_s^k - ste{p_{\textbf M}} \cdot {\nabla F} |{{\textbf M}_s},$$
where the superscripts indicate the iterations number;$ste{p_{\textbf M}}$ is the step length; $ {\nabla F} |{{\textbf M}_m}$ and $ {\nabla F} |{{\textbf M}_s}$ represent the gradients of objective function with respect to ${{\textbf M}_m}$ and ${{\textbf M}_s}$, respectively. According to Eqs. (2)–(10), (13) and (14), the gradients based on ACA imaging model can be calculated as
$$ {\nabla F} |{{\textbf M}_m}\textrm{ = } {\nabla F} |{{\textbf M}_s} ={-} 4{a_r} \cdot {a_m} \cdot \{ [(\widetilde {\textbf Z} - {\textbf Z}) \odot {\textbf Z} \odot ({\textbf 1} - {\textbf Z}) \odot ({{\textbf M}_b} \otimes {{\textbf h}_C})] \otimes {\textbf h}_C^\circ{\odot} {{\textbf M}_b} \odot ( {\textbf 1} - {{\textbf M}_b}) \}$$
$$- 4{a_r} \cdot {a_m} \cdot \{ [(\widetilde {\textbf Z} - {\textbf Z}) \odot {\textbf Z} \odot ({\textbf 1} - {\textbf Z}) \otimes {({\textbf h}_I^{\circ })^2}] \odot {\textbf M}_b^2 \odot ( {\textbf 1} - {{\textbf M}_b}) \}, $$
where ${\textbf 1} \in {{\mathbb R}^{N \times N}}$ is the one-valued matrix,${\odot}$ is the element-by-element multiplication,${\textbf h}_C^\circ$ and ${\textbf h}_I^\circ$ mean to rotate ${\textbf h}_C^{}$ and ${\textbf h}_I^{}$ by ${180^\circ }$ in both horizontal and vertical directions. In addition, we can redefine the variables in Eq. (16) as following:
$${{\textbf S}_m} = {{\textbf S}_s} = \delta (x,y),\textrm{ }{{\textbf D}_m} = {{\textbf D}_s} = {{{{{\textbf h}_C}} / {||{{{\textbf h}_C}} ||}}_2},\textrm{ }{\textbf T} = (\tilde{{\textbf Z}} - {\textbf Z}) \odot {\textbf Z} \odot ({\textbf 1} - {\textbf Z}),$$
$${{\textbf W}_m} = {{\textbf W}_s} = {{{{\textbf h}_C^\circ } / {||{{\textbf h}_C^\circ } ||}}_2},\textrm{ }{\textbf B} = {{\textbf M}_b} \odot ({\textbf 1} - {{\textbf M}_b}) ,\textrm{ }{{\textbf P}_m} = {{\textbf P}_s} = {{{{{({\textbf h}_I^\circ )}^2}} / {||{{{({\textbf h}_I^\circ )}^2}} ||}}_2},$$
$$\textrm{and}\,Q = 4{a_r} \cdot {a_m} \cdot ste{p_{\textbf M}}, $$
where $\delta (x,y)$ is the Dirac delta function. According to Eqs. (14)–(17), we can draw the flowchart of the gradient-based algorithm in Fig. 3. The two dashed blocks represent the two channels to update the MFs and SRAFs, respectively. Between the two channels, there is a route to merge MFs and SRAFs into an entire mask pattern.

 figure: Fig. 3.

Fig. 3. The flowchart of gradient-based inverse mask optimization algorithm.

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Next, we can unfold the recursive iteration process of the gradient-based algorithm, and then truncate the first $K$ iterations to construct the PCI-MIDL network. Each iteration can be regarded as one layer in the network. The structure of PCI-MIDL network is shown in Fig. 4(a). The input of the network is the target layout $\tilde{{\textbf Z}}$, which is divided into the MF pattern and SRAF pattern in advance. According to Eq. (17), the convolution kernels and weight parameters in the network can be initialized in a systematical way, rather than assigned using random or heuristic method. Similar to Fig. 3, the PCI-MIDL network also contains two channels, which are used to predict the MF pattern and SRAF pattern, respectively. The data of these two channels are merged to form the entire mask pattern at the end of each layer. The final output of the network is the entire binary mask pattern. Therefore, we can regard the PCI-MIDL network as an encoder, which transforms the target layout into its corresponding OPC solution.

 figure: Fig. 4.

Fig. 4. (a) The structure of the PCI-MIDL network (the encoder), and (b) the decoder used to train the PCI-MIDL network. Channel 1 and Channel 2 are used to predict the solutions of SRAF pattern and MF pattern, respectively.

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Without loss of generality, we take the $k$th ($k$=1, 2, …,$K$) layer as an example to formulate the network in detail. The input data of the $k$th layer include the MF pattern ${\textbf M}_{mb}^k$, SRAF pattern ${\textbf M}_{sb}^k$ and the entire mask pattern ${\textbf M}_b^k$. The three input data are all approximate binary matrices. At the end of the layer, the three updated patterns are denoted as ${\textbf M}_{mg}^k$,${\textbf M}_{sg}^k$ and ${\textbf M}_g^k$, which are all grey-scaled matrices. The updated patterns can be calculated as

$${\textbf M}_{mg}^k = {\textbf M}_{mb}^k \otimes {\textbf S}_m^k + [{{\textbf T}^k} \odot ({\textbf M}_b^k \otimes {\textbf D}_m^k)] \otimes {\textbf W}_m^k \odot {{\textbf B}^k} \odot Q + {\textbf M}_b^k \odot ({{\textbf T}^k} \otimes {\textbf P}_m^k) \odot {{\textbf B}^k} \odot Q, $$
$${\textbf M}_{sg}^k = {\textbf M}_{sb}^k \otimes {\textbf S}_s^k + [{{\textbf T}^k} \odot ({\textbf M}_b^k \otimes {\textbf D}_s^k)] \otimes {\textbf W}_s^k \odot {{\textbf B}^k} \odot Q + {\textbf M}_b^k \odot ({{\textbf T}^k} \otimes {\textbf P}_s^k) \odot {{\textbf B}^k} \odot Q, $$
$${\textbf M}_g^k = {\textbf M}_{sg}^k + {\textbf M}_{mg}^k, $$
where ${\textbf S}_m^k \in {{\mathbb R}^{{N_f} \times {N_f}}},\,{\textbf S}_s^k \in {{\mathbb R}^{{N_f} \times {N_f}}},\,{\textbf D}_m^k \in {{\mathbb R}^{{N_f} \times {N_f}}},\,{\textbf D}_s^k \in {{\mathbb R}^{{N_f} \times {N_f}}},\,{\textbf W}_m^k \in {{\mathbb R}^{{N_f} \times {N_f}}},\,{\textbf W}_s^k \in {{\mathbb R}^{{N_f} \times {N_f}}},\,{\textbf P}_m^k \in {{\mathbb R}^{{N_f} \times {N_f}}}$, and ${\textbf P}_s^k \in {{\mathbb R}^{{N_f} \times {N_f}}}$ are the convolution kernels that can be optimized during the training process. ${{\textbf T}^k} \in {{\mathbb R}^{N \times N}}$ is the transmission matrix defined as
$${{\textbf T}^k} = {\textbf T} = (\tilde{{\textbf Z}} - {\textbf Z}) \odot {\textbf Z} \odot ({\textbf 1} - {\textbf Z}). $$

In Eq. (21), the term $(\tilde{{\textbf Z}} - {\textbf Z})$ represents the regions where the target layout $\tilde{{\textbf Z}}$ is different from the actual print image ${\textbf Z}$. The term ${\textbf Z} \odot ({\textbf 1} - {\textbf Z})$ depicts the contour of print image. Therefore, ${{\textbf T}^k}$ represents the overlapped regions between $(\tilde{{\textbf Z}} - {\textbf Z})$ and ${\textbf Z} \odot ({\textbf 1} - {\textbf Z})$, that is, the boundary regions of ${\textbf Z}$ that are not on the target layout $\tilde{{\textbf Z}}$.

