A recursive cost-based approach to fracturing

In microlithography, mask patterns are first fractured into trapezoids and then written with a variable shaped beam machine. The efficiency and quality of the writing process is determined by the trapezoid count and external slivers. Slivers are trapezoids with width less than a threshold determined by the mask-writing tool. External slivers are slivers whose length is along the boundary of the polygon. External slivers have a large impact on critical dimension (CD) variability and should be avoided. The shrinking CD, increasing polygon density, and increasing use of resolution enhancement techniques create new challenges to control the trapezoid count and external sliver length. In this paper, we propose a recursive cost-based algorithm for fracturing which takes into account external sliver length as well as trapezoid count. We start by defining the notion of Cartesian convexity for rectilinear polygons. We then generate a grid-based sampling as a representation for fracturing. From these two ideas we develop two recursive algorithms, the first one utilizing a natural recurrence and the second one a more complex recurrence. Under Cartesian convexity conditions, the second algorithm is shown to be optimal, but with a significantly longer runtime than the first one. Our simulations demonstrate the natural recurrence algorithm to result in up to 60% lower external sliver length than a commercially available fracturing tool without increasing the polygon count.