Analytical optimization of high-transmission attenuated phase-shifting reticles

For sub-50-nm semiconductor patterning processes, extremely coherent illumination and high-transmission (Hi-T) attenuated background phase-shifting reticles are becoming increasingly important in delivering high-quality aerial images. Here, a Hi-T reticle is defined as any attenuated phase-shifting mask with a background transmission intensity of 12% or higher of incident light. While numerical simulation has served as a powerful tool in reticle design, the analytical relation of image log slope (ILS) and mask error enhancement factor (MEEF) are calculable in concise, exact form to reveal additional insight into reticle optimization. The properties of ILS and MEEF can be derived analytically in predicting the optimal range of mask bias in terms of image contrast and sensitivity for improved critical dimension (CD) control. These derived relations also demonstrate the capability for analysis beyond extensive numerical simulation and can be easily scaled and applied to various pitches and mask CDs. These relations are also verified in this paper with simulation of realistic process parameters and thus prove the applicability in optimization of Hi-T reticles.

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