Note that the updated patterns in Eqs. (18)–(20) are grey-scaled, which cannot be used as inputs for the next layer directly. Thus, we add the sigmoid functions at the end of each layer to convert the grey-scaled patterns to the approximate binary patterns. From the perspective of deep learning, these sigmoid functions can be regarded as nonlinear activation functions, which can improve the prediction capacity of the nonlinear mapping between the input and output. After the sigmoid functions, the inputs of the $(k + 1)$th layer can be expressed as

$${\textbf M}_{mb}^{k + 1} = {\textrm{sig} _m}({\textbf M}_{mg}^k,{t_m}) = \frac{1}{{1 + \exp [ - {a_m}({\textbf M}_{mg}^k - {t_m})]}}, $$
$${\textbf M}_{sb}^{k\textrm{ + }1} = {\textrm{sig} _m}({\textbf M}_{sg}^k,{t_m}) = \frac{1}{{1 + \exp [ - {a_m}({\textbf M}_{sg}^k - {t_m})]}}, $$
$${\textbf M}_b^{k\textrm{ + }1} = {\textrm{sig} _m}({\textbf M}_g^k,{t_m}) = \frac{1}{{1 + \exp [ - {a_m}({\textbf M}_g^k - {t_m})]}}. $$

In the training stage, the final output of PCI-MIDL network is an approximate binary mask pattern $\hat{{\textbf M}} = {\textrm{sig} _m}({\textbf M}_g^K,{t_m})$. After training, we need to hardly threshold $\hat{{\textbf M}}$ to obtain the standard binary mask as the predicted OPC solution.

It is noted that the sigmoid function in Eq. (2) is used at the end of each layer of PCI-MIDL network. That is because the output layouts of some layers in PCI-MIDL may include overlapped MFs and SRAFs. In order to binarize the outputs of all layers, the sigmoid function is needed at the end of each layer.

It is worth noting that both of the ACA model and FSE model mentioned in Section 2 are scalar imaging models of PCI lithography system. In order to consider the polarization of illumination, we could generalize the proposed deep learning method based on the vector imaging model. Based on the Abbe’s method, the vector imaging model of PCI lithography system is given by [24]

$${\textbf I} = \sum\limits_{{x_s},{y_s}} {\left[ {{\textbf J}({x_s},{y_s})\sum\limits_{p = x,y,z} {{{|{({{\textbf M}_b} \odot {{\textbf B}^{{x_s},{y_s}}}) \otimes {\textbf H}_p^{{x_s},{y_s}}} |}^2}} } \right]}, $$
where ${\textbf J} \in {{\mathbb R}^{{N_s} \times {N_s}}}$ represents the source pattern,${\textbf J}({x_s},{y_s})$ is the light intensity of the source point at the coordinate $({x_s},{y_s})$, ${{\textbf B}^{{x_s},{y_s}}}$ denotes the mask diffraction matrix associated with the point source $({x_s},{y_s})$, and ${\textbf H}_p^{{x_s},{y_s}}(p = x,y,z)$ denotes the equivalent point spread functions of the x, y, and z components. Substituting Eq. (25) into Eqs. (9) and (13), we can derive the gradients of cost function as following
$${ {\nabla F} |_{{{\textbf M}_m}}} = { {\nabla F} |_{{{\textbf M}_s}}} ={-} 4{a_r} \cdot {a_m} \cdot \sum\limits_{{x_s},{y_s}} {\textrm{Real}\left\{ {{\textbf J}({x_s},{y_s})\sum\limits_{p = x,y,z} {(\widetilde {\textbf Z} - {\textbf Z}) \odot {\textbf Z} \odot ({\textbf 1} - {\textbf Z}) \odot ({{\textbf M}_b} \odot {{\textbf B}^{{x_s},{y_s}{\ast }}})} } \right.}$$
$$\left. { \otimes {\textbf H}_p^{{x_s},{y_s}{\ast }} \otimes {{({\textbf H}_p^{{x_s},{y_s}})}^{{\ast}^\circ }} \odot {{\textbf B}^{{x_s},{y_s}\ast }} \odot {{\textbf M}_b} \odot (\mathop {\mathop {\textbf 1}\limits_{} }\limits_{}^{} - {{\textbf M}_b})} \right\}. $$

According to Eq. (26), we can define the following variables:

$${{\textbf S}_m} = {{\textbf S}_s} = \delta (x,y),\textrm{ }{\textbf D}_m^{{x_s}{y_s}} = {\textbf D}_s^{{x_s}{y_s}} = {({\textbf H}_p^{{x_s},{y_s}})^{{\ast\circ }}},\textrm{ }{\textbf T} = (\tilde{{\textbf Z}} - {\textbf Z}) \odot {\textbf Z} \odot ({\textbf 1} - {\textbf Z}),\textrm{ }{\textbf W}_m^{{x_s}{y_s}} = {\textbf W}_s^{{x_s}{y_s}} = {\textbf H}_p^{{x_s},{y_s}\ast },$$
$${\textbf G}_m^{{x_s},{y_s}} = {\textbf G}_s^{{x_s},{y_s}} = {{\textbf B}^{{x_s},{y_s}\ast }},\textrm{ and }Q = 4{a_r} \cdot {a_m} \cdot step, $$
where the defined notations represent the convolution kernels and weighting parameters within the PCI-MIDL network. Similar to the scalar imaging model, we can construct the PCI-MIDL network based on the vector imaging model by unfolding and truncating the iteration process. In the future, we also plan to extend the PCI-MIDL method based on the sum of coherent system (SOCS) model [25] and the vector Abbe model [9]. Due to the length limit of this paper, these topics will be studied in the future.

The proposed PCI-MIDL method can be also generalized to take into account the optical aberrations and defocus. According to [26], when the defocus and aberrations are present, the point spread function of lithography system can be modified as:

$${\textbf h^{\prime}} = \frac{1}{\pi }\int_0^1 {\rho {e^{j\delta {\rho ^2}}} \times \left\{ {\int_0^{2\pi } {{e^{j\Phi (\rho ,\theta + \varphi )}} \times {e^{2\pi j\rho r \cdot \cos \theta }}d\theta } } \right\}d\rho }, $$
where $j = \sqrt { - 1} ;\,\Phi $ is the aberration function; $\delta$ is the defocus variation; the $\rho$ and $\theta$ are the polar coordinates in the exit pupil function; and the r and $\varphi$ are the polar coordinates in the image plane. Based on the modified point spread function, we can calculate the print images under different aberration and defocus conditions. Let ${{\textbf Z}_i} = {\textbf Z}({\Phi _i},{\delta _i})$, where $({\Phi _i},{\delta _i})$ are the ith set of process variation parameters. Then, the overall cost function can be defined as the sum of the pattern errors under different process variation conditions: $F^{\prime} = \sum\nolimits_i {\textrm{||}\tilde{{\textbf Z}} - {{\textbf Z}_i}\textrm{||}_2^2}$. Using the same method, the PCI-MIDL network can be derived based on the modified cost function to consider different aberration and defocus conditions. In the future, the proposed PCI-MIDL approach will be generalized to compensate for the effects of process variations.

