臺灣博碩士論文加值系統

為了推進製程極限，需要發展下世代微影技術。根據2015年國際半導體技術發展藍圖，定向自我組裝技術、電子束微影技術、極紫外光微影技術與多圖樣微影技術是可進一步推進製程極限的候選技術。要注意的是由於極紫外光微影技術主要的障礙為其光源電源的問題，需要的是物理設備方面的提升，而非製造可行性設計或演算法可處理。因此，本論文主要探討其餘三項技術。首先，定向自我組裝技術已在被認可為能實行先進電路製造的候選技術之一，因為定向自我組裝技術於奈米尺寸下仍有良好的圖樣製造能力。在此技術中，一種被稱作嵌段共聚物的特殊聚合分子結構，可以被一些拓樸圖樣所引導，來自我生成高解析度的電路圖樣。其中一種常見的自我組裝技術乃是用一些引導模板來定義自我組裝圖樣生成的區域。由於自我生成的圖樣具有高解析度與擺置高準確度，此技術可被用於一維晶格電路的切割光罩製造，其中切割光罩於一維晶格電路製造過程中被用於製造線路圖樣。然而，自我組裝圖樣的偏移準確度與引導模板的可製造性會隨引導模板的拓樸特性而變化。除此之外，兩個相近的模板會互相影響使得自我組裝圖樣無法順利生成。因此切割圖樣需重新分配位置，使得自我組裝技術能以適當的模板引導嵌段共聚物生成所需圖樣。除此之外，傳統單一光罩光學製程可能無法於奈米級電路設計中正常曝光引導模板。因此統合多圖案微影與自我組裝技術實乃必要，以雙圖案微影曝光引導模板。除了以引導模板為主的自我組裝技術外，二維自我組裝技術是另一有競爭力的次10奈米製程實現技術。二維自我組裝技術中，嵌段共聚物可被晶格點狀圖樣引導形成奈米級二維走向金屬線，並能夠因此減少繞線層數使用。然而，點狀圖樣的走向必須先決定後才能引導嵌段共聚物形成二維金屬線。除此之外，某些二維金屬線圖樣良率過低。因此二維自我組裝的繞線規則與傳統繞線規則之間有顯著差異，並且需要有一個新的方法以處理二維自我組裝的可繞性。再來，為了製造拓樸引導圖樣，電子束微影技術可被用於高解析度圖樣製造。與傳統光學製程不同，電子束不須經過光罩(光經過光罩孔隙會繞射)，而是利用特殊透鏡與磁場控制電子束，所以電子束可有高解析度。然而，因高能量電子束通常以鄰近連續的方式做電子束直寫，此直寫方式易累積大量熱能於小部分區域，進而造成關鍵尺寸失真。針對其熱量殘留問題，電子束子領域直寫順序應被重新排序。有鑑於此，為了要充分利用這些下世代微影技術之好處並解決它們的潛在問題，必須要有突破的方法。本論文提出嶄新的演算法來應用這些下世代微影技術於現代積體電路設計流程。對於二維自我組裝技術，我們提出第一個於精細電路擺置階段考慮二維自我組裝可繞性的演算法。對於引導模板導向自我組裝技術，我們首先提出一個在特殊情況下能得到最佳解的線性時間演算法。此特殊情況為每條一維線路列上最多只會有一段多餘線段。對於一般情況，我們則是提出以二分配演算法來將電路分解成數個符合此特殊情況的子問題(因此每個子問題能被最佳解決)。除此之外，多圖案微影可與引導模板導向自我組裝技術結合以提升良率。在此論文中，我們首先提出一個線性時間最佳演算法來解決在一維線路列數量有限的情況下之考慮引導模板之切割光罩重新分配問題。對於一般情況下之電路，我們則以能充分利用雙圖案微影優勢之線性時間分解電路演算法，來將電路分解成數個符合前述條件之子問題。除此之外，針對其電子束熱量殘留問題，我們利用圖論演算法提出一種子領域排程演算法將子領域直寫順序重新排序。為了考慮熱量會在子領域殘存一段時間的問題，我們不只同時考慮每個子域的封鎖區域以減緩熱問題，並且將問題用一個推銷員問題的變種做處理。我們嘗試將此問題分解成數個子問題來解決，之後再將他們的解答合併。每個子問題都由兩條平行線組成，我們以一個被證明高品質之線性近似演算法來解決子問題。再來，在實驗結果中，我們所提出的演算法們不但能充分解決可製造性問題，還有良好效率。

To further shrink layout features in advanced circuit designs, it is necessary to develop next-generation-lithography (NGL) technologies. According to the 2015 international technology roadmap for semiconductors (ITRS), directed self-assembly (DSA), electron beam lithography (EBL), extreme ultraviolet lithography (EUVL), and multiple patterning extensions may push the limits of lithography. Note that the main challenge of EUVL is its source power, which cannot be solved by methods of design for manufacturing. Therefore, this dissertation is focused on DSA, EBL, and multiple patterning. First, in the 2015 ITRS, DSA was recognized as a promising candidate for advanced circuit designs because DSA can have robust patterning capability in nanoscale. In DSA, specialized polymer molecules, called block copolymers (BCPs), can be directed by some guiding topographical features to form high-resolution self-assembled features. One common category of DSA involves using guiding templates to define the rough regions within which high-resolution features should be located. In particular, DSA with guiding templates has shown its capability for cut-mask fabrication in 1-D gridded designs (cut masks are responsible for creating wire patterns for 1-D layouts). However, the overlay accuracy of the self-assembled features and the printability of templates may vary with different topologies of templates. Moreover, two close templates might interfere each other and might not pattern desired cuts. Therefore, cut patterns of 1-D gridded designs might need to be redistributed such that the corresponding fabricated cut masks could pattern DSA guiding templates. Furthermore, traditional single 193i patterning may not be sufficient to create guiding templates for DSA without distortion in a nanoscale circuit design. Therefore, it is necessary to adopt hybrid lithography incorporating double patterning and DSA (using double patterning to create guiding templates for DSA) in cut-mask fabrication. In addition, the two-dimensional directed self-assembly (2D-DSA) technology with a square lattice of topographic features, denoted by posts, is another competitive option for advanced circuit designs. Directed by the posts, specialized BCPs can create two-dimensional metal wires, which can reduce the number of used metal layers. However, the orientations of posts must be determined before creating metal patterns, and some metal patterns have low yields in 2D-DSA. Therefore, the routing rules of 2D-DSA can be much different from those of traditional routing technologies, and thus the routability of 2D-DSA should be addressed in a new methodology. Then, to fabricate guiding topographical features, EBL can be used in high-resolution pattern fabrication. Unlike traditional optical technologies, where light diffracts when passing through the apertures on a photomask, maskless electron beams (e-beams) can be concentrated to the nanometer scale by using electromagnetic or electrostatic lenses. However, high-voltage beams often deposit a considerable amount of heat in a region, causing critical dimension (CD) distortion. To avoid the heating problem, subfield scheduling which reorders a sequence of subfields in the writing process is executed before layout fabrication. As a result, to utilize the benefits of these NGL technologies and address their difficulties, it is necessary to have solutions for a breakthrough. This dissertation proposes new methodologies to apply these NGL technologies. For 2D-DSA, we present the first detailed placement algorithm for 2D-DSA with a routability model based on the post-orientation probability. For DSA with guiding templates, we first propose a linear-time optimal dynamic-programming-based algorithm for a special case of the template guided cut redistribution problem, where there is at most one dummy wire segment on a track. We then extend our algorithm to general cases by applying a bipartite matching algorithm to decompose a general problem. Furthermore, for hybrid lithography, we first develop a linear-time optimal algorithm for a special case, consisting of a limited number of rows, of the template guided cut redistribution problem. Then, we develop a linear-time double-patterning aware partitioning method to decompose a general problem. Moreover, for EBL, to consider longer-range heat dissipation, we model a subfield scheduling problem with blocked region consideration as a constrained max-min m-neighbor traveling salesman problem (called constrained m-nTSP). To solve the problem, we decompose it into subproblems conforming to a special case with points on two parallel lines, solve each of them with a provably good linear-time approximation algorithm, and merge them into a complete scheduling solution. Experimental results show the effectiveness and efficiency of our proposed algorithms.

Acknowledgements vi
Abstract (Chinese) vii
Abstract x
List of Tables xvii
List of Figures xix
Chapter 1. Introduction 1
1.1 Introduction to NGL Technologies . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Directed Self-Assembly Technology . . . . . . . . . . . . . . . . . . 3
1.1.2 Hybrid Lithography with DSA and Multiple Patterning . . . . . . 7
1.1.3 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Design Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Routability Estimation for 2D-DSA . . . . . . . . . . . . . . . . . 11
1.2.2 Guiding-Template Design Rules for DSA Cut-Mask Fabrication . . 11
1.2.3 Hybrid Lithography Integration of DSA and Multiple Patterning
for Cut-Mask Fabrication . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.4 The Heating Problem of EBL . . . . . . . . . . . . . . . . . . . . . 13
1.3 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Detailed Placement for 2D-DSA . . . . . . . . . . . . . . . . . . . 16
1.3.2 Cut Redistribution for Advanced 1-D Gridded Designs with DSA . 16
1.3.3 Cut Redistribution for Advanced 1-D Gridded Designs with Hybrid
Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.4 Sub eld Scheduling for EBL . . . . . . . . . . . . . . . . . . . . . 17
1.4 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 2. Detailed Placement for 2D-DSA 19
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Routability Estimation . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 Routability Cost Computation . . . . . . . . . . . . . . . . . . . . 22
2.1.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Global Moving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Single-row Dynamic Programming . . . . . . . . . . . . . . . . . . 30
2.2.3 Cell Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.4 Cell Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 3. Cut Redistribution for Advanced 1-D Gridded Designs
with DSA 37
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Layouts with DSA Template Guided Cuts . . . . . . . . . . . . . . 41
3.1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Algorithm for the Special Case . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Feasible Gap Solution Construction . . . . . . . . . . . . . . . . . 46
3.2.2 Best Predecessor Determination . . . . . . . . . . . . . . . . . . . 53
3.2.3 Solution Path Construction . . . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Template Distribution Construction . . . . . . . . . . . . . . . . . 56
3.2.5 Optimality and Complexity . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Algorithm for the General Layouts . . . . . . . . . . . . . . . . . . . . . . 59
3.3.1 General Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.2 Quality Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1 General Design Rules . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.2 Simple Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.3 Dummy Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Results of Con
ict and Wire Costs . . . . . . . . . . . . . . . . . . 73
3.5.2 Results on Large Benchmarks . . . . . . . . . . . . . . . . . . . . . 74
3.5.3 Results under General Design Rules . . . . . . . . . . . . . . . . . 74
3.5.4 Results for Simple Templates and Dummy Cuts . . . . . . . . . . . 76
Chapter 4. Cut Redistribution for Advanced 1-D Gridded Designs
with Hybrid Lithography 79
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.1 1-D Layouts with Double-Patterning Assisted DSA Template Guided
Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Overall Algorithm Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Algorithm for the Special Case . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Column Construction . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.2 Column Solution Graph Construction . . . . . . . . . . . . . . . . 88
4.3.3 Optimal Column Solution Path Construction . . . . . . . . . . . . 90
4.3.4 Template Distribution Derivation . . . . . . . . . . . . . . . . . . . 91
4.3.5 Time Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Subproblem Candidate Selection . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.1 Cost Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.2 Quality and E ciency . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter 5. Sub eld Scheduling for EBL 101
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.1 Thermal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.2 Blocked Boxes of Sub elds . . . . . . . . . . . . . . . . . . . . . . 107
5.1.3 Constrained Max-Min m-neighbor Travelling Salesman Problem . . 108
5.1.4 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Graph-Based Sub eld Scheduling . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Algorithm for the Max-Min m-neighbor TSP with Vertices on Two Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.1 Subgroup Generation . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.2 2-approximation Subgroup Concatenation . . . . . . . . . . . . . . 118
5.3.3 Cost Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3.4 Further Improvement . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.1 Comparison of Thermal E ect . . . . . . . . . . . . . . . . . . . . 128
5.4.2 Comparison of Average Distance and Runtime . . . . . . . . . . . 130
5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Chapter 6. Concluding Remarks and Future Work 139
6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Bibliography 147
Vita 155
Publication List 157

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