Machine Learning (ML)-Guided OPC Using Basis Functions of Polar Fourier Transform Suhyeong Choia , Seongbo Shimab , and Youngsoo Shina a School
of Electrical Engineering, KAIST, Daejeon 34141, Korea b Samsung Electronics, Hwasung 18448, Korea ABSTRACT
With shrinking feature size, runtime has become a limitation of model-based OPC (MB-OPC). A few machine learning-guided OPC (ML-OPC) have been studied as candidates for next-generation OPC, but they all employ too many parameters (e.g. local densities), which set their own limitations. We propose to use basis functions of polar Fourier transform (PFT) as parameters of ML-OPC. Since PFT functions are orthogonal each other and well reflect light phenomena, the number of parameters can significantly be reduced without loss of OPC accuracy. Experiments demonstrate that our new ML-OPC achieves 80% reduction in OPC time and 35% reduction in the error of predicted mask bias when compared to conventional ML-OPC. Keyword: Optical proximity correction (OPC), polar Fourier transform, ML-OPC
1. INTRODUCTION As layout feature size shrinks down, traditional MB-OPC becomes more time consuming. MB-OPC iterates lithography simulation and mask image correction by inspecting simulation result. A lithography simulation takes longer time due to denser polygons, larger ambit, and more kernels. In addition, smaller feature size requires more iterations of lithography simulations to achieve more accurate OPC result. In metal 1 layer of logic devices in modern technology, for instance, runtime of MB-OPC is about 180 times of that in 40nm technology. Recently, ML-OPC has been proposed as a promising solution to overcome the limitation of MBOPC. In ML-OPC, a segment of interest (and its surroundings) is represented by some parameters, e.g. local pattern densities measured around the segment, which are arranged as a vector as shown in Figure 1(a). The vector becomes an input of ML-OPC model that outputs a desired mask bias of the segment, which is then used to synthesize a mask image as shown in Figure 1(b). ML-OPC model is trained in advance using many test segments, so that a variety of layout patterns can all be corrected. The method of model training and the choice of parameters are important in the accuracy and runtime of ML-OPC. A few existing studies adopt hierarchical Bayesian model (HBM),1 multi-layer perceptron (MLP),2 and support vector regression (SVR)3 for model training. They use local pattern densities or pixel values of rasterized layout as parameters., but the number of parameters is typically very large (e.g. some hundreds), which negatively affects runtime as well as accuracy. We propose to use polar Fourier transform (PFT) signals as parameters of ML-OPC. A PFT signal is obtained by multiplying a PFT basis function with layout image near a segment of interest. Due to orthogonality of PFT basis functions, the PFT signals have little redundancy which allows us to reduce the number of parameters. Reduced number of parameters also helps in accuracy provided that Optical Microlithography XXIX, edited by Andreas Erdmann and Jongwook Kye, Proc. of SPIE Vol. 9780, 97800H · © 2016 SPIE · CCC code: 0277-786X/16/$18 · doi: 10.1117/12.2219073
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Figure 1. (a) Parameterization and (b) mask bias prediction in ML-OPC.
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Figure 2. Parameters of conventional ML-OPC: (a) binary pixel value and (b) local pattern density.
unnecessarily large number of parameters in the conventional approach often causes overfitting that negatively affects accuracy. The remainder of this paper is organized as follows. In Section 2, we review previous works of ML-OPC. Our proposed ML-OPC is presented in Section 3, in which we study PFT signals and their application for new ML-OPC. In Section 4, we conduct experiments to assess new ML-OPC in the number of parameters and accuracy, and to investigate the impact of illumination complexity and design types on new ML-OPC. Section 5 summarizes this paper.
