375 Correspondence An Iterative Cutting Procedure for Determining the Optimal Wafer Exposure Pattern Chen-Fu Chien, Shao-Chung Hsu, and Chih-Ping Chen Abstract—Conventionally, people focus on defect reduction to improve yield rate. Little research has been done to deal with the problem of optimizing wafer exposure patterns. This paper develops a computer- based procedure to maximize the number of dies possibly produced from a wafer. A program has been developed and implemented in a 6-in wafer fabrication factory in Taiwan. The results validate the practical viability of the proposed procedure. Index Terms—Algorithms, cutting, optimization methods, wafer expo- sure, yield optimization. I. INTRODUCTION It is crucial for wafer fabs to compete with others by lowering die cost, because the capital investment and operating costs of wafer fabs are rising significantly. Hence, many studies have focused on defect reduction to increase the yield rate. This study has taken the innovative approach of optimizing the wafer exposure pattern to increase the number of dies produced from a wafer (i.e., gross dies). Also, we have developed an expert system to assist the (inexperienced) engineer in determining the wafer exposure pattern. The present problem is, under certain constraints, to determine the wafer exposure pattern so as to maximize the number of gross dies per wafer (i.e., minimize the waste of unused wafer area). This problem has important industrial and commercial application. Ferris- Prabhu [1] presents an algebraic expression that explicitly relates the die number to the wafer diameter and to the geometric parameters of the die. However, little research has been done to deal with the wafer exposure problem. This problem is a special type of cutting and packing problem, e.g., [2]–[4]. On one hand, because of the following characteristics, this problem may be easier. First, all the exposed areas (i.e., cells) are the same sized rectangles. One cell usually contains multiple dies. Second, the number of exposed cells has no upper bound (i.e., unconstrained). Third, the profit of one die is the same as another (i.e., unweighted). On the other hand, several constraints differentiate this problem from other two-dimensional cutting problems. The shape of a 6-in wafer is circular. Its flat bottom, designed for data recording and alignment, cannot be patterned. In this study, the alignment machine is an ASM wafer stepper, and the two alignment marks that are required for alignment cannot be patterned. Furthermore, because of the requirement of the wafer sawing process, all scribe channels must be aligned in both the horizontal and vertical directions. The borders of the wafer cannot be used to produce good dies. However, an exposure field (cell) contains multiple dies, so when lying partly on Manuscript received September 10, 1998; revised February 22, 1999. This work was supported in part by the National Science Council, R.O.C. under Grant NSC87-2213-E-007-047. C.-F. Chien is with the Department of Industrial Engineering & Engineering Management, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C. (e-mail: cfchien@faculty.nthu.edu.tw). S.-C. Hsu and C.-P. Chen are with the Macronix International Co., Ltd., Hsinchu 30077, Taiwan, R.O.C. Publisher Item Identifier S 0894-6507(99)06376-9. the border it may still produce some dies inside the effective radius. There is obviously a cost-effective ratio to determine whether such a cell should be exposed or not. II. PROCEDURE Instead of constructing a mathematical model to solve this problem exactly, we develop an iterative computational procedure to determine the optimal wafer exposure patterns. Before proceeding with further discussion, we specify the notation used throughout this paper as follows. Number of total iterations. Starting point of the th iteration in area , Length where may shift horizontally. Shifted distance of the -axis coordinate of each iteration. Length of a die. Width of a die. Length of a cell. Note that will be a multiple of . Let . Width of a cell. Note that will be a multiple of . Let . Number of dies exposed within a cell. That is, Diameter of the wafer. Effective radius of the wafer, i.e., —the border width. Left alignment mark on the wafer. Right alignment mark on the wafer. Length of the alignment mark. Width of the alignment mark. Length between the wafer center and the bottom. [] A Gaussian function. Number of cell columns of area one (i.e., the max- imum number of the columns of cells possibly ex- posed between two alignment marks). Cost-effective ratio for determining whether a cell lying partly in the borders should be exposed or not. Thus, the ratio of dividing the number of dies that produced from such a cell by should be no less than . First, we divide the wafer into five areas (the central area in between the two alignment marks and the other four areas of the four wafer corners) as shown on Fig. 1 and specify the criteria for placing the starting points. To reduce the waste area, the starting point of area 1 should be placed as close to the bottom as possible. However, the starting point is allowed to shift horizontally within a length that is the smaller one between one half of the cell width and one half of the remained length by taking columns of cells from the area between two alignment marks, i.e., . Therefore, let where Note that is the shifted distance of the -axis coordinate of each iteration, i.e., . The other four areas should be exposed as close to the area 1 and the two alignment marks as possible to reduce the wasted wafer area. Thus, 0894–6507/99$10.00  1999 IEEE