Automatic hotspot classification using pattern-based clustering

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1 Automatic hotspot classification using pattern-based clustering Ning Ma *a, Justin Ghan a, Sandipan Mishra a, Costas Spanos a, Kameshwar Poolla a Norma Rodriguez b, Luigi Capodieci b a University of California at Berkeley, CA 947 b Advanced Micro Devices, Sunnyvale, CA 9488 ABSTRACT This paper proposes a new design check system that works in three steps. First, hotspots such as pinching/bridging are recognized in a product layout based on thorough process simulations. Small layout snippets centered on hotspots are clipped from the layout and similarities between these snippets are calculated by computing their overlapping areas. This is accomplished using an efficient, rectangle-based algorithm. The snippet overlapping areas can be weighted by a function derived from the optical parameters of the lithography process. Second, these hotspots are clustered using a hierarchical clustering algorithm. Finally, each cluster is analyzed in order to identify the common cause of failure for all the hotspots in that cluster, and its representative pattern is fed to a pattern-matching tool for detecting similar hotspots in new design layouts. Thus, the long list of hotspots is reduced to a small number of meaningful clusters and a library of characterized hotspot types is produced. This could lead to automated hotspot corrections that exploit the similarities of hotspots occupying the same cluster. Such an application will be the subject of a future publication. Keywords: hotspot classification, DRC, pattern-matching, hierarchical clustering. INTRODUCTION Below the 9nm technology node, printability problems inherent in the process technology have become the major yield detractors of integrated circuit manufacture. Due to increasing process sensitivity to variations, for instance focus and exposure excursions in the optical lithography process, specific layout configurations might become yield detractors, exhibiting bridging, necking, line-end shortening outside a limited process window. These layout configurations can therefore be defined as design hotspots, for a particular RET/OPC (Resolution Enhancement Technique / Optical Proximity Correction) process choice. The design rule checking (DRC) engine was developed to help circuit designers to detect such potential yield detractors by specifying certain geometric and connectivity restrictions which ensure sufficient margins to account for process variations. To continuously achieve high yield and design reliability at the 65nm node and below, DRC has evolved into a complex system that involves a very large number of complex design rules, in addition to so-called recommended and conditional rules [-]. This solution is becoming cumbersome and thus more expensive at sub-45nm nodes for multiple reasons. First, it is becoming increasingly difficult to maintain the ever-expanding design rule library to cover all possible cases. Second, rule enforcement often leads to new problematic geometries. Finally, manual correction of the often large number of hotspots found by DRC is time consuming. This paper proposes a method which classifies design hotspots based on geometric similarity, identifies the common cause of failure of each hotspot class, and stores a compact description of each hotspot class in a library. This library can then be fed to a pattern-matching tool that can detect these types of hotspots in new design layouts. This approach offers several advantages over standard DRC. Generation of the library of hotspot classes is completely automatic and offline, and it can be automatically updated for variant processes or more advanced technology nodes. The library can be embedded into current design tools and function together with standard DRC. Detection of problematic patterns in a layout using pattern-matching is fast. Finally, this approach could lead to automated hotspot corrections that exploit the similarities of hotspots occupying the same cluster. In this paper, the process of generating a library of hotspot classes is described. Section describes the collection of the data from which the library is generated. Section 3 discusses the metric which indicates the similarity between two * ning@eecs.berkeley.edu; U.C. Berkeley Design for Manufacturability through Design-Process Integration II, edited by Vivek K. Singh, Michael L. Rieger, Proc. of SPIE Vol. 695, 6955, (8) X/8/$8 doi:.7/ Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

