Graph-Based Subfield Scheduling for Electron-Beam Photomask Fabrication

Electron beam lithography has shown great promise in photomask fabrication; however, its successive heating process centralizing in a small region may cause a severe problem of critical dimension (CD) distortion. Consequently, subfield scheduling that reorders the sequence of the writing process is needed to avoid successive writing of neighboring subfields. In addition, the writing process of a subfield raises the temperature of neighboring regions and may block other subfields for writing. This paper presents the first work to solve the subfield scheduling problem while considering blocked regions by formulating the problem into a constrained maximum scatter traveling salesman problem (constrained MSTSP). To tackle the constrained MSTSP that can be shown to be NP-complete in general, we identify a special case thereof with points on two parallel lines and solve it optimally in linear time. We then decompose the constrained MSTSP into subproblems conforming to the special case, solve each subproblem optimally and efficiently by a graph-based algorithm, and then merge the subsolutions into a complete scheduling solution. We also extend our algorithm to handle the cases when the moving time of an e-beam writing head is comparable with the writing time of a subfield. Experimental results show that our algorithms are effective and efficient in finding good subfield scheduling solutions that can alleviate the successive heating problem (and thus reduce CD distortion) for e-beam photomask fabrication.