Machine Learning-Based Resist 3D Model Seongbo Shim†‡ , Suhyeong Choi‡ and Youngsoo Shin‡ † Samsung

‡ School

Electronics, Hwasung 18448, Korea of Electrical Engineering, KAIST, Daejeon 34101, Korea

ABSTRACT Accurate prediction of resist profile has become more important as technology node shrinks. Non-ideal resist profiles due to low image contrast and small depth of focus affect etch resistance and post-etch result. Therefore, accurate prediction of resist profile is important in lithographic hotspot verification. Standard approaches based on a single- or multiple-2D image simulation are not accurate, and rigorous resist simulation is too time consuming to apply to full-chip. We propose a new approach of resist profile modeling through machine learning (ML) technique. A position of interest are characterized by some geometric and optical parameters extracted from surroundings near the position. The parameters are then submitted to an artificial neural network (ANN) that outputs predicted value of resist height. The new resist 3D model is implemented in commercial OPC tool and demonstrated using 10nm technology metal layer.

1. INTRODUCTION Accurate prediction of resist profile is important in lithographic hotspot verification. Non-ideal resist profiles due to low image contrast and small depth of focus, e.g. footing, T-topping, and top-loss (Figure 1), affect etch resistance and post-etch results.1 A standard resist prediction relies on 2D resist model, and a contour image at pre-determined resist height is estimated,2 which cannot distinguish between ideal and non-ideal profiles as shown in Figure 1. Rigorous simulation of resist profile (Figure 2(a)3 ) in full-chip level is computationally very expensive. Alternative straightforward approach is to build individual 2D models at some different image heights (Figure 2(b)). However, the relevant image heights need to be predetermined by engineers themselves; furthermore, the individual 2D models

Resist

CD Substrate (a)

(b)

(c)

(d)

Figure 1. (a) Ideal resist profile; examples of non-ideal profiles with (b) footing, (c) T-topping, and (d) top-loss. Note that all the profiles yield the same 2D contour and bottom CD. Optical Microlithography XXX, edited by Andreas Erdmann, Jungwook Kye, Proc. of SPIE Vol. 10147, 101471D · © 2017 SPIE · CCC code: 0277-786X/17/$18 · doi: 10.1117/12.2257904 Proc. of SPIE Vol. 10147 101471D-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 04/02/2017 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx

validation pattern (2D End -to -End)

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Figure 2. (a) Rigorous resist simulation result and (b) multiple 2D images at some different image heights. !

Artificial neural network (ANN)

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Parameter 2

p2

Parameter n pn Position of interest

Figure 3. ANN-based 3D resist model: a number of parameters are extracted from a layout position, ANN is constructed so that it outputs predicted resist height using input parameters.

may deviate from each other during the separate calibration processes,4 which may result in abrupt change of 2D contours at the adjacent heights. Our approach to resist 3D model is illustrated in Figure 3. At particular position of a layout, a number of parameters (e.g. local densities and optical kernel signals) are extracted by scanning nearby mask patterns. These parameters are submitted to an artificial neural network (ANN), which outputs predicted height of remaining resist after a lithography process. Optimizing ANN so that the difference between predicted- and actual-resist height is minimized is important. This is performed by refining predicted resist height by comparing it to actual resist height that is obtained by a rigorous simulation; the process is repeated for each layout position and using a number of sample layout patterns. The final ANN corresponds to our machine learning-based resist 3D model (ML-R3D model). Due to discontinuity of the predicted resist heights, an image processing such as interpolation is subsequently required to

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Layout of real circuits

DLS

Iso-line

DLS

E2E

E2E (a)

(b)

Figure 4. (a) Traditional synthetic patterns and (b) complicated patterns extracted from real layouts.

reconstruct resist profile. Accuracy and efficiency of our approach are determined by the parameters that are chosen and the structure of ANN that are employed. A few parameter sets, e.g. local densities (Figure 6(a)), optical kernel signals (Figure 6(b)), and their combinations, are tried and their impact on accuracy and efficiency are assessed. ANN structure is optimized by varying the numbers of layers and nodes and assessing the corresponding accuracy achieved from the structure. The remainder of this paper is organized as follows. In Section 2, we describe how to prepare test patterns and extract sample points for ANN training. Section 3 presents how we characterize the sample point and its surroundings using two types of parameters. In Section 4, we present two types of ANN and how we optimize the networks. In Section 5, we show how ML-R3D model performs on some test patterns of metal layer in 10nm technology. In Section 6, we draw some conclusions.

