Design of exact solutions for the manufacturing of "vias" using DSA technology

Dehia Ait Ferhat 1
1 G-SCOP_OC - OC
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : Controlling the manufacturing costs of integrated circuits while increasing their density is of a paramount importance to maintain a certain degree of profitability in the semi-conductor industry. Among various components of a circuit, we are interested in vertical metallic connections known as “vias”. During manufacturing, a complex lithography process is used to form an arrangement of vias on a silicon wafer support, using an optical mask. For manufacturing reasons, a minimum distance between the vias must be respected. Whenever this is not the case, we are talking about a “conflict”. In order to eliminate these conflicts, the industry uses a technique that decomposes an arrangement of vias in several subsets, where minimum distance constraints are respected: the formation of the individual subsets is done, in sequence, on a silicon wafer using one optical mask per subset. This technique is called Multiple Patterning (MP). There are several ways to decompose an arrangement of vias, the goal being to assign the vias to a minimum number of masks, since the masks are expensive. Minimizing the number of masks is equivalent to minimizing the number of colors in a unit disk graph. This is a NP-hard problem however, a number of “good” heuristics exist. A recent and promising technique is based on the direction and self-assembly of the molecules called Directed Self Assembly (DSA), allows to group vias in conflict according to certain conditions. The main challenge is to find the best way of grouping vias to minimize the number of masks while respecting the constraints related to DSA. This problem is a graph coloring problem where the vertices within each color define a set of independent paths of length at most k also called a k-path coloring problem. During the graph modeling, we distinguished two k-path coloring problems: a general problem and an induced problem. Both problems are known to be NP-hard, which explains the use of heuristics in the industry to find a valid decomposition into subsets. In this study, we are interested in exact methods to design optimal solutions and evaluate the quality of heuristics developed in the industry (at Mentor Graphics). We present different methods: an integer linear programming (ILP) approach where we study several formulations, a dynamic programming approach to solve the induced case when k=1 or k=2 and when the graphs have small tree-width; finally, we study a particular case of line graphs. The results of the various numerical studies show that the naïve ILP formulations are the best, they list all possible paths of length at most k. Tests on a snippet of industrial instances of at most 2000 vertices (a largest connected component among those constituting an instance) have shown that the two problems, general and induced, are solved in less than 6 seconds, for k=1 and k=2. Dynamic programming, applied to the induced k-path coloring when k=1 and k=2, shows results equivalent to those of the naïve ILP formulation, but we expect better results by dynamic programming when the value of k increases. Finally, we show that the particular case of line graphs can be solved in polynomial time by exploiting the properties of Edmonds’ algorithm and bipartite matching.
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Dehia Ait Ferhat. Design of exact solutions for the manufacturing of "vias" using DSA technology. Modeling and Simulation. Université Grenoble Alpes, 2018. English. ⟨NNT : 2018GREAM094⟩. ⟨tel-02171255⟩

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