Analytical Placement Considering the Electron-Beam Fogging Effect

Abstract

Modern electron-beam lithography (EBL) suffers from the long-range fogging effect that incurs undesired excessive exposure and thus layout pattern distortions. In this article, we propose an analytical 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 derive a fogging source modeling and further develop an efficient, accurate evaluation scheme to estimate the fogging effect by the fast Gauss transform with the Hermite expansion. The scheme achieves a 30.2X speedup over the traditional convolution computation, with only about 2.35% absolute average errors, which enables the iterative evaluation and variation minimization of the effect during global placement. We also develop fogging-aware legalization and detailed placement to further optimize the placement quality, while limiting fogging variation. The experimental results show that our algorithm can effectively reduce the fogging variation by 35.4%, while maintaining high wirelength quality, at reasonable runtime.