4. Unsupervised training method for the PCI-MIDL network

The training method has a significant impact on the performance of deep learning neural network. The supervised learning methods require to spend lots of computing resources to label the training samples. For the PCI-MIDL problem, we need to calculate the OPC solutions for all of the training layouts using existing algorithm or software. Here, we introduce an unsupervised training method to avoid the time-consuming labelling manipulations thus reducing the computational cost of training process. As mentioned above the PCI-MIDL network in Fig. 4(a) is regarded as an encoder to transform the target layout $\tilde{{\textbf Z}}$ into the OPC solution $\hat{{\textbf M}}$. According to the principle of computational lithography, the purpose of PCI-MIDL is to make the print image of $\hat{{\textbf M}}$ close to the target layout as much as possible. Therefore, we can use the lithography imaging model as the decoder, which transforms $\hat{{\textbf M}}$ to its print image on the wafer. As shown in Fig. 4(b), we choose the FSE model in Section 2.1 to build up the decoder since it has higher accuracy than the ACA model [19]. Although the FSE model will increase the computational complexity, the training process is carried out only once, and the optimized network parameters can be repetitively used in the following simulations.

The output mask pattern from PCI-MIDL network is used as the input of the decoder. According to Eq. (5), the aerial image calculated by the decoder is

$${\textbf I^{\prime}} = {\sum\limits_{\textbf m} {{{\mathbf \Gamma }_{\textbf m}}|{\hat{{\textbf M}} \otimes {{\textbf h}^{\textbf m}}} |} ^2}. $$

According to Eqs. (9) and (29), the print image of $\hat{{\textbf M}}$ can be calculated as

$${\textbf Z^{\prime}} \approx {\textrm{sig} _r}({\textbf I^{\prime}},{t_r}) = \frac{1}{{1 + \exp [ - {a_r}({\textbf I^{\prime}} - {t_r})]}}. $$

Therefore, the objective function of lithography image fidelity can be formulated as

$$F^{\prime} = ||{\tilde{{\textbf Z}} - {\textbf Z^{\prime}}} ||_2^2 \approx ||{\tilde{{\textbf Z}} - {{\textrm{sig} }_r}({\textbf I^{\prime}},{t_r})} ||_2^2. $$

The goal of training process is to optimize the convolution kernels ${\textbf S}_m^k,{\textbf D}_m^k,{\textbf W}_m^k,{\textbf P}_m^k,{\textbf S}_s^k,{\textbf D}_s^k,{\textbf W}_s^k$ and ${\textbf P}_s^k$, such that the actual print image ${\textbf Z^{\prime}}$ is as close to the target layout as possible. Therefore, the training problem of the network can be expressed as

$$\{ \hat{{\textbf S}}_m^k,\hat{{\textbf D}}_m^k,\hat{{\textbf W}}_m^k,\hat{{\textbf P}}_m^k,\hat{{\textbf S}}_s^k,\hat{{\textbf D}}_s^k,\hat{{\textbf W}}_s^k,\hat{{\textbf P}}_s^k\} = \arg\mathop {\min }\limits_{\hat{{\textbf S}}_m^k,\hat{{\textbf D}}_m^k,\hat{{\textbf W}}_m^k,\hat{{\textbf P}}_m^k,\hat{{\textbf S}}_s^k,\hat{{\textbf D}}_s^k,\hat{{\textbf W}}_s^k,\hat{{\textbf P}}_s^k} E, $$
where $E$ is the loss function given by
$$E = F^{\prime}\textrm{ + }{\gamma _Q} \cdot {R_Q}\textrm{ = }||{\tilde{{\textbf Z}} - {\textbf Z^{\prime}}} ||_2^2 + {\gamma _Q} \cdot {R_Q}, $$
where ${\gamma _Q}$ is the weight of quadratic penalty. The quadratic penalty term ${R_Q}$ is used to enforce the network outputs to be close to binary masks [2,5]. The quadratic penalty term is
$${R_Q} = {\textbf 1}_{N \times 1}^T \cdot 4\hat{{\textbf M}} \odot ({\textbf 1} - \hat{{\textbf M}}) \cdot {{\textbf 1}_{N \times 1}}, $$
where ${{\textbf 1}_{N \times 1}}$ is the one-valued vector with dimension of $N \times 1$. In the unsupervised training procedure, we only need to feed all training samples into the network, and the network parameters will be learned automatically.

In this paper, the back-propagation algorithm is used to optimize the network parameters based on Eq. (32) [27]. In the training stage, we need to calculate the gradient of loss function with respect to all convolution kernels. Let ${\textbf A}_m^k$ represent the convolution kernels ${\textbf S}_m^k$,${\textbf D}_m^k$,${\textbf W}_m^k$ or ${\textbf P}_m^k$, and let ${\textbf A}_s^k$ represent the convolution kernels ${\textbf S}_s^k$,${\textbf D}_s^k$,${\textbf W}_s^k$ or ${\textbf P}_s^k$. According to the Chain rule, the partial derivatives of loss function to ${\textbf A}_m^k$ and ${\textbf A}_s^k$ can be calculated as follows:

$$\frac{{\partial E}}{{\partial {\textbf A}_m^k}} = \frac{{\partial E}}{{\partial \hat{{\textbf M}}}} \cdot \frac{{\partial \hat{{\textbf M}}}}{{\partial {\textbf M}_g^K}} \cdot \left( {\frac{{\partial {\textbf M}_{mg}^K}}{{\partial {\textbf M}_{mb}^K}} \cdot \frac{{\partial {\textbf M}_{mb}^K}}{{\partial {\textbf M}_{mg}^{K - 1}}} + \frac{{\partial {\textbf M}_{mg}^K}}{{\partial {\textbf M}_b^K}} \cdot \frac{{\partial {\textbf M}_b^K}}{{\partial {\textbf M}_{mg}^{K - 1}}} + \frac{{\partial {\textbf M}_{sg}^K}}{{\partial {\textbf M}_b^K}} \cdot \frac{{\partial {\textbf M}_b^K}}{{\partial {\textbf M}_{mg}^{K - 1}}}} \right) \cdots \frac{{\partial {\textbf M}_{mg}^k}}{{\partial {\textbf A}_m^k}}, $$
$$\frac{{\partial E}}{{\partial {\textbf A}_s^k}} = \frac{{\partial E}}{{\partial \hat{{\textbf M}}}} \cdot \frac{{\partial \hat{{\textbf M}}}}{{\partial {\textbf M}_g^K}} \cdot \left( {\frac{{\partial {\textbf M}_{sg}^K}}{{\partial {\textbf M}_{sb}^K}} \cdot \frac{{\partial {\textbf M}_{sb}^K}}{{\partial {\textbf M}_{sg}^{K - 1}}} + \frac{{\partial {\textbf M}_{sg}^K}}{{\partial {\textbf M}_b^K}} \cdot \frac{{\partial {\textbf M}_b^K}}{{\partial {\textbf M}_{sg}^{K - 1}}} + \frac{{\partial {\textbf M}_{mg}^K}}{{\partial {\textbf M}_b^K}} \cdot \frac{{\partial {\textbf M}_b^K}}{{\partial {\textbf M}_{sg}^{K - 1}}}} \right) \cdots \frac{{\partial {\textbf M}_{sg}^k}}{{\partial {\textbf A}_s^k}}. $$

It is noted that the inner parenthesis in both Eq. (35) and Eq. (36) include three terms. That means the residual error in the training stage will be propagated back through three paths. This mechanism can be also observed from the network structure in Fig. 4(a). The multiple-path structure will help to overcome the vanishing gradient problem and to improve the prediction capacity of PCI-MIDL network.

