True Three-Dimensional Proximity E?ect Correction in Electron-beam Lithography: Control of Critical Dimension and Sidewall Shape by Qing Dai A dissertation submitted to the Graduate Faculty of Auburn University in partial ful?llment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 4, 2013 Keywords: Electron-beam Lithography, Proximity E?ect Correction, Three-dimensional, Resist Pro?le, Critical Dimension, Sidewall Shape Copyright 2013 by Qing Dai Approved by Soo-Young Lee, Professor of Electrical and Computer Engineering Stanley Reeves, Professor of Electrical and Computer Engineering Bogdan Wilamowski, Professor of Electrical and Computer Engineering Abstract One of the major limiting factors in electron-beam (e-beam) lithography is the geometric distortion of written features due to electron scattering, i.e., proximity e?ect, which puts a fundamental limit on the minimum feature size and maximum pattern density that can be realized. A conventional approach to proximity e?ect correction (PEC) is, through two- dimensional (2-D) simulation, to determine the dose distribution and/or shape modi?cation for each feature in a circuit pattern such that the written pattern is as close to the target pattern as possible. For circuit patterns with nanoscale features, it is not unusual that the actual written pattern is substantially di?erent from the written pattern estimated by a 2-D PEC method. One of the reasons for this deviation is that the 2-D model ignores the exposure variation along the resist depth dimension. Earlier, it was shown that three-dimensional (3-D) PEC which considers the variation of exposure along the resist depth dimension would be required for nanoscale features, espe- cially for feature size well below 100 nm. In 3-D PEC, the resist pro?le estimated through simulation of resist development process was employed instead of the exposure distribution, in order to obtain more realistic results. However, It has been demonstrated that in order to minimize any deviation from a target resist pro?le, a 3-D PEC scheme must check the estimated resist pro?le during the dose optimization procedure. One practical issue of such an approach to 3-D PEC is that a time-consuming resist development simulation needs to be carried out in each iteration of the dose optimization for each feature. For the case of large-scale circuit patterns, such a feature-by-feature correction procedure would be too time-consuming to be practical. Also, it has been shown that the dose distribution of ?V-shape?, which is used by most 2-D PEC ii schemes, is not optimal for realizing a vertical sidewall of the resist pro?le, especially when the total dose is to be minimized. In this dissertation, the characteristics of our 3-D exposure model are further analyzed and utilized for accurate estimation of resist pro?les. Our true 3-D PEC method is also extended to applications for large-scale circuit patterns. A variety of optimizations are developed in order to improve the correction e?ciency, such as path-based resist development simulation, critical-location-based dose control, etc. New types of dose distributions are derived under our 3-D exposure model and 3-D PEC method for achieving better sidewall shapes with total dose being minimized. For improving the correction e?ciency without sacri?cing the correction quality, a dose optimization scheme with a systematic type-based dose updating procedure is developed which is adaptable to di?erent dose distribution types. A dose determination scheme is also developed which can adaptively determine the optimal dose distribution type and the minimum of required total dose based on a given circuit pattern and substrate system settings. The results from extensive simulation along with experiments are provided for performance analysis. iii Acknowledgments First of all, I would like to express my gratitude to my advisor Dr. Soo-Young Lee for leading me into this exciting research area of proximity e?ect correction in electron- beam lithography and providing me with valuable support throughout my doctoral program. Without his detailed guidance and helpful suggestions this dissertation would not have been possible. I thank him for his patience and encouragement that carried me on through di?cult times. I appreciate his vast knowledge and skills in many areas that greatly contributed to my research and this dissertation. Special thanks are also given to the other members of my committee, Dr. Stanley Reeves and Dr. Bogdan Wilamowski, for their valuable assistance and comments. I also bene?ted a lot from the courses taught by them. I would like to extend my gratitude to former and current members of our research group. I would like to thank Pengcheng Li, Xinyu Zhao, and Rui Guo for our memorable collaborations in learning and improving PYRAMID. Thanks are also due to Samsung Elec- tronics Co., Ltd. for funding this research. Finally, I am greatly indebted to my parents, Xichao Dai and Xuyan Xie, for their support to me through my entire life, and in particular to my wife, Rong Zhang, for her love, patience, and understanding. iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Problem De?nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Dose Modi?cation PEC Techniques . . . . . . . . . . . . . . . . . . . 3 1.2.2 Shape Modi?cation PEC Techniques . . . . . . . . . . . . . . . . . . 10 1.2.3 Distinctive PEC Techniques . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.4 Hybrid PEC Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Essential Models in Electron-beam Lithography . . . . . . . . . . . . . . . . . . 21 2.1 Electron-beam Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Point Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Exposure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Development Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Exposure-to-rate Conversion Formula . . . . . . . . . . . . . . . . . . 27 2.4.2 Cell Removal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.3 Proposed Path-based Method . . . . . . . . . . . . . . . . . . . . . . 29 2.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 36 v 3 Estimation of Resist Pro?le for Line/Space Patterns . . . . . . . . . . . . . . . . 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Conventional Estimation Approach . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Proposed Estimation Approach . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.1 Base Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.2 2-D Exposure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.3 Layer-based Exposure Model . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.1 Line/Space Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.3 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 Critical Dimension Control for Large-scale Uniform Patterns . . . . . . . . . . . 62 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Large-scale Uniform Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Correction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 2-D Exposure Correction (2-D PEC) . . . . . . . . . . . . . . . . . . 64 4.3.2 3-D Resist Pro?le Correction (3-D PEC) . . . . . . . . . . . . . . . . 65 4.4 Proposed Correction Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4.1 Correction of a Single Feature . . . . . . . . . . . . . . . . . . . . . . 67 4.4.2 Feature-wise Global Adjustment . . . . . . . . . . . . . . . . . . . . . 67 4.4.3 Critical-location-based Dose Control . . . . . . . . . . . . . . . . . . 71 4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.1 Test Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vi 4.5.3 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5 Sidewall Shape Control: Vertical Sidewall . . . . . . . . . . . . . . . . . . . . . 82 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2 Dose Distribution Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.1 Conventional Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.2 Proposed Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 Dose Optimization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.1 Fast Exposure Computation . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.2 Critical-path-based Resist Development Simulation . . . . . . . . . . 93 5.3.3 Type-based Dose Updating . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4 Dose Determination Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 Sidewall Shape Control for Large-scale Uniform Patterns . . . . . . . . . . . 100 5.6 Sidewall Shape Control for Large-scale Nonuniform Patterns . . . . . . . . . 103 5.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.7.1 Performance Comparison for Single Features . . . . . . . . . . . . . . 106 5.7.2 Performance Comparison for Large-scale Uniform Patterns . . . . . . 114 5.7.3 Performance Comparison for Large-scale Nonuniform Patterns . . . . 126 5.7.4 Dependency on Feature and Lithographic Parameters . . . . . . . . . 139 5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 vii List of Figures 2.1 Illustration of the e-beam binary lithographic process. . . . . . . . . . . . . . . 22 2.2 PSF for the substrate system of 300 nm PMMA on Si with the beam energy of 50 KeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Monte Carlo simulation of (a) electron paths (cross-section) and (b) energy dis- tribution (cross-section). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Illustration of the substrate system where H is the initial thickness of resist. . . 25 2.5 Illustration of the remaining resist pro?le after the resist development process. . 26 2.6 The experiment-based nonlinear exposure-to-rate conversion formula. . . . . . . 27 2.7 Illustration of the cell removal method where the dashed curve represents a cross- section of resist pro?le. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 Illustration of the path-based method where the dashed curve represents a cross- section of resist pro?le. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9 A path turns to (a) only the positive direction of the X-axis at the point (x;y;z) if r(x;y;z) > r(x + ?x;y;z) but r(x;y;z) ? r(x ? ?x;y;z) and (b) both the positive and negative directions of the X-axis at the point (x;y;z) if r(x;y;z) > r(x+ ?x;y;z) and r(x;y;z) > r(x? ?x;y;z). . . . . . . . . . . . . . . . . . . . 30 viii 2.10 The development paths with one and two turns where the black and gray lines are the vertical and lateral path segments, respectively, and the dashed curve represents a cross-section of resist pro?le. The Z-axis corresponds to the resist depth dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.11 The computation of each path terminates when the sum of the time spent on its path segments is equal to the given developing time T. . . . . . . . . . . . . . . 32 2.12 (a) All the points in the resist are marked as ?undeveloped? before the resist development simulation. (b) After one path is computed, the points which it passes through are marked as ?developed?. (c) After another path is computed, more points are marked as ?developed?. (d) The ?nal resist pro?le is determined by tracing the boundaries between those developed points and those which are not. 33 2.13 Flowchart of the computation of a two-turn path, where ?tx, ?ty, and ?tz (?tx+?ty+?tz = T) are the time spent on the path segment in the X-dimension, Y-dimension, and Z-dimension, respectively, and p is the vector representation of the path in the 3-D space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.14 The layout of (a) Pattern I and (b) Pattern II. . . . . . . . . . . . . . . . . . . 37 2.15 Contours (top view) of resist pro?les for Pattern I at the (a) top, (c) middle, and (e) bottom layers of resist by the cell removal method, and at the (b) top, (d) middle, and (f) bottom layers of resist by the path-based method. . . . . . . . . 38 2.16 Contours (top view) of resist pro?les for Pattern II at the (a) top, (c) middle, and (e) bottom layers of resist by the cell removal method, and at the (b) top, (d) middle, and (f) bottom layers of resist by the path-based method. . . . . . . 39 3.1 Illustration of the conventional estimation approach where both exposing step and developing step are simulated. . . . . . . . . . . . . . . . . . . . . . . . . . 44 ix 3.2 Cross-section of the resist pro?le of a base pattern. . . . . . . . . . . . . . . . . 45 3.3 Di?erence in estimation of remaining resist pro?les between 2-D model and 3-D model through (a) linear mapping (r = e) and (b) nonlinear mapping (r = e2). . 46 3.4 Distribution curves of layer-based exposure of three arbitrary points in (a) the exposed area and (b) the unexposed area on the substrate system of 500 nm PMMA on Si. The curves were obtained through computing the 3-D exposure distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 The developing process can be modeled as two separate parts, i.e., vertical de- velopment and lateral development. . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Depth at x, d(x), may be modeled as a combination of vertical component dV (x) and lateral component dL(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7 Estimation of the lateral component dL(x): (a) the initial resist pro?le d(x), (b) after the lateral component for point x1 is estimated, and (c) after the estimation for all points is completed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.8 Flowchart of the proposed estimation method. . . . . . . . . . . . . . . . . . . . 52 3.9 Cross-section of the resist pro?le of a line/space pattern. . . . . . . . . . . . . . 53 3.10 Remaining resist pro?les of an 8-line pattern with (a) L = 100 nm/S = 100 nm and (b) L = 100 nm/S = 200 nm on the substrate system of 300 nm PMMA on Si. 54 3.11 Remaining resist pro?les of an 8-line pattern with (a) L = 100 nm/S = 100 nm and (b) L = 100 nm/S = 200 nm on the substrate system of 500 nm PMMA on Si. 55 3.12 Remaining resist pro?les of an 8-line pattern with (a) L = 100 nm/S = 100 nm and (b) L = 100 nm/S = 200 nm on the substrate system of 1000 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 x 3.13 Average and maximum percent depth errors with respect to (a) the ratio of L to S with N = 8 on the substrate system of 1000 nm PMMA on Si; (b) the number of lines with L = 100 nm/S = 100 nm on the substrate system of 500 nm PMMA on Si; (c) the developed depth (controlled by the developing time) with L = 100 nm/S = 100 nm and N = 8 on the substrate system of 1000 nm PMMA on Si; (d) the resist thickness (PMMA thickness) with L = 100 nm/S = 100 nm and N = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.14 Experimental results of (a) resist pro?le of set I measured from the SEM image; (b) resist pro?le of set II measured from the SEM image; (c) resist pro?le of set I estimated from that of set II; (d) resist pro?le of set II estimated from that of set I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1 The deviation of resist pro?les at corner, edge and center of a large-scale uniform pattern without PEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 The target 2-D exposure distribution is illustrated for a line feature. . . . . . . . 65 4.3 The cross-section of resist pro?le is illustrated for a line feature where pj and qj are the actual and target widths at the j-th layer, respectively, where the resist is modeled by 10 layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Illustration of (a) the region-wise dose distribution within the feature before cor- rection, (b) the region-wise dose distribution within the feature after correction, (c) the corresponding resist pro?le before correction, and (d) the corresponding resist pro?le after correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.5 For each location (m;n) of the feature (line segment), d(i) is weighted by the deconvolution surface A(m;n;). . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.6 The exposure distributions obtained using the global (weighted) dose distribution. 70 xi 4.7 The corresponding resist pro?les at (a) corner, (b) edge, and (c) center of the pattern from the exposure distributions obtained using the global (weighted) dose distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.8 Line segments only at the critical locations of the pattern are corrected individually. 71 4.9 The adjusted dose distributions for (a) corner, (b) edge, and (c) center of the pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.10 (a) The corresponding exposure distributions and (b) the corrected resist pro?le using the adjusted dose distributions. . . . . . . . . . . . . . . . . . . . . . . . . 72 4.11 Flowchart of the proposed correction procedure. . . . . . . . . . . . . . . . . . . 73 4.12 Critical locations (marked by ?) and test locations (marked by ?) in a test pattern. 74 4.13 Cross-section resist pro?les without correction (i.e., uniform dose) at the critical locations ((a) corner, (b) edge and (c) center; refer to Fig. 4.12) and the test locations ((d), (e) and (f); refer to Fig. 4.12). . . . . . . . . . . . . . . . . . . . 76 4.14 Cross-section resist pro?les achieved by 2-D PEC method at the critical locations ((a) corner, (b) edge and (c) center; refer to Fig. 4.12) and the test locations ((d), (e) and (f); refer to Fig. 4.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.15 Cross-section resist pro?les achieved by 3-D PEC method at the critical locations ((a) corner, (b) edge and (c) center; refer to Fig. 4.12) and the test locations ((d), (e) and (f); refer to Fig. 4.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1 The region-wise feature partition with a uniform dose distribution. . . . . . . . 84 5.2 The target 2-D exposure distribution is illustrated for a line feature. . . . . . . . 85 xii 5.3 The cross-section of resist pro?le is illustrated for a line feature where pj and qj are the actual and target widths at the j-th layer, respectively, where the resist is modeled by 10 layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4 Comparison of (a) the uniform dose distribution and (b) the corresponding dis- tribution curves of 3-D exposure along Z-dimension (resist depth), with (c) a non-uniform dose distribution and (d) the corresponding distribution curves of 3-D exposure along Z-dimension (resist depth) at three points (A, B, and C) on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . 88 5.5 The resist pro?le during development process (a) at time T1, (b) at time T2, and (c) at time T for Type-V case, respectively, where T1 < T2 < T. . . . . . . . . . 89 5.6 The resist pro?le during development process (a) at time T1, (b) at time T2, and (c) at time T for Type-M case, respectively, where T1 < T2 < T. . . . . . . . . . 90 5.7 The resist pro?le during development process (a) at time T1, (b) at time T2, and (c) at time T for Type-A case, respectively, where T1 < T2 < T. . . . . . . . . . 91 5.8 The exposure distribution of the top, middle and bottom layers of resist when only (a) the two edge regions, (b) the two middle regions, and (c) the center region are exposed with a unit dose. . . . . . . . . . . . . . . . . . . . . . . . . 92 5.9 The outer width errors and the inner width errors de?ned at the cross-section of resist pro?le. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.10 The resist pro?le (a) before dose updating and (b) after dose updating for Type-V case, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.11 The resist pro?le (a) before dose updating and (b) after dose updating for Type-M case, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 xiii 5.12 The resist pro?le (a) before dose updating and (b) after dose updating for Type-A case, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.13 Flowchart of the proposed dose optimization scheme. . . . . . . . . . . . . . . . 