Directed self-assembly cut mask assignment for unidirectional design

Abstract

Recently, directed self-assembly (DSA) has emerged as a promising lithography solution for cut manufacturing. We perform a comprehensive study on the DSA aware mask optimization problem to provide a DSA friendly design on cut layers. We first formulate the problem as an integer linear programming (ILP) to assign cuts to different guiding templates, targeting both conflict minimization and line-end extension minimization. As ILP may not be scalable for very large size problems, we then propose two speed-up strategies. The first one is to decompose the initial problem into smaller ones and solve them separately, followed by solution merging without much loss of quality. The second one is using the set cover algorithm to decide the DSA guiding pattern assignment, and then legalize the template placement. Our approaches can be naturally extended to handle arbitrary DSA guiding template patterns with complicated shapes. Experimental results show that our methodologies can significantly improve the DSA friendly, i.e., both the unresolved pattern number and the lineend extensions can be reduced.

Combining lithography and Directed Self Assembly for the manufacturing of vias: Connections to graph coloring problems, integer programming formulations, and numerical experiments

“…Heuristics and exact approaches for multiple patterning with DSA (and variants) have been investigated in [5,12,26,37,38,43]. Note that in all studies, the number of patterning steps is fixed and the goal is to group vias into feasible guiding patterns so as to minimize the number of conflicts remaining (allowing sometimes for the insertion of redundant vias).…”

“…Variable λ i ∈ {0, 1} indicates whether color i is chosen, and x i g ∈ {0, 1} indicates whether a group g ∈ V is colored with color i. Constraint (37) imposes that each via is assigned to exactly one group and one color. Constraint (38) imposes that groups in conflict (for which there are two vias that are too close) have to receive different colors.…”

“…In principle, it would be possible to use the exact methods developed in some prior studies in parallel: run the algorithm for a given number of patterning steps and then verify for which number of patterning steps zero coloring conflicts emerge. However, most of these methods either employ heuristics to accelerate finding a coloring solution at a large scale (and hence no longer guaranteeing optimality) [37,43], propose formulations that do not work for any number of patterning steps [5,26], or exploit additional structures and/or placement options [12,26,38]. In our case, given that the objective is to formally find the minimum number of patterning steps required for (large scale) layouts, we do not build on the methods developed in prior research.…”

Toward Unidirectional Routing Closure in Advanced Technology Nodes

“…With guiding template constraints, DSA has been adopted for cut mask design and line-end extensions with a greedy scheme [61]. Mixed integer linear programming formulations and dynamic programming algorithms have been proposed to enable DSA-friendly cut mask designs [35], [36], [48], [49]. Given a set of design rules, line-end extension problems aforementioned are similar to the layout migration issue [20], where the objective is to minimize the amount of dummy patterns and the design rules are formulated as linear constraints.…”