2. DATA PREPARATION Two types of pattern are considered to prepare training data: traditional synthetic patterns and more complicated patterns that are extracted from real layout. The traditional synthetic patterns (see Figure 4(a)) are presented by a few geometrical parameters and can be analyzed to see changes of resist profile in response to changes in the geometry, e.g. a space of DSL patterns. For example, abrupt changes of resist width (or space) in response to a gradual change in a certain geometry parameter imply that trend may imply that there are errors in the data, and that we need to reexamine the experiment. We also consider complex patterns (see Figure 4(b)) in test layouts of real circuits. The test layouts may be prepared by shrinking some actual layouts produced with earlier technologies or synthesizing designs from a preliminary version of the new technology library. Some sample points are then extracted from each test pattern as shown in Figure 5(a). Sampling precision is determined by Nyquist sampling theory in a way that sampling pitch should be smaller than at least a half of the minimum pitch defined by design rule or optical resolution. Note that we only pick sample points that are placed apart from the boundary of the simulation region, so that

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Region of rigorous simulation

Interesting location (sample point)

Region of extracting parametes (surroundings)

(a) p1

Representative sample point Cluster

...

p2 pn (b)

Figure 5. (a) Sample points are extracted from each test pattern, and (b) they are mapped to points in ndimensional parameter space and clustered.

undesired effect due to the simulation boundary can be avoided. Each sample point in the test pattern is then represented by n parameters such as local pattern densities and optical signals, which are extracted from surroundings of the point. The sampled points are mapped to corresponding points in n-dimensional parameter space as shown in Figure 5(b). Some close points are then grouped as a cluster through modified K-mean method.5, 6 A few representative points from each cluster are picked, and their corresponding sample points will be used as training data for constructing our resist 3D model.

3. PARAMETERIZATION Two types of parameter are used to characterize a position of interest and its surroundings. Local pattern densities are the first type of parameter. The local pattern density is measured at points sampled around the interesting position as shown in Figure 6(a). We draw a few concentric circles around the position of interest with some lines passing through the interesting position. At each point where the circles and lines intersect (measurement point), a pattern density within local circular region (region of density measurement) is measured. Since density is measured along the concentric circles, circular propagation of diffracted light from a mask pattern is represented well.7–9 The numbers of the concentric circles and lines that are used determine the number of parameters. This parameter represents geometry information of nearby patterns, which affect acid and base distribution determining remaining resist after develop process. To take account of optical property, we choose an optical kernel signal φnm as the second parameter

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Region of density measurement

Measurment point

Optical kernel

Position of interest (a)

(b)

Figure 6. Parameters: (a) local pattern densities and (b) optical kernel signals.

type, which is given by φnm =

X

L(x, y)Ψnm (x, y),

(1)

∀(x,y)

where L is a pixelated image of the layout, in which the value of a pixel is 1 if it is within a layout polygon, and 0 otherwise. Ψnm is an optical kernel function, centered on an interesting segment as shown Figure 6(b), and expressed as follows: Ψnm (r, ϕ) = Jn (r)cos(mϕ),

(2)

where Jn is nth Bessel function, which is a radial component of the optical kernel function; and the value of n is the number of critical points in the radial direction. The angular component of the optical kernel function is represented by the cosine function, where ϕ is the angle and m is the number of cycles through this angle. The optical kernel function becomes more complicated as n and m increase. Optical kernel signals model constructive or destructive interference of light at the interesting position, and thus are associated with intensity, which affect the amount of acid that are generated in the resist due to exposure. The number of parameters is determined by how many functions we use. Since each function are orthogonal to each other, thus make a separate contribution to the resulting light intensity, we choose only a few functions that make the largest contribution.