Supplement 1 provides the detailed calculation process of Eqs. (35) and (36). Then, the network parameters can be iteratively updated as follows:

$${\textbf A}_m^{k(n + 1)} = {\textbf A}_m^{k(n)} - ste{p_A} \cdot \nabla E|{\textbf A}_m^k, $$
$${\textbf A}_s^{k(n + 1)} = {\textbf A}_s^{k(n)} - ste{p_A} \cdot \nabla E|{\textbf A}_s^k, $$
where $ste{p_A}$ is the step size, $\nabla E|{\textbf A}_m^k$ and $\nabla E|{\textbf A}_s^k$ are the gradients of loss function to ${\textbf A}_m^k$ and ${\textbf A}_s^k$, respectively. After every iteration, the convolution kernels are enforced to be centrosymmetric, and are normalized based on their l2-norm such that ${\textbf A}_m^{k(n + 1)} = {\textbf A}_m^{k(n + 1)}\textrm{/||}{\textbf A}_m^{k(n + 1)}\textrm{|}{\textrm{|}_2}$. When the training process is completed, the decoder is removed from the network, and the encoder is used to predict the OPC solutions of other layouts.

5. Simulation and analysis

This section provides a set of simulations to demonstrate the validity of the proposed PCI-MIDL method. In Section 5.1, a set of simple layout patterns are used to train and test the PCI-MIDL network. In Section 5.2, the PCI-MIDL network is validated based on a set of complex layout patterns.

5.1. Simulations based on simple layout patterns

In this section, we train and test the PCI-MIDL network based on a set of simple layout patterns. We performed the following simulations using a DUV partially coherent lithography system with the illumination wavelength of $\lambda \textrm{ = 193nm}$. The light source of lithography system is an annular illumination with the inner and outer partial coherence factors of ${\sigma _{\textrm{inner}}}\textrm{ = }0.8$ and ${\sigma _{\textrm{outer}}}\textrm{ = }0.975$. The critical dimension of target layout is 45nm. The numerical aperture (NA) is set to be 1.35. The lateral size of matrix ${\textbf M}$ is $N = 150$, and the pixel size is 1.875nm × 1.875nm. The lateral size of the convolution kernels is ${N_f} = 21$. The steepness indices are ${a_r} = {a_m} = \textrm{10}$. The threshold of photoresist is ${t_r} = 0.07$. The weight of the quadratic penalty is ${\gamma _Q} = 0.05$. The step size to update mask is $ste{p_{\textbf M}} = 1$. The layer number of the PCI-MIDL is 5.

The nine target layouts in Fig. 5(a) are used as the training samples, and the three target layouts in Fig. 5(b) are used as the testing layouts. The training samples are selected from the representative clip features on the IC layouts, including the two-bar shape, T-shapes, L-shapes, lines, and contacts. These basic blocks are extensively used in the large scaled layout patterns. In the training process of network, we input the training samples into the network in turn. The step size to update convolution kernels is $ste{p_A} = 4 \times {10^{ - 8}}$. Please note that most of IC layout patterns are composed of normalized Manhattan geometries, and any regular Manhattan geometry can be decomposed into three basic features, namely concave corners, convex corners, and straight edges [28]. Although only 12 layout patterns were used in the simulations, these layouts include most of typical basic features of IC layout patterns. In the simulations, we only consider the layout patterns that consist of normalized Manhattan geometries with 90-degree angles and 270-degree angles. Since the mask patterns are treated as pixelated images in this work, the proposed method can be generalized to optimize the all-angled mask patterns.

 figure: Fig. 5.

Fig. 5. The simple training layouts and testing layouts for the PCI-MIDL network.

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When we input the training layouts into the encoder during the training phase, the network will automatically initialize the MFs and SRAFs. Taking the two-bar shaped mask as an example. The target layout including two parallel bars is used as the initial MF pattern, which is represented by the white regions in Fig. 6(a). The shadow area in Fig. 6(b) is defined as the initial SRAF pattern, which is the narrow contour around the target layout. In the proposed method, the width of the initial SRAF pattern is set to 3 pixels, and the distance from the initial SRAF pattern to the initial MF pattern is 2 pixels.

 figure: Fig. 6.

Fig. 6. Examples of (a) the initial MF pattern (white regions), and (b) the initial SRAF pattern (shadow regions).

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The simulation results of the traditional OPC method based on steepest descent (SD) algorithm [23] are illustrated in the left half of Fig. 7. The details of SD method can be found in [23]. It is noted that the SD method directly optimize the mask patterns based on inverse lithography framework without the training or learning operations. The maximum iteration number of SD method is set to 100, and other parameters are the same as those in PCI-MIDL. The simulation results of the proposed PCI-MIDL method are illustrated in the right half of Fig. 7. Figure 7 shows the simulation results for the three different testing layouts. In both figures, the first row and the second row respectively present the optimized mask patterns and the corresponding print images on wafer. Pattern error (PE) and root mean square error (RMSE) are two key metrics to evaluate the image fidelity of lithography systems. In this paper, the PE is defined as the square of Euclidean distance between the print image and target layout, that is $PE = \textrm{||}\tilde{{\textbf Z}} - \Gamma \textrm{\{ }{\textbf I} - {t_r}\textrm{\} ||}_2^2$. The RMSE is defined as $\textrm{RMSE} = \sqrt {{{\textrm{||}\tilde{{\textbf Z}} - \Gamma \textrm{\{ }{\textbf I} - {t_r}\textrm{\} ||}_2^2} / {{N^2}}}}$, where N is the lateral size of the mask pattern. The values of PEs and RMSEs for all the print images are provided under the figures.

 figure: Fig. 7.

Fig. 7. Simulation results of SD and PCI-MIDL methods based on simple layout patterns.

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According to Fig. 7, we can obtain the following observations on the comparison between the SD method and PCI-MIDL method. First, the OPC solutions obtained by SD method only include a few tiny SRAFs, but the PCI-MIDL successfully inserts remarkable SRAFs, which are curvilinear and deviate from the MFs. As we all know, SRAF is a key technology to improve the imaging performance and robustness of lithography patterning [29]. This advantage of PCI-MIDL is mainly due to the unique dual-channel structure that is capable of optimizing the MFs and SRAFs simultaneously, as well as the SRAF seeds inserted in the initial mask pattern as shown in Fig. 6(b). Second, compared to the SD method, the PCI-MIDL can respectively reduce the PEs and RMSEs of print images by 24% and 13% on average, and thus further improving the imaging performance of lithography system. It should be noted that for simple layouts, the PCI-MIDL method can obtain promising OPC solutions directly without any subsequent processing.