96 5.14 The critical path for (a) Type-V case, (b) Type-M case, and (c) Type-A case, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.15 Flowchart of the proposed dose determination scheme for a speci?c dose distri- bution type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.16 Flowchart of the improved dose optimization scheme with the proposed dose determination scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.17 Flowchart of the correction procedure for large-scale uniform patterns. . . . . . 102 5.18 Flowchart of the correction procedure for large-scale nonuniform patterns. . . . 104 5.19 A feature is partitioned into 5 regions (a) along X-dimension and (b) along Y- dimension, respectively. The region-wise dose distribution (c) in X-dimension (dm;n(x)) and (d) in Y-dimension (dm;n(y)) after correction, respectively. . . . . 105 5.20 Illustration of the 2-D interpolation for computing the 2-D region-wise dose dis- tribution dm;n(x;y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.21 Cross-section resist pro?les: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 100 nm PMMA on Si. . . . . . . . . . . 107 5.22 Cross-section resist pro?les: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . 108 xiv 5.23 Cross-section resist pro?les: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 500 nm PMMA on Si. . . . . . . . . . . 109 5.24 Cross-sectionresistpro?les: (a)theuniformdosedistribution(totaldose430 C=cm2), (b) the Type-V dose distribution (total dose 320 C=cm2), (c) the Type-M dose distribution (total dose 250 C=cm2), and (d) the Type-A dose distribution (total dose 230 C=cm2) on the substrate system of 300 nm PMMA on Si. . . . . . . . 111 5.25 Cross-section resist pro?les at the corner location of Pattern I: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distri- bution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of background exposure. . . . . 115 5.26 Cross-section resist pro?les at the edge location of Pattern I: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of background exposure. . . . . . . . . . 116 5.27 Cross-section resist pro?les at the center location of Pattern I: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distri- bution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of background exposure. . . . . 117 5.28 Cross-section resist pro?les at the corner location of Pattern I: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distri- bution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xv 5.29 Cross-section resist pro?les at the edge location of Pattern I: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.30 Cross-section resist pro?les at the center location of Pattern I: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distri- bution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.31 Cross-section resist pro?les at the corner location of Pattern II: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distri- bution, and (d) the Type-A dose distribution on the substrate system of 100 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.32 Cross-section resist pro?les at the edge location of Pattern II: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 100 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.33 Cross-section resist pro?les at the center location of Pattern II: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distri- bution, and (d) the Type-A dose distribution on the substrate system of 100 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.34 Five test locations in the test nonuniform pattern. . . . . . . . . . . . . . . . . . 126 xvi 5.35 Cross-section resist pro?les in X-dimension at the test location 1 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of pattern density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.36 Cross-section resist pro?les in Y-dimension at the test location 1 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of pattern density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.37 Cross-section resist pro?les in X-dimension at the test location 5 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of pattern density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.38 Cross-section resist pro?les in Y-dimension at the test location 5 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si without considering the variation of pattern density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.39 Cross-section resist pro?les in X-dimension at the test location 1 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . 133 xvii 5.40 Cross-section resist pro?les in Y-dimension at the test location 1 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . 134 5.41 Cross-section resist pro?les in X-dimension at the test location 4 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . 135 5.42 Cross-section resist pro?les in Y-dimension at the test location 4 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . 136 5.43 Cross-section resist pro?les in X-dimension at the test location 5 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . 137 5.44 Cross-section resist pro?les in Y-dimension at the test location 5 of the nonuni- form pattern: (a) the uniform dose distribution, (b) the Type-V dose distribution, (c) the Type-M dose distribution, and (d) the Type-A dose distribution on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . . . . . . . . . 138 xviii List of Tables 2.1 Pro?le di?erence between the cell removal method and the path-based method. 40 2.2 Simulation time of the cell removal method and the path-based method. . . . . 40 2.3 Speedup of the path-based method to the cell removal method. . . . . . . . . . 40 3.1 Average and maximum percent depth errors with respect to the ratio of L to S with N = 8 on the substrate system of 1000 nm PMMA on Si. . . . . . . . . . . 57 3.2 Average and maximum percent depth errors with respect to the number of lines with L = 100 nm/S = 100 nm on the substrate system of 500 nm PMMA on Si. 58 3.3 Average and maximum percent depth errors with respect to the developed depth (controlled by the developing time) with L = 100 nm/S = 100 nm and N = 8 on the substrate system of 1000 nm PMMA on Si. . . . . . . . . . . . . . . . . . . 58 3.4 Average and maximum percent depth errors with respect to the resist thickness (PMMA thickness) with L = 100 nm/S = 100 nm and N = 8. . . . . . . . . . . 58 4.1 Average and maximum percent width errors in resist pro?les of Pattern I for no correction, 2-D PEC method and 3-D PEC method. . . . . . . . . . . . . . . . . 79 4.2 Average and maximum percent width errors in resist pro?les of Pattern II for no correction, 2-D PEC method and 3-D PEC method. . . . . . . . . . . . . . . . . 80 5.1 Average and maximum percent width errors in resist pro?les of 50 nm feature size for uniform, Type-V, Type-M and Type-A dose distributions. . . . . . . . . 112 5.2 Average and maximum percent width errors in resist pro?les of 100 nm feature size for uniform, Type-V, Type-M and Type-A dose distributions. . . . . . . . . 112 5.3 The best dose distribution type with the minimum of required total dose under a variety of system parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.4 Average and maximum percent width errors in resist pro?les of Pattern I on the substrate system of 300 nm PMMA on Si for uniform, Type-V, Type-M and Type-A dose distributions without considering the variation of background exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 xix 5.5 Average and maximum percent width errors in resist pro?les of Pattern I on the substrate system of 300 nm PMMA on Si for uniform, Type-V, Type-M and Type-A dose distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.6 Average and maximum percent width errors in resist pro?les of Pattern II on the substrate system of 100 nm PMMA on Si for uniform, Type-V, Type-M and Type-A dose distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7 Average and maximum percent width errors in X-dimension of resist pro?les of the nonuniform pattern for uniform, Type-V, Type-M and Type-A dose distri- butions on the substrate system of 300 nm PMMA on Si without considering the variation of pattern density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.8 Average and maximum percent width errors in Y-dimension of resist pro?les of the nonuniform pattern for uniform, Type-V, Type-M and Type-A dose distri- butions on the substrate system of 300 nm PMMA on Si without considering the variation of pattern density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.9 Average and maximum percent width errors in X-dimension of resist pro?les of the nonuniform pattern for uniform, Type-V, Type-M and Type-A dose distri- butions on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . 139 5.10 Average and maximum percent width errors in Y-dimension of resist pro?les of the nonuniform pattern for uniform, Type-V, Type-M and Type-A dose distri- butions on the substrate system of 300 nm PMMA on Si. . . . . . . . . . . . . . 139 5.11 Region-wise dose contrast for the Type-V dose distribution. . . . . . . . . . . . 141 5.12 Region-wise dose contrast for the Type-M dose distribution. . . . . . . . . . . . 142 5.13 Region-wise dose contrast for the Type-A dose distribution. . . . . . . . . . . . 143 xx Chapter 1 Introduction Electron-beam (e-beam) lithography plays an important role in nanofabrication, being able to transfer high-resolution circuit patterns onto the resist. However, as the feature size decreases well below micron into nanoscale, blurring in the written pattern caused by the proximity e?ect due to electron scattering in the resist puts a fundamental limit on the min- imum feature size and maximum pattern density that can be realized. The importance of developing e?ective schemes for correcting the proximity e?ect has been well recognized and extensively investigated for a long time, and various methods were proposed and implemented by many researchers. This dissertation addresses a speci?c issue in true three-dimensional (3-D) proximity e?ect correction (PEC) in e-beam lithography, i.e., control of critical di- mension (CD) and sidewall shape. The main goal of our work is to design practical and e?ective methods for minimizing the deviation of resist pro?le from the target one in terms of CD error and sidewall shape, while minimizing the total dose. This chapter provides the problem de?nition and background review along with the motivation and objectives of this dissertation. 1.1 Problem De?nition The proximity e?ect in e-beam lithography is an e?ect due to forward-scattering of elec- trons in the resist and backscattering of electrons from the substrate. The electron scattering leads to undesired exposure (e-beam energy deposited) of the resist in the unexposed regions adjacent to those exposed by the e-beam, which in turn causes changes in the dissolution rate of the resist. Therefore, the unexposed regions receiving the scattered electrons are 1 also partially developed, which results in circuit patterns with dimensions di?erent from the target ones. In general, the scattering of electrons causing the proximity e?ect can be modeled as a convolution of the dose (e-beam energy given) distribution of a circuit pattern with a proximity function [1]. The proximity function, usually referred to as the point spread function (PSF), is radially symmetric and shows how the electron energy is distributed throughout the resist when a single point is exposed. While the resist is inherently 3-D, most PEC schemes developed so far employ a two- dimensional (2-D) model where the exposure variation along the resist depth dimension is not taken into account. In other words, a 3-D PSF is not directly used in correction but averaged along the resist depth dimension to obtain a 2-D PSF. In the previous work [94, 95, 96], the limitations of 2-D PEC were analyzed and the need for 3-D PEC was well demonstrated for circuit patterns with nanoscale features. In [95], the idea of true 3-D PEC was proposed and its ?rst implementation for a single line and a small number of lines was reported. In this implementation, the resist pro?le estimated through simulation of resist development process was employed instead of the exposure distribution, in order to obtain more realistic results. In this dissertation, an extension of the true 3-D PEC method for speci?c applications is described. A challenge is that a 3-D PEC method requires a tremendous amount of com- putation due to the increased dimensionality and simulation of resist development process. 1.2 Background Review It is well known that the proximity e?ect can be reduced or corrected by appropriate measures, including physical techniques and software-based techniques [2]-[102]. The physical techniques are to modify the physical conditions of the e-beam lithography systems or the physical characteristics of resists, such as high beam energy technique [5], low 2 beam energy technique [5, 47, 51], substrate material optimization technique [6], multilayer resists technique [15], intermediate layer technique [56], etc. Experimental results show that these physical techniques provide a possible way to reduce the proximity e?ect, however they can only reduce it to some extent, with drawbacks and limitations. For example, the multilayer resists technique requires a complicated resist process step which introduces extra complexity and does not work for circuit patterns with feature size of 1 m or less. The high beam energy technique and the low beam energy technique are also a?ected by the backscattering range and the resist thickness, respectively. Therefore, most of previous work was focused on developing practical and e?ective software-based correction schemes in order to eliminate the proximity e?ect completely, which is presented in detail as below. 1.2.1 Dose Modi?cation PEC Techniques Self-consistent Method The ?rst work in this speci?c type of PEC schemes was done by Parikh in 1978 [2, 7, 8, 9], who developed a self-consistent PEC method for resist exposure. The purpose of this PEC method is to compute the dose which must be applied in order to obtain identical average absorbed (incident plus backscattered) exposure in the resist in each written shape of the pattern addressed by the e-beam. The self-consistent PEC method can be formulated mathematically by solving the fol- lowing set of linear equations: ? ??? ??? ?? ??? ??? ?? ET = I11 ?D1 +I12 ?D2 + ???I1N ?DN ET = I21 ?D1 +I22 ?D2 + ???I2N ?DN ... ET = IN1 ?D1 +IN2 ?D2 + ???INN ?DN ; (1.1) 3 where N is the number of shapes, ET is the required target exposure for each shape, Di is the dose given for shape i, and Iij is the proximity interaction between shape i and shape j. The integral de?ning the proximity interaction Iij between shape i and shape j with area Ai and area Aj is given by: Iij = ? Ai ? Aj psf(r)dAjdAi; (1.2) where psf(r) is the PSF. In general, the integral in Eq. 1.2 cannot be evaluated explicitly for two arbitrary shapes, i.e., a general formula does not exist. However, Parikh derived an analytical expression for evaluating the proximity interaction between two rectangular shapes. For any other types of shape, such as trapezoid, it can be approximated by a set of rectangles, which has the same total area as the original shape. Then the proximity interaction between two arbitrary shapes can be approximated by a summation of the proximity interactions between these rectangular components. Since the proximity interaction between two rectangular shapes can be computed rather exactly, one can split the whole pattern into rectangles before correction and set up the proximity interaction matrix I in Eq. 1.1. Once the proximity interaction matrix I is computed, the N linear equations in Eq. 1.1 can be uniquely solved for each Di, i.e., the corrected dose for each shape. Later many researchers developed various PEC methods based on the self-consistent PEC method to further improve its correction accuracy and e?ciency. One limitation of the self-consistent PEC method is that it relies only on the constraint that the average exposure in the resist is identical within the exposed area of each written shape. Therefore, it does not take the exposure in the regions not addressed by the e- beam into account. In order to solve this issue, Parikh developed another PEC method, i.e., the unaddressed-region compensation method [7], which attempts to account for the exposure in the unaddressed regions between shapes. The unaddressed regions, also known as the unexposed area, receive undesired exposure due to electron scattering, which has to 4 be minimized. Therefore, the purpose of this PEC method is to make the exposure below a speci?c value in the unexposed area while the identical average exposure is still received in each exposed shape. Note that this PEC method can be considered as an expansion of the self-consistent PEC method as the latter one is a special case of the former one when the exposure threshold for the unexposed area becomes large enough. In Parikh?s self-consistent PEC method, average exposure in each exposed shape is computed by integrating the exposure over the shape. However, Phang and Ahmed pointed out that the exposure value at the edge of the exposed shape is more important for obtaining accurate pattern ?delity of the fabricated structure. Based on this conclusion, they developed a PEC method for line patterns [4]. In this PEC method, for an exposed line element, a dose is required such that the edge of the exposed line element has an exposure equal to ET, i.e., the threshold of exposure for the dissolution of the resist. This is the fundamental principle involved in obtaining pattern ?delity between the designed and the actual fabricated structure. Similar to the self-consistent PEC method, an exposure equation to compute the the exposure at the edge of a line element is set up for each exposed line element. Then the dose of each line element can be derived by solving a set of linear equations. Carroll pointed out that Parikh?s self-consistent PEC method is not general as its model is based on equality constraints which requires exactly as many constraints as variables. He developed a PEC method based on inequality constraints which is far more general and can have arbitrarily many constraints [11, 16]. In this PEC method, N pixels on the resist surface is chosen for exposure computation. For those pixels in the exposed area, their exposure values are required to be not smaller than a target threshold ET1 (but not have to be equalized), while for those pixels in the unexposed area, their exposure values are required to be not larger than another target threshold ET2. Through this way, N inequality constraints can be formed with N dose values as variables. In general, a system of this form is usually underdetermined, thus a cost function is employed to help solving this underdetermined 5 system. In the cost function, the total dose which represents the total exposure time, is to be minimized. The best solution can be derived through linear programming approach. As mentioned before, the computation of average exposure results in a single exposure value for each shape, i.e., each exposed shape is considered as a whole in Parikh?s self- consistent PEC method. Kratschmer developed a new PEC method in which each exposed shape is partitioned into elements in order to gain more ?exibility in dose control and thus achieve better exposure distribution [12]. In this PEC method, each shape is ?rst partitioned to account for the intra proximity e?ect, then the subshapes at the edges of each shape are further partitioned to account for the inter proximity e?ect. The setting up and solving of exposureequationsarethesameasintheself-consistentPECmethod, buthereeachexposure equation corresponds to a partitioned element rather than a whole shape. By appropriate partitioning, the number of linear equations can be limited to reduce the computation time. For a pattern of N shapes, the implementation of the self-consistent PEC method re- quires the computation of N2 proximity interaction coe?cients Iij. This number can be reduced to N(N + 1)=2 as matrix I is symmetric (Iij = Iji) according to the reciprocity principle, which however is still a signi?cant amount of computation. A series of approaches was employed in order to reduce the computation time, including Otto and Gri?th?s parallel processing approach [28, 29] and Vermeulen et al.?s clustering approach [33]. By applying those approaches, the linear equation set in Eq. 1.1 is decomposed into a number of indepen- dent linear equation sets of lower order which are handled independently, thus the correction speed can be remarkably increased. Transform-based Method In 1983, Chow et al. proposed that the PEC problem can be solved by employing conventional image processing approaches, based on which they developed a transform-based PEC method [18]. 