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Parameter 1

Weight

Parameter 2

Node

Threshold

...

...

Predicted resist height

Parameter n Input layer (n input nodes)

Output layer (single output node)

Hidden layers (and hidden nodes) (a)

Parameter 1 Parameter 2

...

0 or 1

...

0 or 1

Parameter n Input layer (n input nodes)

Hidden layers (and hidden nodes)

Output layer (Two output nodes)

(b)

Figure 7. (a) ANN for regression and (b) ANN for classification.

4. ANN CONSTRUCTION ANN is a model inspired by biological neural network, which consists of neurons (corresponding to nodes in Figure 7) and synapses (corresponding to edges). Input nodes receive n parameters that are extracted from layout near the interesting position. Their values are propagated to every hidden nodes in the first hidden layer; each edge has a weight, which is multiplied by a propagated signal value. The weighted signals that are received by a hidden node are summed up; if the total is larger than a certain threshold, the hidden node outputs 1, and it outputs 0 otherwise. Thresholding is often approximated by using a sigmoid function h(x), which outputs a floating value between 0 and 1: h(x) =

1 1+

e−(x−xo )

,

(3)

where x is the sum of the weighted signals, and xo is the threshold. Each node in the first hidden layer then propagates its output value to every node in the next layer in the same manner as the nodes in the input layer, and this process is repeated until the output layer is reached. An ANN can be trained to perform regression or classification. We constructed an ANN to perform

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regression, in which the output layer consists of one node, as shown in Figure 7(a). The weighted signals that are received by this output node are summed up, and the result value is the predicted resist height. We also constructed an ANN for classification, with an output layer consisting of two nodes, as shown in Figure 7(b). Each output node corresponds to whether resist remains at the interesting position or not at all. An input point of interest is mapped to a single output node, which returns 1 (or very close value to 1) while another return very small value near 0. We evaluated each ANN over all the training segments, and optimized the edge weights and threshold values to minimize the cost, which is the difference between the predicted and the desired value. In case of ANN for regression, the resist height obtained from a rigorous resist simulation is used as the desired value; and in case of ANN for classification, 0 or 1 is assigned to each node as the desired value. Standard steepest descent method is too time consuming because it requires too many evaluation of ANN for all training data to compute the derivative of the cost function. Therefore, the method of stochastic gradient descent is employed to speed up this minimization. ANN is evaluated for only one training data; the variables (such as weight and threshold) are optimized to minimize the difference between the predicted value (or values) and the desired value of that training data; this process is repeated for next training data that are randomly selected until the iteration reaches user-defined number. Fortunately, the the sigmoid function (3) has a simple derivative, dh(x) = h(x)(1 − h(x)), dx

(4)

the minimization is fast for each training data. The resulting ANN is our ML-R3D model. Since many variables are involved in the network training, the ML-R3D model may become too specific to the training data, which may degrade prediction accuracy for unknown data; this is called over-fitting. As the number of layers and nodes increase, the ANN is more likely to be specific to training data. To avoid this, we should carefully determine the complexity of ANN, i.e. the numbers of hidden layers and nodes. We can adjust them through k-fold cross-validation. The set of training data is randomly divided into k subsets of equal size; k-1 of these sets are used to construct an ANN while the remaining set is used to assess the proportion of correctly predicted data. The accuracy is averaged over k iterations, and the whole iterations are repeated for another ANN with different numbers of nodes and layers. This adjustment procedure is embedded into an optimization that determines the numbers of nodes and layers that give the highest accuracy.