Next, we compare the computational efficiency of the SD and PCI-MIDL methods. Table 1 shows the average runtimes of the two methods over the three simple testing layouts. The traditional SD method does not include the training and testing process, so its iteration time is equal to the mask optimization time. On the other hand, the runtime of PCI-MIDL method includes two parts, the training time and testing time. It is noted that the network needs to trained only once, so the testing time of PCI-MIDL is regarded as the mask optimization time. In this paper, the optimization time is used to evaluate the computational efficiency of each method. Compared with SD method, PCI-MIDL method can significantly improve the computational efficiency by about 247.5 fold. Therefore, the PCI-MIDL not only achieves better the image fidelity, but also greatly improves the computational efficiency.

Tables Icon

Table 1. Average runtimes of different OPC methods based on simple layout patterns.

5.2. Simulations based on complex layout patterns

In this section, we train and test the PCI-MIDL network based on a set of complex layout patterns. The lateral size of the matrix ${\textbf M}$ is $N = 1\textrm{84}$, and the pixel size is 5.625nm × 5.625nm. The lateral size of convolution kernels is ${N_f} = \textrm{1}1$, the photoresist threshold is ${t_r} = 0.\textrm{1}$, the step size to update mask is $ste{p_{\textbf M}} = \textrm{4}$, and the PCI-MIDL includes 19 layers. Other parameters are the same as those in Section 5.1.

The nine complex layouts in Fig. 8(a) are used as the training samples, and the three layouts in Fig. 8(b) are used as the testing layouts. Similar to Section 5.1, the training samples are input in turn to train the PCI-MIDL network, and the step size to update the convolution kernels is $ste{p_A} = \textrm{2} \times {10^{ - 8}}$. The width of initial SRAF pattern is 1 pixel, and the distance from the initial SRAF pattern to the initial MF pattern is set to be 1 pixel.

 figure: Fig. 8.

Fig. 8. The complex training layouts and testing layouts for the PCI-MIDL network.

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The left half of Fig. 9 provides the simulation results of the SD method based on the complex testing layouts, where the maximum iteration number is 100. The right half of Fig. 9 shows the simulation results of the PCI-MIDL method. It is observed that the PCI-MIDL can automatically generate sufficient SRAFs for the OPC solutions. However, the PEs and RMSEs achieved by PCI-MIDL are much larger than those of SD. It is noted that the geometrical environments and feature characteristics in complex layouts are much more complicated than those in simple layouts. Although the training layouts in Fig. 8(a) include most of the typical geometrical features in the complex layouts, it is hard for PCI-MIDL to directly obtain the accurate OPC solutions for the complex cases. This is because the prediction capacity of deep learning is always limited by the network architecture and the training data sets. It was shown that most of existing mask optimization methods based on deep learning are difficult to directly achieve the final OPC solutions [15,30]. Therefore, we propose to use the gradient-based algorithm as the post-processing method to further refine the OPC solutions obtained by PCI-MIDL. This method is referred to as PCI-MIDL + SD for short. The general principle is to initialize the mask through the deep learning network, and then apply the SD method to further refine the mask pattern. In particular, the output of PCI-MIDL is obtained as the initial guess of OPC solution. Then, we use the initial OPC solution as the initial mask, and run the SD algorithm for 40 iterations to further refine the OPC pattern and improve the image fidelity.

 figure: Fig. 9.

Fig. 9. Simulation results of SD and PCI-MIDL methods based on complex testing layout patterns.

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The simulation results of PCI-MIDL + SD method based on complex layout patterns are illustrated in Fig. 10. It is shown that PCI-MIDL + SD method can generate sufficient SRAFs on the masks, and achieve higher image fidelity compared to the SD method. Table 2 lists the average runtimes of the different OPC methods over the three testing layouts. It is noted that the runtime of PCI-MIDL + SD method is composed of three parts, i.e., the training time, test time, and the iteration time of the mask refine process. Since the network training is only performed once, the mask optimization time of PCI-MIDL + SD method is the sum of testing time and iteration time. We evaluate the computational efficiency of the three OPC methods based on the mask optimization time. The PCI-MIDL is the fastest, but the resulting PEs and RMSEs are much higher than the other two methods. Compared with the SD method, the PCI-MILD + SD method improves the computational efficiency by 2.2 fold, and reduces the PEs and RMSEs by about 14% and 7% on average, respectively. Therefore, the PCI-MIDL + SD method can effectively improve the computational efficiency and the imaging performance simultaneously.

 figure: Fig. 10.

Fig. 10. Simulation results of PCI-MIDL + SD method based on complex layout patterns.

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Tables Icon

Table 2. Average runtimes of different OPC methods based on complex layout patterns.

It is above simulations, the depths of networks for the simple and complex layouts are 5 and 19, respectively. It is observed that the shallow PCI-MIDL network is adequate to learn the mapping relationship between the target layouts and OPC solutions for the simple layouts. However, a deeper network is required to learn the geometrical environments and feature characteristics in the complex layouts. It is known that deeper networks may bring better results, but may also lead to overfitting problem and vanishing gradient problem. The number of network layers is usually regarded as a hyper-parameter in deep learning approaches. In this work, we properly adjust the number of layers to avoid the overfitting problem and achieve the promising prediction capacity. As the result, the prediction capacity of the PCI-MIDL has been proved by the simulations mentioned above. In order to overcome the vanishing gradient problem, the proposed PCI-MIDL uses two channels to optimize the MFs and SRAFs, respectively. In addition, each channel includes multiple branches as shown in Fig. 4(a). An important merit of this network structure is to propagate the residual error back through multiple paths during the training process, thus effectively alleviate the vanishing gradient problem.

Our current simulations do not take into account the near-field calculation introduced by the thick mask effect. It was shown that the thick mask effect can be involved in the lithography imaging model by using the convolution-based compact model of three-dimensional (3D) mask [31,32]. Then, the traditional SD method and the proposed PCI-MIDL method can be generalized to solve for the optimization problem of 3D mask patterns. Since the proposed deep learning approach aims at accelerating the iteration process of SD method, it can effectively improve the computational efficiency even if the mask near-field calculation is taken into account. However, the consideration of 3D mask effect falls outside the scope of this paper, and is a topic for future work.

The proposed PCI-MIDL method can be combined with source optimization to establish an efficient source-mask-optimization (SMO) method, which is another widely used computational lithography technique. Due to the length limit of this paper, the SMO approach based on model-informed deep learning will be studied in our future work. In the future, more realistic factors will be considered in the PCI-MIDL model, including the optical aberrations, defocus, and mask manufacturability.

