STOMA: Simultaneous Template Optimization and Mask Assignment for Directed Self-Assembly Lithography With Multiple Patterning

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).…”

“…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.…”

“…We again develop an integer model in the same vein as the previous one for k ≥ 2. Constraints (25), (26), (27), and (28) have the same meaning as in the previous model. Constraints (29) and (30) identify the path each node belongs to by propagating connectivity, e.g., if u is in a path of color i starting in o and (u, v) is taken in color i (in the direction from u to v), then v is in the path of color i starting in o. Constraint (31) imposes consistency for the value taken by y i v,o , (a vertex is in a path in color i if and only if it is in color i).…”

“…We do not use the true values for Litho dist , L0, and U0 for confidentiality reasons. Instead, following [26,43], we use three different values of Litho dist (31nm, 41nm, 49nm note that distance is border to border), and set L0 = 20nm and U0 = 40nm (note the distance is center to center here). We re-scaled first the layout so that the minimum distance -border to border -between any two vias corresponds to a targeted pitch size of 10nm and so that the diameter of the vias is also 10nm, as Name clip1 clip2 clip3 clip4 clip5 clip6 clip7 clip8 clip9 clip10 Number of vias 200850 173741 236979 184486 214697 248375 225795 126029 200850 173741 Table 1: Number of vias for each instance.…”

“…in [26] (we assume here for simplicity that vias are disks, hence the distances border to border and center to center are easily obtained). We focus on 193 immersion and thus only consider DSA edges that are parallel to the x or y axis.…”