A Local Optimal Method on DSA Guiding Template Assignment with Redundant/Dummy Via Insertion Xingquan Li 1 , Bei Yu 2 , Jianli Chen 1 , Wenxing Zhu 1 , 24th Asia and South Pacific Design T h e p i c Automation Conference t u r e c a n 't b e d Tokyo Japan 2019 i s p l a y e d . 1 Fuzhou University 2 The Chinese University of Hong Kong 1
Outline Introductions Problem Formulation Algorithm Experimental Results Conclusions 2
Outline Introductions Problem Formulation Algorithm Experimental Results Conclusions 3
Block Copolymers Directed Self-Assembly (DSA) l Block copolymer (BCP) ¾ A unique string of two types of polymer. ¾ One type of polymer is hydrophilic and another is hydrophobic. l Nanostructures ¾ Cylinders, spheres, and lamellae. ¾ The cylindrical nanostructure is suitable for patterning contacts and vias. Polymer-A Polymer-B (PS) (PMMA) Lamellae Cylinders (A=B) (A>B) 4
DSA Process l Vias are not regularly placed in practical layout. l A simple regular hole array generated by standard DSA is not suitable for IC fabrication. l Topological template guided DSA process has been proposed to support patterning irregularly vias layout. l Closed vias are grouped; And a guiding template is identified for each group. Guiding template Via Via group v 1 v 2 v 3 Metal DSA pattern 5
DSA Process l Given a vias layout, we should assign the guiding templates for every via. l Guiding templates are patterned on a wafer through optical lithography. l Each guiding template is filled with BCPs. l DSA can be controlled by thermal annealing process. Guided by template Pattern template by 193i DSA design process 6
Guiding Templates l Pre-defined DSA pattern set to improve robust. l Within-group contact/via distance. l Complex shapes are difficult to print. l Unexpected holes and placement error of holes for some patterns. l The distance of any two guiding templates should larger than minimum optical resolution spacing d s. d s v 1 v 2 v 3 7
Redundant Via Insertion (RVI) l Insert an extra via near a single via. l Prevent via failure, improve circuit yield and reliability. v 2 v 1 v 3 r v r r r 2 v 2 r v 2 r 3 r 2 v 1 v 3 8
Dummy Via Due to the characteristic of DSA, vias in a group must match some l specific patterns so that they can be assigned to the same guiding template. Increase the choices to form guiding templates with the help of l dummy via insertion. v 2 r 3 r 2 v 2 r 3 r 2 v 1 v 3 v 1 v 3 Case 3 ✓ ✘ ✘ v 2 v 2 v 2 r 3 r 3 r 3 r 2 r 2 r 2 v 1 v 3 v 1 v 3 v 1 v 3 d 1 Case 1 Case 2 Case 4 9
Outline Introductions Problem Formulation Algorithm Experimental Results Conclusions 10
DSA Guiding Template Assignment with Redundant/Dummy Via Insertion (DRDV) l Input ¾ Post-routing layout v 1 v 2 ¾ Usable DSA guiding templates ¾ Optical resolution limit space v 3 v 5 v 4 v 6 Post-routing layout Usable DSA guiding templates d s Optical resolution limit space 11
DSA Guiding Template Assignment with Redundant/Dummy Via Insertion (DRDV) l Input ¾ Post-routing layout v 1 v 2 r 3 ¾ Usable DSA guiding templates ¾ Optical resolution limit space r 1 r 2 v 3 l Output ¾ Redundant via insertion for every via ¾ Guiding template assignment with v 5 r 4 d 1 suitable dummy vias for every via and redundant via v 4 v 6 r 6 12
DSA Guiding Template Assignment with Redundant/Dummy Via Insertion (DRDV) l Input ¾ Post-routing layout ¾ Usable DSA guiding templates ¾ Optical resolution limit space l Output ¾ Redundant via insertion for every via ¾ Guiding template assignment with suitable dummy vias for every via and redundant via l Constraints ¾ Inserted redundant vias should be legal ¾ The spacing between neighboring guiding template should larger than the optical resolution limit space l Objectives ¾ Maximize the number (ratio) of inserted redundant vias ¾ Maximize the number (ratio) of patterned vias by DSA 13
Outline Introductions Problem Formulation Algorithm Experimental Results Conclusions 14
Solution Flow DSA Guiding Routing Optical resolution Template Layout limit spacing d s Preprocessing Find All Redundant/Dummy Via Candidates Detect Building Blocks Construct Conflict Graph Local Optimal Solver Integer Linear Programming Formulation Initial Solution Generation Unconstrained Nonlinear Programming Solver Redundant/Dummy Via Insertion with Template Assignment 15
Preprocessing DSA Guiding Routing Optical resolution Template Layout limit spacing d s Preprocessing Find All Redundant/Dummy Via Candidates Detect Building Blocks Construct Conflict Graph Local Optimal Solver Integer Linear Programming Formulation Initial Solution Generation Unconstrained Nonlinear Programming Solver Redundant/Dummy Via Insertion with Template Assignment 16