2. PREVIOUS WORKS OF ML-OPC Popular parameters of ML-OPC are pixel values of a rasterized layout. As shown in Figure 2(a), a local layout with its center of the segment of interest is extracted and rasterized; if more than half of pixel area is overlapped with layout, 1 is assigned to the pixel and, otherwise, 0 is assigned. As layout
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Figure 3. Bessel function.
size increases and pixel size decreases, the number of parameters drastically increases; this increases OPC runtime and often causes overfitting due to many unnecessary parameters. Small layout with large pixels, on the other hand, yields small number of parameters, but it causes loss of geometry information, thereby resulting in inaccurate OPC result. Another popular parameters are local pattern densities .1 As shown in Figure 2(b), a few concentric circles with their auxiliary lines are drawn around the segment of interest. At each point where the circles and lines intersect (measurement point), a pattern density within local region (region of density measurement) is measured. The number of parameters is determined by the numbers of the concentric circles and lines that are used. Since measurement positions are rather sparsely distributed, the number of parameters is smaller than that of the pixel value parameters. In both methods, the region where parameters are extracted is usually smaller than optical influence range, a circular region of about 1µm radius. If we expand the region for the benefit of accuracy so that we extract parameters within optical influence range, the number of parameters becomes unrealistic, e.g. 32 times in pixel value parameters and twice in local density parameters.
3. PROPOSED ML-OPC 3.1 PFT Signal PFT decomposes a spatial distribution of light using PFT basis functions. It is popular in modeling optical diffraction and interference. PFT basis function is given by Ψnm (r, ϕ) = Jn (r)cos(mϕ),
(1)
where Jn is n-th Bessel function as shown in Figure 3, which is a radial component of a PFT basis function; n corresponds to the number of critical points along the radial direction. The angular component of PFT basis function is represented by a cosine function∗ , where ϕ is angle and m is the number of periods along the angular direction. PFT basis function becomes more complex as n and m increase as shown in Figure 4. Note that PFT basis functions are orthogonal each other due to the orthogonality of Bessel functions and cosine functions. ∗
A complete form of Bessel function is Ψnm (r, ϕ) = Jn (r)eimϕ , but we only consider a real part in this paper.
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Figure 5. PFT signal
A PFT signal is now given by φnm =
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where L(x, y) is a binary function whose value is 1 if (x, y) is within a layout polygon and 0 otherwise, thereby representing a layout image around the segment of interest; the center of Ψnm is assumed at the segment as shown in Figure 5; note that PFT basis function is now represented in Cartesian coordinate. PFT signal as expressed by (2) is sum of Ψnm (x, y) values within layout polygon region, which is associated with the amount of light interference at the center of the segment due to its surroundings. There are three benefits if we use PFT signals as parameters of ML-OPC. • Efficiency: PFT basis functions are orthogonal. Therefore, there is little redundancy in corresponding PFT signals, implying that smaller number of parameters (compared to using local pattern densities or pixel values) can be used without loss of accuracy. • Ease of implementation: A commercial OPC package4, 5 usually has internal optical simulator, which uses PFT basis functions. This allows us to implement our method on top of commercial OPC tool with ease. • Accuracy: PFT basis functions are widely used to model optical diffraction and interference5, 6 because a light on a wafer is concentrically distributed due to the circular scanner optics. A mask
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bias is determined by optical characteristic, so using PFT basis functions should offer higher accuracy of OPC.
3.2 Construction of ML-OPC Model PFT signals obtained from each segment are arranged as a vector, which is submitted to ML-OPC model. We employ multi-layer perceptron (MLP) to build our ML-OPC model. MLP consists of a few layers containing some nodes (neurons), and nodes in the adjacent layers are fully connected by edges (synapses) as shown in Figure 6. MLP maps the input vector to one output scalar, which is a predicted mask bias. Each element of the vector is assigned to input nodes one by one, and propagated to every node in the next hidden layer. A node in the hidden layer receives multiple signals from every node in previous layer via connected edges; while edge carries a signal, the signal is multiplied by the edge weight. If summation of received signals at a node is larger than some threshold, the node outputs 1, otherwise, it outputs 0. Similarly, the output values from nodes of the first hidden layer are propagated through some subsequent hidden layers to the output node, which in turn has a scalar. Mask bias is more accurately predicted if weights and threshold in each node are appropriately adjusted in advance. Sample segments are extracted from test layouts, and each sample is associated with a vector of PFT signals and predicted mask bias result from MB-OPC. MLP predicts mask bias once the vector is applied, and the difference of mask bias from MB-OPC and ML-OPC is defined as an error. Cost function is a sum of squared errors for all segments used in training process. Gradient descent method is applied to minimize the cost function by adjusting edge weights and threshold; the function is differentiable if thresholding is done by using sigmoid or tanh function. The numbers of layers and neurons in each layer are determined in empirical fashion.