2 hotspot patterns. Section 4 explains how the hotspot data is clustered and each cluster becomes a class of hotspots with a compact description. Results of clustering and examples of hotspot classes are presented in Section 5. Section 6 presents some concluding remarks and discusses future work.. DATA COLLECTION A large set of hotspots must be analyzed in order to perform the hotspot classification. The accuracy and completeness of the resulting library of hotspot classes depends upon the size and diversity of the set of hotspots used to generate it. For the results in this paper, the hotspots were extracted from a single layer of a pre-opc layout at the 45 nm node, with dimensions 755 µm 6 µm. A full lithography simulation was carried out to obtain the nominal printed contour as well as the process variation bands for focus and exposure. A script was then run to locate and tag various types of hotspots based on the process variation bands. Our results are based on those hotspots violating the minimum width check (MWC). These were divided into two categories: high-severity (HS) for those which fail to meet specification, and low-severity (LS) for printability distortions which are in specification, but could become marginal, due to higher sensitivity to process variation. We used the 78 MWC HS hotspots as our first dataset, and the 3 MWC LS hotspots as our second dataset. For each hotspot found, a clipping tool was used to extract the pattern contained within a square-shaped region centered on the hotspot. These layout snippets were stored and served as the hotspot datasets to be clustered. To determine an appropriate size for the snippets to be used, the complex degree of coherence for the lithography system associated with the layout was examined [3]. The complex degree of coherence, µ, gives the degree of optical interaction between the hotspot center located at (,) and a point on the mask at (x,y). It was found for this particular lithography system that µ(x,y) was less than % for points further than 565 nm from the hotspot center. Therefore a snippet size of.3 µm.3 µm was used. 3. HOTSPOT METRIC In this section, various metrics which describe the distance between two snippets of the same size are defined. The aim of these metrics is to capture the geometric similarity between hotspots based on their local surrounding context. 3. Pattern difference metric Every point of the snippet is either light or dark. If we overlay two snippets then, at every point, the patterns either match (if they are both light or both dark) or differ (if one is light and one is dark). If the snippets have similar patterns, then the total area where they differ will be small. We can then define a metric ρ as the square root of the total area where the two snippets differ: ρ( Θ, Θ ) = d A () Θ Θ The motivation behind this definition of the metric is that it is equivalent to the Euclidean metric, if the snippets were pixilated and represented as vectors of s and s for each pixel. Figure shows an example of the differing area between two snippets. Fig.. Two examples of snippets and, on the right, the area where the two snippets differ Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

3 3. Rotations and reflections A limitation of the metric defined above is that two snippets may be considered far apart with respect to the metric, even if they have very similar patterns except that one is a rotation or a reflection of the other. Assuming that the lithography system treats the horizontal and vertical directions equally (as is the case for most illumination types except off-axis and dipole sources), this is undesirable. Since the two snippets will print similarly and have a similar sensitivity to process variation, they should be considered close together. Therefore, the following pseudometric is defined: P H P _ ( ) ˆ( ρ Θ, Θ ) = min ρ Θ, τ( Θ ), () τ D4 where D 4 is the set of eight transformations (corresponding to the eight symmetries of a square): the four rotations of the pattern, and the four rotations of the reflection of the pattern (see Figure ). Thus, if Θ is a reflection or rotation of Θ, then ˆ( ρ Θ, Θ ) will be zero. (D n is the dihedral group associated with a polygon with n sides.) Fig.. The eight transformations of a snippet 3.3 Weighting functions Snippets of finite size are used in our comparison of hotspots because features which are several wavelengths or more away from the hotspot have negligible effect on the hotspot, so they do not need to be considered. However, features closer to the hotspot have a greater impact than features at the edges of the snippet. Therefore, a more accurate gauge of the similarity between snippets can be obtained by weighting the center of the snippet more heavily than the outside. To do this, a weighting function is chosen which describes the magnitude of the effect of each point of the snippet on the hotspot at the center. The weighting function chosen for our work was the square of the complex degree of coherence, µ, defined above, wxy (, ) = µ ( xy, ). The new weighted metric, ρ w, is then defined as the square root of the total weighted area where the two snippets differ: Then, considering rotations and reflections, we obtain the pseudometric: ρ w ( Θ, Θ ) = wxy (, ) d A. (3) Θ Θ ˆ ρ ( Θ, Θ ) = min ρ Θ, τ( Θ ). (4) w w( ) τ D4 3.4 Implementation The algorithm for computing the distance metrics is presented below and each step will be elaborated after: ) The weighting function is pre-integrated over the domain of the snippet. ) Each snippet pattern is decomposed into a set of disjoint rectangles. 3) The (weighted) area of each snippet is computed. Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

