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Chapter 4:
Modeling and Thin-Film Effects

Imagine if the only way to determine whether bridges were designed with sufficient strength to support the weight of fully loaded trucks was to build each bridge in its desired location according to an engineer's best guess, drive trucks across to see if the bridge collapsed, and redesign the bridge if it failed. This approach to bridge design is clearly impractical. What has made it possible to build large bridges in a practical way are the mathematical theories of beams and plates that enable civil engineers to predict whether their designs will work or not, prior to construction. Similarly, advancements in lithographic technology have been facilitated by the availability of predictive theoretical models and tools. Today, imaging performance can be simulated on personal computers and engineering workstations in order to determine what the design parameters should be for lenses that cost millions of dollars to fabricate. Process options can be explored extensively without building large numbers of extremely expensive photomasks. In short, modeling of the lithographic process has contributed substantially to the progress in microelectronic technology.

Lithography simulations involve several key steps:

(1) The calculation of optical images. These are intensities I(x, y) in the plane of the wafer that are applicable for low-NA optics, or they can be fully three-dimensional intensities I(x, y, z) that may be needed for accurate simulation of exposures using high-NA lenses.

(2) Prediction of the photochemical reactions resulting from the exposure of photoresist to the previously calculated light distributions. This provides a calculated state-of-exposure at every point (x, y, z) of interest in the resist film.

(3) Computation of changes in chemical distributions within the resist as a consequence of chemical reactions and diffusion that occur during postexposure bakes.

(4) Calculation of resist profiles following resist development. Models for all of these key steps are discussed in this chapter.

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