Provably Good Max–Min- m -Neighbor-TSP-Based Subfield Scheduling for Electron-Beam Photomask Fabrication
Abstract: Electron beam lithography (EBL) has been used for high-resolution photomask fabrication; its successive heating process in a certain region, however, may cause critical dimension (CD) distortion. As a result, subfield scheduling, which reorders a sequence of subfields in the writing process, is desirable to avoid the heating problem and thus CD distortion. To consider longer range heat dissipation, this paper models a subfield scheduling problem with blocked region consideration as a constrained max–min m -neighbor traveling salesman problem (called constrained m -nTSP). To solve the constrained m -nTSP which is NP-complete in general, we decompose a constrained m -nTSP into subproblems conforming to a special case with points on two parallel lines, solve each of them with a provably good linear-time approximation algorithm, and merge them into a complete scheduling solution. In particular, our algorithm can also minimize the distances between successive subfields to alleviate the throughput degradation of EBL writing due to moving a writing head, while minimizing the heating problem. Average reductions of 10% in the maximum temperature and 14% in the distances between successive subfields over the state-of-the-art work can be achieved.
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