三維超大型積體電路多層連線的新模擬策略

Abstract

在今日深次微米積體電路設計中，物理連線效應扮演著重要角色。隨 著操作頻率的增加，類似微波的寄生效應將愈顯著，因此萃取寄生參數和 驗證的工作更形重要。在本論文中，我們研究超大型積體電路多層連線的 寄生參數萃取與精確和有效率模式，其中包括提出目前最佳之寄生電容、 電感及電阻參數的萃取技術和一些暫態模式的簡化技巧，這些努力將應用 於電腦輔助系統設計中。 第一章包含有關於我們研究動機的概括論述及介紹本論文的組織架構。在 第二章中，我們以一個有效率的二維多層連線寄生電容模式為基礎，繼而 發展出往後數章中所提出的三維多層連線寄生電容模式。當工作頻率很低 ，電磁場的波長遠大於物質的長度，時變的磁場可忽略，Maxwell's 方程 式在擬似靜態的電場假設下，簡化成Poisson's方程式來描述電位分佈， 本章用任意線性有限元素配合適當的多層格林函數來近似多層介質中的導 體表面電荷分佈，並採用變分技巧將其有效和正確的萃取出來，其中所有 的表面電荷分佈積分將被解析評估（包括奇異積分），進而計算出多層連 線寄生電容。 在第三章中,採用邊界有限元素法和 矩陣運算，避免在第二章中導體表面電荷分佈多餘的計算步驟，直接求出 多層連線寄生電容。本章用適當的多層格林函數來近似多層介質中的導體 表面電荷分佈和低階多項式當作形狀函數，配合Galerkin原理於邊界任意 三角元素上，其中大部分的表面電荷分佈積分將被解析評估和奇異積分將 藉著適當的座標轉換被消除，進而有效率和精確的計算出任意結構的多層 連線寄生電容，甚至更複雜的結構如：交叉線、角落導體、金屬接觸區域 、多層介電質和他們之間的綜合體，亦可有效率及正確的計算出來。在任 意結構導體表面中任意三角元素上，使用低階多項式當作形狀函數，這些 努力將成功的推廣邊界有限元素法成為有效率和精確的電腦輔助設計工具 。 緊接著第三章後，我們專注於一個可能應用於未來的 問題上，發展出不規則形狀的介電質中之任意結構多層連線寄生電容模式 。在第四章裡，我們提出結合邊界元素法，使用任意三角元素在導體表面 和非均勻不規則介質的界面上，評估導體表面電荷和束縛電荷的分布。再 者，藉著修正後的結合邊界元素法來加速修正電荷係數矩陣和電容係數， 使用格林函數處理多層介質和常數元素當作形狀函數，配合Galerkin原理 於邊界任意三角元素上，其中所有的表面電荷和束縛電荷分佈積分將被解 析評估（包括奇異積分），進而計算出多層連線寄生電容。另一項優點是 處理不規則介質上的界面區域變得十分簡單容易，因為對於此複雜結構只 須要考慮邊界和界面的格子點分佈即可。在任意結構導體表面和不規則形 狀介電質界面中的任意三角元素上，使用常數元素當作形狀函數，這些努 力將成功的推廣結合邊界元素法之應用到更複雜的結構如：交叉線、角落 導體、金屬接觸區域、不規則形狀的介電質和他們之間的綜合體，成為適 用範圍更廣泛之有效率和精確的電腦輔助設計工具。 前面的幾章僅考慮穩態的參數萃取，而暫態的特性分析將在接下來的章節 中討論。在第五章裡，我們將廣泛的探討線性網路的暫態分析技巧如： moment-matching methods ( AWE、CFH ) 和Krylov subspace-based methods ( PVA、PVL )，這些方法將被驗證，對於分析大型系統上比用傳 統或直接的方法 ( SPICE ) 模擬時間快二至三個次方。但由於moment- matching methods會產生不良和趨近奇異矩陣，所以Krylov subspace- based methods配合Arnoldi和Lanczos過程比moment-matching methods更 穩定、精確。然而對於分析低階的系統， moment-matching methods和 Krylov subspace-based methods有相同的精確度，但是對於分析大型系 統，當高階的近似不可忽略時，Krylov subspace-based methods將顯出 其優越性。這些分析將有助於往後章節對於複雜的三維分散式之傳輸線的 暫態分析。 在第六、七章裡，我們將專注於發展複雜的三維分散式電阻-電容之多層 傳輸線的暫態簡化模式。在第七章裡，根據邊界元素法和 Pade`-via- Lanczos過程配合適當的多層格林函數，可以避免體積切割成格子點和有 限差分法的暫態分析之冗長工作，有效的大量節省記憶體和時間。一個適 當的多層完整形式格林函數被用來探討多層導線之間有趣的電荷轉移物理 機制。為了改善有限差分法的暫態分析，Pade`-via-Lanczos過程被用來 簡化物理模式，得到一些主要的極點，以有效分析複雜的三維分散式電 阻-電容之多層傳輸線的暫態特性。繼而在第七章裡，為了推廣第六章的 應用範圍，一個適當的多層封閉形式格林函數被用來探討三維任意層介電 質中的複雜結構導線之間有趣的電荷轉移物理機制，由於封閉形式格林函 數是有限項的組合，除了可以計算任意層的介電質複雜結構外，還可有效 的降低計算電位所費時間。所以對於更複雜的結構如：交叉線、角落導體 、金屬接觸區域、任意層介電質和他們之間的綜合體，我們成功的發展出 暫態簡化模式，有效率及正確的計算出一些主要的極點來分析其暫態特性 ；總言之，這些努力將可配合現有的電腦輔助設計工具，有效率和精確的 分析更複雜的大型系統之暫態響應。 當晶 片上的信號頻率穩定增加，電磁場的波長將近於物質的長度，晶片上寄生 電感將顯著增加而不可忽略。在第八章中藉著部分元素等效電路方法，發 展出三維分散式電阻-電感模式來探討任意數目的三維複雜導體結構之頻 率相依的電阻和電感。在擬似靜態的磁場假設下，根據Maxwell's方程式 推導出部分元素等效電路方法，將有效的運用在三維複雜導體結構的電感 分析上，在此基礎上配合邊界條件和電荷守恆，我們成功的發展出電阻- 電感模式來探討任意數目三維複雜導體結構中的趨膚效應和鄰近效應；簡 言之，頻率相依的電感及電阻可以被重新組合成三維分散式電阻-電感模 式。 第九章將本論文的重要貢獻作一總結, 並提出後續的研究方向。 Physical interconnect effects have a dominant impact on today's deep submicron integration circuit designs. Due to increasing operation frequencies, microwave-like effects will become important. Therefore, stronger demands are put on extraction and verification tools.In this thesis, we investigate concepts and techniques for accurately and efficiently modeling and extracting multi-level interconnect parasitic parameters in Very-Large-Scale Integration (VLSI) or even Ultra-Large-Scale Integration (ULSI) designs. We propose the state-of-the art for capacitance, inductance, resistance extractions and several transient model reduction techniques related to simulation and implementation in a computer-aided design system. Chapter 1 includes a general introduction concerned with the motivation and the organization of this thesis. In Chapter 2, an efficient method is presented to model the parasitic capacitance of 2-D VLSI multi-level interconnections. Beginning with the formalism and improvement of the efficiency, we develop the parasitic capacitance of 3-D VLSI multi-level interconnection modeling, as derived in the following chapters. Assuming that the frequencies at which IC's working are not so high that the wavelength of electromagnetic wave length longer than the length of conductor, we can neglect the temporal derivative of the magnetic field. Thus, the complete set of Maxwell's equations can be reduced to the electrostatic case governed by Poisson's equation while still accurately describing the field distribution. Based on the