Heuristic Method for Phase-Conflict Minimization in Automatic Phase-Shift Mask Design

Abstract

A shifter placement program has been developed for alternate-type phase-shift masks that require a 180°-phase difference between adjacent aperture pairs. Using a heuristic approach, this program minimizes the number of phase conflicts that arise when adjacent apertures are forced to have the same phase. Adjacent aperture relationships are expressed by an undirected graph; the phase conflicts are found in the closed loops containing an odd number of apertures. Minimization is achieved by listing these loops and identifying the adjacent aperture pairs that occur in more than one loop. The validity of this algorithm was demonstrated using test patterns.