4 4) For each pair of snippets, the (weighted) area of the overlap is computed for each rotation/reflection. 5) The maximum overlap is chosen and used to obtain the value of the distance metric between them. In order to perform clustering, the distance metric needs to be calculated for many pairs of snippets. In fact, for the hierarchical clustering algorithm used in this work, the complete matrix of distances between all pairs of snippets must be computed. Therefore it is necessary to make this calculation as efficient as possible. To calculate the weighted metric Eq. (3), definite integrals of the weighting function must be calculated over various rectangles. However, even rough approximations of the definite integrals are computationally expensive, and calculating them for every pair in a set of thousands of snippets becomes prohibitively slow. This problem can be resolved by pre-integrating the weighting function, as follows. A grid of points is created over the snippet. (For this work, the points were equally spaced by 5 nm in both directions.) For each point (x',y'), the definite integral x y W( x, y ) = w( x, y) dydx (5) is calculated and stored in a look-up table. (Note that the number of definite integrals calculated here is far less than the number which would be calculated if the distance for each pair of snippets was calculated separately.) Then, using this table, definite integrals of the weighting function can be easily calculated over arbitrary rectangles within the snippet, since they can be expressed as a sum/difference of values of W in the look-up table: x y (, ) d d (, ) (, ) (, ) (, ). x wxy y x= W x y W x y + W x y W x y (6) y If the vertices of the rectangle do not coincide with the grid points at which the weighting function was pre-integrated, the values of W can be approximated using linear interpolation of the values at the closest grid points. Next, we need to be able to determine the regions where two snippets differ. Each snippet is stored in memory as a set of polygons, with each polygon being stored as a list of vertex coordinates. Each polygon represents the outside boundary of the dark field portions of the pattern within the snippet, and the remaining regions are light field. In fact, we can equivalently represent each snippet as a set of disjoint rectangles, by decomposing each polygon into rectangles. An example is shown in Figure 3. (This is possible since the polygons in the layout are all right-angled. There are many algorithms to perform such decompositions, such as Gourley and Green s [].) Fig. 3. Decomposition of a snippet into rectangles Suppose snippet Θ is represented by rectangles {A, A,, A m }. The weighted area of each rectangle, by Eq. (6), so we can find the total weighted area of the snippet as m i= A i, can be found Θ = A (7) (where represents weighted area). Next suppose snippet Θ is represented by rectangles {B, B,, B n }, and we want to find the total weighted area where the two snippets differ. First we find the weighted area of the overlap between the rectangles of snippet Θ and the rectangles of snippet Θ (see Figure 4), This is demonstrated diagrammatically in Figure 4. m n i= j= i Θ Θ = Ai Bj. (8) Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

5 HH Fig. 4. Visual explanation of Eq. (8) For each pair (i,j), it can be easily decided from the coordinates of the rectangles whether A i and B j overlap. If they do, the coordinates of the rectangle Ai Bj can be found and the weighted area can be found using Eq. (6). If they do not, Ai Bj =. Finally, the total weighted area where the two snippets differ is given by the XOR of the two sets of rectangles (see Figure 5), This is demonstrated diagrammatically in Figure 5. Θ Θ =Θ +Θ Θ Θ. (9) Fig. 5. Visual explanation of Eq. (9) Then the distance metric between Θ and Θ is w (, ) [ ] () ρ Θ Θ = Θ Θ If we are considering rotations and reflections, then ˆ ρ ( Θ, Θ ) = min ρ Θ, τ( Θ ) w w τ D4 = min Θ + τ( Θ ) Θ τ( Θ ) τ D4 ( ) [ ] = [ Θ + Θ max Θ τ ( Θ )] τ D4 () so the weighted area of the overlap needs to be calculated for all eight transformations of snippet Θ, and the maximum is used in the final distance computation. 4. CLUSTERING Given a set of hotspot patterns and a distance metric, a clustering algorithm is used to form groups of similar hotspot patterns. These clusters are then analyzed to yield a compact description of a class of hotspots which can easily be detected using a pattern-matching tool. 4. Clustering algorithms Clustering algorithms can be classified into two major types: partitioning methods (such as k-means) and hierarchical methods (such as single-link and complete-link). Partitioning methods generally require that the number of desired clusters k must be known in advance, whereas hierarchical methods produces a hierarchical decomposition of the set of objects which can then be divided into any number of clusters. For our hotspots dataset, it is unknown what number of clusters is most suitable before clustering, so hierarchical clustering is used. The complete-link hierarchical algorithm Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