6. Conclusion

This paper developed a PCI-MIDL method to effectively improve the efficiency and imaging performance of computational lithography in partially coherent lithography system. First, the PCI-MIDL network structure was derived based on the inverse mask optimization model. The dual-channel feature was introduced to simultaneously predict MFs and SRAFs, and to reduce the vanishing gradient problem. In addition, an unsupervised training method was proposed based on the partially coherent lithography imaging model. In particular, the ACA imaging model was used to establish the PCI-MIDL framework to reduce the complexity of network structure. The FSE imaging model was used as the decoder to train the network, which benefits to improve the accuracy and stability of training process. The superiority of proposed methods was verified by simulations on different kinds of layout patterns. In the future, we will study new kinds of neural networks to improve the mask manufacturability, and to improve the robustness of OPC solutions with respect to the process variations.

Funding

National Natural Science Foundation of China (61675021); Fundamental Research Funds for the Central Universities (2018CX01025, 2020CX02002).

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

References

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30. C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020). [CrossRef]  

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References

  • View by:

  1. G. E. Moore, “Cramming more components onto integrated circuits,” Proc. IEEE 36(1), 82–85 (1998).
    [Crossref]
  2. X. Ma and G. R. Arce, Computational Lithography (John Wiley and Sons, 2010).
  3. F. Schellenberg, “A little light magic,” IEEE Spectrum 40(9), 34–39 (2003).
    [Crossref]
  4. A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001).
  5. A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. on Image Process. 16(3), 774–788 (2007).
    [Crossref]
  6. J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(22), 23331–23342 (2010).
    [Crossref]
  7. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolithogr., MEMS, MOEMS 5(4), 043002 (2006).
    [Crossref]
  8. X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19(3), 2165–2180 (2011).
    [Crossref]
  9. 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(7), 1300–1312 (2012).
    [Crossref]
  10. S. Choi, S. Shim, and Y. Shin, “Machine learning (ML)-guided OPC using basis functions of polar Fourier transform,” Proc. SPIE 9780, 97800H (2016).
    [Crossref]
  11. R. Luo, “Optical proximity correction using a multilayer perceptron neural network,” J. Opt. 15(7), 075708 (2013).
    [Crossref]
  12. K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
    [Crossref]
  13. H. Yang, S. Li, Y. Ma, B. Yu, and E. Young, “GAN-OPC: mask optimization with lithography-guided generative adversarial nets,” in Proceedings of IEEE Conference on Design Automation Conference, 1–6 (2018).
  14. Y. Zhang and W. Ye, “Deep learning based inverse method for layout design,” https://arxiv.org/abs/1806.03182 (2018).
  15. S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
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  16. X. Ma, Q. Zhao, H. Zhang, Z. Wang, and G. R. Arce, “Model-driven convolution neural network for inverse lithography,” Opt. Express 26(25), 32565–32584 (2018).
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  17. X. Ma, X. Zheng, and G. R. Arce, “Fast inverse lithography based on dual-channel model-driven deep learning,” Opt. Express 28(14), 20404–20421 (2020).
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  22. X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50(28), 5567–5576 (2011).
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  24. 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(1), 112–123 (2013).
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  30. C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
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2020 (2)

X. Ma, X. Zheng, and G. R. Arce, “Fast inverse lithography based on dual-channel model-driven deep learning,” Opt. Express 28(14), 20404–20421 (2020).
[Crossref]

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

2019 (2)

X. Su, P. Gao, Y. Wei, and W. Shi, “SRAF rule extraction and insertion based on inverse lithography technology,” Proc. SPIE 10961, 23 (2019).
[Crossref]

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

2018 (2)

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
[Crossref]

X. Ma, Q. Zhao, H. Zhang, Z. Wang, and G. R. Arce, “Model-driven convolution neural network for inverse lithography,” Opt. Express 26(25), 32565–32584 (2018).
[Crossref]

2017 (1)

X. Ma, S. Jiang, J. Wang, B. Wu, Z. Song, and Y. Li, “A fast and manufacture-friendly optical proximity correction based on machine learning,” Microelectron. Eng. 168, 15–26 (2017).
[Crossref]

2016 (1)

S. Choi, S. Shim, and Y. Shin, “Machine learning (ML)-guided OPC using basis functions of polar Fourier transform,” Proc. SPIE 9780, 97800H (2016).
[Crossref]

2013 (3)

R. Luo, “Optical proximity correction using a multilayer perceptron neural network,” J. Opt. 15(7), 075708 (2013).
[Crossref]

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(1), 112–123 (2013).
[Crossref]

P. Liu, X. Xie, W. Liu, and K. Gronlund, “Fast 3D thick mask model for full-chip EUVL simulations,” Proc. SPIE 8679, 86790W (2013).
[Crossref]

2012 (1)

2011 (3)

2010 (2)

Y. Shen, N. Wong, and E. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 77481U (2010).
[Crossref]

J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(22), 23331–23342 (2010).
[Crossref]

2008 (1)

2007 (1)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. on Image Process. 16(3), 774–788 (2007).
[Crossref]

2006 (1)

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolithogr., MEMS, MOEMS 5(4), 043002 (2006).
[Crossref]

2003 (1)

F. Schellenberg, “A little light magic,” IEEE Spectrum 40(9), 34–39 (2003).
[Crossref]

1998 (1)

G. E. Moore, “Cramming more components onto integrated circuits,” Proc. IEEE 36(1), 82–85 (1998).
[Crossref]

1996 (1)

1994 (1)

1982 (2)

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982).
[Crossref]

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29(12), 1828–1836 (1982).
[Crossref]

1951 (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A 208(1093), 263–277 (1951).
[Crossref]

Adam, K.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Arce, G.

Arce, G. R.

Bang, N.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Baron, S.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Boone, R.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Burbine, A.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Chang, J.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Choi, S.

S. Choi, S. Shim, and Y. Shin, “Machine learning (ML)-guided OPC using basis functions of polar Fourier transform,” Proc. SPIE 9780, 97800H (2016).
[Crossref]

Chung, N.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Dong, L.

Fenger, G.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Ganjugunte, S.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Gao, P.

X. Su, P. Gao, Y. Wei, and W. Shi, “SRAF rule extraction and insertion based on inverse lithography technology,” Proc. SPIE 10961, 23 (2019).
[Crossref]

Granik, Y.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolithogr., MEMS, MOEMS 5(4), 043002 (2006).
[Crossref]

Gronlund, K.

P. Liu, X. Xie, W. Liu, and K. Gronlund, “Fast 3D thick mask model for full-chip EUVL simulations,” Proc. SPIE 8679, 86790W (2013).
[Crossref]

Gupta, R.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Han, C.

Hopkins, H. H.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A 208(1093), 263–277 (1951).
[Crossref]

Jiang, S.

X. Ma, S. Jiang, J. Wang, B. Wu, Z. Song, and Y. Li, “A fast and manufacture-friendly optical proximity correction based on machine learning,” Microelectron. Eng. 168, 15–26 (2017).
[Crossref]

Kailath, T.

Kim, C.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Kim, J.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Kim, M.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Kim, Y.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Lam, E.

Y. Shen, N. Wong, and E. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 77481U (2010).
[Crossref]

Lam, M.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Lan, S.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
[Crossref]

Lee, S.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29(12), 1828–1836 (1982).
[Crossref]

Li, D.

D. Li and D. Yu, Deep Learning: Methods and Applications (Now Publishers2014).

Li, J.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
[Crossref]

Li, P.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Li, S.

