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  • 學位論文

汲取複雜結構金屬-介電質互連電容之新隨機算法

A New Stochastic Solver for Complex-Structured Metal-Dielectric Interconnected Capacitance Extraction

指導教授 : 張建成

摘要


本研究目的為利用隨機漫步法發展一快速、準確且能計算二維及三維含有斜邊金屬與多層介電質互連電容的新隨機演算法。其中本研究是以方形隨機漫步法為基礎,並結合Chang,C.C研究團隊獨創的停留介面法(始於Lin,Y.B碩士論文)、Coz文獻中:在介面上時,往介面兩側漫步之機率及單層介電質方形隨機漫步以及Yu文獻中的多層介面格林函數數值特徵化,以處理含有斜邊導體及多層介電質結構之問題。而在電場計算方面同樣採用Chang,C.C研究團隊獨創的口字型積分法,將電位值利用解析的方式來求取電場,其具有高精準度的特色。 斜邊金屬導體的研究資料在文獻中十分少見,在本論文中,我們擴展和改進了所有上述算法,以計算具有斜邊之金屬和多層電介質的互連的電容。本文中拓展以及改進之處有以下幾點: (i) 在取樣點布置方面,為因應因斜邊所造成的狹縫問題,因此採用沿著導體布置取樣點的方式。在導體邊上時,取樣點會與導體邊相互平行;而在導體轉角處時,則以頂點為圓心,取樣點以極座標的方式布置。如此高斯面才可完全包覆導體。 (ii) 由於隨機漫步時之最大正方形需與最接近之導體或介面相切,因此正方形通常需要做旋轉。因此本研究提出一新方法以計算正方形旋轉且有一界面通過正方形中心之機率分布。 (iii) 由於導體轉角處之取樣點是以極座標方式布置,因此無法使用原先之口字型積分法計算由取樣點構成的網格的中心點電場,因此導體轉角處之取樣點網格之中心點電場改以有限差分的方法計算。 (iv) 本研究也採用Cubic Spline的方法以提升演算法計算之速度。 除了開發適用於計算斜邊金屬導體的演算法之外,我們更是對斜邊結構計算時之取樣點間距、重複計算次數、取樣點與導體之距離以及最大步長限制…等參數做詳細的分析以及探討。 (1)由結果顯示,取樣點間距是決定電容值計算結果精準度最重要的參數,取樣點間距越小,電位值分布的情況就能計算的越詳細,藉此以提高電容計算的精準度。 (2) 重複計算次數則是決定電位值計算結果的準確度的重要參數。重複計算次數越高,電位值結果越準確,但在達到一定次數之後結果便會收。 (3) 依照高斯定律,高斯面之大小並不會影響電容的計算結果,但由於演算法的特性,其計算結果誤差會隨著取樣點與導體之距離增大而增加。 (4) 最大步長限制對於電容計算結果並沒有明確的正面或負面的影響,但對於演算法計算時間卻有著很大的影響,當最大步長限制到達一定大小時,計算時間便會劇烈增加。 (5) 本研究利用Cubic Spline內插法,將少數精準計算點之結果做內插,以達到減少取樣點計算量。其中一般導體都可以利用此方法將計算量減少到原本的1/8並且保有1%左右的誤差大小。 (6) 最後比較二維以及三維的模擬結果,其中二維電容值計算結果需要較高的重複計算次數才會收斂,三維則反之。 總結而論,本研究之演算法在計算二維及三維多層介電質-斜邊導體結構之計算結果已經有十分不錯的精準度,而未來則需增加薄膜計算的功能、程式碼的優化以及平行化等…加速計算的演算法,使本研究之演算法能更加完善。

