廖運炫臺灣大學：機械工程學研究所全湘偉Chyuan, Shiang-WoeiShiang-WoeiChyuan2007-11-282018-06-282007-11-282018-06-282004http://ntur.lib.ntu.edu.tw//handle/246246/61581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。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.摘要…………………………………………………………………….... Ⅰ 英文摘要………………………………………………………………….. Ⅲ 目錄……………………………………………………………………….. Ⅴ 符號說明………………………………………………………………….. Ⅷ 表目錄…………………………………………………………………….. ⅩⅡ 圖目錄…………………………………………………………………….. ⅩⅢ 第一章· 緒論…………………………………………………………….. 1 1-1. 引言………………………………………………………………... 1 1-2. 文獻回顧…………………………………………………………... 2 1-3. 本文目的…………………………………………………………... 6 1-4. 本文內容…………………………………………………………... 6 第二章·對偶邊界元素法之基本理論…………………………………… 8 2-1. 引言………………………………………………………………... 8 2-2. 對偶邊界積分方程式……………………………………………... 9 2-3. 退化邊界(Degenerate boundary)問題之處理…………………….. 15 2-3-1. 傳統BEM之處理方式………………………………………... 15 2-3-2. DBEM之處理方式…………………………………………….. 16 2-4. 多層介電質材料之模擬…………………………………………... 17 2-5. 弱奇異(Weakly singular)矩陣秩數不足(Rank-deficiency)問題之處理………………………………………………………………... 18 2-5-1. 奇異或病態(Ill-conditioned)矩陣與非奇異矩陣之研判……... 20 2-5-2. 奇異或病態矩陣之求解法……………………………………. 20 2-5-2-1. RBM解法………………………………………………….. 21 2-5-2-2. CHEEF與CHIEF解法…………………………………….. 21 2-5-2-3. 超強奇異(Hypersingular)矩陣之使用……………………. 22 2-6. BEM與FEM之比較……………………………………………….. 23 2-7. DBEM與傳統BEM之比較……………………………………….. 24 2-7-1. 角奇異問題……………………………………………………. 25 2-7-2. 邊界附近切向通量問題………………………………………. 25 2-8. 結語………………………………………………………………... 26 第三章· 電學基本理論暨對偶邊界元素法在電子元件上之應用…….. 27 3-1. 引言………………………………………………………………... 27 3-2. 基本電學理論……………………………………………………... 29 3-2-1. 靜電學之基本參數定義與原理………………………………. 29 3-2-2. 導體與介電質…………………………………………………. 32 3-2-3. 接觸界面邊界條件……………………………………………. 34 3-2-4. 電容與電容器…………………………………………………. 34 3-2-5. 靜電能與靜電力………………………………………………. 35 3-3. 求解靜電問題之解析法與數值法介紹…………………………... 36 3-4. DBEM在靜電學分析模擬之精確度驗證………………………… 38 3-4-1. 例題3-1………………………………………………………... 38 3-4-2. 例題3-2………………………………………………………... 38 3-4-3. 例題3-3………………………………………………………... 39 3-5. DBEM在電子元件上之應用實例………………………………… 40 3-5-1. 退化邊界問題…………………………………………………. 40 3-5-1-1. Microstrip問題…………………………………………….. 40 3-5-1-2. 平行板電容器問題………………………………………... 41 3-5-1-3. 多層介電質材料內含金屬片問題………………………... 43 3-5-2. 矩陣秩數不足問題……………………………………………. 46 3-6. 結語………………………………………………………………... 49 第四章· 微機電系統分析模擬基本理論……………………………….. 50 4-1. 引言………………………………………………………………... 50 4-2. MEMS簡介………………………………………………………… 52 4-2-1. MEMS之特色與優點………………………………………….. 52 4-2-2. 微機械加工製造技術簡介……………………………………. 53 4-3. 靜電學於MEMS致動元件上之應用…………………………….. 55 4-3-1. 平行板靜電致動器……………………………………………. 56 4-3-2. Combdrive靜電致動器………………………………………... 58 4-3-2-1. Combdrive元件之介紹……………………………………. 58 4-3-2-2. Combdrive之驅動(Driving)與飄浮(Levitating)負荷……... 59 4-3-2-2-1. Combdrive之驅動負荷………………………………... 59 4-3-2-2-2. Combdrive之飄浮負荷………………………………... 60 4-3-2-3. Combdrive之飄浮控制……………………………………. 61 4-4. MEMS分析設計流程與數值分析模擬之挑戰…………………… 62 4-4-1. MEMS分析設計流程之簡介………………………………….. 62 4-4-2. MEMS設計數值分析模擬之挑戰…………………………….. 63 4-5. 結語………………………………………………………………... 63 第五章· 對偶邊界元素法在微機電系統上之應用…………………….. 65 5-1. 引言………………………………………………………………... 65 5-2. DBEM與FEM於MEMS元件分析適用性之比較……………….. 65 5-3. DBEM求解MEMS combdrive之靜電場問題……………………. 66 5-3-1. 例題5-1………………………………………………………... 67 5-3-2. 例題5-2………………………………………………………... 69 5-3-3. 例題5-3………………………………………………………... 70 5-4. 間距與飄浮力之關係探討………………………………………... 71 5-4-1. 可移動指(Movable finger)與基板之間距影響……………….. 71 5-4-2. 可移動指與固定指(Fixed finger)之間距影響………………... 72 5-4-3. 不同行進距離(Traveled distance)狀況下之結構力學反應….. 72 5-4-4. 不同間距狀況下之結構力學反應……………………………. 74 5-5. 間距暨指寬比(Finger width ratio)及行進距離與驅動力之關係探討………………………………………………………………... 75 5-5-1. 不同行進距離之影響…………………………………………. 75 5-5-2. 可移動指與固定指之間距影響………………………………. 76 5-5-3. 可移動指與固定指之指寬比影響……………………………. 76 5-6. 指寬比暨深寬比(Aspect ratio)與飄浮負荷間之關係探討………. 77 5-6-1. 不同指寬比之影響……………………………………………. 77 5-6-2. 不同深寬比之影響……………………………………………. 78 5-6-3. 不同指寬比狀況下之結構力學反應…………………………. 78 5-6-4. 不同深寬比狀況下之結構力學反應…………………………. 79 5-7. 結語………………………………………………………………... 79 第六章· 結論與展望…………………………………………………….. 82 參考文獻………………………………………………………………….. 86 附表……………………………………………………………………….. 100 附圖……………………………………………………………………….. 107 發表文章目錄…………………………………………………………….. 1702177533 bytesapplication/pdfen-US對偶邊界元素法微機電電子元件電磁dual BEMelectromagneticsMEMSelectronic devicethesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/61581/1/ntu-93-D88522017-1.pdf