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Figure 7. (a) The number of parameters and (b) RMSE of conventional and new ML-OPC.
4. EXPERIMENTS We implemented our new ML-OPC using commercial OPC package (Progen and Proteus4, 5 ) for PFT signal computation and Python for MLP construction. We assumed ArF immersion lithography (1.2NA) with annular illumination, which corresponds to optical influence range of about 1µm. Our MLP consists of 3 hidden layers, each of which contains 10 nodes. The MLP was trained for 20,000 segments extracted from metal 1 layouts in 20nm logic technology. Other test layouts that contain about 2 million segments were prepared for testing the trained MLP.
4.1 Assessment of New Parameters We also implemented a standard ML-OPC, which uses local pattern densities, as a reference of comparison. It uses 80 local pattern densities as parameters, which can be compared to only 10 parameters in our ML-OPC as illustrated in Figure 7(a). Runtime of MLP construction is reduced by 26% in our ML-OPC due to smaller number of parameters. There is even larger reduction of runtime in actual OPC, by 80%. The accuracy of ML-OPC is assessed by using root mean square error (RMSE) of predicted mask bias, with mask bias from MB-OPC as a reference. As shown in Figure 7(b), new ML-OPC achieves 3.5nm RMSE, which is 1.9nm smaller than that of standard one, even though smaller number of parameters are used. This is due to less overfitting that occurs in our ML-OPC.
4.2 Illumination Complexity Optical proximity effect is affected by illumination types. To investigate the impact of illumination complexity on new ML-OPC, we performed the same experiment assuming cross-pole illumination. As shown in Figure 8, RMSE increases to 3.9nm, because of increased complexity of cross-pole (compared to basic annular), which causes more complex optical proximity effects. We tried to reduce RMSE to 3.5nm, the same quantity if annular illumination is assumed, which required 3 more PFT signals.
4.3 Layer and Device We tried our ML-OPC using metal 1 and contact layouts. Metal 1 causes larger RMSE as shown in Figure 9(a) since its layout is typically more complex than contact layout. For the same reason, our ML-OPC causes larger RMSE in logic devices than in memory devices as shown in Figure 9(b).
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Figure 8. RMSE of new ML-OPC with different illumination type.
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Figure 9. RMSE of new ML-OPC with different design type: layer and device.
5. CONCLUSION In ML-OPC, the choice of parameters is very important in its accuracy as well as runtime. We have proposed to use PFT signals as parameters. Since they are orthogonal while they model light interference very well, the number of parameters can significantly be reduced without loss of accuracy, which has be demonstrated through experiments.
6. ACKNOWLEDGMENT We would like to thank Mr. Junghoe Choi and Mr. Munhoe Do from Synopsys Korea for technical support. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A2A2A01008037).
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REFERENCES 1. T. Matsunawa, B. Yu, and D. Z. Pan, “Optical proximity correction with hierarchical Bayes model,” in SPIE Advanced Lithography, Mar 2015, pp. 1–10. 2. R. Luo, “Optical proximity correction using a multilayer perceptron neural network,” Journal of Optics, vol. 15, no. 7, pp. 075 708–075 713, Jun 2013. 3. K.-S. Luo, Z. Shi, X.-L. Yan, and Z. Geng, “SVM based layout retargeting for fast and regularized inverse lithography,” Journal of Zhejiang University SCIENCE C, vol. 15, no. 5, pp. 390–400, Apr 2014. 4. Synopsys, “Proteus,” Dec. 2013. 5. ——, “Progen,” Dec. 2013. 6. M. Born and E. Wolf, Principles to Optics, 4th ed. Pergamon Press, 1970.
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