6 In this PEC method, the required dose distribution d(x) is solved through direct de- convolution of the target exposure distribution e(x) with the PSF psf(x), which is given by: d(x) = e(x)~?1 psf(x): (1.3) The above deconvolution can also be implemented in the frequency domain using the forward and inverse Fourier transforms, which is given by: d(x) = F?1 [ F[e(x)] F[psf(x)] ] : (1.4) Note that unlike the above-mentioned self-consistent PEC methods in which the cor- rected dose is constant within each shape or partitioned element, the transform-based PEC method is based on the pixel level, thus derives an extremely accurate solution for the dose distribution which is almost exact. However, this simple PEC method su?ers from two major problems. The ?rst problem is that the solved dose distribution has negative values at some locations due to the Gibbs phenomenon from the Fourier transform, which cannot be physically realized. To solve this problem, the method applies corner rounding to the target exposure distribution to minimize the negative dose values in the solution. Then the remaining negative values is further removed by adding a constant to the solved dose distribution to su?ciently guarantee the positive condition. The second problem is that the solved dose distribution contains too much details from rapid oscillations, thus results in an extremely large data base, which is unmanageable and ine?cient. The method solve this problem based on an approximation process by introducing the Walsh transform thinning algorithm to reduce the detail part while preventing the deterioration of the e?ective dose. Through Walsh transform, the data base compression is achieved, and the ?nal data is stored in the form of its Walsh coe?cients. 7 Since the Fourier transform can be implemented using its fast algorithm on special purpose computing hardware, the computation time of this method can be greatly reduced. Haslam et al. later further improved this PEC method and veri?ed it through exper- iments [22, 25, 27]. The most notable improvement was to extent the application of this method from simple one-dimensional line patterns to two-dimensional general patterns. In order to achieve further data compression, the data thinning algorithm employs the more powerful two-dimensional Haar transform instead of the previous Walsh transform. An ad- ditional bene?t of using the Haar transform is that its fast algorithm can be implemented by the same computing hardware which is used for the fast Fourier transform. Eisenmann et al. later achieved a further reduction of the computation time by separating the calculation into correction related and pattern related steps [55, 61]. Approximate Formula-based Method In 1989, Abe et al. found that the forward-scattered electron range can be assumed to be negligibly small especially for the cases of high beam energy or thin resist thickness. Based on this assumption, they developed a PEC method which utilizes an approximate dose correction formula [35]. In this PEC method, the corrected dose D(x) for a speci?c feature is derived by the following approximate formula: D(x) = D0(1 ?kU(x)); (1.5) where D0 is the initial dose for each feature, and k is a parameter selected to minimize the relative error of exposure. The function U(x) corresponds to the exposure caused by the backscattered electrons, which is given by: U(x) = 1 2 ? A exp [ ?(x?x ?)2 2 ] dx?; (1.6) 8 where is as described in the double-Gaussian PSF (refer to Eq. 2.1), and the region A is the exposed area within 3 from the center of the feature. The ?rst and second terms of Eq. 1.5 corresponds to the initial dose and the background dose by the backscattered electrons, respectively. Therefore, the corrected dose D(x) for a speci?c feature is approximated by subtracting its background dose from the initial dose D0. In order to reduce the computation time, the function U(x) can be modi?ed into another form which contains only the mathematical error function erf(z) by dividing or approximating all the features of the circuit pattern into rectangles, where the values of the error function table can be predetermined for fast referring. Abe et al. later further accelerate the correction speed by adding the pattern inversion method and the data compaction method, and installed it into a high beam energy e-beam direct writing system [32]. A more powerful representative ?gure method is later employed to further reduce the computation time with negligible error being introduced [42, 63]. Pattern Area Density Map Method In 1992, Murai et al. developed a PEC method adjusting dose by referring to a pattern area density map which is calculated by utilizing the ?xed size of square meshes [46]. In this PEC method, the circuit pattern is partitioned with a ?xed sized mesh. The mesh size is chosen such that the variation of exposure by the backscattered electrons within a single mesh site is negligible. The pattern area density , which is de?ned as the ratio of the exposed area to the total area in a region, is computed in each mesh site, giving a pattern area density map for the circuit pattern. The map is then convolved with a smoothing ?lter, giving a smoothed map, i.e., sm map. Each circuit feature is then partitioned into rectangles, and a ? value is assigned to each rectangle which is the linear interpolation of the sm values of the four nearest mesh sites to the rectangle. Finally, a corrected dose is assigned to each rectangle which is given by: 9 D = C2(1 + )1 + 2 ? ; (1.7) where is as described in the double-Gaussian PSF (refer to Eq. 2.1), and C is a constant depending on the speci?c resist and beam energy used. The determination of the mesh size is important because the correction error depends on the mesh size and the smoothing range. In order to reduce the error, a small mesh size and a wide smoothing range are desirable, which however needs to be compromised with the amount of map size. The smoothing ?lter is based on a form of template convolution where the forward scattering is neglected for fast computation purpose. Kasuga et al. later improved the correction accuracy of this PEC method by employing an adaptive partition and dose adjustment algorithm based on the gradient vector from the pattern area density map [64]. Ea and Brown further enhanced this PEC method by reducing the correction error using an iterative algorithm and a framing procedure [76], and incorporating it with a corner correction scheme [82]. Other Methods Other dose modi?cation PEC methods include Greeneich?s dose compensation curve method [14], Pavkovich?s integral equation approximate solution method [23], Gerber?s split- ting equation exact solution method [26], Frye?s adaptive neural network method [40], Aristov et al.?s simple compensation method [49], Rau et al.?s nonlinear optimization method [62], Watson et al.?s inherent forward scattering correction method [68], etc. 1.2.2 Shape Modi?cation PEC Techniques Empirically-determined Method The history of the shape modi?cation PEC techniques began as early as the dose modi?- cation PEC techniques. In 1978, Sewell developed the ?rst PEC method in this speci?c area 10 [3]. The core part of this PEC method is that although the change in the pattern dimensions during fabrication depends on parameters such as beam current, beam scan frequency, devel- oping time, developing temperature, etc., there is a basic relationship between the designed pattern dimensions and the pattern dimensions after development. Therefore, the changes in the designed pattern dimensions to compensate the in?uence of proximity e?ect could yield patterns with the correct dimensions in the resist. By analyzing the changes in the actual measured pattern dimensions for di?erent amount of dose values from experimental results, a set of empirical curves is derived to set up design tables. For a design table, a speci?c feature is chosen as the control feature to set the exposure condition such that all resist exposed above a speci?c value is developed, and the rest of the pattern dimensions are all adjusted using this exposure threshold. Based on the design table, the pattern dimensions after development can be predicted, and the designed pattern dimensions are adjusted to compensate the proximity e?ect. Analytically-determined Method In 1979, Parikh analytically modeled the relationship between the designed pattern dimensions and the pattern dimensions after development, and pointed out that this leads to an underdetermined system of nonlinear equations, whose solution is extremely di?cult and impractical for an arbitrary shape [7]. However, for simple shapes such as squares and in?nitely long lines, the solution is possible to derive as only one dimension variable (e.g., width) needs to be considered for each shape. In 1980, he developed a shape-dimension adjustment method in which exact solutions can be obtained for simple shapes based on analytical calculation instead of the empirical method [10]. In this PEC method, the formulation of intra proximity e?ect and inter proximity e?ect lead to a set of simultaneous nonlinear equations, which is individually given by. ET = 12(1 + ) [f( ) + f( )]; (1.8) 11 where , , and are as described in the double-Gaussian PSF (refer to Eq. 2.1), ET is the threshold of exposure for the dissolution of the resist, and the function f(x) is given by: f(x) = erf (wa 2x )[ erf (w a ?wd 2x ) + erf (w a +wd 2x )] ; (1.9) where wd is the designed width of the pattern while wa is the actual width of the pattern after development, and erf(z) is the error function as de?ned in mathematics. For the case of isolated features which consider only intra proximity e?ect, only one nonlinear equation for the exposure at the edge of a square or line is required to solve for the designed width of the square or line, i.e., wd. For the case of interacting features which consider both intra proximity e?ect and inter proximity e?ect, a set of nonlinear equations for the exposure at di?erent locations are required which are much more di?cult in solving. For a simple case such as an in?nitely long line adjacent to a large area, both the exposure at the left and right edge of the line are calculated to set up nonlinear equations to solve for the designed width of the line, i.e., wd. 1.2.3 Distinctive PEC Techniques Background Exposure Equalization Method In 1983, Owen and Rissman developed a PEC method, GHOST, which does not su?er from the lengthy and costly computation required in the dose and shape modi?cations [19]. The correction scheme of GHOST is based on equalization of the background exposure, i.e., backscattered electron dose, received by all points within a pattern. The pattern is ?rst exposed (to generate a pattern exposure) using a focused beam with a dose De. Then the reverse ?eld of the pattern is exposed (to generate a correction exposure) using a defocused electron beam with a diameter dc: dc = (1 + )1=4; (1.10) 12 and a reduced dose Dc: Dc = De ? 1 + ; (1.11) where and are as described in the double-Gaussian PSF (refer to Eq. 2.1). By making a correction exposure in addition to the pattern exposure for equalizing the background exposure, the compensation for the proximity e?ect is achieved. As a result, all the features in the pattern will develop out in a more uniform manner in the resist development stage. Note that the implementation of GHOST only requires image reversal of the pattern data without any further computation. Since the correction scheme of GHOST is simple and general enough to be implemented on any type of e-beam lithography systems, its application was further extended with various patterns and e-beam parameter settings [21, 30, 31, 34, 36, 37, 38, 54]. Watson et al. later improved GHOST by applying the pattern exposure and the correction exposure at the same time to eliminate the throughput drawback of the direct-write GHOST [58, 71]. Hierarchical Rule-based Method In 1991, Lee et al. developed a PEC method, PYRAMID, which has a hierarchical rule-based correction scheme [41, 74, 75]. One of the most distinct features of PYRAMID is its two-level hierarchy in exposure estimation and correction, which is not restricted to either dose modi?cation or shape modi?cation. The ?rst version and early improvements of PYRAMID adopted a shape modi?cation for the correction part. However, there is nothing inherent in PYRAMID that requires shape modi?cation, later the same overall correction hierarchy was extended to dose modi?cation and hybrid (dose/shape) modi?cation [69, 84, 87, 88]. Based on the digital image processing model of e-beam lithography, the exposure esti- mation of PYRAMID is implemented by calculating the total exposure as the sum of two separated components, global exposure and local exposure. The reason that the exposure 13 can be separated into two components stems directly from the fact that the PSF can be considered as consisting of the sum of two distinct components, i.e., a sharp, short range local component and a ?at, long range global component. The global exposure is an approx- imation of the exposure due to circuit features located far away from the critical point and is calculated through a coarse grain convolution. The process of calculating global exposure begins by producing a coarse image of the circuit pattern by dividing it into large pixels or global exposure blocks, where the value of each block is the circuit area contained within the corresponding global exposure block in the circuit pattern. The global exposure can then be found by convolving the coarse image with a 2-D sampled version of the PSF, which has a sampling pixel size equal to the global exposure block size. The local exposure, as opposed to the global exposure, is exact and considers circuit area located close to the critical point. The local exposure at a given point is calculated by applying exact convolution to all circuit area falling within a small window centered about the global exposure block containing the point. This window is termed the local exposure window and serves to separate circuit area contributing to global exposure (area outside the window) from area contributing to local exposure (area inside the window). Similarly, the correction procedure of PYRAMID is also divided into two parts, local correction and global correction. The local correction attempts to adjust the local dose distribution (for dose modi?cation) or the pattern shape (for shape modi?cation) to com- pensate for intra proximity e?ect and inter proximity e?ect caused by all the features within a small window. The local correction itself is very systematic, using two levels of correction to minimize intra proximity e?ect and inter proximity e?ect, respectively. Rule tables are used to dictate correction modes for di?erent situations. Special consideration is given to correction of features which are in close proximity or connected at junctions. While the local correction ignores interactions between widely separated features, the global correction takes general characteristics of the entire circuit pattern into account to makes adjustments to the 14 local corrections based on di?erences in exposure values at various geometric locations in the circuit pattern. Later, Lee et al. made various improvements to enhance the correction accuracy and e?ciency of PYRAMID, including an extension of PYRAMID for circuit patterns of arbitrary size to allow the rectangles to be partitioned arbitrarily [45], an extension of PYRAMID for thicker resists which have a much greater proximity e?ect than do thinner resists [53], an e?cient convolution method based on a table of cumulative distribution function values for the local exposure calculation [53], an interior area removal method by allowing PYRAMID to remove area from circuit features not only from the edges, but also from the interior as well [57], a region-wise correction method for heterogeneous substrates in which exposure estimation and correction are more complicated than in homogeneous substrates [65], an application of global exposure estimation and adjustment factor correction for reducing the recursive e?ect due to the sequential process adopted by PYRAMID [70], an adaptive method for selection of control points for high-density ?ne-feature circuit patterns [72], a distributed implementation of PYRAMID on a network of workstations which is highly scalable for realistic size of circuit patterns [78], a neural network based correction method for reducing the recursive e?ect by correcting circuit features in a group [81], a new format for hierarchical representation of circuit patterns which maintains compactness of data structure and allows e?cient searching [85], a two-step hierarchical procedure for representation of nonrectangular features and four di?erent methods for exposure estimation of nonrectangular features [89], etc. 1.2.4 Hybrid PEC Techniques In order to achieve better correction accuracy and e?ciency, several hybrid PEC meth- ods have been proposed. In a hybrid PEC method, two or more independent PEC methods are usually contained, e.g., a combination of dose modi?cation method and shape modi?ca- tion method. 15 In Groves?s hybrid PEC method, the analytically determined shape modi?cation method is adopted to o?set forward scattering while the self-consistent dose modi?cation method is adopted to o?set backscattering [52]. Later, Cook and Lee developed a hybrid PEC method for PYRAMID [69]. In this hybrid PEC method, a circuit feature is partitioned into regions for region-wise dose control, where each region is assigned a di?erent dose using the self-consistent dose modi?cation method and the hierarchical rule-based shape modi?cation method is carried out within each region. Wind et al. developed a suite of PEC programs incorporating several dose modi?cation methods, where the self-consistent dose modi?cation method is used for forward-scattering correction and the pattern area density map dose modi?cation method is used for backscat- tering correction [73]. Takahashi et al. separated the forward-scattering correction and the backscattering correction such that the analytically-determined shape modi?cation method is used for the forward-scattering and is performed only once for repeated features while the pattern area density map dose modi?cation method is used for backscattering correction and is iterated for reducing error [80, 83, 86]. Recently, Klimpel et al. developed a model-based hybrid PEC method, where the PSF (the proximity function) is modeled in di?erent forms for the forward-scattering correction and the backscattering correction, respectively [102]. Based on this model, an iterative shape modi?cation method is adopted for forward-scattering correction while an iterative dose modi?cation method is adopted for backscattering correction. 