variation technique, arbitrary linear elements on the surface of conductors for charge distribution are used to efficiently calculate capacitances of both parallel conductors and complex configurations such as a stratified dielectric medium. Using an adaptive multi-level Green's function and a low-order polynomial as the distribution of surface charge density, we apply the variation technique for surface charges over finite elements, and all of the surface integrals of charge distribution can be evaluated analytically and the singular integrals can be eliminated. In Chapter 3, an efficient method is presented to model the parasitic 3-D capacitance of VLSI multi-level interconnections, which can avoid redundant work in calculating the surface charge distribution of conductors, as shown in Chapter 2. Based on the boundary-finite-element method (BFEM) of integral formulation, arbitrary triangular elements on the surface of conductors for charge distribution are used to efficiently calculate capacitances of both parallel conductors and complex configurations such as crossing lines, corners, contacts, multilayers and their combinations. Using an adaptive multi-level Green's function and low-order polynomials as shape function, we apply the Galerkin principle over finite elements, and most of the surface integrals of charge distribution can be evaluated analytically and the singular integrals can be eliminated by choosing proper coordinate transformation. Moreover, an even less complex and more general method for arbitrary geometry configuration of multi-level interconnection lines is proposed in order to link with the finite-element pre- processor in present CAD tools. After Chapter 3, we dedicate our attention to the development of the 3-D parasitic capacitance modeling for complex multi-level interconnection in irregular dielectric layers. In Chapter 4, an efficient method is proposed to model the parasitic 3-D capacitances of ULSI multi-level interconnections on complex topography. Based on the merged boundary-element method (MBEM) of integral formulation, arbitrary triangular elements on the surface of conductors for charge distribution and the interface of irregular permittivity for bounded charge distribution are used to efficiently calculate the free charge and bounded charge distributions of complex structure based on actual topography/processes. Furthermore, the modified charge coefficient matrices and the coefficients of capacitances have been obtained by modification of MBEM. Using the Green's function for a stratified/layered medium and the constant element as shape functions, we apply the Galerkin principle over boundary elements, and all of the surface and interface integrals of charge distribution have been evaluated analytically. Accordingly, another important advantage of the proposed method is that the grid partition for the merged regions of different dielectrics is very simple and easy, because we only need to consider the boundary and interface of complex structure. The above chapters only focus on the extraction techniques of quasi-static parasitic parameters including capacitance, inductance, and resistance, the analysis of transient characteristics will be described in the following chapters. In Chapter 5, the transient modeling for linear networks is introduced.The techniques based on the Pade synthesis of linear lumped networks, such as the moment-matching methods (AWE, CFH) and the Krylov subspace-based methods (PVA, PVL), give a two or three order magnitude saving in simulation time over the traditional methods or direct simulation methods (SPICE) when a large system is analyzed. The Krylov subspace- based methods associated with Arnoldi and Lanczos algorithms compare favorably to the moment-matching methods when analyzing linear lumped networks. For low-order systems, the moment- matching methods and the Krylov subspace-based methods have identical accuracy. But for large and stiff systems, the high- order approximations can't be neglected, the Krylov subspace- based methods are superior. These concepts will benefit us to develop