離子植入之蒙地卡羅模擬：物理模式、數值技術、高效能計算及其他運用

Abstract

在本研究中，吾人發展了一些高效能離子植入蒙地卡羅模擬程式，這些 程式使用了精確而有效率的方法來計算非晶質單元素及多元素單層及多層 靶在離子植入過程中之離子及回彈原子的分佈。為了決定使用何種物理模 式於離子植入模擬中，吾人分別比較了各種核子及電子阻擋模式。對核子 阻擋而言，吾人發現莫里耳電位最重，ZBL電位最輕，而Kr-C電位則介於 其間。對非局部電子阻擋而言，以LSS模式最簡單，若選擇適當的修正參 數值，可以得到不錯的結果。ZBL模式則依實驗值而設，必須使用與實驗 結果相吻合的八個參數。ABS模式在許多離子與靶的組合中，通常具有最 大的電子能量損失，只適用於中低能量之範圍。為了建立一個較具物理意 義的電子阻擋模式，吾人採用KPT局部能量損失模式，使用RKF積分法計算 離子在靶中的路徑，並使用cubic-spline匹配法求出靶原子在不同位置的 電子電荷密度，而與局部電子濃度相依的電子能量損失，則依照ABS模式 來計算。此研究所得到的電子阻擋與其他局部及非局部電子阻擋相比較， 吾人發現本研究所得到的結果與實驗具有良好的一致性。為了節省模擬時 間並得到精確的結果，吾人發展了一些新方法計算散射角、時間積分及電 子能量損失。不同於傳統積分和硬球體近似，吾人精確定義時間積分為離 子到靶原子的距離與離子路徑長度之差。為了精確及有效地計算散射角和 時間積分，吾人採用了較平緩的積分式，並以一種調適辛浦森積分法求出 。在本研究中，與局部電子濃度相依的電子能量損失是藉由追蹤離子路徑 而得，為了節省計算時間，吾人推導了一種描述離子運動的二階微分方程 式。藉助於具有向量處理能力的CONVEX C-220超級電腦以及使用平行虛擬 機(PVM)軟體環境於快速工作站上，吾人成功地發展出一些向量化與平行 化的蒙地卡羅模擬程式。為了節省計算時間，吾人預先建立電子能量損失 及散射角對離子能量及入衝參數的二維表列，而在離子植入的模擬過程中 ，再使用一種二維內插法來估算各個碰撞的對應數值。吾人曾記錄硼離子 在不同入射能量植入矽中的表列建立及全部模擬時間。向量化程式的速度 要比原來的純量計算提昇三到六倍；而平行化程式的效率也接近於一的理 想值。為了避免不同離子間的模擬結果之可能相關聯，吾人曾細心地將亂 數產生器加以向量化及平行化。 In this study, we have developed some high performance Monte Carlo simulation programs for ion implantation. These simulation programs can calculate the ion and recoil distributions in amorphous multi-layer and multi-element targets using accurate and efficient methods. To determine physical models to be used in ion implantations, we have compared various models of nuclear and electronic stopping. For nuclear stopping, we find that the Moliere potential is heaviest, the ZBL potential is lightest and the Kr-C potential is intermediate. For nonlocal electronic stopping, the LSS model is the simplest model, and good results can be obtained by properly adjusting the correction factor. The ZBL model is most elaborate, eight parameters have to be used to fit the experimental results. The ABS model has largest stopping power for various ion-target combinations and is usable only at low or intermediate energy. A physical model of electronic stopping is established based on the local energy loss proposed by Klein, Park, and Tasch (KPT). An adaptive Runge-Kutta- Fehlberg algorithm is used to compute the trajectory of an ion in collision with a target atom, and the cubic spline fitting method is used to interpolate the electronic charge density of a target atom at different positions. Electron-concentration- dependent energy loss is evaluated according to the ABS model. The electronic stopping power obtained in this work is compared with other local and nonlocal electronic stopping power models. We find that our results are agreed well with the experimental results. To reduce computation times and obtain accurate results, we have developed some new methods for calculating the scattering angle, time integral and electronic energy loss. In contrast to the conventional integration and the hard sphere approximations, new time integral is defined as the difference between the distance to the target atom and the ion path length. To compute the scattering angle and time integral accurately and efficiently, an adaptive Simpson quadrature using a smoother integrand is adopted. The energy loss arising from the local electron-concentration-dependent electronic stopping power is obtained by tracing the ion trajectory. To reduce the computation time, a second-order differential equation for the ion motion is developed. By taking advantage of the vector processing capability of the CONVEX C220 mini-supercomputer and the parallel processing environment of high speed workstations using the Parallel Virtual Machine (PVM) software, we have successfully developed vectorized and parallelized Monte Carlo simulation programs. Two-dimensional tables of the scattering angle and electronic energy loss as functions of ion energy and impact parameter are constructed in advance and a two- dimensional interpolation technique is devised to evaluate the corresponding values for each binary collision during the particle simulation. The CPU times used for the table constructions and overall simulations for boron implanted into silicon as functions of incident energy have been recorded. The speedup factor of vectorization ranges from three to six and the efficiency of parallelization is nearly equal to unity. To avoid the possible correlation of simulated results between different ions, we have vectorized and parallelized the random number generators carefully.