1.2.5 Summary In sum, most existed PEC techniques can be classi?ed into two types: (1) Adjustment of incident electron dose, i.e., dose modi?cation. This is achieved by appropriate variations in the dwell time or the beam intensity of a e-beam scanning system. (2) Adjustment of pattern dimensions, i.e., shape modi?cation. This is achieved by appropriate changes in 16 the pattern dimensions such that patterns with the desired dimensions are obtained after exposure and development. The dose modi?cation PEC technique has the advantage that it provides a mathemati- cally unique solution which depends only on the form and the magnitude of the PSF. This PEC technique is general enough to apply to an arbitrary de?nition of a region, from the smallest electron beam de?ned element to an entire shape. However, this PEC technique requires a lengthy and costly computation to evaluate the correction and a large database to store the dose value of each region. Furthermore, it cannot be applied on some certain kinds of e-beam lithography systems, such as the e-beam projection systems, which do not have the freedom of being able to change the dose for each shape as the pattern is being exposed. The shape modi?cation PEC technique has the advantage that it does not require sophisticated dose adjustment, thus is much more simple than the dose modi?cation PEC technique. Also in the case of e-beam projection systems, the shape modi?cation PEC technique is a better choice as it requires a relatively simple pattern generator in terms of not changing the beam frequency and a relatively simple computer programs for correcting pattern dimensions. The main disadvantage of this PEC technique is similar to the dose modi?cation PEC technique, i.e., requiring a lengthy and costly computation to evaluate the correction. Its correction accuracy is lower than the dose modi?cation PEC technique. Furthermore, the minimum magnitude of the shape changes is limited by the resolution of e-beam lithography systems. Other PEC techniques, such as the GHOST method, has the advantage that it is simple and general enough to be implemented on any type of e-beam lithography systems and does not su?er from the time-consuming computation and large database storage required in the dose and shape modi?cations. The main disadvantage of this PEC technique is that the throughput of the lithographic process is reduced, typically by a factor of 2, due to compensating the proximity e?ect by a correction exposure. Another drawback is the 17 contrast degradation between the pattern exposure and the background exposure which leads to a dull resist pro?le. 1.3 Motivation and Objectives From the above review, it is noticed that a typical approach to PEC employs a 2-D model of resist layer, i.e., the exposure variation along the resist depth dimension is not taken into account. In a 2-D PEC, the exposures at the selected points in a circuit pattern are estimated by convolution between a circuit pattern and a 2-D PSF, which is obtained by averaging the corresponding 3-D PSF along the resist depth dimension. The dose amount or shape adjustment for each circuit feature is determined such that a desired exposure distribution is achieved. Recently, there were claims of 3-D PEC being proposed [90, 98, 101]. However, they were claimed to be 3-D due to considering the thickness control of the resist pro?les while still employed a 2-D exposure model. For circuit patterns with nanoscale features, it is not unusual that the actual written pattern is substantially di?erent from the written pattern estimated by a 2-D PEC method. One of the reasons for this deviation is that the 2-D model ignores the exposure variation along the resist depth dimension. In an earlier study [94], it was shown that the remaining resist pro?le estimated using a 2-D model can be signi?cantly di?erent from the one based on a 3-D model which considers the depth-dependent exposure variation. Another reason is that the actual resist pro?le cannot be derived directly from the exposure distribution, e.g., the developing rate is not linearly proportional to the exposure at a given point. Therefore, the PEC methods which consider the exposure only may su?er from substantial CD errors or deviation from the target resist pro?le in general. In addition, with a 2-D model, it is not possible to consider the lateral development of resist which a?ects the resist pro?le, in particular the sidewall shape, signi?cantly. Therefore, a 3-D PEC, which employs the resist pro?le estimation through simulation of resist development process instead of the exposure distribution, is required in order to obtain 18 more realistic results. It has been demonstrated that in order to minimize any deviation from a target resist pro?le, a 3-D PEC scheme must check the estimated resist pro?le during the dose optimization procedure [95]. One practical issue of such an approach to 3-D PEC is that a time-consuming resist development simulation needs to be carried out in each iteration of the dose optimization for each feature. For the case of large-scale patterns, such a feature-by- feature correction procedure would be too time-consuming to be practical. Also, it has been shown that the dose distribution of ?V-shape?, which is used by most 2-D PEC schemes, is not optimal for realizing a vertical sidewall of the resist pro?le, especially when the total dose is to be minimized. Note that a higher total dose worsens the charging e?ect and lengthens the exposing time. The main objectives of this dissertation are: ? To develop a fast method for 3-D resist development simulation suitable to be employed in an iterative PEC scheme. ? To develop an accurate and e?cient approach for estimation of remaining resist pro?les for line/space patterns. ? To develop a practical and e?cient 3-D PEC method for handling correction of large- scale uniform patterns. ? To derive new types of dose distributions for achieving the target resist pro?le of vertical sidewall for nanoscale features with the minimum total dose. ? To develop adaptive dose optimization and dose determination schemes for our 3-D PEC method. 1.4 Organization of Dissertation The rest of this dissertation is organized as follows: 19 ? Chapter 2 introduces the essential models in e-beam lithography and a fast path-based method for 3-D resist development simulation employed in this study. ? Chapter 3 describes an adaptive approach for estimation of remaining resist pro?les for line/space patterns using a layer-based exposure model which does not require exposure computation and resist development simulation [103]. ? Chapter 4 discusses a critical-location-based 3-D PEC method for correction of large- scale uniform patterns avoiding the intensive computation without sacri?cing the cor- rection quality [104]. ? Chapter 5 describes two new types of dose distributions being derived under our 3-D exposure model and 3-D PEC method and two adaptive schemes for dose optimization and dose determination being developed for improving the correction e?ciency without sacri?cing the correction quality [105]. ? Chapter 6 presents conclusions of this dissertation. 20 Chapter 2 Essential Models in Electron-beam Lithography 2.1 Electron-beam Lithography E-beam lithography is a lithographic process used to transfer circuit patterns onto silicon or other substrates. It employs a focused beam of electrons to expose a circuit pattern into the electron-sensitive resist applied to the surface of the substrate. The main advantage of e- beam lithography is that it can o?er much higher patterning resolution than the conventional optical lithography which is limited by the di?raction of light, thus it is extremely suitable for fabrication of nanoscale features. A typical e-beam binary lithographic process is illustrated in Fig. 2.1. In the ?rst step, the areas containing circuit features of a circuit pattern are exposed by the e-beam with di?erent dose amounts, depositing energy in the resist (refer to Fig. 2.1(a)). After the exposing step, a solvent developer is used to selectively remove either the exposed regions or the unexposed regions of the resist depending on the type of resist (refer to Fig. 2.1(b)). For a positive resist, the most common type, the exposed regions become soluble in the developer, while the unexposed regions become soluble in the developer for a negative resist. After the resist development process, it can be seen that the substrate is covered by the undeveloped resist with di?erent thickness. Finally, a corrosive etchant is applied to selectively etch the substrate, transferring the circuit pattern onto the substrate (refer to Fig. 2.1(c)). In this step, the remaining resist pro?le serves as a mask to control the etched depth. 21 Expose Substrate Resist Dose Dose Dose (a) Develop Substrate (b) Etch Substrate (c) Figure 2.1: Illustration of the e-beam binary lithographic process. 2.2 Point Spread Function The proximity e?ect in e-beam lithography is usually modeled by the convolution be- tween the dose distribution of a circuit pattern and a proximity function or point spread function (PSF), which is radially symmetric and shows how the electron energy is distributed throughout the resist when a single point is exposed. Fig. 2.2 shows a typical PSF for the 22 substrate system of 300 nm PMMA on Si with the beam energy of 50 KeV, as a function of the distance from the exposed point. 10?3 10?2 10?1 100 101 10210 ?4 10?2 100 102 104 106 108 Radius (?m) Exposure (eV/ ?m 3 ?electron) Figure 2.2: PSF for the substrate system of 300 nm PMMA on Si with the beam energy of 50 KeV. In general, PSFs have the similar overall shape as illustrated in Fig. 2.2, which de- pends on resist thickness, substrate composition, beam energy, beam diameter, etc., and is independent of the dose given to the exposed point. One key property of the PSF is that it can be decomposed into two separate components, the local (or short range) component and the global (or long range) component. The local component, which has relatively large magnitude and is very sharp, describes the electron?s forward scattering, while the global component, which has relatively low magnitude and is very ?at, describes the electron?s backward scattering. One common way to model PSFs is through analytical approximation by a double- Gaussian function, which was proposed by Chang in 1975 [1], where one Gaussian function models the deposited energy due to the forward-scattering of electrons and the other Gaus- sian function models that due to the backscattering of electrons. The normalized double- Gaussian PSF is given by: 23 psf(r) = 1 (1 + ) [ 1 2 exp( ?r2 2 ) + 2 exp( ?r2 2 ) ] ; (2.1) where the parameter represents the forward-scattering range, the parameter represents the backscattering range, and the parameter is the ratio of the backscattered energy level to the forward-scattered energy level. Later more complicated forms of functions were proposed in order to enhance the ac- curacy of the Gaussian function ?tting model, such as the triple-Gaussian function, the double Gaussian function plus an exponential term, etc [106, 107]. Furthermore, various methods for determination of the proximity e?ect parameters ( , , and ) were developed [108, 109, 110, 112, 113, 116, 120]. (a) (b) Figure 2.3: Monte Carlo simulation of (a) electron paths (cross-section) and (b) energy distribution (cross-section). The Monte Carlo simulation is another widely used approach to deriving PSFs [111, 114, 115, 117, 121]. A substrate system including the resist layer is modeled as a 3-D array of cubic cells, and the random path of each individual electron is traced with the electron energy being deposited in the cells along the path which is stochastically determined based on the phenomena and e?ects theoretically modeled, as illustrated in Fig. 2.3(a). The ?nal 24 electron energies deposited in the cells are computed along each path, to generate a 3-D PSF sampled at the interval of cell size, as illustrated in Fig. 2.3(b). Recently, some researchers proposed experiment-based methods to generate much more realistic PSFs using experimental data, which are not constrained by a curve-?tting function and do not require a time-consuming simulation [118, 119]. 2.3 Exposure Model In a typical substrate system employed in this study, a resist layer with initial thickness of H is on top of the substrate, as illustrated in Fig. 2.4 where the X-Y plane corresponds to the top surface of resist and the resist depth is along the Z-dimension. The 3-D PSF is denoted by psf(x;y;z). Let d(x;y;0) represent the e-beam dose given to the point (x;y;0) on the surface of the resist for writing a circuit feature or pattern (refer to Fig. 2.4). For example, in the case of a uniform dose distribution, d(x;y;0) = ? ?? ?? D if (x;y;0) is within a feature 0 otherwise ; (2.2) where D is a constant dose. H D Figure 2.4: Illustration of the substrate system where H is the initial thickness of resist. 25 Let us denote the exposure at the point (x;y;z) in the resist by e(x;y;z). Then, the 3-D spatial distribution of exposure can be expressed by the following convolution: e(x;y;z) = ? ? d(x?x?;y ?y?;0)psf(x?;y?;z)dx?dy?: (2.3) From Eq. 2.3, it can be seen that the exposure distribution at a certain depth z0 can be computed by the 2-D convolution between d(x;y;0) and psf(x;y;z0) in the corresponding plane z = z0, i.e., e(x;y;z) may be estimated layer by layer. Note that the PSF, psf(x;y;z), re?ects all the phenomena a?ecting energy deposition including the e-beam blur. The goal of PEC is to determine d(x;y;0) such that a target pattern is successfully transferred onto the resist layer. 2.4 Development Method Although the 3-D exposure model provides the complete information on how electron energy is distributed in the resist, it does not directly depict the remaining resist pro?le after development, as illustrated in Fig. 2.5. Therefore, it is necessary to take the resist development process also into account in order to obtain a realistic correction result. H Figure 2.5: Illustration of the remaining resist pro?le after the resist development process. 26 2.4.1 Exposure-to-rate Conversion Formula First, the developing rate at each point in the resist is computed based on the exposure at the point and developing conditions. Most resists are nonlinear in nature when exposed by the e-beam, i.e., the developing rate is not linearly proportional to the exposure. Therefore, the developing rate r(x;y;z) at each point in the resist is calculated from its correspond- ing exposure e(x;y;z) through a nonlinear exposure-to-rate conversion formula, which is obtained experimentally. 0 1 2 3 4 5 x 1010 0 500 1000 1500 2000 Exposure (eV/?m3) Developing Rate (nm/min) Figure 2.6: The experiment-based nonlinear exposure-to-rate conversion formula. A long line with width of 100 nm is exposed with various dose levels. After resist development, the center depth is measured in the cross-section of remaining resist pro?le obtained for each dose level. Note that the resist is developed only vertically at the center of line when the dose distribution within the line is uniform. The center depth is also obtained for each dose level through simulation based on our 3-D exposure model. By comparing the two sets of depth measurements obtained experimentally and via simulation, the following conversion formula has been derived (also refer to Fig. 2.6): 27 r(x;y;z) = F[e(x;y;z)] = 3700 ?e?(e(x;y;z)?1:0e115:6e10 )2 ? 80 ?e?(e(x;y;z)?9:0e99:0e9 )2 ? 123; (2.4) where r(x;y;z) is in nm/minute and e(x;y;z) in eV= m2. 2.4.2 Cell Removal Method Then, a development method is employed to derive the remaining resist pro?le. In the past, several 2-D and 3-D methods were developed, such as the the string method, the ray- tracing method, and the cell removal method. The string and ray-tracing methods usually have di?culties in the treatment of boundaries and the elimination of loops. In contrast, the cell removal method, e.g., PEACE [122], is quite stable and robust as it does not su?er from such problems. Figure 2.7: Illustration of the cell removal method where the dashed curve represents a cross-section of resist pro?le. 28 In this method, each resist layer is partitioned into cubic cells, and the resist development process is traced on a cell-by-cell basis (refer to Fig. 2.7). This kind of cell-based method is generally easy to implement which makes it widely used. However, the main drawback of the cell removal method is a long computation time required, specially for estimating 3-D resist pro?les. When an iterative PEC scheme requires the resist development simulation in each iteration, it would not be practical to employ such a time-consuming method. 