Computational Lithography

A Unified Summary of the Models and Optimization Methods Used in Computational Lithography Optical lithography is one of the most challenging areas of current integrated circuit manufacturing technology. The semiconductor industry is relying more on resolution enhancement techniques (RETs), since their implementation does not require significant changes in fabrication infrastructure. Computational Lithography is the first book to address the computational optimization of RETs in optical lithography, providing an in-depth discussion of optimal optical proximity correction (OPC), phase shifting mask (PSM), and off-axis illumination (OAI) RET tools that use model-based mathematical optimization approaches. The book starts with an introduction to optical lithography systems, electric magnetic field principles, and the fundamentals of optimization from a mathematical point of view. It goes on to describe in detail different types of optimization algorithms to implement RETs. Most of the algorithms developed are based on the application of the OPC, PSM, and OAI approaches and their combinations. Algorithms for coherent illumination as well as partially coherent illumination systems are described, and numerous simulations are offered to illustrate the effectiveness of the algorithms. In addition, mathematical derivations of all optimization frameworks are presented. The accompanying MATLAB® software files for all the RET methods described in the book make it easy for readers to run and investigate the codes in order to understand and apply the optimization algorithms, as well as to design a set of optimal lithography masks. The codes may also be used by readers for their research and development activities in their academic or industrial organizations. An accompanying MATLAB® software guide is also included. An accompanying MATLAB® software guide is included, and readers can download the software to use with the guide at ftp://ftp.wiley.com/public/sci_tech_med/computational_lithography. Tailored for both entry-level and experienced readers, Computational Lithography is meant for faculty, graduate students, and researchers, as well as scientists and engineers in industrial organizations whose research or career field is semiconductor IC fabrication, optical lithography, and RETs. Computational lithography draws from the rich theory of inverse problems, optics, optimization, and computational imaging; as such, the book is also directed to researchers and practitioners in these fields.

Table of Contents:
Preface xi Acknowledgments xiii Acronyms xv 1 Introduction 1 1.1 Optical Lithography 1 1.1.1 Optical Lithography and Integrated Circuits 2 1.1.2 Brief History of Optical Lithography Systems 3 1.2 Rayleigh’s Resolution 5 1.3 Resist Processes and Characteristics 7 1.4 Techniques in Computational Lithography 10 1.4.1 Optical Proximity Correction 11 1.4.2 Phase-Shifting Masks 11 1.4.3 Off-Axis Illumination 14 1.4.4 Second-Generation RETs 15 1.5 Outline 16 2 Optical Lithography Systems 19 2.1 Partially Coherent Imaging Systems 19 2.1.1 Abbe’s Model 19 2.1.2 Hopkins Diffraction Model 22 2.1.3 Coherent and Incoherent Imaging Systems 24 2.2 Approximation Models 25 2.2.1 Fourier Series Expansion Model 25 2.2.2 Singular Value Decomposition Model 29 2.2.3 Average Coherent Approximation Model 32 2.2.4 Discussion and Comparison 34 2.3 Summary 36 3 Rule-Based Resolution Enhancement Techniques 37 3.1 RET Types 37 3.1.1 Rule-Based RETs 37 3.1.2 Model-Based RETs 38 3.1.3 Hybrid RETs 39 3.2 Rule-Based OPC 39 3.2.1 Catastrophic OPC 40 3.2.2 One-Dimensional OPC 40 3.2.3 Line-Shortening Reduction OPC 42 3.2.4 Two-Dimensional OPC 43 3.3 Rule-Based PSM 44 3.3.1 Dark-Field Application 44 3.3.2 Light-Field Application 45 3.4 Rule-Based OAI 46 3.5 