H. Yang, S. Li, Y. Ma, B. Yu, and E. Young, “GAN-OPC: mask optimization with lithography-guided generative adversarial nets,” in Proceedings of IEEE Conference on Design Automation Conference, 1–6 (2018).

Li, Y.

Liu, J.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
[Crossref]

Liu, P.

P. Liu, X. Xie, W. Liu, and K. Gronlund, “Fast 3D thick mask model for full-chip EUVL simulations,” Proc. SPIE 8679, 86790W (2013).
[Crossref]

P. Liu, “Accurate prediction of 3D mask topography induced best focus variation in full-chip photolithography applications,” Proc. SPIE 8166, 38 (2011).
[Crossref]

Liu, W.

P. Liu, X. Xie, W. Liu, and K. Gronlund, “Fast 3D thick mask model for full-chip EUVL simulations,” Proc. SPIE 8679, 86790W (2013).
[Crossref]

Luo, R.

R. Luo, “Optical proximity correction using a multilayer perceptron neural network,” J. Opt. 15(7), 075708 (2013).
[Crossref]

Ma, X.

Ma, Y.

H. Yang, S. Li, Y. Ma, B. Yu, and E. Young, “GAN-OPC: mask optimization with lithography-guided generative adversarial nets,” in Proceedings of IEEE Conference on Design Automation Conference, 1–6 (2018).

Milanfar, P.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. on Image Process. 16(3), 774–788 (2007).
[Crossref]

Moore, G. E.

G. E. Moore, “Cramming more components onto integrated circuits,” Proc. IEEE 36(1), 82–85 (1998).
[Crossref]

Moyroud, C.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Oh, M.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Park, S.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Pati, V.

Poonawala, A.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. on Image Process. 16(3), 774–788 (2007).
[Crossref]

Rabbani, M.

Rosen, J.

Saleh, B. E. A.

Salik, B.

Schellenberg, F.

F. Schellenberg, “A little light magic,” IEEE Spectrum 40(9), 34–39 (2003).
[Crossref]

Shchehlik, K.

K. Adam, S. Ganjugunte, C. Moyroud, K. Shchehlik, M. Lam, A. Burbine, G. Fenger, and Y. Granik, “Using machine learning in the physical modeling of lithographic processes,” Proc. SPIE 10962, 14 (2019).
[Crossref]

Shen, Y.

Y. Shen, N. Wong, and E. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 77481U (2010).
[Crossref]

Shi, W.

X. Su, P. Gao, Y. Wei, and W. Shi, “SRAF rule extraction and insertion based on inverse lithography technology,” Proc. SPIE 10961, 23 (2019).
[Crossref]

Shim, S.

S. Choi, S. Shim, and Y. Shin, “Machine learning (ML)-guided OPC using basis functions of polar Fourier transform,” Proc. SPIE 9780, 97800H (2016).
[Crossref]

Shin, Y.

S. Choi, S. Shim, and Y. Shin, “Machine learning (ML)-guided OPC using basis functions of polar Fourier transform,” Proc. SPIE 9780, 97800H (2016).
[Crossref]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29(12), 1828–1836 (1982).
[Crossref]

Song, Z.

X. Ma, S. Jiang, J. Wang, B. Wu, Z. Song, and Y. Li, “A fast and manufacture-friendly optical proximity correction based on machine learning,” Microelectron. Eng. 168, 15–26 (2017).
[Crossref]

Su, X.

X. Su, P. Gao, Y. Wei, and W. Shi, “SRAF rule extraction and insertion based on inverse lithography technology,” Proc. SPIE 10961, 23 (2019).
[Crossref]

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29(12), 1828–1836 (1982).
[Crossref]

Wang, J.

X. Ma, S. Jiang, J. Wang, B. Wu, Z. Song, and Y. Li, “A fast and manufacture-friendly optical proximity correction based on machine learning,” Microelectron. Eng. 168, 15–26 (2017).
[Crossref]

Wang, Y.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
[Crossref]

Wang, Z.

Wei, Y.

X. Su, P. Gao, Y. Wei, and W. Shi, “SRAF rule extraction and insertion based on inverse lithography technology,” Proc. SPIE 10961, 23 (2019).
[Crossref]

Wong, A. K.

A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001).

Wong, N.

Y. Shen, N. Wong, and E. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 77481U (2010).
[Crossref]

Wu, B.

X. Ma, S. Jiang, J. Wang, B. Wu, Z. Song, and Y. Li, “A fast and manufacture-friendly optical proximity correction based on machine learning,” Microelectron. Eng. 168, 15–26 (2017).
[Crossref]

Xie, X.

P. Liu, X. Xie, W. Liu, and K. Gronlund, “Fast 3D thick mask model for full-chip EUVL simulations,” Proc. SPIE 8679, 86790W (2013).
[Crossref]

Yang, H.

H. Yang, S. Li, Y. Ma, B. Yu, and E. Young, “GAN-OPC: mask optimization with lithography-guided generative adversarial nets,” in Proceedings of IEEE Conference on Design Automation Conference, 1–6 (2018).

Yariv, A.

Ye, J.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Ye, W.

Y. Zhang and W. Ye, “Deep learning based inverse method for layout design,” https://arxiv.org/abs/1806.03182 (2018).

Young, E.

H. Yang, S. Li, Y. Ma, B. Yu, and E. Young, “GAN-OPC: mask optimization with lithography-guided generative adversarial nets,” in Proceedings of IEEE Conference on Design Automation Conference, 1–6 (2018).

Yu, B.

H. Yang, S. Li, Y. Ma, B. Yu, and E. Young, “GAN-OPC: mask optimization with lithography-guided generative adversarial nets,” in Proceedings of IEEE Conference on Design Automation Conference, 1–6 (2018).

Yu, D.

D. Li and D. Yu, Deep Learning: Methods and Applications (Now Publishers2014).

Yu, J.

Yu, P.

Zhang, H.

Zhang, Y.

Y. Zhang and W. Ye, “Deep learning based inverse method for layout design,” https://arxiv.org/abs/1806.03182 (2018).

Zhao, K.

S. Lan, J. Liu, Y. Wang, K. Zhao, and J. Li, “Deep learning assisted fast mask optimization,” Proc. SPIE 10587, 17 (2018).
[Crossref]

Zhao, Q.

Zheng, X.

Zhou, X.

C. Kim, S. Lee, S. Park, N. Chung, J. Kim, N. Bang, S. Lee, S. Lee, R. Boone, P. Li, J. Chang, X. Zhou, Y. Kim, M. Oh, M. Kim, R. Gupta, J. Ye, and S. Baron, “Machine learning techniques for OPC improvement at the sub-5 nm node,” Proc. SPIE 11323, 1132317 (2020).
[Crossref]

Appl. Opt. (2)

IEEE Spectrum (1)

F. Schellenberg, “A little light magic,” IEEE Spectrum 40(9), 34–39 (2003).
[Crossref]

IEEE Trans. Electron Devices (1)

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29(12), 1828–1836 (1982).
[Crossref]

IEEE Trans. on Image Process. (1)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. on Image Process. 16(3), 774–788 (2007).
[Crossref]

J. Micro/Nanolithogr., MEMS, MOEMS (1)

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolithogr., MEMS, MOEMS 5(4), 043002 (2006).
[Crossref]

J. Opt. (1)

R. Luo, “Optical proximity correction using a multilayer perceptron neural network,” J. Opt. 15(7), 075708 (2013).
[Crossref]

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Supplementary Material (1)

NameDescription
Supplement 1       The detailed calculation process of Eqs. (35) and (36) in the primary manuscript.