2.4.3 Proposed Path-based Method In this study, a fast path-based method for 3-D resist development simulation which avoids the time-consuming computation without sacri?cing the simulation accuracy has been proposed. The proposed method employs the concept of ?development paths?, which start from the top surface of resist toward the boundaries of the ?nal resist pro?le, to model the development process. Figure 2.8: Illustration of the path-based method where the dashed curve represents a cross- section of resist pro?le. 29 Conventionally, one should model the development process in an isotropic way as the resist is developed in all possible directions in reality. However, in the proposed method, the development process is simply modeled as two separate parts, i.e., vertical development and lateral development. Similarly, each path is also modeled as consisting of two orthogonal types of path segments, i.e., vertical (to depict vertical development) and lateral (to depict lateral development) path segments (refer to Fig. 2.8). More speci?cally, each path has one and only one vertical path segment (along the Z-dimension) and may have one or more lateral path segments (along the X-dimension or Y-dimension). xy x + ?xyx -?xy (a) xy x + ?xyx -?xy (b) Figure 2.9: A path turns to (a) only the positive direction of the X-axis at the point (x;y;z) if r(x;y;z) > r(x + ?x;y;z) but r(x;y;z) ? r(x ? ?x;y;z) and (b) both the positive and negative directions of the X-axis at the point (x;y;z) if r(x;y;z) > r(x + ?x;y;z) and r(x;y;z) > r(x? ?x;y;z). In order to model the resist pro?le in a more accurately way, the concept of ?turns? is employed to compute each path based on its path segments. A path makes a turn when it switches from one vertical path segment to one lateral path segment or one lateral path segment to another lateral path segment. Whether and in what direction a path is to be turned at a point depends on the relative developing rates of adjacent points in X-Y plane 30 compared to that of the point. This is based on the fact that the developer passes a point with relatively higher developing rate faster than its adjacent points with relatively lower developing rates, then the developer will keep developing laterally toward those adjacent points. It is obvious that a larger lateral development takes place when there is a larger di?erence between the developing rates of adjacent points. Figure 2.10: The development paths with one and two turns where the black and gray lines are the vertical and lateral path segments, respectively, and the dashed curve represents a cross-section of resist pro?le. The Z-axis corresponds to the resist depth dimension. Let the distances between adjacent points in the X-dimension, Y-dimension, and Z- dimension be denoted by ?x, ?y, and ?z, respectively. A turn is allowed only from one point (x;y;z) with relatively higher developing rate to its adjacent points with relatively lower developing rates, i.e., r(x;y;z) > r(x + ?x;y;z) or r(x;y;z) > r(x ? ?x;y;z) or r(x;y;z) > r(x;y +?y;z) or r(x;y;z) > r(x;y ??y;z). For example, a path turns to only the positive direction of the X-axis at the point (x;y;z) if r(x;y;z) > r(x + ?x;y;z) but r(x;y;z) ? r(x ? ?x;y;z), as illustrated in Fig. 2.9(a), while it turns to both the positive 31 and negative directions of the X-axis at the same point if r(x;y;z) > r(x + ?x;y;z) and r(x;y;z) > r(x? ?x;y;z), as illustrated in Fig. 2.9(b). It is obvious that the more turns allowed, the more accurate the simulation result becomes, but the longer the simulation time is. It is found that in many cases, one turn is su?cient to achieve high accuracy of simulation, e.g., when one is interested only in cross- sections of resist pro?le for a line feature. Even for more general shapes of feature, two turns are su?cient to obtain accurate 3-D resist pro?les at the expense of a longer simulation time compared to the case where only one turn is allowed (refer to Fig. 2.10). X ZY Substrate Resist (x0,y0,0) (x0,y0,z0) (x1,y',z0) (x1,y0,z0) (x0,y0,z') (x',y0,z0) Figure 2.11: The computation of each path terminates when the sum of the time spent on its path segments is equal to the given developing time T. The computation of each path terminates when the sum of the time spent on its path segments is equal to the given developing time T, which is given by (refer to Fig. 2.11): z?? z=0 ?z r(x0;y0;z) = T; (2.5) for no-turn (vertical-only) paths, z0? z=0 ?z r(x0;y0;z) + x?? x=x0 ?x r(x;y0;z0) = T; (2.6) 32 for one-turn paths, and z0? z=0 ?z r(x0;y0;z) + x1? x=x0 ?x r(x;y0;z0) + y?? y=y0 ?y r(x1;y;z0) = T; (2.7) for two-turn paths, respectively. (a) (b) (c) (d) Figure 2.12: (a) All the points in the resist are marked as ?undeveloped? before the resist development simulation. (b) After one path is computed, the points which it passes through are marked as ?developed?. (c) After another path is computed, more points are marked as ?developed?. (d) The ?nal resist pro?le is determined by tracing the boundaries between those developed points and those which are not. 33 In order to derive the ?nal resist pro?le, a mark is used to represent the status of each point during the resist development simulation. Initially, all the points in the resist are marked as ?undeveloped?. Then, after one path is computed, the points which it passes through are marked as ?developed?. It is obvious that more and more points are marked as ?developed? as more and more paths are computed. The ?nal resist pro?le is determined by tracing the boundaries between those developed points and those which are not (refer to Fig. 2.12). The complete procedure of the path-based method is depicted below (also refer to the ?owchart in Fig. 2.13). Step 1: Compute the exposure distribution e(x;y;z), and convert it into the developing rate distribution r(x;y;z) through the conversion formula F[?] in Eq. 2.4. Step 2: Compute a vertical path segment along the Z-dimension starting from a speci?c point (x0;y0;0) on the top surface of resist using the developing rate distribution r(x;y;z) and the given developing time T, such that z?? z=0 ?z r(x0;y0;z) = T. Step 3: Compute a lateral path segment along the X-dimension starting from a speci?c point (x0;y0;z0) (0 < z0 < z?) on the vertical path segment in Step 2 (if r(x0;y0;z0) > r(x0+ ?x;y0;z0) or r(x0;y0;z0) > r(x0 ? ?x;y0;z0)) using the developing rate distribution r(x;y;z) and the given developing time T, such that z0? z=0 ?z r(x0;y0;z)+ x?? x=x0 ?x r(x;y0;z0) = T. Step 4: Compute a lateral path segment along the Y-dimension starting from a speci?c point (x1;y0;z0) (x0 < x1 < x?) on the lateral path segment in Step 3 (if r(x1;y0;z0) > r(x1;y0+?y;z0) or r(x1;y0;z0) > r(x1;y0??y;z0)) using the developing rate distribu- tionr(x;y;z)andthegivendevelopingtimeT, suchthat z0? z=0 ?z r(x0;y0;z)+ x1? x=x0 ?x r(x;y0;z0)+ y?? y=y0 ?y r(x1;y;z0) = T. 34 t 0 0 0 0 0 , , z z z z zt r x y z p z '' ?r xyz t t?tz t t?tx r xyz r xyz t t?ty pz px pz px py pz 1 0 0 0 1 0 , , x x x x x xt r x y z p x x ''  ? 0 ' 1 0 0 , , ' y y y y y yt r x y z p y y ''  ? Figure 2.13: Flowchart of the computation of a two-turn path, where ?tx, ?ty, and ?tz (?tx + ?ty + ?tz = T) are the time spent on the path segment in the X-dimension, Y- dimension, and Z-dimension, respectively, and p is the vector representation of the path in the 3-D space. Step 5: Store the vector representation of the path in the 3-D space (considering all the above three path segments) as p = (px;py;pz), where px = |x1 ? x0|, py = |y? ? y0|, and pz = z0 are the three path segments, respectively. 35 Step 6: Mark the points which the path passes through as ?developed? using p = (px;py;pz). Step 7: If all the possible paths are computed, proceed to Step 8. Otherwise, go back to Step 2. Step 8: Determine the ?nal resist pro?le by tracing the boundaries between those developed points and those which are not. It can be seen that compared with the cell removal method which employs a time- consuming iterative procedure to update the status of each cell, the proposed method is more straightforward as it does not require any iterations in computing each path. Furthermore, the simulation of each cell is correlated in the cell removal method, which means that the ?nal status (the portion being developed) of each cell cannot be determined until all the cells are simulated. However, each path is computed individually in the proposed method, thus avoids this kind of issue and reduces the simulation time. In order to minimize the simulation time, an adaptive approach is incorporated into the simulation procedure. Before the ?nal stage of simulation, the resist pro?le would be usually smooth without sharp shapes such as corners and junctions. Therefore, a developing period (time) is partitioned into two phases where only one turn is allowed in the ?rst phase and two turns in the second phase. This adaptive approach reduces the simulation time greatly without sacri?cing simulation accuracy substantially. 2.4.4 Results and Discussion The proposed path-based method for 3-D resist development simulation has been im- plemented and its performance has been compared with the cell removal method in terms of simulation accuracy and time. Eight di?erent test patterns (Patterns I-VIII) including Pattern I (shown in Fig. 2.14(a)) and Pattern II (shown in Fig. 2.14(b)) are considered in the comparison. The largest pattern size for simulation is 1 m by 1 m. 36 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (a) 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (b) Figure 2.14: The layout of (a) Pattern I and (b) Pattern II. The PSFs employed in the simulation are generated by a Monte Carlo simulation method, SEEL [111]. The substrate system assumed in this comparison is composed of 300 nm PMMA on Si. The beam energy is set to 50 KeV with the beam diameter of 5 nm. The developing time is 40 sec. The resist pro?les simulated by the proposed path-based method are compared with those simulated by the cell removal method with the constraint that the resolution of the resist pro?le is the same for both methods. The contours of resist pro?les at the top, middle and bottom layers, obtained for Pattern I and Pattern II, are provided in Fig. 2.15 and Fig. 2.16. By comparing the above resist pro?les, it can be seen that the resist pro?les by the proposed method are well matched with those by the cell removal method. The pro?le di?erence between the two methods is provided in Table 2.1. The same high accuracy is observed in all of the other test patterns. 37 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (a) 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (b) 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (c) 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (d) 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (e) 0 100 200 300 400 5000 100 200 300 400 500 X (nm) Y (nm) (f) Figure 2.15: Contours (top view) of resist pro?les for Pattern I at the (a) top, (c) middle, and (e) bottom layers of resist by the cell removal method, and at the (b) top, (d) middle, and (f) bottom layers of resist by the path-based method. 38 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (a) 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (b) 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (c) 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (d) 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (e) 0 150 300 450 600 7500 150 300 450 600 750 X (nm) Y (nm) (f) Figure 2.16: Contours (top view) of resist pro?les for Pattern II at the (a) top, (c) middle, and (e) bottom layers of resist by the cell removal method, and at the (b) top, (d) middle, and (f) bottom layers of resist by the path-based method. 39 Pattern Pro?le Di?erence ID Size (nm ? nm) Top (%) Middle (%) Bottom (%) I 500 ? 500 0.45 0.53 1.19 II 750 ? 750 0.42 0.49 1.54 III 1000 ? 1000 0.48 0.78 2.89 IV 150 ? 1000 0.70 0.78 3.37 V 250 ? 1000 0.70 0.84 2.94 VI 350 ? 1000 0.69 1.00 3.28 VII 450 ? 1000 0.69 0.95 3.08 VIII 550 ? 1000 0.69 1.04 2.41 Table 2.1: Pro?le di?erence between the cell removal method and the path-based method. Pattern Cell Removal Path-based Method (sec) ID Size (nm ? nm) Method (sec) One-Turn Two-Turn Adaptive I 500 ? 500 403.86 6.96 32.52 18.73 II 750 ? 750 1869.36 19.09 64.55 53.58 III 1000 ? 1000 5332.55 27.01 116.07 88.97 IV 150 ? 1000 160.72 4.89 19.50 12.21 V 250 ? 1000 594.32 10.92 39.48 24.94 VI 350 ? 1000 1050.71 13.92 59.99 38.56 VII 450 ? 1000 1902.56 20.03 84.31 52.30 VIII 550 ? 1000 2744.31 22.48 101.25 66.04 Table 2.2: Simulation time of the cell removal method and the path-based method. Pattern Speedup ID Size (nm ? nm) One-Turn Two-Turn Adaptive I 500 ? 500 58.03 12.42 21.56 II 750 ? 750 97.92 28.96 34.89 III 1000 ? 1000 197.43 45.94 59.93 IV 150 ? 1000 32.82 8.24 13.16 V 250 ? 1000 54.42 15.05 23.83 VI 350 ? 1000 75.48 17.51 27.25 VII 450 ? 1000 94.99 22.57 36.38 VIII 550 ? 1000 122.08 27.11 41.56 Table 2.3: Speedup of the path-based method to the cell removal method. In Table 2.2, the simulation time measured on a PC with a 2.53 GHz CPU (Intel Core i5-540M) and 2 GB memory is provided for the test patterns. It is clear that the proposed 40 method takes much less time for simulation, compared to the cell removal method. Note that the simulation time of the cell removal method with respect to the pattern size is close to a quadratic relationship while that of the proposed method is almost linearly proportional to the pattern size. From Table 2.3, it can be seen that the speedup of the proposed method to the cell removal method increases signi?cantly for larger patterns. Therefore, the proposed path-based method has a good potential to be a practical and e?cient alternative to the existing methods such as the time-consuming cell removal method. 41 Chapter 3 Estimation of Resist Pro?le for Line/Space Patterns 3.1 Introduction The e-beam lithographic process consists of selectively exposing resist by e-beam and subsequently developing the resist for pattern transfer. For applications such as predicting the remaining resist pro?le (just resist pro?le hereafter) in grayscale lithography and PEC, both steps are often simulated. In the ?rst step, the exposure distribution is computed by convolution between a circuit pattern and a PSF. In the second step, the developing rate at each point in the resist is computed based on the exposure at the point and developing conditions, and a development method is employed to derive the remaining resist pro?le. While such simulations are widely used, it is not unusual that the estimated pro?le of the remaining resist is substantially di?erent from the actual pro?le obtained in experiment. The reasons for this deviation may include (i) the actual parameters such as beam diameter are di?erent from those in the system speci?cations, (ii) certain e?ects may not be considered in simulation, and (iii) the actual remaining resist pro?le varies with the developing condition. Also, such simulations are very time-consuming. Hence, it is worthwhile to develop a new method which does not require exposure calculation and development simulation, in order to provide an alternative to the conventional simulation-based methods. Resist pro?les obtained in experiments re?ect all e?ects and actual parameters involved in the e-beam lithographic process. Therefore, an experiment-based method has a potential to generate an estimated pro?le close to the actual one. A new method is proposed for estimating the resist pro?les using a set of experimental results without exposure estimation anddevelopmentsimulation. Thisnewmethodshouldbedistinguishedfromtheconventional approaches, where exposure distribution and/or resist pro?le is obtained through simulation. 42 The idea of the method is to adopt the concept of ?base patterns.? The resist pro?les of the base patterns are obtained through experiments and are used in estimating the resist pro?le of a given pattern consisting of the base patterns. In this chapter, an implementation of the method which utilizes the base pattern of line to estimate resist pro?les of line/space patterns is presented to demonstrate the feasibility of the method through computer simulation and experiments. The rest of the chapter is organized as follows. The conventional estimation approach is brie?y reviewed in Section 3.2. The proposed estimation approach is described in Section 3.3. Details of the estimation procedure are presentedin Section 3.4. Simulationand experimental results are discussed in Section 3.5, followed by a summary in Section 3.6. 3.2 Conventional Estimation Approach A conventional estimation approach is to compute the exposure distribution for a target pattern through convolution with the PSF and then to obtain the resist pro?le via simulation of development process based on the exposure distribution (refer to Section 2.3 and Section 2.4). Fig. 3.1 shows this approach being applied on a three-line pattern. It can be seen that the estimated pro?le of the remaining resist from this approach is completely dependent on the accuracy of the PSF which is usually obtained through modeling or simulation (refer to Section 2.2). Furthermore, this approach requires a tremendous amount of simulation time for patterns containing a large number of lines which is common in practical applications. 3.3 Proposed Estimation Approach 3.3.1 Base Patterns The proposed estimation method is based on the approach where a minimal set of experiments is carried out in order to extract a minimally su?cient amount of information on the resist development process in a certain experimental set-up. The patterns employed in these experiments are referred to as base patterns, which are long rectangles in this study. 43 Figure 3.1: Illustration of the conventional estimation approach where both exposing step and developing step are simulated. Through these experiments, the dose and width of rectangle are varied to collect the data needed for estimation of the remaining resist pro?le of a target pattern. The more data are collected, the more accurate estimation is possible. However, the number of experiments is to be minimized from the viewpoint of practicality. Therefore, it is required to sample a limited number of base patterns to collect enough data for the estimation. It is assumed that a base pattern is su?ciently long along the Y-dimension such that any variation along the Y-dimension can be ignored, or only the cross-section of resist per- pendicular to the Y-dimension is considered (refer to Fig. 3.2). In the cross-section, e(x;y;z) and r(x;y;z) are replaced by e(x;z) and r(x;z), respectively. 44 Figure 3.2: Cross-section of the resist pro?le of a base pattern. 3.3.2 2-D Exposure Model In an earlier e?ort, a method which takes the above-mentioned (refer to Section 3.3.1) approach to estimation of remaining resist pro?les was developed, but based on a 2-D ex- posure model. This method, referred to as ?2-D Mod,? is brie?y described below, to be compared to the proposed method. In the 2-D exposure model, exposure is assumed not to change along the resist depth dimension (Z-dimension), i.e., e(x;z) in the cross-section is averaged over 0 ? z ? H (see Fig. 2.4), resulting in e(x). Suppose that the target pattern consists of two identical lines and the center-to-center distance between the two lines is I. The base pattern for estimating the resist pro?le of the target pattern is a single line. Let the resist (depth) pro?le of the base pattern be represented by d1(x). The resist pro?le of the target pattern, d2(x), is estimated from d1(x) as follows. The developing rate is estimated as r1(x) = d1(x)T where T is the developing time, and the exposure distribution for the base pattern is computed by e1(x) = F?1[r1(x)] (refer to Eq. 2.4). Then, noting the linearity of exposure, the developing rate distribution for the target pattern can be obtained as: r2(x) = F[e2(x)] = F[e1(x) +e1(x?I)]: (3.1) Finally, the resist pro?le of the target pattern is estimated to be: 45 d2(x) = r2(x) ?T: (3.2) A fundamental problem of this method is that the exposure variation along the resist depth dimension is not taken into account, which can lead to a signi?cant estimation er- ror [94, 95, 96]. Furthermore, the nonlinear mapping F[?] makes the error even larger, as illustrated in Fig. 3.3. Also, the lateral development of resist is not explicitly considered in the estimation procedure though d1(x) itself includes the lateral component of resist devel- opment. Therefore, in this chapter, an estimation method which employs a 3-D exposure model and explicitly accounts for lateral development is presented. ex rx ex T exz rxz exz T ?dx dxd2Dx d2Dx dx (a) ?dx dxd2Dx d2Dx dx ex rx e2x T exz rxz e2xz T (b) Figure 3.3: Di?erence in estimation of remaining resist pro?les between 2-D model and 3-D model through (a) linear mapping (r = e) and (b) nonlinear mapping (r = e2). 3.3.3 Layer-based Exposure Model In this study, a new method which adopts the idea of layer-based exposure modeling has been developed in order to improve estimation accuracy. By analyzing the simulation results for several base patterns on di?erent substrate systems, it is shown that the distribution of e(xi;z) with respect to z is similar to that of e(xj;z), where xi and xj are any two points within the exposed area, as shown in Fig. 3.4(a). Therefore, e(x;z) can be modeled using a family of normalized decreasing functions with parameters including the resist thickness, the feature size (width of exposed area), and the distance from x to the center of a feature. 46 In this way, given a certain point x0 within the feature, a normalized distribution curve of e(x0;z) can be modeled, which is denoted by N(x0;z). The actual values of e(x0;z) can be computed by: e(x0;z) = e(x0) ?N(x0;z); (3.3) where e(x0) is initiated by: e(x0) = F?1[r(x0)] = F?1[d(x0)T ]: (3.4) Then, for the developing time T, the estimated depth d?