Fogging Effect Aware Placement in Electron Beam Lithography.

Modern electron beam lithography (EBL) suffers from the long-range fogging effect which incurs undesired excessive exposure and thus layout pattern distortions. In this paper, we propose the first placement algorithm to tackle the fogging effect. The underlying idea is to place standard cells, guided by our efficient, yet reasonably accurate fogging effect model to minimize the fogging variation during placement, and thus the effect can be corrected by reducing dosage uniformly over the chip. We use fast Gauss transform with Hermite expansion for convolution approximation to estimate the effect. This approximation achieves a 26.5X speedup over traditional convolution computation, with only about 0.33% absolute average errors. We derive a fogging source model and further develop an efficient, accurate evaluation scheme to estimate the fogging effect by fast Gauss transform. The scheme achieves a 30.2X speedup over traditional convolution computation, with only about 2.35% absolute average errors, which enables the iterative evaluation and variation minimization of the effect during analytical global placement. We also develop fogging-aware legalization and detailed placement to further optimize the placement quality, while maintaining fogging variation. Experimental results show that our algorithm can effectively reduce the fogging variation by 15.5%, while maintaining high wirelength quality, at reasonable runtime.