汲取多金屬-介電質互連電容之新隨機算法

Abstract

本研究目的為發展一快速且準確度高，可用於汲取二維及三維多金屬與介電質互連電容的新隨機算法。本文發展之新隨機算法是以方形隨機漫步法為基礎， 結合Chang,C.C. 研究團隊獨創的停留介面法(於Lin,Y.B.的論文中提及)，以及Yu文獻中的多層介面格林函數數值特徵化的方法以處理多層電質問題; 而電場積分的方式同樣使用由Chang,C.C.研究團隊獨創的口字型積分法，因其具有高準確度的優勢。 網格大小、重覆次數以及有限差分的網格數目為隨機漫步的主要參數， 但於文獻中較少評估這些參數的影響程度， 因此於本文中加以探討這三個參數對於隨機漫步法的準確度以及計算時間的影響。 結果顯示，參數對於準確度影響按大小依序為網格大小、重覆次數、有限差分的網格數目。網格越小越能清楚捕捉勢能分佈情形，準確度也因此提高;重覆次數達到一定數量即可使計算結果收斂，更多的重覆次數並不能提升準確度;有限差分的網格數目影響甚微，更多的網格數目並不能提升計算的準確度，且會因網格越小或重覆次數越多而越小。最後比較二維與三維模擬結果，並從結果觀察出三維較二維易於收斂，此結果乃是因二維計算點少每個點皆對整體計算結果相當重要，因此需要將每個點經過多次重覆計算，才能得到準確度高的結果，而三維計算點多，因此每個點的重要性相對小，因此僅需少量重覆次數即可達到一定準確度。 總結而論，新隨機算法可用於計算比商用軟體更為複雜的二維及三維結構，且模擬結果準確度已達一定水準，未來，研究之主要課題將朝向優化程式以及平行運算進行。

This study aims at developing a fast and accurate stochastic solver for extracting 2D and 3D multi metal-dielectric interconnect capacitances. The new stochastic solver is fundamentally based on the squared-shaped random walk, combing Stop at Interface method proposed by Chang,C.C. et al. and technique of numerical characterization of Green's function method proposed by Yu in order to solve the multi-dielectric problems. As for the purpose to enhance accuracy, we use a novel square like integration method to calculate the electric field proposed by Chang,C.C. et al. Factors such as the grid sizes, realizations and the grid points of the finite difference method are three major parameters for the random walk. However, little literature has been published on the issue of evaluating the influence of each parameters. Thus, in this study, it is also illustrated the errors and computation costs by adjusting these three factors. Numerical results show that the grid size has the dominant influence on accuracy, followed by realizations and grid points of finite difference. The smaller the grid size is, the more possibility to capture the potential distribution clearly. Hence, the accuracy have a great improvement. In addition, once the realizations up to a specific value, the computational results converge. It is, therefore indicating the accuracy cannot be improved by more realizations. Further, the grid points of finite difference is proven to have less influence on the accuracy and become even independent with the accuracy when grid size is decreasing of realizations is increasing. In the last part, by comparing the results of 2D and 3D, we discover that it is prone to converge in 3D cases. This is because the 3D simulation has an extra dimension than 2D which implies more realizations superimposing on the model. This is because 2D simulations compute less grid points such that high realizations of each point will lead to accuracy result. In contrast, Therefore, in 3D cases, it is simply using little realizations to achieve high accuracy. To sum up, the proposed new stochastic method can solve more complicated 2D and 3D cases than the commercial software. In addition, the accuracy of the proposed method has been proven to meet up with a specific extent. It is hope to providing an optimized programming with parallel computation.