(x0) may be determined by: ? d?(x0)=d0 0 d0 F[e(x0;z)]dz = T; (3.5) where d0 is the thickness of each layer of the resist de?ned in the model. A smaller d0 leads to a higher estimation accuracy. Up to now a percentage depth error is derived by: ?d(x0) = d ?(x0) ?d(x0) d(x0) : (3.6) This error is used for controlling the increment of r(x0), or equivalently e(x0) to re- compute e(x0;z) until the error converges. Through this kind of iteration, e(x0;z) can be modeled accurately enough to avoid either underestimation or overestimation. From the analysis of 3-D exposure distribution, it is also found that the distribution of e(x;z) along the Z-dimension changes abruptly when crossing the boundary between the exposed and unexposed areas without a gradual transition region. The e(x;z)?s for di?erent x in the unexposed area are di?erent from those in the exposed area, but similar among themselves, as shown in Fig. 3.4(b). Therefore, the same approach to modeling e(x;z) (as the one described above for the exposed area) can be taken also for the points in the 47 unexposed area, i.e., outside features. This layer-based modeling is essential to the proposed estimation method, and is proved to be e?ective. 0 20 40 60 80 1005 6 7 8 9 10x 109 Layer Index Exposure (eV/ ?m3 ) (a) 0 20 40 60 80 1000 0.5 1 1.5 2 2.5x 109 Layer Index Exposure (eV/ ?m3 ) (b) Figure 3.4: Distribution curves of layer-based exposure of three arbitrary points in (a) the exposed area and (b) the unexposed area on the substrate system of 500 nm PMMA on Si. The curves were obtained through computing the 3-D exposure distribution. 3.4 Estimation Procedure In Section 3.3, it is implied that d(x0) is only dependent on r(x0;z). However, the resist developing process is isotropic and progresses in all possible directions. Therefore, d(x0) depends on not only r(x0;z) but also r(x;z) in the domain which can be expressed by: {r(x;z)|0 < |x?x0| ? w}; (3.7) where w is a certain width within which development of resist interacts laterally. Though resist is developed in all possible directions, the developing process can be simply modeled as two separate parts, i.e., vertical development and lateral development, as illustrated in Fig. 3.5. Therefore, d(x) may be modeled as a combination of vertical component dV (x) (de?ned as the depth increment due to vertical development) and lateral component dL(x) (de?ned as the depth increment due to lateral development), as illustrated 48 in Fig. 3.6. Note that the increased width due to lateral development makes the depth larger and dL(x) refers to this depth increment. Figure 3.5: The developing process can be modeled as two separate parts, i.e., vertical development and lateral development. For a feature as described in Section 3.3.1 with constant dose, the exposure is always highest in the center and decreases monotonically toward the tail part, which indicates the lateral development is non-existent in the center point. Hence, the exposure of the center of the feature can be estimated using the layer-based model (refer to Section 3.3.3) without considering any lateral development. For any other point in the exposed and unexposed areas, its exposure should be estimated by additionally considering its lateral development due to exposures or equivalently rates of all the points from the center point to the point 49 dVx x dLx dx dVx dLx Figure 3.6: Depth at x, d(x), may be modeled as a combination of vertical component dV (x) and lateral component dL(x). itself. In the current implementation, an iterative procedure is employed in this estimation, which can be expressed by (also refer to Fig. 3.7): dL(xi) = G[T;r(xk)|k = 0;1;2;???;i]; (3.8) where GL[?] represents the process of estimating lateral development. For simplicity, the estimation procedure is described for a target pattern consisting of two lines su?ciently long along the Y-dimension and separated by distance I. In this case, the base pattern is a single line of which remaining resist pro?le obtained through experiment is denoted by d1(x), composed of vertical component d1V (x) and lateral component d1L(x). The remaining resist pro?le of the target pattern, which is to be estimated, is denoted by d2(x). The developing rate distributions of the base and target patterns are denoted by 50 (a) (b) (c) Figure 3.7: Estimation of the lateral component dL(x): (a) the initial resist pro?le d(x), (b) after the lateral component for point x1 is estimated, and (c) after the estimation for all points is completed. r1(x;z) and r2(x;z), respectively. The estimation steps are depicted below (also refer to the ?owchart in Fig. 3.8). Step 1: For the base pattern, set an initial value for its exposure e1(x) (without considering change along the resist depth dimension), i.e., e1(x) = F?1[r1(x)] = F?1[d1(x)T ]. Step 2: Compute the layer-based exposure e1(x;z) based on e1(x) using Eq. 3.3, and convert it into developing rate r1(x;z) using the nonlinear mapping in Eq. 2.4. Step 3: Compute the vertical component d1V (x) using Eq. 3.5, then additionally compute the lateral component d1L(x) using Eq. 3.8. Step 4: Check the percentage depth error ?d1(x) = d1V (x)+d1L(x)?d1(x)d1(x) . If it is smaller than a certain threshold, proceed to Step 5. Otherwise, use it to control the increment of r1(x), or equivalently e1(x), and go back to Step 2. Step 5: Compute the layer-based exposure e2(x;z), i.e., e2(x;z) = e1(x;z) + e1(x ? I;z), and convert it into developing rate r2(x;z) using the nonlinear mapping in Eq. 2.4. 51 d1V(x) + d1L(x) d1V(x) ?d1(x) IncrementControlling r1(x) e1(x) e1(x,z) Layer-basedDevelopingLateralDevelopment Estimation r1(x,z) d2V(x) + d2L(x) e2(x,z) = e1(x,z) + e1(x-I,z) d1(x) ForwardMapping ? InverseMapping ForwardMapping d2V(x) r2(x,z)Layer-based Developing LateralDevelopment Estimation Layer-basedExposure Modeling d2(x) Figure 3.8: Flowchart of the proposed estimation method. Step 6: Compute the vertical component d2V (x) using Eq. 3.5, then additionally compute the lateral component d2L(x) using Eq. 3.8. Step 7: Derive the ?nal depth pro?le d2(x) by taking both vertical and lateral components into account, i.e., d2(x) = d2V (x) +d2L(x). It should be clear that the above estimation procedure can be easily generalized for N-line target patterns where N > 2. 52 3.5 Results and Discussion 3.5.1 Line/Space Patterns In order to evaluate the performance of the proposed estimation method, an extensive simulation has been performed with line/space patterns and di?erent substrate systems. The line width (L), space between lines (S) and the number of lines (N) are varied (refer to Fig. 3.9). The substrate system is composed of PMMA on Si where the three di?erent PMMA thicknesses, 300 nm, 500 nm, and 1000 nm, are considered. In the current model, the resist consists of 100 layers. LX Z H Substrate Resist L LS S Figure 3.9: Cross-section of the resist pro?le of a line/space pattern. 3.5.2 Simulation Results The proposed method and the 2-D Mod method (refer to Section 3.3.2) have been compared to the conventional method (Exp-Dev, refer to Section 3.2) which requires exposure computation and resist development simulation. A set of typical remaining resist pro?les estimated by the three methods is provided in Figs. 3.10, 3.11, and 3.12. Compared to the 2-D Mod method, it can be seen that the proposed method can achieve remaining resist pro?les much closer to those by the Exp-Dev method in most cases. However, it should be noticed that the 2-D Mod method never overestimates remaining resist 53 0 200 400 600 200 220 240 260 280 X (nm) Resist Depth (nm) Exp?Dev 2?D Mod Proposed (a) 0 200 400 600 800 200 220 240 260 280 X (nm) Resist Depth (nm) Exp?Dev 2?D Mod Proposed (b) Figure 3.10: Remaining resist pro?les of an 8-line pattern with (a) L = 100 nm/S = 100 nm and (b) L = 100 nm/S = 200 nm on the substrate system of 300 nm PMMA on Si. 54 0 200 400 600 300 340 380 420 460 X (nm) Resist Depth (nm) Exp?Dev 2?D Mod Proposed (a) 0 200 400 600 800 300 340 380 420 460 X (nm) Resist Depth (nm) Exp?Dev 2?D Mod Proposed (b) Figure 3.11: Remaining resist pro?les of an 8-line pattern with (a) L = 100 nm/S = 100 nm and (b) L = 100 nm/S = 200 nm on the substrate system of 500 nm PMMA on Si. 55 0 200 400 600 500 620 740 860 980 X (nm) Resist Depth (nm) Exp?Dev 2?D Mod Proposed (a) 0 200 400 600 800 500 620 740 860 980 X (nm) Resist Depth (nm) Exp?Dev 2?D Mod Proposed (b) Figure 3.12: Remaining resist pro?les of an 8-line pattern with (a) L = 100 nm/S = 100 nm and (b) L = 100 nm/S = 200 nm on the substrate system of 1000 nm PMMA on Si. 56 pro?les since it employs the 2-D exposure model while the proposed method leads to a slight overestimation in some cases. 3.5.3 Performance Comparison The 2-D Mod method and the proposed method are also compared quantitatively in terms of the percent depth errors: Average Percent Depth Error = 1n n? i=1 |dest(i) ?dsim(i)| dsim(i) ? 100%; (3.9) and Max Percent Depth Error = nmax i=1 |dest(i) ?dsim(i)| dsim(i) ? 100%; (3.10) where dest denotes the depth estimated by the 2-D Mod method or the proposed method, and dsim denotes the depth simulated by the Exp-Dev method, respectively, and n is the number of points along the X-dimension. Simulation Settings 2-D Mod Method Proposed Method Ratio of Number of Developing Resist Average Maximum Average Maximum L to S Lines (N) Time (sec) Thickness (nm) (%) (%) (%) (%) 0.33 8 40 1000 10.78 11.47 0.89 2.57 0.5 8 40 1000 11.47 11.89 1.03 2.79 1.0 8 40 1000 13.08 13.58 1.02 2.74 2.0 8 40 1000 15.65 16.56 1.41 2.91 Table 3.1: Average and maximum percent depth errors with respect to the ratio of L to S with N = 8 on the substrate system of 1000 nm PMMA on Si. 57 Simulation Settings 2-D Mod Method Proposed Method Ratio of Number of Developing Resist Average Maximum Average Maximum L to S Lines (N) Time (sec) Thickness (nm) (%) (%) (%) (%) 1.0 8 40 500 6.08 6.61 1.51 2.21 1.0 16 40 500 7.23 7.91 1.69 2.82 1.0 32 40 500 7.69 9.08 4.03 6.36 Table 3.2: Average and maximum percent depth errors with respect to the number of lines with L = 100 nm/S = 100 nm on the substrate system of 500 nm PMMA on Si. Simulation Settings 2-D Mod Method Proposed Method Ratio of Number of Developed Resist Average Maximum Average Maximum L to S Lines (N) Depth (nm) Thickness (nm) (%) (%) (%) (%) 1.0 8 333 1000 7.73 8.78 0.27 0.97 1.0 8 514 1000 8.85 9.66 0.29 1.12 1.0 8 716 1000 11.30 12.10 1.26 2.31 1.0 8 914 1000 13.08 13.58 1.02 2.74 Table 3.3: Average and maximum percent depth errors with respect to the developed depth (controlled by the developing time) with L = 100 nm/S = 100 nm and N = 8 on the substrate system of 1000 nm PMMA on Si. Simulation Settings 2-D Mod Method Proposed Method Ratio of Number of Developing Resist Average Maximum Average Maximum L to S Lines (N) Time (sec) Thickness (nm) (%) (%) (%) (%) 1.0 8 40 300 4.57 5.14 1.26 3.05 1.0 8 40 500 6.08 6.61 1.51 2.21 1.0 8 40 1000 13.08 13.58 1.02 2.74 Table 3.4: Average and maximum percent depth errors with respect to the resist thickness (PMMA thickness) with L = 100 nm/S = 100 nm and N = 8. This percent depth error is de?ned for general resist pro?les (binary or grayscale lithog- raphy). But for binary lithography where feature width is of main concern, percent width error may be adopted. The average and maximum percent depth errors are provided with respect to the ratio of L to S, the number of lines, the developed depth (controlled by the developing time), and the resist thickness in Tables. 3.1, 3.2, 3.3, and 3.4, which are also plotted in Fig. 3.13. The results show that the proposed method outperforms the 2-D 58 Mod method by reducing the average error up to about 14.24%. And the reduction of the maximum error is about 13.65%. 0 0.5 1 1.5 20 24 68 1012 1416 1820 Ratio of L to S Error (%) 2?D Mod Mean Error 2?D Mod Max ErrorProposed Mean Error Proposed Max Error (a) 8 16 24 320 2 4 6 8 10 12 Numer of Lines Error (%) 2?D Mod Mean Error 2?D Mod Max ErrorProposed Mean Error Proposed Max Error (b) 200 400 600 800 10000 2 4 6 8 10 12 14 16 Developed Depth (nm) Error (%) 2?D Mod Mean Error 2?D Mod Max ErrorProposed Mean Error Proposed Max Error (c) 200300400500600700800900100011000 2 4 6 8 10 12 14 Resist Thickness (nm) Error (%) 2?D Mod Mean Error 2?D Mod Max ErrorProposed Mean Error Proposed Max Error (d) Figure 3.13: Average and maximum percent depth errors with respect to (a) the ratio of L to S with N = 8 on the substrate system of 1000 nm PMMA on Si; (b) the number of lines with L = 100 nm/S = 100 nm on the substrate system of 500 nm PMMA on Si; (c) the developed depth (controlled by the developing time) with L = 100 nm/S = 100 nm and N = 8 on the substrate system of 1000 nm PMMA on Si; (d) the resist thickness (PMMA thickness) with L = 100 nm/S = 100 nm and N = 8. From the above tables and plots of errors, it is observed that as each of the ratio of L to S, the number of lines, the developed depth (controlled by the developing time), and the resist thickness increases, the advantage of the proposed method becomes more and more visible. For a larger ratio of L to S or a larger number of lines, interaction among lines increases and therefore a less accurate estimation method, 2-D Mod, su?ers more. For a 59 thicker resist, the exposure variation along the resist depth dimension is larger, which makes the 2-D Mod cause larger errors since it ignores the variation. 3.5.4 Experimental Results The proposed estimation scheme has been tested also via experiment. Si wafer was spin-coated with 300 nm PMMA and soft-baked at 160 oC for 1 minute. Two separated long lines of 100 nm wide were exposed with di?erent doses (200 C=cm2 and 225 C=cm2) using ELIONIX ELS-7000 e-beam lithography system (50 KeV) and the sample was developed in MIBK:IPA=1:2 for 40 seconds. (a) (b) (c) (d) Figure 3.14: Experimental results of (a) resist pro?le of set I measured from the SEM image; (b) resist pro?le of set II measured from the SEM image; (c) resist pro?le of set I estimated from that of set II; (d) resist pro?le of set II estimated from that of set I. 60 The resist pro?les of each set are measured from the cross-section SEM image, which are provided in Fig. 3.14(a) (denoted by set I) and Fig. 3.14(b) (denoted by set II). Using the resist pro?le of one set as the source, that of the other is estimated through the proposed method, which are provided in Fig. 3.14(c) and Fig. 3.14(d). The average percent depth errors between the measured and estimated resist pro?les of set I and set II are 5.25% and 3.81%, respectively, demonstrating high accuracy of the proposed estimation method. 3.6 Summary In this chapter, a method for estimation of remaining resist pro?les of line/space pat- terns, which does not require exposure calculation and resist development simulation, is proposed. By analyzing the simulation results from a set of base patterns on di?erent sub- strate systems, a layer-based model is formulated. An adaptive procedure is employed in estimation of the lateral development in remaining resist pro?les. Through simulation and also experiments, it is shown that the proposed method estimates almost the same resist pro- ?les as those by a typical conventional method like the Exp-Dev method in most cases, and outperforms the 2-D Mod method by further reducing the estimation error. Therefore, the proposed method has a good potential to provide an alternative way for estimation of resist pro?les where the conventional methods are not applicable and/or are too time-consuming. 61 Chapter 4 Critical Dimension Control for Large-scale Uniform Patterns 4.1 Introduction For circuit patterns with nanoscale features, it is not unusual that the actual written pattern is substantially di?erent from the written pattern estimated by a 2-D PEC method. One of the reasons for this deviation is that the 2-D model ignores the exposure variation along the resist depth dimension. In an earlier study [94], it was shown that the remaining resist pro?le estimated using a 2-D model can be signi?cantly di?erent from the one based on a 3-D model which considers the depth-dependent exposure variation. Another reason is that the actual resist pro?le cannot be derived directly from the exposure distribution, e.g., the developing rate is not linearly proportional to the exposure at a given point. Therefore, the PEC methods which consider the exposure only may su?er from substantial CD errors or deviation from the target resist pro?le in general. In addition, with a 2-D model, it is not possible to consider the lateral development of resist which a?ects the resist pro?le, in particular the sidewall shape, signi?cantly. In the previous work [94, 95, 96], the limitations of 2-D PEC were analyzed and the need for 3-D PEC was well demonstrated for circuit patterns with nanoscale features. In [95], the idea of true 3-D PEC was proposed and its ?rst implementation for a single line and a small number of lines was reported. In this implementation, the resist pro?le estimated through simulation of resist development process was employed instead of the exposure distribution, in order to obtain more realistic results. In this chapter, an extension of the 3-D PEC method for large-scale uniform patterns is described. A challenge is that a 3-D PEC method requires a tremendous amount of computation due to the increased dimensionality and resist development simulation. The 62 feature-by-feature correction procedure would be too time-consuming to be practical. Hence, a new 3-D PEC method has been developed which cuts down the required computation greatly and still achieves the resist pro?le closely matched to the target one. It explicitly corrects the features only at the critical locations in a circuit pattern and employs a fast correction procedure for other features exploiting the uniformity of feature distribution. A distinct aspect of the method is that it attempts to adjust the exposures cross (inside and outside) the feature boundary in order to control the location and shape of sidewall in the resist pro?le. The rest of the chapter is organized as follows. The large-scale uniform patterns are described in Section 4.2. The 2-D and 3-D correction methods are brie?y reviewed in Section 4.3. The proposed 3-D correction procedure is presented in detail in Section 4.4. Simulation results are discussed in Section 4.5, followed by a summary in Section 4.6. 4.2 Large-scale Uniform Patterns A large-scale uniform pattern consists of a large number of same features (line, square, etc.) which is replicated uniformly throughout the pattern. One practical issue for this kind of patterns is that the proximity e?ect is considerable due to a very high pattern density. Especially, compared to the corner and edge of the pattern, the center of the pattern su?ers from much higher proximity e?ect which results in a severe deviation of resist pro?les at di?erent locations (refer to Fig. 4.1). 4.3 Correction Methods Let?s assume that a target written pattern is speci?ed by the corresponding resist pro?le (e.g., line widths at the top, middle and bottom layers) and the cross-section of resist pro?le is in the X-Z plane (refer to Fig. 2.4). During correction, at each location (x;y) in a circuit pattern, the cross-section of resist pro?le needs to be examined. Then, e(x;y;z) and r(x;y;z) can be replaced by e(x;z) and r(x;z), respectively. 