5. EXPERIMENTS Our methods are implemented in Matlab and Proteus10 for parameter extraction and Python to construct the ANN. We extracted 1000 test patterns consisting of 500 synthetic- and 500 complex-patterns in 10nm-node metal layer. Each test pattern is submitted to a rigorous resist simulation,3 which yields distribution of remaining resist height for the simulation area. We then extracted 400 (20 × 20) sample points with 10nm regular intervals from each test pattern, thus 400000 sample points were in turn prepared; 300000 points were used for network training and other 100000 were used for test. For each sample point, we extracted 49 densities using 8 lines with 6 concentric circles and 20 optical kernel

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Table 1. Errors with ANN for regression

Synthetic pattern Complex pattern Overall

Training RMSE (nm) Max error (nm) 4.1 7.7 7.5 11.9 6.1 11.9

RMSE (nm) 6.5 9.3 7.4

test Max error (nm) 11.6 15.1 15.1

Table 2. Prediction accuracy of ANN for binary classification

Synthetic pattern Complex pattern Overall

Accuracy (%) Training test 98.9 97.2 93.1 91.7 96.1 94.5

signals. We performed 10-fold cross-validation of two alternative ANNs. In case of ANN for regression, ANN in turn had 69 input nodes and 5 hidden layers, each of which consists of 7 hidden nodes; thus, 35 thresholds and 686 weights were used as variables in ANN training. Each sample point corresponds to a resist height obtained from the rigorous simulation as its desired value. In case of ANN for classification, ANN had 69 input nodes, two output nodes, and 3 hidden layers, each of which consists of 10 hidden nodes; thus, 32 thresholds and 892 weights were used as variables. Each sample point corresponds to a resist height obtained from the rigorous simulation as its desired value; in case of ANN for classification, each point corresponds to a desired value of 1 or 0 according to the simulation result.

5.1 Accuracy of ML-R3D Model The accuracy of ANN for regression (Figure 7(a)) is evaluated in terms of root mean square error (RMSE) for about 298000 training points and about 99300 test points, respectively. Note that about 2700 points were excluded because the desired resist height cannot be defined as a single value due to T-topping resist profile (see Figure 1(c)). For the training points (see the 2nd column of Table 1), overall RMSE for all training points from both synthetic- and complex patterns is 6.1nm, which is less than 5% of initial resist height. RMSE for the points from the synthetic pattern is 3.4nm smaller than that for the points from the complex pattern due to variety of the complex pattern. For the test points (see the 4th column), overall RMSE slightly increases by 1.3nm (by 2.4nm for the synthetic pattern and by 1.8nm for the complex pattern), which is understandable consequence of performing the cross-validation; the effect of the cross-validation will be analyzed in Section 5.2. We also evaluated the accuracy of ANN for classification (Figure 7(b)) in terms of the prediction

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Table 3. Errors with Different Parameter Sets

Parameter sets

# Parameters

Den only Opt only Den(49)+Opt(20) Den(65)+Opt(50)

49 20 69 115

Accuracy Training 83.2 81.3 96.1 97.2

(%) Test 81.7 79.1 94.5 94.9

accuracy, which is defined as follows: Accuracy =

#Correct × 100 (%), #All

(5)

where #Correct is the number of points where the existence of resist is correctly predicted, and #All indicates the number of all points that are involved in the training or test. Note that, in this time, no sample point was excluded; the points of T-topping resist profile (that were excluded in the previous experiment) were classified in a group of points where resist remains. Overall prediction accuracy for the 300000 training points was 96.1% (98.9% and 93.1% for synthetic- and complex-pattern, respectively) as shown in Table 2. For the test points, the overall prediction accuracy decreases only slightly, which is also due to the cross-validation. Rigorous simulation for one pattern, where 400 sample points reside, in fact took about 30 seconds, thus total simulation time for the 100000 test points turns out to be about 2 hours. However, our approaches took only a few seconds to predict the resist height or whether resist remains for the whole test points.