the transient modeling for the 3-D complex distributed transmission lines. Chapters 6-7, we dedicate our attention to the development of transient model-order reduction techniques for 3-D multi-level interconnections on complex topography. In Chapter 6, an efficient method is presented to model the transient characteristics of 3-D distributed resistor-capacitor of ULSI multi-level interconnections on complex topography, in which the reformulation of the boundary-element method (BEM) and the Pade-via-Lanczos (PVL) algorithm associated with multilayer Green's function can avoid the redundant works on both volume mesh and transient analysis associated with the finite- difference method. An adaptive multilayer Green's function is adopted to investigate several cases that have revealed interesting physical mechanisms in charge transfer between conductors on multilayer topography. To improve the timing analysis efficiency of the finite-difference method, the dominant poles are obtained by introducing the Pad\`{e}-via- Lanczos (PVL) algorithm for model-order reduction. Hence it is easy to calculate the transient characteristics of both parallel conductors and complex configurations such as crossing lines, corners, contacts, multilayers and their combinations. Chapter 7 is devoted to the efficient modeling for the transient characteristics of 3-D distributed resistor-capacitor of ULSI multi-level interconnections on more complex topography. By introducing the closed-form spatial Green's function, we can efficiently deal with the number of layer more than three, as compared with the method using the full-form Green's function in Chapter 6. An adaptive multilayer closed-form spatial Green's function for BEM is developed to examine the voltage and current responses of the multi-level conductor system by using the boundary-element method of integral formulation, in which arbitrary triangular elements on the surface of conductors are used to efficiently calculate the free-charge distributions of complex structure based on actual topography/processes. Several interesting physical mechanism cases have been investigated in charge transfer between coIt is greatly reduce computation time and memory, because each matrix element is consisted of finite series of surface integral using multilayer closed-form spatial Green's functions. Major improvements are the reformulation of the time-dependent BEM and the PVL algorithm associated with multilayer closed-form spatial Green's function to model the transient characteristics, which are proven to be applicable to even more complex configuration. Therefore, a simple and more general transient model is proposed for solving the combinations of complex structures based on actual topography/processes and arbitrary geometric configurations of multi-level interconnection lines in order to link with the present CAD tools. As the typical frequencies of on-chip signals are steadily increasing and the wavelengths are approaching the physical dimensions. Consequently, the parasitic on-chip inductances also become important. In Chapter 8, the partial element equivalent circuit has been first developed as the 3-D distributed resistor-inductor modeling to handle arbitrary number of the complex geometric conductors. For the 3-D interconnection system, based on the quasi-magnetostatic approximation, the Maxwell's equations can be reformulated as the partial element equivalent circuit to overcome complicated inductance problem of the 3-D complex interconnection configuration. Under boundary condition of potential and conservation of charge, the existing partial inductances of integral formulation are modified and improved. In order to capture efficiently the skin effect and the proximity effect, the proposed method associated with sinusoidal weighting scheme can reduce redundant work on volume mesh. In brief, the frequency-dependent resistance and inductance can be reconstructed as the interconnection-distributed resistor- inductor modeling. Chapter 9 summarizes this thesis, in which the major contributions and the future works are presented.