63 Figure 4.1: The deviation of resist pro?les at corner, edge and center of a large-scale uniform pattern without PEC. 4.3.1 2-D Exposure Correction (2-D PEC) In the 2-D exposure model, the exposure is assumed not to vary along the resist depth dimension (Z-dimension), i.e., e(x;z) in the cross-section is averaged over 0 ? z ? H (refer to Fig. 2.4), resulting in e(x). Let h(x) represent the depth distribution (i.e., depth at x) in the target resist pro?le. A target exposure distribution Et(x) is derived from h(x) by: Et(x) = F?1[h(x)T ]; (4.1) where T is the developing time. 64 The exposure Et(x) can be considered as a threshold for a certain point to be fully developed (refer to Fig. 4.2). 0 50 100 150 200 250 3000 1 2 3 4x 10 10 X (nm) Ex pos ure (eV /Pm 3 ) Et x Figure 4.2: The target 2-D exposure distribution is illustrated for a line feature. Then, the dose distribution d(x) required to achieve the target resist pro?le (depth distribution) h(x) may be determined iteratively by minimizing the following error: Error = ? x |e(x) ?Et(x)|: (4.2) In each iteration, e(x) is computed from d(x) according to Eq. 2.3. The objective of this 2-D correction scheme is to control the dose distribution d(x) such that e(x) is as high as the threshold Et(x) in the exposed area, and as low as possible in the unexposed area. 4.3.2 3-D Resist Pro?le Correction (3-D PEC) Unlike the above 2-D exposure correction, this 3-D correction incorporates the estima- tion of remaining resist pro?le into the correction procedure and utilizes the resist pro?le to determine the dose distribution based on a 3-D model. The overall correction procedure is very similar to that of the 2-D correction. However, the error, which is to be minimized 65 determining the corrected dose distribution, is computed based on the estimated remaining resist pro?le rather than the exposure distribution, and is given by: Error = max j |p(j) ?q(j)|; (4.3) where p(j) and q(j) denote the width distribution of the estimated remaining resist pro?le and that of the target resist pro?le, respectively (refer to Fig. 4.3). 1 1 6 10 6 10 Figure 4.3: The cross-section of resist pro?le is illustrated for a line feature where pj and qj are the actual and target widths at the j-th layer, respectively, where the resist is modeled by 10 layers. 4.4 Proposed Correction Procedure The proposed 3-D PEC method minimizes the deviation of resist pro?le from the target one, avoiding the feature-by-feature correction without sacri?cing the correction quality. It consists of three steps: correcting a single feature in isolation, global adjustment of feature- wise dose, and intra-feature dose control at critical locations. 66 4.4.1 Correction of a Single Feature In the ?rst step of the proposed approach, a single instance of the repeated feature in the pattern is corrected in isolation. The feature is partitioned into regions for each of which a dose is determined using the 3-D resist pro?le correction (refer to Section 4.3.2). The cost function to determine the dose distribution is given by (also refer to Fig. 4.3): Error = max j=1;5;10 |p(j) ?q(j)|; (4.4) where j is the layer index (Z-dimension) from 1 to 10. p(j) and q(j) are considered as p(z) and q(z) (refer to Section 4.3.2) sampling at j-th layer. In the current implementation, the error de?ned in Eq. 4.3 is computed only considering the top, middle and bottom layers of resist. The maximum of the three errors is minimized such that the pro?le as close to the target resist pro?le as possible is obtained. The region- wise dose distributions along with the resist pro?les before and after correction are illustrated for a line feature partitioned into 5 regions in Fig. 4.4. The dose distribution within the feature is denoted by d(i) and the corresponding expo- sure distribution by e(i) where i is the region index (refer to Fig. 4.4). The spatial averages of d(i) and e(i) are denoted by D and E, respectively, i.e., D = 1R R? i=1 d(i) and E = 1R R? i=1 e(i), where R is the number of regions. Note that all of the notations, d(i), e(i), D and E (i.e., without subscripts) are for a single feature in isolation. 4.4.2 Feature-wise Global Adjustment In the second step, the global distribution of feature-wise dose throughout the pattern is derived by the following deconvolution: A = Et ~?1 psf; (4.5) 67 (a) (b) (c) (d) Figure 4.4: Illustration of (a) the region-wise dose distribution within the feature before correction, (b) the region-wise dose distribution within the feature after correction, (c) the corresponding resist pro?le before correction, and (d) the corresponding resist pro?le after correction. where Et(m;n) is the target (average) exposure for the m;n-th location of the feature and is assigned the value of E (refer to Section 4.4.1), and psf is the normalized PSF (refer to Section 2.2) sampled at the feature interval. The output of the deconvolution, i.e., matrix A, which speci?es the (average) feature- wise dose distribution required to achieve E at all locations of the feature, is referred to as deconvolution surface. This deconvolution is not computationally intensive since the spatial resolution involved is coarse (feature size). For the m;n-th location of the pattern, d(i) is weighted (scalar multiplication) by the deconvolution surface A to results in (refer to Fig. 4.5): 68 Weight 0 10 20 30 40 50 010 2030 4050 0.7 0.8 0.9 1 X (Pm)Y (Pm) d(1)d(2)d(3)d(4)d(5) d(1)d(2)d(3)d(4)d(5) di dm,n i Am,n Figure 4.5: For each location (m;n) of the feature (line segment), d(i) is weighted by the deconvolution surface A(m;n;). dm;n(i) = d(i) ?A(m;n): (4.6) The spatial average of dm;n(i) is denoted by Dm;n. Now, em;n(i), corresponding to dm;n(i), is well balanced to be E throughout the pattern, but the exposure in the unexposed area (referred to as background exposure) may vary with location (refer to Fig. 4.6). Then, as illustrated in Fig. 4.7, the resist pro?les at di?erent locations would be di?erent. This vari- ation is due to the di?erence in the background exposure even though the internal exposure is equalized by the deconvolution surface. In order to achieve the resist pro?le throughout 69 0 50 100 150 2000 0.5 1 1.5 2 x 10 10 Radius (nm) Ex po su re (e V /Pm 3 ) C orner Edge C enter Figure 4.6: The exposure distributions obtained using the global (weighted) dose distribu- tion. the pattern, one may further adjust the internal exposure to compensate for the di?erence in the background exposure among locations. (a) (b) (c) Figure 4.7: The corresponding resist pro?les at (a) corner, (b) edge, and (c) center of the pattern from the exposure distributions obtained using the global (weighted) dose distribu- tion. 70 4.4.3 Critical-location-based Dose Control In a large-scale uniform pattern, the same feature such as line, square, etc., is repli- cated uniformly throughout the pattern. Then, the global spatial distribution of exposure is smooth, and in turn the global distribution of feature-wise dose required for correction must be smooth, which is indicated by the deconvolution surface (refer to Section 4.4.2). Therefore, it is su?cient to consider only the features at the critical locations, e.g., corner, edge and center (refer to Fig. 4.8) instead of a feature-by-feature correction. Figure 4.8: Line segments only at the critical locations of the pattern are corrected individ- ually. Therefore, in the third step, dm;n(i) at each of the three critical locations is adjusted based on the same scheme in Section 4.4.1. One di?erence is that the background expo- sure at each critical location, which is computed using the deconvolution surface (refer to Section 4.4.2), should be considered in the iterative correction. For each critical location, the total exposure is the sum of the base exposure and the background exposure, where the base exposure depends on dm;n(i) within each iteration, and the background exposure only 71 depends on Dm;n, i.e., constant among iterations. Since the resist pro?le after the second step is already close to the target one, dm;n(i) can be used as an initial distribution in order to minimize the number of iterations. (a) (b) (c) Figure 4.9: The adjusted dose distributions for (a) corner, (b) edge, and (c) center of the pattern. 0 50 100 150 2000 0.5 1 1.5 2 x 10 10 Radius (nm) Ex po su re (e V /Pm 3 ) C orner Edge C enter (a) (b) Figure 4.10: (a) The corresponding exposure distributions and (b) the corrected resist pro?le using the adjusted dose distributions. The corrected dose distributions at the critical locations are illustrated in Fig. 4.9 and the corresponding exposure distributions in Fig. 4.10(a). Achieving the identical exposure distribution at all locations in a large pattern, which is attempted by most of the conventional 72 PEC methods, would be extremely di?cult if not impossible. The proposed correction method lets the di?erence in the internal exposure compensate for that in the background exposure such that the deviation from the target resist pro?le is minimized at the critical locations (refer to Fig. 4.10(b)). After the ?nal dm;n(i) is obtained at each critical location, a 2-D global interpolation is employed to compute the dm;n(i) at all other locations based on the deconvolution surface. Through the above correction procedure (also refer to the ?owchart in Fig. 4.11), the dose distribution is well adjusted throughout the pattern with accuracy and e?ciency. Large-scaleUniform Pattern PSF Single Feature A(m,n) d(i) dm,n(i) Interpolation Deconvolution Single FeatureDose Control Feature-wiseGlobal Adjustment Critical LocationDose Control Figure 4.11: Flowchart of the proposed correction procedure. 73 4.5 Results and Discussion 4.5.1 Test Patterns The proposed approach to 3-D PEC for large-scale uniform patterns has been imple- mented and its performance has been analyzed through simulation. Two kinds of the uniform pattern consisting of lines shown in Fig. 4.12 are employed for performance analysis. In the one, to be referred to as Pattern I, each line is 51 m long and 50 nm wide, and the gap between lines is 50 nm. In the other pattern, to be referred to as Pattern II, each line is 51 m long and 100 nm wide, and the gap between lines is 100 nm. Both patterns cover the full range of electron scattering (for 50 KeV). The correction program partitions each line into segments of 3 m for dose control. The three critical and three test locations are shown in Fig. 4.12. Figure 4.12: Critical locations (marked by ?) and test locations (marked by ?) in a test pattern. 74 4.5.2 Simulation Results The PSFs employed in the simulation are generated by a Monte Carlo simulation method, SEEL [111]. The substrate systems employed in the simulation are composed of PMMA on Si where the three di?erent PMMA thicknesses, 100 nm, 300 nm, and 500 nm, are considered. The beam energy is set to 50 KeV with the beam diameter of 5 nm. These three substrate systems along with the two patterns provide six di?erent combinations so that the proposed approach can be thoroughly tested. The patterns are corrected by the 2-D PEC method (refer to Section 4.3.1) and the proposed 3-D PEC method for the target resist pro?le of vertical sidewall. No correction (a uniform dose for all features), the 2-D PEC method, and the 3-D PEC method are compared with a constraint that the total dose for a pattern must be the same for all the methods. Also in all cases, only 80% of the line width is exposed in order to compensate the high pattern density (50%). The results (cross-section resist pro?les) of Pattern I for 300 nm PMMA on Si are provided in Figs. 4.13, 4.14, and 4.15. From the pro?les obtained without correction (Fig. 4.13), it can be seen that the resist pro?le signi?cantly varies with location. The line width is not uniform and the sidewall is clearly overcut. A distinct aspect of the 3-D PEC method is that it explicitly equalizes the resist pro?le (rather than the exposure) of a feature throughout a pattern. Though the resist pro?les (Fig. 4.14) obtained by the 2-D PEC method are better than those without correction, the variation of resist pro?le among locations is still substantial. However, as can be seen in Fig. 4.15, the proposed 3-D PEC method greatly improves the uniformity of resist pro?le among not only the critical locations but also the test locations, and also achieves the resist pro?les much closer to the target one in terms of CD error and sidewall shape (vertical). 75 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (a) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (b) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (c) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (d) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (e) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (f) Figure 4.13: Cross-section resist pro?les without correction (i.e., uniform dose) at the critical locations ((a) corner, (b) edge and (c) center; refer to Fig. 4.12) and the test locations ((d), (e) and (f); refer to Fig. 4.12). 76 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (a) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (b) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (c) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (d) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (e) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (f) Figure4.14: Cross-sectionresistpro?lesachievedby2-DPECmethodatthecriticallocations ((a) corner, (b) edge and (c) center; refer to Fig. 4.12) and the test locations ((d), (e) and (f); refer to Fig. 4.12). 77 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (a) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (b) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (c) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (d) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (e) 50 100 0 50 100 150 200 250 300 X (nm) Resist Depth (nm) (f) Figure4.15: Cross-sectionresistpro?lesachievedby3-DPECmethodatthecriticallocations ((a) corner, (b) edge and (c) center; refer to Fig. 4.12) and the test locations ((d), (e) and (f); refer to Fig. 4.12). 78 4.5.3 Performance Comparison The correction methods are also compared quantitatively in terms of the percent width errors (refer to Section 4.3.2): Average Percent Width Error = 1n n? j=1 |p?(j) ?q(j)| q(j) ? 100%; (4.7) and Max Percent Width Error = nmax j=1 |p?(j) ?q(j)| q(j) ? 100%; (4.8) where p?(j) is the width of j-th layer in the ?nal resist pro?le after correction, q(j) is the target resist pro?le as de?ned in Section 4.4.1, and n is the number of resist layers. The average and maximum percent width errors for PatternI and Pattern II are provided in Table 4.1 and Table 4.2, respectively. Note that the maximum error reaches 100% in some cases, due to the underdevelopment in resist pro?les (e.g., the bottom layer is not developed at all, leading to a zero width). The results show that the errors for the 2-D PEC method are signi?cantly smaller than those when no correction is done (i.e., a uniform dose). More importantly, compared to the 2-D PEC method, the proposed 3-D PEC method further reduces both average and maximum errors greatly. PMMA No Correction 2-D PEC Method 3-D PEC Method Thickness Average Maximum Average Maximum Average Maximum (nm) (%) (%) (%) (%) (%) (%) 100 5.20 12.33 4.40 8.33 2.33 4.33 300 44.39 100.00 36.07 94.33 8.70 13.17 500 69.08 100.00 65.45 100.00 41.30 60.33 Table 4.1: Average and maximum percent width errors in resist pro?les of Pattern I for no correction, 2-D PEC method and 3-D PEC method. 79 PMMA No Correction 2-D PEC Method 3-D PEC Method Thickness Average Maximum Average Maximum Average Maximum (nm) (%) (%) (%) (%) (%) (%) 100 1.12 1.67 0.50 1.08 0.32 1.00 300 13.90 32.27 10.00 17.00 2.57 4.42 500 45.85 100.00 33.88 100.00 9.50 13.58 Table 4.2: Average and maximum percent width errors in resist pro?les of Pattern II for no correction, 2-D PEC method and 3-D PEC method. From the above tables of errors, it is observed that as the resist thickness increases, the improvement (reduction in the total percent errors) by the proposed method becomes larger. For a thicker resist, the exposure variation along the resist depth dimension is larger, which makes the 2-D PEC method su?er from larger errors since it ignores the variation. Also, the increase of exposure variation makes the resist pro?le more sensitive to the dose distribution. Therefore, incorporating the estimation of remaining resist pro?le into the correction procedure can achieve a more realistic correction, which lacks in the 2-D PEC method. When the feature size decreases, the improvement by the proposed method tends to be larger (except for the resist thickness of 500 nm). For the same substrate system, a smaller feature is a?ected relatively more by the proximity e?ect and the development process, therefore has a larger room for improvement. However, for a very thick resist, the PEC itself becomes harder for a smaller feature. 4.6 Summary A new practical method for true 3-D PEC, which can handle large-scale uniform patterns e?ciently, has been developed. It performs true 3-D PEC by taking into account the exposure variation along the resist depth dimension (in addition to the lateral dimensions). The cost (error) function employed in correction is not the conventional (2-D) CD error, but the 3-D CD error, i.e., the CD errors at multiple layers of resist are considered. Also, the method incorporates the resist development simulation into the correction procedure to 80 achieve realistic results. In order to alleviate the computational requirement for large-scale patterns, a critical-location-based correction approach is taken along with the deconvolution surface for deriving the corrected doses for other locations. Through an extensive simulation, it has been shown that the proposed 3-D PEC method can signi?cantly improve correction results, compared to the 2-D PEC method, in terms of resist pro?le, i.e., not only CD error but also sidewall shape. 81 Chapter 5 Sidewall Shape Control: Vertical Sidewall 5.1 Introduction For the most common sidewall shape, i.e., the vertical sidewall, a typical 2-D target exposure distribution is constant within and zero outside a feature. Then the required dose distribution obtained by the deconvolution is of ?V-shape,? i.e., the dose is highest at the edge and gradually decreases toward the center of a feature. However, our earlier studies have shown that such 2-D PEC does not lead to realistic results especially for nanoscale features and therefore true 3-D PEC is needed [94, 95, 96]. In 3-D PEC, the resist pro?le estimated through simulation of resist development was employed instead of the exposure distribution, in order to obtain more realistic results. Moreover, it has been demonstrated that in order to minimize any deviation from a target resist pro?le, a 3-D PEC scheme must check the estimated resist pro?le during the dose optimization procedure [95]. One practical issue of such an approach to 3-D PEC is that a time-consuming resist development simulation needs to be carried out in each iteration of the dose optimization. Also, it is shown that the dose distribution of ?V-shape? is not optimal for realizing a vertical sidewall of the resist pro?le, especially when the total dose is to be minimized. Note that a higher total dose worsens the charging e?ect and lengthens the exposing time. In this chapter, an e?cient dose optimization scheme which does not require a direct resist development simulation while achieving a target 3-D resist pro?le has been developed. The systematic dose updating procedure coupled with the fast resist development simulation makes the dose optimization scheme fast and e?ective. Besides, an adaptive scheme has been developed which determines the optimal dose distribution type and the minimum of 82 required total dose based on a given circuit pattern and substrate system settings. The most noteworthy result from this study is that in order to achieve a vertical sidewall of nanoscale feature with the minimum total dose, one has to employ a dose distribution di?erent from the conventional one of ?V-shape.? Based on our 3-D exposure model and 3-D PEC method, the new dose distributions of ?M-shape? and ?A-shape? have been derived for achieving the target resist pro?le of vertical sidewall minimizing the total dose. Through an extensive simulation, they are compared with other distributions, uniform and ?V-shape,? in terms of deviation from the target resist pro?le of vertical sidewall. The rest of the chapter is organized as follows. Details of the conventional and proposed dose distribution types are presented in Section 5.2. The proposed dose optimization scheme and dose determination scheme for single features are described in Section 5.3 and Section 5.4, respectively. The complete correction procedures of sidewall shape control for large- scale uniform patterns and large-scale nonuniform patterns are presented in Section 5.5 and Section 5.6, respectively. Simulation results are discussed in Section 5.7, followed by a summary in Section 5.8. 