5.2 Effect of Cross-Validation To investigate the effect of cross-validation, we constructed two more ANNs for regression. One is a simple ANN, which consists of a single hidden layer with 10 nodes. This ANN produced very high RMSEs for both training (14.7nm) and test points (21.8nm) due to its simplicity. Another ANN is very complicated, with 10 hidden layers, each consisting of 15 nodes. Fitting RMSE for the training points decreased by 48% (decreased to 3.2nm), which is understandable consequence of large degrees of freedom in ANN optimization. However, its error for test points is 12.6nm, which is 1.7 times larger than the errors produced by an ANN constructed with the cross-validation (shown in Table 1). We therefore attribute this decline to over-fitting.

5.3 Changing the Parameter Sets We tried four different parameter sets with ANNs for classification. Table 3 shows the accuracy of the ANNs for the training and test points. If only one type of parameter is considered (see the second and third rows), the prediction accuracy is very low for both training and test points. The accuracy is much improved when we consider both density and optical parameters (see the fourth row). We further increased the numbers of density and optical parameters by 16 and 30, respectively (see the last row);

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the ANN in turn had 460 (about 1.5 times) more variables than those of the ANN used in Section 5.1. Despite of such large number of variables, the prediction accuracy increased only slightly, because the contribution of density decreases as its measurement point is located more apart from the interesting position, and only small number of optical kernels highly contribute to light intensity.

6. SUMMARY We have presented a new resist 3D model based on machine learning using ANN. The key components of our approach are preparation of training points, parameterization, and ANN construction. Training points are systematically prepared to ensure sufficient coverage of patterns in real layouts, while the number of the points can be kept as small as possible. An artificial neural network (ANN) is used as the resist 3D model, which is optimized with cross-validation to avoid over-fitting. We have demonstrated our methods using 10nm-node metal layer, and about 95% prediction accuracy were observed. We expect that this new ML-R3D model could be applied to detecting assist feature printing and lithographic defect in full-chip, which could be a topic for further study.

REFERENCES 1. A. N. Samya, R. Seltmanna, F. Kahlenberga, J. Schramm, B. Ku.chlerb, and U. Klostermannb, “Role of 3D photo-resist simulation for advanced technology nodes,” in Proc. SPIE Advanced Lithography, Mar. 2013, pp. 1–9. 2. C. Wu, J. Chang, H. Song, and J. Shiely, “AF printability check with a full chip 3D resist profile model,” in Proc. SPIE Advanced Lithography, Mar. 2013, pp. 1–9. 3. Synopsys, “Sentaurus Lithography,” Dec. 2013. 4. Y. Fan, C. R. Wu, Q. Ren, H. Song, and T. Schmoeller, “Improving 3D resist profile compact modeling by exploiting 3D resist physical mechanisms,” in Proc. SPIE Advanced Lithography, Mar. 2014, pp. 1–11. 5. J. B. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proc. Fifth Berkeley Symp. on Mathematical Statistics and Probability, vol. 1, Dec. 1967, pp. 281–297. 6. R. Ng and J. Han, “CLARANS: a method for clustering objects for spatial data mining,” IEEE Trans. on Knowledge and Data Engineering, vol. 14, no. 5, pp. 1003–1016, Sep. 2002. 7. T. Matsunawa, B. Yu, and D. Z. Pan, “Optical proximity correction with hierarchical Bayes model,” in SPIE Advanced Lithography, Mar 2015, pp. 1–10. 8. X. Xu, T. Matsunawa, S. Nojima, C. Kodama, T. Kotani, and D. Z. Pan, “A machine learning based framework for sub-resolution assist feature generation,” in Proc. Int. Symp. Phys. Des., Apr. 2016, pp. 161–168. 9. S. Shim and Y. Shin, “Etch proximity correction through machine-learning-driven etch bias model,” in Proc. SPIE Advanced Lithography, Mar. 2016, pp. 1–10. 10. Synopsys, “Proteus,” Dec. 2013.

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