波茲曼傳輸方程式離子植入模擬-物理模式及數值技術

Abstract

在本研究中﹐我們提出一些新的物理模式及數值技術以改善現有的波 茲曼傳輸方程式離子植入模擬程式。我們發展出有理式近似的核子散設射 截面函數和直接的核子散射截面計算方法。對於在波茲曼傳輸方程式中記 錄離子能量大小及方向角度的動量矩陣﹐則首次採用不等分離子能量分格 的動量矩陣。並且成功地運用於非晶質離子植入及回彈原子分佈的模擬運 算。 在波茲曼傳輸方程式中﹐散射效應均以散射截面來表示。在 本研究中﹐我們首先針對核子散射截面的計算方法做了深入的探討。先前 ﹐Linhard等人提出近似的核子散射截面函數﹐但我們發現現存的散射截 面函數運算式均不夠準確或過於繁複﹐因此我們發展出較有效率的有理式 表示式。然而﹐為了較精確地描述核子散射現象﹐我們更進一步發展出一 套直接計算核子散射截面的方法。由Biersack計算散射角度的魔術公式﹐ 我們反向計算而建立壹組入射參數對應能量及散射角的關係﹐在模擬過程 中﹐再採用內插法找出入射參數﹐並直接計算出核子散射截面。經與實驗 及蒙地卡羅(Monte Carlo)模擬程式結果比較後﹐我們發現採用直接算法 可以得到比較正確的模擬結果﹐以硼離子植入矽標靶為例﹐與蒙地卡羅結 果比較下﹐投射距離的誤差可降至10%以下。 對於較重的離子入射問 題﹐我們發現採用不等分離子能量分格的動量矩陣將可以得到較正確的結 果﹐這是因為較嚴謹地考量了低能量的部份和精準地計算停止繼續運動離 子的個數的關係。以100 keV鉍離子植入矽標靶為例﹐與蒙地卡羅結果比 較下﹐投射距離的誤差由14%降為10%。 我們同時也發展了多次往復的 波茲曼傳輸方程式模擬程式﹐以運用於較輕的離子入射和回彈原子的模擬 問題。除了單元素標靶外﹐我們亦考慮到多元素標靶的問題﹐與實驗結果 比較發現﹐採用我們所發展的直接核子散射截面計算和不等分離子能量分 格的動量矩陣﹐對中低能量的重離子入射多元素標靶的模擬上﹐可得到較 準確的結果。以鉺離子植入碳化矽標靶為例﹐與實驗結果比較下﹐投射距 離的誤差由15%降為11%。 此外﹐我們也藉助具有多顆處理器的CONVEX SPP-1000超級電腦以及平行虛擬機(PVM)的軟體環境﹐我們成功地發展出 平行處理的多元素標靶的波茲曼傳輸方程式離子植入模擬程式。以砷離子 植入五元素標靶AZ1350的問題為例﹐平行化程式的計算速度要比原先程式 快上3.3倍。 In this dissertation, we have developed some physical models and numerical techniques for stepwise Boltzmann transport equation (BTE) simulations. They are the rational function fitting of the nuclear scattering cross-section function, the exact nuclear scattering cross-section calculation and the non- uniform energy grid momentum matrix which records the ion energy and direction angle. They are successfully applied in ion implantation simulations including ion and recoil distributions. The stopping power is expressed in terms of the differential scattering cross-section. The nuclear differential scattering cross-section is examined in great detail in this work. In order to simplify the scattering cross-section, Lindhard, Nielson and Scharff (LNS) had proposed a nuclear scattering cross-section function to reduce the nuclear cross-section into one-variable. However, current existing cross-section function equations are too tedious or not accurate enough. A more efficient and precise rational function fitting is devised for the LNS nuclear scattering cross-section function. In order to describe the nuclear stopping effects correctly, a systematic evaluation of the exact nuclear differential scattering cross-section is presented. It is composed of a two-dimensional table construction and a two-dimensional divided difference interpolation. The two-dimensional table is synthesized of the impact parameter as a function of ion energy and scattering angle. It is obtained by the iterative reverse magic formula method. The integrals involving the nuclear scattering cross- section are carefully evaluated with our interpolation. After comparing to conventional methods, we find our new scheme can produce a better agreement with the experiment and Monte Carlo (MC) simulations. For example, the relative errors of projected ranges for B in Si are reduced to be lower than 10% compared to MC. In addition, a non-uniform energy grid momentum matrix is proposed to replace the conventional uniform grid matrix for heavy ion implantations. The non-uniform grid can consider the low energy part strictly and the number of stopped particles is obtained exactly. As an example, the relative errors of projected ranges for 100 keV Bi in Si are reduced from 14% to 10% compared to MC. Our program is also applied to other implantation problems, a multi-pass BTE is developed for the calculations of the light ion and recoil distributions. Besides, the multi-component target implantations are considered. It is proved that our program can give a more correct outcome, for instance, the errors of projected ranges for Er in SiC are reduced from 15% to 11% compared to the experiment. A parallelized BTE program for the multi-component target implantation simulation is developed on CONVEX SPP-1000 with PVM (Parallel Virtual Machine) environment. A speed-up factor of 3.3 has been achieved for the simulation of AZ1350 of five components.