5.2 Dose Distribution Types In this chapter, only a single feature of line is considered for simplicity. Suppose that a target written feature is speci?ed by the corresponding resist pro?le (e.g., line widths at the top, middle and bottom layers) in 3-D PEC. In our simulation model, the line is su?ciently long in the Y-dimension such that the dimensional variation along the Y-dimension may be ignored. Then, one only needs to consider the cross-section of resist pro?le in the X-Z plane (refer to Fig. 2.5). During correction, at each location (x;y) in a circuit pattern, the cross-section of resist pro?le needs to be examined. Therefore, e(x;y;z) and r(x;y;z) can be replaced by e(x;z) and r(x;z), respectively. 83 In order to avoid a high complexity of the optimization procedure and also have a su?cient spatial control of dose distribution, the line feature is partitioned into 5 regions along its length dimension and a dose is determined for each region, as shown in Fig. 5.1. Figure 5.1: The region-wise feature partition with a uniform dose distribution. The main goal of 3-D PEC is to adjust the spatial distribution of dose in order to minimize the deviation of resist pro?le from the target one. The focus of this study is on investigating the e?ectiveness of dose distribution type (shape) in achieving the target resist pro?le, in particular vertical sidewall. 5.2.1 Conventional Type A typical shape of spatial dose distribution cross a line feature is such that the dose is highest in the edge regions and gradually decreases toward the center region (to be referred to as Type-V according to its shape). This type of dose distribution is usually obtained by an exposure-based correction where a target 2-D exposure distribution is speci?ed (to be referred to as 2-D exposure correction [96]). In the 2-D exposure model, the exposure is assumed not to vary along the resist depth dimension (Z-dimension), i.e., e(x;z) in the cross-section is averaged over 0 ? z ? H (refer 84 to Fig. 2.4), resulting in e(x). Let h(x) represent the depth distribution (i.e., depth at x) in the target resist pro?le. A target exposure distribution Et(x) is derived from h(x) by: Et(x) = F?1[h(x)T ]; (5.1) where T is the developing time. The exposure Et(x) can be considered as a threshold for a certain point to be fully developed. Therefore, the objective of this 2-D exposure correction is to control the dose distribution d(x) such that e(x) is as high as the threshold Et(x) in the exposed area, and as low as possible in the unexposed area, i.e., ideally a rectangular window function with width of the feature size and height of Et(x) (refer to Fig. 5.2). 0 50 100 150 200 250 3000 1 2 3 4x 10 10 X (nm) Ex pos ure (eV /Pm 3 ) Et x Figure 5.2: The target 2-D exposure distribution is illustrated for a line feature. Then, the dose distribution d(x) required to achieve this kind of e(x) can be determined through iterations of dose adjustment, or equivalently deconvolution of e(x) with the PSF. In general, d(x) derived from deconvolution of a rectangular window function with the PSF is of Type-V. 85 The Type-V dose distribution seems quite reasonable according to the above analysis. However, when the feature size decreases into nanoscale, it usually results in an undesired 3-D resist pro?le in terms of CD error and sidewall shape, and requires a relatively higher total dose for full development (refer to Section 5.2.2 for more details). This is mainly due to the drawbacks of the 2-D exposure correction. Most importantly, in the 2-D model, the exposure variation along the resist depth dimension is not taken into account while the resist pro?le depends on the variation to great extents. Also, the actual resist pro?le cannot be derived directly from the exposure distribution, e.g., the resist pro?le derived from the above-mentioned exposure distribution (a rectangular window function) cannot have a vertical sidewall. The 2-D exposure correction does not consider the resist development process and therefore is likely to end up with an unrealistic result. For example, the lateral development of resist, which signi?cantly a?ects the resist pro?le in particular the sidewall shape of ?ne features, is not taken into account in the 2-D exposure correction. 5.2.2 Proposed Types In order to overcome the drawbacks of the 2-D exposure correction, a resist-pro?le-based correction was developed along with a 3-D model (to be referred to as 3-D resist pro?le cor- rection [96]). Unlike the above 2-D exposure correction, this 3-D correction incorporates the estimation of remaining resist pro?le into the correction procedure and utilizes the resist pro?le to determine the dose distribution based on a 3-D model. The overall correction pro- cedure is similar to that of the 2-D correction. However, the error, which is to be minimized determining the corrected dose distribution, is computed based on the estimated remaining resist pro?le rather than the exposure distribution, and is given by: Error = max j |p(j) ?q(j)|; (5.2) where p(j) and q(j) denote the width distribution of the estimated remaining resist pro?le and that of the target resist pro?le, respectively (refer to Fig. 5.3). 86 1 1 6 10 6 10 Figure 5.3: The cross-section of resist pro?le is illustrated for a line feature where pj and qj are the actual and target widths at the j-th layer, respectively, where the resist is modeled by 10 layers. Based on the 3-D resist pro?le correction, two other types of dose distributions are proposed and analyzed in order to achieve a target resist pro?le with the minimal total dose. The two new dose distribution types are one with the highest dose in the two middle regions (to be referred to as Type-M according to its shape) and the other with the highest dose in the center region and monotonically decreasing toward the edge regions (to be referred to as Type-A according to its shape). The e?ectiveness of the Type-M and Type-A dose distributions in achieving a vertical sidewall stems from the 3-D distributions of exposure that they result in. In Section 3.3.3, it was noticed that the distribution of exposure along the resist depth dimension behaves quite di?erently between the exposed area and the unexposed area, as shown in Fig. 5.4(a) and Fig. 5.4(b). For the exposed area, the exposure usually decreases along the resist depth dimension, while it changes abruptly to an increasing behavior in the unexposed area. 87 (a) 0 50 100 150 200 250 3000 1 2 3 4x 1010 Z (nm) Exposure (eV/ ?m 3 ) A B C (b) (c) 0 50 100 150 200 250 3000 1 2 3 4x 1010 Z (nm) Exposure (eV/ ?m 3 ) A B C (d) Figure 5.4: Comparison of (a) the uniform dose distribution and (b) the corresponding distribution curves of 3-D exposure along Z-dimension (resist depth), with (c) a non-uniform dose distribution and (d) the corresponding distribution curves of 3-D exposure along Z- dimension (resist depth) at three points (A, B, and C) on the substrate system of 300 nm PMMA on Si. For a given point xi, the distribution of its self-exposure (denoted as es(xi;z)) decreases along the resist depth dimension. But the distribution of the sum of exposure contributions from its neighboring points (denoted as eo(xi;z)) increases along the resist depth dimension. 88 It is obvious that the overall behavior of exposure distribution for that point, i.e., e(xi;z) = es(xi;z) +eo(xi;z), depends on the dose distribution. If the region which the point belongs to has a relatively lower dose while its neighboring regions have high doses, then eo(xi;z) becomes the dominant component and e(xi;z) tends to increase with z, and vice versa (refer to Fig. 5.4(c) and Fig. 5.4(d)). Based on this observation, the resist developing behaviors of di?erent dose types can be analyzed, which are shown in Figs. 5.5, 5.6, and 5.7 where a thicker arrow corresponds to a higher developing rate. The developing time is given by T, while T1 and T2 are two intermediate points in time with T1 < T2 < T. (a) (b) (c) Figure 5.5: The resist pro?le during development process (a) at time T1, (b) at time T2, and (c) at time T for Type-V case, respectively, where T1 < T2 < T. For the vertical sidewall shape, take the Type-A dose distribution for an example (also refer to Fig. 5.7). The top layer of resist develops earlier than the bottom layer of resist in the center region. However, the exposure distribution of the edge and middle regions increases in the Type-A. As a result, the developing rate of the bottom layer of resist is higher than that of the top layer for the edge and middle regions. Therefore, when the developer reaches 89 (a) (b) (c) Figure 5.6: The resist pro?le during development process (a) at time T1, (b) at time T2, and (c) at time T for Type-M case, respectively, where T1 < T2 < T. the bottom layer of resist and begins to develop laterally from the center region toward the edge regions, it ?nally catches up with the developing process of the top layer of resist at the boundaries of the feature (the boundaries between the exposed area and the unexposed area) and achieves a vertical sidewall. Note that the exposure distribution in the top layer has a larger contrast over the feature boundaries. From Fig. 5.5, it is also seen that the Type-V cannot achieve the target vertical sidewall. In the Type-V, the top layer of resist always develops earlier than the bottom layer of resist within the edge regions. As a result, the developing process of the bottom layer of resist can never catch up with that of the top layer before reaching the boundaries, thus resulting in an overcut sidewall. This problem may be solved by introducing shape correction, i.e., exposing an area smaller than the feature. However, in that way, the Type-V becomes an implicit version of the Type-M. Another solution is to give a higher total dose for the Type-V case to let the bottom layer develop longer, which however requires a higher total dose and tends to make the width of written feature greater than the target width. 90 d1 d2 d3 d4 d5 (a) d1 d2 d3 d4 d5 (b) d1 d2 d3 d4 d5 (c) Figure 5.7: The resist pro?le during development process (a) at time T1, (b) at time T2, and (c) at time T for Type-A case, respectively, where T1 < T2 < T. Note that the new ?nding of Type-M and Type-A dose distributions would not have been possible without using our 3-D exposure model and 3-D PEC method. 5.3 Dose Optimization Scheme An e?cient iterative optimization scheme has been developed which derives the dose distribution of each type minimizing the deviation from the target resist pro?le, i.e., vertical sidewall, and total dose. Each iteration consists of three steps: fast exposure computation, critical-path-based resist development simulation, and type-based dose updating. 5.3.1 Fast Exposure Computation In the ?rst step, the three base exposures (region-wise) (ei(x;z) | i = 1;2;3) are precal- culated using the 3-D exposure model (refer to Section 2.3). 91 d1 d5 50 100 150 200 2500 2 4 6 8 10x 109 X (nm) Exp osur e (eV /Pm3 ) TopMiddle Bottom (a) d2 d4 50 100 150 200 2500 2 4 6 8 10x 109 X (nm) Exp osur e (eV /Pm3 ) TopMiddle Bottom (b) d3 50 100 150 200 2500 2 4 6 8 10x 109 X (nm) Exp osur e (eV /Pm3 ) TopMiddle Bottom (c) Figure 5.8: The exposure distribution of the top, middle and bottom layers of resist when only (a) the two edge regions, (b) the two middle regions, and (c) the center region are exposed with a unit dose. Note that ei(x;z) is the exposure when a unit dose is given only to the i-th and (6?i)-th regions, i.e., d(i) = 1:0 and d(6 ? i) = 1:0 while the other regions are not exposed (refer to Fig. 5.8). The total exposure e(x;z) can be computed by: 92 e(x;z) = 3? i=1 d(i) ?ei(x;z); (5.3) where d(i) is the actual dose distribution given to the the i-th region. The exposure is then converted into developing rate through Eq. 2.4. 5.3.2 Critical-path-based Resist Development Simulation The main goal is to achieve the minimal deviations from the target 3-D resist pro?le at the top, middle and bottom layers of resist. Therefore, in the second step, the simulation of resist development process is employed to generate the resist pro?le. Figure 5.9: The outer width errors and the inner width errors de?ned at the cross-section of resist pro?le. For this stage, it is not required to run the whole simulation to get a complete resist pro?le since only the widths at the top, middle and bottom layers of resist need to be considered. Therefore, our path-based method (refer to Section 2.4) is further simpli?ed by 93 tracing the developing process along several critical paths required to obtain those desired widths only. Note that the critical paths are not ?xed but depend on the speci?c dose distribution type (refer to Figs. 5.5, 5.6, and 5.7). Then, the width information derived are used for computing width errors. The width errors are de?ned at the top, middle and bottom layers of resist (refer to Fig. 5.3). One improvement is that, besides the outer width error (deviation from the feature boundary), now the inner width error (deviation from the feature centerline) is also de?ned, as shown in Fig. 5.9. These errors are measured for each layer in order to guarantee not only CD error and sidewall shape but also fully-developed condition. (a) (b) Figure 5.10: The resist pro?le (a) before dose updating and (b) after dose updating for Type-V case, respectively. 5.3.3 Type-based Dose Updating In the third step, the dose of each region is updated based on the current resist pro?le. Depending on the relationship among the width errors (positive or negative, larger or smaller, etc.), the dose either increases or decreases. The overall dose updating mechanism is common 94 (a) (b) Figure 5.11: The resist pro?le (a) before dose updating and (b) after dose updating for Type-M case, respectively. (a) (b) Figure 5.12: The resist pro?le (a) before dose updating and (b) after dose updating for Type-A case, respectively. for di?erent dose distribution types (Type-V, Type-M, Type-A), but the details are di?erent. 95 The dose distribution is adjusted such that it is guided toward a given type and the maximum of the width errors is minimized. For example, if the current resist pro?le has a positive outer width error in the top layer but a negative outer width error in the bottom layer (refer to Fig. 5.9), i.e., an overcut sidewall, and intersects with the feature boundary, the dose of the two edge regions increases in the Type-V while it decreases in the Type-M or Type-A (refer to Figs. 5.10, 5.11, and 5.12). In this way, the dose distribution can be guided toward the preset dose distribution type. The procedure goes back to the ?rst step and continues through iterations until the maximum of the width errors is smaller than a certain threshold. Exposure-to-rate Mapping r(x,z) = F[e(x,z)] Dose Distribution d(i) = d(i) + ?d(i) Resist Development Simulation ?w(j) = p(j) -q(j) Type-based Dose Updating ?d(i) ~?w(j) Precalculated Region-wise Exposure Distribution ei(x,z) Exposure Computation 3 1 , ,i i e x z d i e x z ?? Figure 5.13: Flowchart of the proposed dose optimization scheme. Note that since the base exposures are precalculated, the cost for exposure computation in the ?rst step during iterations can be greatly reduced. Also, the critical-path-based 96 method in the second step saves a lot of simulation time for the resist development process. These improvements make the proposed dose optimization scheme (also refer to the ?owchart in Fig. 5.13) an e?ective and e?cient alternative to the conventional PEC approach. 5.4 Dose Determination Scheme An adaptive scheme has been developed which determines the optimal dose distribution type (Type-V, Type-M, Type-A) and the minimum of required total dose based on a given circuit pattern and substrate system settings. The proposed scheme utilizes the concept of critical path in dose determination. First, a preset region-wise dose ratio q(i), i.e., d(1) : d(2) : d(3) : d(4) : d(5), is employed for each dose distribution type, which is set to 4 : 2 : 1 : 2 : 4 for Type-V, 1 : 5 : 1 : 5 : 1 for Type-M, and 1 : 2 : 7 : 2 : 1 for Type-A. Note that this dose ratio is only used to determine the best dose distribution type and minimum total dose, but not the actual dose distribution (actual dose ratio) for correction which is determined by the proposed dose optimization scheme. The total exposure e(x;z) can be computed by: e(x;z) = D ? 3? i=1 q(i) ?ei(x;z) 5? i=1 q(i) ; (5.4) where ei(x;z) are the base exposures (refer to Section 5.3.1) and D is an initial total dose. The exposure is then converted into developing rate through Eq. 2.4. Then, those developing rates along a speci?c critical path are used to compute the required developing time. The critical path is de?ned as the developing path in the resist development process which plays the most important role in determining the resist pro?le, as illustrated in Fig. 5.14. 97 Note that the critical path is di?erent for each dose distribution type (Type-V, Type- M, Type-A). For example, in the Type-V case, the critical path starts from one of the edge regions at the top layer of resist, then goes vertically down to the bottom layer of resist, then goes laterally toward the center and stops at the center point of the exposed area (also refer to Fig. 5.14(a)). But in the Type-A case, the critical path starts from the center region at the top layer of resist, then goes vertically down to the bottom layer of resist, then goes laterally toward the left (right) and stops at the boundary between the exposed area and unexposed area (also refer to Fig. 5.14(c)). (a) (b) d1 d2 d3 d4 d5 (c) Figure 5.14: The critical path for (a) Type-V case, (b) Type-M case, and (c) Type-A case, respectively. The total dose D is updated such that the required developing time T spent on the critical path in the resist is no larger than the given developing time T0 (the one speci?ed for correction). The solved total dose which satis?es this developing time condition is the minimum of required total dose for a speci?c dose distribution type. 98 Exposure-to-rate Mapping r(x,z) = F[e(x,z)] Critical-path-based Simulation Exposure Computation 3 1 5 1 , , ii i D q i e x z e x z q i ? ? ?? Total Dose Updating ?D~?T D=D+?D Time Comparing ?T=T-T0 ei(x,z) q(i) D0 T D T0 ?T ConvolutionPSF Circuit Pattern ?T