Lithographic modeling has played very critical roles in advancing lithographic science and technology. These roles span many aspects of the patterning process: as a research tool for gaining insight into the outcomes of experiments that are difficult or impossible to execute any other way, as a development tool for evaluating options and optimizing processes, as a manufacturing tool for troubleshooting process problems and determining optimal yield settings, and as a learning tool for studying different aspects of the lithographic process. All of these functions of lithographic modeling are today readily performed on personal computers and engineering workstations. How did all of these come about? What is involved in lithographic modeling? Specifically, what physical models are employed in a typical lithographic simulator? How good is lithographic modeling in predicting lithographic results? And how is lithographic modeling currently employed in the semiconductor industry? These questions and more are addressed in this chapter.

12.2 Historical Background

The roots of lithographic modeling date to the early 1970s when Dill and co-workers at IBM set out to describe the basic steps of the lithographic process with mathematical equations, resulting in a set of publications in 1975, now commonly referred to as the "Dill papers."1 Together, the Dill papers not only gave birth to the field of lithographic modeling, they also marked the first time a serious attempt was made to describe lithography not as an art, but as a science. In the papers, Dill and his co-workers presented a simple model for image formation with incoherent illumination, namely, "Dill's first order exposure kinetic model," and an empirical model for development coupled with a cell algorithm for photoresist profile calculations.

In order to make their task more tractable, they broke up the lithographic process into a sequence of calculations designed to determine the intensity of light inside the resist (calculation of the standing wave), the chemical concentration of exposure products resulting from this light, the impact of this chemistry on the development rate, and, finally, the integration of the development rate through time to predict the resist thickness after development. A most auspicious event at this time was the development of an automated thin-film thickness measurement tool at IBM; this tool could be used to monitor the effects of lithographic process steps on an exposed resist film.