電子束石版術蒙地卡羅模擬之研究

Abstract

吾人利用蒙地卡羅方法 (Monte Carlo Method) 發展出電子束石版術之模擬程式，電子和核子之間的碰撞截面是利用拉塞福方程式 (Rutherford Equation) 來決定，碰撞時的能量損失則以貝茲方程式 (Bethe Equation) 來計算。利用此程式我們可以追蹤電子在光阻及基材中的三維運動軌跡，並紀錄下能量沉積的三維分佈、能量的橫向擴展及分佈。由數值模擬的結果吾人可以了解電子束石版術中的漸近臨效應 (Proximity Effect)。此外我們亦計算長程 (Long Range) 參數，利用此參數可以定量地描述電子從基材彈回至光阻的情形。 本程式的能量沉積紀錄係採用電子雲 (Electron Cloud) 的策略，以便降低模擬時所需要取樣的電子數量、縮短計算時間及提高準確度。同時採用迴旋積分 (Convolution Integral) 的方法，將能量沉積密度分佈予以擴充，如此可以降低直接計算所耗費的時間。最後吾人亦順便探討電子入射角度對能量沉積分佈的影響。

A Monte Carlo simulation program for the electron-beam lithography has been developed. The screened Rutherford equation for the differential scattering cross section and the Bethe equation for the energy loss between elastic scatterings are used. Three dimensional electron trajectories in the resist and substrate have been followed. The three dimensional distribution, lateral spreading range and longitudinal penetration depth of energy deposition have been obtained. The spreading range has been fitted as a function of incident energy and target density. The proximity effect is ascribed to electron backscattered from the substrate. The long-range (LR) parameters have been defined to indicate the backscattering character of electrons. An “electron cloud” scheme has been adopted in this program to reduce the simulated particles and computational times. A convolution method has been devised to extend the simulated energy deposition distribution. In addition, the effect of incident angle on distribution of energy deposition has also been shown.