3.3.1 Die-Size Simulation of Optical Proximity Correction

The most important application of large area or die-size simulators is optical proximity correction (OPC). The proximity effect refers to the situation that the actual image of a pattern depends on its proximity to other features. Examples for that are line width variation between dense and isolated lines, line shortening at the end of lines, and corner rounding in the case of contact holes. Although the proximity effect is primarily caused by diffraction, it is not a pure optical phenomenon as its actual magnitude and relevance depends on other lithographic steps, whereby the resist process has the greatest influence [56]. Proximity effects are not the only type of situation in which features print differently from the mask depending on their dimensions. Print bias, i.e., the difference between mask and the printed dimension, may be different for different feature sizes. This problem is commonly referred to as linearity. Both linearity and proximity need to be considered in IC fabrication, since geometries of many different shapes and sizes exist on a typical masking level. The above discussion shows that only large area simulations can give insight to the encountered problem since the phenomena do not occur in case of a single feature.

OPC provides a technique to compensate or at least reduce the disturbing effects by either modifying the shape of the features or by adding adjacent subresolution geometries like serifs on the corners to improve image quality [57,58]. Simulation has proved to be a valuable tool to investigate, improve or even automatically correct optical proximity effects. Simple rule-based approaches show promise for first order optical effects and enable restricted process corrections [59,60]. More accurate models are based on Hopkins' method of partial coherent imaging (cf. Section 4.3.2) and make use of expansion techniques similar to the Fast Fourier Transform, but which are much faster and allow mask edges and sizes to have arbitrary location and sizes [61]. For additional speed in OPC a technique of reusing precalculated lens effects over a small window and then scanning that window over the sizeable mask area of interest has been developed [62,63]. Using such advanced algorithms the computation of the aerial image for dense patterns in areas up to 400 $\mu$m x 400 $\mu$m has been reported [64].

Some second order effects are also of concern in aerial image simulation. One is accounting for propagation through lenses with high numerical apertures of 0.5 and above [65,66,67]. Here it becomes important to consider the large ray angles beyond the paraxial assumption in formulating ray path effects, to introduce obliquity factors for flux at various takeoff angles, and to investigate the rotation of the electric field component vectors with propagation angle in the transverse magnetic polarization case. The complexity of such models increases considerably and the simulation areas are therefore restricted to smaller areas.