Summary 47 4 Fundamentals of Optimization 48 4.1 Definition and Classification 48 4.1.1 Definitions in the Optimization Problem 48 4.1.2 Classification of Optimization Problems 49 4.2 Unconstrained Optimization 50 4.2.1 Solution of Unconstrained Optimization Problem 50 4.2.2 Unconstrained Optimization Algorithms 52 4.3 Summary 57 5 Computational Lithography with Coherent Illumination 58 5.1 Problem Formulation 59 5.2 OPC Optimization 62 5.2.1 OPC Design Algorithm 62 5.2.2 Simulations 64 5.3 Two-Phase PSM Optimization 65 5.3.1 Two-Phase PSM Design Algorithm 65 5.3.2 Simulations 68 5.4 Generalized PSM Optimization 72 5.4.1 Generalized PSM Design Algorithm 72 5.4.2 Simulations 75 5.5 Resist Modeling Effects 79 5.6 Summary 82 6 Regularization Framework 83 6.1 Discretization Penalty 84 6.1.1 Discretization Penalty for OPC Optimization 84 6.1.2 Discretization Penalty for Two-Phase PSM Optimization 86 6.1.3 Discretization Penalty for Generalized PSM Optimization 87 6.2 Complexity Penalty 93 6.2.1 Total Variation Penalty 93 6.2.2 Global Wavelet Penalty 94 6.2.3 Localized Wavelet Penalty 98 6.3 Summary 100 7 Computational Lithography with Partially Coherent Illumination 101 7.1 OPC Optimization 102 7.1.1 OPC Design Algorithm Using the Fourier Series Expansion Model 102 7.1.2 Simulations Using the Fourier Series Expansion Model 105 7.1.3 OPC Design Algorithm Using the Average Coherent Approximation Model 107 7.1.4 Simulations Using the Average Coherent Approximation Model 111 7.1.5 Discussion and Comparison 111 7.2 PSM Optimization 115 7.2.1 PSM Design Algorithm Using the Singular Value Decomposition Model 116 7.2.2 Discretization Regularization for PSM Design Algorithm 118 7.2.3 Simulations 118 7.3 Summary 122 8 Other RET Optimization Techniques 123 8.1 Double-Patterning Method 123 8.2 Post-Processing Based on 2D DCT 128 8.3 Photoresist Tone Reversing Method 131 8.4 Summary 135 9 Source and Mask Optimization 136 9.1 Lithography Preliminaries 137 9.2 Topological Constraint 140 9.3 Source–Mask Optimization Algorithm 141 9.4 Simulations 141 9.5 Summary 145 10 Coherent Thick-Mask Optimization 146 10.1 Kirchhoff Boundary Conditions 147 10.2 Boundary Layer Model 147 10.2.1 Boundary Layer Model in Coherent Imaging Systems 147 10.2.2 Boundary Layer Model in Partially Coherent Imaging Systems 151 10.3 Lithography Preliminaries 153 10.4 OPC Optimization 157 10.4.1 Topological Constraint 157 10.4.2 OPC Optimization Algorithm Based on BL Model Under Coherent Illumination 158 10.4.3 Simulations 159 10.5 PSM Optimization 162 10.5.1 Topological Constraint 162 10.5.2 PSM Optimization Algorithm Based on BL Model Under Coherent Illumination 165 10.5.3 Simulations 165 10.6 Summary 170 11 Conclusions and New Directions of Computational Lithography 171 11.1 Conclusion 171 11.2 New Directions of Computational Lithography 173 11.2.1 OPC Optimization for the Next-Generation Lithography Technologies 173 11.2.2 Initialization Approach for the Inverse Lithography Optimization 173 11.2.3 Double Patterning and Double Exposure Methods in Partially Coherent Imaging System 174 11.2.4 OPC and PSM Optimizations for Inverse Lithography Based on Rigorous Mask Models in Partially Coherent Imaging System 174 11.2.5 Simultaneous Source and Mask Optimization for Inverse Lithography Based on Rigorous Mask Models 174 11.2.6 Investigation of Factors Influencing the Complexity of the OPC and PSM Optimization Algorithms 174 Appendix A: Formula Derivation in Chapter 5 175 Appendix B: Manhattan Geometry 181 Appendix C: Formula Derivation in Chapter 6 182 Appendix D: Formula Derivation in Chapter 7 185 Appendix E: Formula Derivation in Chapter 8 189 Appendix F: Formula Derivation in Chapter 9 194 Appendix G: Formula Derivation in Chapter 10 195 Appendix H: Software Guide 199 References 217 Index 223