6 was chosen because it results in relatively compact spherical clusters, which proves useful when it comes to patternmatching. 4. Choosing number of clusters After the complete-link hierarchical algorithm has been carried out, it is still necessary to choose the number of clusters. This can be done by examining the quality of the clustering that results from choosing each number of clusters. The quality of the clustering can be quantitatively evaluated and compared based on indices. In this study, we will use two indices, the C-index and the point-biserial correlation, to assess the validity of the clustering results. The C-index is a measure of how well the closest items have been clustered. The C-index is defined as [5] S Min C = Max Min where S is the sum of the all the intra-cluster distances (say there are m of these), Min is the sum of the m smallest distances between all items and Max is the sum of the m largest distances between all items. For a good clustering result, the C-index will be small (ideally zero). The point-biserial correlation [6] is the correlation coefficient between the distance matrix and a matrix P constructed as if Θi, Θj are in the same cluster, P ij = otherwise. A smaller value of the point-biserial correlation indicates a better clustering result. These indices are calculated for each possible value of k, and a value is chosen for which the C-index and the pointbiserial correlation are both small. 4.3 Hotspot class representation and pattern-matching After hotspot clusters are determined, each cluster must be analyzed to produce a description of a class of hotspots. This description should be in form such that a new design layout can be scanned in an efficient manner for patterns in this class. This can be achieved by representing each class of hotspots by a center pattern and a radius, then using a patternmatching tool. Each hotspot class is the set of all patterns whose distance from the cluster center is less than the cluster radius. The cluster center can be seen as a representative pattern of the cluster, which all of the elements of the cluster are similar to. We want the cluster radius to be as small as possible, while encompassing all of the hotspot patterns in the cluster, to minimize the number of false positives. For a cluster S of hotspots, the center can be chosen as the pattern Θ which minimizes the radius: Θ = arg min ( max ˆ ρw( Θ, Θ i) Θ S ), r max ˆ ρw(, i) Θ S Θ S i = Θ Θ. () The center of a class of hotspots can fed as a target pattern to a pattern-matching tool, which can detect similar hotspots in new layouts using fast D image matching [7]. This pattern-matching tool scans over the new layout and computes, for each point of interest, the distance between the local layout geometry and the representative hotspot pattern (with respect to the metrics defined above). The tool can identify and label snippets across the layout that are within a certain distance of the target pattern. By repeating this for all hotspot classes in the library, hotspots of all types will be detected in the layout. Since every hotspot in the same class share similar geometric shapes, it is reasonable to expect that most of the time these hotspots could be fixed in the same way, that is, by applying the same changes to the patterns. If a fixing solution for the representative center pattern of a hotspot class is known, then it should be possible to apply the same solution to all hotspots in this class. Therefore, the classification method described in this paper may lead to a method for automatically fixing hotspots in new design layouts. i Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

7 5. RESULTS Dataset I contains 78 hotspot patterns. For this dataset, we show results from two cases: first using the unweighted distance metric(), and then using the weighted distance metric(4). 5. Dataset I, without weighting Figure 6 shows the histogram of pairwise distances between snippets in the dataset. Two distinct peaks indicate that this dataset is naturally clusterable, with the first peak indicating intra-cluster distances and the second inter-cluster distances. These peaks can help to suggest a natural size for clusters in the data, and thus aid the choice of the number of clusters, k. Figure 7 shows the two indices used to choose k. We choose k = since both indices indicate good results for this point. Figure 8 shows the sizes of the resulting clusters. Figure 9 presents two examples of clusters with (a) showing 4 randomly picked hotspot patterns from the second largest cluster and (b) showing all hotspot patterns from the third largest cluster. Rotations and reflections have been considered to maximize hotspot pattern similarities, which can be seen well in Figure 9(b). In each example, the representative center hotspot is the top left pattern. number of pairs of patterns histogram of distances Point biserial index C-index cluster indices number of clusters distance Fig. 6. Histogram of pairwise distances between snippets in dataset I. Two peaks indicate a good data clusterability number of clusters Fig. 7. Cluster indices. Above: the smallest value of the point-biserial correlation indicates the best clustering. Below: the smallest C-index indicates the best clustering. 6 5 cluster size cluster ID Fig. 8. Cluster sizes. The number of clusters k = and the size of each cluster is shown above. Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

8 #78 (center) #93 #5 #9 # #489 #556 (center) #43 #548 #549 #55 #55 #339 #54 #57 #7 #369 # #553 #554 #555 #557 #558 #56 #34 #4 #335 #37 #9 #7 #73 #74 #75 #76 #77 #78 #33 #88 #96 #38 # #347 #79 #7 #7 (a) (b) Fig. 9. Two examples of clustered hotspots and their representative center hotspots. 5. Dataset I, with weighting Figure shows the histogram of pairwise distances between snippets in the dataset. In this case, two distinct peaks cannot be clearly seen in the histogram, so the point-biserial correlation and the C-index are used to determine the number of clusters, shown in Figure. We choose k = 6 since both indices indicate good results for this point. number of pairs of patterns x 4 histogram of distances distance Fig.. Histogram of pairwise distances between snippets in dataset II. C-index Point bi-serial index cluster indices number of clusters number of clusters Fig.. Cluster indices. Above: the smallest value of the point-biserial correlation indicates the best clustering. Below: the smallest C-index indicates the best clustering. Figure shows the sizes of the resulting clusters. Using the weighted metric has resulted in much more compact clusters than were found in case. Figure 3 presents two examples of clusters with (a) showing 4 randomly picked hotspot patterns from the fourth largest cluster and (b) showing all 8 hotspot patterns from the sixth largest cluster. The representative center hotspots are the top left patterns. Comparing to the corresponding cluster in Figure 9(b), the cluster in Figure 3(b) excludes 3 hotspot patterns that were classified together with the other 8 patterns when the pattern centering area was not weighted more than peripheral features; the resulting cluster is therefore more compact. Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