Redundant/Dummy Via Candidates l Redundant via candidate ¾ It should be inserted next to every via. ¾ It should not overlap with any metal wire from other nets of wires. l Dummy via candidate ¾ It can make up a multi-hole (not less than three holes) guiding template with other vias or redundant vias. ¾ It can improve the insertion rate or manufacture rate. l Find all redundant/dummy via candidates for every via in time O ( n ). r 2 r 1 v 2 r 2 v 2 v 1 v 1 17
Building-Blocks l building-block1: a original via l building-block2: a redundant via l building-block3: a original via and a redundant via l building-block4: two original vias l building-block5: two redundant vias l building-block6: a original via and a redundant via (diagonal) l building-block7: two original vias (diagonal) l building-block8: two redundant vias (diagonal) l building-block9: six original/redundant vias 1 2 3 4 5 6 7 8 9 18
Combinations of Building-Blocks 1 2 3 4 5 6 7 8 9 l Combinations of building-blocks to form guiding templates 19
Building-Blocks Detection & Conflict Graph l Conflict graph CG (V, E ) ¾ vertex v ∈ V denotes a building-block , ¾ e ij ∈ E is an edge and E = (E C −E T ) ∪ E O . E C , E T and E O are the sets of conflict edges, template edges and overlap edges. v 2 r 1 a c b d e f v 2 r 1 v 2 r 1 v 1 v 2 r 1 v 1 v 1 v 1 b d c Conflict edge Overlap edge Template edge a f e 20
Conflict Edges l The distance between two building-blocks are less than resolution limit space d s. a b c d e f v 2 r 1 v 2 v 1 v 2 r 1 r 1 v 1 v 1 b d v 2 r 1 c v 1 a e f f r 1 v 2 e Conflict edge Overlap edge v 1 Template edge 21
Overlap Edges l Two building-blocks are overlapped. a b c d e f v 2 r 1 v 2 v 1 v 2 r 1 r 1 v 1 v 1 b d v 2 r 1 d f c r 1 v 2 v 1 v 1 a f e Conflict edge Overlap edge Template edge 22
Template Edges l If building-blocks i and j with e ij ∈ E C can be assigned to a guiding template without any design error. a b c d e f v 2 r 1 v 2 v 1 v 2 r 1 r 1 v 1 v 1 b d c v 2 r 1 v 2 r 1 c a v 1 v 1 a f e Conflict edge Overlap edge Template edge 23
Solution Flow DSA Guiding Routing Optical resolution Template Layout limit spacing d s Preprocessing Find All Redundant/Dummy Via Candidates Detect Building Blocks Construct Conflict Graph Local Optimal Solver Integer Linear Programming Formulation Initial Solution Generation Unconstrained Nonlinear Programming Solver Redundant/Dummy Via Insertion with Template Assignment 24
Constraints b d d f c c r 1 v 2 v 2 r 1 a v 1 ✘ v 1 a f ✓ e e f r 1 v 2 ✘ v 1 ∪ E = ( E C − E T ) E O Template edge Conflict edge Overlap edge 25
Conflict Structure Constraint l Template constraint ¾ If two building-blocks i and j are connected by a template edge, then they may be assigned to the same guiding template, but not necessarily. ¾ If both of building-blocks i and l connect with k by template edges, then i, k, l may not be assigned to a same guiding template. l Conflict structure (CS) ¾ Three bblocks i , k and l , in which e ik and e kl are template edges and there does not exist any edge between i and l . l k k i k l i l i 26
Integer Linear Programming (ILP) l Objectives: ¾ Maximize the number of inserted redundant vias ¾ Maximize the number of patterned vias by DSA ¾ Let N v and N r are the numbers of included vias and redundant vias by building-block i, and l ILP Formulation (1) 27
Inequality Constraints l Claim 1. The ILP is equivalent to the DRDV problem. (1) l Transfer inequality constraints to equality constraints. (2) 28
Equality Constraints l Relax equality constraints to objective function. (2) (3) 29
Adjacent Matrix & CS Tensor l Handle adjacent matrix and CS tensor. (3) (4) ! "# = %1, ( "# ∈ * - "./ = 01, (2, 3, 4) ∈ -6 0, ( "# ∉ * 0, (2, 3, 4) ∉ -6 30
Unconstrained Nonlinear Programming (UNP) (4) ! " = $1, ' " ≥ 0 0, ' " < 0 1 = 2 = 4 0.8 = 6 = 8 0.6 = 10 0.4 0.2 (5) 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 31
UNP Solver (5) Greedy selection method Wolfe-Powell inexact line search method 32
Local Optimal Convergence l Lemma 1. Under above Equations, ∑ i wi∆gi ≥0. l Theorem 1. Under above Equations, f(y) does not decrease. l Corollary 1. Strict inequality ∑ i wi∆gi >0 cannot be achieved. l Theorem 2. Our UNP solver converges to a local maximum. 33
Outline Introductions Problem Formulation Algorithm Experimental Results Conclusions 34
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