並列摘要


The study is aimed to develop a new fast and accurate stochastic algorithm for extracting the electric capacitance of-non-rectangular metal-dielectric interconnect. The elements of the new stochastic solver for rectangular interconnects comprise (i) a known block algorithm for the Laplace equation without and with interface (Coz et al.), (Yu et al.) combined with (ii) a pausing algorithm at the Interface between dielectrics to definitely capture the effect of the interface (Chang et al.), and (iii) a new boundary treatment (Chang et al.) based on a Green’s function formulation (Yu et al.). In particular, a special technique of integration using the sampling potentials on the corner points of a boundary square is developed to calculate the electric field near a conductor with high precision. The total charge of each conductor is thus determined, and then all the self- and mutual capacitances of the interconnect can be evaluated. There are few references in the literature to the current issue for oblique metal-dielectric interconnect. In this thesis, we extend and/or improve all the above algorithms/techniques to calculate the capacitances of interconnects that have oblique shapes of metal and multi-layers of dielectrics. The features of the extension and improvements are several: (i) At the sampling point arrangement, the sampling points are distributed parallel to the side of the conductor and distribute in the form of the polar coordinate by taking the vertex as the center of circle when the sampling points are close to the corner of the conductor. (ii) Because of the reason that the block of the random walk must be parallel to the nearest conductor or interface, so the blocks usually needs to be rotated. Here we develop a new way to calculate the probability distribution of the block when the block is rotated and the interface is passing through the block. (iii) Because the sampling points distribute in the form of the polar coordinate by taking the vertex as the center of circle when the sampling points are close to the corner of the conductor, we use Finite difference method to calculate the electric field of the corner of the conductor. (iv) We also use Cubic Spline to reduce the amount of calculation and accelerate the simulation. We also do a detailed analysis and discussion of the parameters such as : Sampling point spacing, The number of repetition calculations, The distance between the sampling point and the conductor, and The restriction of maximum step size. (1) The results show that the sampling point spacing is the most important parameter to determine the accuracy of the ‘capacitance calculation’. The smaller the sampling point spacing we used, the more detailed the description of the potential energy distribution is, and we can improve the accuracy of the capacitance calculation. (2) The number of iterations is an important parameter to determine the accuracy of the potential energy’ calculation results. The higher the number of iterations, the more accurate the potential value is, but the results will converge when the number of iterations reach a certain number of times. (3) According to Gaussian law, the size of the Gaussian surface does not affect the calculation results of capacitance. However, due to the characteristics of the algorithm, the error of calculation result will increase with the increased of the distance between the sampling point and the conductor. (4) The restriction of maximum step size has no positive or negative effect on the capacitance calculation result, but has a significant effect on the computational time of the algorithm. When the size of maximum step reaches a certain size, the computational time of the algorithm will increase dramatically. (5) This study also uses Cubic Spline interpolation, which uses a few results with high precision, to do the interpolation to reduce the amount of calculation. In general, the amount of calculation can be reduced to 1/8 and keep the error about 1%. (6) We also compare the result of the two-dimensional and the result of the three-dimensional. The result of the two-dimensional capacitance calculation requires a higher number of iterations to converge, and the results of three-dimensional does not. In summary, the result of problems, containing oblique edge conductors and multi-dielectric in 2D and 3D, calculated by the algorithm of this study have a good accuracy. To make the algorithm more perfect, we have to add to function of thin film computing, code optimization and parallel operation, which can accelerate the calculation of the algorithm, in the future.

參考文獻


[1] Y. L. Coz and R. Iverson, “A stochastic algorithm for high speed capacitance extraction in integrated circuits,” Solid-State Electronics, vol. 35, no. 7, pp. 1005 – 1012,1992.
[2] J. N. Jere and Y. L. L. Coz, “An improved floating-random-walk algorithm for solving the multi-dielectric Dirichlet problem,” IEEE Transactions on Microwave Theory and Techniques, vol. 41, pp. 325–329, Feb 1993.
[3] G. M. Royer, “Monte Carlo Procedure for Theory Problems Potential,” IEEE Transactions on Microwave Theory and Techniques, vol. 19, pp. 813–818, Oct 1971.
[7] H. Zhuang, W. Yu, G. Hu, Z. Liu, and Z. Ye, “Fast floating random walk algorithm formulti-dielectric capacitance extraction with numerical characterization of Green’s functions,” in 17th Asia and South Pacific Design Automation Conference, pp. 377–382, Jan 2012.
[8] W. Yu and X. Wang, Advanced field-solver techniques for RC extraction of integrated circuits. Springer, 2014.

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