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Figures (10)

Fig. 1.
Fig. 1. Diagram of PCI lithography system, where the commonly used partially coherent illuminations include circular, annular, dipole, quadrupole illuminations and so on.
Fig. 2.
Fig. 2. The imaging process of PCI lithography system.
Fig. 3.
Fig. 3. The flowchart of gradient-based inverse mask optimization algorithm.
Fig. 4.
Fig. 4. (a) The structure of the PCI-MIDL network (the encoder), and (b) the decoder used to train the PCI-MIDL network. Channel 1 and Channel 2 are used to predict the solutions of SRAF pattern and MF pattern, respectively.
Fig. 5.
Fig. 5. The simple training layouts and testing layouts for the PCI-MIDL network.
Fig. 6.
Fig. 6. Examples of (a) the initial MF pattern (white regions), and (b) the initial SRAF pattern (shadow regions).
Fig. 7.
Fig. 7. Simulation results of SD and PCI-MIDL methods based on simple layout patterns.
Fig. 8.
Fig. 8. The complex training layouts and testing layouts for the PCI-MIDL network.
Fig. 9.
Fig. 9. Simulation results of SD and PCI-MIDL methods based on complex testing layout patterns.
Fig. 10.
Fig. 10. Simulation results of PCI-MIDL + SD method based on complex layout patterns.

Tables (2)

Tables Icon

Table 1. Average runtimes of different OPC methods based on simple layout patterns.

Tables Icon

Table 2. Average runtimes of different OPC methods based on complex layout patterns.

Equations (43)

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M b = Γ M g t m } =  Γ { ( M m + M s ) t m ,
M b = sig m ( M g , t m ) = 1 1 + exp [ a m ( M m + M s t m ) ] ,
I ( r ) = M b ( r ) M b ( r ~ ) γ ( r , r ~ ) h ( r ~ , r ) h ( r , r ) d r d r ~ ,
γ ( r ) = m Γ m exp ( j ω 0 m r ) , Γ m = 1 D 2 A γ γ ( r ) exp ( j ω 0 m r ) d r ,
I = m Γ m | M b h m | 2 ,
h m = h exp ( j ω 0 m r ) ,
h = J 1 ( 2 π r N A / λ ) 2 π r N A / λ .
Z = Γ { I t r } ,
Z sig r ( I , t r ) = 1 1 + exp [ a r ( I t r ) ] ,
I = | M b h C | 2  +  | M b 2 h I 2 | ,
h C ( r , r ) = f ( r , r ) 1 / 2 h ( r , r )  and  h I ( r , r ) = [ 1 f ( r , r ) ] 1 / 2 h ( r , r ) .
f ( r , r )  =  | h ( r , r ^ ) | 2 μ ( r , r ^ ) d r ^ | h ( r , r ~ ) | 2 d r ~ and μ ( r , r ^ )  =  γ ( r , r ^ ) [ γ ( r , r ) γ ( r ^ , r ^ ) ] 1 / 2 ,
F = | | Z ~ Z | | 2 2 | | Z ~ sig r ( I , t r ) | | 2 2 ,
{ M ^ m , M ^ s } = arg min M m , M s F ,
M m k + 1 = M m k s t e p M F | M m and M s k + 1 = M s k s t e p M F | M s ,
F | M m  =  F | M s = 4 a r a m { [ ( Z ~ Z ) Z ( 1 Z ) ( M b h C ) ] h C M b ( 1 M b ) }
4 a r a m { [ ( Z ~ Z ) Z ( 1 Z ) ( h I ) 2 ] M b 2 ( 1 M b ) } ,
S m = S s = δ ( x , y ) ,   D m = D s = h C / | | h C | | 2 ,   T = ( Z ~ Z ) Z ( 1 Z ) ,
W m = W s = h C / | | h C | | 2 ,   B = M b ( 1 M b ) ,   P m = P s = ( h I ) 2 / | | ( h I ) 2 | | 2 ,
and Q = 4 a r a m s t e p M ,
M m g k = M m b k S m k + [ T k ( M b k D m k ) ] W m k B k Q + M b k ( T k P m k ) B k Q ,
M s g k = M s b k S s k + [ T k ( M b k D s k ) ] W s k B k Q + M b k ( T k P s k ) B k Q ,
M g k = M s g k + M m g k ,
T k = T = ( Z ~ Z ) Z ( 1 Z ) .
M m b k + 1 = sig m ( M m g k , t m ) = 1 1 + exp [ a m ( M m g k t m ) ] ,
M s b k  +  1 = sig m ( M s g k , t m ) = 1 1 + exp [ a m ( M s g k t m ) ] ,
M b k  +  1 = sig m ( M g k , t m ) = 1 1 + exp [ a m ( M g k t m ) ] .
I = x s , y s [ J ( x s , y s ) p = x , y , z | ( M b B x s , y s ) H p x s , y s | 2 ] ,
F | M m = F | M s = 4 a r a m x s , y s Real { J ( x s , y s ) p = x , y , z ( Z ~ Z ) Z ( 1 Z ) ( M b B x s , y s )
H p x s , y s ( H p x s , y s ) B x s , y s M b ( 1 M b ) } .
S m = S s = δ ( x , y ) ,   D m x s y s = D s x s y s = ( H p x s , y s ) ,   T = ( Z ~ Z ) Z ( 1 Z ) ,   W m x s y s = W s x s y s = H p x s , y s ,
G m x s , y s = G s x s , y s = B x s , y s ,  and  Q = 4 a r a m s t e p ,
h = 1 π 0 1 ρ e j δ ρ 2 × { 0 2 π e j Φ ( ρ , θ + φ ) × e 2 π j ρ r cos θ d θ } d ρ ,
I = m Γ m | M ^ h m | 2 .
Z sig r ( I , t r ) = 1 1 + exp [ a r ( I t r ) ] .
F = | | Z ~ Z | | 2 2 | | Z ~ sig r ( I , t r ) | | 2 2 .
{ S ^ m k , D ^ m k , W ^ m k , P ^ m k , S ^ s k , D ^ s k , W ^ s k , P ^ s k } = arg min S ^ m k , D ^ m k , W ^ m k , P ^ m k , S ^ s k , D ^ s k , W ^ s k , P ^ s k E ,
E = F  +  γ Q R Q  =  | | Z ~ Z | | 2 2 + γ Q R Q ,
R Q = 1 N × 1 T 4 M ^ ( 1 M ^ ) 1 N × 1 ,
E A m k = E M ^ M ^ M g K ( M m g K M m b K M m b K M m g K 1 + M m g K M b K M b K M m g K 1 + M s g K M b K M b K M m g K 1 ) M m g k A m k ,
E A s k = E M ^ M ^ M g K ( M s g K M s b K M s b K M s g K 1 + M s g K M b K M b K M s g K 1 + M m g K M b K M b K M s g K 1 ) M s g k A s k .
A m k ( n + 1 ) = A m k ( n ) s t e p A E | A m k ,
A s k ( n + 1 ) = A s k ( n ) s t e p A E | A s k ,

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