9 cluster size cluster ID Fig.. Cluster sizes. The number of clusters k = 6 and the size of each cluster is shown above. #367 (center) # #6 #3 #345 #7 #56 (center) #548 #549 #55 #55 #365 #9 #5 #353 #35 #7 #553 #554 #555 #556 #557 # #366 #36 # #364 #34 #558 #75 #76 #77 #78 #355 #359 #357 #363 #3 #6 #79 #7 #7 (a) (b) Fig. 3. Two examples of clustered hotspots and their representative center hotspots. 5.3 Computational complexity Both time and space complexities of our current algorithm are on the order of n, where n is the number of hotspots to be clustered. Table shows the amount of time and space required for distance matrix computation and hotspot clustering on a GHz processor with 4 GB of memory. Table. Time and space requirements Dataset I (78 hotspot patterns) Dataset II (3 hotspot patterns) Without weighting With weighting With weighting Distance matrix Time 8. minutes 9.7 minutes 8.8 hours computation Space.4 MB 4.9 MB.4 GB Clustering Time.7 seconds.67 seconds 3. minutes Space 3.7 MB 3.7 MB. GB Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use:

10 6. CONCLUSIONS Our work proposes a design check system that automatically generates a library hotspot classes from a set of hotspot data. This library is fed to a pattern-matching tool for fast detection of hotspots in new design layouts. Such a system is not only easy to use and handle, but also supplies the process information and knowledge for layout designers. Since the detected hotspots in a same class share similar geometric shapes, it is expected that they can all be fixed by a common fixing solution. Thus the design-check cycle before tape-out can be tremendously shortened. In our current algorithms, the specific manufacturing process has been considered during data collection and analysis by choosing a corresponding weighting function for the distance metric. To further consolidate the resulting hotspot classes and validate automatic fixing of each class, in our future work we will include more information such as layer to layer interactions, electrical path location of each hotspot and performance restrictions for fixing solutions and so on. This information will be incorporated into the distance metric and will thus be taken into account to produce the clusters. The clustering indices used in this paper are not always conclusive in indicating the most appropriate number of clusters, so choosing k can be a partially subjective task. In order to fully automate this step, a more definitive index needs to be used. Alternative weighting functions and clustering algorithms will also be explored with the goal of producing tighter hotspot classes which result in a low number of false positives when used by the pattern-matching tool on a new layout. The results of our work here show that our hotspot classification process is successful in taking a number of hotspots and using the data to automatically generate a set of clusters, each containing geometrically similar hotspots. The next stage is to take the hotspot class representations and compare the results of a pattern-matching tool using these hotspot classes with the results from a simulation-based hotspot detection. This will be explored in future work. ACKNOWLEDGEMENT This work is supported by UC Discovery Grant No. 64. REFERENCES [] [] [3] [4] [5] [6] [7] Cote, M. and Hurat, P., Standard Cell Printability Grading and Hot Spot Detection, 6 th Intl. Symp. on Quality of Electronic Design, (5). Dai, V., Yang, J., Rodriguez, N. and Capodieci, L., DRC Plus: Augmenting Standard DRC with Pattern Matching on D Geometries, Proc. SPIE Int. Soc. Opt. Eng. 65, 65A (7) Wong, A. K., [Resolution Enhancement Techniques in Optical Lithography], SPIE Press, () Gourley, K. D. and Green, D. M., A Polygon-to-Rectangle Conversion Algorithm, IEEE Computer Graphics and Applications 3(), 3-36 (983). Hubert, L. and Schultz, J., Quadratic Assignment as a General Data-Analysis Strategy, Br. J. Math. Stat. Psychol. 9, 9 4 (976) Milligan, G. W., A Monte Carlo Study of Thirty Internal Criterion Measures for Cluster Analysis, Psychometrika 46(), (98) Gennari, F. and Neureuther, A. R., A Pattern Matching System for Linking TCAD and EDA, 5 th Intl. Symp. on Quality of Electronic Design, 65-7 (4) Proc. of SPIE Vol Downloaded From: on /8/6 Terms of Use: