Optimization of photomask design for reducing aberration-induced placement error

In semiconductor manufacturing, the accurate placement of circuit components ensures the proper functioning of microelectronic circuits. This is often subject to photolithography, an optical technique that transfers circuit patterns from photomasks to silicon wafers. Sources of placement error include aberration and misalignment between different levels, and we focus on the former. Aberration is an optical phenomenon that often degrades imaging system performance. Since aberration differs from one imaging system to another, a photomask design that minimizes the aberration-induced placement error is desired. In this paper, we discuss the optimization process of a general one-dimensional mask pattern under a general illumination condition. The constraint is a known population mean of the root mean square aberrations for the imaging systems under consideration. To apply the theory, we search for the optimal parameters for two common mask designs: alternating phase-shifting masks (PSMs) and attenuated PSMs. The theoretical results are compared with those from a Monte Carlo analysis on a large set of imaging systems. These results are indicative to mask manufacturers and circuit designers of increasing manufacturability of circuits