Generalized mask error enhancement factor theory

Microlithographic difficulties in design shrinkage are commonly characterized by the k1 resolution factor, depth of focus, exposure latitude, and mask error enhancement factor (MEEF). Though the importance of exploring mask error enhancements under a range of process conditions and for various feature types is well understood, the MEEF theory embraces only simple features like isolated lines, dense lines, or contacts, with a single degree of mask distortion freedom. We introduce a generalized mask error enhancement theory that explores complex 2-D mask distortions. The error enhancement is described by the mask error enhancement matrix (MEEM) that transforms mask distortions into wafer damages. MEEM captures the complex effects of self- and cross-enhancements when neighboring mask features collectively contribute to wafer errors. The concept of generalized MEEF (or G-MEEF) is introduced. G-MEEF is a strictly defined, unambiguous measure of mask error amplifications. We introduce distortion constraints to establish the partial MEEF (P-MEEF) framework that reconciles G-MEEF with "global biasing," "local biasing," and other MEEF variations. Next, a singular value decomposition apparatus is used to conduct spectral analysis of MEEM. This theory is applied to realistic mask regions with complex shapes.