對偶邊界元素法與其在電子元件暨微機電上之分析模擬應用

現代電子元件暨微機電設計較以往複雜許多，傳統之設計方式已不敷應用之需，為有效降低研發設計成本工時，電腦數值分析模擬方式已被融入設計階段，使得設計者能預先了解不同設計參數下之元件物理反應分析值，進而可設計出較優質之產品，而原型通過相關測試工作的機會自然大增，有助於提昇業界之競爭力。本論文即在探討電子元件暨微機電之分析模擬技術。有限元素法、邊界元素法或矩量法為現今電子元件暨微機電電磁場主要之分析模擬技術，其中邊界元素法由於在進行電磁元件外域或無限域分析問題時，有其相對優勢，故邊界元素法已成為近期計算電磁學中相當受矚目的分析技術。然而在以傳統邊界元素法處理部分電子元件之問題(例如：微帶結構、平行板電容器…等)時，定義域內存在退化邊界，若只使用單獨一組核函數無法解題，需配合相當複雜繁瑣之人工邊界模擬方式方能求解，本論文則引用同時利用奇異積分方程與超奇異積分方程兩組核函數的對偶邊界元素法，適切處理相關退化邊界問題，了解不同之薄金屬片間距設計與其帶電量大小間之關係，並快速計算所需之電容係數C值。而在某些特殊退化尺寸狀況，傳統邊界元素法之影響矩陣會還會有矩陣秩數不足問題，產生解題上的困難，在本論文中先以SVD奇異值分解技術研判弱奇異矩陣是否為矩陣秩數不足，再以RBM、CHEEF或CHIEF與超強奇異矩陣等方法，可順利處理上述退化尺寸問題。此外，針對常見於電子或半導體元件設計之多層介電質內含退化邊界，本論文以對偶邊界元素法之相關基本理論為出發點，將單一定義域配合分離技術擴充至多層介電質材料之模擬。採用本論文之分析方式，可取代複雜的Fourier-Bessel transform運算方式，明確掌握電場邊緣效應，了解不同之介電質組成配方設計之變化，進而準確計算出金屬片上之電荷分佈，迅速提供設計者有效之資訊。在微機電元件上分析模擬應用層面方面，本論文以對偶邊界元素法求解微機電系統梳狀電極之靜電場問題、間距與飄浮力之關係探討、間距暨指寬比及行進距離與驅動力之關係探討、指寬比暨深寬比與飄浮負荷間之關係探討等。經由本論文之分析可知：作用於微機電系統梳狀電極可移動指之驅動負荷值在行進初期與末期會因為電場邊緣效應而產生劇烈變化，而在其他行進距離下狀況，尚可維持在一個定值，但驅動負荷值會隨著可移動指與固定指之間距值增加而減少，而可移動指與固定指之指寬比對驅動負荷值的影響則可予以忽略；此外，本論文之分析解可有效改善傳統之簡易解由於無法掌握可移動指尖端電場邊緣效應產生之誤差。針對目前並無簡易解之作用於可移動指之漂浮負荷，以本論文之分析方法，不必如有限元素法般需要建立巨大之分析模型，即可快速了解相關物理機制反應，亦即漂浮負荷會隨著可移動指與固定指之指寬比值增大而變小，而在特定範圍內則會隨著可移動指與固定指之深寬比值增大而變大。除了指寬比與深寬比值之影響外，漂浮負荷也會隨著可移動指與基板之間距值增大而變小，而且亦會隨著可移動指與固定指之間距值變小而增大。

關鍵字

對偶邊界元素法；微機電；電子元件；電磁

並列摘要

For electronic devices and MEMS, the calculations of surface charge density and electrostatic force due to electric field are very important, and an accurate electrostatic analysis is essential. For variable designs of electronic devices and MEMS, the BEM has become a better method than the domain-type FEM because BEM can provide a complete solution in terms of boundary values only, with substantial saving in modeling effort. Although the BEM is a widely used computational technique nowadays because of the superiority for infinite boundary and unbounded domain, the influence matrix is rank deficient or ill-conditioned and numerical results become unstable when the degenerate problems with singularity caused by a degenerate scale or a degenerate boundary are encountered. In this dissertation, the dual BEM and some regularization techniques were used to obtain rapid, precise, and efficient solutions for simulating and modeling of electronic devices and MEMS. In addition to degenerate problem, engineers usually adopt multilayered design for semiconductor and electronic devices, the dual BEM accompanied by subregion technology, instead of tedious calculation of Fourier-Bessel transforms for the spatial Green’s functions, could be used to efficiently simulate the electric effect of diverse ratios of permittivity between arbitrarily multilayered domain and the fringing field around the edge of conductors. Results show that different ratios of permittivity will affect the electric field seriously, and the values of surface charge density on the edge of conductors are much higher than those on the middle part because of fringing effect. In addition, if using the dual BEM to model the fringing field around the edge of conductors, the minimum allowable data of dielectric strength for keeping off dielectric breakdown can be obtained very efficiently. Since the gap size between combdrive fingers and ground plane or movable finger and fixed finger, the width ratio between movable and fixed fingers, and the aspect ratio between the height and width of fingers, can play very important roles for MEMS performance, the studies of the variations in gap, finger ratio and aspect ratio are indispensable. By way of dual BEM, the less the gaps between combdrive fingers and ground plane are, the larger the levitating force acting on the movable finger is, and the levitating force becomes more predominant as the gaps between movable finger and fixed finger decrease. Besides levitating force, the results from dual BEM also show that the driving force is obviously dependent on traveled distance, and the approximate method can’t work well for all traveled positions because there is an apparent error, especially at the beginning and ends of the range of travel. In addition, the smaller the gap between movable and fixed fingers is, the larger the driving force is, and the error of approximate method also becomes more and more predominant as the gap decreases.

並列關鍵字

dual BEM ； electromagnetics ； MEMS ； electronic device

參考文獻

[1] Adams S. G., Design of electrostatic actuators to tune the effective stiffness of microelectromechanical systems, PhD thesis, Cornell University, 1996

[2] Aliabadi M. H., The boundary element method, John Wiely & Sons, Ltd., 2002

[3] Bathe K. J., Finite element procedures in engineering analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1982

[4] Beerschwinger U., Miline N. G., Yang S. J., Reuben R. L., Sangster A. J. and Ziad H., “Coupled electrostatic and mechanical FEA of a micromotor,” IEEE J. Microelectromech. Syst., Vol. 3, No. 4, pp. 162-171, 1994

[5] Brebbia C. A., The boundary element method for engineers, Prentech Press, London, 1978

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對偶邊界元素法與其在電子元件暨微機電上之分析模擬應用

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