A novel curve-fitting procedure for determining proximity effect parameters in electron beam lithography

Accelerating voltage as low as 5 kV for operation of the electron-beam micro-columns as well as solving the throughput problem is being considered for high-throughput direct-write lithography for the 22-nm half-pitch node and beyond. The development of efficient proximity effect correction (PEC) techniques at low-voltage is essential to the overall technology. For realization of this approach, a thorough understanding of electron scattering in solids, as well as precise data for fitting energy intensity distribution in the resist are needed. Although electron scattering has been intensively studied, we found that the conventional gradient based curve-fitting algorithms, merit functions, and performance index (PI) of the quality of the fit were not a well posed procedure from simulation results. Therefore, we proposed a new fitting procedure adopting a direct search fitting algorithm with a novel merit function. This procedure can effectively mitigate the difficulty of conventional gradient based curve-fitting algorithm. It is less sensitive to the choice of the trial parameters. It also avoids numerical problems and reduces fitting errors. We also proposed a new PI to better describe the quality of the fit than the conventional chi-square PI. An interesting result from applying the proposed procedure showed that the expression of absorbed electron energy density in 5keV cannot be well represented by conventional multi-Gaussian models. Preliminary simulation shows that a combination of a single Gaussian and double exponential functions can better represent low-voltage electron scattering.