Filling algorithms and analyses for layout density control

In very deep-submicron very large scale integration (VLSI), manufacturing steps involving chemical-mechanical polishing (CMP) have varying effects on device and interconnect features, depending on local characteristics of the layout. To reduce manufacturing variation due to CMP and to improve performance predictability and yield, the layout must be made uniform with respect to certain density criteria, by inserting "fill" geometries into the layout. To date, only foundries and special mask data processing tools perform layout post-processing for density control. In the future, better convergence of performance verification flows will depend on such layout manipulations being embedded within the layout synthesis (place-and-route) flow. In this paper, we give the first realistic formulation of the filling problem that arises in layout optimization for manufacturability. Our formulation seeks to add features to a given process layer, such that 1) feature area densities satisfy prescribed upper and lower bounds in all windows of given size and 2) the maximum variation of such densities over all possible window positions in the layout is minimized. We present efficient algorithms for density analysis, notably a multilevel approach that affords usertunable accuracy. We also develop exact solutions to the problem of fill synthesis, based on a linear programming approach. These include a linear programming (LP) formulation for the fixeddissection regime (where density bounds are imposed on a predetermined set of windows in the layout) and an LP formulation that is automatically generated by our multilevel density analysis. We briefly review criteria for fill pattern synthesis, and the paper then concludes with computational results and directions for future research.