Optimizing Movement Sequences for Step-and-Scan Lithography Equipment

Abstract

The purpose of this work is to improve the throughput of step-and-scan lithography equipment to shorten the production time of a wafer. For this purpose, we propose a method for solving the MSOP (Movement Sequence Optimization Problem), which is the problem of computing the fastest schedule for visiting all shots on a wafer. It is well-known that the MSOP on step-and-repeat lithography equipment can be modeled as a traveling salesman problem. In contrast to step-and-repeat lithography equipment, a schedule for step-and-scan lithography equipment must also indicate the scanning direction of each shot, in addition to the sequence of the shots. For this reason, the traveling salesman problem formulation for step-and-repeat lithography equipment cannot be applied to solve the MSOP on step-and-scan lithography equipment directly. We overcame this difficulty by introducing auxiliary vertices to model the scanning directions in the traveling salesman problem formulation. By this method, we were able to compute exact optimal sequences considering the scanning directions of shots for several MSOP instances. Our numerical experiments demonstrated that our proposed method was capable of computing exact optimal solutions for real-world MSOP instances having up to 232 shots on a wafer. These optimal solutions gave a 0.25% to 4.66% improvement in productivity over solutions computed by previously known methods.