On the elimination of low-frequency breakdown problem and the development of fast solvers for finite-element-based analysis of high-speed ICS

Jianfang Zhu, Purdue University

Abstract

Integrated circuit (IC) design has been guided by circuit theory for more than three decades. The continuous scaling of feature sizes and frequency necessitates full-wave electromagnetics (EM)-based analysis. However, for EM-based methods to be widely adopted in realistic IC design, they have to overcome two main obstacles: First, the problem of low-frequency breakdown, which is also a major contributor to the passivity, stability, and causality issues in existing frequency-domain models; Second, the limited capacity and efficiency of existing full-wave EM-based methods. In this work, first, we fundamentally solved the low-frequency breakdown problem and hence naturally cured the passivity, stability, and causality violation due to low-frequency inaccuracy for full-wave based analysis of ICs. Our approach provides the very first rigorous full-wave solution that is applicable to both partial-differential-equation and integral-equation based numerical methods, truly from DC to any high frequency. It also works for general 3-D problems involving inhomogeneous lossless/lossy dielectrics and non-ideal conductors. In addition to the aforementioned theoretically rigorous solution, we also provided accurate approximation-based methods to effectively eliminate the low-frequency breakdown problem arising from the analysis of 2.5-D problems, and 3-D problems respectively. Both methods can be incorporated into existing full-wave finite-element-based CAD tool with minimal computational overhead. To overcome the second obstacle, we developed three fast solvers for frequency-domain analysis of ICs. In the first fast solver, we developed a method to use the solution of the mass matrix to construct the solution of the sum of the mass and the stiffness matrix. By doing so, we are able to fully utilize the nice properties of the mass matrix to significantly speed up the computation and reduce the memory cost. The system matrix thus can be efficiently solved by the orthogonal finite-element reduction-recovery method. The second fast solver is to accelerate the low-frequency full-wave solution to Maxwell’s equation. At low frequencies, the solution is only dominated by DC modes for both lossless and combined dielectric/conductor cases. We develop an efficient approach to identify DC modes for both cases based on the fact that all DC modes can be grouped into one solution vector for a given excitation since they share the same eigenvalue in common. With the single DC mode, we transform the original system of O( N) to a system of O(1), from which the low-frequency breakdown problem can be fixed rapidly for any low frequency. The third fast solver is developed based on the modal analysis of general 3-D problems involving inhomogeneous lossy or lossless dielectrics and nonideal conductors. We show that the physically important modes present in an IC only include DC and a few higher order modes. These modes constitute a reduced order model of the original large-scale system. We then overcome the numerical challenge of a quadratic eigenvalue solution to identify these modes correctly. However, the usefulness of this approach is limited by the null-space of the system matrix, which grows linearly with the problem size. We hence develop an efficient method to shrink the dimension of the null-space to be only one. As a result, we find a minimal-order representation of O(k) of the original large-scale problem of O(N), where the O(k) representation only consists of a single DC mode and a few higher orders modes.

Degree

Ph.D.

Advisors

Jiao, Purdue University.

Subject Area

Computer Engineering|Electrical engineering|Electromagnetics

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